Azimuth simple climate model example

A very simple model of Snowball Earth

Let's look at a model described by Gerald R. North in his book Simple Models of Global Climate. This is a 'zero-dimensional energy balance model', meaning that it only involves the average temperature of the earth, the average solar radiation coming in, and the average infrared radiation going out.

The average temperature will be $T$, measured in Celsius. We'll assume the Earth radiates power square meter equal to $$ A + B T $$ where $A = 218$ watts/meter² and $B = 1.90$ watts/meter² per degree Celsius. This is a linear approximation taken from satellite data on our Earth.

We'll assume the Earth absorbs solar energy power per square meter equal to

$$ Q a_p(T) $$

Here $Q = 1370/4$ watts/meter² is called the insolation: it's the average amount of power per square meter hitting the Earth. And here $a_p(T)$ is called the coalbedo: it's the fraction of solar power that gets absorbed. The coalbedo depends on the temperature; we'll have to say how.

Given all this, we get

$$ C \frac{d T}{d t} = - A - B T + Q a_p(T(t)) $$

where $C$ is Earth's heat capacity in joules per degree per square meter. Of course this is a funny thing, because heat energy is stored not only at the surface but also in the air and/or water, and the details vary a lot depending on where we are. But if we consider a uniform planet with dry air and no ocean, we may roughly take $C$ equal to about half the heat capacity at constant pressure of the column of dry air over a square meter, namely 5 million joules per degree Celsius. (Why just half? Gerald North promises to explain.)

The easiest thing to do is find equilibrium solutions, where $\frac{d T}{d t} = 0$, so that

$$ A + B T = Q a_p(T) $$

Now $C$ doesn't matter anymore! We'd like to solve for $T$ as a function of the insolation $Q$, but it's easier to solve for $Q$ as a function of $T$: $$ Q = \frac{ A + B T } {a_p(T) }$$ To go further, we need to guess some forumula for the coalbedo $a_p(T)$. Let's try this: $$ a_p(T) = a_i + \frac{1}{2} (a_f-a_i) (1 + \tanh(\gamma T)) $$ Here $$ a_i = 0.35 $$ is the 'icy' coalbedo, good for low temperatures when there's lots of ice and snow, and $$ a_f = 0.7 $$

is the 'ice-free' coalbedo, good for high temperatures when the Earth is darker. The function $\frac{1}{2}(1 + \tanh(\gamma T)) $ is just some simple function that goes from 0 at low temperatures to 1 at high temperatures. This ensures that the coalbedo is near its icy value $a_i$ at low temperatures, and near its ice-free value $a_f$ at high temperatures. But the fun part here is $\gamma$, a parameter that says how rapidly the coalbedo rises as the Earth gets warmer. Depending on this, we'll get different effects!

Note that $a_p(T)$ rises the most rapidly at $T = 0$, because that's where $\tanh (\gamma T)$ has the biggest slope. We're just lucky that in Celsius, $T = 0$ is the melting point of ice, so this makes a bit of sense!

Now, you can slide this slider to adjust the parameter $\gamma$ to various values between 0 and 1:

$\gamma$ (in 0.01 increments):

You can see how the coalbedo $a_p$ changes as a function of temperature. In this graph the temperature ranges from -50C and 50C; the graph depends on what value of $\gamma$ you choose:

You can also see the insolation $Q$ required to yield a given temperature $T$ between -50C and 50C:

The exciting thing is that when $\gamma$ gets big enough, three different temperatures are compatible with the same amount of insolation! This means the Earth can have a warm and a cold state even when the insolation is fixed. (The intermediate state is unstable, it turns out.)