tag:blogger.com,1999:blog-88906719360208851122024-03-13T02:01:06.629-07:00The Oil ConunDRUMSpin-off of the M O B J E C T I V I S T@whuthttp://www.blogger.com/profile/18297101284358849575noreply@blogger.comBlogger43125tag:blogger.com,1999:blog-8890671936020885112.post-41171077859205133592020-03-18T05:45:00.003-07:002020-03-18T05:45:21.503-07:00Mathematical Geoenergy<div style="background-color: white; border: 0px; box-sizing: border-box; color: #444444; font-family: Lato, sans-serif; font-size: 18px; margin-bottom: 1.7em; outline: 0px; padding: 0px; vertical-align: baseline;">
Our book <a href="https://www.wiley.com/en-us/Mathematical+Geoenergy%3A+Discovery%2C+Depletion%2C+and+Renewal-p-9781119434290" style="border: 0px; box-sizing: border-box; color: #1abc9c; font-family: inherit; font-style: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; text-decoration-line: none; transition: all 0.2s ease-in-out 0s; vertical-align: baseline;">Mathematical Geoenergy</a> presents a number of novel approaches that each deserve a research paper on their own. Here is the list, ordered roughly by importance (IMHO):</div>
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<ol style="background-color: white; border: 0px; box-sizing: border-box; color: #444444; font-family: Lato, sans-serif; font-size: 18px; list-style-image: initial; list-style-position: initial; margin: 0px 0px 1.7em 3em; outline: 0px; padding: 0px; vertical-align: baseline;">
<li style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: 700; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">Laplace’s Tidal Equation Analytic Solution</span>.<br style="box-sizing: border-box;" /><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: 700; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">(<a href="https://agupubs.onlinelibrary.wiley.com/doi/10.1002/9781119434351.ch11" style="border: 0px; box-sizing: border-box; color: #1abc9c; font-family: inherit; font-style: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; text-decoration-line: none; transition: all 0.2s ease-in-out 0s; vertical-align: baseline;">Ch 11</a>, <a href="https://agupubs.onlinelibrary.wiley.com/doi/10.1002/9781119434351.ch12" style="border: 0px; box-sizing: border-box; color: #1abc9c; font-family: inherit; font-style: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; text-decoration-line: none; transition: all 0.2s ease-in-out 0s; vertical-align: baseline;">12</a>)</span> A solution of a Navier-Stokes variant along the equator. Laplace’s Tidal Equations are a simplified version of Navier-Stokes and the equatorial topology allows an exact closed-form analytic solution. This could classify for the Clay Institute Millenium Prize if the practical implications are considered, but it’s a lower-dimensional solution than a complete 3-D Navier-Stokes formulation requires.</li>
<li style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: 700; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">Model of El Nino/Southern Oscillation (ENSO)</span>.<br style="box-sizing: border-box;" /><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: 700; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">(<a href="https://agupubs.onlinelibrary.wiley.com/doi/10.1002/9781119434351.ch12" style="border: 0px; box-sizing: border-box; color: #1abc9c; font-family: inherit; font-style: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; text-decoration-line: none; transition: all 0.2s ease-in-out 0s; vertical-align: baseline;">Ch 12</a>)</span> A tidally forced model of the equatorial Pacific’s thermocline sloshing (the ENSO dipole) which assumes a strong annual interaction. Not surprisingly this uses the Laplace’s Tidal Equation solution described above, otherwise the tidal pattern connection would have been discovered long ago.</li>
<li style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: 700; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">Model of Quasi-Biennial Oscillation (QBO)</span>.<br style="box-sizing: border-box;" /><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: 700; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">(<a href="https://agupubs.onlinelibrary.wiley.com/doi/10.1002/9781119434351.ch11" style="border: 0px; box-sizing: border-box; color: #1abc9c; font-family: inherit; font-style: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; text-decoration-line: none; transition: all 0.2s ease-in-out 0s; vertical-align: baseline;">Ch 11</a>)</span> A model of the equatorial stratospheric winds which cycle by reversing direction ~28 months. This incorporates the idea of amplified cycling of the sun and moon nodal declination pattern on the atmosphere’s tidal response.</li>
<li style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: 700; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">Origin of the Chandler Wobble</span>.<br style="box-sizing: border-box;" /><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: 700; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">(<a href="https://agupubs.onlinelibrary.wiley.com/doi/10.1002/9781119434351.ch13" style="border: 0px; box-sizing: border-box; color: #1abc9c; font-family: inherit; font-style: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; text-decoration-line: none; transition: all 0.2s ease-in-out 0s; vertical-align: baseline;">Ch 13</a>)</span> An explanation for the ~433 day cycle of the Earth’s Chandler wobble. Finding this is a fairly obvious consequence of modeling the QBO.</li>
<li style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: 700; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">The Oil Shock Model.</span><br style="box-sizing: border-box;" /><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: 700; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">(<a href="https://agupubs.onlinelibrary.wiley.com/doi/10.1002/9781119434351.ch5" style="border: 0px; box-sizing: border-box; color: #1abc9c; font-family: inherit; font-style: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; text-decoration-line: none; transition: all 0.2s ease-in-out 0s; vertical-align: baseline;">Ch 5</a>)</span> A data flow model of oil extraction and production which allows for perturbations. We are seeing this in action with the recession caused by oil supply perturbations due to the Corona Virus pandemic.</li>
<li style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: 700; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">The Dispersive Discovery Model.</span><br style="box-sizing: border-box;" /><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: 700; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">(<a href="https://agupubs.onlinelibrary.wiley.com/doi/10.1002/9781119434351.ch4" style="border: 0px; box-sizing: border-box; color: #1abc9c; font-family: inherit; font-style: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; text-decoration-line: none; transition: all 0.2s ease-in-out 0s; vertical-align: baseline;">Ch 4</a>)</span> A probabilistic model of resource discovery which accounts for technological advancement and a finite search volume.</li>
<li style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: 700; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">Ornstein-Uhlenbeck Diffusion Model</span><br style="box-sizing: border-box;" /><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: 700; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">(<a href="https://agupubs.onlinelibrary.wiley.com/doi/10.1002/9781119434351.ch6" style="border: 0px; box-sizing: border-box; color: #1abc9c; font-family: inherit; font-style: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; text-decoration-line: none; transition: all 0.2s ease-in-out 0s; vertical-align: baseline;">Ch 6</a>)</span> Applying Ornstein-Uhlenbeck diffusion to describe the decline and asymptotic limiting flow from volumes such as occur in fracked shale oil reservoirs.</li>
<li style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: 700; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">The Reservoir Size Dispersive Aggregation Model.</span><br style="box-sizing: border-box;" /><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: 700; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">(<a href="https://agupubs.onlinelibrary.wiley.com/doi/10.1002/9781119434351.ch4" style="border: 0px; box-sizing: border-box; color: #1abc9c; font-family: inherit; font-style: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; text-decoration-line: none; transition: all 0.2s ease-in-out 0s; vertical-align: baseline;">Ch 4</a>)</span> A first-principles model that explains and describes the size distribution of oil reservoirs and fields around the world.</li>
<li style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: 700; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">Origin of Tropical Instability Waves (TIW)</span>.<br style="box-sizing: border-box;" /><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: 700; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">(<a href="https://agupubs.onlinelibrary.wiley.com/doi/10.1002/9781119434351.ch12" style="border: 0px; box-sizing: border-box; color: #1abc9c; font-family: inherit; font-style: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; text-decoration-line: none; transition: all 0.2s ease-in-out 0s; vertical-align: baseline;">Ch 12</a>)</span> As the ENSO model was developed, a higher harmonic component was found which matches TIW</li>
<li style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: 700; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">Characterization of Battery Charging and Dischargin</span>g.<br style="box-sizing: border-box;" /><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: 700; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">(<a href="https://agupubs.onlinelibrary.wiley.com/doi/10.1002/9781119434351.ch18" style="border: 0px; box-sizing: border-box; color: #1abc9c; font-family: inherit; font-style: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; text-decoration-line: none; transition: all 0.2s ease-in-out 0s; vertical-align: baseline;">Ch 18</a>)</span> Simplified expressions for modeling Li-ion battery charging and discharging profiles by applying dispersion on the diffusion equation, which reflects the disorder within the ion matrix.</li>
<li style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: 700; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">Anomalous Behavior in Dispersive Transport explained.</span><br style="box-sizing: border-box;" /><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: 700; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">(<a href="https://agupubs.onlinelibrary.wiley.com/doi/10.1002/9781119434351.ch18" style="border: 0px; box-sizing: border-box; color: #1abc9c; font-family: inherit; font-style: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; text-decoration-line: none; transition: all 0.2s ease-in-out 0s; vertical-align: baseline;">Ch 18</a>)</span> Photovoltaic (PV) material made from disordered and amorphous semiconductor material shows poor photoresponse characteristics. Solution to simple entropic dispersion relations or the more general Fokker-Planck leads to good agreement with the data over orders of magnitude in current and response times.</li>
<li style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: 700; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">Framework for understanding Breakthrough Curves and Solute Transport in Porous Materials.</span><br style="box-sizing: border-box;" /><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: 700; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">(<a href="https://agupubs.onlinelibrary.wiley.com/doi/10.1002/9781119434351.ch20" style="border: 0px; box-sizing: border-box; color: #1abc9c; font-family: inherit; font-style: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; text-decoration-line: none; transition: all 0.2s ease-in-out 0s; vertical-align: baseline;">Ch 20</a>)</span> The same disordered Fokker-Planck construction explains the dispersive transport of solute in groundwater or liquids flowing in porous materials.</li>
<li style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: 700; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">Wind Energy Analysis</span>.<br style="box-sizing: border-box;" /><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: 700; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">(<a href="https://agupubs.onlinelibrary.wiley.com/doi/10.1002/9781119434351.ch11" style="border: 0px; box-sizing: border-box; color: #1abc9c; font-family: inherit; font-style: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; text-decoration-line: none; transition: all 0.2s ease-in-out 0s; vertical-align: baseline;">Ch 11</a>)</span> Universality of wind energy probability distribution by applying maximum entropy to the mean energy observed. Data from Canada and Germany. Found a universal BesselK distribution which improves on the conventional Rayleigh distribution.</li>
<li style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: 700; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">Terrain Slope Distribution Analysis.</span><br style="box-sizing: border-box;" /><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: 700; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">(<a href="https://agupubs.onlinelibrary.wiley.com/doi/10.1002/9781119434351.ch16" style="border: 0px; box-sizing: border-box; color: #1abc9c; font-family: inherit; font-style: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; text-decoration-line: none; transition: all 0.2s ease-in-out 0s; vertical-align: baseline;">Ch 16</a>)</span> Explanation and derivation of the topographic slope distribution across the USA. This uses mean energy and maximum entropy principle.</li>
<li style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: 700; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">Thermal Entropic Dispersion Analysis</span>.<br style="box-sizing: border-box;" /><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: 700; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">(<a href="https://agupubs.onlinelibrary.wiley.com/doi/10.1002/9781119434351.ch14" style="border: 0px; box-sizing: border-box; color: #1abc9c; font-family: inherit; font-style: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; text-decoration-line: none; transition: all 0.2s ease-in-out 0s; vertical-align: baseline;">Ch 14</a>)</span> Solving the Fokker-Planck equation or Fourier’s Law for thermal diffusion in a disordered environment. A subtle effect but the result is a simplified expression not involving complex <em style="border: 0px; box-sizing: border-box; font-family: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">errf </em>transcendental functions. Useful in ocean heat content (OHC) studies.</li>
<li style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: 700; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">The Maximum Entropy Principle and the Entropic Dispersion Framework.</span><br style="box-sizing: border-box;" /><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: 700; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">(<a href="https://agupubs.onlinelibrary.wiley.com/doi/10.1002/9781119434351.ch10" style="border: 0px; box-sizing: border-box; color: #1abc9c; font-family: inherit; font-style: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; text-decoration-line: none; transition: all 0.2s ease-in-out 0s; vertical-align: baseline;">Ch 10</a>)</span> The generalized math framework applied to many models of disorder, natural or man-made. Explains the origin of the entroplet.</li>
<li style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: 700; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">Solving the Reserve Growth “enigma”.</span><br style="box-sizing: border-box;" /><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: 700; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">(<a href="https://agupubs.onlinelibrary.wiley.com/doi/10.1002/9781119434351.ch6" style="border: 0px; box-sizing: border-box; color: #1abc9c; font-family: inherit; font-style: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; text-decoration-line: none; transition: all 0.2s ease-in-out 0s; vertical-align: baseline;">Ch 6</a>)</span> An application of dispersive discovery on a localized level which models the hyperbolic reserve growth characteristics observed.</li>
<li style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: 700; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">Shocklets.</span><br style="box-sizing: border-box;" /><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: 700; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">(<a href="https://agupubs.onlinelibrary.wiley.com/doi/10.1002/9781119434351.ch7" style="border: 0px; box-sizing: border-box; color: #1abc9c; font-family: inherit; font-style: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; text-decoration-line: none; transition: all 0.2s ease-in-out 0s; vertical-align: baseline;">Ch 7</a>)</span> A kernel approach to characterizing production from individual oil fields.</li>
<li style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: 700; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">Reserve Growth, Creaming Curve, and Size Distribution Linearization.</span><br style="box-sizing: border-box;" /><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: 700; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">(<a href="https://agupubs.onlinelibrary.wiley.com/doi/10.1002/9781119434351.ch6" style="border: 0px; box-sizing: border-box; color: #1abc9c; font-family: inherit; font-style: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; text-decoration-line: none; transition: all 0.2s ease-in-out 0s; vertical-align: baseline;">Ch 6</a>)</span> An obvious linearization of this family of curves, related to Hubbert Linearization but more useful since it stems from first principles.</li>
<li style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: 700; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">The Hubbert Peak Logistic Curve explained.</span><br style="box-sizing: border-box;" /><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: 700; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">(<a href="https://agupubs.onlinelibrary.wiley.com/doi/10.1002/9781119434351.ch7" style="border: 0px; box-sizing: border-box; color: #1abc9c; font-family: inherit; font-style: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; text-decoration-line: none; transition: all 0.2s ease-in-out 0s; vertical-align: baseline;">Ch 7</a>)</span> The Logistic curve is trivially explained by dispersive discovery with exponential technology advancement.</li>
<li style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: 700; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">Laplace Transform Analysis of Dispersive Discovery.</span><br style="box-sizing: border-box;" /><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: 700; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">(<a href="https://agupubs.onlinelibrary.wiley.com/doi/10.1002/9781119434351.ch7" style="border: 0px; box-sizing: border-box; color: #1abc9c; font-family: inherit; font-style: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; text-decoration-line: none; transition: all 0.2s ease-in-out 0s; vertical-align: baseline;">Ch 7</a>)</span> Dispersion curves are solved by looking up the Laplace transform of the spatial uncertainty profile.</li>
<li style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: 700; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">Gompertz Decline Model.</span><br style="box-sizing: border-box;" /><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: 700; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">(<a href="https://agupubs.onlinelibrary.wiley.com/doi/10.1002/9781119434351.ch7" style="border: 0px; box-sizing: border-box; color: #1abc9c; font-family: inherit; font-style: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; text-decoration-line: none; transition: all 0.2s ease-in-out 0s; vertical-align: baseline;">Ch 7</a>)</span> Exponentially increasing extraction rates lead to steep production decline.</li>
<li style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: 700; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">The Dynamics of Atmospheric CO2 buildup and Extrapolation.</span><br style="box-sizing: border-box;" /><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: 700; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">(<a href="https://agupubs.onlinelibrary.wiley.com/doi/10.1002/9781119434351.ch9" style="border: 0px; box-sizing: border-box; color: #1abc9c; font-family: inherit; font-style: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; text-decoration-line: none; transition: all 0.2s ease-in-out 0s; vertical-align: baseline;">Ch 9</a>)</span> Convolving a fat-tailed CO2 residence time impulse response function with a fossil-fuel emissions stimulus. This shows the long latency of CO2 buildup very straightforwardly.</li>
<li style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: 700; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">Reliability Analysis and Understanding the “Bathtub Curve”.</span><br style="box-sizing: border-box;" /><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: 700; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">(<a href="https://agupubs.onlinelibrary.wiley.com/doi/10.1002/9781119434351.ch19" style="border: 0px; box-sizing: border-box; color: #1abc9c; font-family: inherit; font-style: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; text-decoration-line: none; transition: all 0.2s ease-in-out 0s; vertical-align: baseline;">Ch 19</a>)</span> Using a dispersion in failure rates to generate the characteristic bathtub curves of failure occurrences in parts and components.</li>
<li style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: 700; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">The Overshoot Point (TOP) and the Oil Production Plateau.</span><br style="box-sizing: border-box;" /><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: 700; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">(<a href="https://agupubs.onlinelibrary.wiley.com/doi/10.1002/9781119434351.ch8" style="border: 0px; box-sizing: border-box; color: #1abc9c; font-family: inherit; font-style: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; text-decoration-line: none; transition: all 0.2s ease-in-out 0s; vertical-align: baseline;">Ch 8</a>)</span> How increases in extraction rate can maintain production levels.</li>
<li style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: 700; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">Lake Size Distribution.</span><br style="box-sizing: border-box;" /><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: 700; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">(<a href="https://agupubs.onlinelibrary.wiley.com/doi/10.1002/9781119434351.ch15" style="border: 0px; box-sizing: border-box; color: #1abc9c; font-family: inherit; font-style: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; text-decoration-line: none; transition: all 0.2s ease-in-out 0s; vertical-align: baseline;">Ch 15</a>)</span> Analogous to explaining reservoir size distribution, uses similar arguments to derive the distribution of freshwater lake sizes. This provides a good feel for how often super-giant reservoirs and Great Lakes occur (by comparison).</li>
<li style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: 700; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">The Quandary of Infinite Reserves due to Fat-Tail Statistics.</span><br style="box-sizing: border-box;" /><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: 700; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">(<a href="https://agupubs.onlinelibrary.wiley.com/doi/10.1002/9781119434351.ch9" style="border: 0px; box-sizing: border-box; color: #1abc9c; font-family: inherit; font-style: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; text-decoration-line: none; transition: all 0.2s ease-in-out 0s; vertical-align: baseline;">Ch 9</a>)</span> Demonstrated that even infinite reserves can lead to limited resource production in the face of maximum extraction constraints.</li>
<li style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: 700; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">Oil Recovery Factor Model.</span><br style="box-sizing: border-box;" /><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: 700; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">(<a href="https://agupubs.onlinelibrary.wiley.com/doi/10.1002/9781119434351.ch6" style="border: 0px; box-sizing: border-box; color: #1abc9c; font-family: inherit; font-style: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; text-decoration-line: none; transition: all 0.2s ease-in-out 0s; vertical-align: baseline;">Ch 6</a>)</span> A model of oil recovery which takes into account reservoir size.</li>
<li style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: 700; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">Network Transit Time Statistics.</span><br style="box-sizing: border-box;" /><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: 700; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">(<a href="https://agupubs.onlinelibrary.wiley.com/doi/10.1002/9781119434351.ch21" style="border: 0px; box-sizing: border-box; color: #1abc9c; font-family: inherit; font-style: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; text-decoration-line: none; transition: all 0.2s ease-in-out 0s; vertical-align: baseline;">Ch 21</a>)</span> Dispersion in TCP/IP transport rates leads to the measured fat-tails in round-trip time statistics on loaded networks.</li>
<li style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: 700; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">Particle and Crystal Growth Statistics.</span><br style="box-sizing: border-box;" /><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: 700; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">(<a href="https://agupubs.onlinelibrary.wiley.com/doi/10.1002/9781119434351.ch20" style="border: 0px; box-sizing: border-box; color: #1abc9c; font-family: inherit; font-style: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; text-decoration-line: none; transition: all 0.2s ease-in-out 0s; vertical-align: baseline;">Ch 20</a>)</span> Detailed model of ice crystal size distribution in high-altitude cirrus clouds.</li>
<li style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: 700; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">Rainfall Amount Dispersion.</span><br style="box-sizing: border-box;" /><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: 700; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">(<a href="https://agupubs.onlinelibrary.wiley.com/doi/10.1002/9781119434351.ch15" style="border: 0px; box-sizing: border-box; color: #1abc9c; font-family: inherit; font-style: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; text-decoration-line: none; transition: all 0.2s ease-in-out 0s; vertical-align: baseline;">Ch 15</a>)</span> Explanation of rainfall variation based on dispersion in rate of cloud build-up along with dispersion in critical size.</li>
<li style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: 700; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">Earthquake Magnitude Distribution.</span><br style="box-sizing: border-box;" /><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: 700; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">(<a href="https://agupubs.onlinelibrary.wiley.com/doi/10.1002/9781119434351.ch13" style="border: 0px; box-sizing: border-box; color: #1abc9c; font-family: inherit; font-style: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; text-decoration-line: none; transition: all 0.2s ease-in-out 0s; vertical-align: baseline;">Ch 13</a>)</span> Distribution of earthquake magnitudes based on dispersion of energy buildup and critical threshold.</li>
<li style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: 700; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">IceBox Earth Setpoint Calculation.</span><br style="box-sizing: border-box;" /><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: 700; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">(<a href="https://agupubs.onlinelibrary.wiley.com/doi/10.1002/9781119434351.ch17" style="border: 0px; box-sizing: border-box; color: #1abc9c; font-family: inherit; font-style: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; text-decoration-line: none; transition: all 0.2s ease-in-out 0s; vertical-align: baseline;">Ch 17</a>)</span> Simple model for determining the earth’s setpoint temperature extremes — current and low-CO2 icebox earth.</li>
<li style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: 700; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">Global Temperature Multiple Linear Regression Model</span><br style="box-sizing: border-box;" /><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: 700; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">(<a href="https://agupubs.onlinelibrary.wiley.com/doi/10.1002/9781119434351.ch17" style="border: 0px; box-sizing: border-box; color: #1abc9c; font-family: inherit; font-style: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; text-decoration-line: none; transition: all 0.2s ease-in-out 0s; vertical-align: baseline;">Ch 17</a>)</span> The global surface temperature records show variability that is largely due to the GHG rise along with fluctuating changes due to ocean dipoles such as ENSO (via the SOI measure and also AAM) and sporadic volcanic eruptions impacting the atmospheric aerosol concentrations.</li>
<li style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: 700; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">GPS Acquisition Time Analysis</span>.<br style="box-sizing: border-box;" /><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: 700; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">(<a href="https://agupubs.onlinelibrary.wiley.com/doi/10.1002/9781119434351.ch21" style="border: 0px; box-sizing: border-box; color: #1abc9c; font-family: inherit; font-style: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; text-decoration-line: none; transition: all 0.2s ease-in-out 0s; vertical-align: baseline;">Ch 21</a>)</span> Engineering analysis of GPS cold-start acquisition times. Using Maximum Entropy in EMI clutter statistics.</li>
<li style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: 700; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">1/f Noise</span> <span style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: 700; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">Model</span><br style="box-sizing: border-box;" /><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: 700; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">(<a href="https://agupubs.onlinelibrary.wiley.com/doi/10.1002/9781119434351.ch21" style="border: 0px; box-sizing: border-box; color: #1abc9c; font-family: inherit; font-style: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; text-decoration-line: none; transition: all 0.2s ease-in-out 0s; vertical-align: baseline;">Ch 21</a>)</span> Deriving a random noise spectrum from maximum entropy statistics.</li>
<li style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: 700; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">Stochastic Aquatic Waves</span><br style="box-sizing: border-box;" /><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: 700; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">(<a href="https://agupubs.onlinelibrary.wiley.com/doi/10.1002/9781119434351.ch12" style="border: 0px; box-sizing: border-box; color: #1abc9c; font-family: inherit; font-style: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; text-decoration-line: none; transition: all 0.2s ease-in-out 0s; vertical-align: baseline;">Ch 12</a>)</span> Maximum Entropy Analysis of wave height distribution of surface gravity waves.</li>
<li style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: 700; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">The Stochastic Model of Popcorn Popping.</span><br style="box-sizing: border-box;" /><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: 700; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">(<a href="https://agupubs.onlinelibrary.wiley.com/doi/10.1002/9781119434351.app3" style="border: 0px; box-sizing: border-box; color: #1abc9c; font-family: inherit; font-style: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; text-decoration-line: none; transition: all 0.2s ease-in-out 0s; vertical-align: baseline;">Appx C</a>)</span> The novel explanation of why popcorn popping follows the same bell-shaped curve of the Hubbert Peak in oil production. Can use this to model epidemics, etc.</li>
<li style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: 700; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">Dispersion Analysis of Human Transportation Statistics</span>.<br style="box-sizing: border-box;" /><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-style: inherit; font-weight: 700; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">(<a href="https://agupubs.onlinelibrary.wiley.com/doi/10.1002/9781119434351.app3" style="border: 0px; box-sizing: border-box; color: #1abc9c; font-family: inherit; font-style: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; text-decoration-line: none; transition: all 0.2s ease-in-out 0s; vertical-align: baseline;">Appx C</a>)</span> Alternate take on the empirical distribution of travel times between geographical points. This uses a maximum entropy approximation to the mean speed and mean distance across all the data points.</li>
</ol>
@whuthttp://www.blogger.com/profile/18297101284358849575noreply@blogger.com0tag:blogger.com,1999:blog-8890671936020885112.post-37792599542075809252019-07-21T03:34:00.000-07:002019-07-21T05:09:28.824-07:00Rating of Climate Change blogsScientific blogging is on a decline and that is especially evident with respect to climate change blogs. Nothing really good left apart from forums such as <a href="https://forum.azimuthproject.org/discussions">https://forum.azimuthproject.org/discussions</a> (which allows equation markup, image posting, and freedom to create threaded discussions).<br />
<br />
Here is a grading of blogs that I have on my RSS feed:<br />
<br />
<ul>
<li>WUWT : <b>F-</b><br />A horrible AGW denier blog that pretends to be fair & balanced. RSS feed does not work with Owl.</li>
<li>Tallbloke's Talkshop : <b>F-</b><br />A horrible AGW denier blog that specializes in numerology</li>
<li>Real Climate : <b>C</b><br />Sparse postings and comment moderation has long latencies so the discussion is glacially paced</li>
<li>Open Mind : <b>C</b><br />Not very interesting, mostly from a statistical angle, which is not where progress in analysis occurs</li>
<li>Science of Doom : <b>D</b><br />I don't understand this site, seems to be run by a thinly veiled skeptic. Might as well read books by Pierrehumbert to gain an understanding of the physics instead of struggling along with the topics.</li>
<li>And Then There's Physics : <b>D+</b><br />The moderators are control freaks, and the discussions are safe as milk</li>
<li><a href="http://peakoilbarrel.com/">Peak Oil Barrel</a> : <b>A</b><br />A very good blog that allows both fossil fuel discussion and climate change discussion, separated in distinct threads. Moderated slightly and images allowed, along with short-term editing.</li>
<li>The Blackboard : <b>F-</b><br />An awful blog run by a mechanical engineer which at one point had climate science discussion but now consists of pro-Trump cheerleading.</li>
<li>Clive Best : <b>D+</b><br />The moderator tries hard but then stumbles as he desperately tries to debunk the AGW consensus. Marginally better than Science of Doom because at least the scientific ideas are creative.</li>
<li>Climate Audit : <b>F-</b><br />Awful conspiracy-laden blog run by a former Canadian mining executive.</li>
<li>Climate Etc : <b>F-</b><br />Pointless blog stressing climate science uncertainty run by a now-retired climate science professor J. Curry with a comment section that seems infested with Australian AGW deniers. </li>
<li><a href="https://moyhu.blogspot.com/">Moyhu</a> : <b>B</b><br />Halfway-decent posts by a retired fluid dynamics researcher but an ugly and unstable comment-entry system. </li>
<li>Hotwhopper : <b>C+</b><br />Well-thought out counter-attacks to nonsense at sites such as WUWT, but nothing really about discussions of science</li>
<li>Robert Scribbler : <b>C</b><br />The American version of HotWhopper, with probably too much doom & gloom.</li>
<li>Skeptical Science: <b>D</b><br />Nothing interesting here as they never seem to veer from the consensus. The comments seem to be overly moderated and at one time the RSS feed was broken, but that has recently been fixed.</li>
<li>Roy Spencer, PhD : <b>F-</b><br />Horrible blog by a religious zealot with comments infested by AGW deniers</li>
<li>Rabbet Run : <b>B-</b><br />Below Moyhu because nothing really innovative but occasional insight</li>
<li>More Grumbine Science : <b>B</b><br />By a NASA guy, posts very rarely</li>
</ul>
This is a previous summary I had written two years ago (I had forgotten I had saved it in a draft folder, and so you can see how little has changed)<br />
<br />
<table>
<tbody>
<tr>
<th>Blog
</th><th>Grade
</th><th>Rationale
</th></tr>
<tr>
<td>Real Climate
</td><td>C
</td><td>Too long turnaround for comments
</td></tr>
<tr>
<td>And Then There's Physics
</td><td>D
</td><td>Too much ClimateBall
</td></tr>
<tr>
<td>Skeptical Science
</td><td>D
</td><td>Too insular, won't discuss cutting edge
</td></tr>
<tr>
<td>Science of Doom
</td><td>D-
</td><td>Inflitrated by deniers
</td></tr>
<tr>
<td>Open Mind
</td><td>C
</td><td>Too much on statistics, which does a disservice to unlocking deterministic aspects such as ENSO
</td></tr>
<tr>
<td>Moyhu
</td><td>B
</td><td>Worst comment entry, but the research is quality
</td></tr>
<tr>
<td>This Week in Science: DailyKos
</td><td>C+
</td><td>Nothing in depth
</td></tr>
<tr>
<td>Watt's Up With That
</td><td>F
</td><td>Garbage
</td></tr>
<tr>
<td>Tallbloke's Talkshop
</td><td>F
</td><td>Loony bin
</td></tr>
<tr>
<td>The Blackboard
</td><td>F
</td><td>Nasty people
</td></tr>
<tr>
<td>Climate Etc
</td><td>F
</td><td>Clueless (mainly Aussies) lead by a clueless
</td></tr>
<tr>
<td>Roy Spencer
</td><td>F
</td><td>Zero real science
</td></tr>
<tr>
<td>Rabbet Run
</td><td>C
</td><td>Too much inside posturing
</td></tr>
<tr>
<td>Hot Whopper
</td><td>C
</td><td>Good if you want to see deniers get debunked
</td></tr>
<tr>
<td>Robert Scribbler
</td><td>C
</td><td>Verges on hysterical but who knows
</td></tr>
<tr>
<td>Azimuth Project
</td><td>A
</td><td>A true forum. Allows everyone to create markup and add charts
</td></tr>
</tbody></table>
@whuthttp://www.blogger.com/profile/18297101284358849575noreply@blogger.com0tag:blogger.com,1999:blog-8890671936020885112.post-50367493380264676942013-09-16T21:11:00.003-07:002013-09-16T21:11:58.282-07:00Context/EarthPlease refer to my new Wordpress blog <a href="http://contextearth.com/"><span style="font-size: small;"><b>Context/Earth</b></span></a> for future posts.<br />
<br />
Wordpress has better commenting options such as provisions for pictures and charts.<br />
<br />
The scope of the blog is also more comprehensive, as it will include all environmental and energy topics tied together in a semantic web framework. <br />
<br />
Onward and upward as they say.<br />
<br />@whuthttp://www.blogger.com/profile/18297101284358849575noreply@blogger.com0tag:blogger.com,1999:blog-8890671936020885112.post-10359161852640489762013-07-07T14:01:00.000-07:002013-08-02T08:55:17.870-07:00Expansion of atmosphere and oceanThis is a short tutorial together with some observational evidence explaining how the atmosphere and ocean is expanding measurably in the face of global warming.<br />
<br />
<h3>
Ocean thermal expansion</h3>
The ocean absorbs heat per area according to its heat capacity<br />
<br />
$$ \Delta E = C_p \cdot \Delta T \cdot {Depth}$$<br />
<br />
The linear coefficient of thermal expansion is assumed constant over a temperature range. Multiplying this over a depth:<br />
<br />
$$ \Delta Z = \epsilon \cdot \Delta T \cdot Depth $$ <br />
But now we can substitute the total energy gained from the first equation:<br />
$$ \Delta Z = \epsilon \frac{\Delta E}{C_p} $$<br />
Assume that the linear coefficient of thermal expansion is 0.000207 per <span class="st">°</span>C, and specific heat capacity is 4,000,000 J/m^3/<span class="st">°</span>C.<br />
<br />
If an excess forcing of 0.6 W/m^2 occurs over one year (see the <a href="http://theoilconundrum.blogspot.com/2013/03/ocean-heat-content-model.html">OHC model</a>), then the increase in the level of the ocean is<br />
<br />
0.000207 * 0.6*(365*24*60*60) / 4,000,000 = 0.98 mm<br />
<br />
This is called the steric sea-level rise, and it is just one component of the sea-level rise over time (the others have to do with melting ice). From the figure below, one can see that the thermal rise is close to 1mm/year over the last decade. <br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="http://www.skepticalscience.com/pics/ChurchWhiteMSL.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="400" src="http://www.skepticalscience.com/pics/ChurchWhiteMSL.png" width="283" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">The red line shows the thermal expansion from the ocean heating. Thermal rise is 1 mm/year over the 3 mm/year total sea-level rise.</td></tr>
</tbody></table>
<br />
<h3>
Atmosphere thermal expansion</h3>
The trick here is to infer the atmosphere expansion by looking at an equal pressure point as a function of altitude (a geopotential height isobar) and determine how much that point has increased over time. I was able to find <a href="http://espanol.wunderground.com/earth-day/2013/stu-ostro-my-climate-change">one piece of data</a> from The Weather Channel's senior meteorologist Stu Ostro.<br />
<br />
The geopotential height anomaly is shown below for the 500 mb (1/2 atmosphere) <br />
<br />
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEivUBdE2yagT8ZJg00BlSdJKB5Tv7GlzmXwymzk_Oylec4NKdpYyMoOxQXdTygQP-0Hvjan7K_bnAV_Rd2LZeU0KyvQn0iHkH06GqeJ_q_DPy15e7s73vtkCOGCHLFeJOzeim10NF8RF20/s1600/geopotentialheight.GIF" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="236" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEivUBdE2yagT8ZJg00BlSdJKB5Tv7GlzmXwymzk_Oylec4NKdpYyMoOxQXdTygQP-0Hvjan7K_bnAV_Rd2LZeU0KyvQn0iHkH06GqeJ_q_DPy15e7s73vtkCOGCHLFeJOzeim10NF8RF20/s400/geopotentialheight.GIF" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Geopotential height anomaly @ 500 mb plotted alongside global temperature anomaly.</td></tr>
</tbody></table>
<div class="separator" style="clear: both; text-align: center;">
</div>
In absolute terms it is charted as follows:<br />
<br />
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="http://img825.imageshack.us/img825/1343/cb8.gif" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="256" src="http://img825.imageshack.us/img825/1343/cb8.gif" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Global average geopotential height @ 500 mb plotted alongside global temperature.<br />
<a href="http://icons-ak.wxug.com/graphics/earthweek/geopotential-height-and-air-temperature.png">http://icons-ak.wxug.com/graphics/earthweek/geopotential-height-and-air-temperature.png</a></td><td class="tr-caption" style="text-align: center;"><br /></td></tr>
</tbody></table>
<div class="separator" style="clear: both; text-align: center;">
</div>
<br />
To understand how the altitude has changed, consider the classical barometric formula:<br />
$$ P(H) = P_0 e^{-mgH/RT} $$<br />
<br />
We take the point at which we reached 1/2 the STP of 1 atmosphere at sea-level:<br />
<br />
$$ P(H)/P_0 = 0.5 = e^{-mgH/RT} $$<br />
<br />
or <br />
$$ H = RT/mg \cdot ln(2) $$<br />
Assuming the average molecular weight of the atmospheric gas constituents does not change, the change in altitude (H) should be related to the change in temperature (T) by:<br />
$$ \Delta H = R \Delta T / mg \cdot ln(2) $$<br />
or<br />
$$ \frac{ \Delta H}{\Delta T} = \frac{R}{mg} ln(2) $$<br />
For R = 8314, m=29, and g=9.8, the slope should be 29.25 *ln(2) = 20.3 m/<span class="st">°</span>C.<br />
<br />
