Yesterday I sketched out how I’d constructed a heat flow simulation model of a core extending from the centre of the Earth to its atmosphere at its surface. Today I’d like to explain how I used this model to estimate how far ice sheets were likely to extend from the poles during an ice age.
One of the things that I found with my model was that the air temperature at the surface of the Earth was almost completely dependent on the albedo of the Earth, which is the amount of sunlight that it reflects. Albedo varies from 0 to 1, and when the albedo has a low value, most of the sunlight is absorbed at the surface of the Earth, and serves to warm it up. And when the albedo has a high value, most sunlight is reflected, and the Earth doesn’t warm up much. The albedo of the Earth, as a whole, is about 0.3,
Knowing this, I ran my model to find the air temperature at the surface over a range of latitudes and albedos, and created the table below:
In this table the first column has the solar heat gain in Watts/square metre on the Earth’s surface at different latitudes, with the highest gain at the equator, and the lowest at the pole. The remaining columns give air temperatures at different latitudes for albedos ranging from 0 to 1. I’ve highlighted the column under albedo of 0.3, because these provided my initial air temperatures, with no ice anywhere on the surface of the Earth. Given water freezing point at 273ºK (0ºC), it can be seen that air temperatures are below freezing point above latitudes greater than about 75º N, while they are very hot (nearly 60ºC) at the equator. The initial mean air temperature over the whole planet is 316ºK (43ºC). So given current estimated global mean temperature of 287ºK (14ºC), I start with quite a warm Earth.
The air temperatures in this table are for motionless air above the surface. They are the air temperatures for a planet with no wind. But the Earth’s atmosphere has winds which are made up of moving air masses, usually in spinning high or low pressure systems. And these spinning masses of air act to exchange air between latitudes, with cold northern air being brought down to lower latitudes, and warm southern air brought up. And it seems that in the northern hemisphere, the wind speeds are at present lowest at the poles and equator, and highest at about 45º N.
So I now wrote a new and very simple simulation model which took air at the motionless air temperatures of each latitude, and exchanged some of it with air at higher and lower latitudes. This created a new set of air temperatures at each latitude, made up of mixed air.
And I assumed that where these new air temperatures rose above water freezing point at 273ºK (0ºC), any precipitation would fall as rain, and otherwise as snow. And so in those latitudes where it was raining, the albedo of the Earth’s surface would remain at 0.3, and where it was snowing, the albedo would jump to 0.9. And using my table, new air temperatures at each latitude could then be found.
So starting with an ice-free Earth, I then went through a process of air mixing to find a new set of albedos and air temperatures, and then repeated the process over and over again, so that air temperatures and albedos kept changing.
And the very first thing that my new simulation model did, when it found that air temperatures above latitude 75º N were below freezing point, was to cover them in snow, and so sharply reduce the air temperatures above them. And at 75º N a change of albedo from 0.3 to 0.9 entailed a fall in air temperature from 270.7ºK to 166.8ºK – a fall of over 100ºK. And because the adjacent ice-free latitude had a temperature only slightly above water freezing point, the two adjacent latitudes only needed to exchange a little air for the lower latitude’s air temperatures to be pushed below freezing point, while the higher latitude would remain well below freezing point.
So the predictable result of introducing air mixing was that the ice-covered areas began to extend southwards (and northward in the southern hemisphere, which I assumed to be its mirror image). There was a domino effect, whereby as one latitude fell below freezing point and became covered in snow, the adjacent lower ice-free latitude would rapidly follow it.
The only question remaining was: how far would it get, given different peak wind speeds and different amounts of air mixing? Here are the results I got, with peak wind speeds of 10 m/s, which would allow winds to cross up to 7 degrees of latitude in a single day:
degree of mixing…. 1.00 2.00 3.00 4.00 5.00
lowest ice latitude.. 56.0 40.o 30.0 26.0 34.0
So with a low degree of air mixing, the ice cap extended from the pole to latitude 56º. And with higher degrees of air mixing, the ice cap extended as far as latitude 26º, although with yet higher amounts of mixing its maximum extent was at higher latitudes. And in fact, if there had been perfect mixing of air over the entire planet, given an initial mean planet temperature of 316ºK, there would have been no ice anywhere, not even at the poles. As it was, pretty much the only thing that prevented the whole planet freezing were the low air speeds at the equator, which reduced air mixing there.
The maximum extent of glaciation was as far as latitude 26º. And given an initial ice extent to 74º (roughly what the Earth is right now), the change in the Earth’s appearance is dramatic (In the image at right, blue areas are ice-free latitudes, not water).
Now it’s currently believed that the furthest southern extent of glaciation was to about 51º N in the UK. And in my model this would correspond to a low degree of mixing: about 1.3. And perhaps during the last ice age, winds were indeed fairly light, and there wasn’t much air mixing between adjacent latitudes.
But could the Earth have been covered in ice from the poles all the way to the tropics of Cancer and Capricorn? My model says that it could easily have been, given sufficiently high winds to move large masses of air between latitudes.
So tomorrow I’ll consider such a world. It’s a rather strange one.