Claes Johnson's problem formulation

I just discovered that Claes Johnson has already formulated a version of the atmospheric problem in terms of the compressible Euler equations with heat source/sink:

I few notes are in place for those who (like me) are newcomers to this field. I guess it is essential to consider the compressible Euler/N-S equations, that is, you allow a variable density and solve for the primitive variables (density, velocity, temperature) . The first equation expresses conservation of mass, the second conservation of momentum and the third conservation of energy. Note that thermal conduction and kinetic viscosity are absent here, which may be justified since these are small for air. More important is perhaps the heat source/sink q, which CJ assumes adds energy to the lower layers and removes energy from the top layers. 

In other words, how you specify the function q(z,t) more or less defines the model. The questions is therefore: How does CJ define this function?

Combining Heat Transfer and Navier-Stokes equations

In this document is described how one can combine the heat equations including convection with the Navier-Stokes equations:

However, as far as I can see, for our purposes we also need to include some mechanism by which the energy is lost to space. Otherwise we need to invent some fictitious "Top of the Atmosphere" which is precisely what I want to avoid.

I think this could be accomplished by including an energy sink S at all altitudes which ought to be a function of the density, the temperature and not to forget the total mass above that altitude. In other words, the more mass aloft the less heat is lost directly into space.

So there you have it. Now we only need someone who can solve all these equations..

Navier-Stokes ♥ Heat conduction = True ?

In a recent post by Claes Johnson I get the impression that Claes have now started to entertain an idea that I myself have come to regard as the most promising alternative to the Greenhouse gas dogma: Heat conduction (which includes radiation) combined with fluid dynamics, where the heat successively escapes to outer space as the atmosphere gets thinner. 

The way convection is treated in state of the art climate models seems to be rather rudimentary. Quoting Liou again:

"For applications to one-dimensional climate models, the critical lapse rate,.., is usually assumed to be 6.5 K /km for the globally averaged condition. This number is based on the fact that the climatological atmospheric temperature profile in the troposphere has a lapse rate close to this value."

Surely we ought to be able to do better than this. But the question then goes to Claes: Is it straightforward to combine Navier-Stokes equations with heat conduction, in that case how is it done and have you tried it?

The Tropopause Height

Let's have another look at what Roy Spencer writes about the GGH:

6) The tropospheric temperature lapse rate would not exist without the greenhouse effect. While it is true that convective overturning of the atmosphere leads to the observed lapse rate, that convection itself would not exist without the greenhouse effect constantly destabilizing the lapse rate through warming the lower atmosphere and cooling the upper atmosphere. Without the destabilization provided by the greenhouse effect, convective overturning would slow and quite possible cease altogether. The atmosphere would eventually become isothermal, as the full depth of the atmosphere would achieve the same temperature as the surface through thermal conduction; without IR emission, the middle and upper troposphere would have no way to cool itself in the face of this heating. This scenario is entirely theoretical, though, and depends upon the atmosphere absorbing/emitting absolutely no IR energy, which does not happen in the real world.

The first thing worth to say about this is that it is a correct account of the GGH. For some reason skeptics have had a tremendously hard time to understand the significance of this. The lukewarmers have been very anxious to keep this secret for understandable reasons. Why? Elementary, because it provides a test which could potentially lead to a falsification of the theory.

Expressed in other terms, what he says is that if the atmosphere was depleted of greenhouse gases the tropopause would be at the surface. Remember, the tropopause marks the beginning of the stratosphere which is the isothermal layer above the troposphere. 

Now please have a look at the following fact sheet:

Tropopause Values

	T K	P mBar	gm/cm3	H Km	Vsound m/s	rad years	phase °
Venus	250.0	100.000	2.1173e-04	5.307	246.4	0.026	14.83
Earth	217.0	100.000	1.6051e-04	6.358	295.4	0.043	14.96
Mars	140.0	0.010	3.7809e-08	7.070	185.0	0.000	0.01
Jupiter	110.0	140.000	3.3983e-05	16.984	768.8	2.273	50.28
Saturn	85.0	100.000	3.0224e-05	33.083	676.8	9.677	64.10
Uranus	53.0	110.000	5.7414e-05	21.770	515.1	46.340	73.81
Neptune	54.0	200.000	1.0246e-04	17.579	519.9	63.132	67.26
Titan	70.0	100.000	4.9261e-04	14.949	167.7	9.506	89.96

Almost consistently, we see that the tropopause is situated at a pressure of about or slightly above 100 mB. (Mars beeing a notable exception, on the other hand it has a very thin atmosphere so the data might not be that reliable).

What does this mean. Well, the pressure is a direct measure of the amount of atmospheric mass above that level. Hence, it corroborates the assumption that it is the total mass, rather than the amount of GHGs that determine the amount of heat trapping in the atmosphere.