Part 1: a Very Simple Model of Temperature Variability around an Equilibrium State
A reader recently linked to a presentation by Roy Spencer called Feedback vs. Chaotic Radiative Forcing: “Smoking Gun” Evidence for an Insensitive Climate System? As time goes by, I have less and less inclination to debunk claims that global warming is “no problem,” especially as most of them are so amateurish that there’s little or no insight about the real climate system to be gained from their demolition, and many of them smack of deliberate deception, i.e., not only are they misleading they’re intentionally so. But this presentation strikes me as different. First, we can learn some things about climate change from studying this dissertation, and second, I get the impression that Spencer really believes what he’s saying.
So let’s take a close look at what Spencer has to say. We’ll start by examining the behavior of his “Very Simple Model of Temperature Variability around an Equilibrium State.” The model is this:
In this equation, T is the surface temperature anomaly (it’s difference from some “reference value”), t is the time, F is the climate forcing anomaly, is what Spencer calls the “feedback,” and is the specific heat of the climate system. The temperature anomaly T and climate forcing anomaly F are functions of time, but the other quantities are constants. We’ve seen this simple model before; it’s the oft-discussed zero-dimensional one-component climate model.
First it must be mentioned that what Spencer calls “feedback” is not what’s often meant by feedback. The most common use of the word refers to the response of the climate system above and beyond what it would be if nothing else changed in response to temperature change. This “no-feedback” response is rather well known; if extra energy comes in to the climate system (climate forcing), then the climate warms in response, and from basic physics when the climate (or anything) warms, it emits more infrared (long-wave, or LW) radiation. When it has warmed enough that the outgoing LW energy balances the incoming (both LW and short-wave, or SW) energy, we’ve reached a new equilibrium (energy in = energy out), so temperature will be stable at the new, warmer value. The necessary temperature change for a given climate forcing can be computed from the Stephan-Boltzmann equation, and turns out to be just about 0.3 K/(W/m^2) (0.3 Kelvin for every W/m^2 of climate forcing). This is the no-feedback climate sentivity.
But the climate system does change in response to temperature change. For example, warmer air holds more water vapor. Not only is water vapor a greenhouse gas itself, greater (absolute) humidity alters the lapse rate of the atmosphere, raising the surface temperature even higher. Also, as the earth warms there’s less ice and snow covering the planet, which changes earth’s albedo (reflectivity to incoming solar energy). As albedo declines, more of the incoming solar power is absorbed into the climate system, so again the temperature rises even further. These are some of the classic feedbacks in climate, which make climate sensitivity even higher.
Spencer is using the word “feedback” in a different sense. When earth warms and hence emits more LW radiation to space, this can be called a feedback too (but in a different sense of the word); it’s the “temperature feedback,” i.e., the change in the radiation budget due to a change in temperature. It’s a perfectly legitimate use of the word, but keep in mind that it’s not the same as the more common meaning which refers to the change above and beyond what it would be with no impact due to changes in water vapor, lapse rate, albedo, clouds, or other factors. For a discussion of climate using the word feedback in the sense Spencer uses, see e.g. Soden and Held 2006, J. Climate, 19, 3354-3360; for a discussion of the more common use of the word see this and this.
But back to Spencer’s simple model. For temperature to be stable, we must have , so
, or .
If we make F constant (i.e., climate forcing doesn’t change with time), then equilibrium occurs at the new temperature given by this equation. This tells us the equilibrium temperature change due to a given change in forcing, which tells us the climate sensitivity. So in this model climate sensitivity is .
The simple model can be solved exactly. I’ll define some new variables (just for convenience):
which we can call the “scaled” forcing function, and
Then the simple model becomes
Then the solution is:
Suppose, for instance, that temperature has been stable at its reference value, so , when climate forcing is at its reference value so . At time , let’s suddenly change the forcing to a new value . Then the value of changes from 0 to . The temperature evolves according to
We see that temperature approaches its new equilibrium value, but with exponential decay; the temperature change doesn’t happen instantaneously because it takes time to accumulate the extra energy to warm the climate system.
I’d like to point out an interesting propery of this simple model. Consider the temperature at time , where the time “difference” is very very small, and supposing that the temperature at time zero is . We see that
Because we insist that is so small, we can be sure that the quantity doesn’t show any significant change in the interval from to . Hence we can approximate it (with arbitrary accuracy by making go to zero) by its value at time . This enables us to treat it as a constant under the integral, so we get
Those of you familiar with statistics may recognize this. Generating new values of by repeatedly applying this equation is simply the process of exponential smoothing. In exponential smoothing, each new data value generates a new smoothed value which is computed from the new data value, and the preceding smoothed value by
Treating the forcing term as the “data” , the temperature anomaly as the smoothed values , and equating the term with the term , our approximate formulation of the solution of the model equation is equivalent to exponential smoothing. So, if we choose the “time step” small enough, we see that solving the simple model is equivalent to computing the exponential smooth of the scaled forcing function. In fact that’s why I’ve expounded at such length about this simple model: to show that its solution is equivalent to an exponential smooth with very short time step.
Both the exact solution of the model, and its approximation as an exponential smooth, are characterized by a “time constant,” which is given by . For this first illustrations of the behavior of the model, Spencer uses feedback parameter W/m^2/K, and he states that the specific heat of the climate system is that of a 50-meter deep “swamp ocean.” I don’t know what numbers he’s using, but as near as I can compute the specific heat for a 50-m deep ocean is about 6.6 W-yr/m^2/K. This gives a time coefficient per year, or a characteristic time scale of yr. As far as climate is concerned, that’s way too small a time scale; the actual characteristic time for the climate system is more like 30 years. Later in his presentation, Spencer uses a 1000 m deep ocean, which gives a characteristic time scale of about 32 yr, much more realistic.
The characteristic time which is chosen for the model has profound impact on the behavior of the model in response to various forcing functions. But we’ll see much more about that when we look at how Spencer uses this model, in the next post.