https://www.ipcc.ch/site/assets/uploads/2018/02/WG1AR5_Chapter09_FINAL.pdf

The ultimate proof of the reliability of ocean heat transport in AOGCMs would be for those models to reproduce the temperatures and temperature gradients we observe given different starting conditions. The ocean and atmospheric modules of AOGCMs are “spun up” separately so that slow heat transport in the ocean can be modeled over long periods of time with large time steps. However ocean currents are driven by atmospheric winds, and it is computationally impractical for the full AOGCM to equilibrate the ocean via multiple circuits of the thermohaline circulation. Even after a long spin up, the surface temperature in some AOGCMs is steadily and slowly changing without forcing – suggesting to me that the deep ocean hasn’t equilibrated with surface temperature.

]]>Fick’s Law of Diffusion says that heat flux is the temperature gradient multiplied by the diffusion coefficient. Fourier’s Law says that conductivity is proportional to the temperature difference between two locations divided by the distance between those locations – also temperature gradient. So thermal diffusivity expresses heat flux in the “continuous” terms of a gradient while thermal conductivity expresses heat flux in “discrete” terms between two locations. Box models are inherently discrete. Fourier’s thermal conductivity (measured in units of W/m-K) divided by volumetric heat capacity (J/m3/K, = density times specific heat capacity) gives Fick’s thermal diffusivity (measured in units of m2/s). Conductivity and thermal diffusivity appear to be different names for quantifying heat flow driven by a temperature difference.

The thermal conductivity of water (0.59 W/m-K) and its volumetric heat capacity (4.18 J/cm3-K) affords a thermal diffusivity for stationary water of 0.0014 cm2/s, about 700 times smaller than the thermal diffusivity assumed by Hansen for a global ocean mixed by bulk motion/convection/turbulence. Mechanistically, thermal diffusion explains a negligible fraction of heat transport in the ocean

Isaac Held analyzes the ocean heat uptake of AOGCMs in terms of two-box models: a mixed layer (T) and a colder deep ocean with effectively infinite heat capacity (T_0):

https://www.gfdl.noaa.gov/blog_held/3-transient-vs-equilibrium-climate-responses/

Heat transport from the mixed layer to the deeper ocean in response to the rising temperature of the mixed layer is modeled as being proportional to the temperature difference between these two compartments. Held says that typical AOGCMs send 0.7+/-0.2 W/m2 more heat from the mixed layer to the deeper ocean per 1K of surface warming. He calls this the “ocean heat uptake efficiency”. Let’s assume that the initial temperature difference (T – T_0) between these two compartments is 10 K and rises to 11 K (T’ – T_0). Let’s also assume the distance between these two compartment 1 km. Using Fourier’s Law, the “effective thermal conductivity” would be 700 W/m-K. The volumetric heat capacity of water is 4.18 J/cm3/K, making the “effective thermal diffusivity” in this version of Held’s two-box model 1.67 cm2/s.

So a thermal conductivity for copper of 3.84 W/m-K and a thermal diffusivity of 1 cm2/s and an ocean heat uptake efficiency of 0.7 W/m2/K all could represent roughly similar easy of heat transfer when driven by a temperature gradient.

If I assumed an initial 20, 5, or 2 K difference between the two compartments, I would have calculated exactly the SAME thermal diffusivity associated with an ocean heat uptake efficiency of 0.7 W/m2/K. Only the change in the gradient matters. If I assumed heat transport over 0.5 or 2 km, however, the thermal diffusivity would have been double or half.

However, Held didn’t discuss what MUST BE HAPPENING BEFORE A FORCING IS APPLIED. Postulating ANY TEMPERATURE DIFFERENCE between two compartments connected by effective thermal diffusivity of 1.0 or 1.6 cm2/s, creates a system that is not at steady state. The only way to have a steady-state model for the ocean expressed in terms of thermal diffusion is for there to be NO temperature difference between the surface and the deep ocean or for the thermal diffusivity of the ocean to be near zero. This suggests that we should be using the 0.0014 cm2/s thermal diffusivity of stationary water, not values a thousand-fold bigger.

In order for our planet to have tropical and temperate oceans with surface temperatures 20 K and 10 K warmer than the deep ocean, the local effective thermal diffusivity in these regions must be low enough that the temperature of bottom water formed in polar regions is negligibly changed on its 1500-year trip via the thermohaline circulation. Some heat must be diffusing into bottom water. The thermohaline circulation is the only way for that heat to escape! The mechanisms that rapidly carry heat and tracers somewhat below the mixed layer in equatorial and temperate zones certainly can’t reach all the way to bottom waters.

