Why should the “mean” temperature to increase linearly with time?
This article is motivated by Martin Ringo’s rather important question:
I guess I just don’t understand why the change in temperature should be a constant or a constant plus noise. It would seem to imply an exponential growth, upward or downward, of temperature. What am I missing here?
This question, asked in the context of doing a hypothesis test on the IPCC’s projections, and its answer are important if we are to understand:
- Why anyone might one assume that the earth’s temperature could be treated as having a mean trend and with random “noise” during some particular time frame, and
- What characteristics might one expect for the residuals to a linear regression during that time period.
In this post, I’ll discuss the answer to the first question. The analysis won’t even fully answer that question, but it does provide a framework for relating the data features we see when performing a statistical hypothesis test to the underlying physics. For brevity, I’ll be glossing over the degree of approximation involved in the physical model, except to admit this is the simplest possible physical model for the earth’s climate. (It is also, quite likely the model that results in the green curves in the IPCC TAR figure 9.1 here). I’ll also admit that the real reason for assuming a linear trend is that the IPCC AR4 projections display this trend in the short term.
The remaining post consists of describing the “one lump” climate model, turning that into two equations so I can discuss the “ensemble average” (or mean) behavior separately from the random aspects, and then discussing circumstances that permit me to treat the “average” behavior as increasing linearly with time. (This is essentially the answer to Marty’s question, as he wishes to know why one would chose linear rather than exponential variation with time.)
Later this week, I’ll be discussing aspects of the random signal. (That is currently a feature of greater interest to me, because I want to further explore hypothesis testing, and appropriate fits to the data. So, I need to write that bit up to help statisticians understand questions that are bothering me, so they can guide me. )
The “one lump climate”.
Let us suppose we can describe the earth’s climate as one big “lump” well mixed mass that is heated by some “forcing”. Under this cirumstance, the following approximation might be said to apply.
dΘ/dt = -Θ/τ + Q
where:
- t represents time,
- Θ(t) is the instantaneous global mean temperature anomaly (GMST),
- τ is the thermal response time of the planet and
- Q is some forcing function representing a rate of heat added to the planet above some baseline level; this has been normalized by a heat capacity of the system.
In equation (1), the -Θ/τ describe radiative heat losses from the earth’s surface: it is linearized; the equation is essentially conservation of energy.
Suppose now that the heat addition rate, Q, varies with time. Some of the variation is random, due to unpredictable factors like volcanic eruptions, solar activity, cloudiness or what not), but the average level of forcing increases over systematically over time. We will denote “average level of forcing” using averaging brackets, <>, and call it, <Q>. In the context of the theory of Anthropogenic Global Warming, this average level of forcing might rise as a result of increases in the level of CO2 or other greenhouse gases.
Using this idea, one could partition the instantaneous rate of heat addition into a ensemble average portion, <Q> and a random part, ‘q’, and write this as:
Q= <Q> + q
Some readers might be interested to learn that, in the context of analyzing systems governed by conservation of mass, momentum and energy, breaking variables into “mean” and “fluctuating” components is referred to as “Reynolds decomposition”, and is discussed many places, including Wikipedia. A similar decomposition is discussed in 9.2.2 Simulating Forced Climate Change of the IPCC TAR, where the terminology “signal” is applied to the ensemble averaged quantities, and “noise” applies to fluctuating components.
One can further apply the decomposition the temperature equation (1) into an average and a fluctuation <Θ> has been decomposed into <Θ>=<Θ>+ θ to obtain:
d<Θ>/dt + dθ/dt = {-<Θ>/τ + <Q>} + {-θ/τ + q}
It turns out that due to the properties of averaging, and those of equtaion (3) one can now apply averaging to (3) and obtain two equations, to obtain a equation describing how the average temperature evolves over time, and then using a magical technique called “subtraction” obtain an equation describing how the “fluctuating” portion evolves over time:
d<Θ>/dt = -<Θ>/τ + <Q>
dθ/dt = -θ/τ + q
So, under the assumption that a very simple equation like (1) we can decompose the response of the earth’s climate into signal plus noise, and, to a large extent discuss them separately. (Discussion would be more complicated if we assumed a more complicated model, but this partitioning into “signal” and “noise” is rather common, and understanding what happens with t his simple model is necessary before proceeding to more complex models.)
