## Mathematics of the Environment (Part 6)

Last time we saw a ‘bistable’ climate model, where the temperatures compatible with a given amount of sunshine can form an S-shaped curve like this:

The horizontal axis is insolation, the vertical is temperature. Between the green and the red lines the Earth can have 3 temperatures compatible with a given insolation. For example, the black vertical line intersects the S-shaped curve in three points. So we get three possible solutions: a hot Earth, a cold Earth, and an intermediate Earth.

But last time I claimed the intermediate Earth was unstable, so there are just two stable solutions. So, we say this model is bistable. This is like a simple light switch, which has two stable positions but also an intermediate unstable position halfway in between.

(Have you ever enjoyed putting a light switch into this intermediate position? If not, you must not be a physicist.)

Why is the intermediate equilibrium unstable? It seems plausible from the light switch example, but to be sure, we need to go back and study the original equation:

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

We need see what happens when we push $T$ slightly away from one of its equilibrium values. We could do this analytically or numerically.

Luckily, Allan Erskine has made a wonderful program that lets us study it numerically: check it out!

Here’s you’ll see a bunch of graphs of temperature as a function of time, $T(t).$ To spice things up, Allan has made the insolation a function of time, which starts out big for some interval $[0,\tau]$ and then drops to its usual value $Q = 341.5.$ So, these graphs are solutions of

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

where $Q(t)$ is a step function with

$Q(t) = \left\{ \begin{array}{ccl} Q + X & \mathrm{for} & 0 \le t \le \tau \\ Q & \mathrm{for} & t > \tau \end{array} \right.$

The different graphs show solutions with different initial conditions, ranging from hot to cold. Using sliders on the bottom, you can adjust:

• the coalbedo transition rate $\gamma,$

• the amount $X$ of extra insolation,

• the time $\tau$ at which the extra insolation ends.

I urge you to start by setting $\tau$ to its maximum value. That will make the insolation be constant as a function of time. Then you if $\gamma$ and $X$ are big enough, you’ll get bistability. For example:

I get this with $\gamma$ about 0.08, $X$ about 28.5. You can see a hot stable equilibrium, a cold one, and a solution that hesitates between the two for quite a while before going up to the hot one. This intermediate solution must be starting out very slightly above the unstable equilibrium.

When $X$ is zero, there’s only one equilibrium solution: the cold Earth.

I can’t make $X$ so big that the hot Earth is the only equilibrium, but it’s possible according to our model: I’ll need to change the software a bit to let us make the insolation bigger.

All sorts of more interesting things happen when we move $\tau$ down from its maximum value. I hope you play with the parameters and see what happens. But essentially, what happens is that the hot Earth is only stable before $t = \tau,$ since we need the extra insolation to make that happen. After that, the Earth is fated to go to a cold state.

Needless to say, these results should not be trusted when it comes to the actual climate of our actual planet! More about that later.

We can also check the bistability in a more analytical way. We get an equilibrium solution of

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

whenever we find a number $T$ obeying this equation:

$- A - B T + Q c(T) = 0$

We can show that for certain values of $\gamma$ and $Q,$ we get solutions for three different temperatures $T.$ It’s easy to see that $- A - B T + Q c(T)$ is positive for very small $T$: if the Earth were extremely cold, the Sun would warm it up. Similarly, this quantity is negative for very large $T$: the Earth would cool down if it were very hot. So, the reason

$- A - B T + Q c(T) = 0$

has three solutions is that it starts out positive, then goes down below zero, then goes up above zero, and then goes down below zero again. So, for the intermediate point at which it’s zero, we have

$\displaystyle{\frac{d}{dT}( -A - B T + Q c(T)) > 0 }$

That means that if it starts out slightly warmer than this value of $T,$ the temperature will increase—so this solution is unstable. For the hot and cold solutions, we get

$\displaystyle{ \frac{d}{dT}(-A - B T + Q c(T)) < 0 }$

so these equilibria are stable.

