## Climate as A Differential Equations Problem

Cross posted at The Air Vent.

I find that when discussing problems of what are ostensibly physical systems, little progress can be made until people formulate their ideas in coherent mathematical terms. At that point there can be no arguing, a statement is either mathematically correct or it isn’t, and it will be completely unambiguous. The only ambiguous questions should be the values of various coefficients and constants, as long as we are sure of equations that characterize the problem. However, much confusion in discussions of climate comes about because the equations that are thought to characterize the problem are rarely stated, and even when they are, they are difficult for most people to understand, partly due to a common lack of numeracy at the necessary level (in this case, understanding of at least some calculus would be helpful, perhaps necessary) and also the fact that no attempt is made to explain the underlying equations to the laymen (I honestly know of zero counter examples). I am going to attempt to explain what I presently understand to be the equations which characterize the climate problem as it is commonly thought about, and try my best explain what the equations mean.

Let’s start off with the equation that relates a function that causes the system to change to the parameter that it changes. First, let’s start off with the common simplifications: we characterize the climate by a single function, the global mean temperature (the natural asymptote, the “equilibrium” is made the zero point here), T(t), the forcing function was the global mean top of the atmosphere radiation flux change caused by some factor, f(t). Well, what is the math that relates these? Let’s start by eliminating obviously wrong relations. To begin with, the equation that is appropriate is not:

Equation (1)

Where lambda is the “sensitivity” that converts forcing to the temperature change. It is not hard to see the problem with this simple equation: it assumes that when there is some force that acts to bring the Earth out of thermal equilibrium (more so than usual!) the Earth would instantaneously get into equilibrium. We know from our daily experience that this can’t be correct: the peak daily temperatures occur an hour or two after local solar noon. Moreover, the seasons lag behind the solstices significantly, depending on which season, the latitude, and whether the location is continental or on/near the ocean: the lag for mid-latitude continental climates averages about a month. To be fair, these examples are not perfectly analogous to our simplified picture of a single global temperature parameter determined by a single global mean forcing, and also, the individual locations experience energy and even mass flow of the atmosphere from other places, not just a flux to space. Even so, it is obvious from other examples that bodies do not warm or cool instantaneously in reaction increased energy input. Consider a cool room, now place a space heater in it, and turn it on. Now, the device itself does not reach its full power instantly, even so, it takes much longer for the temperature of the room to heat up most of the way. So it is obvious that equation (1) cannot be correct. Now, the kind of system we are describing can be modeled as something called a Linear Time Invariant system. Now, for our purposes, getting into what that term means, instead, let’s look at what a first order equation of this type typically looks like:

Equation (2)

Equation (2) is a generic, first order LTI system. What it means is that if you add the rate of change to a constant multiple (inverse tau) of the value of something relative to it equilibrium value, the result is the input function to the system that is causing the change. In the case of a system which begins at a state F0 different from its natural equilibrium, and receives zero input, the solution to Equation (2) is:

Equation (3)

Now what that equation means is that the system, absent an input, will tend to approach equilibrium with exponential decay. Also, this means that unless the initial value happened to be the equilibrium value, climate would be an initial value problem. That initial values are expected to decay to a negligible amount after thirty years perhaps gives us a clue as to the commonly assumed response time of the system. Tau is the “response time” of the system, or the time constant, and represents the time t at which the system has reached e-1 of the initial value, or about 37%. Many electrical engineers must by now (and probably earlier) recognize the kind of equation we are talking about, it is of the same form as equations that describe simple RL and RC circuits. Specifically, tau is the ratio of the inductance to the resistance in the former, and the product of the resistance and the capacitance in the latter. This makes for a good jumping off point for talking about the “doubling CO2” input function to the climate equations, which is how I will derive the specific function I understand to characterize current understanding of climate. The step response of and RC circuit looks like this. Basically, the value that will eventually be taken, after infinite time following a step input, is a tau times the constant A by which the Heaviside function u(t) was multiplied in the input. In other words, the solution to equation (2) when g is an arbitrary instantaneous step input with initial value set to zero is:

Equation (4)

So the new equilibrium value is A times tau. Now, in the case of climate, the common hypothetical is the step function from a doubling of CO2, which is said to have a value of about 3.7 Watts per square meter (the value isn’t really important to our discussion just yet) and the equilibrium response is that value multiplied by lambda, the “sensitivity.” Now, in this particular case we get these equations:

Equation (5)

Equation (6)

Equation (6) must be generalized for situations where the forcing function is arbitrary, so we must isolate the constants that characterize the system (lambda and tau) from the CO2 doubling function (3.7 times the Heaviside function). The result:

Equation (7)

Equation (7) is at last the form of the climate equation, or what seems to be about the form usually used. But what does this equation mean? Well, for smaller response times for a given sensitivity, the Temperature more closely resembles the time evolution of the forcing, for longer response times the forcing function more closely resembles the time derivative of the temperature.