So, give equations 4(a) and 4(b) I can discuss the circumstances where it is appropriate to think of the earth’s surface temperature as consisting of a) A mean component that varies linear with time and b) a random component. This is what is done when hypothesis testing.
It turns out that sort of treatment makes sense if we believe the mean trend is approximately linear.
When might we expect the mean trend to be linear?
There are many circumstances where one can treat the mean trend as linear. I’ll discuss two:
- The mean forcing is thought to have increased suddenly to some new level very recently, but we are examining a period of time that is short compared to the response time of the climate, τ and
- The mean forcing is increasing nearly linearly with time and has been doing so for some time that is relatively long compared to the response time of the climate, τ
Mean forcing increases suddenly
When the mean forcing is thought to increase suddenly, it’s possible to show that the temperature of this simple system would be to increase, at first rapidly, but later more slowly. I discuss this in Linear Thinking. The solution in this circumstance might look like this.
Here, yellow indicates the forcing, <Q>, which is held at “zero” for until time zero, and jumps to 1 at time equals zero. The shape of the orange curve represents the temperature response, which takes on the shape of a decaying exponential. This is sort of behavior Marty is describing in his question, where he said “It would seem to imply an exponential growth, upward or downward, of temperature”.
For this case, we would rarely assume the temperature varied as a linear trend plus noise, except in a particular circumstance. If the time constant of the climate system τ in equation (1) were very long compared to the time period of analysis, we might apply a series expansion and treat the temperature as approximately constant during that period. But, otherwise, treating the mean temperature as having a linear trend would be a poor assumption for this type of change in forcing.
Suppose the mean forcing increases linearly?
As it happens though, the SRES scenarios do not assume forcing due to CO2 increases all at once. Instead, they assume the concentration of CO2 and other greenhouse increases continuously, as for example, in Figure 6.8a in the TAR. As you can see, the forcing increases nearly linearly since the ’50s. So, one might consider it useful to assume the climate has been under the influence of a linearly increasing forcing for roughly 50 years.
Now, let us suppose that the forcing rate increase as <Q>~C t with C describing some rate of increase in forcing as a function of time, t. This sort of forcing is illustrated in yellow below:
Once again, the response of the simplified “climate model” is illustrated with the orange curve. Initially, the temperature increases slowly with time. However, after a while, the temperature is seen to rise linearly with time, as illustrated by the orange curve approaching the behavior described by the red line– a line which shows the temperature increasing at a rate of α τ (which has dimensions temperature per unit time.)
Examining the curve, it’s possible to see that the orange curve describing the expected temperature of the system closely approximates the red line after two multiples of the time constant τ have passed. So, if, for example, we thought the time constant of the climate was 5 years (as in SE Schwartz’ analysis, ) we would expect the mean surface temperature to increase at a nearly linear rate roughly 10 years after the forcing began to increase linearly. In contrast, if the time constant of the climate is 100 years, we would expect it to take much longer before the trend began to rise linearly.
Yet another circumstance where the trend is linear
But, even if the time constant of the climate is long, and we are stuck in the “curving” region for the climate response, it’s still possible to consider the temperature variation linear with time.
Suppose one imagines that time constant for the climate τ is very long– say for purposes of this small discussion, τ=100 years. In this case, it will take a very long time before the temperature trend for the climate will increase at a linear rate. Rather, the temperature would follow that curved portion of the orange line for a very long time.
But, in such a case, 10 or 20 years used during which we analyze data is short compared to the response time of the climate, τ.
Consequently, we could expand the solution for the climates temperature into a Taylor series; this is the equivalent of drawing a line tangent to the orange curve– as shown above. Then, we can treat the mean temperature as having a linear trend for the purposes of statistical evaluation, provided provided we limit investigations to time that are short compared to the time constant of the climate. (In fact, if the time span for analyzing data is relatively short compared to the response time of the climate, we can nearly always assume the trend is linear, with small degrees of approximation.)
