Can you please comment on the similitudes and differences of your model and the one used by Stephen Schwartz in his 2007 paper linked below:

http://www.ecd.bnl.gov/steve/pubs/HeatCapacity.pdf

Let us assume that the climate system consists of the troposphere and the mixed layer of the ocean. External to the climate system are the stratosphere and the deep ocean including the thermocline. We also neglect the effect of the land mass.

Incoming power to this system is mainly the incoming solar radiation that is absorbed and not reflected = II_At S*dA W where II_At is the symbol for the double integral over the earth’s total surface area, At m2, and S W/m2 is the locally absorbed solar radiation.

Outgoing from the climate system is the long-wave radiation = lambda_r *II_At (T_r – T_sp)*dA W where T_r is the temperature of the radiating layer in the atmosphere and T_sp is the temperature of the space (2.7 K). lambda_r W/m2/K is a transfer coefficient for radiative heat transfer and assumed constant.

Outgoing is also the heat flow from the mixed ocean layer through the thermocline to the deep ocean = lambda_do*II_Ado (T_ml – T_do)*dA W where we integrate over the area between the mixed layer and the deep ocean. T_ml is the temperature of the mixed layer and T_do of the deep ocean. lambda_do W/m2/K is a transfer coefficient for convective heat transfer between the mixed layer and the deep ocean.

Let us look at how the energy content of the climate system, E J, changes from a time zero when E = E0 J. Then
dE/dt – dE0/dt = dE/dt = II_At (S-S0)*dA – lambda_r*II_At (T_r – T_r0)*dA – lambda_do*II_Ado (T_ml – T_ml0)*dA (1)

This can be written as integral mean values (symbolized with _m) over the respective areas:
dE/dt = (S-S0)_m*At – lambda_r*(T_r – T_r0)_m*At – lambda_do*(T_ml – T_ml0)*Ado (2)

We now assume that the temperature anomalies are the same in the atmosphere and in the mixed layer (or alternatively that they are proportional):
dE/dt = (S-S0)_m*At – (lambda_r*At + lambda_do*Ado)*(T – T0)_m (3)

Let us apply this equation for the hypothetical case that we go from one so-called equilibrium state of the climate system to another. This means that we start from one steady state with the energy content E0 J, not varying with time, we perturb the system with a forcing Delta S W/m2 and await the new steady state with a constant energy content E J. At the new steady state dE/dt = 0 giving:

(T – T0)_m = (S-S0)_m/(lambda_r + lambda_do*Ado/At) or Delta T = Delta S/lambda (4)

The denominator in this equation is the feedback parameter and we can see that the deep ocean acts as a negative feedback decreasing the effect of the forcing in the numerator.

Obviously the first two terms in equation (2) are measured by the ERBE. If the change in the forcing is known, as during the Pinatubo eruption, the radiative feedback parameter lambda_r can be determined by using the ERBE data and for example the UAH MSU temperature anomalies, as carried out by Dr. Spencer. This is done without having to use equation (2). However, what equations (2) and (4) tell us is that the total feedback parameter lambda is greater than lambda_r because of the negative feedback from the deep ocean.

Thanks for the reply. Now I really am puzzled. I think the simplest way of clarifying is to ask whether you used the relationship

ERBE = F – lambda*T to do your calculations (I apologies if I have the sign of ERBE wrong).

If you do, the precise nature of the term represented by dE/dt (e.g. Cp*[dT/dt] or something accounting for more complicated response) is irrelevant.

I think that your approach is insensitive to the nature of the thermal model (which is a big plus point).

Why do I say a plus point? Because it is insensitive to arguments like:

the lack of cooling in year 2 was due to an “El Nino” type of response being triggered.

Now for that to hold one could not argue that extra heat that came from “somewhere” and that heat is significant ,as that does not impact the argument (or the equation).

(That would still leave arguments of the form that sudden cooling gives a different value for Lambda to gentle warming; I make no such argument but someone could.)

Personally, I think your argument could be put in stronger terms, in that is it is not subject to debate about heat storage/retention. The calculation of an effective depth is useful in the 40 metres is not unreasonable and adds creadence but is not central to the argument, as I see it.

But to get back on track:

Do you use:

ERBE = F – lambda*T to do your calculations

If so it is insensitive to a choice of a thermal model and hence a very strong line of argument.

Now you say:

“if you assume no change in heat content, then there is no change in temperature, and thus no feedback to measure.”

What I am saying is that using ERBE makes the model WYSIWYG:

If dE/dt = 0
then ERBE = 0
and F = lambda*T

lambda = F/T

Now that is not what happened, but the equation would still work if it had. By that I mean that the equation is formally correct irrespective of heat retention. If the system had no thermal mass dE/dt would indeed be zero but that would not imply that the temperature would not change, so I guess I do not understand your point:

“if you assume no change in heat content, then there is no change in temperature, and thus no feedback to measure.”.

Personally I think the approach is terrific, but for some reason I seem to think it is more terrific than you seem to do.

But to clear things up, as I asked above:

Do you drop the oceanic heat content term and set ERBE = F – lambda*T (as your diagrams indicate to me), if you do then I am not saying anything new just saying it louder.

I hope I have been clear.

Best Wishes

Alex

Alex

Now:

Cp*[dT/dt] = F – lambda*T

is correct as stated, but all the arguments would work if it was put as:

dE/dt = F – lambda*T

{where E is the energy retained or stored by the system}

This would be true no matter how the system was modelled, (slab ocean, upwelling- diffusive ocean, with or without atmospherice heat content, moisture, whatever).

Now dE/dt is also what the combined (SW/LW) ERBE tries to measure so whatever the value of dE/dt one replaces it with the ERBE data. The energy balance combined (SW/LW) is just the rate of enrgy storage by the system, there is nowhere else for the heat to go if dE/dt is defined as rate of energy storage inside a system defined as everything below the altitude at which the ERBE tries to measure the balance.

So one does not worry about the rate of heat storage per se; one balances the right hand side:

F – lambda*T

directly against the combined (SW/LW)ERBE data.

What I am trying to say is that the method used gives the correct result; it does not need further corrections for oceanic heat uptake.

This is to be contrasted to cases were ERBE data is not used and the heat balance has to be implied from a thermal model of the land/oceanic/atmospheric system or from OHC and a model of the land/atmospheric system, or some such methods.

The line of argument used is insensitve to the exact nature of the processes behind dE/dt as this value does not appear in the final calculations.

Now I may have this very wrong, but that is how it seems to me. It is I feel what gives the approach additional clarity. All that one needs to consider is whether F – lambda*T correlates strongly with the combined ERBE data, given ones choice of how to observe T.

All that is being required of the model is the function F – lambda*T is a fair account of how the real world behaves, e.g. the real world reaction is linear in both F and T (including no time lag in the response).

Alex

http://www.drroyspencer.com/2009/07/new-study-in-science-magazine-proof-of-positive-cloud-feedback/