What might we say about the response time of the climate system? Well, climate usually is defined as a thirty year average, presumably because scientists expect there to be very little remaining of decaying initial values per equation (3). If negligible contribution from those decaying values is 10%, then that happens in about 2.3 time constants, so the thirty year average implies an assumed time constant of about 13 years. If a negligible contribution is 25% it happens after about 1.4 time constants and the implied time constant assumed is a little less than 22 years. So we see that the thirty year average implies climate scientists think the time constant is on the order of several years. This doesn’t mean they are right, and what level of averaging is necessary depends on the amount of attenuation desired as well as the response time (also note that, contrary to what is commonly asserted, the unforced solution of (7) does not directly depend on sensitivity, since it can be canceled out when f(t) is zero, thought it does depend the response time, in terms of its time scale, but not it’s magnitude).

Can we estimate the response time of the system? Only if we know the climate forcing, the climate evolution, and either the sensitivity or some relationship between the sensitivity and the response time. But we can look at some lines of evidence that might imply certain things about the response time. Consider the ice core data: in them, the temperature appears to vary in proportion to the greenhouse gas forcing, which would imply that, relative to the time scale of the glaciation cycles, the response time is short. Only one problem with this line of reasoning: We know that the earliest temperature changes preceded the CO2 changes, implying a positive CO2 feedback (and we shall discuss the issue of feedback in a bit) and it is the CO2 which has a short response time to temperature relative to length of a glaciation cycle. What about the forcing that actually causes the change to begin with? It turns out that his implies the exact opposite! According to Roe (2006), the changes in insolation track the rate of change of climate. This result would seem to require a very long response time. However, as with the seasons example, Milankovitch climate forcing is something which cannot fit into our simplified model of the climate system, as it is not a globally averaged top of the atmosphere flux change: in fact, the global insolation change is essentially tiny or nonexistent compared to the change in July insolation near the Arctic Circle that Roe correlates with the rate of change in ice volume. So this example cannot tell us what kind of response time to expect from a global forcing like CO2 (although later we will discuss some reasons why this response time may be longer than the one with which we are concerned anyway). Likewise, the seasons cannot establish a very short response time. But there is an example of a global mean flux change forcing which the nature of the response to which may imply something useful about the response time. Lindzen (1995) found that the response time has an interesting effect on the response of the climate system to relatively close in time volcanic eruptions: if the response time is long, the cooling impact of closely spaced together in time volcanic eruptions will build on one another, leading to long term cooling absent a significant warming forcing to offset this (a finding built on later by Lindzen and Gianitsis (1998) in a context of the climate’s sensitivity) in particular the response time which lead to accumulating cooling was sixteen years or longer by Lindzen’s calculations. Note that, if climate scientists really do believe the time constants I think are implied by thirty year average representing forced climate, then values close to this threshold but slightly above or below are considered reasonable. But at least according to Lindzen’s calculations, the assumed response times are generally significantly longer than 16 years (which would implied much less attenuation of unforced variability at thirty years) and thus would imply volcanic cooling building on volcanic cooling that is incompatible with the temperature record. Of course, it could be that the temperature record back then is not good enough to capture the real variation in climate associated with those volcanoes, or it could be that Lindzen’s calculation of the threshold at which volcanic cooling build up is an underestimate. It could even be the case that a large warming forcing canceled the cooling, although both solar variability and greenhouse gases seem inadequate. Still, very long response times seem unreasonable based on the available evidence-middle of the road estimates might be about right, not as low as say, Willis Eschenbach recently estimated at WUWT (months or less) but not as high as models assume (decades or even centuries, evidently).

