The time scale is quicker than I had expected, this is because the heat capacity C is quite low. In particular, the latent heat of fusion of a 1cm layer of ice appears to be of the same order of magnitude as C… one might try to incorporate this in some way, e.g. with an “ice quantity” variable driving c, and the “ice quantity” evolving according to temperature.

In the homepage of Kerry Emanuel (http://eaps4.mit.edu/faculty/Emanuel/) there are a lot of things. In particular there are presentations of a lot of courses. In one of them (Tropical meteorology) there is a model for the equilibrium radiation. The simplest model has two layers, the atmosphere and the surface. The atmosphere is completely transparent to visible light and completely opaque to infrared. Let be the temperature of equilibrium (250 K), then the equilibrium is achieved at at in the atmosphere and in the surface.

A 3 layer model predicts in the outer part and $3^{\frac{1}{4}} T_e$ in the surface. This is way to hot and the explanation is partially because it is not completely opaque and because of convection. Something else should be wrong because if we do the same thing with n layers, we would get a surface temperature of But it is funny that the outer part is always the equilibrium temperature. Finally there is a complete calculation, that unfortunately is not clear how it is done, that gives a vertical profile of the temperature if only radiation is taken into account. The profile is very hot in the surface, again the explanation is the lack of the convection in the model.

To see the stability or instability of a particular equilibrium is convenient to look at the graph of .

Thanks for explaining this! I hope all the students in my class read your explanation, which is better than mine.

I have a couple of questions and comments. About the values of and , the value of is the emission of a body at -23, very close to the radiate equilibrium of -18, I guess this is not a coincidence. Or is it? The derivative of at this value is 3.5, B is in this order of magnitude, but not as close one could imagine. Is there some reason for that?

Good question! It would be nice to understand this in detail.

The main reason the observed infrared power emission of the Earth doesn’t match the power emission of a blackbody is the greenhouse effect. Quite a bit of infrared doesn’t make it through the atmosphere. So, the simplest thing we can say is that we should have

for temperatures in the range we see on Earth. But it would be nice to predict the (linearized) observed average power emission per square meter, starting from a simple model.

Could Simpson’s model in Part 3 be good enough to approximately calculate and ? Or is it too simple?

In its simplest form, this model amounts to saying that the fraction of infrared radiation that escapes to space is a function

of the wavelength , given as follows. To a crude approximation, the atmosphere is:

• completely opaque from 5.5 to 7 micrometers (due to water vapor),

• partly transparent from 7 to 8.5 micrometers (interpolating between opaque and transparent),

• completely transparent from 8.5 to 11 micrometers,

• partly transparent from 11 to 14 micrometers (interpolating between transparent and opaque),

• completely opaque above 14 micrometers (due to carbon dioxide and water vapor).

Simpson’s function is 0 where the atmosphere is ‘completely opaque’, 1 where it’s ‘completely transparent’, and it linearly interpolates between these values for wavelengths where the atmosphere is ‘partly transparent’.

One could multiply the Planck distribution at temperature by this function of wavelength, then integrate it over wavelengths, to estimate the power emitted per square meter as a function of .

This neglects many effects like ‘water vapor feedback’ (there’s more water vapor when it’s hotter), ‘lapse rate feedback’ and ‘cloud feedback’, as mentioned in this post. But it would still be worth doing!

I have a couple of questions and comments. About the values of and , the value of is the emission of a body at -23, very close to the radiate equilibrium of -18, I guess this is not a coincidence. Or is it? The derivative of at this value is 3.5, B is in this order of magnitude, but not as close one could imagine. Is there some reason for that?

To see the stability or instability of a particular equilibrium is convenient to look at the graph of . In this case . In particular the zeros give the equilibrium points. For large negative values of , the function is positive, and for large positive temperatures the function is negative so there is a zero somewhere, an equilibrium. Almost surely there is an odd number of them: The first time you cross the zero (the coldest equilibrium) you cross it downwards, and if you go up and reach zero you will cross it upwards, afterwards you will have to come down again. So we have either one or three, the only exception is if the zero is double, meaning that the derivative is also zero. The equilibrium points where the derivative is negative, there is local negative feedback effect: if your are slightly above the equilibrium you tend to go down if you are slightly down you tend to go up, so the point is stable, in the points where you have positive derivative -when you cross upwards- there is a positive feedback: if you are a little bit up you tend to go upwards and if you are a little bit down you tend to downwards, hence unstable. So if you have three points, the coldest and the hottest are stable and the middle one is unstable.