From the linear regression agreement between the two, we get a value of 25.7 m/<span class="st">°</span>C.<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgQXdZwWaReowqR0TkA6Fz4Ejrj1KHpuSwISN4Dc_0Gn2BGmpoItabO92q2G_qWvleBHAxJBrf3vY7Km8NvcYmd6xnZZ6IV0aLvNIr2xB0IcNCFwFqC08uXj2qhN8H8cAws9Z5KtKbkNG4/s1600/geopotential_height_regression.GIF" style="margin-left: auto; margin-right: auto;"><img border="0" height="380" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgQXdZwWaReowqR0TkA6Fz4Ejrj1KHpuSwISN4Dc_0Gn2BGmpoItabO92q2G_qWvleBHAxJBrf3vY7Km8NvcYmd6xnZZ6IV0aLvNIr2xB0IcNCFwFqC08uXj2qhN8H8cAws9Z5KtKbkNG4/s400/geopotential_height_regression.GIF" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Linear regression between the geopotential height change and temperature change</td></tr>
</tbody></table>
<br />
Why is this geopotential height change about 26% higher than it should be from the theoretical value considering that the height should track the temperature according to the barometric formula?<br />
<br />
If we use the <a href="http://theoilconundrum.blogspot.com/2013/03/standard-atmosphere-model-and.html">polytropic</a> approximation (<a href="http://en.wikipedia.org/wiki/Barometric_formula">equation 1 in the barometric formula</a>), the altitudinal difference between the low temperature and high temperature 500 mb pressure values remains the same as when we use the classic exponential damped barometric formula:<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjrrw8yefuSTyRqFMiA4DIMSn7EP8Iy8-786cKNSvt2zJetDJqHmkZ6aEltQaSbUZiIihh_IwqAWUHp3BJJFlHSW0lUiHWx28-B6eBNVjtFLf9Vz_UFgY29GPWqfvx_OzsG9z_kxWwICRY/s1600/geopotential_polytropic.GIF" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="242" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjrrw8yefuSTyRqFMiA4DIMSn7EP8Iy8-786cKNSvt2zJetDJqHmkZ6aEltQaSbUZiIihh_IwqAWUHp3BJJFlHSW0lUiHWx28-B6eBNVjtFLf9Vz_UFgY29GPWqfvx_OzsG9z_kxWwICRY/s400/geopotential_polytropic.GIF" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">If we apply the polytropic barometric formula instead of the exponential, we still show a real height change that is higher than theoretically predicted by ~25% at the 500 mb isobar.</td></tr>
</tbody></table>
<br />
This discrepancy could be due to measurement error, as the readings are taken by weather balloons and the accuracy could have drifted over the years.<br />
<br />
It is also possible that the composition of the atmosphere could have changed slightly at altitude. What happens if the moisture increased slightly? This shouldn't make much difference.<br />
<br />
Most likely is the possibility that the baseline sea-level pressure has changed, which shifted the 500 mbar point artificially. See <a href="http://www.ipcc.ch/publications_and_data/ar4/wg1/en/ch10s10-3-2-4.html">http://www.ipcc.ch/publications_and_data/ar4/wg1/en/ch10s10-3-2-4.html</a>.<br />
<br />
<blockquote class="tr_bq">
<span style="font-size: x-small;"><i>"10.3.2.4 Sea Level Pressure and Atmospheric Circulation<br /><br />As a basic component of the mean atmospheric circulations and weather patterns, projections of the mean sea level pressure for the medium scenario A1B are considered. Seasonal mean changes for DJF and JJA are shown in Figure 10.9 (matching results in Wang and Swail, 2006b). Sea level pressure differences show decreases at high latitudes in both seasons in both hemispheres. The compensating increases are predominantly over the mid-latitude and subtropical ocean regions, extending across South America, Australia and southern Asia in JJA, and the Mediterranean in DJF. Many of these increases are consistent across the models. This pattern of change, discussed further in Section 10.3.5.3, has been linked to an expansion of the Hadley Circulation and a poleward shift of the mid-latitude storm tracks (Yin, 2005). This helps explain, in part, the increases in precipitation at high latitudes and decreases in the subtropics and parts of the mid-latitudes. Further analysis of the regional details of these changes is given in Chapter 11. The pattern of pressure change implies increased westerly flows across the western parts of the continents. These contribute to increases in mean precipitation (Figure 10.9) and increased precipitation intensity (Meehl et al., 2005a). "</i></span></blockquote>
I will have to figure out the mean sea-level pressure change over this time period to verify this hypothesis.<br />
<br />
The following recent research article looks into the shifts in geopotential height over shorter time durations:<br />
<br />
<div class="csl-bib-body" style="line-height: 1.35;">
<div class="csl-entry" style="clear: left;">
<div class="csl-left-margin" style="float: left; padding-right: 0.5em; text-align: right; width: 1em;">
<span style="font-size: x-small;">[1]</span></div>
<div class="csl-right-inline" style="margin: 0px 0.4em 0px 1.5em;">
<span style="font-size: x-small;">Y. Y. Hafez and M. Almazroui, “The Role Played by Blocking Systems over Europe in Abnormal Weather over Kingdom of Saudi Arabia in Summer 2010,” <i>Advances in Meteorology</i>, vol. 2013, p. 20, 2013.</span></div>
<span style="font-size: x-small;">
</span></div>
<span style="font-size: x-small;">
<span class="Z3988" title="url_ver=Z39.88-2004&ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzotero.org%3A2&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=The%20Role%20Played%20by%20Blocking%20Systems%20over%20Europe%20in%20Abnormal%20Weather%20over%20Kingdom%20of%20Saudi%20Arabia%20in%20Summer%202010&rft.jtitle=Advances%20in%20Meteorology&rft.volume=2013&rft.aufirst=Y.%20Y.&rft.aulast=Hafez&rft.au=Y.%20Y.%20Hafez&rft.au=M.%20Almazroui&rft.date=2013&rft.pages=20"></span> </span></div>
<div class="csl-bib-body" style="line-height: 1.35;">
<div style="text-align: center;">
<span style="background-color: red;"> <b><span style="color: white;">ADDED</span></b> 7/8/2013 </span> </div>
<br />
One possibility for the larger-than-expected altitude change is that the average lapse rate has changed slightly. We can use the lapse rate variation of the barometric formula, and perturb the lapse rate, <i>L</i>, around its average value:<br />
<br />
$$ P(H) = P(0) (1- \frac{LH}{T_0})^{\frac{gm}{LR}} $$<br />
<br />
Granted, the error bars on this calculation are significant but we can see how subtle the effect is.<br />
<br />
Sea-level pressure, P(0) = 1013.25 mb<br />
Gas constant, R = 8.31446 J/K/mol<br />
Earth's gravity, g = 9.807 m/s^2<br />
Avg molecular weight, m = 0.02896 kg/m<br />
<br />
In 1960, the temperature was about 14.65<span class="st">°</span>C and 500 mb altitude was 5647 m.<br />
In 2010, the temperature was about 15.4<span class="st">°</span>C and 500 mb altitude was 5667 m.<br />
<br />
All we need to do is invert the <i>P(H)</i> formula for each pair of values, modifying <i>L</i> slightly.<br />
<br />
If we select a lapse rate, <i>L</i>, for 1960 of 0.005<span class="st">1°</span>C/m, we calculate <i>H</i> = 5647.9m for <i>P(H)</i>=500 mb.<br />
If we select a lapse rate, <i>L</i>, for 2010 of 0.0050<span class="st">°</span>C/m, we calculate <i>H</i> = 5668.4m for <i>P(H)</i>=500 mb.<br />
<br />
The difference in the two altitudes for a change in L of -0.0001<span class="st">°</span>C/m is 20.5m, about what the geopotential height chart shows.<br />
<br />
If we leave the <i>L</i> at 0.005<span class="st">1°</span>C/m for both 1960 and 2010, the difference of the 500mb altitudes is only 14.7m.<br />
<br />
To the extent that we can trust the numbers on the charts from Ostro, the change in geopotential height is suggesting a feedback effect in the lapse rate due to global warming. The lapse rate is decreasing over time, which implies that the heat capacity of the atmosphere is increasing (likely due to higher specific humidity), thereby buffering changes in temperature with altitude.<br />
<br />
This means that a given temperature increase at a particular altitude (where the CO2 IR window can achieve a radiative balance) will be reflected as a scale-modified temperature at sea level<br />
<br />
In 1960, the temperature difference at 500mb altitude is 0.005<span class="st">1°</span>C/m * 5647.9m = 28.8<span class="st">0°</span>C<br />
In 2010, the temperature difference at 500mb altitude is 0.0050<span class="st">°</span>C/m * 5668.4m = 28.34<span class="st">°</span>C<br />
<br />
The difference at sea-level from the chart is 15.4<span class="st">°</span>C-14.65<span class="st">°</span>C = 0.75<span class="st">°</span>C whereas the difference at the 500mb altitude assuming the modified lapse rate is 0.75<span class="st">°</span>C - 0.46<span class="st">°</span>C = 0.29<span class="st">°</span>C. If the lapse rate didn't change then this sea-level difference would maintain at a constant atmospheric pressure isobar in altitude.<br />
<br />
<span class="reference-text"><span class="citation book">For implications in the interpretation, see page 24 of National Research Council Panel on Climate Change Feedbacks (2003). <a class="external text" href="http://books.google.com/?id=X13aXHpni2MC&printsec=frontcover" rel="nofollow"><i>Understanding climate change feedbacks</i></a> : National Academies Press. <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="http://en.wikipedia.org/wiki/Special:BookSources/978-0-309-09072-8" title="Special:BookSources/978-0-309-09072-8">978-0-309-09072-8</a>.</span></span> They caution that the measurements require some precision, otherwise the errors can multiply due to the differences between two large numbers. <br />
<br />
<br />
<br /></div>
<br />@whuthttp://www.blogger.com/profile/18297101284358849575noreply@blogger.com0tag:blogger.com,1999:blog-8890671936020885112.post-64190721442813528142013-06-09T19:32:00.001-07:002013-06-09T19:43:47.330-07:00Characterization of Battery Charging and DischargingI had the good fortune of taking a week long <a href="http://training.sae.org/academies/acad06/">Society of Automotive Engineers (SAE) Academy</a> class on hybrid/electric vehicles. The take-home message behind HEV and EV technology is to remember that a quality battery <i>plus </i>optimizing the management of battery cycling remain the keys to success. That is not surprising -- we all know that gasoline has long been "king", and since current battery technology has nowhere near the energy density of gasoline, the battery has turned into a "diva". In other words, it will perform as long as it is in charge (so to speak) and the battery is well maintained.<br />
<br />
I can report that much class time was devoted to the electrochemistry of Lithium-ion batteries. Lithium is an ideal elemental material due to its position in the upper left-hand corner of the periodic table -- in other words a very lightweight material with a potentially high energy density.<br />
<br />
What was surprising to me was the sparseness of detailed characterization of the material properties. One instructor stated that the lack of measured diffusion properties for battery cell specifications was a pet peeve of his. Having these properties at hand allows a design engineer to better model the charging and discharging characteristics of the battery, and thus to perhaps to develop better battery management schemes. Coming from the semiconductor world, it is almost unheard of to do design without adequate device characteristics such as mobility and diffusivity.<br />
<br />
From my perspective, this is not necessarily bad. Any time I see an anomalous behavior or missing piece, it opens the possibility I can fill a modeling niche.<br />
<br />
<h2>
Introduction</h2>
Modern rechargeable battery technology still relies on the principles of electro-chemistry and a reversible process, which hasn’t changed in fundamental terms since the first lead-acid battery came to market in the early 1900’s. What has changed is the combination of materials that make a low-cost, lightweight, and energy-efficient battery which will serve the needs of demanding applications such as electric and hybrid-electric vehicles (EV/HEV). <br />
<br />
As energy efficient operation is dependent on the properties of the materials being combined, it is well understood that characterizing the materials is important to advancing the state-of-the-art (and in increasing EV acceptance). <br />
<br />
Of vital importance is the characterization of diffusion in the electrode materials, as that is the rate-limiting factor in determining the absolute charging and discharging speed of the material-specific battery technology. Unfortunately, because of the competitive nature of battery producers, many of the characteristics are well-guarded and treated as trade secrets. For example, it is very rare to find diffusion coefficient characteristics on commercial battery specification sheets, even though this kind of information is vital for optimizing battery management schemes [7].<br />
<br />
<div class="Body">
In comparison to the relatively simple diffusional mechanisms of silicon oxide<span style="font-size: x-small;"></span> growth, the engineered structure of well-designed battery cell presents a significant constraint to the diffusional behavior. In <b>Figure 1</b> below we show a schematic of a single lithium-ion cell and the storage particles that charge and discharge. The disordered nature of the storage particles in <b>Figure 2</b> is often described by what is referred to as a tortuosity measure. </div>
<br />
<table><tbody>
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<tr><td style="text-align: center;"><a href="http://img534.imageshack.us/img534/4997/lithiumcartoon.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="143" src="http://img534.imageshack.us/img534/4997/lithiumcartoon.jpg" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><b>Figure 1</b>: <!--[if gte mso 9]><xml>
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<![endif]--></td></tr>
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</td><td><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="http://img521.imageshack.us/img521/5460/lithiumdisorder.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="130" src="http://img521.imageshack.us/img521/5460/lithiumdisorder.png" width="200" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><b>Figure 2</b> : <!--[if gte mso 9]><xml>
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The constraints on the diffusion is that it is limited in scale to that of the radius of the storage particle. The length scale is limited essentially to the values <i>L</i> to <i>L<span style="font-size: x-small;">max</span></i> shown in <b>Figure 3</b> below.<br />
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="http://img109.imageshack.us/img109/7131/churikovschematic.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="244" src="http://img109.imageshack.us/img109/7131/churikovschematic.png" width="640" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><b>Figure 3 </b>: Diffusion of ions takes place through the radial shell of the LiFePO<span style="font-size: xx-small;">4</span> spherical particle [1]. During the discharge phase, the ions need to migrate outward through shell and through the SEI barrier before reaching the electrolyte. At this point they can contribute to current flow.</td></tr>
</tbody></table>
The size of the particles also varies as shown in <b>Figure 4</b> below. The two Lithium-ion materials under consideration, LiFePO<span style="font-size: x-small;">4</span> and LiFeSO<span style="font-size: x-small;">4</span>F, have different materials properties but are structurally very similar (matrixed particles of mixed size) so that we can use a common analysis approach. This essentially allows us to apply uncertainty in the diffusion coefficient and uncertainties in the particle size to establish a common diffusional behavior formulation.<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="http://img708.imageshack.us/img708/1729/lithiumparticlesizedist.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="265" src="http://img708.imageshack.us/img708/1729/lithiumparticlesizedist.png" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><b>Figure 4 </b>: Particle size distribution of FeSO4F spherical granules [2]. The variation in lengths and material diffusivities opens the possibility of applying uncertainty quantification to a model of diffusive growth.</td></tr>
</tbody></table>
<br />
<h2>
Dispersive Diffusion Analysis of Discharging</h2>
The diffusion of ions through the volume of a spherical particle does have similarity to classical regimes such as the diffusion of silicon through silicon dioxide. That process in fact leads to the familiar Fick's law of diffusion, whereby the growing layer of oxide follows a parabolic growth law (in fact this is a square root with time, but was named parabolic by the semiconductor technology industry for historical reasons, <a href="http://books.google.com/books/about/The_Oil_Conundrum.html?id=oY2ZPn5EOTQC">see for example here</a>).<br />
<br />
The model that we can use for Li+ diffusion derives from the classic solution to the Fokker-Planck equation of continuity (neglecting any field driven drift).<br />
$$ \frac{\partial C}{\partial t} - D \nabla^2 C=0 $$ where C is a concentration and D is the diffusion coefficient. Ignoring the spherical orientation, we can just assume a solution along a one dimensional radially outward axis, <i><b>x</b></i>:<br />
<br />
$$ \large C(t,x|D) = \frac{1}{\sqrt{4 \pi D t}} e^{-x^2/{4 D t}} $$<br />
This is a marginal probability which depends on the diffusion coefficient. Since we do not know the variance of the diffusivity, we can apply a maximum entropy distribution across D.<br />
$$ \large p_d(D) = \frac{1}{D_0} e^{-D/D_0} $$<br />
This simplifies the representation to the following workable formulation.<br />
$$ \large C(t,x) = \frac{1}{2 \sqrt{D_0 t}} e^{-x/{\sqrt{D_0 t}}} $$<br />
We now have what is called a kernel solution (i.e. Green's function) that we can apply to specific sets of initial conditions and forcing functions, the latter solved via convolution.<br />
<br />
<b>Fully Charged Initial Conditions</b><br />
Assume the spherical particle is uniformly distributed with a charge density C(0, x) at time t=0.<br />
<br />
<b>Discharging Model</b><br />
For every point along the dimensions of the particle of size L, we calculate the time it takes to diffuse to the outer edge, where it can enter the electrolytic medium. This is simply an integral of the C(t, x) term for all points starting from x' = d to L, where d is the inner core radius.<br />
$$ \large C(t) = \int_{d}^L C(t,L-x) dx $$<br />
this integrates straightforwardly to this concise representation:<br />
$$ \large C(t) = C_0 \frac{ 1 - e^{-(L-d)/{\sqrt{D_0 t}}} } {L - d} $$<br />
The voltage of the cell is essentially the amount of charge available, so as this charge depletes, the voltage decreases in proportion.<br />
<br />
We can test the model on two data sets corresponding to a LiPO<span style="font-size: x-small;">4</span> cell [1] and a LiSO<span style="font-size: x-small;">4</span>F cell [2]. <b>Figure 5</b> below shows the model fit for LiPO<span style="font-size: x-small;">4</span> as the red dotted line, and which should be level-compared to the solid black line labelled 1. The other curves labelled 2,3,4,5 are alternative diffusional model approximations applied by Churikov <i>et al</i> that clearly do not work as well as the dispersive diffusion formulation derived above.<br />
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="http://img5.imageshack.us/img5/6225/lifepo4.gif" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="500" src="http://img5.imageshack.us/img5/6225/lifepo4.gif" width="640" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><b>Figure 5</b> : Discharge profile of LiFePO4 battery cell [1], with the red dotted line showing the parameterized dispersive diffusion model. The curves labelled 1 through 5 show alternative models that the authors applied to fit the data. Only the dispersive diffusion model duplicates the fast drop-off and long-time scale decline.</td></tr>
</tbody></table>
<br />
<b>Figure 6</b> below shows the fit to voltage characteristics of a LiSO4F cell, drawn as a red dotted line above the light gray data points. In this case the diffusional model by Delacourt shown in solid black is well outside acceptable agreement.<br />
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="http://img826.imageshack.us/img826/1001/lifeso4f.gif" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="464" src="http://img826.imageshack.us/img826/1001/lifeso4f.gif" width="640" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><b>Figure 6</b> : Discharge profile of LiFeSO4F battery cell [2], with
the red dotted line showing the parameterized dispersive diffusion
model. The black curve shows the model that the
authors applied to fit the data. </td></tr>
</tbody></table>
The question is why does this simple formulation work so well? As with many similar cases of characterizing disordered material, the fundamentally derived solution needs to be adjusted to take into account the uncertainty in the parameter space. However, this step is not routinely performed and by adding modeling details (see [4]) to try to make up for a poor fit works only as a cosmetic heuristic. In contrast, by performing the uncertainty quantification, like we did with the diffusion coefficient, the first-order solution works surprisingly well with no need for additional detail.<br />
<br />
<b>Constant Current Discharge</b><br />
Instead of assuming that the particle size is L, we can say that the L is an average and apply the same maximum entropy spread in values.<br />
$$ \large C(t) = \int_{0}^{\infty} C(t,x) \frac{1}{L} e^{-x/L} dx $$<br />
this integrates straightforwardly to this concise representation:<br />
$$ \large C(t) = C_0 \frac{1}{ L + {\sqrt{D_0 t}} } $$<br />
The reason we do this is to allow us to recursively define the change in charge to a current. In this case, to get current we need to differentiate the charge with respect to time.<br />
$$ \large I(t) = \frac{dC(t)}{dt} $$<br />
This differentiates to the following expression<br />
$$ \large I(t) = - \frac{C_0}{ (L + {\sqrt{D_0 t}})^2} \frac{1}{2 \sqrt{t}} $$<br />
But note that we can insert C(t) back in to the expression<br />
$$ \large I(t) = \frac{C(t)}{(L + {\sqrt{D_0 t}}) 2 \sqrt{t}} $$<br />
Finally, since I(t) is a constant and we can set that to a value of I_constant. Then the charge has the following profile<br />
$$ C(t) = C(0) - k_c I_{constant} (L + \sqrt{Dt}) 2 \sqrt{t} $$<br />
or as a voltage decline<br />
$$ V(t) = V(0) - k_v I_{constant} (L + \sqrt{Dt}) 2 \sqrt{t} $$<br />
For a set of constant current values, we can compare this formulation against experimental data for LiFePO4 (shown as gray open circles) shown in<b> Figure 7</b> below. A slight constant current offset (which may arise from unspecified shunting and/or series elements) was required to allow for the curves to align proportionally. Even with that, it is clear that the dispersive diffusion formulation works better than the conventional model (solid black lines) except where the discharge is nearing completion.<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="http://img221.imageshack.us/img221/5063/lithiumcurrentconstant.gif" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="408" src="http://img221.imageshack.us/img221/5063/lithiumcurrentconstant.gif" width="640" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><b>Figure 7</b> : Constant current discharge profile [6]. Superimposed as dotted lines are the set of model fits which use the current value as a fixed parameter. </td></tr>
</tbody></table>
<br />
We can also model battery charging but the lack of information on the charging profile makes the discharge behavior a simpler study.<br />
<br />
<b>
Related Diffusion Topics</b><br />
<ul>
<li><a href="http://theoilconundrum.com/">Dispersive Transport</a></li>
<li><a href="http://contextearth.com/">Corrosive Growth</a> </li>
<li><a href="http://theoilconundrum.blogspot.com/2012/07/bakken-dispersive-diffusion-oil.html">Dispersive Flow in Fractured Wells</a></li>
<li><a href="http://theoilconundrum.blogspot.com/2011/09/fat-tail-impulse-response-of-co2.html">Diffusive CO2 Sequestration</a></li>
<li><a href="http://theoilconundrum.blogspot.com/2013/03/ocean-heat-content-model.html">Ocean Heat Content</a></li>
</ul>
<br />
<b>References</b><br />
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<span style="font-size: x-small;">A. Churikov, A. Ivanishchev, I. Ivanishcheva, V. Sycheva, N. Khasanova, and E. Antipov, “Determination of lithium diffusion coefficient in LiFePO< sub> 4 electrode by galvanostatic and potentiostatic intermittent titration techniques,” <i>Electrochimica Acta</i>, vol. 55, no. 8, pp. 2939–2950, 2010.</span></div>
</div>
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<span style="font-size: x-small;">C. Delacourt, M. Ati, and J. Tarascon, “Measurement of Lithium Diffusion Coefficient in Li y FeSO4F,” <i>Journal of The Electrochemical Society</i>, vol. 158, no. 6, pp. A741–A749, 2011.</span></div>
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<span style="font-size: x-small;">M. Park, X. Zhang, M. Chung, G. B. Less, and A. M. Sastry, “A review of conduction phenomena in Li-ion batteries,” <i>Journal of Power Sources</i>, vol. 195, no. 24, pp. 7904–7929, Dec. 2010.</span></div>
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<span style="font-size: x-small;">J. Christensen and J. Newman, “A mathematical model for the lithium-ion negative electrode solid electrolyte interphase,” <i>Journal of The Electrochemical Society</i>, vol. 151, no. 11, pp. A1977–A1988, 2004.</span></div>
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<span style="font-size: x-small;">Q. Wang, H. Li, X. Huang, and L. Chen, “Determination of chemical diffusion coefficient of lithium ion in graphitized mesocarbon microbeads with potential relaxation technique,” <i>Journal of The Electrochemical Society</i>, vol. 148, no. 7, pp. A737–A741, 2001.</span></div>
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<span style="font-size: x-small;">P. M. Gomadam, J. W. Weidner, R. A. Dougal, and R. E. White, “Mathematical modeling of lithium-ion and nickel battery systems,” <i>Journal of Power Sources</i>, vol. 110, no. 2, pp. 267–284, 2002.</span></div>
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@whuthttp://www.blogger.com/profile/18297101284358849575noreply@blogger.com2tag:blogger.com,1999:blog-8890671936020885112.post-80255317968039812312013-05-14T21:02:00.001-07:002013-07-10T06:34:56.045-07:00The homework problem to end all homework problems<div class="separator" style="clear: both; text-align: center;">
</div>
This is a problem that has driven anyone that has studied climate science up the wall.<br />
<br />
<i><b>Premise:</b></i> Venus has an adiabatic index <span class="st">γ (gamma) and a temperature lapse rate </span><span class="st"><span class="st">λ (lambda). Earth also has an adiabatic index and temperature lapse rate. These have been measured, and for the Earth a standard atmospheric profile has been established. The general relationship is based on thermodynamic principles but the shape of the profile diverges from simple applications of adiabatic principles. In other words, a heuristic is applied to allow it to match the empirical observations, both for Venus and Earth. See <a href="http://theoilconundrum.blogspot.com/2013/03/standard-atmosphere-model-and.html">this link for more background</a>. </span></span><br />
<br />
<i><b>Assigned Problem:</b></i> Derive the adiabatic index and lapse rate for both planets, Venus and Earth, using only the planetary gravitational constant, the molar composition of atmospheric constituents, and any laws of physics that you can apply. The answer has to be right on the mark with respect to the empirically-established standards.<br />
<br />
<i><b>Caveat:</b></i> Reminder that this is a tough nut to crack.<br />
<br />
<i><b>Solution:</b></i> The approach to use is concise but somewhat twisty. We work along two paths, the initial path uses basic physics and equations of continuity; while the subsequent path ties the loose ends together using thermodynamic relationships which result in the familiar barometric formula and lapse rate formula. The initial assumption that we make is to start with a sphere that forms a continuum from the origin; this forms the basis of a <i>polytrope</i>, a useful abstraction to infer the generic properties of planetary objects.<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhMg4z3UEydcQ2e20FEzpeAtBBakSJVgC4qutGvpP0_GyYA2hWIC1JkjIGBWpuHxLwZCXiZngjVntsduQpglAPkpbNjACYPhumQA2MGqnEHNx5agCDXIvh86sEThsgzt5zC0_BO7ZLWn7U/s1600/radial-gradient.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="183" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhMg4z3UEydcQ2e20FEzpeAtBBakSJVgC4qutGvpP0_GyYA2hWIC1JkjIGBWpuHxLwZCXiZngjVntsduQpglAPkpbNjACYPhumQA2MGqnEHNx5agCDXIvh86sEThsgzt5zC0_BO7ZLWn7U/s200/radial-gradient.png" width="200" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">An abstracted planetary atmosphere</td></tr>
</tbody></table>
<div class="separator" style="clear: both; text-align: center;">
</div>
The atmosphere has a density <span class="st"> ρ, that decreases outward from the origin. The basic laws we work with are the following:</span><br />
<span class="st"></span><br />
<h4>
<span class="st">Mass Conservation</span></h4>
$$ \frac{dm(r)}{dr} = 4 \pi r^2 \rho $$<br />
<h4>
Hydrostatic Equilibrium</h4>
$$ \frac{dP(r)}{dr} = - \rho g = - \frac{Gm(r)}{r^2}\rho $$<br />
<br />
To convert to purely thermodynamic terms, we first integrate the hydrostatic equilibrium relationship over the volume of the sphere<br />
$$ \int_0^R \frac{dP(r)}{dr} 4 \pi r^3 dr = 4 \pi R^3 P(R) - \int_0^R 12 P(r) \pi r^2 dr $$<br />
on the right side we have integrated by parts, and eliminate the first term as P(R) goes to zero (<span style="font-size: x-small;">note: <i>upon review, the zeroing of P(R) is an approximation if we do not let R extend to the deep pressure vacuum of space, as we recover the differential form later -- right now we just assume P(R) decreases much faster than R^3 increases</i></span> ). We then reduce the second term using the mass conservation relationship, while recovering the gravitational part:<br />
$$ - 3 \int_0^M\frac{P}{\rho}dm = -\int_0^R 4 \pi r^3 \frac{G m(r)}{r^2} dr $$<br />
again we apply the mass conservation<br />
$$ - 3 \int_0^M\frac{P}{\rho}dm = -\int_0^M \frac{G m(r)}{r} dm $$<br />
The right hand side is simply the total gravitational potential energy <span class="st"> Ω while the left side reduces to a pressure to volume relationship:</span><br />
<span class="st"> </span>$$ - 3 \int_0^V P dV = \Omega$$<br />
<span class="st">This becomes a variation of the <i>Virial Theorem</i> relating internal energy to potential energy.</span><br />
<br />
<span class="st">Now we bring in the thermodynamic relationships, starting with the ideal gas law with its three independent variables. </span><br />
<span class="st"><br /></span>
<br />
<h4>
<span class="st">Ideal Gas Law</span></h4>
<span class="st">$$ PV = nRT $$</span><br />
<h4>
<span class="st">Gibbs Free Energy</span></h4>
<span class="st">$$ E = U - TS + PV $$</span><br />
<span class="st"></span><br />
<h4>
<span class="st">Specific Heat (in terms of molecular degrees of freedom)</span></h4>
<span class="st">$$ c_p = c_v + R = (N/2 + 1) R $$</span><br />
<br />
<br />
On this path, we make the assertion that the Gibbs free energy will be minimized with respect to perturbations. i.e. a variational approach.<br />
<br />
$$ dE = 0 = dU - d(TS) + d(PV) = dU - TdS - SdT + PdV + VdP $$<br />
<br />
Noting that the system is closed with respect to entropy changes (an adiabatic or isentropic process) and substituting the ideal gas law featuring a molar gas constant for the last term.<br />
<br />
$$ 0 = dU - SdT + PdV + VdP = dU - SdT + PdV + R_n dT$$<br />
<br />
At constant pressure (dP=0) the temperature terms reduce to the specific heat at constant pressure:<br />
<br />
$$ - S dT + nR dT = (c_v +R_n) dT = c_p dT $$ <br />
<br />
Rewriting the equation<br />
<br />
$$ 0 = dU + c_p dT + P dV $$<br />
<br />
Now we can recover the differential virial relationship derived earlier:<br />
<br />
$$ - 3 P dV = d \Omega $$ <br />
<br />
and replace the unknown <i>PdV</i> term<br />
<br />
$$ 0 = dU + c_p dT - d \Omega / 3 $$<br />
<br />
but dU is the same potential energy term as dΩ, so<br />
<br />
$$ 0 = 2/3 d \Omega+ c_p dT $$<br />
<br />
Linearizing the potential gravitational energy change with respect to radius<br />
<br />
$$ 0 = \frac{2 m g}{3} dr + c_p dT $$<br />
<br />
Rearranging this term we have derived the lapse formula<br />
<br />
$$ \frac{dT}{dr} = - \frac{mg}{3/2 c_p} $$<br />
<br />
Reducing this in terms of the ideal gas constant and molecular degrees of freedom N<br />
<br />
$$ \frac{dT}{dr} = - \frac{mg}{3/2 (N/2+1) R_n} $$<br />
<br />
We still need to derive the adiabatic index, by coupling the lapse rate formula back to the hydrostatic equilibrium formulation.<br />
<br />
Recall that the perfect adiabatic relationship (the <i>Poisson's equation</i> result describing the <a href="http://en.wikipedia.org/wiki/Potential_temperature">potential temperature</a>) does not adequately describe a standard atmosphere -- being 50% off in lapse rate -- and so we must use a more general <i>polytropic process</i> approach.<br />
<br />
Combining the Mass Conservation with the Hydrostatic Equilibrium:<br />
<br />
$$ \frac{1}{r^2} \frac{d}{dr} (\frac{r^2}{\rho} \frac{dP}{dr}) = -4 \pi G \rho $$<br />
<br />
if we make the substitution<br />
$$ \rho = \rho_c \theta^n $$<br />
where <i>n</i> is the <a href="http://en.wikipedia.org/wiki/Polytropic_index"><i>polytropic index</i></a>. In terms of pressure via the ideal gas law<br />
$$ P = P_c \theta^{n+1} $$<br />
if we scale <b><i>r </i></b>as the dimensionless <span class="st">ξ :</span><br />
<br />
$$ \frac{1}{\xi^2} \frac{d}{d\xi} (\frac{\xi^2}{\rho} \frac{dP}{d\xi}) = - \theta^n $$ <br />
<br />
This formulation is known as the <a href="http://en.wikipedia.org/wiki/Lane%E2%80%93Emden_equation"><i>Lane-Emden</i> equation</a> and is notable for resolving to a polytropic term. A solution for <i>n</i>=5 is<br />
$$ \theta = ({1 + \xi^2/3})^{-1/2} $$<br />
<br />
We now have a link to the polytropic process equation<br />
$$ P V^\gamma = {constant} $$<br />
and<br />
$$ P^{1-\gamma} T^{\gamma} = {constant} $$ <br />
or<br />
$$ P = P_0 (\frac{T}{T_0})^{\frac{\gamma}{1-\gamma}} $$<br />
Tieing together the loose ends, we take our lapse rate gradient<br />
$$ \frac{dT}{dr} = \frac{mg}{3/2 (N/2+1) R} $$ <br />
and convert that into an altitude profile, where <i>r = z</i><br />
$$ T = T_0 (1 - \frac{z}{f z_0}) $$<br />
where<br />
$$ z_0 = \frac{R T_0}{m g} $$<br />
and<br />
$$ f = 3/2 (1 + N/2) $$<br />
and the temperature gradient, aka lapse rate<br />
$$ \lambda= \frac{m g}{ 3/2 (1 + N/2) R } $$ <br />
To generate a polytropic process equation from this, we merely have to raise the lapse rate to a power, so that we recreate the power law version of the <a href="http://en.wikipedia.org/wiki/Barometric_formula"><i>barometric formula</i></a>:<br />
$$ P = P_0 (1 - \frac{z}{f z_0})^f $$<br />
which essentially reduces to Poisson's equation on substitution: <br />
$$ P = P_0 (T/T_0)^f $$<br />
where the equivalent adiabatic exponent is<br />
$$ f = \frac{\gamma}{1-\gamma} $$<br />
<br />
Now we have both the lapse rate, barometric formula, and Poisson's equation derived based only on the gravitational constant<i> g</i>, the gas law constant <i>R</i>, the average molar molecular weight of the atmospheric constituents <i>m</i>, and the average degrees of freedom <i>N</i>.<br />
<br />
<i><b>Answer: </b></i>Now we want to check the results against the observed values for the two planets<br />
<br />
<i>Parameters
</i><br />
<table border="0" cellpadding="0" cellspacing="0" style="margin-left: auto; margin-right: auto; text-align: left; width: 383px;"><colgroup><col style="width: 48pt;" width="64"></col>
<col style="mso-width-alt: 3949; mso-width-source: userset; width: 81pt;" width="108"></col>
<col style="mso-width-alt: 3035; mso-width-source: userset; width: 62pt;" width="83"></col>
<col span="2" style="width: 48pt;" width="64"></col>
</colgroup><tbody>
<tr height="20" style="height: 15.0pt;">
<td height="20" style="background-color: black; color: white; font-family: Arial,Helvetica,sans-serif; height: 15pt; width: 48pt;" width="64"><b>Object</b></td>
<td style="background-color: black; color: white; font-family: Arial,Helvetica,sans-serif; text-align: center; width: 81pt;" width="108"><b>Main Gas</b></td>
<td style="background-color: black; color: white; font-family: Arial,Helvetica,sans-serif; text-align: center; width: 62pt;" width="83"><b>N</b></td>
<td style="background-color: black; color: white; font-family: Arial,Helvetica,sans-serif; text-align: center; width: 48pt;" width="64"><b>m</b></td>
<td style="background-color: black; color: white; font-family: Arial,Helvetica,sans-serif; text-align: center; width: 48pt;" width="64"><b>g</b></td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td height="20" style="font-family: Arial,Helvetica,sans-serif; height: 15pt;">Earth</td>
<td style="font-family: Arial,Helvetica,sans-serif; text-align: center;">N2, O2</td>
<td style="font-family: Arial,Helvetica,sans-serif; text-align: center;">5</td>
<td style="font-family: Arial,Helvetica,sans-serif; text-align: center;">28.96</td>
<td style="font-family: Arial,Helvetica,sans-serif; text-align: center;">9.807</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td height="20" style="font-family: Arial,Helvetica,sans-serif; height: 15pt;">Venus</td>
<td style="font-family: Arial,Helvetica,sans-serif; text-align: center;">CO2</td>
<td style="font-family: Arial,Helvetica,sans-serif; text-align: center;">6 </td>
<td style="font-family: Arial,Helvetica,sans-serif; text-align: center;">43.44</td>
<td style="font-family: Arial,Helvetica,sans-serif; text-align: center;">8.87</td>
</tr>
</tbody></table>
<i>Results</i>
<br />
<table border="0" cellpadding="0" cellspacing="0" style="margin-left: auto; margin-right: auto; text-align: left; width: 383px;"><colgroup><col style="width: 48pt;" width="64"></col>
<col style="mso-width-alt: 3949; mso-width-source: userset; width: 81pt;" width="108"></col>
<col style="mso-width-alt: 3035; mso-width-source: userset; width: 62pt;" width="83"></col>
<col span="2" style="width: 48pt;" width="64"></col>
</colgroup><tbody>
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<td height="20" style="background-color: black; color: white; font-family: Arial,Helvetica,sans-serif; height: 15pt; width: 48pt;" width="64"><b>Object</b></td>
<td style="background-color: black; color: white; font-family: Arial,Helvetica,sans-serif; text-align: center; width: 81pt;" width="108"><b>Lapse Rate</b></td>
<td style="background-color: black; color: white; font-family: Arial,Helvetica,sans-serif; text-align: center; width: 62pt;" width="83"><b>observed</b></td>
<td style="background-color: black; color: white; font-family: Arial,Helvetica,sans-serif; text-align: center; width: 48pt;" width="64"><b>f</b></td>
<td style="background-color: black; color: white; font-family: Arial,Helvetica,sans-serif; text-align: center; width: 48pt;" width="64"><b>observed</b></td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td height="20" style="font-family: Arial,Helvetica,sans-serif; height: 15pt;">Earth</td>
<td style="font-family: Arial,Helvetica,sans-serif; text-align: center;">6.506 C/km</td>
<td style="font-family: Arial,Helvetica,sans-serif; text-align: center;">6.5</td>
<td style="font-family: Arial,Helvetica,sans-serif; text-align: center;">21/4</td>
<td style="font-family: Arial,Helvetica,sans-serif; text-align: center;">5.25</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td height="20" style="font-family: Arial,Helvetica,sans-serif; height: 15pt;">Venus</td>
<td style="font-family: Arial,Helvetica,sans-serif; text-align: center;">7.72</td>
<td style="font-family: Arial,Helvetica,sans-serif; text-align: center;"><br />
7.72</td>
<td style="font-family: Arial,Helvetica,sans-serif; text-align: center;">6</td>
<td style="font-family: Arial,Helvetica,sans-serif; text-align: center;">6</td>
</tr>
</tbody></table>
<br />
All the numbers are spot on with respect to the empirical data recorded for both Earth and Venus, with supporting figures <a href="http://theoilconundrum.blogspot.com/2013/03/standard-atmosphere-model-and.html">available here</a>.<br />
<br />
<div style="text-align: center;">
-------</div>
<div style="text-align: center;">
</div>
The rough derivation that I previously <a href="http://theoilconundrum.blogspot.com/2013/03/standard-atmosphere-model-and.html">posted </a>to explain the empirical data was not very satisfying in its thoroughness. The more comprehensive derivation in this post serves to shore up the mystery behind the deviation from the adiabatic derivation. The key seems to be correctly accounting for the internal energy necessary to maintain the gravitational hydrostatic equilibrium. Since the polytropic expansion describes a process, the actual atmosphere can accommodate these constraints (while <i>minimizing Gibbs free energy</i> under <i>constant entropy </i>conditions) by selecting the appropriate polytropic index. The mystery of the profile seems not so mysterious anymore.<br />
<br />
Criticisms welcome as I have not run across anything like this derivation to explain the Earth's standard atmosphere profile nor the stable Venus data (not to mention the less stable Martian atmosphere). The other big outer planets filled with hydrogen are still an issue, as they seem to follow the conventional adiabatic profile, according to the few charts I have access to. The moon of Saturn, Titan, is an exception as it has a nitrogen atmosphere with methane as a greenhouse gas.<br />
<br />
BTW, this post is definitely <b>not</b> dedicated to <a href="http://www.realclimate.org/wiki/index.php?title=Ferenc_Miskolczi">Ferenc Miskolczi</a>. Please shoot me if I ever drift in that direction. It's a tough slog laying everything out methodically but worthwhile in the long run.<br />
<br />
<br />
<div style="text-align: center;">
<span style="background-color: red;"><span style="color: white;"> Added </span></span></div>
<br />
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj5pi3jeTwW4Tg0MkAlooA4qI0zlEtDvNuLb9pte4HwD_8mymf61yAktD0S8xeqxpNYppTecgbsIg3zI_jL_h9IFTBH49-m6WH39C2vNFcg780yO7etcfJDbz6tL9mQKF7SCnflKKrSWuQ/s1600/lapse_rate_contour.GIF" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="261" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj5pi3jeTwW4Tg0MkAlooA4qI0zlEtDvNuLb9pte4HwD_8mymf61yAktD0S8xeqxpNYppTecgbsIg3zI_jL_h9IFTBH49-m6WH39C2vNFcg780yO7etcfJDbz6tL9mQKF7SCnflKKrSWuQ/s400/lapse_rate_contour.GIF" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Added Fig 1 : Lapse Rate on Earth versus Latitude. From <br />
<div class="csl-bib-body" style="line-height: 1.35;">
<div class="csl-entry" style="clear: left;">
<div class="csl-right-inline" style="margin: 0px 0.4em 0px 1.5em;">
<span style="font-size: xx-small;">D. J. Lorenz and E. T. DeWeaver, “Tropopause height and zonal wind response to global warming in the IPCC scenario integrations,” <i>Journal of Geophysical Research: Atmospheres (1984–2012)</i>, vol. 112, no. D10, 2007.</span></div>
</div>
<span class="Z3988" title="url_ver=Z39.88-2004&ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzotero.org%3A2&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Tropopause%20height%20and%20zonal%20wind%20response%20to%20global%20warming%20in%20the%20IPCC%20scenario%20integrations&rft.jtitle=Journal%20of%20Geophysical%20Research%3A%20Atmospheres%20(1984%E2%80%932012)&rft.stitle=Journal%20of%20Geophysical%20Research%3A%20Atmospheres%20(1984%E2%80%932012)&rft.volume=112&rft.issue=D10&rft.aufirst=David%20J&rft.aulast=Lorenz&rft.au=David%20J%20Lorenz&rft.au=Eric%20T%20DeWeaver&rft.date=2007&rft.issn=2156-2202"></span>
</div>
</td></tr>
</tbody></table>
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="http://img46.imageshack.us/img46/1349/lapseratelatitude.gif" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="256" src="http://img46.imageshack.us/img46/1349/lapseratelatitude.gif" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><span class="st">Added Fig 2 : Lapse Rate on Earth versus Latitude. The average was calculated by integrating <br />
with effective cross-sectional area weighting of (sin(Latitude+2.5)-sin(Latitude-2.5)) . Adapted from <br />
</span><br />
<div class="csl-bib-body" style="line-height: 1.35;">
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<div class="csl-right-inline" style="margin: 0px 0.4em 0px 1.5em;">
<span class="st"><span style="font-size: xx-small;">J.
