the Air Vent

Because the world needs another opinion

Climate as A Differential Equations Problem

Posted by Jeff Id on June 12, 2012

I hope to be a blogger someday again but in the meantime Timetochooseagain kindly offered this contribution -  Jeff

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TTCA

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.

44 Responses to “Climate as A Differential Equations Problem”

  1. Thanks Jeff. The labels on the Equations indicating their numbers have been lost, is everyone able to follow what’s what?

  2. Eviul Denier said

    @TTCA
    One wishes that all followed your mathematical rigour. I have my stance (based more on the behaviour of the believers), but would modify it in the face of overwhelming mathematical rigour, based upon evidence, (and I don’t mean models, which output only the prejudices of the modellers [I speak as a{n ex-}modeller])

  3. Andrew said

    2-I actually think mathematical modeling of physical systems is perfectly fine as far as it goes. Equation (7) is in fact a (very simple) climate model. The problem lies in the place where the modeler’s prejudices come in: choice of coefficients. Now, when working with “GCM” results, many will say, the lambda and tau are not actually specified, but emerge from the way that the model treats the ocean and the clouds. Fair enough, except this doesn’t mean that the prejudices of the modelers don’t impact the result: a given choice for various parameters (and GCMs have way more parameters than the simple model of Equation (7)) results in a certain sensitivity, and there are many different options for every parameter that may be realistic (although all parameter choices are simplifications (oversimplifications?) of real processes) so they can tweak dozens of nobs to get an “effective lambda” and “effective tau” about where they a priori expect. It would be better to measure tau and lambda, but it’s not obvious how. It’s not even obvious that they would necessarily be constant.

    Incidentally, if f in Equation (8) is significantly variable, then it almost certainly has a low mean value, since it would have to have miraculously stated below 1 for the entire history of the Earth!

    • Evil Denier said

      Agreed completely. Experiment doesn’t (can’t?) give values to parameters, thus they are too often tweaked (your word) to give a result acceptable to the modeller.

  4. Jim said

    TTCA,

    I hope that you don’t mind my response: I don’t see how the differential equations you posted are at all close to being a model of the climate.

    For any point on the earth, the power in-put to the atmosphere is time variable and is latitude dependent. Incidence of the solar flux varies from equator to poles. At each location, flux varies on day long, year long, and polar axis wobble cycle long, cycles.

    And generalizing; more energy is received by the earth surface at the equatorial regions than is radiated away by the atmosphere at those regions, and more energy is radiated away from the atmosphere than is received by the earth surface in the high latitudes. Atmospheric circulation conveys sensible and latent from the equator to the poles.

    The Earth’s climate system seems *-far more-* complex than the equations you show. You can’t show global disposition of energy, at the least, with the equations above… colder at the poles and warmer at the equator, and vice versa, are not equal, even if ‘T’ (avg temp) is the same. (Is ‘T’ the average of surface temps, the avg of whole atmospheric temps, the avg of whole atmospheric and whole ocean water temps ??)

    Representation of the state of the Earth’s climate by a singular value, some global average ‘surface’ temperature is uselessly simplistic, to the ‘point’ of being ‘pointless’.

    • Jim said

      Correction to the above: …atmospheric circulation conveys sensible and latent *heat* from the equator to the poles.

      I’m sorry for the omitted word mistake.

    • Jim said

      TTCA,

      The curtness the last sentence of the end of my post isn’t aimed at you. It is my irritation with the idea of using average surface temperature as the singular quantifier of the Earth’s “climate”.

      Thanks for the post.

      • timetochooseagain said

        It’s alright, I agree that the model is overly simplistic. But this is the general approach that is taken to the climate problem. I am also quite partial to the legal rhetorical technique of Arguendo so I tend to seem like I am accepting premises when I am merely taking someone else’s to their logical conclusion.

        For those who don’t know:

        http://en.wikipedia.org/wiki/Arguendo

      • Carrick said

        Jim, if you have a field T(x,y,z,t) temperature, there’s nothing to stop you from looking at the field averaged over x and y with z = 0.

        In fact, it’s often possible to start with a set of partial differential equations involving x,y,z and t and reducing them to just “t”. (I call these the “reduced” equations.)

        So I would have to dispute there is no utility to reduced dimensionality, if what you want to model is the global average of T.

