I recently commented for RealClimate on the inapplicability of the simplified model used in Schwartz’s estimate of climate sensitivity. It seems to me that the greatest failure of this model is that it allows only one time scale, or “characteristic time,” for the climate system. But surely the earth has many. The atmosphere reacts quickly to changes in energy input and output; the land heats up a bit more slowly, the ocean more slowly still. Ice and snow grow and diminish. All these systems have their own characteristic time scales, and they all interact to form our climate system.
The model used by Schwartz amounts to this:
where is the temperature, is the heat capacity of the climate system, and is the climate forcing. The quantity is the time scale, and for this model gives the climate sensitivity. We can call this a “1-box” model, because it represents earth’s climate as a single entity with a single temperature, and allows only one time scale. We can make the model more realistic by including more “boxes.” Just to get an idea how multiple “boxes” can affect climate change, I’ll use two: one for the atmosphere/land, another for the oceans.
In this case we’ll have two temperatures: one for each box. We’ll have a separate heat capacity and characteristic time scale for each box, and we can even allow separate climate forcings. We must also allow the two boxes to exchange heat; otherwise we don’t have a 2-box model, we have two 1-box models. We’ll end up with something like this:
Note that all the quantities have subscripts “1″ or “2″ indicating whether they apply to box 1 or 2, except the quantity (the heat exchange coefficient), which is the same for both. It must be, in order not to violate the law of conservation of energy.
These equations can be solved exactly for any forcings . What emerges from the solution is that the system has two time scales, and . The time scales depend on the constants , the heat capacities , and the heat exchange coefficient .
What does it mean to say the system has a “time scale?” Let’s go back, for just a moment, to the 1-box model. Suppose climate forcing has been zero for a long time, and global temperature is zero. Then, let’s raise the forcing by instantaneously doubling atmospheric CO2 and keeping it at that level. Under this circumstance, temperature will evolve according to
From this we see that temperature will rise in response to doubled CO2, and will approach a new equilibrium value with exponential decay. If the new equilibrium value is, say 3 deg.C, and the time scale is yr, the response will look like this:
If, on the other hand, the time scale is yr, it will look like this:
In the first case, with a short time scale, climate adjusts to the new equilibrium value quickly. After 20 years, which is 4 “time scales,” it’s almost reached its new equilibrium value and has only a miniscule amount of warming “still to go.” It’s on this basis Schwartz concludes that if his estimates are correct, almost all of the warming from the greenhouse gases we’ve already emitted has already been felt; there’s almost no warming left “in the pipeline” (warming that still remains even if we stop emitting greenhouse gases entirely). But in the second case 20 years is only 2/3 of a single time scale, and we still have almost half of the warming still in the pipeline. Schartz argues for a short time scale, based on the fact that in the past century climate has responded quickly to forcing changes; with such a short time scale we don’t have to worry about warming still in the pipeline, just about future greenhouse gas emissions.
Now let’s look at the behavior of our 2-box model, and suppose its two time scales are yr and yr. Under not-unrealistic conditions, a sudden doubling of CO2 (with no other changes) might lead to temperature behavior like this:
We can see that the ocean warms slowly, as though it behaved with the longer time scale . But the atmosphere behaves differently; at first it warms quickly, then more slowly. This illustrates how it’s possible to observe what we’ve observed, despite the fact that there’s warming in the pipeline even if we don’t emit any more greenhouse gases. The short time scale for heating the atmosphere makes climate at first respond quickly to changes in forcing, while the long time scale for heating the oceans makes climate institute the full effect of climate forcing much more slowly. The fact that atmospheric temperature responds quickly to climate forcings doesn’t mean that the time scale is short, it only means that one of the time scales is short. But because of the other, longer time scale we can still have substantial warming in the pipeline.
It is the belief of the majority of climate scientists that this is actually the case (not that climate follows a 2-box model, but that it has 2 or more characteristic time scales). The atmosphere responds quickly to climate forcing, but the ocean sluggishly; it takes a long time for heat to penetrate deep into the ocean, so it’s responsible for a second, longer time scale. Because of this, we haven’t yet got even close to the equilibrium response to the climate forcing from the greenhouse gases we’ve already emitted, and there’s quite a bit of warming still in the pipeline — we’ll feel it even if greenhouse gas emissions completely halt today. According to at least one estimate, we still have about 0.6 deg.C warming in the pipeline. That’s if we stop emitting all greenhouse gases instantly.
Some people get the idea that this means we can’t really do anything about global warming. After all, the planet will continue to warm even if we halt all greenhouse gas emissions instantly and completely. This is a false idea; although halting emissions won’t halt global warming, every bit of greenhouse gas emissions leads to more warming. Halting emissions won’t stop the warming that’s already in the pipeline — but that’s even more reason not to put yet more warming in the pipeline.