Return of the simplest climate model

The simplest climate model balances the energy leaving the earth to space with the energy coming in from the sun.  If the climate is not changing, these two will be the equal.  As long as climate is not changing rapidly, a modifier we can make quantitative, they'll be very nearly equal.  It turns out that even for fairly rapid climate changes, by standards of geological history, the earth is very close to that balance.

I'm actually going to take a different approach this time around.  Key to deciding how much solar energy comes in to the earth (more precisely, the climate system) is knowing the albedo -- what fraction of incoming energy gets bounced right back out.  That makes the model unsatisfactory to me on a theoretical basis.  We have to know the earth's albedo to compute its blackbody temperature (the temperature which provides that balance).  The problem with that is that the albedo itself is a climate term.  The state of the climate -- how many clouds we have, how large the sea ice pack is, how large the continental ice sheets and deserts are, how green the forests are -- determines the albedo.  Knowing either the blackbody temperature or the albedo is a climate observation.  Given one, we can compute the other from that simple model.  And, which is a good point, we can compare our computed temperature with the observed.

Is it possible to remove or weaken that restriction on the albedo?  And if so, can we learn anything about the climate system?  Yes, and yes.

I'll do something that would be quite improper if I were to claim that it was exactly true, but which will turn out to be extremely educational.  Namely, I will make up a relationship between albedo and earth's blackbody temperature. 
At very low temperatures, albedo will be something like the albedo of snow and ice.  After all, at very low temperatures, the entire earth freezes over.  That's about 0.8 albedo.  At very high temperatures, the earth becomes cloud-covered, rather like Venus actually is.  That also happens to be an albedo of 0.8.  The earth's current albedo, at a blackbody temperature of 255 K, is about 0.3.  What I'll do is take the decrease in earth's albedo to be a bell curve with magnitude 0.5 (to get from 0.8 to 0.3), peak at 255 K (current temperature -- but, since I've got no particular reason to think we're at the peak, I'll make it a variable and come back to it) and a standard deviation of, oh, 5 K (another made up number, so, again, we will come back and see if it matters).  The earth's albedo, then, looks like figure 1:

The equation is albedo = 0.8 - 0.5 exp( - (T-255)^2/2/sdev^2) where sdev is that 5 K standard deviation.  This is all in a spread sheet with the variables named and marked for you to experiment.  (You can also change that 255 to a temperature you're more interested in; it is labelled T0.)

The simplest model is to look for a temperature which makes this equation true:
T^4 = S*(1-albedo)/4/sigma
where S is the solar constant (1367 W/m^2),  albedo is given by the first equation, and sigma is the Stefan-Boltzmann constant = 5.67e-8.

This is known as an implicit equation.  There's no way to get temperature entirely by itself on one side of the equation, the way we normally try to (an explicit equation).  It can still be solved by hunting for a T which makes the equation true.  The method I use in the spreadsheet is to start off with a guess temperature.  Then see what albedo that gives.  Pretend for a moment that albedo doesn't vary with temperature, and see what temperature the second equation (the simplest model) gives.  Keep repeating this process until consecutive temperature and albedo guesses are pretty much the same.  There's one more thing involved, which I'll save for a later post (after you have a chance to experiment).

So, what to guess for our starting point?  Well, if it's nature that is at hand, it shouldn't matter.  Or, if it does, how it matters should tell us something about climate.  So I'll try 288 K, the current global mean surface temperature.  After a bit, the search settles on 255 K as expected (I did, after all, contrive the function to look like the present climate).  But try something colder than 255.  Say 220.  After a while, the iterations settle down on a temperature, but it's 186 K (-87 C, -124 F).  Almost 90 K colder!  Try it yourself -- any first guess 255 K or warmer leads the iterations to an earthly blackbody temperature of 255 K.  Any first guess colder than 255 heads off to an earth temperature of 186 K.

To come next: analyzing the results and sensitivities.