Summary
So, we see there are at least circumstances where it is possible to think of the global mean surface temperature as exhibiting a linear trend in temperature plus noise. These circumstances are: 1) Forcing has increased linearly for a relatively long time compared to the response time of the climate or 2) Forcing has been increasing linearly for a shorter period of time, but the time in which we do analysis is short relative to the time scale of the climate.
As for why I assume a linear trend plus noise when applying hypothesis tests to the IPCC’s AR4 short term projections, the real reason I do this is simple: The IPCC’s own projections suggest the short term trend is approximately linear. Simple physical models for the earth’s climate suggest that an approximately linear response is plausible, so I use it!
Later this week, I’ll be discussing the random aspect of the simplified model. The reason for this is that I have been looking at the characteristics of residuals in linear regressions between Temperature anomalies and time, and discussing this will help me explain why the “noise” might be expected to look like AR(1) noise with other features.
Update: 4/11/2008 Martin Ringo asked me a question which alerted me to major typos in equatin 4(b). I stripped the avearging symbols, which snuck in as cut and paste errors and which I missed due to poor proof-reading skils!
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16 Responses to “Why should the “mean” temperature to increase linearly with time?”
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George W April 7th, 2008 at 7:03 pm
Lucia,
(I hope I’m not missing something obvious, but) wouldn’t a linear temperature trend be the result of two counter-acting exponential trends?
One trend would would be that each doubling of CO2 produces the same (linear) increase in temperature. I read that (without feedbacks) that is about a 1 deg C per doubling of CO2 concentration.
The other trend would be a constant growth RATE of CO2 concentration - the figure mentions 1% per year.
TomVonk April 8th, 2008 at 2:45 am
Lucia
Your post might be slightly misleading insofar as it seems to make statements about the properties of the MGT when it actually makes statements about the equation (1) .
The equation (1) exhibits trivially asymptotic behaviour so with suitable assumptions it will exhibit linear asymptotic behaviour .
Whether equation (1) represents something really relevant to temperatures (or climate) is another question .
Only a slight modification consisting to write (1) : dΘ/dt = -Θ/τ + Q(Θ,t) wouldn’t allow to conclude anything about linearity (with the exception of the local Taylor expansion that is always possible for any function as long as it has suitable differentiability properties) .
As for the other issues , I will certainly come back to see how you propose to close (4b) .
Geoff Larsen April 8th, 2008 at 4:07 am
Lucia
To reiterate what George w said but expressed slightly differently.
I have always understood the warming effect of CO2 (as in climate model projections) is expected to be approximately linear over long terms as the result of : -
1. an exponential increase in CO2 concentrations resulting from an exponential increase in emissions.
2. a response of temperature (IR radiative adsorption) to increments of CO2 which is logarithmic, meaning that it begins to dampen off as the concentration increases.
The interaction of an exponential increase in CO2 coupled with a logarithmic response of temperature can easily result in a straight line.
lucia April 8th, 2008 at 5:16 am
George W–
I’m not sure what you are saying. But with this very simplified model, if ‘forcing’ rises linearly for a time, the temperature of the planet rises linearly.
TomVonk–
I don’t prose to close 4(b)!
You are right that this post can confuse people. However, I don’t know quite how to get around that. The underlying reason it can confuse people is that most people have questions about phenomenology, but Marty’s question is, largely, motivated by the desire to do appropriate hypothesis testing. The two issues are related and Marty, being a good statistician, knows they are related. So, he a question that’s important with regard to relating statistical inference to physics!
To clarify: this post is more motivated by this question from a statistician: Why fit data to a linear regression of T vs time model for statistical inferences? Why not fit it to an exponential of T vs time?
Your interest tends to fall in this category: “How would you create a physical model that best predicts any and all features of Global Mean temperature over time?”
The two question are related– but oddly not identical.
When performing a hypothesis test to see if a hypothesis has empirical support, one must test that specific hypothesis as it stands against data. In my case, I want to test the IPCC’s projections, which are for linear behavior. Marty in turn wants to know how in the world the IPCC could be projecting linear behavior on some time scale.
You in contrast are reading this in terms of: How might I Lucia go about creating a model that captures “behavior X”? (I’m interested in that question– but it’s just not the topic of today’s post!)