P. Syvitski, S. D. Peckham, R. Hilberman, and T. Mulder, “Predicting
the terrestrial flux of sediment to the global ocean: a planetary
perspective,” <i>Sedimentary Geology</i>, vol. 162, no. 1, pp. 5–24, 2003.</span></span></div>
</div>
</div>
<span class="st">
</span></td></tr>
</tbody></table>
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="http://img822.imageshack.us/img822/214/lapserates.gif" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="320" src="http://img822.imageshack.us/img822/214/lapserates.gif" width="189" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Added Fig 3: This study also suggests an average lapse rate of 6.1C/km over the northern hemisphere.<br />
<div class="csl-bib-body" style="line-height: 1.35;">
<div class="csl-entry" style="clear: left;">
<div class="csl-left-margin" style="float: left; padding-right: 0.5em; text-align: right; width: 1em;">
<span style="font-size: xx-small;"><br /></span></div>
<div class="csl-right-inline" style="margin: 0px 0.4em 0px 1.5em;">
<span style="font-size: xx-small;">I. Mokhov and M. Akperov, “Tropospheric lapse rate and its relation to surface temperature from reanalysis data,” <i>Izvestiya, Atmospheric and Oceanic Physics</i>, vol. 42, no. 4, pp. 430–438, 2006.</span></div>
</div>
<span class="Z3988" title="url_ver=Z39.88-2004&ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzotero.org%3A2&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Tropospheric%20lapse%20rate%20and%20its%20relation%20to%20surface%20temperature%20from%20reanalysis%20data&rft.jtitle=Izvestiya%2C%20Atmospheric%20and%20Oceanic%20Physics&rft.stitle=Izvestiya%2C%20Atmospheric%20and%20Oceanic%20Physics&rft.volume=42&rft.issue=4&rft.aufirst=II&rft.aulast=Mokhov&rft.au=II%20Mokhov&rft.au=MG%20Akperov&rft.date=2006&rft.pages=430-438&rft.spage=430&rft.epage=438&rft.issn=0001-4338"></span> </div>
</td></tr>
</tbody></table>
Since I posted this derivation, I received feedback from several other blogs which I attached as comments below this post. In the original post I concluded by saying that I was satisfied with my alternate derivation, but after receiving the feedback, there is still the nagging issue of why the Venus lapse rate profile can be so linear in the lower atmosphere even though we know that the heat capacity of CO2 varies with temperature (particularly in the high temperature range of greater than 500 Kelvin).<br />
<br />
If we go back and look at the hydrostatic relation derived earlier, we see an interesting identity:<br />
$$ - 3 \int_0^M\frac{P}{\rho}dm = -\int_0^M \frac{G M}{r} dm $$<br />
If I pull out the differential from the integral<br />
$$ 3 \frac{P}{\rho} = \frac{G M}{r} $$<br />
and then realize that the left-hand side is just the Ideal Gas law<br />
$$ 3RT/m = \frac{G M}{r} $$<br />
This is internal energy due to gravitational potential energy.<br />
If we take the derivative with respect to<b> r</b>, or altitude:<br />
$$ 3R \frac{dT}{dr} = - \frac{G M m}{r^2} $$<br />
The right side is just the gravitational force on an average particle. So we essentially can derive a lapse rate directly:<br />
$$ \frac{dT}{dr} = - \frac{g m}{3 R} $$ <br />
This will generate a linear lapse rate profile of temperature that decreases with increasing altitude. Note however that this does not depend on the specific heat of the constituent atmospheric molecules. That is not surprising since it only uses the Ideal Gas law, with no application of the variational Gibbs Free Energy approach used earlier.<br />
<br />
What this gives us is a universal lapse rate that does not depend on the specific heat capacity of the constituent gases, only the mean molar molecular weight, <b>m</b>. This is of course an interesting turn of events in that it could explain the highly linear lapse profile of Venus. However, plugging in numbers for the gravity of Venus and the mean molecular weight (CO2 plus trace gases), we get a lapse rate that is <b><i>precisely twice</i></b> that which is observed.<br />
<br />
The "obvious" temptation is to suggest that half of the value of this derived hydrodynamic lapse rate would position it as the mean of the lapse rate gradient and an isothermal lapse rate (i.e. slope of zero).<br />
$$ \frac{dT}{dr} = - \frac{g m}{6 R} $$ <br />
The rationale for this is that most of the planetary atmospheres are not any kind of equilibrium with energy flow and are constantly swinging between an insolating phase during daylight hours, and then a outward radiating phase at night. The uncertainty is essentially describing fluctuations between when an atmosphere is isothermal <i>(little change of temperature with altitude producing a MaxEnt outcome in distribution of pressures, leading to the classic barometric formula</i>) or isentropic (<i>where no heat is exchanged with the surroundings, but the temperature can vary as rapid convection occurs</i>).<br />
<br />
In keeping with the Bayesian decision making, the uncertainty is reflected by equal an weighting between isothermal (zero lapse rate gradient) and an isentropic (adiabatic derivation shown). This puts the mean lapse rate at half the isentropic value. For Earth, the value of <i>g*m/3R</i> is 11.4 C/km. Half of this value is 5.7 C/km, which is a value closer to actual mean value than the US Standard Atmosphere of 6.5 C/km<br />
<br />
<blockquote class="tr_bq">
<b>J<i>. Levine, The Photochemistry of Atmospheres. Elsevier Science, 1985.</i></b><br />
<i>"The value chosen for the convective adjustment also influences the calculated surface temperature. In lower latitudes, the actual temperature decrease with height approximates the moist adiabatic rate. Convection transports H2O to higher elevations where condensation occurs, releasing latent heat to the atmosphere; this lapse rate, although variable, has an average annual value of <b>5.7 K/km</b> in the troposphere. In mid and high latitudes, the actual lapse rates are more stable; the vertical temperature profile is controlled by eddies that are driven by horizontal temperature gradients and by topography. These so-called baroclinic processes produce an average lapse rate of 5.2 K/km - It is interesting to note that most radiative convective models have used a lapse rate of 6.5 K km - which was based on date sets extending back to 1933. We know now that a better hemispherical annual lapse rate is closer to 5.2 K/km, although there may be significant seasonal variations. </i>"</blockquote>
<div class="csl-bib-body" style="line-height: 1.35;">
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</div>
<span class="Z3988" title="url_ver=Z39.88-2004&ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzotero.org%3A2&rft_id=urn%3Aisbn%3A9780323146630&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The%20Photochemistry%20of%20Atmospheres&rft.publisher=Elsevier%20Science&rft.aufirst=J.&rft.aulast=Levine&rft.au=J.%20Levine&rft.date=1985&rft.isbn=9780323146630"></span>
</div>
BTW, the following references are very interesting presentations on the polytropic approach.<br />
<b><br /></b>
<br />
<b>References</b><br />
<br />
<div class="csl-bib-body" style="line-height: 1.35;">
<div class="csl-entry" style="clear: left;">
<div class="csl-left-margin" style="float: left; padding-right: 0.5em; text-align: right; width: 1em;">
<span style="font-size: x-small;">[1]</span></div>
<div class="csl-right-inline" style="margin: 0px 0.4em 0px 1.5em;">
<span style="font-size: x-small;">“Polytropes.” [Online]. Available: http://mintaka.sdsu.edu/GF/explain/thermal/polytropes.html. [Accessed: 19-May-2013].</span></div>
<span style="font-size: x-small;">
</span></div>
<span style="font-size: x-small;">
</span></div>
<div class="csl-bib-body" style="line-height: 1.35;">
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<div class="csl-left-margin" style="float: left; padding-right: 0.5em; text-align: right; width: 1em;">
<span style="font-size: x-small;">[2]</span></div>
<div class="csl-right-inline" style="margin: 0px 0.4em 0px 1.5em;">
<span style="font-size: x-small;">B. Davies, “Stars Lecture.” [Online]. Available: http://www.ast.cam.ac.uk/~bdavies/Stars2 . [Accessed: 28-May-2013].</span></div>
</div>
</div>
<br />
<br />
<br />
<div style="text-align: center;">
<span style="background-color: red;"><span style="color: white;"> Even More Recent Research </span></span></div>
<br />
A number of Chinese academics [3,4] are attacking the polytropic atmosphere problem from an angle that I hinted at in the original <span class="st"><span class="st"> <a href="http://theoilconundrum.blogspot.com/2013/03/standard-atmosphere-model-and.html">Standard Atmosphere Model and Uncertainty in Entropy</a></span></span> post. The gist of their approach is to assume that the atmosphere is not under thermodynamic equilibrium (which it isn't as it continuously exchanges heat with the sun and outer space in a stationary steady-state) and therefore use some ideas of non-extensible thermodynamics. Specifically they invoke Tsallis entropy and a generalized Maxwell-Boltzmann distribution to model the behavioral move toward an equilibrium. This is all in the context of self-gravitational systems, which is the theme of this post. Why I think it is intriguing, is that they seem to tie the entropy considerations together with the polytropic process and arrive at some very simple relations (at least they appear somewhat simple to me).<br />
<br />
In the non-extensive entropy approach, the original Maxwell-Boltzmann (MB) exponential velocity distribution is replaced with the Tsallis-derived generalized distribution -- which looks like the following power-law equation:<br />
<br />
$$ f_q(v)=n_q B_q (\frac{m}{2 \pi k T})^{3/2} (1-(1-q) \frac{m v^2}{2 k T})^{\frac{1}{1-q}}$$<br />
<br />
The so-called <i>q</i>-factor is a non-extensivity parameter which indicates how much the distribution deviates from MB statistics. As <i>q</i> approaches 1, the expression gradually trasforms into the familiar MB exponentially damped v^2 profile.<br />
<br />
When <i>q</i> is slightly less than 1, all the thermodynamic gas equations change slightly in character. In particular, the scientist Du postulated that the lapse rate follows the familiar linear profile, but scaled by the (1-<i>q</i>) factor:<br />
<br />
$$ \frac{dT}{dr} = \frac{(1-q)g m}{R} $$<br />
<br />
Note that this again has no dependence on the specific heat of the constituent gases, and only assumes an average molecular weight. If <i>q</i>=7/6 or <i>Q </i>= 1-<i>q </i>= -1/6, we can model the<i> f</i>=6 lapse rate curve that we fit to earlier.<br />
<br />
There is nothing special about the value of f=6 other than the claim that this polytropic exponent is on the borderline for maintaining a self-gravitational system [5].<br />
<br />
Note that as <i>q</i> approaches unity, the thermodynamic equilibrium value, the lapse rate goes to zero, which is of course the maximum entropy condition of uniform temperature.<br />
<br />
The Tsallis entropy approach is suspiciously close to solving the problem of the polytropic standard atmosphere. Read Zheng's paper for their take [3] and also Plastino [6]. <br />
<br />
<blockquote class="tr_bq">
<span style="font-size: x-small;"><i>The cut-off in the polytropic distribution (5) is an example of what is known, within the field of non extensive thermostatistics, as “Tsallis cut-off prescription”, which affects the q-maximum entropy distributions when q < 1. In the case of stellar polytropic distributions this cut-off arises naturally, and has a clear physical meaning. The cut-off corresponds, for each value of the radial coordinate r, to the corresponding gravitational escape velocity.</i></span></blockquote>
This has implications for the derivation of the homework problem that we solved at the top of this post, where we eliminated one term of the integration-by-parts solution. Obviously, the generalized MB formulation does have a limit to the velocity of a gas particle in comparison to the classical MB view. The tail in the statistics is actually cut-off as velocities greater than a certain value are not allowed, depending on the value of <i>q</i>. As <i>q</i> approaches unity, the velocities allowed (i.e. escape velocity) approach infinity. <br />
<br />
As Plastino states [6]:<br />
<blockquote class="tr_bq">
<span style="font-size: x-small;"><i>Polytropic distributions happen to exhibit the form of q-MaxEnt distributions, that is, they constitute distribution functions in the (x,v) space that maximize the entropic functional Sq under the natural constraints imposed by the conservation of mass and energy.</i></span></blockquote>
The enduring question is does this describe our atmosphere adequately enough? Zheng and company certainly open it up to another interpretation.<br />
<br />
<div class="csl-bib-body" style="line-height: 1.35;">
<div class="csl-entry" style="clear: left;">
<div class="csl-left-margin" style="float: left; padding-right: 0.5em; text-align: right; width: 1em;">
<span style="font-size: x-small;">[3]</span></div>
<div class="csl-right-inline" style="margin: 0px 0.4em 0px 1.5em;">
<span style="font-size: x-small;">Y. Zheng, W. Luo, Q. Li, and J. Li, “The polytropic index and adiabatic limit: Another interpretation to the convection stability criterion,” <i>EPL (Europhysics Letters)</i>, vol. 102, no. 1, p. 10007, 2013.</span></div>
</div>
<span class="Z3988" title="url_ver=Z39.88-2004&ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzotero.org%3A2&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=The%20polytropic%20index%20and%20adiabatic%20limit%3A%20Another%20interpretation%20to%20the%20convection%20stability%20criterion&rft.jtitle=EPL%20(Europhysics%20Letters)&rft.stitle=EPL%20(Europhysics%20Letters)&rft.volume=102&rft.issue=1&rft.aufirst=Yahui&rft.aulast=Zheng&rft.au=Yahui%20Zheng&rft.au=Wang%20Luo&rft.au=Qinan%20Li&rft.au=Jianjun%20Li&rft.date=2013&rft.pages=10007&rft.issn=0295-5075"></span>
</div>
<div class="csl-bib-body" style="line-height: 1.35;">
<div class="csl-entry" style="clear: left;">
<div class="csl-left-margin" style="float: left; padding-right: 0.5em; text-align: right; width: 1em;">
<span style="font-size: x-small;">[4]</span></div>
<div class="csl-right-inline" style="margin: 0px 0.4em 0px 1.5em;">
<span style="font-size: x-small;">Z. Liu, L. Guo, and J. Du, “Nonextensivity and the q-distribution of a relativistic gas under an external electromagnetic field,” <i>Chinese Science Bulletin</i>, vol. 56, no. 34, pp. 3689–3692, Dec. 2011.</span></div>
</div>
<span class="Z3988" style="font-size: x-small;" title="url_ver=Z39.88-2004&ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzotero.org%3A2&rft_id=info%3Adoi%2F10.1007%2Fs11434-011-4750-2&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Nonextensivity%20and%20the%20q-distribution%20of%20a%20relativistic%20gas%20under%20an%20external%20electromagnetic%20field&rft.jtitle=Chinese%20Science%20Bulletin&rft.volume=56&rft.issue=34&rft.aufirst=ZhiPeng&rft.aulast=Liu&rft.au=ZhiPeng%20Liu&rft.au=LiNa%20Guo&rft.au=JiuLin%20Du&rft.date=2011-12-02&rft.pages=3689-3692&rft.spage=3689&rft.epage=3692&rft.issn=1001-6538%2C%201861-9541"></span>
</div>
<div class="csl-bib-body" style="line-height: 1.35;">
<div class="csl-entry" style="clear: left;">
<div class="csl-left-margin" style="float: left; padding-right: 0.5em; text-align: right; width: 1em;">
<span style="font-size: x-small;">[5]</span></div>
<div class="csl-right-inline" style="margin: 0px 0.4em 0px 1.5em;">
<span style="font-size: x-small;">M. V. Medvedev and G. Rybicki, “The Structure of Self-gravitating Polytropic Systems with n around 5,” <i>The Astrophysical Journal</i>, vol. 555, no. 2, p. 863, 2001.</span></div>
</div>
<span style="font-size: x-small;"><span class="Z3988" title="url_ver=Z39.88-2004&ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzotero.org%3A2&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=The%20Structure%20of%20Self-gravitating%20Polytropic%20Systems%20with%20n%20around%205&rft.jtitle=The%20Astrophysical%20Journal&rft.volume=555&rft.issue=2&rft.aufirst=Mikhail%20V.&rft.aulast=Medvedev&rft.au=Mikhail%20V.%20Medvedev&rft.au=George%20Rybicki&rft.date=2001&rft.pages=863"></span></span>
</div>
<span style="font-size: x-small;">
</span><br />
<div class="csl-bib-body" style="line-height: 1.35;">
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</span><br />
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<div class="csl-left-margin" style="float: left; padding-right: 0.5em; text-align: right; width: 1em;">
<span style="font-size: x-small;">[6]</span></div>
<div class="csl-right-inline" style="margin: 0px 0.4em 0px 1.5em;">
<span style="font-size: x-small;">A. Plastino, “Sq entropy and selfgravitating systems,” <i>europhysics news</i>, vol. 36, no. 6, pp. 208–210, 2005.</span></div>
</div>
<span class="Z3988" title="url_ver=Z39.88-2004&ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzotero.org%3A2&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Sq%20entropy%20and%20selfgravitating%20systems&rft.jtitle=europhysics%20news&rft.stitle=europhysics%20news&rft.volume=36&rft.issue=6&rft.aufirst=AR&rft.aulast=Plastino&rft.au=AR%20Plastino&rft.date=2005&rft.pages=208-210&rft.spage=208&rft.epage=210&rft.issn=0531-7479"></span>
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--@whuthttp://www.blogger.com/profile/18297101284358849575noreply@blogger.com7tag:blogger.com,1999:blog-8890671936020885112.post-65784072683375693602013-05-12T12:22:00.002-07:002013-05-12T12:22:25.509-07:00Airborne fraction of CO2 explained by sequestering modelAs acknowledgement of the <a href="http://news.nationalgeographic.com/news/energy/2013/05/130510-earth-co2-milestone-400-ppm/">atmospheric levels of CO2 reaching 400 PPM</a>, this post is meant to clear up one important misconception (<i>suggested prerequisite reading on fat-tail CO2 sequestration <a href="http://theoilconundrum.blogspot.com/2011/09/fat-tail-impulse-response-of-co2.html">here</a> and the significance of the fat-tail <a href="http://theoilconundrum.blogspot.com/2013/03/stochastic-analysis-of-log-sensitivity.html">here</a></i>)<br />
<br />
A recently active skeptic meme is that the amount of CO2 as an airborne fraction is decreasing over time.<br />
<blockquote class="tr_bq">
<i>"If we look at the data since Mauna Loa started, we see that the
percentage of the CO2 emitted by humans that “remains” in the atmosphere
has averaged around half, but that it has diminished over time, by
around 1% per decade.</i><br />
<i>Over the 30 year period 1959-1989 it was around 55%; over the following 20+ years it was just over 50%.</i><br />
<i>Why is this?"</i></blockquote>
What the befuddled fellow is talking about are the charts being shown below. These are being shown without much context and no supporting documentation, which puts the burden on the climate scientists to explain. Note that the airborne fraction does seem to decrease slightly over the past 50 years, even though the carbon emissions are increasing.<br />
<table><tbody>
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<a href="http://farm9.staticflickr.com/8344/8200196434_ebb7559913_b.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="150" src="http://farm9.staticflickr.com/8344/8200196434_ebb7559913_b.jpg" width="200" /></a></div>
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<a href="http://farm8.staticflickr.com/7225/7217254592_9a13cdbfeb_b.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="150" src="http://farm8.staticflickr.com/7225/7217254592_9a13cdbfeb_b.jpg" width="200" /></a></div>
</td></tr>
</tbody></table>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh1KJAkz0iNA6Uc622SWXBJZAY1Oy1vSBFolihc3l3ah3Vij9wu3uTClWlPLJls7j1JQwPWn_ojMvbP-fInMguO4A3l3HB6RReUcUUjq4CB8WPJR7C5-ueGvNam4wBnJbCl4eTjnDpNxMU/s1600/co2airborneFraction.GIF" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="179" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh1KJAkz0iNA6Uc622SWXBJZAY1Oy1vSBFolihc3l3ah3Vij9wu3uTClWlPLJls7j1JQwPWn_ojMvbP-fInMguO4A3l3HB6RReUcUUjq4CB8WPJR7C5-ueGvNam4wBnJbCl4eTjnDpNxMU/s320/co2airborneFraction.GIF" width="320" /></a></div>
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<br />
<div class="comment-body">
This obviously needs some explaining. The following figure illustrates what the CO2 sequestration model actually does.<br />
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgTbOFQkcH2EGtrhnORw4g2s2E9maPkcp2nHu5taLc-9ALeNkPnxmqEnCOtYGon9EumErw6YiovM9_NP_tmOuapQNZ01r4U10mD89gqOyQtMhlZRyzsHGkqK5hAmQJLP3Cr1gW2TwVEDp0/s1600/co2_fraction.gif" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="289" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgTbOFQkcH2EGtrhnORw4g2s2E9maPkcp2nHu5taLc-9ALeNkPnxmqEnCOtYGon9EumErw6YiovM9_NP_tmOuapQNZ01r4U10mD89gqOyQtMhlZRyzsHGkqK5hAmQJLP3Cr1gW2TwVEDp0/s640/co2_fraction.gif" width="640" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><b>Figure 1: </b> Model airborne fraction of CO2 against actual data</td></tr>
</tbody></table>
<div class="separator" style="clear: both; text-align: center;">
</div>
On the left is the data plotted together with the model of the yearly fraction not
sequestered out. The model is less noisy than the data but it does clearly decline as well. No big surprise as this is a response function, and responses are known to vary depending on the temporal profile of the input and the fat-tail in the adjustment time impulse response function.<br />
<br />
On the right is the model with the incorporation
of a temperature-dependent outgassed fraction. In this case the model is more noisy than the data, as it includes outgassing of CO2 depending on the global temperature for that year. Since the temperature is noisy, the CO2 fraction picks up all of that noise. Still, the airborne fraction shows a small yet perceptible decline, and the model matches the data well, especially in recent years where the temperature fluctuations are reduced.<br />
<br />
Amazing that over 50 years, the mean fraction has not varied much
from 55%. That has a lot to do with the math of diffusional physics.
