It follows my earlier post on Venus and lapse rates, and a post from a while ago on the adiabatic heat pump.
Heat transfer mechanismsThis may seem elementary, but in the atmosphere, some aspects of heat transfer are different:
ConductionMolecular conduction carries very minor flux, in accordance with Fourier's Law.
RadiationTransmission of thermal range IR in an atmosphere like Venus is not like transmission in a vacuum. At most wavelengths the absorption length L is fairly small relative to the depth of the atmosphere. However, heat balance requires that absorption is balanced by emission. This gas-to-gas transmission is very dependent on the temperature gradient.
The Rosseland model of IR transmission works for high opacity gas. If you simplify by neglecting scattering, then
F/F0 = (16/3)*(L/*T0) ∇ T
where F0 is BB (black body) emission at ambient T0, F is flux.
This is a Fourier Law with a conductivity that can be quite high. For example, if L is 1 km, T0=400K, the flux is about 0.13 times BB flux.
You can visualise how this transmission works. Imagine yourself with IR vision in such an atmosphere with a lapse rate. The gas below is hotter than the gas above. If you hold out your hand, it's warmer below than above. There is a nett heat flux upward.
How much? Well, it's proportional to the lapse rate, which determines the temperature difference that you see. But it depends on the absorption length too. If that is higher, you can see further, and see hotter (and colder) gas, because of the linear lapse rate. This effect is proportional to L, hence the Rosseland expression.
As L increases further, things change. Looking down, the gas gets denser, so you can't see as far. The hotness increases less than linearly. Looking up, you see further. The coolness increases more than linearly. They cancel somewhat, so changes to the flux are second order in L. But eventually there is a deviation from Fourier's Law. And if you see far enough, the ground will have an effect too.
Eventually, when L is large relative to the depth of the troposphere, a substantial fraction of IR is transmitted without any absorption. This is an atmospheric window, and the relation between transmission and lapse rate diminishes.
But up to that point, Fourier's Law is a good way of thinking about IR transmission.
This also has a wrinkle. On a room scale, turbulent convection is also often thought of as obeying a Fourier Law. Gravity-forced changes of pressure are negligible. The adiabatic lapse rate is about 0.01 C/m.
However, on the atmospheric scale, compression with gravity is important in modifying the temperature. Compression warms. Fourier's Law applies not to the gradient of temperature, but of potential temperature:
where Î½=R/cp and P0 is a reference pressure.
The important thing here is that unlike with conductive and radiative transport, convective transport is zero at the dry adiabat lapse rate, rather than zero gradient, and is proportional to the difference between the actual lapse rate and the adiabat. Below the adiabat, the gas is convectively stable, and energy is used making it go up and down. Above the adiabat, it is unstable, and the temperature gradient adds energy to the motion, and accelerates the transport.
Now that we've outlined the modes, and agreed to ignore conduction, we have a downflux of sunlight on Venus, after allowing for albedo, of about 160 W/m2. Not much of that actually reaches the surface - it is absorbed at various depths. But once absorbed, the heat has to get out again, and the modes available for it to reach the tropopause to be radiated away are IR transfer and convection.
Suppose the convective component were small. The lapse rate would then be determined by a Fourier Law, as given by the Rosseland model. This might be more or less than the adiabat.
If more, then the gas would be convectively unstable. Motions would be induced which would increase convective transport. Since the overall flux is determined by the sunlight, that means the proportion carried by IR would reduce, lowering the lapse rate. This argument, made by DeWitt in the first SoD thread, gives a mechanism whereby the adiabat lapse rate cannot be much exceeded.
But there is an analogous argument from below. If the IR flux caused a temperature gradient less than the adiabat, then the gas would be convectively stable, and it could remain at that. However, there are likely to be other effects inducing motion. Polar regions (and the long nights on Venus) emit more heat than they receive - sunny regions emit less. This heat has to be transported by the atmosphere, and the temperature difference drives a heat engine. The resulting circulation conveys the heat.
But the motion then affects vertical convection. I described earlier how motion pumps heat downward. This then augments the heat that must come up conveyed by IR. This in turn increases the lapse rate towards to adiabat. In doing so, it extracts KE from the air to drive the pump.
Leonard Weinstein's problem
At SoD, Leonard Weinstein proposed a thought experiment where a shell was placed around Venus at about the altitude where IR is currently emitted to space. This is high in the troposphere, where the temperature is about 230K. The shell is opaque to all radiation, but a good conductor. He contended that the temperature profile in the atmosphere would remain much the same. My initial thought was that it would become isothermal, as did others.
I now think a reasonably quantitative analysis is possible. About 160W/m2 sunlight penetrates the atmosphere, and is balanced by an IR and a convective flux. I don't know how much is radiative, but the total is known.
The shell blocks that 160W/m2. If the atmosphere remains the same, then the same IR transport upwards must continue, because it is determined by the unchanged temperature gradient. The convection would also continue, since it is determined by the gradient and the motion. But there's no longer heat supplied, so something has to change. Net upflux must go to zero.
The lapse rate must drop well below the adiabat, to ensure that convective flow is downward, by the heat pumping mechanism. If it does, there will be a balance when the downflux matches the IR upflux, giving the required zero. But how low would the lapse rate go?
Heat pump arithmetic
In this scenario, the heat pump must pump downward about 160 W/m2 than it did before the shell. It has to pump it to depths comparable to where sunlight formerly penetrated. Figures get rough here, but let's say the average depth is where the temperature has risen from 230K to 460K.
Thermodynamics tells us the energy needed to do that. It's 160*(Î´ T)/T, or about 80 W/m2. That comes from the KE in the gas, which must in turn have come from a heat engine somewhere.
But the equator to pole differential, which is a promising source of energy for a heat engine, is much too small. The actual flux of heat redistributed horizontally can only be a small fraction of the 160 W/m2 arriving. And the temperature differential, even with reduced circulation, is never going to match the factor of 2 difference between surface and depth ( without a shell, it's very small).
In fact, there just isn't that sort of energy available anywhere. It's half of total average incoming solar.
So what would happen?
If the convective heat pump can't replace the blocked sunlight, the lapse rate must just reduce until the IR Fourier's Law flux can be matched by the very limited energy available from a realistic heat engine. This needs a rather elaborate but doable calculation which would consider differential insolation over a sphere, and a plausible temperature differential around the shell. The resulting regional discrepancy between sunlight in and IR out would determine the horizontal fluxes, and the temperatures would determine the energy available as KE to the atmosphere. Then some unknown fraction of that could pump heat down, making possible an IR upflux and a positive lapse rate.
So a SWAG? Dunno, but surface temperatures warmer than the shell, but Earth-like rather than Venus-like.