But, since today’s goal is not to develop a “better model”, I’m limiting myself to explaining “how in the world” the IPCC might come up with a linear trend for T vs time, in the short term.
As you know, the IPCC makes its projections using “simple” models that are tuned to AOGCM’s. (I discussed this to some extent here: http://rankexploits.com/musing.....ound-2000/ )
While you are correct that adding Q(Θ,t) terms to their simple models would change the character of the response of the model planet (particularly over the long term), and this term is likely necessary to predict interesting features (like for example ice ages). However, the IPCC “simple models” don’t include this (unless it’s in the sea ice parameter). So, it turns out that the possible existence of a term Q(Θ,t) is not relevant to a statistical hypothesis test of the recent IPCC projections!
(Though, the term does become relevant if their projections fail the hypothesis test. In that case, we decree that their projections leave out some important physics — and then we can suggest terms that the models need to include.)
In contrast, equation (1) is relevant to a discussion of what behaviors one can expect from IPCC models. That equation is an oversimplification, but it is simplest possible version of the IPCC ’simple’ model, and in fact, dominates the short time response of all ’simple’ IPCC models discussed here:http://www.grida.no/climate/ipcc_tar/wg1/371.htm. (Just set vertical diffusivity to zero, and/or make the ocean zero thickness! Set RLO to 1, I think set LO/NS=1, and I don’t quite know what to do with the sea ice parameter! You’ll get equation 1.)
Many people know the solution to (1) is an exponential. So, it’s natural to ask: If the IPCC uses equations with properties similar to (1), why would it makes sense to expect temperature will vary linearly, ever? And the purpose of this post is to answer that question.
In fact, there are many circumstances where equations like (1) suggest temperature will increase linearly (for longer or shorter times). The IPCC happens to be projecting that over the short time frame. And, so, when testing the IPCC projections we test their projections: which are linear for the short term.
And of course, the whole many paragraphs is the long way to say:
The reason we assume that temperature varies linearly when testing the IPCC’s short term hypothesis is that that is the IPCC’s short term hypothesis. I got this by fitting a line, and reading the text of the AR4. So, whether physical, unphysical, good bad of indifferent, we testing to see if it matches data, we first accept their form, and then only throw it out if it fails.
lucia April 8th, 2008 at 5:26 am
Geoff–
Yes. But I think the posted chart from the IPCC has already converted the CO2 increase into forcing, using the log law, or whatever one they prefer. So, since the 50’s they are suggesting forcing rose more or less linearly, but it hasn’t always. If the forcing rises linearly, a model like (1) suggests temperature rises linearly — eventually.
Still, it is certainly true that with an oversimplification like (1) if you stuff in some “forcing vs. time” you can get some “Temperature vs. time”. If, for example, you thought that the dominant forcing was periodic with time (like the solar cycle), you can get a periodic response. It’s damped, and out of phase. But if you were to force 1 with an 11 year cycle, you get an 11 year cycle.
Alan D. McIntire April 8th, 2008 at 6:21 am
The arguments I’ve read, but don’t agree with, state that temperature is roughly proportional to the logarithm of CO2, but there’ll also be a water vapor feedback. The increase in temperature from an increase in water vapor is almost exponential. By multiplying a logarithmic factor by an exponential factor, you’ll get a linear trend- A. McIntire
TomVonk April 9th, 2008 at 6:07 am
Lucia
I am disappointed but not surprised that you will not try to close 4b) …
As for 1) , I have understood what you were doing and why .
My point was mostly if not solely on the semantic side .
I tend to read “normal” sentences like mathematical statements - call it a mathematical reflex .
So making a statement about temperatures (f.ex GMT) is making a statement about a well defined even if probably irrelevant and uninteresting physical parameter .
Making a statement about the behaviour of some function Θ is making a statement about a solution of a differential equation , namely 1) .
Those are 2 very different statements and I know that you know that it is very different .
What we both do NOT know is how very different it is .
However there are many people out there who think that there is no difference , they think that both statements are identical that temperatures and solutions of differential equations like are always identical while in reality sometimes they ALMOST are and sometimes not at all .
So it is always good to choose formulations that make clearly the difference .