        I’d probably use a “two box” model E.g., see Lucia’s work here, since this includes an ocean heat sink as well as an atmosphere.

        These models can do a good job of representing the change in temperature of ocean and land given a known forcing series. The big challenge for climate science is in nailing down the past history of forcings and predicting the future history of forcings. There are huge uncertainties in these, and this is what allows for such a wide range of estimates of sensitivity to CO2 forcing.

        • timetochooseagain said

          Yes, this is basically a “one box” look at the system with the boundary across which energy may flow being the top of the atmosphere. An additional boundary is almost certainly necessary for many applications, for energy flow between the atmosphere and oceans.Unfortunately, each layer of complexity we add, leads to more unknown coefficients.

          • Carrick said

            To be clear, I think you can get away with the single box model if you are looking at transient behavior in the atmosphere. This is because the long integration time for the oceans integrates into a very small amplitude for the transient response of the oceans.

          • Jim said

            Thanks, Carrick,

            My inclination is to think the the spacial dimension is very important in the description of climate; that climate is circulations, at the first cut. Temp avg, of course, makes on distinction.

        • Jim said

          Carrick,

          I don’t see that T avg over the field is indicative of the processes within the field. The partial differential with respect to latitude seems like it would be useful. Divergence or heat flux within the field seems like it would be very useful, though I don’t think I’ve seen data or modeling results expressed that way.

          Argos temp measurement is the the most interesting to me; Animation of the divergence of a lat-long, depth, and time field would be interesting, I think.

          Same with surface air temps. RSS and UAH atmosphere temps seem more important than the surface temp reconstructions.

  5. DocMartyn said

    I am somewhat confused by this account. It appears to assume that the energy input into the system is constant, that at steady state the efflux from the top of the system is equal to the input (with a possible lag) and the surface temperature is a function of the influx + what ever photonic recycling we get from green house gasses.
    However, the input is not invariant; The planet is rotating and day ‘average’ temperature is calculated from ((Tmax+Tmin)/2). Their is an orbital tilt that changes the light flux in the two hemisphere and finally the orbit is elliptical with 3.3% variation in max and min distance.
    The surface of the oceans show a seasonal Tmax/Tmin range 12/8 degrees. This seasonal oscillation can be observed down to 1 km

    https://sites.google.com/site/climateadj/argo-sine-fitting

    Now these Argos readings are averages of the daily Tmax/Tmin readings. Each point is a pseudo-steady state reading of a dynamic system.
    We know the timescale that the surface temperature changes in response to the seasonal influx, <<< 1 year. The plots show that the surface layer and 1 km down are coupled with a lag of <<< 1 year.
    The exchange of heat between the top and bottom sea water layers most probably due to radiative transfer as the salinity of the 1 km layer has no seasonal signal, so bulk sea water is not exchanges.
    The tmin of the ocean surface is 8 degrees, six months later its 12 degrees, than at 12 months back to 8 degrees. The change is due to the change in the solar flux.
    If we increase the background (GHG induced) flux onto of the signal the response should also be the same order as the seasonal change <<< 1 year.
    What am I missing?
    Why does everyone get such long lags?

    • timetochooseagain said

      I said in the post that I really don’t think you can gauge the system response time to uniform forcing from seasonal cycles-which are not due to a uniform forcing at all. The model in question is a simplification, but takes the form that most of the “consensus” people use for “simple” models. GCMs can be closely approximated by this form, too.

      The reason why I don’t believe that the seasonal cycle can be used is perhaps unclear. I think of it this way:

      The Earth is a thermodynamic system. As a whole, it is “closed” or close enough, since there is negligible mass flow across the system boundary (the “top of the atmosphere”) but there is energy flow. Now, the Earth is not an internally homogeneous system, but as long as the change in the flow across the boundary is approximately the same everywhere, we might approximate it as such. It is also important that the individual parts of the Earth system are open thermodynamic systems, heat flowing from one to the other and mass. So when we look at one place on Earth, and compare the energy flow across the top of the atmosphere, we are ignoring the energy flow to other parts of the system. I think that heat flow from the summer hemisphere to the winter hemisphere acts to damp the response to the summer insolation. Even forgetting that, the individual locations are being exposed to large, but short lived, changes in forcing compared to the global average, and different locations surely have very different feedback processes. Consider the following example that shows how inhomogeneous forcing can mislead you:

      We have an Earth system where half has a positive feedback of f=.5, let’s say because of ice albedo this is the more Arctic-ward half. Now let’s say the “tropical half” of the Earth has an f=-.5, say from a thunderstorm thermostat. Our system exposed to a uniform forcing of 3.7 W/m^2 everywhere:

      The tropical half warms .8 degrees, the higher latitude half warms 2.4 degrees. So the global sensitivity is 1.6 degree for a doubling of CO2.

