There are three pieces of the puzzle regarding this issue, which collectively have to fit together for us to be able to make sense about the net energy flow.
- The surface temperature time series, see the CSALT model
- Heat sinking via ocean heat content diffusion, see the OHC model
- The land and ocean surface temperatures have an interaction where they can exchange energy
We have an excellent start on the first two but the last requires a fresh analysis. Understanding the exchange of energy is crucial to not getting twisted up in knots trying to explain any perceived deficit in heat accounting.
Consider Figure 1 below where the energy fluxes are shown at a level of detail appropriate for book-keeping. On the left side, we have an energy balance of incoming flux perturbation (Li) and outgoing flux (Lo) for the land area (we don't consider the existing balance as we assume that is already in a steady state). On the right side we do the same for the sea or ocean area (Si and So).
The question is how do we proceed if we don't have direct knowledge of all the parameters. The first guess is that they have to be inferred collectively. The complicating factor is that the sea both absorbs thermal energy (heat) into the bulk shown as the OHC arrow, and that some fraction of the latent and radiative heat emitted by the sea surface transfers over to the land. There is little doubt that this occurs as the Pacific Ocean-originating El Nino events do impact the land, while regular seasonal monsoons work to redistribute enormous amounts as rain originating from the ocean and delivered to the land as moist latent energy. So the question mark in the figure indicates where we need to estimate this split.
Solving this problem would make an excellent homework assignment and perfect for a class in climate science. Let's give it a try.
Here is a list of parameters that we need to consider
|f||Forcing flux averaged over the earth||?|
|Li||Incident flux over land||?|
|Lo||Exit flux from land||?|
|Si||Incident flux over sea||?|
|So||Exit flux from sea||?|
|a||Fraction of incident flux over land above the global average||?|
|b||Fraction of exiting flux over ocean that radiates to space||?|
|TL||Average surface temperature anomaly over land||1.36 (estimated from CSALT model using NOAA data)|
|TS||Average surface temperature anomaly over sea||0.71 (estimated from CSALT model using NOAA data)|
|PL||Proportion of land area||?|
|PS||Proportion of sea area||0.7|
|OHC||Average net ocean heat content flux||1 W/m^2 (estimated from OHC model)|
|T||Average global surface temperature anomaly||? but estimated at 0.90 from CSALT model using NOAA data|
|λ||Climate sensitivity||0.3 degrees / W/m^2|
Some of these parameters are measured or estimated but the rest aren't because of the interactions between the fluxes. To try to solve for their values we need to find or develop equations that map to the problem domain. I count 9 variables so assuming a linear solution, we will need to create 9 or more equations.
|Coverage of land and sea|
|Mean global surface temperature land\sea composite|
|Globally averaged forcing perturbation|
|Definition of incoming land flux relative to average forcing|
|Definition of incoming sea flux relative to average forcing|
|No missing heat on land (limited heat sinking)|
|Transfer from sea to land (* fixed see note at bottom)|
|Climate sensitivity definition for land|
|Climate sensitivity definition for sea|
|Conservation of energy flux at the sea surface|
We could solve this with a linear equation solver but the equations appear sparse enough that it is less effort to do the substitutions manually. Take the easy ones first:
which matches the estimate from CSALT. This makes sense because that is also how NOAA does the kriging estimate.
If we continue to make substitutions, we arrive at the expanded form for the transfer equation
rearranging terms and using the proportional area identity, we can solve for So
and f follows from the last equation
and then a and b derived from the climate sensitivities
Based on the values in the first table, we get a value of 0.22 or 22% for a and a value of 0.87 or 87% for b. The incoming flux is therefore increased from approximately 3.7 W/m^2 to over 4.5 W/m^2 over land as it absorbs moisture from the ocean. By the same token 87% of the non-heat-sunk incoming flux over the ocean is diverted to space, with the remainder of this transferred to land. These values seem reasonable as 13% indicates the level of diversion of latent heat from ocean that we would expect, and 22% is reasonable in terms of amplification of warming that we could intuit.
The value of OHC is estimated in the diffusional model based on the limited data of Levitus  and Balmaseda . One can change this number and the energy balance equations would work to correct any deficit of heat. As Trenberth has explained, the OHC numbers are crucial to being able to close the energy budget and also to calibrate the satellite readings such as from the ERBE/CERES experiments (which though precise in relative terms are not that accurate in absolute value).
This analysis resolves a couple of issues. First, it explains why the land surface is warming at twice the rate of the ocean surface -- in spite of the smaller than anticipated OHC rate of increase. Secondly, it explains how the "missing heat" is a confusion in how the excess thermal energy gets redistributed by latent actions due to the significant evapotranspiration mechanism at the oceans surface as shown below in Figure 2 (see ). It is no coincidence that the evapotranspiration rate of 80 divided by the surface radiation of 396 (calculated as 80/(396+80) = 0.17) shown is consistent with the 13% predicted by the above analysis. That is the amount of latent energy that could be transported to the land. Like I said, all these pieces fit tightly as a jigsaw puzzle.
The rest is uncertainty in the calculation of OHC. Yet in store, I will use the CSALT model to drive the OHC dispersive diffusion formulation (see interactive view) to create a comprehensive view of the earth's climate. Fun times ahead in climate science, not "seeming a bit boring", as climatologist Curry has suggested.
 M. A. Balmaseda, K. E. Trenberth, and E. Källén, “Distinctive climate signals in reanalysis of global ocean heat content,” Geophysical Research Letters, 2013.
 It is still possible that negative feedback in the lapse rate can provide further refinement in the analysis, by providing a difference between ocean and land radiative effects, but that would be a second-order correction. Something different between the moisture in the ocean versus moisture over the land would have to occur (possibly due to cloud formation) for this not to be incorporated in the general CO2 control knob theory of climate sensitivity. See the following link for further info: http://www.ipcc.ch/publications_and_data/ar4/wg1/en/ch8s8-6-3-1.html
(*) In the second table, the equation for sea-to-land energy transfer was mis-transcribed by me as:
Commenter DC used this in his own verification and got the wrong result. I left out the implicit part in the diagram that f*p is coming in as a forcing. Two arrows coming in, one from the ocean and one from the external forcing : (1) f*p and (2) f*a*p, so total incoming = f*p+f*a*p, while going out is f*(1+a)*p. So we only take the f*a*p term and equate that to what is coming from the ocean.
Thanks to DC.