I know that it takes some extra writing but that is the reason why in mathematics every single word counts its weight in gold .
I guess what I wanted to say in so many words was that it may not come out clearly for everybody that what you were doing was only taking IPCC by their own word in every last little bit of detail and looking to what place it lead and if this place was consistent with all those initial bits of details (like equations 1)) .
That approach allows you indeed to make economy of a long , difficult and probably frustrating discussion about how far thetas are from real temperatures and I find that checking internal consistency is always a good idea .
P.S
I noticed that you were mentionning multidecadal oscillations somewhere but I don’t know if you consider to include them in the statistical models .
If yes , do you intend or did you already look at non linear statistical treatments (K-S entropy , Lyapounov exponents etc) ?
lucia April 9th, 2008 at 7:57 am
Tom–
For testing hypotheses, ideally we would have enough data that multidecadal cycles just average out. Of course it is true that this doesn’t happen unless we have more than several full cycles.
This is equally glossed over in many empirical “proofs” of AGW I’ve seen on blogs, and “disproof” of IPCC claims! (Though, actually, I do discuss it from time to time.)
With regard to techniques, I’ll look at any technique that seems suitable for the test I’m currently considering. If they can reveal an answer to a question I ask, I’d use it.
So, since you asked about these methods, what questions are you anticipating I might be asking? Because I could more easily tell you what questions I might be thinking of asking than what method I will chose to get the answer!
TomVonk April 10th, 2008 at 2:48 am
Lucia
Well I guess that once you embarked on that trip that I have been following since 11 years , you will finish by seeing that pseudo periodical events in a whole range of frequencies have a demonstrable impact on temperatures among other things .
Seeing that you might begin to wonder about 3 questions :
1) Has the arbitrarily chosen GMT averaging period (15 years , 23 years , 30 years) an impact on the results ?
2) Has the beginning , the length and the end point of the time series an impact on the results ?
3) Can a purely radiative (GH effect) signal be seen in the data ?
1) Is a tractable technicality . It is not necessarily easily interpreted but the numbers can be computed .
2) Asks among others the difficult data density question which is largely treated by the Climate Audit blog . From there follow questions of spatial autocorrelation with appropriate techniques .
It also asks another extremely difficult question if there is a lower limit of the frequencies of the “multidecadal events” . I remember having read a paper several years ago that treated with the variation of ice thickness in northern Siberia over a long period , something around a century if memory serves . The authors concluded that there seemed to be some pseudo periodical event(s) whose period was of the same order of magnitude as the length of the time series but it was impossible to have a clue about what kind of event(s) that might be .
3) Is the most difficult because by definition the data must be first “stripped” of effects of those pseudo periodic events . For that their behaviour must be known . There are 2 extremes . First is that their amplitude and frequency are strictly deterministic and second is that they are purely random .
Between the two is that they randomly fluctuate around some average constant value .
The natural and easiest choice is to take the hypothesis between the 2 extremes .
However by doing that one might do the most common error there is - to confuse randomness with non linear deterministic interaction of several pseudo periodic oscillators .
In the first case the statistics work , in the second case they only sometimes SEEM to work … for a certain time .
To distinguish between the 2 it is necessary to ask the question of predictability .
This is the question of creation of information in the system - if the system is random this creation goes to infinity with the time (any state can appear and will appear) and if the system is deterministic this creation is 0 (the system is completely defined by only the initial conditions and some law) .
Obviously a real sufficiently complex system will be between the 2 and its (un)predictability will depend if its information creation is rather near to 0 or rather near to infinity .
That problem is treated by specific techniques like the K-S entropy (analogy with thermodynamics : information = entropy) , Lyapounov exponent , fractal data dimension etc .
Of course as long as you closely stick to IPCC data and hypothesis you will by definition not be concerned by these questions and consequently these techniques .
However if you should choose to question the hypothesis one day (specifically the point 3) , you will need to go beyond classical regressions .