Essentially a random walk moving into and out of sequestering sites is a
50/50 proposition. That’s the way to intuit the behavior, but the <a href="http://theoilconundrum.blogspot.com/2011/09/fat-tail-impulse-response-of-co2.html">math really does the heavy lifting in predicting the fraction sequestered out</a>.<br />
<br />
It looks like the theory matches the data once again. The skeptics provide a knee-jerk view that this behavior is not well understood, but not having done the analysis themselves, they lose out -- the skeptic meme is simply one of further propagating fear, uncertainty, and doubt (FUD) without concern for the underlying science.<br />
<br />
</div>
@whuthttp://www.blogger.com/profile/18297101284358849575noreply@blogger.com0tag:blogger.com,1999:blog-8890671936020885112.post-50524094577132435132013-05-03T21:41:00.001-07:002013-06-25T19:36:02.335-07:00Proportional Land/Sea Global Warming Model <div class="separator" style="clear: both; text-align: center;">
</div>
I found an interesting global temperature regression exercise that may lead to some insight with respect to understanding ocean heat content . <a href="http://theoilconundrum.blogspot.com/2013/03/ocean-heat-content-model.html">From a previous post</a> we estimated that about half the excess heat produced over the oceans is getting stored (sequestered) in the ocean depths. The intriguing premise is that we can potentially substantiate the flow of heat by comparing the relative growths of <i>global </i>temperatures against the <i>land-only</i> temperatures and the <i>ocean-surface</i> temperatures (i.e. the sea-surface temperatures known as SST).<br />
<br />
The elementary kriging approximation is that the global temperature anomaly (<i>T<sub>G</sub></i>) is a proportional mix of the ocean temperature (<i>T<sub>o</sub></i>) with the land temperature (<i>T<sub>l</sub></i> ):<br />
$$ T_G = p_o T_o + p_l T_l $$<br />
where<br />
$$ 1 = p_o + p_l $$<br />
with the approximate fraction of earth's coverage by the ocean equal to 0.7 (and the land therefore 0.3).<br />
<br />
We use that Hadley center data sets <br />
<table border="1"> <tbody>
<tr><th><span style="color: blue;"><i><span style="background-color: white;">global</span></i></span></th><th><span style="color: blue;"><i><span style="background-color: white;">land</span></i></span></th><th><span style="color: blue;"><i><span style="background-color: white;">ocean</span></i></span></th></tr>
<tr><th>hadcrut4</th><th>crutem4vgl</th><th>hadsst2gl</th></tr>
</tbody></table>
<br />
Shifting the baseline anomaly by at most 0.1C, we come up with the following fit, shown in <b>Figure 1</b>. The composed temperature lines up very closely to the reported global temperature, with the red areas peeking out where the agreement is not perfect.<br />
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhKMnh2yqnibJ9DeTx-mhlfjD-NZ16ftTK03H_gkFPmm_z1RJbRtqOewnFO7F0N0lxI92DbHFucG8inTHvxyIhUpLC0bUPpd6LLR9yvqC3XD7QvhiBuv-A5L4wD8YNodk1A42xc5xPw0Wc/s1600/hadcrut.GIF" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="251" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhKMnh2yqnibJ9DeTx-mhlfjD-NZ16ftTK03H_gkFPmm_z1RJbRtqOewnFO7F0N0lxI92DbHFucG8inTHvxyIhUpLC0bUPpd6LLR9yvqC3XD7QvhiBuv-A5L4wD8YNodk1A42xc5xPw0Wc/s400/hadcrut.GIF" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><b>Figure 1 </b>: Global temperature anomaly is straightforwardly recreated by assuming that the global temperature is composed by a proportion of ocean and land area. The fit works very well apart from a few points (years centered around 1948) that may be traced to systemic errors or discrepancies the database versions.</td></tr>
</tbody></table>
<br />
That by itself is only somewhat interesting, as it merely confirms that the Hadley center researchers know how to do first order proportional mapping (aka <i>kriging</i>). The regression agreement is shown in <b>Figure 2</b>.<br />
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg9g2Jc1-jqvA28KMlZss1dw_Fns_3fqmo7DbSj47zWD5lOS18agQj0Cc53B7LTRri1hQjJtJAR3-Ily8vuNMrgHxsOMAtf80KkVoUEzPmqq9ISyDXf0ntX4AwK3LvBd8ENT6LE462RHNk/s1600/hadcrut_global_composed.GIF" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="237" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg9g2Jc1-jqvA28KMlZss1dw_Fns_3fqmo7DbSj47zWD5lOS18agQj0Cc53B7LTRri1hQjJtJAR3-Ily8vuNMrgHxsOMAtf80KkVoUEzPmqq9ISyDXf0ntX4AwK3LvBd8ENT6LE462RHNk/s400/hadcrut_global_composed.GIF" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><b>Figure 2</b> : The composed temperature maps nearly one-to-one to the actual global temperature.</td></tr>
</tbody></table>
<br />
<br />
The other identity we need to consider involves a fraction of the heat sunk by the ocean. Since the land has essentially no heat sink, but that the ocean has a fraction <i><b>f</b></i> that acts as a heat sink, we can assert:<br />
$$ T_o = f T_g $$<br />
where <a href="http://theoilconundrum.blogspot.com/2013/03/ocean-heat-content-model.html">we determined previously</a> that<i><b> f</b></i> is about 0.5.<br />
<br />
We can plot the linear regression between the two below.<br />
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjkYCrKPvOw1tn9cToUO-nPri287Khj0dPGs_9mbcEFegwV9cuI-JsCkqIQ-FDDjngJA-sQJfY1jj6XwCsxJ0a1Oze-kH_ynntcRU5kanKOyQmCm35wh5DTJYxUG9Co8HxjBOtZ8Z9MXyg/s1600/hadcrut_sea_vs_land.GIF" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="243" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjkYCrKPvOw1tn9cToUO-nPri287Khj0dPGs_9mbcEFegwV9cuI-JsCkqIQ-FDDjngJA-sQJfY1jj6XwCsxJ0a1Oze-kH_ynntcRU5kanKOyQmCm35wh5DTJYxUG9Co8HxjBOtZ8Z9MXyg/s400/hadcrut_sea_vs_land.GIF" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><b>Figure 3 :</b> Linear regression between Land and SST temperature is more noisy.</td></tr>
</tbody></table>
<br />
As this is fit is fairly noisy, we can try to reduce the variance by plotting against the global mean temperature as a multiple regression fit. We take the first equation and replace each of the component temperatures with their fractional equivalents.<br />
$$ T_G = p_o f T_l + p_l T_l $$<br />
$$ T_G = p_o T_o + p_l \frac{T_o}{f} $$<br />
and then apply these as equal weightings<br />
$$ T_G = 1/2 ( p_o f T_l + p_l T_l ) + 1/2 ( p_o T_o + p_l \frac{T_o}{f} )$$<br />
<div class="separator" style="clear: both; text-align: center;">
</div>
rearranging terms<br />
$$ T_G = 1/2 ( f p_o + p_l ) T_l + 1/2 ( p_o + \frac{p_l}{f} ) T_o $$<br />
If we apply a multiple regression of the global temperature data against a linear combination of the ocean and temperature data we get the following tabulated results:<br />
<br />
<table bgcolor="black" border="1" cellpadding="0" cellspacing="0" style="width: 520px;"><colgroup><col style="width: 48pt;" width="64"></col>
<col style="mso-width-alt: 2779; mso-width-source: userset; width: 57pt;" width="76"></col>
<col style="mso-width-alt: 3547; mso-width-source: userset; width: 73pt;" width="97"></col>
<col span="2" style="width: 48pt;" width="64"></col>
<col style="mso-width-alt: 2742; mso-width-source: userset; width: 56pt;" width="75"></col>
<col style="mso-width-alt: 2925; mso-width-source: userset; width: 60pt;" width="80"></col>
</colgroup><tbody>
<tr height="20" style="height: 15.0pt;">
<td class="xl64" height="20" style="height: 15.0pt; width: 48pt;" width="64"></td>
<td class="xl64" style="width: 57pt;" width="76"><i><span style="color: yellow;">Coefficients</span></i></td>
<td class="xl64" style="width: 73pt;" width="97"><i><span style="color: yellow;">Standard Error</span></i></td>
<td class="xl64" style="width: 48pt;" width="64"><i><span style="color: yellow;">t Stat</span></i></td>
<td class="xl64" style="width: 48pt;" width="64"><i><span style="color: yellow;">P-value</span></i></td>
<td class="xl64" style="width: 56pt;" width="75"><i><span style="color: yellow;">Lower 95%</span></i></td>
<td class="xl64" style="width: 60pt;" width="80"><i><span style="color: yellow;">Upper 95%</span></i></td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td height="20" style="height: 15.0pt;"><b><span style="color: white;">Intercept</span></b></td>
<td align="right"><span style="color: white;">0.02225179</span></td>
<td align="right"><span style="color: white;">0.003884172</span></td>
<td align="right"><span style="color: white;">5.728838</span></td>
<td align="right"><span style="color: white;">4.97E-08</span></td>
<td align="right"><span style="color: white;">0.0145802</span></td>
<td align="right"><span style="color: white;">0.02992339</span></td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td height="20" style="height: 15.0pt;"><b><span style="color: white;">Land</span></b></td>
<td align="right"><span style="color: white;">0.29922517</span></td>
<td align="right"><span style="color: white;">0.015995888</span></td>
<td align="right"><span style="color: white;">18.70638</span></td>
<td align="right"><span style="color: white;">6.56E-42</span></td>
<td align="right"><span style="color: white;">0.26763182</span></td>
<td align="right"><span style="color: white;">0.33081852</span></td>
</tr>
<tr height="21" style="height: 15.75pt;">
<td class="xl63" height="21" style="height: 15.75pt;"><b><span style="color: white;">Ocean</span></b></td>
<td align="right" class="xl63"><span style="color: white;">0.65796939</span></td>
<td align="right" class="xl63"><span style="color: white;">0.024637656</span></td>
<td align="right" class="xl63"><span style="color: white;">26.70584</span></td>
<td align="right" class="xl63"><span style="color: white;">1.86E-60</span></td>
<td align="right" class="xl63"><span style="color: white;">0.60930775</span></td>
<td align="right" class="xl63"><span style="color: white;">0.70663103</span></td>
</tr>
</tbody></table>
<br />
With this information, we can solve for <i><b>f </b></i>and the land ocean split.<br />
$$ 1/2 ( f p_o + p_l ) = 0.299 $$<br />
$$ 1/2 ( p_o + \frac{p_l}{f} ) = 0.658 $$<br />
Given two linear equations and two unknowns, we get <i><b>f</b></i> = 0.46 and ocean fraction of 73.5%.<br />
The solution is also shown as the open circles in <b>Figure 1</b>.<br />
<br />
We originally asserted that 1/2 the heat is entering the ocean, and substantiated this with a value of 0.46. We can also compare the generally agreed upon value of 71% of the surface water coverage with the value of 73.5% determined here. Given the confidence interval uncertainty in the coefficients as shown in the table above, we see that this simple analysis substantiates our original premise.<br />
<br />
This is also a subtle effect and can easily be misinterpreted as arising from just the proportional warming of ocean and land. However something has to create the temperature imbalance between the ocean and land, and the fact that the coefficients of proportionality shown in the table come out fairly close to 0.3 and 0.7 (those of land and ocean) is just a coincidence in the math. If some value markedly different from 0.5 for heat sinking was involved, then the ratios would differ more obviously.<br />
<br />
Climate science, like other science disciplines, consists of an array of interlocking parts that need to fit together. If these don't fit, our model will lose its predictive power. In the case of this model, we can help verify that the excess heat is entering the ocean, <i>suppressing the</i> <i>global temperature by about 2/3 from the land temperature</i>. This will continue as long as the ocean acts as a heat sink, a point still some distance in the future. All we can really say is that global temperature anomalies have the potential for increasing by 3/2 or by 50% from the current readings when they eventually and asymptotically reach a near-equilibrium steady-state.<br />
<br />
<div style="text-align: center;">
<span style="background-color: red;"> <span style="color: yellow;"> UPDATE </span> </span></div>
<br />
Using the Eureqa curve fitting software, a linear combination of data sets provides the low complexity fit.<br />
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiH8jPL7R9Y3KgLALXc_LOoul39FrIPErIwO_DK1IjxLnZ2zlqy30vkT8ZoTcHi93eN8gHkYftF1tM8C944n2b1GGuz-wUcF3cFSbBtTJWOnAwP_nnibKAHLEp9THWjZf8J32AMDOkMu8g/s1600/eureqa_sst_land_had.GIF" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="266" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiH8jPL7R9Y3KgLALXc_LOoul39FrIPErIwO_DK1IjxLnZ2zlqy30vkT8ZoTcHi93eN8gHkYftF1tM8C944n2b1GGuz-wUcF3cFSbBtTJWOnAwP_nnibKAHLEp9THWjZf8J32AMDOkMu8g/s400/eureqa_sst_land_had.GIF" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><b>Figure 4:</b> The linear combination of SST and Land with an offset gives a Pareto frontier optimum</td></tr>
</tbody></table>
<br />
<br />
<br />
<div style="text-align: center;">
<span style="background-color: red;"> <span style="color: yellow;"> It's the ocean heat content, stupid </span> </span></div>
<br />
June 22, 2013. This paper has applicability to the proportional land/sea warming model<br />
<blockquote class="tr_bq">
<div class="csl-bib-body" style="line-height: 1.35;">
<div class="csl-entry" style="clear: left;">
<div class="csl-right-inline" style="margin: 0 .4em 0 1.5em;">
M. Watanabe, Y. Kamae, M. Yoshimori, A. Oka, M. Sato, M. Ishii, T. Mochizuki, and M. Kimoto, <a href="http://onlinelibrary.wiley.com/doi/10.1002/grl.50541/abstract">“Strengthening of ocean heat uptake efficiency associated with the recent climate hiatus,”</a> <i>Geophysical Research Letters</i>, 2013.</div>
</div>
</div>
</blockquote>
The research results claim that the ocean has been adjusting its heat uptake in the last few years as a result of transient changes in the large-scale hydrodynamics. This has the effect of suppressing the warming in terms of temperature, although the heat uptake from the AGW forcing still exists. So the implication is that what is lacking in a temperature rise is made up for by the heat sinking of the ocean (also see the <a href="http://www.skepticalscience.com/Kevin-Trenberth-travesty-cant-account-for-the-lack-of-warming.htm">"missing heat" issue</a> studied by Trenberth).<br />
<br />
The ocean heat uptake efficiency measure of Watanabe is related to the ratio <i><b>f</b></i> between ocean and land temperature defined at the top of this post. The idea is that -- similar to the aim of the Japanese research study -- to see if we can detect changes in <i><b>f </b></i>over the last few years. <br />
<br />
To do this we need to take great care with the numbers. Instead of using the WoofForTrees data, I used the <a href="http://www.cru.uea.ac.uk/cru/data/temperature/">CRU data directly</a>. The sets were CRUTEM4 (land), HadCRUT4 (global), and HadSST3 (sea). The composed set looks like the following chart for a value of <i><b>f</b></i> = 0.5, which is the nominal fraction assumed for the original proportional land/sea analysis.<br />
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="http://img534.imageshack.us/img534/1622/5vj.gif" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="258" src="http://img534.imageshack.us/img534/1622/5vj.gif" width="640" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">The composed temperature lies on top of the HadCRUT4 global temperature</td></tr>
</tbody></table>
<br />
<div class="separator" style="clear: both; text-align: center;">
</div>
If we look at the error residual between the HadCRUT4 global temperature and the fractionally composed model, we get the following chart. Note that as an absolute error, the value is obviously decreasing over time, likely attributed to better and more accurate record keeping with current temperature measurement techniques.<br />
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="http://img189.imageshack.us/img189/5863/uvt.gif" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="243" src="http://img189.imageshack.us/img189/5863/uvt.gif" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">The absolute error decreases with more recent records. <br />
(The last data point is 2012, which often undergoes corrections for the next update.)</td></tr>
</tbody></table>
<br />
The high resolution and low error in recent years indicates that perhaps we can try to more accurately fit the fraction <i><b>f</b></i>. So essentially, we want to zero out the error by solving the proportional land/sea warming model for a continuously varying value of <i><b>f</b></i>.<br />
$$ 0 = T_G - 1/2 ( f p_o + p_l ) T_l + 1/2 ( p_o + \frac{p_l}{f} ) T_o $$<br />
This turns into a quadratic equation for <i><b>f</b></i>, which we can solve by the quadratic formula. The set of value calculated by minimizing the error is shown below. Note that the average remains around<i><b> f</b></i> = 0.5, but it shows a distinct decreasing trend in recent years.<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="http://img442.imageshack.us/img442/4509/n5t.gif" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="376" src="http://img442.imageshack.us/img442/4509/n5t.gif" width="640" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">The fraction ratio of ocean to land temperature appears to be decreasing in recent years, leading to an apparent flattening in global temperature rise. Lower values of<b><i> f </i></b>cause the global temperature signal to appear cooler for a given AGW forcing.</td></tr>
</tbody></table>
<br />
If this is a real trend (as opposed to some type of accumulating systemic error or noise) it is telling us that more of the heat is accumulating in the ocean, consistent with the claims of Watanabe <i>et al</i>. It is possible that the fraction is actually decreasing from a past value of around 0.6 to a current value of 0.4. Although this is a subtle effect in terms of the fit (probably the not most robust metric one can imagine), it has significant effect in terms of the global surface temperature signal. <br />
<br />
This is seen if we deconstruct the proportional model in terms of the land temperature alone, assuming the area land/ocean split as 0.71/0.29 : <br />
<br />
$$ T_G = (0.71 \cdot f + 0.29 ) T_l $$<br />
Note that with a slowly increasing land temperature signal<i> T<sub>l</sub></i> , the declining <i><b>f </b></i>can compensate for this value and actually cause the global temperature value <i>T<sub>G</sub></i> to flatten.<br />
<br />
To take an example, reducing the value of <i><b>f</b></i> from 0.6 to 0.4 causes the global temperature to decline from 0.716*<i>T<sub>l</sub></i> to 0.574*<i>T<sub>l</sub></i>. If the land temperature is held constant, the global temperature will decline, while if the land temperature rises by 25%, the global temperature rise will look flat.<br />
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiWEDRSNH8uaoW67mvZneV6gGpFSd7_GXLTpnEHQq7fpIvJiMcmilYvJKuAueMDCWSuPxHVeii_VdnbDfe6-R37JJVqTV0mMRFXxdjkIy6x1ZMis4OOcYstx1ejzNBspN7CUhwyHyldcz4/s1600/3d_fractional_warming.GIF" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="379" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiWEDRSNH8uaoW67mvZneV6gGpFSd7_GXLTpnEHQq7fpIvJiMcmilYvJKuAueMDCWSuPxHVeii_VdnbDfe6-R37JJVqTV0mMRFXxdjkIy6x1ZMis4OOcYstx1ejzNBspN7CUhwyHyldcz4/s640/3d_fractional_warming.GIF" width="640" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Contour plot showing optimal values of <i><b>f</b></i>. This is a log plot and higher negative values indicate low error</td></tr>
</tbody></table>
<br />
That is exactly what Watanabe <i>et al</i> are claiming. Moreover, they assert that this decline can't remain in place for the long term, and eventually the ocean hydrodynamics will stabilize or even reverse, with a concomitant rebound in global temperature.<br />
<br />
To review, the essential premise of the proportional land/ocean model is:<br />
<ol>
<li>The land surface reaches the steady-state temperature quickly</li>
<li>The ocean sinks excess heat, thus moderating the sea surface temperature rise.</li>
<li>The fractional ratio of ocean temperature to land temperature is given by <i><b>f</b></i>.</li>
<li>The global surface temperature is determined as combination of land and sea surface temperatures prorated according to the land/sea areal split.</li>
</ol>
From this set of premises, we can algebraically estimate the amount of ocean heat sinking from global temperature records as gleaned from the <a href="http://www.cru.uea.ac.uk/cru/data/temperature/">Climatic Research Unit</a>.<br />
<br />
(also see I. Held's <a href="http://www.gfdl.noaa.gov/blog/isaac-held/2013/06/14/38-nh-sh-differential-warming-and-tcr">blog post </a>on this topic [2])<br />
<br />
<u><span style="font-family: Arial,Helvetica,sans-serif;"><b>References</b></span></u><br />
<br />
<div class="csl-bib-body" style="line-height: 1.35;">
<span style="font-size: x-small;">
</span><br />
<div class="csl-entry" style="clear: left;">
<span style="font-size: x-small;">
</span><br />
<div class="csl-left-margin" style="float: left; padding-right: 0.5em; text-align: right; width: 1em;">
<span style="font-size: x-small;">[1]</span></div>
<div class="csl-right-inline" style="margin: 0px 0.4em 0px 1.5em;">
<span style="font-size: x-small;">D. Dommenget, “The ocean’s role in continental climate variability and change,” <i>Journal of Climate</i>, vol. 22, no. 18, pp. 4939–4952, 2009.</span></div>
</div>
</div>
<blockquote class="tr_bq">
<span style="font-size: x-small;"><i>“The land–sea warming ratio in the ECHAM–HadISST holds also for the
warming trend over the most recent decades, despite the fact that no
anthropogenic radiative forcings are included in the simulations. The
temperature trends during the past decades as observed and in the
(ensemble mean) model response (Fig. 4) are roughly consistent with each
other, which indicates that much of the land warming is a response to
the warming of the oceans. The simulated land warming, however, is
weaker than that observed in many regions, with an average land–sea
warming ratio of 1.6, amounting to about 75% <b>of the observed ratio of 2.1</b> .”</i></span></blockquote>
<div class="csl-bib-body" style="line-height: 1.35;">
<div class="csl-entry" style="clear: left;">
<div class="csl-left-margin" style="float: left; padding-right: 0.5em; text-align: right; width: 1em;">
<span style="font-size: x-small;">[2]</span></div>
<div class="csl-right-inline" style="margin: 0px 0.4em 0px 1.5em;">
<span style="font-size: x-small;">I. Held, “38. NH-SH differential warming and TCR « Isaac Held’s Blog.” [Online]. Available: http://www.gfdl.noaa.gov/blog/isaac-held/2013/06/14/38-nh-sh-differential-warming-and-tcr/#more-5774. [Accessed: 25-Jun-2013].</span></div>
</div>
</div>
<br />@whuthttp://www.blogger.com/profile/18297101284358849575noreply@blogger.com0tag:blogger.com,1999:blog-8890671936020885112.post-10501649629920214922013-04-28T19:55:00.000-07:002013-04-28T19:55:11.796-07:00Greenland Red NoiseTamino analyzed a <a href="http://tamino.wordpress.com/2012/04/18/nothin-but-noise/" target="_blank">paper by Chylek</a> [1] who claimed to be able to extract a periodic signal out of Greenland ice core data (<a href="ftp://ftp.ncdc.noaa.gov/pub/data/paleo/icecore/greenland/gisp/dye3/dye3-1yr.txt" target="_blank">Dye3</a>). <br />
<br />
Tamino's post is comprehensive, but I wanted to check to see if I could extract any signal from the data myself.<br />
<br />
The data is in the upper right of the following figure, and the FFT-based power spectral density (PSD) is in the lower left. <br />
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="http://imageshack.us/a/img199/4329/greenlandcorepsd.gif" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="212" src="http://imageshack.us/a/img199/4329/greenlandcorepsd.gif" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">FFT PSD lower left, Data upper right</td></tr>
</tbody></table>
<br />
The PSD is nondescript with a red noise envelope shown by the solid line. Red noise is the embodiment of the <a href="http://theoilconundrum.blogspot.com/2013/04/ornstein-uhlenbeck-diffusion.html" target="_blank">Ornstein-Uhlenbeck process</a>. This red noise shows very short correlation, at most a lag of one or two time units, as shown in the autocorrelation below.<br />
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="http://img28.imageshack.us/img28/2100/acdye3.gif" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="226" src="http://img28.imageshack.us/img28/2100/acdye3.gif" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Autocorrelation function, centered at 1/2 time series length</td></tr>
</tbody></table>
<br />
Hard pressed to find any real signal in such a time series. A simulation of Ornstein-Uhlenbeck red noise is shown to the left below, with a slice of the Greenland data to the right.<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjoTjVYPsukzy1cvM6wGFIcEl-asvhCDVKPw00lznoV_etObfrZve3wr4PS57JR9ADhzZktMaTyU-gos5HrkY-fU7jhxbQOLzelzg66OR-lSafgZFIPiAEx3762NwC7ZwZ9xIjt7PNZX7o/s1600/ou-noise-greenland.gif" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="160" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjoTjVYPsukzy1cvM6wGFIcEl-asvhCDVKPw00lznoV_etObfrZve3wr4PS57JR9ADhzZktMaTyU-gos5HrkY-fU7jhxbQOLzelzg66OR-lSafgZFIPiAEx3762NwC7ZwZ9xIjt7PNZX7o/s400/ou-noise-greenland.gif" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Ornstein-Uhlenbeck red noise time series (left) and slice of data (right)</td></tr>
</tbody></table>
Chylek claims that he can see 20 year AMO (Atlantic Ocean) oscillations in the data. Like Tamino, I see nothing but garden variety noise.<br />
<br />
<br />
<b>References</b><br />
<br />
<div class="csl-bib-body" style="line-height: 1.35;">
<div class="csl-entry" style="clear: left;">
<div class="csl-left-margin" style="float: left; padding-right: 0.5em; text-align: right; width: 1em;">
<span style="font-size: x-small;">[1]</span></div>
<div class="csl-right-inline" style="margin: 0px 0.4em 0px 1.5em;">
<span style="font-size: x-small;">P. Chylek, C. Folland, L. Frankcombe, H. Dijkstra, G. Lesins, and M. Dubey, “Greenland ice core evidence for spatial and temporal variability of the Atlantic Multidecadal Oscillation,” <i>Geophysical Research Letters</i>, vol. 39, no. 9, 2012.</span></div>
</div>
<span class="Z3988" title="url_ver=Z39.88-2004&ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzotero.org%3A2&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Greenland%20ice%20core%20evidence%20for%20spatial%20and%20temporal%20variability%20of%20the%20Atlantic%20Multidecadal%20Oscillation&rft.jtitle=Geophysical%20Research%20Letters&rft.stitle=Geophysical%20Research%20Letters&rft.volume=39&rft.issue=9&rft.aufirst=Petr&rft.aulast=Chylek&rft.au=Petr%20Chylek&rft.au=Chris%20Folland&rft.au=Leela%20Frankcombe&rft.au=Henk%20Dijkstra&rft.au=Glen%20Lesins&rft.au=Manvendra%20Dubey&rft.date=2012&rft.issn=1944-8007"></span> </div>
<div class="csl-bib-body" style="line-height: 1.35;">
</div>
@whuthttp://www.blogger.com/profile/18297101284358849575noreply@blogger.com0tag:blogger.com,1999:blog-8890671936020885112.post-89092172741980342882013-04-24T21:34:00.000-07:002013-04-25T15:30:12.522-07:00Filtering Sea Level RiseMeasuring global sea-level rise is not for the faint-of-heart. The levels are averaged over the earth's oceans and likely are processed to remove tidal and other effects.<br />
<br />
At one of the climate blogs, a gang of climate change skeptics tried to convince their fellow skeptics that the sea level rise had turned the tide (so to speak) and started to decrease. The skeptic Rob (Ringo) Starkey kindly provided the data and pointed to a recent dip in the data: <br />
<blockquote class="tr_bq">
<a href="http://sealevel.colorado.edu/files/2013_rel3/sl_ns_global.txt">http://sealevel.colorado.edu/files/2013_rel3/sl_ns_global.txt</a></blockquote>
BDD smelled something fishy and pointed out that these are at best seasonal dips and labelled it rubbish, and reasoned that they were likely trying to save face over some other wrong they had committed. What I usually find is that the more that the fake skeptics howl about
something, the more likely that there are creepy crawly things they are
trying to hide under the rock. Well, in this case the sea-level rise
data that Ringo pointed us to shows a clear 2 month oscillation
in the time series. Sorry Ringo, "steven", and Captain Tuna, easy enough to put a 2-month
notch filter in the time series data and just look at how much smoother
it gets.<br />
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="http://img546.imageshack.us/img546/3079/sealevelrise.gif" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="225" src="http://img546.imageshack.us/img546/3079/sealevelrise.gif" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><b>Figure 1 :</b> Upper left inset is a FFT of the detrended sea-level rise data shown in the lower right inset. A clear two month cycle shows up in the FFT and by eyeballing any yearly interval. After performing a simple 2 month notch filter on the data and adding back in the trend, the red curve shows significant noise reduction.</td><td class="tr-caption" style="text-align: center;"><br /></td></tr>
</tbody></table>
<br />
The oscillation is possibly identified as a meridional current fluctuation of 2 months located in the Pacific described by this Woods Hole investigation: <a href="http://www.whoi.edu/cicor/page.do?pid=19476&tid=282&cid=39812" target="_blank">"Air-Sea Interaction in the Eastern Tropical Pacific ITCZ/Cold Tongue Complex"</a> page. It also is described by a member of the same team in this PhD thesis [1] :<br />
<blockquote class="tr_bq">
<span style="font-size: x-small;"><i>"The period of the oscillations can be seen to be about <span style="font-size: small;"><b>two months</b></span> from October 1997 to June 1998, and the oscillations rapidly intensify during the first few months of 1998. The amplitude of the oscillations approximately doubles between December 1997 and February 1998, and it doubles again between February and April 1998. At its peak strength in April 1998, the signal is associated with nearly a 7 cm peak-to-peak change in the thickness of the upper 110 m of the water column. </i></span>"</blockquote>
One can see the peaks in late 1997 and early 1998 from <b>Figure 1</b>, enlarged and highlighted below<br />
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEguArAPEoYnWB-VxXN6eMd8ijUEF5VzmnEU5vzhxNpSwxgYazDLk6826UKpwtxD2qQaMQbHIDuzRxCwx6lb4H62sbBUFwEXWFkmnyOI189thyphenhyphenYJLpVJL6T24-E-y0hbMxQU7ImUUTySNLI/s1600/sea-level-rise-2008.GIF" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEguArAPEoYnWB-VxXN6eMd8ijUEF5VzmnEU5vzhxNpSwxgYazDLk6826UKpwtxD2qQaMQbHIDuzRxCwx6lb4H62sbBUFwEXWFkmnyOI189thyphenhyphenYJLpVJL6T24-E-y0hbMxQU7ImUUTySNLI/s320/sea-level-rise-2008.GIF" width="224" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><b>Figure 2: </b>Identification of water column thickness increase</td></tr>
</tbody></table>
The main point is that these fluctuations are oscillatory and don't contribute to a persistent rise in the sea-level. Like tides, this phenomena is more similar to the sloshing of water in a bucket. (This particular two month cycle is not a tide as lunar tides such as neap and spring tides have monthly or biweekly periods). <br />
<br />
<h3>
References</h3>
<div class="csl-bib-body" style="line-height: 1.35;">
<div class="csl-entry" style="clear: left;">
<div class="csl-left-margin" style="float: left; padding-right: 0.5em; text-align: right; width: 1em;">
<span style="font-size: x-small;">[1]</span></div>
<div class="csl-right-inline" style="margin: 0px 0.4em 0px 1.5em;">
<span style="font-size: x-small;">J. Farrar, “Air-sea interaction at contrasting sites in the Eastern Tropical Pacific : mesoscale variability and atmospheric convection at 10°N,” PhD Thesis, MIT, Woods Hole, 2007.<br /> <a href="http://dspace.mit.edu/handle/1721.1/39009">http://dspace.mit.edu/handle/1721.1/39009</a></span></div>
<div class="csl-right-inline" style="margin: 0 .4em 0 1.5em;">
<br />
<br />
<h3 style="text-align: center;">
<span style="background-color: red;"><span style="color: yellow;"> <b><span style="color: white;"> EDIT </span></b> </span> </span></h3>
<br />
I noticed that the sea level research group at U of Colorado had already placed a 60 day (2 month) smoothing filter when they display the data (see figure below). This is similar to the 2-point notch filter that I use, which I set specifically to pair up data points so that they are out-of-phase at the 30-day mark. A general smoothing filter will accomplish the same thing but will also filter out shorter periods in the data set.<br />
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="http://sealevel.colorado.edu/files/2013_rel3/sl_ns_global.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="278" src="http://sealevel.colorado.edu/files/2013_rel3/sl_ns_global.png" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><br />
<span style="font-size: small;"><a href="http://sealevel.colorado.edu/content/2013rel3-global-mean-sea-level-time-series-seasonal-signals-removed">http://sealevel.colorado.edu/content/2013rel3-global-mean-sea-level-time-series-seasonal-signals-removed</a></span></td></tr>
</tbody></table>
<br /></div>
</div>
<span class="Z3988" title="url_ver=Z39.88-2004&ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzotero.org%3A2&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Adissertation&rft.title=Air-sea%20interaction%20at%20contrasting%20sites%20in%20the%20Eastern%20Tropical%20Pacific%20%3A%20mesoscale%20variability%20and%20atmospheric%20convection%20at%2010%C2%B0N&rft.aufirst=JT&rft.aulast=Farrar&rft.au=JT%20Farrar&rft.date=2007"></span>
</div>
<br /><br />
<br />@whuthttp://www.blogger.com/profile/18297101284358849575noreply@blogger.com0tag:blogger.com,1999:blog-8890671936020885112.post-43166212840373522322013-04-20T14:36:00.000-07:002013-04-27T05:43:45.968-07:00The Rise and Decline of UK Crude Oil<br />
This is updated information from <a href="http://mobjectivist.blogspot.com/2005/10/uk-north-sea-simulation.html" target="_blank">2005 </a>and <a href="http://mobjectivist.blogspot.com/2006/08/north-sea-update.html" target="_blank">2008</a> concerning projections of North Sea UK Oil production. <br />
<br />
The following <b>Figure 1</b> is a screenshot of an interactive <a href="http://theoilconundrum.com/" target="_blank">oil shock model</a> session, which is part of a larger environmental modeling framework. <br />
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="http://img593.imageshack.us/img593/9130/uknorthseaview.gif" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="215" src="http://img593.imageshack.us/img593/9130/uknorthseaview.gif" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><b>Figure 1: </b> Shock Model for UK North Sea Oil. The oil production output is the profile with two humps, with model and data shown together. Extraction rate applied to the extrapolated reserves gives the production.</td></tr>
</tbody></table>
<br />
The discovery curve follows a dispersive profile which allows us to project future availability along a declining tail. The extraction perturbations show a temporary glitch corresponding to the Piper Alpha platform disaster, but otherwise shows around a 6% extraction rate from reserves over the years. This rate isn't expected to move upwards much (contrary to my previous projection) because any increase in technological advancements will be compensated by fields that are tougher to extract from.<br />
<br />
Also included in the interactive session is an ability to pull up individual fields (all part of a semantic web server), shown in <b>Figure 2</b>. The <a href="http://en.wikipedia.org/wiki/Forties_Oil_Field" target="_blank">"Forties" oil field</a> is one of the earliest discoveries and commenced production at the start of the UK North Sea era. <br />
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="http://img10.imageshack.us/img10/4522/forties.gif" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="160" src="http://img10.imageshack.us/img10/4522/forties.gif" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><b>Figure 2:</b> Individual plot of a UK platform, the "Forties" oil field. The downward glitch at the end is a -1 value indicating production values are not yet available for that year.</td></tr>
</tbody></table>
The UK Prime Ministers during this time (see <b>Figure 3</b>) likely benefited politically from the oil production bonanza, starting with Margaret Thatcher's leadership of the conservative party in 1975.<br />
<div class="separator" style="clear: both; text-align: center;">
<br /></div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg3oHPracfAF3ybV6ukmK6toBF1_205gEVgzK9wPad8qPWc5y859VVOHaOG4pnlGAVU19-Do78WW04PCZYrocFm8Xp_fDND7IpZA2z9pT07X9Yi9cy8lk1wSXvv7wPcvk6eleLkU30KjmE/s1600/uk_oil_thatcher.gif" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="351" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg3oHPracfAF3ybV6ukmK6toBF1_205gEVgzK9wPad8qPWc5y859VVOHaOG4pnlGAVU19-Do78WW04PCZYrocFm8Xp_fDND7IpZA2z9pT07X9Yi9cy8lk1wSXvv7wPcvk6eleLkU30KjmE/s400/uk_oil_thatcher.gif" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><b>Figure 3:</b> PM Margaret Thatcher was at the right place at the right time.
Leader of the conservative party starting in 1975, and then became Prime
Minister in 1979, she reigned during the huge buildup in oil production,
which was temporarily halted by the Piper Alpha explosion in 1988.</td></tr>
</tbody></table>
<br />
<br />
The Piper production halted for several years and pulled down other platform production levels as better maintenance policies were instituted. This had the unintended benefit of extending the UK production level before the eventual decline set in.<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhHt-0bKkwox38S3us1xOpE6EKaLT2nXBsyI2Sgmlv0nAL7NVGaRq9SoD4CoPuN3LdlvhKlsJpuXNT8LmNdTGIP-RrR6_h6DJhPqGMUTrm3gUD-_eR6rkou5-0xCy5Koif_0_uHWQQ1rxQ/s1600/piper.GIF" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="238" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhHt-0bKkwox38S3us1xOpE6EKaLT2nXBsyI2Sgmlv0nAL7NVGaRq9SoD4CoPuN3LdlvhKlsJpuXNT8LmNdTGIP-RrR6_h6DJhPqGMUTrm3gUD-_eR6rkou5-0xCy5Koif_0_uHWQQ1rxQ/s320/piper.GIF" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><b>Figure 4:</b> Piper Alpha oil production halted for several years</td></tr>
</tbody></table>
As I have always said, having good numbers for production data allows use to apply sophisticated analyses such as the oil shock model to project and predict future trends. The UK government has done a fine job in making the data publicly available, allowing lots of interesting analysis which I documented in <a href="http://theoilconundrum.com/" target="_blank">The Oil Conundrum book</a>.<br />
<br />
<br />
<div style="text-align: center;">
<span style="color: white;"><span style="background-color: #cc0000;"> Updates </span></span></div>
<br />
Feedback from others<br />
<a href="http://www.theoildrum.com/node/9946#comment-958176" target="_blank">http://www.theoildrum.com/node/9946#comment-958176</a><br />
<a href="http://transportblog.co.nz/2013/04/12/oil-dependancy-and-the-wealth-of-nations/" target="_blank">http://transportblog.co.nz/2013/04/12/oil-dependancy-and-the-wealth-of-nations/</a><br />
<br />
Sources<br />
<a href="http://www.worldreview.info/content/uk-north-sea-oil-and-gas-resurgence-activity">http://www.worldreview.info/content/uk-north-sea-oil-and-gas-resurgence-activity</a><br />
<br />
<br />
<br />
<br />
<br />@whuthttp://www.blogger.com/profile/18297101284358849575noreply@blogger.com2tag:blogger.com,1999:blog-8890671936020885112.post-7645993161714402212013-04-18T08:43:00.001-07:002013-04-18T08:49:59.762-07:00Ornstein-Uhlenbeck DiffusionThe Ornstein-Uhlenbeck correction to diffusion can be used in applications ranging from modeling <a href="http://theoilconundrum.blogspot.com/2012/07/bakken-dispersive-diffusion-oil.html" target="_blank">Bakken oil production</a> to modeling <a href="http://theoilconundrum.blogspot.com/2013/03/stochastic-analysis-of-log-sensitivity.html" target="_blank">CO2 sequestration</a>.<br />
<br />
Due to its origins as a random walk process, a pure diffusion model of particles will show unbounded excursions given a long enough time duration. This is characterized by the unbounded Fickian growth law showing a <math xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math" xmlns:mml="http://www.w3.org/1998/Math/MathML"><msqrt><mi>t</mi></msqrt></math> dependence for a pure random walk with a single diffusivity.<br />
<br />
In practice, the physical environment of a particle may prevent unbounded excursions. It is physically possible that the environment may impose limiting effects on the extent of motion, or that it will place some form of drag on the particle’s hopping rate the further it moves away from a mean starting value. <br />
<br />
The rationale for this limiting process within a producing hydrofractured reservoir may
arise from a barrier to diffusion beyond a certain range.<br />
<br />
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Similarly, a sequestration barrier may exist preventing CO2 from permanently exiting the active carbon cycle; this makes the fat temporal tail even fatter.<br />
<br />
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<br />
We can use the Ornstein-Uhlenbeck process to model mathematically how this pure random walk becomes bounded. The Ornstein-Uhlenbeck process has its origins in the modeling of Brownian motion with a special “reversion to the mean” property in motion excursions (specifically, it was first
formulated to describe Brownian motion in the presence of drag on
particle velocities, extending Einstein's work). The following expression shows the stationary marginal probability given a stochastic differential equation<br />
<br />
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<br />
which models a drag on an excursion [1]<br />
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<math xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mi>P</mi><mfenced close="|" separators="|"><mrow><mi>X</mi><mfenced separators="|"><mrow><mi>t</mi><mo>+</mo><mi>s</mi></mrow></mfenced><mo>=</mo><mi>x</mi><mi> </mi></mrow></mfenced><mi> </mi><mi>X</mi><mfenced separators="|"><mrow><mi>s</mi></mrow></mfenced><mo>=</mo><mi> </mi><mn>0</mn><mo>)</mo><mo>=</mo><mi> </mi><mfrac><mrow><mn>1</mn></mrow><mrow><msqrt><mn>2</mn><mi>π</mi><mi>τ</mi></msqrt></mrow></mfrac><msup><mrow><mi>e</mi></mrow><mrow><mo>-</mo><mfrac><mrow><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow><mrow><mn>2</mn><mi>τ</mi></mrow></mfrac></mrow></msup><mi> </mi><mi>d</mi><mi>x</mi></math> </div>
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<math xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mi>w</mi><mi>h</mi><mi>e</mi><mi>r</mi><mi>e</mi><mi> </mi><mi> </mi><mi> </mi><mi>τ</mi><mo>=</mo><mfrac><mrow><mn>1</mn><mo>-</mo><msup><mrow><mi>e</mi></mrow><mrow><mo>-</mo><mn>2</mn><mi>a</mi><mi>t</mi></mrow></msup></mrow><mrow><mn>2</mn><mi>a</mi></mrow></mfrac></math>
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<br />
<br />
The O-U correction is straight-forward to apply on our dispersive growth formulation, we only need apply a non-linear transformation to the time-scale. <br />
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<div style="text-align: center;">
<math xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mi> </mi><mover accent="false"><mo>→</mo><mrow><mi>O</mi><mo>-</mo><mi>U</mi></mrow></mover><mi>τ</mi></math>
</div>
<br />
<br />
and then apply this to a dispersive growth term such as the following:<br />
<br />
<br />
<div style="text-align: center;">
<math xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mfenced separators="|"><mrow><mi>τ</mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mfenced><mo>=</mo><msqrt><mi>D</mi><mi>τ</mi><mo>(</mo><mi>t</mi><mo>)</mo></msqrt><mfrac><mrow><msqrt><mfrac><mrow><mi>D</mi><mi>τ</mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow></mfrac></msqrt></mrow><mrow><mn>1</mn><mo>+</mo><mi> </mi><msqrt><mfrac><mrow><mi>D</mi><mi>τ</mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow></mfrac></msqrt></mrow></mfrac><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mover accent="false"><mo>→</mo><mrow><mi>w</mi><mi>h</mi><mi>e</mi><mi>r</mi><mi>e</mi></mrow></mover><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mi>τ</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>=</mo><mo>(</mo><mn>1</mn><mo>-</mo><msup><mrow><mi>e</mi></mrow><mrow><mo>-</mo><mn>2</mn><mi>a</mi><mi>t</mi></mrow></msup><mo>)</mo><mo>/</mo><mn>2</mn><mi>a</mi><mi> </mi><mi> </mi></math>
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<br />
<br />
This has the equivalent effect of appearing to slow down time at an exponential rate. This exponential rate turns out to be much faster than the Fickian growth law can sustain, so that an asymptotic limit is achieved in the diffusional growth extent.<br />