Most heat transport in the ocean is by fluid flow. Downward fluid flow is OPPOSED by the local temperature/density gradient – NOT speeded up by a steeper temperature gradient. Bulk motion of water against the local temperature/density gradient requires doing work against gravity. The concept of thermal diffusion mistakenly predicts that the greatest heat transfer will occur in the tropics, where the steepest temperature gradients are found. It wrongly predicts that there should be no temperature difference between the deep ocean and the surface. The concept of heat transfer by fluid flow correctly predicts that most heat uptake after forcing occurs in polar regions and that the temperature of the deep ocean is controlled by the temperature in polar regions. Thermal diffusion proportional to a temperature gradient is a conceptually flawed approach to heat transfer in the ocean.

]]>As one who did diffusion models in the semiconductor industry, being able to deduce the shortcuts taken is not hard. Sorry, no shortcuts are allowed in characterizing diffusion during wafer fab.

]]>Frank replied with a long comment amateurishly discussing mechanisms by which bulk motion of water transports heat from the surface into the ocean. The reference linked below analyzed six separate mechanisms in three AOGCMs: advection, convection, mixed layer turbulence, eddy-induced advection, isopycnal diffusion and diapycnal diffusion. Figure 1 shows each model uses different mechanisms to different extents at different depths. In CMIP5 models, total heat uptake in historic runs from 1971 to 2005 ranged from and 8 to 36*10^22 J. (AR5 WG1 Figure 9.17) Ocean heat uptake – and the related, more-critical ocean uptake of CO2 – appear to be another aspect of “settled climate science” with large uncertainties

https://journals.ametsoc.org/doi/pdf/10.1175/JCLI-D-14-00235.1

In a separate comment, I noted that the energy balance models used by dozens of researchers (including Lewis) relied on observations of warming of the ocean, but AOGCM’s did need to get all of these mechanisms right and some require parameterization. And, in 1981, Hansen didn’t model any mechanisms of ocean heat uptake.

Now geoenergymath claims I misrepresented Hansen’s work: “Wrong. By incorporating diffusion with a 1 cm^2/s term, James Hansen did an infinite slab model and did it correctly, instead of this one-box or two-box junk that Nic Lewis is doing. You don’t “connect” one box with another via a diffusivity term. When you decide to use D then you either do the complete slab model numerically or you can use the analytical form, involving erf formulations or the simplified form that I have described elsewhere.”

Hansen’s publication dealt with several 1-dimensional models for the planet. 1-D models do have infinite slabs in the horizontal directions. However, Hansen himself referred using “box diffusion model[s]” on p 595. “Figure 1 : “Heat is rapidly mixed in the upper 100 m of the ocean and diffused to 1000 m with diffusion coefficient k”. k is either 1 cm2/s or infinity. That figure had two 1-box models for the ocean and two 2-box models for the ocean. So the brilliant James Hansen is using the same box models as the disdained Nic Lewis. And Hansen’s diffusion coefficient of 1 cm2/s, was derived (with significant uncertainty) from box diffusion models of analyzing ocean uptake of C-14 carbon dioxide and tritium.

]]>“In 1981, however, Jim Hansen took none of this into account. He simply used one-box (for the heat content of a simple mixed layer) or a two-box model for the heat content of the mixed layer and thermocline (with a thermal diffusivity of 1 cm2/s connecting them). See Fig 1.”

Wrong. By incorporating diffusion with a 1 cm^2/s term, James Hansen did an infinite slab model and did it correctly, instead of this one-box or two-box junk that Nic Lewis is doing.

You don’t “connect” one box with another via a diffusivity term. When you decide to use D then you either do the complete slab model numerically or you can use the analytical form, involving erf formulations or the simplified form that I have described elsewhere.

]]>And continued: “Obviously there isn’t one value of effective diffusion coefficient that’s operational, and doing the math for a range of values (according to maximum entropy) actually simplifies the formulation (which is a nasty erf result). I have that described here

https://agupubs.onlinelibrary.wiley.com/doi/10.1002/9781119434351.ch14

Nic Lewis and dozens of supporters of the consensus use energy balance models (conservation of energy) to convert observed transient warming (TCR), estimated anthropogenic and natural forcings perturbing our planet, and the observed rate of ocean heat uptake into an effective ECS. Unlike the parameterized ocean heat uptake in AOGCMs (which can be tuned so that transient warming will agree with observed warming), EBM’s rely on real measurements of ocean heat uptake by ARGO. So your complaints are relevant to AOGCMs, not EBMs.

