 Okay, so what happened here? We found that this system has a very interesting and non-intuitive property that for a given value of the governing parameters, for example, for a solar constant of 1370 watts per meter squared, Earth's temperature can potentially have two very different values. It doesn't have a unique value. You can either have a value similar to what we see today, or a temperature below 30 degrees below zero Celsius for the current day solar constant. And what determines which of those two temperatures it's likely to have is the prior history. The properties of the system depend on its prior history. If we had started out with a frozen Earth and increased the solar constant to its current day value, we would still have a frozen Earth. And that's because the ice, once it accumulates and increases the albedo, which is reflecting away most of the solar radiation, it's fairly difficult to get rid of that ice. And only when you turn up the solar constant to a very high value are you able to melt away the ice, suddenly lower Earth's albedo from the albedo of an essentially frozen planet, and Earth's temperature can warm quite a bit. On the other hand, if you start out with a very warm Earth and slowly lower the solar constant, you're not going to form ice until Earth's average temperature gets well below zero degrees Celsius, and the various latitude zones now start dropping below that critical threshold temperature of minus 10 degrees Celsius, which is the temperature we specified represents when a latitude zone becomes ice covered. And then at that point, of course, the albedo increases dramatically, and Earth's surface temperature cools rapidly. So the temperature of the Earth depends not only on the value of the governing parameters, in this case the solar constant, but the prior history of the system. That is a property that's known as hysteresis, when the properties of a system depend on its prior history, not just on the governing parameters. It is an intrinsically nonlinear property. It's a property of complex systems which exhibit nonlinear behavior, and in particular in this case, by stable behavior that for a given value of the parameter, there are two stable points in terms of Earth's surface temperature. And under some circumstances in systems like this, it's possible to undergo transitions rapidly between these two stable states. That is one theory for explaining the dramatic changes in the meridional overturning circulation of the North Atlantic Ocean that has taken place in the past. And we've talked a little bit about that in the past. We'll talk a little bit more about that in the future. This is an example of a nonlinear, and in this case, a by stable system. And in general, as our models become more complex, in this case, we've generalized the model from zero dimensions to one dimension. We've allowed for the possibility of a temperature dependent albedo. As soon as we start to make our models more complex, more detailed, more sophisticated, we introduce the possibility of very unusual, potentially unusual behavior of the system. This sort of nonlinear behavior. And this is something that we'll talk quite a bit more about in this course. It's relevant to the issue of thresholds and the possibility that as we continue Earth's climate, we will pass certain critical thresholds where the climate system will undergo rapid transitions from its current state to some new state. Finally, it's worth noting that this original model constructed by a Russian scientist named Budiko decades ago in the mid-20th century represented a real conundrum for climate scientists because it suggested that once Earth goes into a frozen ice-covered state, which it has in the past, it is very difficult to get out of that state. Even at current values of the solar constant, the Earth cannot come out of this frozen state. The solution, it turns out, to this problem has to do with carbon cycle feedbacks and what happens with this climate, carbon cycle when this happens. We'll talk more about that later on in the course.