<br />
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<br />
<br />
A physical model of an attractor or potential well which “tugs” on the random walker to bring it back to the mean state (see Figure below).<br />
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgUYzTb1PKZVujSmwwdX6CT6h7phBkd9Qn4JzTHlCccC8gtMPghpdJMPOD90rHXwX_gTaLmLs5UbCSngsOhw8mY-Z0EixGccgU98pZXIUD2MRPlu5E0CMZ2gHVUin_h88srUAoLweVkEBo/s1600/ornstein-uhlenbeck.gif" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="244" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgUYzTb1PKZVujSmwwdX6CT6h7phBkd9Qn4JzTHlCccC8gtMPghpdJMPOD90rHXwX_gTaLmLs5UbCSngsOhw8mY-Z0EixGccgU98pZXIUD2MRPlu5E0CMZ2gHVUin_h88srUAoLweVkEBo/s320/ornstein-uhlenbeck.gif" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><b>Figure 4: </b>Representation of an Ornstein-Uhlenbeck random walk process. The hopping
rate works similarly to a potential well, with a greater resistance to
hopping the further excursion ways from the mean.</td></tr>
</tbody></table>
<span id="goog_337498659"></span><span id="goog_337498660"></span><br />
<h3>
Algorithmic Code</h3>
<br />
The following pseudo-code snippet sets up an Ornstein-Uhlenbeck random walk model with a reversion-to-the-mean term. The diffusion term is the classical Markovian random walk transition rate. The drag term places an attractor which opposes large excursions in the term Z. <br />
<br />
<br />
<blockquote class="tr_bq">
<span style="font-family: "Courier New",Courier,monospace;">-- Ornstein-Uhlenbeck random walk = ou</span><br />
<span style="font-family: "Courier New",Courier,monospace;"></span>
<span style="font-family: "Courier New",Courier,monospace;"><br /></span>
<span style="font-family: "Courier New",Courier,monospace;">ou(X1, X2, Z) </span><br />
<span style="font-family: "Courier New",Courier,monospace;"> random(R)</span><br />
<span style="font-family: "Courier New",Courier,monospace;"> if R < 0.5 then</span><br />
<span style="font-family: "Courier New",Courier,monospace;"> Z = Z*X1 + X2</span><br />
<span style="font-family: "Courier New",Courier,monospace;"> else</span><br />
<span style="font-family: "Courier New",Courier,monospace;"> Z = Z*X1 - X2</span><br />
<span style="font-family: "Courier New",Courier,monospace;"> end</span></blockquote>
<br />
<br />
<blockquote class="tr_bq">
<span style="font-family: "Courier New",Courier,monospace;">-- This is how it gets parameterized</span></blockquote>
<br />
<blockquote class="tr_bq">
<span style="font-family: "Courier New",Courier,monospace;">ou_random_walker (dX, Diffusion, Drag, Z) </span><br />
<span style="font-family: "Courier New",Courier,monospace;"> X1 = exp(-2*Drag*dX)</span><br />
<span style="font-family: "Courier New",Courier,monospace;"> X2 = sqrt(Diffusion*(1-exp(-2*Drag*dX))/2/Drag)</span><br />
<span style="font-family: "Courier New",Courier,monospace;"> ou(X1, X2, Z)</span></blockquote>
<br />
<h3>
Variance</h3>
To determine whether an Ornstein-Uhlenbeck process is apparent on a set of data, one can apply a simple multiscale variance to the result of a Z array of length N :<br />
<br />
<blockquote class="tr_bq">
<span style="font-family: "Courier New",Courier,monospace;">variance(Z,N) { </span><br />
<span style="font-family: "Courier New",Courier,monospace;"> L = N/2</span><br />
<span style="font-family: "Courier New",Courier,monospace;"> while(L > 1) {</span><br />
<span style="font-family: "Courier New",Courier,monospace;"> Sum = 0.0</span><br />
<span style="font-family: "Courier New",Courier,monospace;"> for(i=1; i<N/2; i++) {</span><br />
<span style="font-family: "Courier New",Courier,monospace;"> Val = Z[i] - Z[i+L]</span><br />
<span style="font-family: "Courier New",Courier,monospace;"> Sum += Val * Val</span><br />
<span style="font-family: "Courier New",Courier,monospace;"> }</span><br />
<span style="font-family: "Courier New",Courier,monospace;"> print L “ “ sqrt(Sum/(N/2))</span><br />
<span style="font-family: "Courier New",Courier,monospace;"> L = 0.95*L;</span><br />
<span style="font-family: "Courier New",Courier,monospace;"> }</span><br />
<span style="font-family: "Courier New",Courier,monospace;">}</span></blockquote>
<span style="font-family: "Courier New",Courier,monospace;"><br /></span>
<br />
So that for a given random walk simulation, the asymptotic variance will tend to saturate at longer correlation length scales. A typical multiscale variance plot will look like those described here: <a href="http://theoilconundrum.blogspot.com/2011/11/multiscale-variance-analysis-and.html">http://theoilconundrum.blogspot.com/2011/11/multiscale-variance-analysis-and.html</a>. <br />
<br />
<h3>
Autocorrelation and Spectral Representation</h3>
The autocorrelation of the Ornstein-Uhlenbeck process is, where Theta=Drag:<br />
<br />
<div style="text-align: center;">
<math xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mfenced separators="|"><mrow><mi>x</mi></mrow></mfenced><mo>=</mo><mi> </mi><mfrac><mrow><mi>D</mi></mrow><mrow><mi>θ</mi></mrow></mfrac><mi> </mi><msup><mrow><mi>e</mi></mrow><mrow><mo>-</mo><mi>θ</mi><mo>|</mo><mi>x</mi><mo>|</mo></mrow></msup></math>
</div>
<br />
Even though this shows a saturation level, the power spectrum still obeys a 1/S^2 fall-off.<br />
<br />
<div style="text-align: center;">
<math xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mi>I</mi><mfenced separators="|"><mrow><msub><mrow><mi>S</mi></mrow><mrow><mi>x</mi></mrow></msub></mrow></mfenced><mo>=</mo><mi> </mi><mfrac><mrow><mi>D</mi></mrow><mrow><mi>π</mi></mrow></mfrac><mo>∙</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msup><mrow><msub><mrow><mi>S</mi></mrow><mrow><mi>x</mi></mrow></msub></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><msup><mrow><mi>θ</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac></math>
</div>
<br />
<br />
The Orenstein-Uhlenbeck process is often referred to as red noise and the two parameters of Diffusion and Drag can be determined either from the autocorrelation function or from the PSD. For the PSD, on a log-log plot, this involves reading the peak near S=0 and then determining the shoulder of the power-law roll-off. Between these two measures, one can infer both parameter values.<br />
<br />
<h3>
References</h3>
<br />
<div class="csl-bib-body" style="line-height: 1.35;">
<div class="csl-entry" style="clear: left;">
<div class="csl-left-margin" style="float: left; padding-right: 0.5em; text-align: right; width: 1em;">
<span style="font-size: x-small;">[1]</span></div>
<div class="csl-right-inline" style="margin: 0px 0.4em 0px 1.5em;">
<span style="font-size: x-small;">D. Mumford and A. Desolneux, <i>Pattern Theory: The Stochastic Analysis Of Real-World Signals</i>. A K Peters, Ltd., 2010.</span></div>
</div>
<span class="Z3988" title="url_ver=Z39.88-2004&ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzotero.org%3A2&rft_id=urn%3Aisbn%3A978-1-56881-579-4&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Pattern%20Theory%3A%20The%20Stochastic%20Analysis%20Of%20Real-World%20Signals&rft.publisher=A%20K%20Peters%2C%20Ltd.&rft.aufirst=D.&rft.aulast=Mumford&rft.au=D.%20Mumford&rft.au=A.%20Desolneux&rft.date=2010&rft.tpages=375&rft.isbn=978-1-56881-579-4"></span>
</div>
<br />
<br />
<br />@whuthttp://www.blogger.com/profile/18297101284358849575noreply@blogger.com1tag:blogger.com,1999:blog-8890671936020885112.post-54091170986492246702013-03-28T06:23:00.004-07:002013-04-29T05:54:05.020-07:00Ocean Heat Content ModelThe ocean heat content continues to increase and perhaps accelerate [1], as expected due to global warming. In an <a href="http://theoilconundrum.blogspot.com/2012/01/thermal-diffusion-and-missing-heat.html" target="_blank">earlier post</a>, I analyzed the case of the "missing heat", which wasn't missing after all, but the result of the significant heat sinking characteristics of the oceans [2].<br />
<br />
The objective is to create a simple model which tracks the transient growth as shown in the recent paper by Balmaseda, Trenberth, and Källén (BTK)[1] and described in<b> Figure 1 </b>below<br />
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="http://www.skepticalscience.com//pics/BTK13Fig1.jpg" style="margin-left: auto; margin-right: auto;"><img border="0" height="214" src="http://www.skepticalscience.com//pics/BTK13Fig1.jpg" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><b>Figure 1:</b> Ocean Heat Content from BTK [1] (<a href="http://www.skepticalscience.com//pics/BTK13Fig1.jpg" target="_blank">source</a>)</td></tr>
</tbody></table>
We will assume a diffusive flow of heat as described in James Hansen’s 1981 paper [2]. In general the diffusion of heat is qualitatively playing out according to the way Fick’s law would apply to a heat sink. Hansen also volunteered an effective diffusion that should apply, set to the nice round number of 1 cm^2/second.<br />
<br />
In the following we provide a mathematical explanation which works its way from first principles to come up with an <i>uncertainty-quantified</i> formulation. After that we present a first-order sanity check to the approximation.<br />
<br />
<h4>
Solution </h4>
We have three depths that we are looking at for heat accumulation (in addition to a surface layer which gives only a sea-surface temperature or SST). These are given as depths to 300 meters, to 700 meters, and down to infinity (or 2000 meters from another source), as charted on <b>Figure 1</b>.<br />
<br />
We assume that excess heat (mostly infrared) is injected through the ocean's surface and works its way down through the depths by an effective diffusion coefficient. The kernel transient solution to the planar heat equation is this:<br />
<br />
$$\Delta T(x,t) = \frac{c}{\sqrt{D_i t}} \exp{\frac{-x^2}{D_i t}} $$<br />
where<i> D</i>_i is the diffusion coefficient, and <i>x</i> is the depth (note that often a value of 2<i>D </i>instead of <i>D</i> is used depending on the definition of the random walk). The delta temperature is related to a thermal energy or heat through the heat capacity of salt water, which we assume to be constant through the layers. Any scaling is accommodated by the prefactor, <i>c</i><br />
<br />
The first level of uncertainty is a <i>maximum entropy prior</i> (<a href="http://en.wikipedia.org/wiki/Prior_probability" target="_blank">MaxEnt</a>) that we apply to the diffusion coefficient to approximate the various pathways that heat can follow downward. For example, some of the flow will be by eddy diffusion and other paths by conventional vertical mixing diffusion. If we apply a maximum entropy probability density function, assuming only a mean value for the diffusion coefficient:<br />
$$ p(D_i) = \frac{1}{D} e^{\frac{-D_i}{D}} $$<br />
<br />
then we get this formulation:<br />
$$\Delta T(t|x) = \frac{c}{\sqrt{D t}} e^{\frac{-x}{\sqrt{D t}}} (1 + \frac{x}{\sqrt{D t}})$$<br />
<br />
The next uncertainty is in capturing the heat content for a layer. The incremental heat across a layer, <i>L</i>, we can approximate as<br />
$$ \int { \Delta T(t|x) p(x) dx} = \int { \Delta T(t|x) e^{\frac{-x}{L}} dx} $$<br />
<br />
Which gives as the excess heat response the following concise equation. <br />
$$ I(t) = \frac{ \frac{\sqrt{Dt}}{L} + 2 }{(\frac{\sqrt{Dt}}{L} + 1)^2}$$<br />
<br />
This is also the response to a delta forcing impulse, but for a realistic situation where a growing aCO2 forces the response (<a href="http://theoilconundrum.blogspot.com/2013/03/stochastic-analysis-of-log-sensitivity.html" target="_blank">explained in the previous post</a>), we simply apply a convolution of the thermal stimulus with the thermal response. The
temporal profile of the increasing aCO2 generates a growing thermal forcing
function. <br />
<br />
$$ R(t) = F(t) \otimes I(t) = \int_0^t { F(\tau) I(t-\tau) d\tau} $$<br />
If the thermal stimulus is a linearly growing heat flux, which roughly matches the GHG forcing function (see <a href="http://theoilconundrum.blogspot.com/2013/03/stochastic-analysis-of-log-sensitivity.html" target="_blank">Figure 7 in the previous post</a>)<br />
$$ F(t) = k \cdot (t-t_0) $$<br />
then assuming a starting point t_0=0.<br />
$$ R(t)/k = \frac{2 L}{3} {(D t)}^{1.5}- L^2 D t+2 L^3 \sqrt{D t}-2 L^4 \ln(\frac{\sqrt{D t}+L}{L})) $$<br />
<br />
A good approximation is to assume the thermal forcing function started kicking in about 50 years ago, circa1960. We can then plot the equation for various values of the layer thickness, <i>L</i>, and a value of <i>D</i> of 3 cm^2/s.<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="http://img819.imageshack.us/img819/7860/ohc.gif" style="margin-left: auto; margin-right: auto;"><img border="0" height="286" src="http://img819.imageshack.us/img819/7860/ohc.gif" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><b>Figure 2 :</b> Thermal Dispersive Diffusion Model applied to the OHC data</td></tr>
</tbody></table>
<br />
Another view of the OHC includes the thermal mass of the non-ocean regions (see <b>Figure 3</b>). This data appears smoothed in comparison to the raw data of <b>Figure 2</b>.<br />
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjha_HJKaA-QM6x6jgnSJqDEl42WdB2qjAP4ag56udBJXb-EVRD9qPAcV_Ud92oaXsvqO0_ueNh4_GaJShR38FCEEDFsmEtAP4oa1PFj7E4m3dqfqpAt-oNIESVTjpbwhGxyL4DsrNxPTs/s1600/SkS_OHC.GIF" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="250" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjha_HJKaA-QM6x6jgnSJqDEl42WdB2qjAP4ag56udBJXb-EVRD9qPAcV_Ud92oaXsvqO0_ueNh4_GaJShR38FCEEDFsmEtAP4oa1PFj7E4m3dqfqpAt-oNIESVTjpbwhGxyL4DsrNxPTs/s400/SkS_OHC.GIF" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><b>Figure 3 </b>: Alternate view of growing ocean heat content. This also includes non-ocean.</td></tr>
</tbody></table>
In the latter figure, the agreement with the uncertainty-quantified theory is more striking. A single parameter, Hansen's effective diffusion coefficient<i> D</i>, along with the inferred external thermal forcing function is able to reproduce the temporal profile accurately.<br />
<br />
The strength of this modeling approach is to lean on the maximum entropy principle to fill in the missing gaps where the variability in the numbers are uncertain. In this case, the diffusion and ocean depths hold the uncertainty, and we use first-order phyiscs to do the rest.<br />
<br />
<br />
<h4>
Sanity Check</h4>
The following is a sanity check for the above formulation; providing essentially a homework assignment that a physics professor would hand
out in class.<br />
<br />
An application of Fick’s Law is to approximate the amount of material that has diffused (with thermal diffusion coefficient <i>D</i>) at least a certain distance, <i>x</i>, over a time duration, <i>t</i>, by<br />
<br />
$$ e^{\frac{-x}{\sqrt{Dt}}}= \frac{Q}{Q_0}$$<br />
<ul>
<li>For greater than 300 meters, <i>Q/Q_0</i> is 13.5/20 read from <b>Figure 1</b>.</li>
<li>For greater than 700 meters, <i>Q/Q_0</i> is 7.5/20</li>
</ul>
Where <i>Q_0</i>=20 is the baseline for the total heat measured over all depths (i.e. between x=0 and <i>x</i>=infinite depth) reached at the current time. No heat will diffuse to infinite depths so at that point <i>Q/Q_0</i> is 0/20.<br />
<br />
First, we can check to see how close the value of <i>L</i>=sqrt(<i>Dt</i>) scales, by fitting the <i>Q/Q_0 </i>ratio at each depth..<br />
<ul>
<li>For <i>x</i>=300 meters, we get <i>L</i>=763</li>
<li>For <i>x</i>=700 meters, we get <i>L</i>=713</li>
</ul>
These two are close enough to maintaining invariance that the Fick’s law scaling relation holds and we can infer that the flow is by an effective diffusion, just as was surmised by Hansen et al in 1981.<br />
<br />
We then use an average elapsed diffusion time of <i>t</i>=40 years and assume an average diffusion depth of 740, and <i>D</i> comes out to 4.5 cm^2/s.<br />
<br />
Hansen in 1981 [2] used an estimated value of diffusion of 1 cm^2/s, which is within an order of magnitude of 4.5 cm^2/s<br />
<br />
This is a scratch attempt at an approximate solution to the more general solution, which is the equivalent of
generating an impulse function in the past and watching that evolve. The general solution which we formulated earlier involves a modulated forcing function and we compute the
convolution against the thermal impulse response (i.e. Greens function)
and compare that against <b>Figure 1</b> and <b>Figure 2</b>, and the full temporal profile.<br />
<br />
<h4>
Discussion</h4>
Reading Hansen and looking at the BTK paper’s results, we should note that nothing looks out of the ordinary for the OHC trending if we compare it to what conventional thermal physics would predict. The total heat absorbed by the ocean is within range of that expected by a 3C doubling sensitivity. BTK has done the important step in determining the amount of total heat absorbed, which allows us to estimate the effective diffusion coefficient by evaluating how the heat contents modulate with depth.We add a maximum uncertainty modifier to the coefficient to model the disorder in diffusivity (some of it eddy diffusivity, some vertical, etc) and that allows us to match the temporal profile accurately.<br />
<br />
<br />
<br />
<b>References</b><br />
<br />
<div class="csl-bib-body" style="line-height: 1.35;">
<div class="csl-entry" style="clear: left;">
<div class="csl-left-margin" style="float: left; padding-right: 0.5em; text-align: right; width: 1em;">
<span style="font-size: x-small;">[1]</span></div>
<div class="csl-right-inline" style="margin: 0px 0.4em 0px 1.5em;">
<span style="font-size: x-small;">M. A. Balmaseda, K. E. Trenberth, and E. Källén, “Distinctive climate signals in reanalysis of global ocean heat content,” <i>Geophysical Research Letters</i>, 2013.</span></div>
<span style="font-size: x-small;">
</span></div>
<span style="font-size: x-small;">
<span class="Z3988" title="url_ver=Z39.88-2004&ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzotero.org%3A2&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Distinctive%20climate%20signals%20in%20reanalysis%20of%20global%20ocean%20heat%20content&rft.jtitle=Geophysical%20Research%20Letters&rft.aufirst=Magdalena%20A&rft.aulast=Balmaseda&rft.au=Magdalena%20A%20Balmaseda&rft.au=Kevin%20E%20Trenberth&rft.au=Erland%20K%C3%A4ll%C3%A9n&rft.date=2013&rft.issn=1944-8007"></span>
</span></div>
<span style="font-size: x-small;"><br /></span>
<br />
<div class="csl-bib-body" style="line-height: 1.35;">
<div class="csl-entry" style="clear: left;">
<div class="csl-left-margin" style="float: left; padding-right: 0.5em; text-align: right; width: 1em;">
<span style="font-size: x-small;">[<span style="font-size: x-small;">2</span>]</span></div>
<div class="csl-right-inline" style="margin: 0px 0.4em 0px 1.5em;">
<span style="font-size: x-small;">J. Hansen, D. Johnson, A. Lacis, S. Lebedeff, P. Lee, D. Rind, and G. Russell, “Climate impact of increasing atmospheric carbon dioxide,” <i>Science</i>, vol. 213 (4511), pp. 957–966.</span><br />
<span style="font-size: x-small;"><a href="http://pubs.giss.nasa.gov/docs/1981/1981_Hansen_etal.pdf">http://pubs.giss.nasa.gov/docs/1981/1981_Hansen_etal.pdf</a> </span></div>
</div>
<br /></div>
<br />
<a name='more'></a><br />
<blockquote class="tr_bq">
<h3 style="text-align: center;">
<span style="background-color: red;"><span style="color: white;"> EDI<span style="background-color: red;"></span>TED </span></span></h3>
</blockquote>
More data from the NOAA site using the same fit. Levitus et al [3] apply an alternative characterization to the above. I apply the same model as dashed green line.<br />
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiUAyohpArFHVuQAt_Yqz0XmU-8_rZDHtHxQd24kCNzDC6w_7lhCnHHiivIIwZ7CmqWGBHzz4R7yiwrtUH7ataEVN45fUMH4yXNRNZn8veXGkGHbKMEZzp2tvZgusUualF_WoHuIe4Ul1E/s1600/noaa.GIF" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="640" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiUAyohpArFHVuQAt_Yqz0XmU-8_rZDHtHxQd24kCNzDC6w_7lhCnHHiivIIwZ7CmqWGBHzz4R7yiwrtUH7ataEVN45fUMH4yXNRNZn8veXGkGHbKMEZzp2tvZgusUualF_WoHuIe4Ul1E/s640/noaa.GIF" width="468" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><b>Figure 4 :</b> From the NOAA site, with the same model<br />
<a href="http://www.nodc.noaa.gov/OC5/3M_HEAT_CONTENT/">http://www.nodc.noaa.gov/OC5/3M_HEAT_CONTENT/</a></td></tr>
</tbody></table>
<br />
And here is a plot of OHC using the effective forcing described by Hansen [4]. Instead of using a ramp forcing function which gives the analytical result of <b>Figure 2</b>, a realistic forcing (which takes into account perturbations due to volcanic events) is numerically convolved with the diffusive response function, <i>I(t)</i>.<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhOoDbEbjrjh_fl_Uacn6pR81t_cZEyeO5VHGjh1bcVpE6TwxwLToyvEBPJE11Twx4GasuhVtddroLL719_IQT9_M3RGbCWzqcNuh6ga0ANQpYqxAHcfHouYfzFNPFZMIKhXPBUYUH4W0k/s1600/hansen_forcing_diffusive_response.GIF" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="344" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhOoDbEbjrjh_fl_Uacn6pR81t_cZEyeO5VHGjh1bcVpE6TwxwLToyvEBPJE11Twx4GasuhVtddroLL719_IQT9_M3RGbCWzqcNuh6ga0ANQpYqxAHcfHouYfzFNPFZMIKhXPBUYUH4W0k/s640/hansen_forcing_diffusive_response.GIF" width="640" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"> <b>Figure 5:</b> Numerical convolution of effective forcing (lower left) is convolved with the diffusive transfer function to give the response (upper right). The three curves are < 300 m, < 700 m, and < 2000 m.</td></tr>
</tbody></table>
Note that the volcanic disturbances are clearly visible in the response, although they do not show as sharp a transient decrease after the events, perhaps half of what<b> Figure 1</b> shows. The suppression due to the 1997-1998 El Nino is also not observable, but that of course is not a effective forcing and so won't appear.<br />
<br />
<br />
<div class="csl-bib-body" style="line-height: 1.35;">
<div class="csl-entry" style="clear: left;">
<div class="csl-left-margin" style="float: left; padding-right: 0.5em; text-align: right; width: 1em;">
<span style="font-size: x-small;">[<span style="font-size: x-small;">3</span>]</span></div>
<div class="csl-right-inline" style="margin: 0px 0.4em 0px 1.5em;">
<span style="font-size: x-small;">S. Levitus, J. Antonov, T. Boyer, O. Baranova, H. Garcia, R. Locarnini, A. Mishonov, J. Reagan, D. Seidov, and E. Yarosh, “World ocean heat content and thermosteric sea level change (0–2000 m), 1955–2010,” <i>Geophysical Research Letters</i>, vol. 39, no. 10, 2012.</span></div>
<span style="font-size: x-small;">
</span></div>
<span style="font-size: x-small;">
<span class="Z3988" title="url_ver=Z39.88-2004&ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzotero.org%3A2&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=World%20ocean%20heat%20content%20and%20thermosteric%20sea%20level%20change%20(0%E2%80%932000%20m)%2C%201955%E2%80%932010&rft.jtitle=Geophysical%20Research%20Letters&rft.stitle=Geophysical%20Research%20Letters&rft.volume=39&rft.issue=10&rft.aufirst=S&rft.aulast=Levitus&rft.au=S%20Levitus&rft.au=JI%20Antonov&rft.au=TP%20Boyer&rft.au=OK%20Baranova&rft.au=HE%20Garcia&rft.au=RA%20Locarnini&rft.au=AV%20Mishonov&rft.au=JR%20Reagan&rft.au=D%20Seidov&rft.au=ES%20Yarosh&rft.date=2012&rft.issn=1944-8007"></span>
</span></div>
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<div class="csl-left-margin" style="float: left; padding-right: 0.5em; text-align: right; width: 1em;">
<span style="font-size: x-small;">[4]</span></div>
<div class="csl-right-inline" style="margin: 0px 0.4em 0px 1.5em;">
<span style="font-size: x-small;">J. Hansen, M. Sato, P. Kharecha, and K. von Schuckmann, “Earth’s energy imbalance and implications,” <i>Atmospheric Chemistry and Physics</i>, vol. 11, no. 24, pp. 13421–13449, Dec. 2011.</span>
</div>
<span class="Z3988" title="url_ver=Z39.88-2004&ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzotero.org%3A2&rft_id=info%3Adoi%2F10.5194%2Facp-11-13421-2011&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Earth's%20energy%20imbalance%20and%20implications&rft.jtitle=Atmospheric%20Chemistry%20and%20Physics&rft.volume=11&rft.issue=24&rft.aufirst=J.&rft.aulast=Hansen&rft.au=J.%20Hansen&rft.au=M.%20Sato&rft.au=P.%20Kharecha&rft.au=K.%20von%20Schuckmann&rft.date=2011-12-22&rft.pages=13421-13449&rft.spage=13421&rft.epage=13449&rft.issn=1680-7324"></span>
</div>
<br />
<a href="http://www.vliz.be/imisdocs/publications/218046.pdf">Ocean map of depths</a><br />
</div>
@whuthttp://www.blogger.com/profile/18297101284358849575noreply@blogger.com13tag:blogger.com,1999:blog-8890671936020885112.post-35128732763073514632013-03-21T12:55:00.000-07:002013-03-22T09:08:29.553-07:00Stochastic analysis of log sensitivity to CO2<div class="separator" style="clear: both; text-align: center;">
</div>
<br />
Often both skeptics and non-skeptics of AGW will suggest that climate models and their simulations are becoming too unwieldy to handle. Stochastic models of the climate are always an option to consider, as Marston points out in his paper "<i>Looking for new problems to solve? Consider the climate</i>", where he recommended to<br />
<blockquote class="tr_bq">
"<i>look at the sort of question a statistical description of the climate system would be expected to answer</i>" [5] . </blockquote>
<br />
It is no wonder that applying stochasticity is one of the grand challenges in science. In the past few years, DARPA put together a list of <a href="http://science.dodlive.mil/2012/08/08/23-mathematical-challenges-and-you/" target="_blank">23 mathematical challenges</a> inspired by <a href="http://en.wikipedia.org/wiki/Hilbert%27s_problems" target="_blank">Hilbert's original list</a>. Several of them feature large scale problems which would benefit from the stochastic angle that both Marston and David Mumford [4] have recommended:<br />
<ul>
<li><span style="font-size: small;"><i><b style="color: blue; font-family: Arial,Helvetica,sans-serif;">Mathematical Challenge 3: Capture and Harness Stochasticity in Nature</b></i></span><span style="font-family: Arial,Helvetica,sans-serif; font-size: small;">
</span><br />Address David Mumford’s call for new mathematics for the 21st century. Develop methods that capture persistence in stochastic environments.
<b> </b></li>
<li><span style="font-size: small;"><i><b style="color: blue; font-family: Arial,Helvetica,sans-serif;">Mathematical Challenge 4: 21st Century Fluids</b></i></span><span style="font-family: Arial,Helvetica,sans-serif; font-size: x-small;"><span style="font-size: small;">
</span></span>Classical fluid dynamics and the Navier-Stokes Equation were extraordinarily successful in obtaining quantitative understanding of shock waves, turbulence and solitons, but new methods are needed to tackle complex fluids such as foams, suspensions, gels and liquid crystals.</li>
<li><span style="font-size: small;"><i><b style="color: blue; font-family: Arial,Helvetica,sans-serif;">Mathematical Challenge 7: Occam’s Razor in Many Dimensions</b></i>
</span><br />As data collection increases can we “do more with less” by finding lower bounds for sensing complexity in systems? This is related to questions about entropy maximization algorithms.</li>
<li><span style="font-size: small;"><i><b style="color: blue; font-family: Arial,Helvetica,sans-serif;">Mathematical Challenge 20: Computation at Scale</b></i>
</span><br />How can we develop asymptotics for a world with massively many degrees of freedom?</li>
</ul>
<br />
The common theme is one of reducing complexity, which is often at odds with the mindset of those convinced that only more detailed computations of complex models will accurately represent a system.<br />
<br />
A representative phenomena of what stochastic processes are all about is the carbon cycle and the sequestering of CO2. Much work has gone into developing box models with various pathways leading to sequestering of industrial CO2. The response curve according to the BERN model [7] is series of damped exponentials showing the long term sequestering.
Yet, by a more direct statistical interpretation of what is involved in the process of CO2 sequestration, we can model the adjustment time of CO2 decay by a dispersive diffusional model derived via maximum entropy principles.<br />
<br />
$$ I(t) = \frac{1}{1+\sqrt{t/\tau}} $$<br />
<br />
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="http://imageshack.us/a/img18/8127/normalizeddecayofco2.gif" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="458" src="http://imageshack.us/a/img18/8127/normalizeddecayofco2.gif" width="640" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><b>Figure 1</b> : Impulse response of atmospheric CO2 concentration due to excess carbon emission.</td></tr>
</tbody></table>
<div class="separator" style="clear: both; text-align: center;">
</div>
<br />
A main tenet of the AGW theory presupposes the evolution of excess CO2.
The essential carbon-cycle physics says that the changing CO2 is
naturally governed by a base level which changes with ambient
temperature, but with an additional impulse response governed by a
carbon stimulus, either natural (volcano) or artificial (man-made
carbon). The latter process is described by a convolution, one of the
bread and butter techniques of climate scientists:<br />
<br />
$$CO_2(t,T) = CO_2(0,T) + \kappa \int_0^t C(\tau) I(t-\tau) d\tau $$<br />
<br />
With the impulse response of <b>Figure 1</b> convolved against the historical carbon outputs as archived at the
<a href="http://cdiac.ornl.gov/" target="_blank">CO2 Information Analysis Center</a>, it
matches the measured CO2 from the <a href="http://climexp.knmi.nl/getindices.cgi?WMO=CDIACData/co2_annual&STATION=CO2&TYPE=i&id=someone@somewhere&NPERYEAR=1" target="_blank">KNMI Climate Explorer</a> with the following agreement:<br />
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="http://img706.imageshack.us/img706/3041/carbonmodelco2.gif" style="margin-left: auto; margin-right: auto;"><img border="0" height="521" src="http://img706.imageshack.us/img706/3041/carbonmodelco2.gif" width="640" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><b>Figure 2</b> : CO2 impulse response convolution</td></tr>
</tbody></table>
<br />
The match to the CO2 measurements is very good after the year 1900. No inexplicable loss of carbon to be seen. This is all
explainable by diffusion kinetics into sequestration sites. Diffusional
physics is so well understood that it is no longer arguable.<br />
<br />
If on the other hand, you do this analysis <b>incorrectly</b> and use a naive damped
exponential response ala Segalstad [1] and Salby [unpublished] and other contrarian climate scientists, you do end up apparently believing that half of the
CO2 has gone missing. If that were indeed the case, this is what the response looks like, given the
skeptics view of a short 6.5 year CO2 residence time:<br />
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="http://img812.imageshack.us/img812/8065/co2expresponse.gif" style="margin-left: auto; margin-right: auto;"><img border="0" height="521" src="http://img812.imageshack.us/img812/8065/co2expresponse.gif" width="640" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><b>Figure 3</b>: Incorrect impulse response showing atmospheric CO2 deficit due to a short residence time. <br />
This is clearly not observed in the data</td></tr>
</tbody></table>
<br />
<br />
The obvious conclusion is that if you don't do the statistical physics correctly, you end up with nonsense numbers.<br />
<br />
<br />
Note that this does not contradict findings of the mainstream climate researchers, but it places our understanding on a potentially different footing, and one with potentially different insight.<br />
<br />
Consider next the sensitivity of temperature to the log of atmospheric CO2. In 1863, Tyndall discovered the properties of CO2 and other gases via an experimental apparatus [3];he noticed that light radiation absorption was linear up to a point but beyond that the absorptive properties of the gas showed a diminishing effect:<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="http://img32.imageshack.us/img32/9624/tyndall.gif" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="320" src="http://img32.imageshack.us/img32/9624/tyndall.gif" width="273" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><b>Figure 4</b>: Tyndall's description of absorption of light rays by gases [6]</td></tr>
</tbody></table>
<br />
The
non-linear nature derives from the logarithmic sensitivity of forcing
to CO2 concentration. The way to understand this is to consider
differential increases of gas concentration. In the test chamber
experiment, this is just a differential increase in the length of the
tube. The scattering cross-section is derived as a proportional increase
to how much gas is already there so it integrates as the ratio of delta
increase to the cumulative length of travel of the photon:<br />
<br />
$$ \int_{L_0}^L \frac{dX}{X} = ln(\frac{L}{L_0}) \sim ln(\Delta{[CO_2]}) $$<br />
<br />
So
as the photon travels a length of tube, it gets progressively more
difficult to increase the amount of scattering, not because it isn't physically possible but
because the photon has had a large probability of being intercepted already as it makes its way out of the atmosphere. That generates the logarithmic sensitivity.<br />
<br />
Note that the
lower bound of the integral has to exist to prevent a singularity in the proportional
increase. To represent this another way, we can say that there is a
minimum length at which the absorption cross-section occurs.<br />
<br />
<br />
$$ \int_{0}^L \frac{dX}{X_0+X} = ln(\frac{X_0+L}{X_0}) \sim ln(1+\Delta{[CO_2]}) $$<br />
<br />
<br />
The analogy to the test chamber is the height of the
atmosphere. The excess CO2 that we emit from burning fossil fuels
increases the concentration at the top of the atmosphere, and that
increases the length of the travel of the outgoing IR photons, and the
same logarithmic sensitivity results. This is incorrectly labeled as
saturating behavior, instead of an asymptotic logarithmic behavior.<br />
<br />
I
haven't seen this derived in this specific way but the math is
high-school level integral calculus. The actual model used by Lacis and other climate scientists is to configure the cross-sections
by compositing the atmosphere into layers or "slabs" and then
propagating the interception of photons by CO2 by differential numerical
calculations. In addition, there is a broadening of the spectral
lines as concentration increases, contributing to the first order result of logarithmic sensitivity. In general the more that the infrared parts of the photonic spectrum get trapped by CO2 and H20, the higher the temperature has to be to make up for the missing propagated wavelengths while not violating the earth's strict steady-state incoming/outgoing energy balance.<br />
<br />
To get a sense of this logarithmic climate sensitivity, the temperature response
is shown to the below right for a 3°C sensitivity to CO2 doubling, mapped against the <a href="http://berkeleyearth.org/faq/" target="_blank">BEST land-based temperature record</a>. Observe that trending temperature shifts are
already observed in the early 20th century. <br />
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="http://img811.imageshack.us/img811/4829/bestco2model.gif" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="268" src="http://img811.imageshack.us/img811/4829/bestco2model.gif" width="640" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><b>Figure 5:</b> CO2 model applied to AGW via a log sensitivity and <br />
compared against the fast rsponse of BEST land-based records.</td></tr>
</tbody></table>
If we choose a 2.8°C sensitivity from Hansen's 1981 paper [2] and compare the above model to Hansen's projection, one can see that we are on the right track in duplicating his analysis.<br />
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="http://imageshack.us/a/img802/3918/hansen1981.gif" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="278" src="http://imageshack.us/a/img802/3918/hansen1981.gif" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><b>Figure 6: </b>Hansen's original model projection circa 1981 [2] and the 2.8C model used here.</td></tr>
</tbody></table>
<br />
Perhaps the only real
departure from the model is a warming around 1940 not accounted for by CO2. But even this is likely due to a stochastic characteristic in the form of noise riding along with the trend.. This noise could be volcanic disruptions or ocean upwelling fluctuations of a random nature, which only requires the property that upward excursions match downward.<br />
<br />
So if we consider the difference between the green line model and blue line data below:<br />
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="http://img856.imageshack.us/img856/5774/bestco2fit.gif" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="345" src="http://img856.imageshack.us/img856/5774/bestco2fit.gif" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><b>Figure 7:</b> Plot used for regression analysis. <br />
Beyond 1900, the RMS error is less than 0.2 degrees</td></tr>
</tbody></table>
<br />
and take the model/data residuals over time, we get the following plot.<br />
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="http://imageshack.us/a/img442/9962/bestresidual.gif" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="258" src="http://imageshack.us/a/img442/9962/bestresidual.gif" width="640" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><b>Figure 8: </b>Regression difference between log sensitivity model and BEST data </td></tr>
</tbody></table>
Note that the yearly excursions never exceed about 0.2°C from the underlying trend. The dotted line red curve above the solid blue residual curve is an Ornstein-Uhlenbeck model (red noise) of a yearly random walk with a reversion-to-the-mean drag that prevents fluctuations beyond approximately 0.2°C. At this kind of scale and considering the Markov process which will allow short-term fluctuations in the tenths of a degree, the bump of increased temperature in the 1940's is not a rare nor completely unlikely occurrence.<br />
<br />
A red noise model on this scale can not accommodate both the short-term fluctuations and the large upward trend of the last 50 years. That is why the CO2-assisted warming is the true culprit, as evidenced by a completely stochastic analysis.<br />
<br />
<b>[EDIT]</b><br />
Here is a sequence of red noise time series (of approximately 0.2°C RMS excursion) placed on top of an accelerating monotonically warming temperature profile (the no-noise accelerating profile is in the lower left panel). The actual profile is shown in the lower right panel. An averaging window of 30 years is placed on the profiles to give an indication of where plateaus, pauses, and apparent cooling can occur.<br />
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="http://img822.imageshack.us/img822/8060/ourwtemperatureprofiles.gif" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="640" src="http://img822.imageshack.us/img822/8060/ourwtemperatureprofiles.gif" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><b>Figure 9</b>: Hypothetical red noise series placed on an accelerating warming curve.</td></tr>
</tbody></table>
"Noise riders" are people that look at time series and read too much into each fluctuation. The temptation to place too much emphasis on recent readings is strong but ultimately misguided. Dead reckoning models are built-in as a human intuition mechanism but don't serve us well in the face of noise.<br />
<br />
The baseball analogy to the red noise of <b>Figure 9</b> is the knuckle-ball pitcher. Being able to hit a knuckle-baller means that you have quick reflexes and an ability to suppress the dead reckoning urge.<br />
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="http://sports.cbsimg.net/images//visual/whatshot/R.A.-Dickey-knuckle.gif" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="240" src="http://sports.cbsimg.net/images//visual/whatshot/R.A.-Dickey-knuckle.gif" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">R.A. Dickey's Cy Young Knuckleball Pitch (<a href="http://www.sportsgrid.com/mlb/watching-r-a-dickeys-knuckleball-in-slow-motion-proves-how-difficult-it-is-to-hit/" target="_blank">SOURCE</a>)</td></tr>
</tbody></table>
<br />
<br />
<br />
<h3>
References</h3>
<div class="csl-bib-body" style="line-height: 1.35;">
<div class="csl-entry" style="clear: left;">
<div class="csl-left-margin" style="float: left; padding-right: 0.5em; text-align: right; width: 1em;">
<span style="font-size: x-small;">[1]</span></div>
<div class="csl-right-inline" style="margin: 0 .4em 0 1.5em;">
<span style="font-size: x-small;">T. V. Segalstad, “Carbon cycle modelling and the residence time of natural and anthropogenic atmospheric CO2,” <i>BATE, R.(Ed., 1998): Global Warming</i>, pp. 184–219, 1998.</span></div>
</div>
<span class="Z3988" style="font-size: x-small;" title="url_ver=Z39.88-2004&ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzotero.org%3A2&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Carbon%20cycle%20modelling%20and%20the%20residence%20time%20of%20natural%20and%20anthropogenic%20atmospheric%20CO2&rft.jtitle=BATE%2C%20R.(Ed.%2C%201998)%3A%20Global%20Warming&rft.aufirst=Tom%20V&rft.aulast=Segalstad&rft.au=Tom%20V%20Segalstad&rft.date=1998&rft.pages=184-219&rft.spage=184&rft.epage=219">
</span></div>
<span style="font-size: x-small;"></span><br />
<span style="font-size: x-small;"><br /></span>
<div class="csl-bib-body" style="line-height: 1.35;">
<div class="csl-entry" style="clear: left;">
<div class="csl-left-margin" style="float: left; padding-right: 0.5em; text-align: right; width: 1em;">
<span style="font-size: x-small;">[2]</span></div>
<div class="csl-right-inline" style="margin: 0 .4em 0 1.5em;">
<span style="font-size: x-small;">J. Hansen, D. Johnson, A. Lacis, S. Lebedeff, P. Lee, D. Rind, and G. Russell, “Climate impact of increasing atmospheric carbon dioxide,” <i>Science, 2l3</i>, pp. 957–966.</span></div>
</div>
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<span style="font-size: x-small;">[3]</span></div>
<div class="csl-right-inline" style="margin: 0 .4em 0 1.5em;">
<span style="font-size: x-small;">A. A. Lacis, G. A. Schmidt, D. Rind, and R. A. Ruedy, “Atmospheric CO2: principal control knob governing Earth’s temperature,” <i>Science</i>, vol. 330, no. 6002, pp. 356–359, 2010.</span></div>
</div>
<span class="Z3988" style="font-size: x-small;" title="url_ver=Z39.88-2004&ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzotero.org%3A2&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Atmospheric%20CO2%3A%20principal%20control%20knob%20governing%20Earth%E2%80%99s%20temperature&rft.jtitle=Science&rft.stitle=Science&rft.volume=330&rft.issue=6002&rft.aufirst=Andrew%20A&rft.aulast=Lacis&rft.au=Andrew%20A%20Lacis&rft.au=Gavin%20A%20Schmidt&rft.au=David%20Rind&rft.au=Reto%20A%20Ruedy&rft.date=2010&rft.pages=356-359&rft.spage=356&rft.epage=359&rft.issn=0036-8075">
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<div class="csl-left-margin" style="float: left; padding-right: 0.5em; text-align: right; width: 1em;">
<span style="font-size: x-small;">[4]</span></div>
<div class="csl-right-inline" style="margin: 0 .4em 0 1.5em;">
<span style="font-size: x-small;">D. Mumford, “The dawning of the age of stochasticity,” <i>Mathematics: Frontiers and Perspectives</i>, pp. 197–218, 2000.</span></div>
</div>
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</span></div>
<span style="font-size: x-small;"></span><br />
<span style="font-size: x-small;"><br /></span>
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<div class="csl-left-margin" style="float: left; padding-right: 0.5em; text-align: right; width: 1em;">
<span style="font-size: x-small;">[5]</span></div>
<div class="csl-right-inline" style="margin: 0 .4em 0 1.5em;">
<span style="font-size: x-small;">B. Marston, “Looking for new problems to solve? Consider the climate,” <i>Physics</i>, vol. 4, p. 20, 2011.</span></div>
</div>
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<span style="font-size: x-small;"><br /></span>
<span style="font-size: x-small;">[6] Tyndall, John, 1861. On the Absorption and Radiation of Heat by Gases
and Vapours, and on the Physical Connection of Radiation, Absorption,
and Conduction. 'Philosophical Magazine<i> ser. 4, vol. 22, 169–94, 273–85.</i></span> <br />
<span style="font-size: x-small;"><br /></span>
<span style="font-size: x-small;">[7] J. Golinski, “Parameters for tuning a simple carbon cycle model,” United Nations Framework Convention on Climate Change. [Online]. Available: http://unfccc.int/resource/brazil/carbon.html. [Accessed: 12-Mar-2013].