      Now let’s expose the same system to no net forcing, but -3.7 W/m^2 in the tropical half and 3.7 W/m^2 in the high latitude half

      The tropics cools .8 degrees, the high latitude half warms 2.4 degrees, the average change is .8 degrees. The average flux change is zero. The diagnosed sensitivity is infinite!

      • DocMartyn said

        I wasn’t trying to be nasty or anything; it is just I have a problem with using classical equilibrium theory to explain steady state systems.
        I have always though that a proper examination of the return to steady state following a large vulcanic explosion would be the way to go. I cannot believe that lags greater than a year or so can be real.

      • Jim said

        TTCA,

        What you say above is what I was thinking.

        “I think that heat flow from the summer hemisphere to the winter hemisphere acts to damp the response to the summer insolation. ”

        I’ve been scolded by experts that there isn’t inter-hemispheric heat flow, only intra-hemi flow, FWIW. Heat flow from low latitudes to higher latitudes in each hemisphere would be dampimg, as you say. Seems to me there is a radiative balance net influx at low latitudes and net outflux at higher latitudes. Intuitively, the over-time change, or rate change, of TOA imbalance as a function of latitude and time would be indicators of climate CO2 sensitivity.

        • Carrick said

          Jim:

          I’ve been scolded by experts that there isn’t inter-hemispheric heat flow, only intra-hemi flow, FWIW.

          Who are these “experts”?

          Have they never looked at real climate data? I think if they did, I’m pretty sure they’d find they’re wrong. (For two reasons, I believe, one is due to the shift in the inter tropical convergence zone, and the second is ocean transport of heat energy in the mixed layer.)

          Here’s an example of a
          simple” climate model:

          8.8.2 Simple Climate Models

          As in the TAR, a simple climate model is utilised in this report to emulate the projections of future climate change conducted with state-of-the-art AOGCMs, thus allowing the investigation of the temperature and sea level implications of all relevant emission scenarios (see Chapter 10). This model is an updated version of the Model for the Assessment of Greenhouse-Gas Induced Climate Change (MAGICC) model (Wigley and Raper, 1992, 2001; Raper et al., 1996). The calculation of the radiative forcings from emission scenarios closely follows that described in Chapter 2, and the feedback between climate and the carbon cycle is treated consistently with Chapter 7. The atmosphere-ocean module consists of an atmospheric energy balance model coupled to an upwelling-diffusion ocean model. The atmospheric energy balance model has land and ocean boxes in each hemisphere, and the upwelling-diffusion ocean model in each hemisphere has 40 layers with inter-hemispheric heat exchange in the mixed layer.

          Sounds to me like you need to substitute these “experts” for a new set. ;-)

          • Jim said

            Yes, I don’t have expertise. And fashions change; C14 dating, atmospheric nuclear testing, and CFC arguments are old history.

            In the quote you posted, the bolded ‘heat exchange in the mixing layer’ refers to the equatorial oceans? Meaning “mixing” between north and south?

          • timetochooseagain said

            I’m interested to know the magnitude of the seasonal atmospheric energy exchange across the equator (preferably observed, if possible. In Willis Eschenbach’s recent WUWT posts, he ignores this factor in doing a one box model of the seasonal cycle. I asked him about it, and well, see lower in the thread for his reaction to this factor being brought up. If it is large, his ignoring it is unjustifiable.

        • Carrick said

          * one is due to the [seasonal] shift in the intertropical convergence zone

    • Jim said

      DocMartyn,

      Good point!

      It’s interesting to try to imagine what the climate of the earth would be if the Earth rotated much faster or slower; a day of a few hours or a few hundred hours. Also, what the climate would be if the polar axis of the Earth were normal to the orbital plane; ie an Earth with a permanent Arctic ice cap.