### A moral

What morals can we extract from this model?

As far as climate science goes, one moral is that it pays to spend some time making sure we understand simple models before we dive into more complicated ones. Right now we’re looking at a very simple one, but we’re already seeing some interesting phenomena. The kind of model we’re looking at now is called a Budyko-Sellers model. These have been studied since the late 1960’s:

• M. I. Budyko, On the origin of glacial epochs (in Russian), Meteor. Gidrol. 2 (1968), 3-8.

• M. I. Budyko, The effect of solar radiation variations on the climate of the earth, Tellus 21 (1969), 611-619.

• William D. Sellers, A global climatic model based on the energy balance of the earth-atmosphere system, J. Appl. Meteor. 8 (1969), 392-400.

• Carl Crafoord and Erland Källén, A note on the condition for existence of more than one steady state solution in Budyko-Sellers type models, J. Atmos. Sci. 35 (1978), 1123-1125.

• Gerald R. North, David Pollard and Bruce Wielicki, Variational formulation of Budyko-Sellers climate models, J. Atmos. Sci. 36 (1979), 255-259.

I should talk more about some slightly more complex models someday.

It also pays to compare our models to reality! For example, the graphs we’ve seen show some remarkably hot and cold temperatures for the Earth. That’s a bit unnerving. Let’s investigate. Suppose we set $\gamma = 0$ on our slider. Then the coalbedo of the Earth becomes independent of temperature: it’s 0.525, halfway between its icy and ice-free values. Then, when the insolation takes its actual value of 342.5 watts per square meter, the model says the Earth’s temperature is very chilly: about -20 °C!

Does that mean the model is fundamentally flawed? Maybe not! After all, it’s based on very light-colored Earth. Suppose we use the actual albedo of the Earth. Of course that’s hard to define, much less determine. But let’s just look up some average value of the Earth’s albedo: supposedly it’s about 0.3. That gives a coalbedo of $c = 0.7.$ If we plug that in our formula:

$\displaystyle{ Q = \frac{ A + B T } {c} }$

we get 11 °C. That’s not too far from the Earth’s actual average temperature, namely about 15 °C. So the chilly temperature of -20 °C seems to come from an Earth that’s a lot lighter in color than ours.

Our model includes the greenhouse effect, since the coeficients $A$ and $B$ were determined by satellite measurements of how much radiation actually escapes the Earth’s atmosphere and shoots out into space. As a further check to our model, we can look at an even simpler zero-dimensional energy balance model: a completely black Earth with no greenhouse effect. We discussed that earlier.

As he explains, this model gives the Earth a temperature of 6 °C. He also shows that in this model, lowering the albedo to a realistic value of 0.3 lowers the temperature to a chilly -18 ° C. To get from that to something like our Earth, we must take the greenhouse effect into account.

This sort of fiddling around is the sort of thing we must do to study the flaws and virtues of a climate model. Of course, any realistic climate model is vastly more sophisticated than the little toy we’ve been looking at, so the ‘fiddling around’ must also be more sophisticated. With a more sophisticated model, we can also be more demanding. For example, when I said 11 °C is “is not too far from the Earth’s actual average temperature, namely about 15 °C”, I was being very blasé about what’s actually a big discrepancy. I only took that attitude because the calculations we’re doing now are very preliminary.

### 4 Responses to Mathematics of the Environment (Part 6)

1. Brilliant stuff as usual, John. I hadn’t run across the B-S model before; it makes for an interesting acronym :).

2. Last time we saw how the ice albedo effect could, in theory, make the Earth’s climate have two stable states. In a very simple model, we saw that a hot Earth might stay hot since it’s dark and absorbs lots of sunlight, while a cold Earth might stay cold—since it’s icy and white and reflects most of the sunlight.

3. In Part 5 and Part 6, we studied a model where the Earth can be bistable. [...]

4. [...] Part 6 – A model showing bistability of the Earth’s climate due to the ice albedo effect: dynamics. [...]