</span>@whuthttp://www.blogger.com/profile/18297101284358849575noreply@blogger.com2tag:blogger.com,1999:blog-8890671936020885112.post-52626769997346013202013-03-18T23:26:00.001-07:002013-03-19T08:05:30.210-07:00Spatial and Temporal Correlations of WindIn terms of climate statistics, we always have more data than we know what to do with. The challenge is in reducing the data into meaningful bits of information which can be characterized and modeled succinctly.<br />
<br />
So this is a case of where enough data exists that we can get an understanding of the spatial and temporal correlations of prevailing winds.<br />
<br />
One source of data comes from measurements of winds at altitude, taken by jet airliners as part of their regular flight routes and then post-processed and analyzed[1]. This gives an indication of spatial correlation over a large dynamic range, stretching a little over three orders of magnitude. Once collected, the post-processing involved applying an autocorrelation to the data over all spatial scales. The power spectral density (PSD) is the Fourier transform of the autocorrelation, and that is shown in <b>Figure 1</b> below, along with a suggested model achieving a good fit to the actual meridonal wind PSD.<br />
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiq_rUeoAZg-zP8bPUSSDCqBdiL5KCiMnJTTkduCYMmOH69-Okc77JOwh_TttXnSVTLiH8SROIOTssjdoUebWq8g1Mda3d98PKPhfgd1WKO88PR1wClP10ZiCAXCVcXbVSuIip80drwXmA/s1600/zonal_meridional.gif" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="321" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiq_rUeoAZg-zP8bPUSSDCqBdiL5KCiMnJTTkduCYMmOH69-Okc77JOwh_TttXnSVTLiH8SROIOTssjdoUebWq8g1Mda3d98PKPhfgd1WKO88PR1wClP10ZiCAXCVcXbVSuIip80drwXmA/s400/zonal_meridional.gif" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><b>Figure 1</b> : Zonal and Meridional wind speed PSD. The dashed line model fit works well apart from deviations near the noisy shorter wavelengths.</td></tr>
</tbody></table>
The chosen model is simply the PSD of an exponentially damped cosine autocorrelation (see [2] for examples in meteorology).<br />
<br />
$$ C(x) = e^{-\alpha |x|} \cdot cos(\beta x) $$<br />
<br />
The Fourier transform of this autocorrelation gives a PSD that is best described as a shifted Cauchy or Lorentzian profile. In <b>Figure 1</b>, the shape is obvious as it shows a peak and then a concave-upward inflection following the peak.<br />
<br />
$$ I(S_x) = \frac{\alpha}{(S_x-\beta)^2 + \alpha^2} $$<br />
<br />
This specific profile is generated by a semi-Markovian process. The "semi" part captures the periodic portion while the Markov contributes the memory for close-proximity coupling. Specifically, the Markov term is the alpha-factor damping which gives the overall 1/S^2 roll-off, while the beta-spatial shift indicates a semi-Markov ordering feature indicating some underlying longer-range spatial periodicity.<br />
<br />
$$ \beta = 2 \pi/L = \text{1.3e-6 rad/m} $$ <br />
$$ \alpha = \text{7.63e-7 rad/m} $$<br />
<br />
This gives a weak periodicity of L=4833 km. For a prevailing wind-stream of 30 MPH, this spans approximately 4 days of flow between a peak and a lull in wind speed. <br />
<br />
The spatial correlation intuitively leads to the concept of a temporal correlation in wind speed, and the question of whether this characteristic can be measured and contrasted to the spatial correlation.<br />
<br />
For temporal correlation, I used the same data from the BPA which I had previously applied to a <a href="http://theoilconundrum.blogspot.com/2012/02/wind-speeds-of-world.html" target="_blank">probability density function (PDF) analysis</a>. <br />
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgedTsLf2lD7DZMhbkzPnZ-XMHfCO4FJEzjb2eAD_fDxPmKGI2cCMQNumr5-olZnrIsWfdd0QjUYdxVVEUIy4KggtuFN_TY8ZLcUU8X30YMcKMHTCx-4jre-cBgIOiikmwhSVNrwliKVVc/s1600/roosevelt_island_wind_autocorrelation.gif" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="278" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgedTsLf2lD7DZMhbkzPnZ-XMHfCO4FJEzjb2eAD_fDxPmKGI2cCMQNumr5-olZnrIsWfdd0QjUYdxVVEUIy4KggtuFN_TY8ZLcUU8X30YMcKMHTCx-4jre-cBgIOiikmwhSVNrwliKVVc/s400/roosevelt_island_wind_autocorrelation.gif" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><b>Figure 2 :</b> Temporal autocorrelation of wind speed</td></tr>
</tbody></table>
The temporal autocorrelation of the Roosevelt Island site shows some periodic fine structure. Using the Eureqa equation fitting software, I get the following empirical autocorrelation match.<br />
<br />
$$ c(t)= 0.02934 \cdot sin(0.5012 - 1.626 t) + 0.01668 \cdot cos(0.8645 + 6.203 t) + exp(-1.405 t) - 0.02134 \cdot sin(1.088 - 0.599 t) $$<br />
<br />
The first and second terms are periodicities of 3.86 and 1 days, the latter easily explained by daily (diurnal) variations. There is also a strong damping factor with a half-life less than a day. The last term is a 10 day period which generates a longer term modulation (and also has more uncertainty in its weighting).<br />
<br />
The 3.86 day periodicity is likely due to a principle oscillation pattern in unstable meandering baroclinic <a href="http://en.wikipedia.org/wiki/Rossby_wave" target="_blank">Rossby waves</a> as detected through a separate statistical analysis [3]<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgR-gbTxEMLVYcRGzQz8qdOjbIOkmWO1hTPi0BHryF1XZkfL2Vkcvdy2y8CzxbW_dU4Zw1Xlykw4t3t15Ede_LKNz3AN8k3pTjiR2z6k8KqJe7sHEoXmhgxfjRVQdql0tC87fmF0cQxfPI/s1600/storch_rossby.GIF" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="640" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgR-gbTxEMLVYcRGzQz8qdOjbIOkmWO1hTPi0BHryF1XZkfL2Vkcvdy2y8CzxbW_dU4Zw1Xlykw4t3t15Ede_LKNz3AN8k3pTjiR2z6k8KqJe7sHEoXmhgxfjRVQdql0tC87fmF0cQxfPI/s640/storch_rossby.GIF" width="328" /></a></div>
<br />
This demonstrates how the spatial correlation relates to the temporal correlation. As the tropospheric wave patterns propagate and disperse, they influence the local wind patterns. These are quite subtle effects yet they can be detected and perhaps can be put to good use in battling the intermittency in wind power.<br />
<br />
<br />
<h4>
References</h4>
[1] G. Nastrom and K. S. Gage, “A climatology of atmospheric wavenumber spectra of wind and temperature observed by commercial aircraft,” <i>Journal of the atmospheric sciences</i>, vol. 42, no. 9, pp. 950–960, 1985.<br />
<br />
[2] H. J. Thiebaux, “Experiments with correlation representations for objective analysis,” <i>Monthly Weather Review</i>, vol. 103, p. 617, 1975. <br />
<br />
[3] H. Von Storch and F. W. Zwiers, <i>Statistical analysis in climate research</i>. Cambridge University Press, 2002.
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@whuthttp://www.blogger.com/profile/18297101284358849575noreply@blogger.com0tag:blogger.com,1999:blog-8890671936020885112.post-79467950968656029242013-03-17T09:16:00.000-07:002013-05-16T21:25:53.651-07:00Standard Atmosphere Model and Uncertainty in Entropy<div style="text-align: center;">
(see <a href="http://theoilconundrum.blogspot.com/2013/05/the-homework-problem-to-end-all.html">this more recent derivation</a> as well)</div>
<div style="text-align: center;">
<br /></div>
The lower atmosphere known as the troposphere is neither completely isentropic (constant entropy) nor completely isothermal (constant temperature).<br />
<br />
An isentropic process will not exchange energy with the environment and therefore will maintain a constant entropy (also describing an adiabatic process).<br />
<br />
If it was isothermal, all altitudes would be at the same temperature, realizing a disordered equilibrium state.This is maximum entropy as all the vertical layers are completely mixed.<br />
<br />
Yet, when convective buoyancy results from evaporation, energy is exchanged up and down the column. And when green-house gases change the radiative properties of the layers, the temperature will also change to some degree.<br />
<br />
So, how would we naively estimate what the standard atmospheric profile of such a disordered state would be? The standard approach is to assume maximum uncertainty between the extremes of isentropic and isothermal conditions. This is essentially a mean estimate of half the entropy of the isothermal state, and a realization of Jaynes recommendation to assume maximum ignorance when confronted with the unknown.<br />
<br />
We start with the differential representation<br />
<br />
$$dH = V dp + T ds $$<br />
<br />
Add in the ideal gas law assuming the universal gas constant and one mole.<br />
$$ pv = RT $$<br />
<br />
Add in change in enthalpy applying the specific heat capacity of air (at constant pressure).<br />
$$dH = c_p dT$$<br />
<br />
Substitute the above two into the first equation:<br />
$$c_p dT = \frac{RT}{p} dp + T ds $$<br />
<br />
Rearrange in integrable form<br />
$$c_p \frac{dT}{T} = \frac{R}{p} dp + ds $$<br />
<br />
Integrate between the boundary conditions<br />
$$c_p \cdot ln(\frac{T_1}{T_0} ) = R \cdot ln(\frac{p_1}{p_0}) + \Delta s $$<br />
<br />
The average delta entropy change (loss) from a low temperature state to a high temperature state is: <br />
$$ \Delta s = - \frac{1}{2} c_p \cdot ln(\frac{T_1}{T_0} ) $$<br />
<br />
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<br />
The factor of 1/2 is critical because it represents an average entropy between zero entropy change and maximum entropy change. So this is halfway in between an isentropic and an isothermal process (somewhere in the continuum of a <a href="http://en.wikipedia.org/wiki/Polytropic_process" target="_blank">polytropic process</a>, categorized as a quasi-adiabatic process 1 < <i>n</i> < γ, where <i>n</i> is the <i>polytropic index</i> and γ is known as the <i>adiabatic index</i> or <i>isentropic expansion factor</i> or <i>heat capacity ratio</i>).<br />
<blockquote class="tr_bq">
<b><i>Added Note</i></b>: The loss or dispersion of energy is likely due to the molecules having to fight gravity to achieve a MaxEnt distribution as per the barometric formula. The average loss is half the potential energy of the barometric height (an average troposphere height) and this is reflected as half again of the <i>dH</i> kinetic heat term <span style="font-family: Arial,Helvetica,sans-serif; font-size: 10pt;">½ </span><i>Cp</i>Δ<i>T</i> = <i>mg</i> <span style="font-family: Arial,Helvetica,sans-serif; font-size: 10pt;">½</span><!--[if gte mso 9]><xml>
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<![endif]--><span style="mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> Δ<i>Z</i></span>. This is a hypothesis to quantify the dispersed energy defined by the hidden states of Δ<i>S.</i></blockquote>
Substituting and rearranging: <br />
$$ \frac{3}{2} c_p \cdot ln(\frac{T_1}{T_0} ) = R \cdot ln(\frac{p_1}{p_0}) $$<br />
<br />
which leads to the representation<br />
<br />
$$ T_1 = T_0 \cdot (\frac{p_1}{p_0})^{R/{1.5 c_p}} $$<br />
<br />
which only differs from the adiabatic process (below) in its modified power law (since <i>n</i><γ it is a quasi-adiabatic process):<br />
<br />
$$ T_1 = T_0 \cdot (\frac{p_1}{p_0})^{R/c_p} $$<br />
<br />
<br />
The <a href="http://www.atmoscuhttp//www.atmosculator.com/The%20Standard%20Atmosphere.html?lator.com/The%20Standard%20Atmosphere.html?" target="_blank">Standard Atmosphere model</a> represents a typical barometric profile<br />
<br />
<blockquote class="tr_bq">
<span style="font-family: Lucida Grande, Arial, Helvetica, sans-serif; font-size: x-small;">The
"Standard Atmosphere" is a hypothetical vertical distribution of
atmospheric properties which, by international agreement,
is roughly representative of year-round, mid-latitude conditions.
Typical usages include altimeter calibrations and aircraft
design and performance calculations. It should be recognized that
actual conditions may vary considerably from this standard.</span> </blockquote>
<blockquote class="tr_bq">
<span style="font-family: Lucida Grande, Arial, Helvetica, sans-serif; font-size: x-small;"></span><span style="font-family: Lucida Grande, Arial, Helvetica, sans-serif; font-size: x-small;">The
most recent definition is the "US Standard Atmosphere, 1976" developed
jointly by NOAA, NASA, and the USAF. It is an idealized,
steady state representation of the earth's atmosphere from the
surface to 1000 km, as it is assumed to exist during a period
of moderate solar activity. The 1976 model is identical with the
earlier 1962 standard up to 51 km, and with the International
Civil Aviation Organization (ICAO) standard up to 32 km.</span></blockquote>
<br />
Using R = 287 J/K/kg, cp =1004 J/K/kg, we fit the equation that we derived above to the US Standard Atmosphere data set in the figure below:<br />
<br />
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj268HA5NV6aqcgF-A3POyECY3BiEeNooj7PjBQBHd4hcGsDOqrMDvPZaetta2FyI9hCUKU77AHqB1Fq25glH_Pm1tA_SlfM9OxTCn1hYsr9UDggHrHHX7Z4r6GAW-TRKTWQbG0A4VhOB4/s1600/standard_atmosphere_model.gif" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="520" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj268HA5NV6aqcgF-A3POyECY3BiEeNooj7PjBQBHd4hcGsDOqrMDvPZaetta2FyI9hCUKU77AHqB1Fq25glH_Pm1tA_SlfM9OxTCn1hYsr9UDggHrHHX7Z4r6GAW-TRKTWQbG0A4VhOB4/s640/standard_atmosphere_model.gif" width="640" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Standard Atmosphere Model (circles) compared to average entropy derivation (line)</td></tr>
</tbody></table>
<div class="separator" style="clear: both; text-align: center;">
<a href="http://www.blogger.com/blogger.g?blogID=8890671936020885112" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"></a></div>
<br />
The correlation coefficient between the standard atmospheric model and the derived expression is very close to one, which means that the two models likely converge to the same reduced form.<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="http://www.blogger.com/blogger.g?blogID=8890671936020885112" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"></a></div>
How was the US Standard Atmosphere 1975 model chosen? Was this empirically derived from measured temperature gradients (i.e. lapse rates) and barometric pressure profiles? Or was it estimated in an equivalent fashion to that derived here?<br />
<br />
The globally averaged temperature gradient is 6.5 C/km, which is exactly g/(1.5cp). Compare this to the adiabatic gradient of 9.8 C/km = g/cp.<br />
<br />
<div style="background-color: #e06666; color: white; text-align: center;">
<b>[EDITS below]</b></div>
<br />
The exponent R/(1.5*cp) in the standard atmosphere derivation described above is <br />
R/(1.5*cp) = 286.997/(1.5*1004.4) = <span style="background-color: yellow;">α = 0.190493</span>. Or, since cp = 7/2*R for an ideal diatomic gas, then R/(1.5*7/2*R) = <span style="background-color: yellow;">α = 4/21 = 0.190476</span> .<br />
<br />
Note from reference [1] that this same value of <span style="background-color: white;">α </span>is apparently a <i>best fit</i> to the data.<br />
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhqLNyLyccC5v4PVkH6ImOXKVfArWoD3cG7OnLbFAPZU-MxYlFmrbMuRox-dl22y8bsq6aZBDIv-QrANZuMcvQ-77vUpGwysSWAZzBdantQWA-HyqBtiU_pLsD6c_eNjdjkW3V7ahRZHBQ/s1600/sorokhtin.GIF" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhqLNyLyccC5v4PVkH6ImOXKVfArWoD3cG7OnLbFAPZU-MxYlFmrbMuRox-dl22y8bsq6aZBDIv-QrANZuMcvQ-77vUpGwysSWAZzBdantQWA-HyqBtiU_pLsD6c_eNjdjkW3V7ahRZHBQ/s1600/sorokhtin.GIF" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">from <span style="font-size: x-small;">O. G. Sorokhtin, G. V. Chilingarian, and N. O. Sorokhtin, <br /><i>Evolution of Earth and its Climate: Birth, Life and Death of Earth</i>. Elsevier Science, 2010.</span></td></tr>
</tbody></table>
What is the probability that a value fitted to empirically averaged data would match to a derived quantity to essentially 4 decimal places after rounding? That would be quite a coincidence unless there is some fundamental maximum entropy truth (or perhaps the <a href="http://en.wikipedia.org/wiki/Virial_theorem" target="_blank">Virial theorem</a>, <span style="font-family: Arial,Helvetica,sans-serif;">K.E = </span><!--[if gte mso 9]><xml>
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<![endif]--><span style="font-family: Arial,Helvetica,sans-serif; font-size: 10pt;">½ </span><span style="font-family: Arial,Helvetica,sans-serif;">P.E.</span>) hidden in the derived relation.<br />
<br />
So essentially we have the T vs P relation for the US Standard Atmosphere as<br />
<br />
$$ T_1 = T_0 \cdot (\frac{p_1}{p_0})^{4/21} $$<br />
<br />
with very precise resolution. <br />
<br />
Martin Tribus recounted his discussion with Shannon [2] :<br />
<br />
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<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgEfWzDpitJFSUtUKF1Ks1PptpCMm0SO50mzvdCOCXnpdxbSVm1gyAePFjaTgaHBoOHt2kM-w-pIObkmNOVE65EjBj-A2pfukRa8qsrs1JDP-R51RHLdYSmUu8LB_-pfK6lwr7mSd80TnA/s1600/shannon_quote.GIF" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="200" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgEfWzDpitJFSUtUKF1Ks1PptpCMm0SO50mzvdCOCXnpdxbSVm1gyAePFjaTgaHBoOHt2kM-w-pIObkmNOVE65EjBj-A2pfukRa8qsrs1JDP-R51RHLdYSmUu8LB_-pfK6lwr7mSd80TnA/s400/shannon_quote.GIF" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">from <span style="font-size: x-small;">M. Tribus, “An engineer looks at Bayes,” in <i><br />Maximum-Entropy and Bayesian Methods in Science and Engineering</i>, Springer, 1988.</span></td></tr>
</tbody></table>
I don't completely understand entropy either, but it seems to understand us and the world we live in. Entropy defined as <a href="http://entropysite.oxy.edu/" target="_blank">dispersion of energy at a specific temperature</a> often allows one to reason about the analysis through some intuitive notions.<br />
<br />
<br />
<br />
<b>References </b><br />
<br />
<div class="csl-bib-body" style="line-height: 1.35;">
<div class="csl-entry" style="clear: left;">
<div class="csl-left-margin" style="float: left; padding-right: 0.5em; text-align: right; width: 1em;">
<span style="font-size: x-small;">[1]</span></div>
<div class="csl-right-inline" style="margin: 0px 0.4em 0px 1.5em;">
<span style="font-size: x-small;"> O. G. Sorokhtin, G. V. Chilingarian, and N. O. Sorokhtin, <i>Evolution of Earth and its Climate: Birth, Life and Death of Earth</i>. Elsevier Science, 2010.</span></div>
</div>
<span class="Z3988" title="url_ver=Z39.88-2004&ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzotero.org%3A2&rft_id=urn%3Aisbn%3A9780444537584&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Evolution%20of%20Earth%20and%20its%20Climate%3A%20Birth%2C%20Life%20and%20Death%20of%20Earth&rft.publisher=Elsevier%20Science&rft.series=Developments%20in%20Earth%20and%20Environmental%20Sciences&rft.aufirst=O.G.&rft.aulast=Sorokhtin&rft.au=O.G.%20Sorokhtin&rft.au=G.V.%20Chilingarian&rft.au=N.O.%20Sorokhtin&rft.date=2010&rft.isbn=9780444537584">
</span></div>
<span style="font-size: x-small;"><br /></span>
<br />
<div class="csl-bib-body" style="line-height: 1.35;">
<div class="csl-entry" style="clear: left;">
<div class="csl-left-margin" style="float: left; padding-right: 0.5em; text-align: right; width: 1em;">
<span style="font-size: x-small;">[2]</span></div>
<div class="csl-right-inline" style="margin: 0px 0.4em 0px 1.5em;">
<span style="font-size: x-small;"> M. Tribus, “An engineer looks at Bayes,” in <i>Maximum-Entropy and Bayesian Methods in Science and Engineering</i>, Springer, 1988, pp. 31–52.</span></div>
</div>
<span class="Z3988" title="url_ver=Z39.88-2004&ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzotero.org%3A2&rft_id=urn%3Aisbn%3A9401078718&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=An%20engineer%20looks%20at%20Bayes&rft.publisher=Springer&rft.aufirst=Myron&rft.aulast=Tribus&rft.au=Myron%20Tribus&rft.date=1988&rft.pages=31-52&rft.spage=31&rft.epage=52&rft.isbn=9401078718">
</span></div>
<br />
<br />
<div style="background-color: #e06666; color: white; text-align: center;">
<b>[UPDATE]</b></div>
<br />
<br />
The lapse rate of temperature with altitude (i.e. gradient) is a characteristic that is easily measured for various advective planetary atmospheres.<br />
<br />
If we refer to the polytropic factor 4/21 by <i>f</i> then the lapse rate<br />
<br />
$$ LR = f \cdot g \cdot MW / R $$<br />
<br />
where <i>g </i>is the acceleration due to gravity on a planet, and <i>MW</i> is the average molar molecular weight of the atmospheric gas composition on the planet. For the planets, Earth, Venus, Mars and the sun, the value of gravity and molecular weight are well characterized, with lapse rates documented in the references below.<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="http://img580.imageshack.us/img580/4503/lapserateplanets.gif" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="273" src="http://img580.imageshack.us/img580/4503/lapserateplanets.gif" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Observed lapse rate gathered for Mars [3,4], Venus [5,6,7], Sun [8].<br />
Venus seems to have the greatest variability with altitude</td></tr>
</tbody></table>
<div style="text-align: center;">
<br /></div>
<div style="text-align: center;">
</div>
<table border="0" cellpadding="0" cellspacing="0" style="margin-left: auto; margin-right: auto; text-align: left; width: 383px;"><colgroup><col style="width: 48pt;" width="64"></col>
<col style="mso-width-alt: 3949; mso-width-source: userset; width: 81pt;" width="108"></col>
<col style="mso-width-alt: 3035; mso-width-source: userset; width: 62pt;" width="83"></col>
<col span="2" style="width: 48pt;" width="64"></col>
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<tr height="20" style="height: 15.0pt;">
<td height="20" style="background-color: black; color: white; font-family: Arial,Helvetica,sans-serif; height: 15pt; width: 48pt;" width="64"><b>Object</b></td>
<td style="background-color: black; color: white; font-family: Arial,Helvetica,sans-serif; text-align: center; width: 81pt;" width="108"><b>Theory</b></td>
<td style="background-color: black; color: white; font-family: Arial,Helvetica,sans-serif; text-align: center; width: 62pt;" width="83"><b>Observed</b></td>
<td style="background-color: black; color: white; font-family: Arial,Helvetica,sans-serif; text-align: center; width: 48pt;" width="64"><b>MW</b></td>
<td style="background-color: black; color: white; font-family: Arial,Helvetica,sans-serif; text-align: center; width: 48pt;" width="64"><b>g</b></td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td height="20" style="font-family: Arial,Helvetica,sans-serif; height: 15pt;">Earth</td>
<td style="font-family: Arial,Helvetica,sans-serif; text-align: center;">6.506</td>
<td style="font-family: Arial,Helvetica,sans-serif; text-align: center;">6.5</td>
<td style="font-family: Arial,Helvetica,sans-serif; text-align: center;">28.96</td>
<td style="font-family: Arial,Helvetica,sans-serif; text-align: center;">9.807</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td height="20" style="font-family: Arial,Helvetica,sans-serif; height: 15pt;">Venus</td>
<td style="font-family: Arial,Helvetica,sans-serif; text-align: center;">8.827</td>
<td style="font-family: Arial,Helvetica,sans-serif; text-align: center;">8.5 ref [5]</td>
<td style="font-family: Arial,Helvetica,sans-serif; text-align: center;">43.44</td>
<td style="font-family: Arial,Helvetica,sans-serif; text-align: center;">8.87</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td height="20" style="font-family: Arial,Helvetica,sans-serif; height: 15pt;">Mars</td>
<td style="font-family: Arial,Helvetica,sans-serif; text-align: center;">3.703</td>
<td style="font-family: Arial,Helvetica,sans-serif; text-align: center;">3.7 ref [4]</td>
<td style="font-family: Arial,Helvetica,sans-serif; text-align: center;">43.56</td>
<td style="font-family: Arial,Helvetica,sans-serif; text-align: center;">3.711</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td height="20" style="font-family: Arial,Helvetica,sans-serif; height: 15pt;">Sun</td>
<td style="font-family: Arial,Helvetica,sans-serif; text-align: center;">16.01</td>
<td style="font-family: Arial,Helvetica,sans-serif; text-align: center;">16 ref [8]</td>
<td style="font-family: Arial,Helvetica,sans-serif; text-align: center;">2.55</td>
<td style="font-family: Arial,Helvetica,sans-serif; text-align: center;">274.0</td>
</tr>
</tbody></table>
<div style="text-align: center;">
<br /></div>
The US Standard Atmosphere references the lapse<b> </b>rate as 6.5 while the International Civil Aviation Organization as 6.49. If we take the latter and adjust the composition of water vapor in the atmosphere to match, we come up with an 18% relative humidity at sea level extending to higher altitudes.<br />
<br />
This change in humidity will cause a subtle change in lapse rate with global warming. As the relative humidity goes up with average temperature, the lapse rate will slightly decline, as water vapor is less dense than either molecular oxygen and nitrogen. So this means that an increase in temperature at the top of the atmosphere due to excess GHG will be slightly compensated by a change in tropospheric height, i.e. that point where the barometric depth is equal to the optical depth. What magnitude of change this will make on surface temperature is worthy of further thought, but as has been known since Manabe, the shift toward heating is as shown in the figure below. Note how the profile shifts right (warmer) with additional greenhouse gases.<br />
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj2Ch5sBvSErNVxl16u3CvcJYxG2OxIZJVZp8MSh-cKox9yYruisCLRC2Il2nNPjk0BKkMNm5TkGT2eHJvfK5DGubHCBQR0JfD9tD4egJ5qlQbERe7vBIdftCGIg1kW8QAyDOlZzjUQ3vw/s1600/lapse_rate_manabe.GIF" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="400" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj2Ch5sBvSErNVxl16u3CvcJYxG2OxIZJVZp8MSh-cKox9yYruisCLRC2Il2nNPjk0BKkMNm5TkGT2eHJvfK5DGubHCBQR0JfD9tD4egJ5qlQbERe7vBIdftCGIg1kW8QAyDOlZzjUQ3vw/s400/lapse_rate_manabe.GIF" width="390" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Change in lapse rate with atmospheric composition is subtle. The larger effect is in radiative trapping of heat thus leading to an overall temperature rise. Taken from [9]</td></tr>
</tbody></table>
<br />
<br />
The temperature of the earth is ultimately governed by the radiative physics of the greenhouse gases and the general application of Planck''s law. Based on how the other planets behave (and even the sun) the thermodynamic aspects seem more set in stone.<br />
<br />
<b>Lapse Rate References </b><br />
<br />
<div class="csl-bib-body" style="line-height: 1.35;">
<div class="csl-entry" style="clear: left;">
<div class="csl-left-margin" style="float: left; padding-right: 0.5em; text-align: right; width: 1em;">
<span style="font-size: x-small;">[3]</span></div>
<div class="csl-right-inline" style="margin: 0px 0.4em 0px 1.5em;">
<span style="font-size: x-small;"> R. Cess, V. Ramanathan, and T. Owen, “The Martian paleoclimate and enhanced atmospheric carbon dioxide,” <i>Icarus</i>, vol. 41, no. 1, pp. 159–165, 1980.</span></div>
</div>
<span style="font-size: x-small;"><span class="Z3988" title="url_ver=Z39.88-2004&ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzotero.org%3A2&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=The%20Martian%20paleoclimate%20and%20enhanced%20atmospheric%20carbon%20dioxide&rft.jtitle=Icarus&rft.volume=41&rft.issue=1&rft.aufirst=RD&rft.aulast=Cess&rft.au=RD%20Cess&rft.au=V%20Ramanathan&rft.au=Tobias%20Owen&rft.date=1980&rft.pages=159-165&rft.spage=159&rft.epage=165&rft.issn=0019-1035">
</span></span></div>
<span style="font-size: x-small;"><br /></span>
<br />
<div class="csl-bib-body" style="line-height: 1.35;">
<div class="csl-entry" style="clear: left;">
<div class="csl-left-margin" style="float: left; padding-right: 0.5em; text-align: right; width: 1em;">
<span style="font-size: x-small;">[4]</span></div>
<div class="csl-right-inline" style="margin: 0px 0.4em 0px 1.5em;">
<span style="font-size: x-small;"> C. Fenselau, R. Caprioli, A. Nier, W. Hanson, A. Seiff, M. Mcelroy, N. Spencer, R. Duckett, T. Knight, and W. Cook, “Mass spectrometry in the exploration of Mars,” <i>Journal of mass spectrometry</i>, vol. 38, no. 1, pp. 1–10, 2003.</span></div>
</div>
<span style="font-size: x-small;"><span class="Z3988" title="url_ver=Z39.88-2004&ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzotero.org%3A2&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Mass%20spectrometry%20in%20the%20exploration%20of%20Mars&rft.jtitle=Journal%20of%20mass%20spectrometry&rft.volume=38&rft.issue=1&rft.aufirst=Catherine&rft.aulast=Fenselau&rft.au=Catherine%20Fenselau&rft.au=Richard%20Caprioli&rft.au=AO%20Nier&rft.au=WB%20Hanson&rft.au=A%20Seiff&rft.au=MB%20Mcelroy&rft.au=NW%20Spencer&rft.au=RJ%20Duckett&rft.au=TCD%20Knight&rft.au=WS%20Cook&rft.date=2003&rft.pages=1-10&rft.spage=1&rft.epage=10&rft.issn=1096-9888">
</span></span></div>
<span style="font-size: x-small;"><br /></span>
<br />
<div class="csl-bib-body" style="line-height: 1.35;">
<div class="csl-entry" style="clear: left;">
<div class="csl-left-margin" style="float: left; padding-right: 0.5em; text-align: right; width: 1em;">
<span style="font-size: x-small;">[5]</span></div>
<div class="csl-right-inline" style="margin: 0px 0.4em 0px 1.5em;">
<span style="font-size: x-small;"> E. Palomba, “Venus surface properties from the analysis of the spectral slope@ 1.03-1.04 microns,” 2008.</span></div>
</div>
<span style="font-size: x-small;"><span class="Z3988" title="url_ver=Z39.88-2004&ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzotero.org%3A2&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Venus%20surface%20properties%20from%20the%20analysis%20of%20the%20spectral%20slope%40%201.03-1.04%20microns&rft.aufirst=E&rft.aulast=Palomba&rft.au=E%20Palomba&rft.date=2008">
</span></span></div>
<br />
<div class="csl-bib-body" style="line-height: 1.35;">
<div class="csl-entry" style="clear: left;">
<div class="csl-left-margin" style="float: left; padding-right: 0.5em; text-align: right; width: 1em;">
<span style="font-size: x-small;">[6]</span></div>
<div class="csl-right-inline" style="margin: 0px 0.4em 0px 1.5em;">
<span style="font-size: x-small;"> A. Sinclair, J. P. Basart, D. Buhl, W. Gale, and M. Liwshitz, “Preliminary results of interferometric observations of Venus at 11.1-cm wavelength,” <i>Radio Science</i>, vol. 5, no. 2, pp. 347–354, 1970.</span></div>
</div>
<span style="font-size: x-small;"><span class="Z3988" title="url_ver=Z39.88-2004&ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzotero.org%3A2&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Preliminary%20results%20of%20interferometric%20observations%20of%20Venus%20at%2011.1-cm%20wavelength&rft.jtitle=Radio%20Science&rft.volume=5&rft.issue=2&rft.aufirst=ACE&rft.aulast=Sinclair&rft.au=ACE%20Sinclair&rft.au=John%20P%20Basart&rft.au=David%20Buhl&rft.au=WA%20Gale&rft.au=M%20Liwshitz&rft.date=1970&rft.pages=347-354&rft.spage=347&rft.epage=354&rft.issn=0048-6604">
</span></span></div>
<span style="font-size: x-small;"><br /></span>
<br />
<div class="csl-bib-body" style="line-height: 1.35;">
<div class="csl-entry" style="clear: left;">
<div class="csl-left-margin" style="float: left; padding-right: 0.5em; text-align: right; width: 1em;">
<span style="font-size: x-small;">[7]</span></div>
<div class="csl-right-inline" style="margin: 0px 0.4em 0px 1.5em;">
<span style="font-size: x-small;"> G. Fjeldbo, A. J. Kliore, and V. R. Eshleman, “The neutral atmosphere of Venus as studied with the Mariner V radio occultation experiments,” <i>The Astronomical Journal</i>, vol. 76, p. 123, 1971.</span></div>
</div>
<span class="Z3988" title="url_ver=Z39.88-2004&ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzotero.org%3A2&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=The%20neutral%20atmosphere%20of%20Venus%20as%20studied%20with%20the%20Mariner%20V%20radio%20occultation%20experiments&rft.jtitle=The%20Astronomical%20Journal&rft.volume=76&rft.aufirst=Gunnar&rft.aulast=Fjeldbo&rft.au=Gunnar%20Fjeldbo&rft.au=Arvydas%20J%20Kliore&rft.au=Von%20R%20Eshleman&rft.date=1971&rft.pages=123&rft.issn=0004-6256">
</span></div>
<span style="font-size: x-small;"><br /></span>
<br />
<div class="csl-bib-body" style="line-height: 1.35;">
<div class="csl-entry" style="clear: left;">
<div class="csl-left-margin" style="float: left; padding-right: 0.5em; text-align: right; width: 1em;">
<span style="font-size: x-small;">[8]</span></div>
<div class="csl-right-inline" style="margin: 0px 0.4em 0px 1.5em;">
<span style="font-size: x-small;"> M. Molodenskii and L. Solov’ev, “Theory of equilibrium sunspots,” <i>Astronomy Reports</i>, vol. 37, pp. 83–87, 1993.</span></div>
</div>
<span style="font-size: x-small;"><span class="Z3988" title="url_ver=Z39.88-2004&ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzotero.org%3A2&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Theory%20of%20equilibrium%20sunspots&rft.jtitle=Astronomy%20Reports&rft.volume=37&rft.aufirst=MM&rft.aulast=Molodenskii&rft.au=MM%20Molodenskii&rft.au=LS%20Solov'ev&rft.date=1993&rft.pages=83-87&rft.spage=83&rft.epage=87&rft.issn=1063-7729">
</span></span></div>
<span style="font-size: x-small;"><br /></span>
<br />
<div class="csl-bib-body" style="line-height: 1.35;">
<div class="csl-entry" style="clear: left;">
<div class="csl-left-margin" style="float: left; padding-right: 0.5em; text-align: right; width: 1em;">
<span style="font-size: x-small;">[9]</span></div>
<div class="csl-right-inline" style="margin: 0px 0.4em 0px 1.5em;">
<span style="font-size: x-small;"> D. L. Hartmann, <i>Global Physical Climatology</i>. Elsevier Science, 1994.</span></div>
</div>
<span style="font-size: x-small;"><span class="Z3988" title="url_ver=Z39.88-2004&ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzotero.org%3A2&rft_id=urn%3Aisbn%3A9780080571638&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Global%20Physical%20Climatology&rft.publisher=Elsevier%20Science&rft.series=International%20Geophysics&rft.aufirst=D.L.&rft.aulast=Hartmann&rft.au=D.L.%20Hartmann&rft.date=1994&rft.isbn=9780080571638">
</span></span></div>
<br />
<br />
<div style="background-color: #e06666; color: white; text-align: center;">
<b>[FINAL UPDATE]</b></div>
<br />
CO2 is a polyatomic molecule and so (neglecting the vibrational modes) has 6 degrees of freedom (3 translation and 3 rotational) instead of the 5 (3 translational +2 rotational) of N2 and O2 in the gas phase. Understanding that, we can adjust the polytropic factor from <b><i>f</i></b> = (1+5/2)*(3/2)=21/4=5.25 to <b><i>f</i></b> = (1+6/2)*3/2=6 for planets such as Venus and Mars.<br />
<br />
Accessing more detailed atmospheric profiles of Venus, we can test the following thermo models for f=6, where T0 and P0 are the surface temperature and pressure.<br />
<br />
<b>Lapse Rate </b><br />
$$ T_1 = T_0 \cdot (1 - z/(f z_0)) $$<br />
<br />
<b>Barometric Formula</b><br />
$$ p_1 = p_0 \cdot (1 - z/(f z_0))^f $$<br />
<br />
with<br />
$$ z_0 = \frac{R T_0}{MW g} $$<br />
<br />
<b>P-T Curves</b><br />
<br />
$$ p_1 = p_0 \cdot (\frac{T_1}{T_0})^f $$<br />
<br />
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="http://imageshack.us/a/img839/3073/magellanvenusdual.gif" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="243" src="http://imageshack.us/a/img839/3073/magellanvenusdual.gif" width="640" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Barometric Formula Profile (left) and Lapse Rate Profile (right) as determined by the Magellan Venus probe. The model fit for f=6, and the estimated surface pressure is shown by the green line. T0 is the only adjustable parameter. The zig-zag green line is a deliberate momentary shift of f=6 to values of f=6.5, 6.7, and 6.85 to indicate how the profile is transitioning to a more isothermal regime at higher altitude. <br />
Adapted from charts on Rich Townsend's MadStar web site: <br />
http://www.astro.wisc.edu/~townsend/resource/teaching/diploma/venus-p.gif<br />
http://www.astro.wisc.edu/~townsend/resource/teaching/diploma/venus-t.gif</td><td class="tr-caption" style="text-align: center;"><br /></td></tr>