  6. uc said

    Some time ago I tried very simple model with local solar energy as input:

    \DeltaT=A+BSol(t)+CTemp(t-1)^4

    http://www.climateaudit.info/data/uc/T414_1.png

    http://www.climateaudit.info/data/uc/T414_2.png

    http://www.climateaudit.info/data/uc/T478_1.png

    http://www.climateaudit.info/data/uc/T478_2.png

    http://www.climateaudit.info/data/uc/T4m21_1.png

    http://www.climateaudit.info/data/uc/T4m21_2.png

    It seems to work quite well, but I have to fit three parameters per location (A,B,C), so it might be some kind of overfit.. What is the official equation that explains the season lag?

    • Andrew said

      If we take out the annual mean temp from the temp at each location, and reinterpret f(t) in equation (7) to no longer be the change in energy flow in and out of the atmosphere to/from space, but instead the change in energy flow into/out of the locations “box” at any given moment (heat exchange with the surrounding areas, the heat exchange with the ocean if you are over it, etc.) then equation (7) should work, providing we can neglect mass flow (wind) or at least I think it should. The main reason you need an extra parameter, I think, is that you are using the energy flow across the boundary with space only, but when looking at individual locations there is energy flow (and mass flow) horizontally to.

      I told Willis this is why I don’t think much of his analysis of the seasonal cycles in the Hemispheres. He told me:

      “As far as I know, although there is an exchange of atmosphere from one hemisphere to the other, it is both slow and not all that large. I’m happy to be corrected, but when you look at say the lag between the CO2 concentration in the NH and SH, it has a time scale of years. I find it difficult to believe that the effect would make much difference to what I am considering here.”

      To be honest, I don’t know what either the mass exchange or the heat exchange horizontally is, but I really doubt it is safe to ignore it.

  7. KR said

    TTCA – Very nice, a clear derivation.

    A single time constant, however, is not a very good model. The climate has multiple components (atmosphere, ocean, vegetation, etc) with different time constants for responding to forcings, and a single tau model makes for a poor fit.

    Tamino discusses this at some length on http://tamino.wordpress.com/2011/05/14/fake-forcing/, where he uses two time constants (lambda1, tau1 2 year for short responses, such as volcanic aerosols, lambda2, tau2 30 year for oceanic responses), and obtains a much better fit. [The title refers to avoiding ad hoc adjustments that would be required for a single tau model.] Some of the ocean circulation models I’ve seen require at least four boxes/compartments, with rates and tau’s for each, in order to match up with observational data.

    Again, a very clear derivation – I would just caution against too simple a model.

    • timetochooseagain said

      Oh, no doubt it’s overly simplistic. Most (all?) models inevitably are simpler than reality-and the simplest models are usually too simple to give accurate answers. That being said, Tamino is hardly the only person to propose altering the model to have a secondary response time. See Carrick’s post above linking to lucia’s blog posts on “two box” models. Much of it focuses on some oddities with Tamino’s particular attempt to create one. Lots of discussion, too!

      But you have to understand, this is meant to be an introduction to the basic mathematical nature of the climate, not an attempt to create “the” model for the climate.

      • Carrick said

        Yep, it ends up getting Lucia tossed from Tamino’s blog for having the temerity to point out that Tamino’s model violates the 1st law of thermodynamics and to point to Tamino’s on waffling on this topic. This, apparently, is not something one does on Tamino’s blog. Not and remain unbanned anyway.

        • KR said

          I’ve looked at some of that discussion, in particular http://arthur.shumwaysmith.com/life/content/heat_transfer_in_the_two_box_model – It’s possible to put in values for a two-box model that are physically inconsistent (particularly if climate sensitivities for the two tau’s are significantly different), but if they are even close to the same value there are no such issues:

          “This suggests that a real physical constraint is associated with requiring the long-term (or steady-state) sensitivities for the “fast” and “slow” boxes to be comparable.”

          Which only makes sense – as the climate warms/cools, all components will (eventually) reach the same temperatures, so sensitivities to forcings should be very close if not identical.

          I would, in fact, consider that a constraint of physics (the climate will at equilibrium have a fairly uniform temperature in all components at any particular location), and hence quite reasonable. I do not, therefore, consider Lucia’s objections valid.

