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

**Figure 1**below

Figure 1: Ocean Heat Content from BTK [1] (source) |

In the following we provide a mathematical explanation which works its way from first principles to come up with an

*uncertainty-quantified*formulation. After that we present a first-order sanity check to the approximation.

#### Solution

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**Figure 1**.

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:

$$\Delta T(x,t) = \frac{c}{\sqrt{D_i t}} \exp{\frac{-x^2}{D_i t}} $$

where

*D*_i is the diffusion coefficient, and

*x*is the depth (note that often a value of 2

*D*instead of

*D*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,

*c*

The first level of uncertainty is a

*maximum entropy prior*(MaxEnt) 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:

$$ p(D_i) = \frac{1}{D} e^{\frac{-D_i}{D}} $$

then we get this formulation:

$$\Delta T(t|x) = \frac{c}{\sqrt{D t}} e^{\frac{-x}{\sqrt{D t}}} (1 + \frac{x}{\sqrt{D t}})$$

The next uncertainty is in capturing the heat content for a layer. The incremental heat across a layer,

*L*, we can approximate as

$$ \int { \Delta T(t|x) p(x) dx} = \int { \Delta T(t|x) e^{\frac{-x}{L}} dx} $$

Which gives as the excess heat response the following concise equation.

$$ I(t) = \frac{ \frac{\sqrt{Dt}}{L} + 2 }{(\frac{\sqrt{Dt}}{L} + 1)^2}$$

This is also the response to a delta forcing impulse, but for a realistic situation where a growing aCO2 forces the response (explained in the previous post), 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.

$$ R(t) = F(t) \otimes I(t) = \int_0^t { F(\tau) I(t-\tau) d\tau} $$

If the thermal stimulus is a linearly growing heat flux, which roughly matches the GHG forcing function (see Figure 7 in the previous post)

$$ F(t) = k \cdot (t-t_0) $$

then assuming a starting point t_0=0.

$$ R(t)/k = \frac{2 L}{3} {(D t)}^{1.5}- L^2 D t+2 L^3 \sqrt{D t}-2 L^4 \ln(\frac{\sqrt{D t}+L}{L})) $$

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,

*L*, and a value of

*D*of 3 cm^2/s.

Figure 2 : Thermal Dispersive Diffusion Model applied to the OHC data |

Another view of the OHC includes the thermal mass of the non-ocean regions (see

**Figure 3**). This data appears smoothed in comparison to the raw data of

**Figure 2**.

Figure 3 : Alternate view of growing ocean heat content. This also includes non-ocean. |

*D*, along with the inferred external thermal forcing function is able to reproduce the temporal profile accurately.

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.

#### Sanity Check

The following is a sanity check for the above formulation; providing essentially a homework assignment that a physics professor would hand out in class.An application of Fick’’s Law is to approximate the amount of material that has diffused (with thermal diffusion coefficient

*D*) at least a certain distance,

*x*, over a time duration,

*t*, by

$$ e^{\frac{-x}{\sqrt{Dt}}}= \frac{Q}{Q_0}$$

- For greater than 300 meters,
*Q/Q_0*is 13.5/20 read from**Figure 1**. - For greater than 700 meters,
*Q/Q_0*is 7.5/20

*Q_0*=20 is the baseline for the total heat measured over all depths (i.e. between x=0 and

*x*=infinite depth) reached at the current time. No heat will diffuse to infinite depths so at that point

*Q/Q_0*is 0/20.

First, we can check to see how close the value of

*L*=sqrt(

*Dt*) scales, by fitting the

*Q/Q_0*ratio at each depth..

- For
*x*=300 meters, we get*L*=763 - For
*x*=700 meters, we get*L*=713

We then use an average elapsed diffusion time of

*t*=40 years and assume an average diffusion depth of 740, and

*D*comes out to 4.5 cm^2/s.

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

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. Green’s function) and compare that against

**Figure 1**and

**Figure 2**, and the full temporal profile.

#### Discussion

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.**References**

[1]

M. A. Balmaseda, K. E. Trenberth, and E. Källén, “Distinctive climate signals in reanalysis of global ocean heat content,”

*Geophysical Research Letters*, 2013.
[2]

J. Hansen, D. Johnson, A. Lacis, S. Lebedeff, P. Lee, D. Rind, and G. Russell, “Climate impact of increasing atmospheric carbon dioxide,”

http://pubs.giss.nasa.gov/docs/1981/1981_Hansen_etal.pdf

*Science*, vol. 213 (4511), pp. 957–966.http://pubs.giss.nasa.gov/docs/1981/1981_Hansen_etal.pdf