</tbody></table>
<br />
<br />
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="http://imageshack.us/a/img18/758/magellanwikivenus.gif" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="298" src="http://imageshack.us/a/img18/758/magellanwikivenus.gif" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">The P-T profile which combines the Magellan data above with data provided on the "Atmosphere of Venus" Wikipedia page. The polytropic factor of 6 allows a good fit to the data to 3 orders of magnitude of atmospheric pressure. This includes the super-critical gas phase of CO2.</td><td class="tr-caption" style="text-align: center;"><br /></td></tr>
</tbody></table>
<br />
<br />
The lapse rate perhaps can be best understood by the following differential formulation<br />
<br />
$$ \Delta E = m g \Delta z + c_p \Delta T $$<br />
<br />
If 1/3 of the gravitational energy term is lost as convective kinetic energy, and cp=4R then<br />
<br />
$$ \Delta E = 1/3 m g \Delta z = m g \Delta z + 4 R \Delta T $$<br />
and this allows 2/3 of the gravitational energy to go through a quasi-adiabatic transition.<br />
<br />
Whether this process describes precisely what happens (see <i><b>Added Note</b></i> in the first part of this post), it accurately matches the standard atmospheric profiles of Venus and Earth. It also follows Mars well, but the variability is greater on Mars, as can be see seen in the figures below. The CO2-laden atmosphere of Mars is thin so the temperature varies quickly from day to night, and the polytropic profile is constantly adjusting itself to match the solar conditions.<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="http://img707.imageshack.us/img707/2126/lapsevenusmars.gif" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="208" src="http://img707.imageshack.us/img707/2126/lapsevenusmars.gif" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Venus has a lower atmospheric lapse rate that is more uniform, while Mars shows higher fluctuations on top of the mean slope. From <a href="http://www.lpl.arizona.edu/~showman/climate/venus-mars-pdflatex.pdf">http://www.lpl.arizona.edu/~showman/climate/venus-mars-pdflatex.pdf</a></td></tr>
</tbody></table>
<br />
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEisG-fL_wK03OPLAlbucTRCtyWgwxWkdnwlnHmLFw8PNTZPEHp4CuLXcCpOCJiJRngTlvrKZIaMmbN_d-9R3KBGPhNIe5L_UC8BX4SGtVdx7yMcVL05cXiIoUEP81KtEP_LG_f7FOPOMn8/s1600/mariner-hunten-mars-lapse-rate.gif" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="388" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEisG-fL_wK03OPLAlbucTRCtyWgwxWkdnwlnHmLFw8PNTZPEHp4CuLXcCpOCJiJRngTlvrKZIaMmbN_d-9R3KBGPhNIe5L_UC8BX4SGtVdx7yMcVL05cXiIoUEP81KtEP_LG_f7FOPOMn8/s640/mariner-hunten-mars-lapse-rate.gif" width="640" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Within the variability, Mars will show good agreement with the polytropic lapse rate = <br />
(1/6)MW*g/R = (1/6)*43.56*3.711/8.314 = 3.24 K/km.<br />
Chart adapted from<br />
<div class="csl-bib-body" style="line-height: 1.35;">
<div class="csl-entry" style="clear: left;">
<div class="csl-left-margin" style="float: left; padding-right: 0.5em; text-align: right; width: 1em;">
<br /></div>
<div class="csl-right-inline" style="margin: 0 .4em 0 1.5em;">
D. M. Hunten, “Composition and structure of planetary atmospheres,” <i>Space Science Reviews</i>, vol. 12, no. 5, pp. 539–599, 1971.</div>
</div>
<span class="Z3988" title="url_ver=Z39.88-2004&ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzotero.org%3A2&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Composition%20and%20structure%20of%20planetary%20atmospheres&rft.jtitle=Space%20Science%20Reviews&rft.stitle=Space%20Science%20Reviews&rft.volume=12&rft.issue=5&rft.aufirst=Donald%20M&rft.aulast=Hunten&rft.au=Donald%20M%20Hunten&rft.date=1971&rft.pages=539-599&rft.spage=539&rft.epage=599&rft.issn=0038-6308"></span>
</div>
</td></tr>
</tbody></table>
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="http://img593.imageshack.us/img593/3849/atlantalapserate.gif" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="313" src="http://img593.imageshack.us/img593/3849/atlantalapserate.gif" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">The Earth is described by the "Standard Atmosphere" polytropic profile which is shallower than the adiabatic.</td></tr>
</tbody></table>
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiP9QQ9eE_umnVSWRIWjkuyKg__odd9ALGRmMg7RM0t3HuMimX9q7_CapyrCKzVLn6uU3S9TDr96_EpISLuFjNgx-sQqXWyuAQX0HW1GlCx-0C6aqfHYqpg3e2jW4MnWKmYUgwRHynJ6Wc/s1600/venera.GIF" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="380" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiP9QQ9eE_umnVSWRIWjkuyKg__odd9ALGRmMg7RM0t3HuMimX9q7_CapyrCKzVLn6uU3S9TDr96_EpISLuFjNgx-sQqXWyuAQX0HW1GlCx-0C6aqfHYqpg3e2jW4MnWKmYUgwRHynJ6Wc/s400/venera.GIF" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Data from the Russian Venera probes to Venus, with model overlaid. Adapted from <br />
<div class="csl-bib-body" style="line-height: 1.35;">
<div class="csl-entry" style="clear: left;">
<div class="csl-left-margin" style="float: left; padding-right: 0.5em; text-align: right; width: 1em;">
</div>
<div class="csl-right-inline" style="margin: 0 .4em 0 1.5em;">
M. I. A. Marov and D. H. Grinspoon, <i>The planet Venus</i>. Yale University Press, 1998.</div>
</div>
<span class="Z3988" title="url_ver=Z39.88-2004&ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzotero.org%3A2&rft_id=urn%3Aisbn%3A9780300049756&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The%20planet%20Venus&rft.publisher=Yale%20University%20Press&rft.series=Yale%20planetary%20exploration%20series&rft.aufirst=M.I.A.&rft.aulast=Marov&rft.au=M.I.A.%20Marov&rft.au=D.H.%20Grinspoon&rft.date=1998&rft.isbn=9780300049756"></span>
</div>
</td></tr>
</tbody></table>
<div class="separator" style="clear: both; text-align: center;">
</div>
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<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjV1pUSw93GfpWIHYVJxRHlXrRHX9aiRzm_ACxjkEkdKzrgU-LYV0BgqLwnW_HkiyIPXsqAQIpwj-GG5WSF8JZiO0Gl7RrcPmcWxKi8VfOkOc1JLa0JHc-jpVp-ZiMdgskKd0DjnCF3-bE/s1600/pioneer-venus.gif" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="280" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjV1pUSw93GfpWIHYVJxRHlXrRHX9aiRzm_ACxjkEkdKzrgU-LYV0BgqLwnW_HkiyIPXsqAQIpwj-GG5WSF8JZiO0Gl7RrcPmcWxKi8VfOkOc1JLa0JHc-jpVp-ZiMdgskKd0DjnCF3-bE/s400/pioneer-venus.gif" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">The Venus lapse rate accurately follows from LR= (1/6) g MW/R, where 6 is the polytropic factor. Adapted from:<br />
<div class="csl-bib-body" style="line-height: 1.35;">
<div class="csl-entry" style="clear: left;">
<div class="csl-left-margin" style="float: left; padding-right: 0.5em; text-align: right; width: 1em;">
<br /></div>
<div class="csl-right-inline" style="margin: 0 .4em 0 1.5em;">
V. Formisano, F. Angrilli, G. Arnold, S. Atreya, K. H. Baines, G. Bellucci, B. Bezard, F. Billebaud, D. Biondi, M. I. Blecka, L. Colangeli, L. Comolli, D. Crisp, M. D’Amore, T. Encrenaz, A. Ekonomov, F. Esposito, C. Fiorenza, S. Fonti, M. Giuranna, D. Grassi, B. Grieger, A. Grigoriev, J. Helbert, H. Hirsch, N. Ignatiev, A. Jurewicz, I. Khatuntsev, S. Lebonnois, E. Lellouch, A. Mattana, A. Maturilli, E. Mencarelli, M. Michalska, J. Lopez Moreno, B. Moshkin, F. Nespoli, Y. Nikolsky, F. Nuccilli, P. Orleanski, E. Palomba, G. Piccioni, M. Rataj, G. Rinaldi, M. Rossi, B. Saggin, D. Stam, D. Titov, G. Visconti, and L. Zasova, “The planetary fourier spectrometer (PFS) onboard the European Venus Express mission,” <i>Planetary and Space Science</i>, vol. 54, no. 13–14, pp. 1298–1314, Nov. 2006.</div>
</div>
</div>
</td></tr>
</tbody></table>
<br />
<br />
<b>Final Caveat</b>: Not all planets follow this polytropic quasi-adiabatic profile. The "big planets" of our solar system seem to follow the dry adiabatic very well at low atmospheric altitudes, see figure below. All of these consist mainly of H2 and a scattering of Helium (mono-atomic gas with Cp=5/2R) and so are slightly below 7/2R for a value of cp. Convective losses don't seem to play a role on these planets, in contrast to the sun, which is special in that the radiative temperature gradient can easily exceed the adiabatic lapse rate. In this case, the sun's atmosphere becomes collectively unstable and convective energy transport takes over (see T. I. Gombosi, <i>Physics of the space environment</i>. Springer, 1998, p.217). <br />
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="http://imageshack.us/a/img850/6898/bigplanets.gif" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="370" src="http://imageshack.us/a/img850/6898/bigplanets.gif" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">The "Big Planets" of our solar system have lower atmospheres that follow the adiabatic profiles closely.<br />
Adapted from : <br />
L. A. McFadden, P. Weissman, and T. Johnson,<a href="http://books.google.com/books?id=G7UtYkLQoYoC&pg=PA385&lpg=PA385&dq=%22lapse+rate%22+of+jupiter&source=bl&ots=jETl82q_FS&sig=TXj8hIblFacX1HIzBn8P-iXHNhA&hl=en&sa=X&ei=0et1UYCbCMnGrgGG6IHwBQ&ved=0CGsQ6AEwCA" target="_blank"> Encyclopedia of the Solar System</a>. Elsevier Science, 2006.</td></tr>
</tbody></table>
<br />
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgnGaYhLpN_5wGq5E4SmeagzPVcs9jnQLrpLPM40khZZeWbjwM0DVcG-nu6kiwe96i_coAcdNQIiXQtJrF2-EJ1u2vcCWhUBKj-ULNQtFptAmm4CEn2q4LZCuZ_MM-Qagk6JizjEflrTJc/s1600/planets_mars_earth_big.gif" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="305" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgnGaYhLpN_5wGq5E4SmeagzPVcs9jnQLrpLPM40khZZeWbjwM0DVcG-nu6kiwe96i_coAcdNQIiXQtJrF2-EJ1u2vcCWhUBKj-ULNQtFptAmm4CEn2q4LZCuZ_MM-Qagk6JizjEflrTJc/s640/planets_mars_earth_big.gif" width="640" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">All the planets (Venus not shown) follow either polytropic or adiabatic P-T profiles in the lower atmosphere. <br />
Adapted from :<br />
<div class="csl-bib-body" style="line-height: 1.35;">
<div class="csl-entry" style="clear: left;">
<div class="csl-right-inline" style="margin: 0 .4em 0 1.5em;">
F. Bagenal, “Planetary Atmospheres ASTR3720 course notes.” [Online]. Available: http://lasp.colorado.edu/~bagenal/3720/index.html. [Accessed: 01-May-2013].</div>
</div>
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<br />
<br />@whuthttp://www.blogger.com/profile/18297101284358849575noreply@blogger.com0tag:blogger.com,1999:blog-8890671936020885112.post-61055903358634392002013-03-04T22:31:00.000-08:002013-03-14T07:48:00.224-07:00Climate Sensitivity and the 33C DiscrepancyThe connection between greenhouse gases such as CO2 and climate change is solid.
The global land temperature record is easily on track for a 3° C swing due to a doubling of CO2 levels;
this also happens to be the consensus mean for climate sensitivity over a set of peer-reviewed models.<br />
<br />
The breakdown of the 3°C doubling is that it is equal parts due to CO2 doubling, water vapor (which comes along for a ride),
and a set of miscellaneous positive feedbacks
<br />
<blockquote>
<span style="font-size: x-small;">"If Earth were a blackbody without climate feedbacks the equilibrium response to 4 W/m2 forcing would be about 1.2°C (Hansen et al., 1981, 1984; Lacis et al., 2010), implying that the net effect of all fast feedbacks is to amplify the equilibrium climate response by a factor 2.5. GISS climate models suggest that water vapor and sea ice feedbacks together amplify the sensitivity from 1.2°C to 2-2.5°C. The further amplification to 3°C is the net effect of all other processes, with the most important ones probably being aerosols, clouds, and their interactions. "</span></blockquote>
<span style="font-family: Times,"Times New Roman",serif;"><span style="font-size: x-small;"> <i>J. E. Hansen and M. Sato, <a href="http://arxiv.org/ftp/arxiv/papers/1105/1105.0968.pdf">“Paleoclimate implications for human-made climate change,”</a> Climate Change, pp. 21–47, 2012. </i></span></span><br />
<br />
All the pieces to the climate change puzzle are interlocking. Every new piece of evidence has to be consistent with prior understanding, otherwise it weakens the consensus view.
I have not been very successfull at finding any real holes in the GHG AGW argument myself. That is likely more do to my lack of detailed knowledge of the intricacies of the models, but it doesn't prevent me from trying other angles. For example, in lieu of my finding any breakthrough debunking arguments, I seek to find simplifications in the overall argument.<br />
<br />
This post describes an interesting derivation that ties together the climate sensitivity of CO2 and water vapor along with the 33° C discrepancy between the no-GHG earth and our current average global temperature.<br />
<br />
We start with an assumption that the baseline radiatively steady-state temperature is T0 and the temperature raises due to the climate sensitivity:
<br />
$$ T = T_0 + \alpha \log(C/C_0) $$
<br />
We assume that the second term is mainly due to water vapor, catalyzed by introduction of CO2.
This is enough to raise the temperature from a ground-state snowball earth which would occur as the water would normally condense out at 255° K.
Thereafter, the water vapor undergoes a bootstrapping process, whereby a fraction of the outgassed vapor serves to feed on itself,
and through the GHG process raises the temperature further, leading to more outgassing.<br />
<br />
The log dependence of water vapor concentration, <i>C</i>, on temperature, <i>T</i>, reflects the doubling sensitivity of a GHG. Water vapor works much like CO2 in this regard, only that it needs a boost from a non-condensing gas to sustain its elevated concentration.<br />
<br />
Next, we assume the outgassing of water vapor follows the Clausius-Clapeyron activation relation,
whereby the partial pressure is activated by a Boltzmann factor with an activation energy corresponding to the heat of vaporization.
<br />
$$ H = 0.42\:{ eV} = 4873\:{ Kelvins} $$
<br />
Then
<br />
$$ C/C_0 = \beta e^{-H/T} $$
<br />
We can make the assumption that the temperature, <i>T</i>, reaches a steady-state such that the positive feedback limits its extent as the Boltzmann acts as a rather weak feedback term.
(This is not runaway warming we are talking about)<br />
<br />
When we make the algebraic substitution for <i>C/C0</i> in the first equation.
<br />
$$ T = T_0 + \alpha \log(\beta e^{-H/T}) $$
<br />
Because of the <i>log(exp(X))=X</i> identity this reduces to:
<br />
$$ T = T_0 + \alpha \log(\beta) - \alpha H / T $$
<br />
In terms of solution space, this is a quadratic with respect to temperature.
For real valuations, we have a low temperature and high temperature solution.<br />
<br />
Graphically, the intersection points look like the following.<br />
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><img src="http://img832.imageshack.us/img832/9558/watervaporsolution.gif" style="margin-left: auto; margin-right: auto;" /></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><b>Figure 1:</b> Graphical solution to constrained positive feedback.<br />
The red dots show the intersection points.</td></tr>
</tbody></table>
<br />
<br />
Mapping this to the real process that is occurring, the low T point likely corresponds to the no-GHG temperature of 255° K, while the high T point is intuitively the steady-state temperature of 289° K. In other words, the gap between the two intersection points is 33° C.
<br />
The quadratic equation is
<br />
$$ T^2 - (T_0 + \alpha \beta) T - \alpha H $$
<br />
with solution
<br />
$$ T=\frac{-\gamma\pm\sqrt{\gamma^2-4\alpha H}}{2} $$
<br />
where<br />
$$ \gamma = T_0 + \alpha \log(\beta) $$
<br />
Safely assuming that the cold and hot solutions are symmetric about the midway point (255+289)/2 :
<br />
$$ \Delta T = \pm \sqrt{\gamma^2 - 4 \alpha H}/2 = \pm 16.5° C$$
<br />
where
<br />
$$ \gamma = T_{low} + T_{high} $$
<br />
We can now invert to determine alpha:
<br />
$$ \alpha = \frac {\gamma^2 -{2 \Delta T}^2}{4 H} $$
<br />
Inserting the values for the 3 fixed values:
<br />
$$ \alpha = 15.1° C $$
<br />
The climate sensitivity for a doubling of water vapor is
<br />
$$ 10.5° C = 15.1 log(2) $$
<br />
Finally, we can calculate how much a CO2-induced temperature change will trigger a corresponding rise in water vapor-induced temperature, i.e. it acts as the control knob as Lacis phrased it.<br />
<br />
From the differential Clausius-Clapeyron
$$ dC/C = {H/T^2} dT $$
<br />
with the standard dT for CO2 doubling given by Lacis [1] of 1.23°C.
Then dC/C = 0.072 at T=289°K, which gives a sensitivity
<br />
$$ \alpha \log(1+dC/C) = 1.05° C $$
<br />
for the water vapor rise carried along by the CO2 doubling.
<br />
To get to the 3.0° C value we add together 1.23° C from CO2, 1.05° from H20, and approximately 0.7° C from other GHGs (see <a href="http://www.esrl.noaa.gov/gmd/aggi/" target="_blank">The NOAA Annual Greenhouse Gas Index</a>) and albedo positive feedbacks.<br />
<br />
<br />
Comparing to the Berkeley BEST land-based temperature values with two different sensitivities, one can see how well the consensus value tracks the historical data.<br />
<br />
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjOPyk-PzC4gvAvTvl7Fj9mUxyO8OeITNvqXA0EiJAfz1ECD3lkvR4wfcG5PLik_TI_kUbO4_Ip-7iRNhyphenhyphenuYD-jJyyjPB-gmfeG1zZ8BTKtPrDAy_wLIBk3p99ccJ6emINFI-eOAdUxkw0/s1600/climate_sensitivity.gif" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="217" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjOPyk-PzC4gvAvTvl7Fj9mUxyO8OeITNvqXA0EiJAfz1ECD3lkvR4wfcG5PLik_TI_kUbO4_Ip-7iRNhyphenhyphenuYD-jJyyjPB-gmfeG1zZ8BTKtPrDAy_wLIBk3p99ccJ6emINFI-eOAdUxkw0/s320/climate_sensitivity.gif" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><b>Figure 2:</b> Consensus CO2 GHG model against historical temperature data.<br />
Lower curve is sensitivity of 3°C for a doubling of CO2</td></tr>
</tbody></table>
<br />
<br />
<hr />
[1]<span style="font-family: Times,"Times New Roman",serif;"> A. A. Lacis, G. A. Schmidt, D. Rind, and R. A. Ruedy,<i> “Atmospheric CO2: principal control knob governing Earth’s temperature,”</i> <b>Science</b>, vol. 330, no. 6002, pp. 356–359, 2010.
</span><br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="clear: left; margin-bottom: 1em; margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><br /></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><br /></td></tr>
</tbody></table>
<br />@whuthttp://www.blogger.com/profile/18297101284358849575noreply@blogger.com5tag:blogger.com,1999:blog-8890671936020885112.post-50862817803962839652013-02-17T09:22:00.000-08:002013-02-17T09:22:59.927-08:00Climate Change and the Energy ProblemA recent book by David Goodstein called <b>"Climate Change and the Energy Problem: physical science and economics perspective"</b> has slipped under the radar. Goodstein is a professor of physics at CalTech and a disciple of Richard Feynman, the AGW skeptics' favorite quote-machine.<br />
<br />
This is the follow-on to Goodstein's earlier book <b>"Out of Gas"</b> that ties together the hydrocarbon depletion challenge with the climate change problem. In <a href="http://ianmcpherson.com/blog/?p=3025">interviews, Goodstein agrees</a> that climate denialism, at its root, is a desire not to face the energy problem. He says that the people seriously working on peak oil are not at the margins but are at the forefront of change.<br />
<br />
Goodstein has serious credentials, and is one of the top thermodynamics and condensed matter physicists in the world. He treats the AGW problem as obvious:
<br />
<blockquote>
<i>"Fortunately for us, that is not all there is to it. If the average surface temperature of the Earth were 0°F. we probably would not have been here. The Earth has a gaseous atmosphere, largely transparent to sunlight, but nearly opaque to the planet's infrared radiation. The blanket of atmosphere traps and reradiates part of the heat that the Earth is trying to radiate away. The books remain balanced, with the atmosphere radiating into space the sonic amount of energy the Earth receives, but also radiating heat back to the Earth's surface, warming it to a comfortable average temperature of 57°F. That is what is known as the greenhouse effect. Without the greenhouse effect and the global warming that results, we probably would not be alive. "</i></blockquote>
And Goodstein is also formidable when it comes to dealing with crackpots. His true skeptical credentials are revealed in his book <b>"On Fact and Fraud: Cautionary Tales from the Front Lines of Science"</b>. This is a fascinating read as it deals with the Pons/Fleischmann cold fusion debacle as well as the Schon affair which I am very familiar with.<br />
<br />
The interview linked above is good. Goodstein sounds like a Brooklynite and delivers answers to the questions in short, no-nonsense replies. The only chuckle that I heard was when the interviewer remarked that coal-power was not used to mine and transport the coal. <br />
<br />
Throughout Goodman stresses the significance of liquid fuel, revealing the difficulty in boot-strapping the lower-grade forms of fossil fuels. <br />
<br />
<br />
<br />
@whuthttp://www.blogger.com/profile/18297101284358849575noreply@blogger.com0tag:blogger.com,1999:blog-8890671936020885112.post-28681363801005484252013-02-09T12:57:00.000-08:002013-02-15T22:46:55.858-08:00The Peak WarmersThe question is whether we can use use solar energy to process all the oil shale or whether this is mind boggling in scope. If you apply your intuition, consider what it will take to collect solar energy in the form of electricity and then use that electricity to (1) dig out that shale and process it or (2) in situ process the shale via heat and refine something approaching a liquid from the kerogen. And then to deliver it to its destination.<br />
<br />
The fear is that is is also possible that we will figure out how to bootstrap the entire oil shale process, whereby we use the energy from the oil shale to "extract itself". That obviously is the case with crude oil, as all the energy going to extract the oil comes from oil-powered machinery and transportation.<br />
<br />
I think that occurred also in the early days of coal extraction, but at some point the returns start to diminish. Remember that coal is barely refined before it is used.<br />
<br />
That is the most frightening prospect in all this, that well more than half of the hydrocarbon energy becomes a kind of waste heat. This is energy that isn't necessarily wasted because it is used for processing (see the concepts of EROEI and emergy), but that is essentially wasted as overhead and not directly contributing to propelling the world's economy.<br />
<br />
Suddenly 80+ million barrels a day turns into 200 million equivalent barrels because 120 million barrels is used to process the 80. And that is just to keep in place with the needs of a growing global economy.<br />
<br />
That leads into Pierrehumbert's reference to the Red Queen scenario in his Slate article. The Red Queen is about running faster just to keep in place. But oil shale makes it worse, as it turns the Red Queen into a voracious cannibal, while eating any seed corn and feedstock we have left.<br />
<br />
Pierrehumbert states at the end of his article <i>"Temporarily cheap and abundant gas buys us some respite—which we should be using to put decarbonized energy systems in place." </i> <br />
<br />
Can we be patient with the use of solar energy or will the second law be insurmountable?<br />
<br />
<br />
The dispersion in wind speeds already follows the second law of thermodynamics. Given a mean wind speed, applying the maximum entropy principle results in the observed variability.<br />
Same goes for aquatic wave height variation.<br />
Same goes for the areal coverage of clouds, which in turn will periodically obscure the sunlight.<br />
<br />
Here is a pic that illustrates how nature follows the Maximum Entropy Principle:<br />
<a href="http://img339.imageshack.us/img339/1530/maxentropydist.gif"><img src="http://img5.imageshack.us/img5/6784/maxentropydistsmall.gif" /></a><br />
click to magnify <br />
<br />
That is the hurdle in dealing with the second law from a source perspective. Everything is variable because nature tries to fill up all available states, mixing the low-likelihood high energy states with the higher-likelihood low-energy states.<br />
<br />
Through a freak of nature, our crude oil supplies were given to us in a very low entropy highly ordered configuration. But even there, the second law applies, as the volume distribution of reservoir sizes follows the maximum entropy principle. The tails in the distribution ultimately become the dispersed pockets we are now essentially mining. We used up all the higher-energy configurations first, and now are left with the lower energy configurations.<br />
<br />
The other hurdle is one of entropic losses as we convert one energy form to another, which is needed to do all the processing of oil shales, etc.<br />
<br />
So not only does entropy barely let us in, but it kicks our butt as we try to get out the door.<br />
The objective really should be in how to sustainably harness the stochasticity in nature and not try to outdo it and burn ourselves into oblivion.<br />
<br />
<div class="content">
Patience is the key. Collect the highly
dispersed energy sources from the sun, wind, etc. into a more
concentrated form and then work with that. After all, isn't that how oil
reservoirs formed in the first place?<br />
<br />
However, growing economies have no room for patience.</div>
<br />
<br />
<br />
<br />
<br />
This innocuous comment of mine was deleted from The Oil Drum today. Never can understand why they decide to delete what they do.<br />
<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjVqdtUR4sQQFzrC07CSWNGX9EtHmK7mvptlC8OOW9N6B1OwaqwjtzTxYJee0wfWZcIoLt_GAGQYW4c0GuPT0WN8WbIt31ZSiKeSEXIs7TwEOsrZeGLDgazbvKP43fkpb7ohavLpFYKQ3s/s1600/pierrehumbert.GIF" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="266" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjVqdtUR4sQQFzrC07CSWNGX9EtHmK7mvptlC8OOW9N6B1OwaqwjtzTxYJee0wfWZcIoLt_GAGQYW4c0GuPT0WN8WbIt31ZSiKeSEXIs7TwEOsrZeGLDgazbvKP43fkpb7ohavLpFYKQ3s/s320/pierrehumbert.GIF" width="320" /></a></div>
<br />
<br />
On the other hand, the blogger <b>Willis Eschenbach </b>has to be the most wrong-headed blowhard that has ever graced the internet. If you ever want to do science properly, read what he writes, try to figure out how he approaches science, and then do the exact opposite. Oh, and think a little bit, not spew every idea that comes into your head, because the sycophantic followers that you will attract will not be able to discriminate between garbage and something worthwhile.<br />
<br />
This is how <a href="http://wattsupwiththat.com/2013/02/15/home-invasion/" target="_blank">inflated a sense of worth</a> he possesses:<br />
<blockquote class="tr_bq">
<span style="font-size: x-small;"><i>"Here on WUWT, I put out my scientific ideas up in the public forum as
clearly as I can explain them, and I hand around the hammers, and
people do their best to demolish my claims. That is science at its
finest, nothing hidden, everything visible, all the relevant data and
code available for any reader to either check my work, or to tear it to
shreds, or to pick it up and take it further.</i></span>
<br />
<span style="font-size: x-small;"><i>This gradual scientific migration to the web is well underway, moved
forwards by things like journals with open review, and by other blogs.
Science done in the dark by a few learned boffins is already dead in the
21st century, the practitioners just didn’t notice when they ran past
their use-by dates, and as a result that dark corner of the scientific
world is populated more and more by zombies. Zombies with PhD’s to be
sure, but zombies nonetheless, everyone else is emerging into the light.
Good news is, it’s somewhat of a self-limiting phenomenon, the best
authors say that zombies can’t reproduce …"</i></span></blockquote>
<br />
<br />
<br />
<br />@whuthttp://www.blogger.com/profile/18297101284358849575noreply@blogger.com0tag:blogger.com,1999:blog-8890671936020885112.post-4423106407581518992013-02-07T06:09:00.001-08:002013-02-07T06:09:36.106-08:00The Luke OilersIn the climate science world, those who side with consensus science and agree that anthropogenic global warming is real are at a minimum referred to as "lukewarmers".<br />
These people may not be as rabid as the true-believers, yet they don't dismiss the scientific theory and evidence as do the so-called "climate deniers".<br />
<br />
I ran across a similar type of minimal acceptance, though very muted and disguised, when I participated in a blog comment discussion at <a href="http://judithcurry.com/2013/02/01/another-hockey-stick" target="_blank">Climate Etc.</a> The top-level post concerned Maugeri's wrong-headed analysis and conclusion of near-cornucopian oil availability.<br />
<br />
In the ensuing discussion, it was clear that the climate skeptics, who would otherwise not admit that Peak Oil was real, would nevertheless continue to push alternatives such as nuclear and unconventional oil, and suggest that BAU could continue. This contradiction pointed to the fact that they implicitly agree in the Peak Oil concept while denying that the progressives and technocrats (such as Hubbert) were correct in their overall assessment.<br />
<br />
I suggest these implicit Peak Oil believers need to be referred to as "luke-oilers", distinct from the explicit Peak Oilers. To be a luke-oiler, all it takes for you is to admit that the Bakken or the Tar Sands or nuclear will meet our future energy needs. Its actually not that high a bar that you have to clear to be a luke-oiler, but it wasn't high for a lukewarmer either -- just an admission to the facts on the the ground. The earth is warming due to man, and the oil is depleting due to man.<br />
<br />
---<br />
<br />
<div class="sl-art-head-dek">
At the cross-roads of peak oil and climate science we see a world of dogs and cats, living together. On occasion this gets stirred up as in <a href="http://www.slate.com/articles/health_and_science/science/2013/02/u_s_shale_oil_are_we_headed_to_a_new_era_of_oil_abundance.html" target="_blank">this Slate opinion piece</a> by noted climate scientist and atmospheric physicist Raymond T. Pierrehumbert. The title is <b>'<i>The Myth of “Saudi America”: Straight talk from geologists about our new era of oil abundance.</i>'</b></div>
<div class="sl-art-head-dek">
</div>
<div class="sl-art-head-dek">
In this piece Pierrehumbert discusses the issue of Bakken oil and acknowledges Rune Likvern's analysis of Red Queen behavior in shale oil. At the end, he suggests a kind of "No Regrets" policy in that we move rapidly toward alternatives to oil, using the oil that we have right now to solve both the predicaments of oil depletion and AGW.</div>
<div class="sl-art-head-dek">
<br /></div>
<div class="sl-art-head-dek">
<br /></div>
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<br />@whuthttp://www.blogger.com/profile/18297101284358849575noreply@blogger.com0tag:blogger.com,1999:blog-8890671936020885112.post-83153892110657902762013-01-19T18:12:00.000-08:002013-01-19T18:34:21.262-08:00Deleted commentI responded indirectly to a post
<a href="http://earlywarn.blogspot.co.nz/2013/01/bakken-well-stats.html#more">http://earlywarn.blogspot.co.nz/2013/01/bakken-well-stats.html</a>
and <a href="http://www.sciencedirect.com/science/article/pii/S0160412012000566" target="_blank">Bakken data</a>
as a comment on <a href="http://theoildrum.com/" target="_blank">The Oil Drum</a>. The comment showed up and then disappeared.<br />
<br />
It was still in my cache when I discovered <a href="http://www.theoildrum.com/node/9785#comment-940968" target="_blank">it was deleted</a>, and reproduced here (click to enlarge)<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgDeQboG71u3YMMpNtk2x2LlTRLL19MffiLbFSu9_Io19yAQxReiPUf9r3W6tAgInfc1Qn2Vb-n__4eGL1GeSEg7w6lOTWMJPy9uGq-YYCZ_DP-6JMHz02nWjdi1R-qcDWCImfKNMzLTGg/s1600/TOD-deleted.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="632" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgDeQboG71u3YMMpNtk2x2LlTRLL19MffiLbFSu9_Io19yAQxReiPUf9r3W6tAgInfc1Qn2Vb-n__4eGL1GeSEg7w6lOTWMJPy9uGq-YYCZ_DP-6JMHz02nWjdi1R-qcDWCImfKNMzLTGg/s640/TOD-deleted.gif" width="640" /></a></div>
This was nothing new, but a rephrasing of analysis work from last year:
<a href="http://theoilconundrum.blogspot.com/2012/05/bakken-growth.html">http://theoilconundrum.blogspot.com/2012/05/bakken-growth.html </a><br />
<br />
Both Rockman's and my comment was apparently deleted, with no reason why. I have learned that one can't complain publicly about why a comment would get deleted on The Oil Drum, as that is grounds for a temporary banishment from posting any further comments.<br />
<br />
But I can complain all I want here because this is my space. It sucks because I spend time doing the analysis and then it goes into a black hole.<br />
<br />
BTW, Stuart Staniford does not seem to add anything as an analyst. He is no Kevin Drum,<br />
who wrote <a href="http://www.motherjones.com/environment/2013/01/lead-crime-link-gasoline" target="_blank">this piece</a>. Interesting that one can use the convolution algorithms of the oil shock model to model the crime rate variation as it follows the gasoline lead content over the last century. Crime rate tracks the convolution of lead content over time with a delay function describing a distribution of adult maturation times (peaking around 20 years of age). I bet Drum is right in the correlation and cause.<br />
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="http://ars.els-cdn.com/content/image/1-s2.0-S0160412012000566-gr3.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="211" src="http://ars.els-cdn.com/content/image/1-s2.0-S0160412012000566-gr3.jpg" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">From http://www.sciencedirect.com/science/article/pii/S0160412012000566</td></tr>
</tbody></table>
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<br />@whuthttp://www.blogger.com/profile/18297101284358849575noreply@blogger.com0tag:blogger.com,1999:blog-8890671936020885112.post-35217735566318748112013-01-15T19:07:00.000-08:002013-01-19T19:16:16.126-08:00Field Guide to Climate ClownsThe climate science blog known as <a href="http://judithcurry.com/" target="_blank">Climate Etc</a> is essentially infested with cranks, crackpots, and wackos, each with their own pet theory on why the consensus AGW science is wrong or an alternate view is preferred over the basic greenhouse-gas-based physics.<br />
<br />
As someone mentioned, crackpot theories on global warming are almost fractal in nature -- in other words, wrong on almost every scale that you can interpret them.<br />
<br />
I compiled the following "Field Guide" in response to my experiences commenting at that site. The most unusual statistical anomaly concerns the relative abundance of crackpots from Down Under, who also seem to be the most rabid, a trait that one might trace to the Oz tradition of mocking authority, known as <a href="http://en.wikipedia.org/wiki/Larrikinism" target="_blank">Larrikinism</a>.<br />
<br />
Whatever compelled me to keep track of these clowns (who are vaguely similar to the fossil fuel cornucopians on oil depletion blogs) I hope it provides some levity.<br />
<br />
<iframe height="400" src="https://docs.google.com/document/d/1TbosA_JLgwcjj6SfOqmvQu4oEgkWNQOw7XgvxvUZoJE/pub?embedded=true" width="550"></iframe>
I want to add that that I have largely stopped commenting at Climate Etc because the editorial policies of the blog site's owner do not allow singling out of crackpots, but instead allow the crackpots themselves free reign (and the blog's proprietor never engages with the crackpot theorists themselves, therefore essentially condoning the pseudo-scientific ideas. Kind of counter-effective to advancing science, in my opinion).