          • Carrick said

            Really don’t follow your argument, which had to do with Tamino’s implementation and parameter choice for the two box model, not Arthur’s later perambulations. Of course it’s the case you can find values that are physically possible (lambda1=lambda2=0 for example). The issue is with setting useful limits that are both physically plausible and still give useful information about climate..

  8. Hehe, looking at Lindzen and Gianitsis again, it looks like they used a model with multiple boxes. Curious that, AFAICT, Lindzen has only mentioned a single tau from the beginning, even though he has been working with a multiple box model to explore these effects.

    • You would need multiple boxes. The thermo of the moist air envelope is different than the radiant envelope which is different than the planetary envelope. It is at least a three box problem. You can use temperature or energy boundaries for each box, -1.9C is the freezing point of salt water, -1.9C @ 50% RH and 306.6Wm-2 is your moist air box or heat engine. Assuming 50% entropy, the radiant air box boundary would be 240.2K 188.8Wm-2 sink temperature, the third box would be 188.8/2=94.4Wm-2 @ 202K. That results in a maximum entropy model (50% entropy-50%work) that considers latent energy correctly. Note that the “average” source temperature for the moist air envelope is 294.25K not 288K, That has a considerable impact on sensitivity. Cuts it in half.

  9. omanuel said

    Differential equations in indeed a useful tool for solving the problems that usually confront engineers and scientists, but the global climate problem may be better understood by studying psychology, sociology, creativity and ethics.

    http://omanuel.wordpress.com/about/#comment-31

    I found three books especially helpful:

    1. “1984” by George Orwell (1949)
    http://www.online-literature.com/orwell/1984/

    2. “The Naked Ape” by Desmond Morris (1967)
    http://evolbiol.ru/large_files/naked_ape.pdf

    3. “The Road Less Travelled” by M. Scott Peck (1978)
    http://tinyurl.com/6qaus5l

    With kind regards,
    Oliver K. Manuel
    Former NASA Principal
    Investigator for Apollo
    http://www.omatumr.com

  10. [...] variations in cloud coverage would be my first guess for a natural factor. Also note that, as I previously stated, one cannot really test “global mean” sensitivity on the basis of spatially [...]

  11. I got into trouble for this, but to eliminate T in order to make a more linear model I used dF/dT~aF/T, and as energy is fungible allows dF/dT~(aF1+bF2+…zFz)/T. Since that is a little tough to work with multiply both sides by T and you get TdF/dT=aF1+bF2+…zFz, that way each type of energy flux or portion of the radiant spectrum can be considered independently. a,b..z would be the specific resistance to energy flow for each energy flux. Kinda neat and it works well if you know the actual resistance for at least one flux.

    • timetochooseagain said

      I think the problem is there isn’t any obvious reason why the system should care what the “kind” of flux is. As long as the fluxes are spatially uniform, energy is energy and has the same effect. That being said, I’m not entirely clear how you got your model to look like it does, so I can’t say for sure how it would relate to mine.

      • The only reason the system cares is because the moisture cycle is not really closed. Ice mass can change. The other reason is that you can estimate the CO2 spectrum impact at different layers. If you consider a moist air layer versus a dry air layer, there is a huge change in the spectrum. The energy “sees” that gradient, so you can fine tune regional impacts. That is kinda import for determining an “average” surface impact. As it is you can predict an impact at an arbitrary effective radiant layer, about 1.5 to 1.6, but you cannot weed out uncertainties below that layer.

  12. Nullius in Verba said

    Regarding the time constant for GCMs, Isaac Held’s post here is very interesting.

    http://www.gfdl.noaa.gov/blog/isaac-held/2011/04/16/8-the-recalcitrant-component-of-global-warming/

    If the CO2 forcing undergoes a step change, the response occurs within about 5 years.

  13. Brian H said

    I’m not sure your intended project worked at all. In fact, to understand your “explanations” requires a substantial familiarity with mathematical notation and conventions, to the degree that your descriptions are redundant for those who know the conventions, but still opaque to those who don’t. It seems that it’s impossible for those who think in algebra/calculus to actually step back far enough to use English to clarify what the formulae are saying.

    • timetochooseagain said

      I’m sorry to hear that. I have had some ideas how to improve the model, and I’m looking at some climate problem where one might be able to overcome some of the difficulties I mentioned. Sadly, I’m finding I’m in over my own head mathematically pretty fast…

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