The beauty of EBMs is their simplicity and their reliance on measurements, not parameters. The heat capacity of the atmosphere and land surface is low enough that they would begin warming in response to imbalance at the TOA at an initial rate of more than 2 K/year if there were no ocean heat uptake. Basically, their heat capacity is negligible. So, at any point in time, the heat being retained somewhere in our climate system because of a forcing is going two places – 1) into the ocean and 2) out to space due to higher temperature increasing OLR and modified reflected SWR. The change in net flux across the TOA with temperature is the climate feedback parameter measured in W/m2/K. F_2x (ca 3.5 W/m2/doubling) plus the climate feedback parameter gives climate sensitivity (K/doubling). The only limitation is the assumption that unforced warming/cooling is negligible and observed warming is all forced warming. Fortunately, estimates of TCR don’t seem to vary much with time, suggesting that unforced temperature change is a minor problem.

]]>And later: “Well if you want to do the math correctly then you have to separate motion by diffusion vs motion by advection. This is related to the reason why all these first-order models of OHC fail — because they don’t incorporate the divergence term and never see the fat-tail of heat uptake.”

If I understand correctly, the eddy diffusion associated with large ocean gyres can be correctly modeled with the grid cells in today’s models. The eddy diffusion associated with eddies roughly the size of a grid cell and smaller must be parameterized. The units on these parameters (cm2/sec) are the same as used for thermal diffusivity in solids – even though eddy diffusion is the result of bulk convection and thermal diffusivity (in solids and fluids) is normally mediated by molecular collisions. Thermal diffusivity in liquids without bulk flow is closely related to molecular diffusivity in liquids. However, molecular collusions and molecular diffusion are negligible processes within or between grid cells compared with bulk flow. Since the ocean is stably stratified by density, eddy diffusion is first calculated on slanted surfaces of equal density (isopycnal) and then transformed into a vertical flux.

The MOC obviously mixes the shallow and deep oceans, but there may not a significant amount of heat transport associated with the MOC because global warming may not have raised the temperature of the water that is subsiding in polar regions (or the temperature of water that is upwelling). Finally there is turbulence that transports heat vertically associated with ocean currents and tides flowing over the irregular ocean floor.

If I understand correctly, vertical heat flux in AOGCMs from large scale eddy diffusion, parameterized eddy diffusion, turbulent mixing due to the ocean floor and possibly the MOC are combined to produce one overall parameter for the average vertical heat flux in the ocean. That parameter varies by a factor of 2 between models. In 1981, however, Jim Hansen took none of this into account. He simply used one-box (for the heat content of a simple mixed layer) or a two-box model for the heat content of the mixed layer and thermocline (with a thermal diffusivity of 1 cm2/s connecting them). See Fig 1.

http://climate-dynamics.org/wp-content/uploads/2016/06/hansen81a.pdf

Beginning with CMIP3, historical emissions of CFCs have been added to the atmosphere and followed as they dissolve into the ocean (much more in colder water) and then are transported by the same bulk-flow processes discussed above that transport heat. The concentrations of CFCs predicted by models have been compared to observations. The uptake of CO2 by the ocean is controlled by the same principles as CFCs, but is pH dependent and depends on the ocean’s buffer capacity (total alkalinity). CO2 is also sequestered by photosynthesis. Now that RCPs have replaced emissions scenarios, projections are driven by postulated forcing, not postulated emissions net of uptake by the ocean and other sinks.

https://journals.ametsoc.org/doi/full/10.1175/JCLI3758.1

Fig. 6. CFC-11 penetration depth = the column inventory divided by the surface concentration: (a) Observations, (b) ocean-alone Ocn run [AOGCM forced with rising SST (c) mean of the ESb, ESg, and ESh runs, [forced with rising GHGs].

“There isn’t one value of this “parameter”. In fact, there is no useful theoretical value, or set of values, that can be derived.”

Tell me about it. Obviously there isn’t one value of effective diffusion coefficient that’s operational, and doing the math for a range of values (according to maximum entropy) actually simplifies the formulation (which is a nasty *erf* result). I have that described here

https://agupubs.onlinelibrary.wiley.com/doi/10.1002/9781119434351.ch14