<br />
<blockquote>
<a href="http://www.blogger.com/judithcurry.com/2013/01/10/trusting-the-experts/#comment-286139"> curryja | January 16, 2013 at 5:42 am</a>
"Very large number of comments (approaching 10% of total CE comments) plus too many insults. I will take you off moderation if you can calm down the insults. Also, anyone that mentioned ‘BBD’ in their comment also went into moderation, so I could assess both sides of these exchanges."</blockquote>
The <span style="font-size: small;">commenter </span>BBD happens to be the most sensible commenter on the site. No wonder the site is such a magnet for <a href="http://www.amazon.com/People-Believe-Weird-Things-Pseudoscience/dp/0805070893">Why People Believe Weird Things</a>. It's not quite as bad and one-sided as the infamously insane "Best Scientific Blog" WUWT, but that's not saying much.<br />
<br />
<span style="font-size: large;"><b>Edit:</b></span><br />
<a href="http://judithcurry.com/2013/01/18/blog-commenting-etiquette/#comment-287233" target="_blank">This bit </a>explains everything, and essentially provides a rationale for why my documentation of these climate clowns is needed.<br />
<div id="comment-287233">
<blockquote class="tr_bq">
<div class="comment-author vcard">
<cite class="fn"><a class="url" href="http://curryja.wordpress.com/" rel="external nofollow">curryja</a></cite>
<span class="comment-meta commentmetadata">
|
<a href="http://judithcurry.com/2013/01/18/blog-commenting-etiquette/#comment-287233">
January 19, 2013 at 8:31 am</a><a class="comment-reply-link" href="http://judithcurry.com/2013/01/18/blog-commenting-etiquette/?replytocom=287233#respond"></a> </span>
</div>
<div class="comment-body">
It has nothing to do with their research
and their views. <b> I tolerate what I view to be scientific crackpottery.</b>
I tolerate people talking about Nazis and commies. I do not tolerate
one person saying the same thing over again. I do not tolerate insults
to other commenters.</div>
</blockquote>
Why would anyone, let alone a scientist, tolerate scientific crackpottery? </div>
<div id="comment-287233">
<br /></div>
<div id="comment-287233">
<span style="font-size: small;">I don't tolerate it, and given the fact that I have no control over scientific discourse at most levels, </span>my choice is to <i>document the atrocities</i>. </div>
<div id="comment-287233">
<blockquote class="tr_bq">
<h1 class="heading">
<span style="font-size: small;"><a href="http://arstechnica.com/science/2013/01/dont-read-the-comments-online-communities-shape-risk-perception/" target="_blank">Don't read the comments! Online communities shape risk perception</a> <br /><i>More people get science news from blogs, where commentary shapes opinions.</i></span></h1>
<h1 class="node-title">
<span style="font-size: small;"><a href="http://www.fastcompany.com/3004580/how-blog-comments-google-autocomplete-reinforce-scientific-bias" target="_blank">How Blog Comments, Google Autocomplete Reinforce Scientific Bias</a></span><span style="font-size: x-small;"><span style="font-size: small;"><br /><i>A new journal article claims that blog comments and Google autocomplete influence<span style="font-size: small;"> </span>the public on new scientific research. </i></span></span></h1>
</blockquote>
<span style="font-size: small;">Cli<span style="font-size: small;">mate Et<span style="font-size: small;">c <span style="font-size: small;">does not help the situation by cond<span style="font-size: small;">oning crackpot commentar<span style="font-size: small;">y. It gets indexe<span style="font-size: small;">d by Google <span style="font-size: small;">just </span>like <span style="font-size: small;">everything else.</span></span></span></span></span></span></span></span></div>
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</div>
<div id="comment-287233">
<span style="font-size: small;"><span style="font-size: small;"><span style="font-size: small;"><span style="font-size: small;"><span style="font-size: small;"><span style="font-size: small;"><span style="font-size: small;"><span style="font-size: small;">Someone</span></span></span></span></span></span></span> recommended to try to avoid the filter bubble. Let<span style="font-size: small;">'</span>s try out the filter-free https://duckduckgo.com/</span></div>
<div id="comment-287233">
<span style="font-size: x-small;"><span style="font-size: x-small;"><span style="font-size: x-small;"><span style="font-size: x-small;"> <span style="font-size: x-small;"> </span></span></span></span></span></div>
<div id="comment-287233">
<span style="font-size: x-small;"><span style="font-size: x-small;"><span style="font-size: x-small;"><span style="font-size: x-small;"><span style="font-size: x-small;">wind "maximum<span style="font-size: x-small;"> entropy"</span></span> </span></span></span></span></div>
<div id="comment-287233">
<span style="font-size: x-small;"><span style="font-size: x-small;"><span style="font-size: x-small;"><span style="font-size: x-small;"><a href="https://duckduckgo.com/?q=wind+%22maximum+entropy%22">https://duckduckgo.com/?q=wind+%22maximum+entropy%22</a></span></span></span></span></div>
<div id="comment-287233">
</div>
<div id="comment-287233">
<span style="font-size: x-small;"><span style="font-size: x-small;">"dispersive transport"</span> </span></div>
<div id="comment-287233">
<a href="https://duckduckgo.com/?q=%22dispersive+transport%22"><span style="font-size: x-small;">https://duckduckgo.com/?q=%22dispersive+transport%22</span></a></div>
<div id="comment-287233">
</div>
<div id="comment-287233">
<span style="font-size: x-small;">"oil shock<span style="font-size: x-small;">" model</span> </span></div>
<div id="comment-287233">
<a href="https://duckduckgo.com/?q=%22oil+shock%22+model"><span style="font-size: x-small;">https://duckduckgo.com/?q=%22oil+shock%22+model</span></a></div>
<div id="comment-287233">
</div>
<div id="comment-287233">
<span style="font-size: x-small;">"hyperbolic decline" </span></div>
<div id="comment-287233">
<span style="font-size: x-small;"><a href="https://duckduckgo.com/?q=%22hyperbolic+decline%22" target="_blank"><span style="font-size: x-small;">https://duckduckgo.com/?q=%22hyperbolic+decline%22</span> </a></span></div>
<div id="comment-287233">
<span style="font-size: x-small;"><br /></span></div>
<div id="comment-287233">
<span style="font-size: x-small;"><span style="font-size: x-small;">CO2 diffusion "adjustment time"</span></span></div>
<div id="comment-287233">
<a href="https://duckduckgo.com/?q=CO2+diffusion+%22adjustment+time%22"><span style="font-size: x-small;"><span style="font-size: x-small;">https://duckduckgo.com/?q=CO2+diffusion+%22adjustment+time%22</span></span></a></div>
<div id="comment-287233">
<span style="font-size: x-small;"><span style="font-size: x-small;"> </span> </span></div>
<div id="comment-287233">
<span style="font-size: small;">For each of these search phrases,<span style="font-size: small;"> which are kind of obscu<span style="font-size: small;">re but not that odd, </span></span>the top hit goes to either my mobjectivist blog or this blog.</span></div>
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<br /></div>
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<br />@whuthttp://www.blogger.com/profile/18297101284358849575noreply@blogger.com5tag:blogger.com,1999:blog-8890671936020885112.post-12488607822969012202012-09-29T06:12:00.000-07:002013-05-02T05:47:33.441-07:00Bakken approaching diffusion-limited kineticsFrom <a href="http://www.greatbearpetro.com/appliance-of-science.html" target="_blank">
Great Bear Petro</a> (image recovered 5/2/2013)<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjY2YBYqWBgLp3nWk56N_sGLlIwE-KmDofpJLyPQEfVAx3a3Lbl4YSS4X8CPgAzg3RarluLULqEG4aR0h-sYTRu49HZeV2FHNQ9ulDDxBkkGbVwvduf6cV6x-bb4h1Jji-GXXMSpqsTOnI/s1600/wellcompletiontechtime.GIF" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjY2YBYqWBgLp3nWk56N_sGLlIwE-KmDofpJLyPQEfVAx3a3Lbl4YSS4X8CPgAzg3RarluLULqEG4aR0h-sYTRu49HZeV2FHNQ9ulDDxBkkGbVwvduf6cV6x-bb4h1Jji-GXXMSpqsTOnI/s1600/wellcompletiontechtime.GIF" /></a></div>
<br />
<br />
What is interesting in that GreatBear graphic is the progression of the permeability of the reservoirs.
The permeability is going down by an order of magnitude for each new technology introduced.
I don't understand all the intricacies of geology but I do understand the mathematics and physics of diffusion. See <a href="http://theoilconundrum.blogspot.com/2012/07/bakken-dispersive-diffusion-oil.html" target="_blank">here</a>.<br />
<br />
What decreasing permeability means is that the production rates of oil are now becoming completely diffusion-limited. In other words, the flow of oil is essentially a random walk from the source to the destination. All these new technologies are doing is exploiting the capabilities of diffusion-limited capture. This is the bottom-of-the-barrel stuff, kind of like driving your car off of fumes, or keeping your maple syrup bottle upside down, to make a more intuitive analogy out of it.
The Bakken rates are likely all diffusion limited and I will be willing to bet this based on some of the data from Mason.<br />
<br />
<a href="http://www.sbpipeline.com/images/pdf/Mason_Oil%20Production%20Potential%20of%20the%20North%20Dakota%20Bakken_OGJ%20Article_10%20February%202012.pdf" target="_blank">James Mason 2012 paper</a><br />
<br />
From Mason's data, the flow of oil out of a hydraulically fractured well appears to be controlled by diffusional dynamics.
This is what an average Bakken well decline looks like if one uses Mason's charts.<br />
<br />
<img src="http://img4.imageshack.us/img4/9282/bakkenmasondiffusionalm.gif" /><br />
<br />
The cumulative is the important part of the curve I believe because he plotted the instantaneous production incorrectly (which I tried to correct with the black dots).<br />
<br />
But then if we look at Brackett's analysis of Bakken (see below), I can better fit the average well to a hyperbolic decline model. A hyperbolic decline is an ensemble average of exponential declines of different rates, assuming maximum entropy in the distribution in rates (this works to describe lots of physical phenomena).<br />
<br />
<img src="http://img40.imageshack.us/img40/8281/bakkenhyperbolicdecline.gif" /><br />
<br />
That conflicts with the diffusional model that better describes Mason's data.<br />
<br />
Now, I believe it's possible that Brackett simply took the 1/e decline point on each well and then tried to extrapolate that to an average production. That's the easy way out and is definitely wrong as this will always approximate a hyerbolic decline; of course I can check this if I can get access to the 3,694 samples that Brackett says goes into his analysis.<br />
<br />
Mason and Brackett can't both be right, as there are sufficient differences between diffusional flow decline and hyperbolic decline to impact projections. The former is steeper at first but has a fatter tail, whereas the latter will definitely decline more in the long term. Brackett says the average well will generate 250,000 barrels of oil while Mason shows twice that and still increasing.<br />
<br />
Rune Likvern has a lot of the data that he painstakingly scraped from PDF files.<br />
<a href="http://fractionalflow.wordpress.com/2012/09/05/er-skiferolje-en-game-changer-del-1-av-2/" target="_blank">Likvern 1</a> | <a href="http://fractionalflow.wordpress.com/2012/09/08/er-skiferolje-en-game-changer-del-2-av-2/" target="_blank">Likvern 2</a> (in Swedish)<br />
<br />
<br />
There will be more data forthcoming in the next few years. We will see how it pans out.@whuthttp://www.blogger.com/profile/18297101284358849575noreply@blogger.com7tag:blogger.com,1999:blog-8890671936020885112.post-55830655777132425922012-09-17T18:38:00.001-07:002012-09-17T19:15:15.717-07:00Lake ice-out dates earlier and earlier ...With all the interest in the <a href="http://neven1.typepad.com/" target="_blank">Arctic sea-ice extent</a> reaching new minimums in area and volume, it seems instructive to point out a similar phenomena occurring in habitable areas.<br />
<br />
Let's take the situation of Minnesota lakes and track the "ice-out" calendar dates. The premise is that if the earth is warming, the ice-out dates should occur earlier and earlier in the season. A similar situation occurs for "first-ice" later in the season, but the "ice-out" date occurs very abruptly on a given day, and therefore has less uncertainty.<br />
<br />
The time of ice-out actually occurs so suddenly on a typical lake that it takes patient observation skills to wait it out. If one is not paying attention, the ice breaks up and within a few hours it's completely melted and gone. But this abruptness is useful in terms of precision, as the timing is certain to within a day for a given lake.<br />
<br />
Minnesota is a good test-case because it has many lakes and a hard freeze is guaranteed to occur every winter. <br />
<br />
For this reason, "ice-out" records have a combination of qualitative knowledge and calibrated precision. The qualitative knowledge lies in the fact that it takes only one observer who knows how to read a calendar and record the date. The precision lies in the fact that the ice-out date is unambiguous, unlike other historical knowledge [1]. Since ice-out is also a natural integral averaging technique, the dates have a built-in filter associated with it and the measure is less susceptible to single-day extremes; in other words, real ice-out conditions require a number of warm days.<br />
<br />
The data can be collected from the <a href="http://www.dnr.state.mn.us/ice_out/index.html" target="_blank">Minnesota DNR web site</a>. As presented, the data has been processed and expressed in a user-friendly geo-spatial graphic showing the ice-out dates for a sampling of lakes of a given year. First, I pulled out an animated GIF below (see <i><b>Figure 1</b></i> ). If you look closely one can see a noisy drift of the <b><span style="color: red;"><span style="color: #f9cb9c;"><b>tan</b><span style="color: black;">/</span></span>red</span>/<span style="color: #f1c232;">orange</span>/<span style="color: yellow;">yellow</span></b><span style="color: #f9cb9c;"><b></b> </span>colors corresponding to March and early April moving northward.<br />
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjlaDhyubqK4L1yHxnEDT79S2N8e0Ae_t2oN_KiFjB1HGZ6xoGhieWOx-mtPHIoSGUJKzU5P8InL3IHFeNyugWS5K_VCQ23qcpaKwZa3O83rAXb3hdvCYxs-LUFGjOwxeHlbiL19JQD8Tc/s1600/mn-ice-out-animated.gif" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="219" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjlaDhyubqK4L1yHxnEDT79S2N8e0Ae_t2oN_KiFjB1HGZ6xoGhieWOx-mtPHIoSGUJKzU5P8InL3IHFeNyugWS5K_VCQ23qcpaKwZa3O83rAXb3hdvCYxs-LUFGjOwxeHlbiL19JQD8Tc/s320/mn-ice-out-animated.gif" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><i><b>Figure 1 :</b></i> Animated GIF of ice-out dates in Minnesota.</td></tr>
</tbody></table>
<div class="separator" style="clear: both; text-align: center;">
</div>
<br />
<br />
Fortunately, underneath the graphics is a server that generates the processed data from a JSON-formatted data stream. By directly reading from the JSON and processing, we can come up with the linear regression plots for various geographic latitudes as shown in <i><b>Figure 2</b></i>. The "ice-out" day on the vertical axis is given by the number of days since the first of the year. Trying not to be too pedantic, but the lower this number, the earlier the ice-out day occurs in the year.<br />
<br />
This essentially pulls the underlying data out of the noise and natural fluctuations. Note the trend toward earlier ice-out dates with year, and of course, a later-in-the-season ice-out day with increasing latitude. Interesting as well is the observation that greater variance in the ice-out date occurs in recent years -- in other words, the highs and lows show more extremes [2].<br />
<div class="separator" style="clear: both; text-align: center;">
</div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgTgitLlg-2r05VqbW0a-GkvNn7XKhsjkaN9AJFnzdP9j4ObJYuOYbXN_1HVl6CtUiHeE1ATv1KHNvrFS-FGdJFAYr2-2hMwKtjfF_6L78FefvB9qEUoz1NI2iDhrOvDXtcij1FVQ9olCg/s1600/minnesota_ice_out_days.gif" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgTgitLlg-2r05VqbW0a-GkvNn7XKhsjkaN9AJFnzdP9j4ObJYuOYbXN_1HVl6CtUiHeE1ATv1KHNvrFS-FGdJFAYr2-2hMwKtjfF_6L78FefvB9qEUoz1NI2iDhrOvDXtcij1FVQ9olCg/s1600/minnesota_ice_out_days.gif" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><b><i>Figure 2</i>:</b> Ice-out dates for lakes of a given latitude occur earlier in the season <br />
according to a linear regression model.</td></tr>
</tbody></table>
<br />
In the inset below is logic code for retrieving and analyzing the ice-out dates from the Minnesota DNR site. The call-out to <span style="font-size: x-small;"><b><span style="font-family: "Courier New",Courier,monospace;">rplot </span></b></span>interfaces to an <i><b>R</b></i> package linear model plot and curve fit. My processing is two-step, first a call to <b><span style="font-size: x-small;"><span style="font-family: "Courier New",Courier,monospace;">get_all_records, which</span></span></b> stores the data in memory, then a call to <b><span style="font-size: x-small;"><span style="font-family: "Courier New",Courier,monospace;">lat_list</span></span></b> which retrieves the ice-out dates for a given latitude. As an example, all lake latitudes for 45N are between 45N and 46N degrees.<br />
<blockquote class="tr_bq" style="background-color: #d9ead3;">
<span style="font-size: xx-small;"><span style="font-family: "Courier New",Courier,monospace;">minnesota_dnr_ice_out('http://www.dnr.state.mn.us/services/climatology/ice_out_by_year.html?year=').</span></span><br />
<br style="font-family: "Courier New",Courier,monospace;" />
<span style="font-size: xx-small;"><span style="font-family: "Courier New",Courier,monospace;">:- dynamic</span></span><br />
<span style="font-size: xx-small;"><span style="font-family: "Courier New",Courier,monospace;"> temperature/6.</span></span><br />
<br style="font-family: "Courier New",Courier,monospace;" />
<span style="font-size: xx-small;"><span style="font-family: "Courier New",Courier,monospace;">assert_temperature(Name, Lat, Year, Month, Date, Days) :-</span></span><br />
<span style="font-size: xx-small;"><span style="font-family: "Courier New",Courier,monospace;"> not(temperature(Name, Lat, Year, Month, Date, Days)),</span></span><br />
<span style="font-size: xx-small;"><span style="font-family: "Courier New",Courier,monospace;"> asserta(temperature(Name, Lat, Year, Month, Date, Days)),</span></span><br />
<span style="font-size: xx-small;"><span style="font-family: "Courier New",Courier,monospace;"> !.</span></span><br />
<span style="font-size: xx-small;"><span style="font-family: "Courier New",Courier,monospace;">assert_temperature(_,_,_,_,_,_).</span></span><br />
<br style="font-family: "Courier New",Courier,monospace;" />
<span style="font-size: xx-small;"><span style="font-family: "Courier New",Courier,monospace;">store_record(Term) :-</span></span><br />
<span style="font-size: xx-small;"><span style="font-family: "Courier New",Courier,monospace;"> Term=json(</span></span><br />
<span style="font-size: xx-small;"><span style="font-family: "Courier New",Courier,monospace;"> [ice_out_first_year=_IceOutFirstYear,</span></span><br />
<span style="font-size: xx-small;"><span style="font-family: "Courier New",Courier,monospace;"> ice_out_last_year=_IceOutLastYear,</span></span><br />
<span style="font-size: xx-small;"><span style="font-family: "Courier New",Courier,monospace;"> lat=Lat,</span></span><br />
<span style="font-size: xx-small;"><span style="font-family: "Courier New",Courier,monospace;"> name=Name,</span></span><br />
<span style="font-size: xx-small;"><span style="font-family: "Courier New",Courier,monospace;"> ice_out_earliest=_IceOutEarliest,</span></span><br />
<span style="font-size: xx-small;"><span style="font-family: "Courier New",Courier,monospace;"> ice_out_latest=_IceOutLatest,</span></span><br />
<span style="font-size: xx-small;"><span style="font-family: "Courier New",Courier,monospace;"> ice_out_date=IceOutDate,</span></span><br />
<span style="font-size: xx-small;"><span style="font-family: "Courier New",Courier,monospace;"> sentinel_lake=_SentinelLake,</span></span><br />
<span style="font-size: xx-small;"><span style="font-family: "Courier New",Courier,monospace;"> ice_out_number_of_entries=_IceOutNumberOfEntries,</span></span><br />
<span style="font-size: xx-small;"><span style="font-family: "Courier New",Courier,monospace;"> id=_Id,</span></span><br />
<span style="font-size: xx-small;"><span style="font-family: "Courier New",Courier,monospace;"> lon=_Lon,</span></span><br />
<span style="font-size: xx-small;"><span style="font-family: "Courier New",Courier,monospace;"> ice_out_median_since_1950=_IceOutMedianSince1950]</span></span><br />
<span style="font-size: xx-small;"><span style="font-family: "Courier New",Courier,monospace;"> ),</span></span><br />
<span style="font-size: xx-small;"><span style="font-family: "Courier New",Courier,monospace;"> atomic_list_concat([Year,Month,Date], '-', IceOutDate),</span></span><br />
<span style="font-size: xx-small;"><span style="font-family: "Courier New",Courier,monospace;"> parse_time(IceOutDate, Stamp),</span></span><br />
<span style="font-size: xx-small;"><span style="font-family: "Courier New",Courier,monospace;"> atom_concat(Year, '-01-01', YearStart),</span></span><br />
<span style="font-size: xx-small;"><span style="font-family: "Courier New",Courier,monospace;"> parse_time(YearStart, Stamp0),</span></span><br />
<span style="font-size: xx-small;"><span style="font-family: "Courier New",Courier,monospace;"> Days is (Stamp-Stamp0)/24/60/60,</span></span><br />
<span style="font-size: xx-small;"><span style="font-family: "Courier New",Courier,monospace;"> print([Name, Lat, Year, Month, Date, Days]), nl,</span></span><br />
<span style="font-size: xx-small;"><span style="font-family: "Courier New",Courier,monospace;"> assert_temperature(Name, Lat, Year, Month, Date, Days).</span></span><br />
<br style="font-family: "Courier New",Courier,monospace;" />
<span style="font-size: xx-small;"><span style="font-family: "Courier New",Courier,monospace;">get_ice_out(Year) :-</span></span><br />
<span style="font-size: xx-small;"><span style="font-family: "Courier New",Courier,monospace;"> minnesota_dnr_ice_out(URL),</span></span><br />
<span style="font-size: xx-small;"><span style="font-family: "Courier New",Courier,monospace;"> atom_concat(URL, Year, U),</span></span><br />
<span style="font-size: xx-small;"><span style="font-family: "Courier New",Courier,monospace;"> http_client:http_get(U, R, []),</span></span><br />
<span style="font-size: xx-small;"><span style="font-family: "Courier New",Courier,monospace;"> atom_json_term(R, J, []),</span></span><br />
<span style="font-size: xx-small;"><span style="font-family: "Courier New",Courier,monospace;"> J=json([status='OK', results=L, message='']),</span></span><br />
<span style="font-size: xx-small;"><span style="font-family: "Courier New",Courier,monospace;"> maplist(store_record,L).</span></span><br />
<br style="font-family: "Courier New",Courier,monospace;" />
<span style="font-size: xx-small;"><span style="font-family: "Courier New",Courier,monospace;">temperature_lat(Lat_Range,Year,Time) :-</span></span><br />
<span style="font-size: xx-small;"><span style="font-family: "Courier New",Courier,monospace;"> temperature(_Name,Lat,Y,_Month,_Day,Time),</span></span><br />
<span style="font-size: xx-small;"><span style="font-family: "Courier New",Courier,monospace;"> atom_number(Lat, Lat_N),</span></span><br />
<span style="font-size: xx-small;"><span style="font-family: "Courier New",Courier,monospace;"> L is floor(Lat_N),</span></span><br />
<span style="font-size: xx-small;"><span style="font-family: "Courier New",Courier,monospace;"> Lat_Range = L,</span></span><br />
<span style="font-size: xx-small;"><span style="font-family: "Courier New",Courier,monospace;"> atom_number(Y, Year).</span></span><br />
<br style="font-family: "Courier New",Courier,monospace;" />
<span style="font-size: xx-small;"><span style="font-family: "Courier New",Courier,monospace;">lat_list(Lat, Years, Times, N) :-</span></span><br />
<span style="font-size: xx-small;"><span style="font-family: "Courier New",Courier,monospace;"> findall(Y, temperature_lat(Lat,Y,T), Years),</span></span><br />
<span style="font-size: xx-small;"><span style="font-family: "Courier New",Courier,monospace;"> findall(T, temperature_lat(Lat,Y,T), Times),</span></span><br />
<span style="font-size: xx-small;"><span style="font-family: "Courier New",Courier,monospace;"> format(atom(Title), '"Minnesota Latitude = ~D North"', [Lat]),</span></span><br />
<span style="font-size: xx-small;"><span style="font-family: "Courier New",Courier,monospace;"> rplot(Years,Times,Title, '"year"', '"iceOutDay"'),</span></span><br />
<span style="font-size: xx-small;"><span style="font-family: "Courier New",Courier,monospace;"> length(Years,N).</span></span><br />
<br />
<br />
<span style="font-size: xx-small;"><span style="font-family: "Courier New",Courier,monospace;">get_all_records(From, To) :-</span></span><br />
<span style="font-size: xx-small;"><span style="font-family: "Courier New",Courier,monospace;"> findall(Year, between(From, To, Year), List),</span></span><br />
<span style="font-size: xx-small;"><span style="font-family: "Courier New",Courier,monospace;"> maplist(get_ice_out, List).</span></span></blockquote>
<br />
According to the data, ice-out dates have gotten earlier by about a week since 1950 (and assume that via symmetry the first-ice could be a week later on average). The table below shows the slope on a per year basis, so that -0.1 would be an average 0.1 day per year earlier ice-out. Note that the slope has increased more rapidly since 1950.<br />
<br />
<table style="margin-left: auto; margin-right: auto; text-align: left;">
<tbody>
<tr align="center"><th style="background-color: cyan;">Latitude</th><th style="background-color: cyan;">Slope since<br />
1843</th><th style="background-color: cyan;">Slope since<br />
1950</th></tr>
<tr align="center"><td style="background-color: #ea9999;">43N</td><td style="background-color: #ea9999;">-0.080</td><td style="background-color: #ea9999;">-0.19</td></tr>
<tr align="center"><td style="background-color: #ea9999;">44N</td><td style="background-color: #ea9999;">-0.056</td><td style="background-color: #ea9999;">-0.16</td></tr>
<tr align="center"><td style="background-color: #ea9999;">45N</td><td style="background-color: #ea9999;">-0.099</td><td style="background-color: #ea9999;">-0.15</td></tr>
<tr align="center"><td style="background-color: #ea9999;">46N</td><td style="background-color: #ea9999;">-0.040</td><td style="background-color: #ea9999;">-0.078</td></tr>
<tr align="center"><td style="background-color: #ea9999;">47N</td><td style="background-color: #ea9999;">-0.090</td><td style="background-color: #ea9999;">-0.13</td></tr>
<tr align="center"><td style="background-color: #ea9999;">48N</td><td style="background-color: #ea9999;">-0.22</td><td style="background-color: #ea9999;">-0.29</td></tr>
<tr align="center"><td style="background-color: #ea9999;">49N</td><td style="background-color: #ea9999;">-0.25</td><td style="background-color: #ea9999;">-0.25</td></tr>
</tbody></table>
<div style="text-align: center;">
slope in fractional days/year</div>
<br />
The smallest decrease occurs in the center of the state where Mille
Lacs Lake is located. No urban heat island effect is apparent, with a
state-wide average of -0.138 days/year since 1950.<br />
<div style="text-align: center;">
<br /></div>
Besides this direct climate evidence, we also see more ambiguous and circumstantial evidence for warmer winters across the state -- for example we regularly see opossum in central Minnesota, which was very rare in the past. Something is definitely changing with our climate; this last winter had a very early ice-out, showing a record for the northern part of the state.<br />
<br />
<h3>
References </h3>
[1] See the <a href="http://judithcurry.com/2011/12/01/the-long-slow-thaw/" target="_blank">ramblings of Tony Brown</a>, who claims qualitative data from such ambiguous sources such as interpretations of medieval and Renaissance paintings of landscapes.<br />
[2] See J.Hansen on the <a href="http://blogs.nasa.gov/cm/blog/whatonearth/posts/post_1344022702866.html" target="_blank">New Climate Dice</a>, and the NASA site<a href="http://www.nasa.gov/topics/earth/features/warming-links.html" target="_blank"> http://www.nasa.gov/topics/earth/features/warming-links.html</a><br />
<br />
<br />@whuthttp://www.blogger.com/profile/18297101284358849575noreply@blogger.com5tag:blogger.com,1999:blog-8890671936020885112.post-77009213289303690762012-07-04T22:07:00.001-07:002012-07-05T17:12:11.556-07:00The Bakken Dispersive Diffusion Oil Production Model<br />
This post continues from <a href="http://theoilconundrum.blogspot.com/2012/05/bakken-growth.html" target="_blank">Bakken Growth</a>.<br />
<br />
<span style="font-size: large;">The Model </span><br />
Intuition holds that oil production from a typical Bakken well is driven by diffusion. The premise
is that a volume of trapped oil diffuses outward along the fractures. After the initial fracturing, oil close by the collection point will
quickly diffuse through the new paths. This does not last long,
however, as this oil is then replenished by oil from further away and
since it takes longer to diffuse, the flow becomes correspondingly
reduced. Eventually, the oil flow is based entirely on diffusing oil
from the furthest points in the effective volume influenced by the
original fractured zone. This shows the classic law of diminishing
returns, characteristic of Fickian diffusion.<br />
<br />
This class of problems is very straightforward to model. The bookkeeping is that the diffusing oil has to travel various
distances to reach the collection point. One integrates all of these
paths and gets the production profile. I call it dispersive because the
diffusion coefficient is actually smeared around a value.<br />
<br />
One can start from the master diffusion equation, also known as the Fokker-Planck equation.<br />
$$ \frac{\partial f(x,t)}{\partial t} = \frac{D_0}{2} \frac{\partial^2 f(x,t)}{\partial x^2} $$<br />
<br />
Consider that a plane of oil will diffuse outward from a depth at position<i> x</i>. The symmetric kernel solution is given by:<br />
$$ f(x,t) = {1\over{2\sqrt{D_0 t}}}e^{-x/\sqrt{D_0 t}} $$<br />
If we assume that the diffusion coefficient is smeared around the value<i> D<span style="font-size: x-small;">0</span></i> with maximum entropy uncertainty, integrate from all reasonable distances from the collection point, the cumulative solution becomes<br />
<br />
$$ P(t) = \frac{P_0}{1+ \frac{1}{\sqrt{D t}}} $$<br />
<br />
The reasonable distances are defined as a mean distance from the collection point and with a distribution around the mean with maximum entropy uncertainty. <i>P<span style="font-size: x-small;">0</span></i> is the effective asymptotic volume of oil collected and the diffusion coefficient turns into a spatially dimensionless effective value <i>D</i>. The details of the derivation are found in the text <a href="http://theoilconundrum.com/" target="_blank">The Oil Conundrum</a> and is what I refer to as a general dispersive growth solution; in this case the dispersive growth follows a fundamental Fickian diffusive behavior proportional to the square root of time. This is all very basic statistical mechanics applied to a macroscopic phenomena, and the only fancy moves are in simplifying the representation through the use of maximum entropy quantification. <br />
<pre> </pre>
<br />
<span style="font-size: large;">Some Data and a Model Fit </span><br />
More recent data on oil production is available from an article <a href="http://solarplan.org/Research/Mason_Oil%20Production%20Potential%20of%20the%20North%20Dakota%20Bakken_OGJ%20Article_10%20February%202012.pdf" target="_blank">Oil Production Potential of the North Dakota Bakken</a> in the Oil&Gas Journal written by James Mason.<br />
<br />
<b>Figure 1 </b>below shows the averaged monthly production values from Bakken compiled by Mason. The first panel shows in <span style="color: blue;">blue</span> his yearly production and his cumulative.
I also plotted the dispersive diffusion model in <span style="color: red;">red</span> with two parameters, an
effective diffusion coefficient and an equivalent scaled swept oil
volume. Note that the model is shifted to the left compared to the blue line, indicating that the
fit may be bad. But after staring at this for awhile, I discovered that Mason did not transcribe his early year
numbers correctly. The panel on the bottom is the production data for the
first 12 months and I moved those over as black markers on the first
panel, which greatly improved the fit. The dashed cumulative is the verification as the diffusive model fits very well over the entire range.<br />
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjy-tJW8Sd23Q4moFKAitc1h7yba_H7io2zzFktFaK1xn9ygEh4TsqZF8YrWvYnRzYgn3dvAq6YxA0kTNwPmKIujUiEyLXGXLhxF4i5SgoyUWEY2Q_xWGxI6TY2bwRcHPkY5yZiTl_kATM/s1600/bakken_mason_diffusional_model.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjy-tJW8Sd23Q4moFKAitc1h7yba_H7io2zzFktFaK1xn9ygEh4TsqZF8YrWvYnRzYgn3dvAq6YxA0kTNwPmKIujUiEyLXGXLhxF4i5SgoyUWEY2Q_xWGxI6TY2bwRcHPkY5yZiTl_kATM/s1600/bakken_mason_diffusional_model.png" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><b>Figure 1</b>: Model adapted to Mason Bakken Data. Top is yearly and bottom is monthly data.</td></tr>
</tbody></table>
<br />
<div class="separator" style="clear: both; text-align: center;">
</div>
For this model, the asymptotic cumulative is set to 2.6 million barrels. This is a deceptive number since the fat-tail is largely responsible for reaching this value asymptotically. In other words, we would have to wait for an infinite amount of time to collect all the diffused oil -- such is the nature of a random walk. Even to collect 800K barrels will take 100 years from extrapolating the curve. After 30 years, the data says 550K barrels, so one can see that another 70 years will lead to only 250K barrels, should the well not get shut-in for other reasons.<br />
<br />
If these numbers that Mason has produced are high quality, and that is a
big if (considering how he screwed up the most important chart) this
may become a de facto physical model describing oil production for
fractured wells. I can guarantee that you won't find a better fit than
this considering it is only two parameters, essentially describing a
rate and a volume. This is likely the actual physical mechanism as
diffusional laws are as universal as entropy and the second law.<br />
<br />
The connection to the <a href="http://theoilconundrum.blogspot.com/2012/05/bakken-growth.html" target="_blank">previous post</a> is that the substantial production
increase is simply a result of gold-rush dynamics and the acceleration
of the number of new wells. Wait until these new starts stop accelerating. All the declines will
start to take into effect, as one can see from the steep decline in the dispersive diffusion profiles. We may still get returns from these wells for many years, but like the 5 barrel/day stripper wells that dot the landscape in Texas and California, they don't amount to much more than a hill of beans. The peak oil problem has transformed into a flow problem, and unless thousands of new wells are added so that we can transiently pull from the initial production spike or to continuously pull from the lower diminishing returns, this is what Bakken has in store -- a few states with thousands and thousands of wells cranking away, providing only a fraction of the oil that we demand to keep the economy running.<br />
<br />
If someone comes up with a way to increase diffusion, it might help increase flow, but diffusion is a sticky problem. That is what nature has laid out for us, and we may have gotten as far as we can by applying hydraulic-fracturing to lubricate the diffusion paths.<br />
<br />
This analysis fits in perfectly with the mathematical analysis laid out in <a href="http://theoilconundrum.com/" target="_blank">The Oil Conundrum</a> book, and will likely get added in the next edition.<br />
<br />
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<br />@whuthttp://www.blogger.com/profile/18297101284358849575noreply@blogger.com3