Now to the question of including feedback in this consideration. At my own blog, I have discussed the equations used to estimate lambda from considering changes in the radiation flux with temperature (and also estimated the relationship between changes in flux and temperature). Let’s consider those equations for a moment:

Equation (8)

Equation (9)

Now, in my original posts, I did not really discuss the nature of these equations. For instance, why does equation (8) look like it does? Well, the reason is because it is describing a process which essentially acts behaves like a Geometric series. To see why this is the case, let’s consider a feedback process at work: Let’s say I add a unit value to the system, and the system responds to that change by adding half of that to the value over again, to which it responds again with half of that. This particular geometric sum is convergent, meaning it approaches value less than infinity as the number of times the process repeats approaches infinity. That particular sum is equal to two. It turns out there is a general formula for feedbacks less than adding the full value added again, and even negative feedbacks. That formula is:

Equation (10)

Now, equation (10) is interesting, since it carries out an infinite sum. The like appears in many areas, for example, in economics (please try to contain yourselves) it is the form of the infamous Keynesian “multiplier”, which also involves an infinitely repeated process (of partial spending and partial saving of marginal income). Now, I believe the use of such formulas in economics acts more to obscure than it does to elucidate, and the Keynesian models are a load of nonsense. But back to climate, where this formula is acceptable because we are dealing with a physical system. It is crucial, for this formula to be useful, that the timescale associated with the feedback processes be very short compared to the system response time. The climate feedbacks we are generally interested in involve cloud and water vapor processes that are very fast and probably are a lot shorter than reasonable estimates of the response time we are interested in. But recall when I said of the Milankovitch response time “we will discuss some reasons why this response time may be longer than the one with which we are concerned anyway”—well, now I intend to. It is obvious that the feedbacks at the glaciation cycle timescale are much longer than the reasonable estimates of the response time: the CO2 feedback and the formation and melting of ice age continental ice sheets are slow process, taking hundreds or thousands of years. For this reason both the sensitivity and response time to Milankovitch forcing appear to be very different from what we might reasonably expect from a doubling of CO2 and so that timescale being looked at for clues to the sensitivity and response time of present interest, will be extremely misleading. Finally, note the nature of the functional form implies that feedback factors close to one imply very large sensitivities, and a feedback factor of one corresponds to infinity sensitivity (greater than one, bizarrely, leads to a system with negative sensitivity, but such values are obviously unphysical). This means that as long as f is positive, slight variations in its value lead to large variations in sensitivity, which is why the range of sensitivities typically given is usually quite large, but if the feedback is negative, even large relative errors in the size of the value of f lead to a small uncertainty band about a low sensitivity.

Finally, can we relate the climate sensitivity to the response time? That is, might it be possible to reduce this problem from one of identifying the value of two parameters to the value of a single parameter? Actually, it turns out there may be a relationship between the two. According to Lindzen (1995) the response time is in fact basically proportional to the gain (or one over one minus f) of the system (following Hansen et al (1985) so this rough relationship for values of tau that are plausible is not controversial, although the exact proportion may not be so). His proportionality implies that the typically assumed sensitivity range implies response times of 80 years or more. This is much longer than I estimated from the attenuation of natural variability by the thirty year average climate. Can climate scientists really believe in such long response times? But the feedback necessary to double the no feedback sensitivity apparently requires such a long response time, and that merely constitutes the “low end” of the IPCC’s sensitivities. Yet perhaps this proportionality is not right. Schwartz (2007) originally claimed a response time of about five years and a sensitivity approximately equal to the no feedback value (gain of one) which seems to suggest a gain of two for response times of only about ten years (note here that it is not relevant that Schwartz revised his estimates of both, only that his proportionality between the two appears to be very different from Lindzen’s). This is the issue in this post about which I know the least and welcome input as to what, precisely, determines the relationship between the sensitivity and the response time.

I certainly hope that I have provided a useful framework for moving discussion of the climate problem forward in a mathematical framework. Commenters, if you find an error in my calculations, I welcome it, for that is how mathematics (as opposed to the usual arm waving climate arguments) works. If you have suggestions for improvements on the assumptions and simplifications necessary to develop the mathematical climate theory, I welcome those too, provided they are mathematically rigorous and logically and scientifically justifiable.

References:

Hansen, J., G. Russell, A. Lacis, I. Fung, and D. Rind (1985) Climate response times: dependence on climate sensitivity and ocean mixing. Science, 229, 857-859

Lindzen R.S. (1995) Constraining possibilities versus signal detection. pp 182-186 in Natural Climate Variability on Decade-to-Century Time Scales, Ed. D.G. Martinson, National Academy Press, Washington, DC

Lindzen, R.S. and C. Giannitsis (1998) On the climatic implications of volcanic cooling. J. Geophys. Res., 103, 5929-5941.

Roe, G. (2006) In defense of Milankovitch. Geophys. Res. Ltrs., 33, L24703, doi:10.1029/2006GL027817

Schwartz, S. E. (2007), Heat capacity, time constant, and sensitivity of Earth’s climate system, J. Geophys. Res., 112, D24S05, doi:10.1029/2007JD008746.