JamesG April 10th, 2008 at 5:34 am
Your response is well damped. Imagine though an under-damped response to a step change - it would ramp up rapidly, oscillate for a while and then plateau. In fact, just like the real data does from 1975 to the present. Just something I noticed. It doesn’t have to mean anything.
rxc April 11th, 2008 at 10:14 am
The problem with assuming that the temperature trend is that it is just an assumption. In order to determine what curve to apply to the data, you have to understand the underlying phenomena, and how they interact with one another. If you start with basic physics F=ma, this is a linear relation that has been tested extensively, and until about 100 years ago, was completely accepted (consensus). Then, some people named Michelson and Morely did some experiments and found out that the speed of light was a constant no matter the frame of reference, and Einstein took this data and created special relativity. F=ma is good for most stuff people do, but under certain circumstances, there are what are called “relativistic effects” that mean that F is not always equal to ma. It becomes a very non-linear relationship at high speeds. It also leads to e=mc^2 (also a linear relationship).
In climate “science” there are a LOT of important phenomena that derive from the Navier-Stokes equation for fluid flow. This equation is NOT amenable to direct solution, and as a result, there are LOTS of approximations, SOME of which display SOME linear characteristics in CERTAIN flow regimes. The models are too complicated to deal with otherwise, so the modelers use linear approximations, or they use non-linear approximations, or they use simple tables of correlations because they are faster to use in the computation. When you do this in engineering, you must compare the calculations against real experiments to ensure that your simplified assumption is valid. If you cannot do that, you have to include large margins to account for the uncertainty.
The AGW models do not have any data to compare to their calculations. They have developed simple correlations to match past data, where is is available, and reconstructions of what they think the world was like, based on indirect measurements like tree rings. They certainly do not have the ability to make any real predictions of experiments that have yet to be done. The closest to this situation is the analysis that Lucia has done, which shows that the earlier predictions do not model the climate that has developed since the calculations were performed. This MAY be just coincidence, but it is only with more time and more data that this situation will be resolved.
The Greens desparately want a legal system put into place before that data appears, because if the data keeps going in the wrong direction, they will not convince people that a crisis is at hand.
Lucia needs to understand that her assumption of a linear trend is exactly that - an assumption. It is certainly not strictly correct, because the underlying phenomena are not linear. The question is whether linear is good enough, but since there is also no agreement on whether the ultimate metric (global temperature) is important for the small variations that we are talking about, it is difficult to evaluate the linear assumption.
You need to talk to thermal-hydraulic modelers, not EEs who look at signal noise. This is not a noise issue, it involves non-linear variations and interactions among a LOT of physical phenomena that are NOT noise. They are known, but modeling them is difficult, and treating them like noise is not appropriate.
lucia April 11th, 2008 at 10:58 am
rxc– I agree with some of what you said, and disagree with other things. But, I can’t help laughing. Lucia has a Ph.D. in mechanical engineering. The two books sitting on her desk this very moment are “Turbulence” by Hinze and “Gas Dynamics” by Zucrow and Hoffman, and the reason they are their is she is trying to figure out something about poisonous gases diluting as they emit from a compressed gas cylinder. (Hinze is useful once the jet finishes doing all the odd things it must do because it emits from the tank at mach 1.)
And even more importantly Lucia is quite familiar with thermal hydraulics code, including ancient broo-ha-has about various mathematical formulations. (And has contributed to the brouhaha’s herself with papers like “Ensemble-average equations of a particulate mixture” http://cat.inist.fr/?aModele=a.....dt=2691563, which applies to both dilute gas-solid and dilute gas-liquid flows.
So….. no. I’m not learning this from EE’s who look at noise. I don’t like the word noise applied to weather as I consider the weather real. But that terminology seem widespread at climate blogs. Not being a “mrs. language person” sort, I’m wiling to use it.
As for the equation: I need to get on to the fluctuation (aka noise) discussion. But my time was scarce this week!
rxc April 11th, 2008 at 12:55 pm
Lucia,
Good. Thank you for clarifying your background to me. I have just jumped into the middle of your work, and did not know that you were familiar with fluid mechanics. I did not mean to put you down - I was just trying to educate someone who seemed to be quite well informed, but maybe not in fluid system modeling.
I will just sit back and watch the debate emerge, as the greens say. It is interesting that when the nuclear industry claimed that nuclear safety issues were resolved, as demonstrated by the consensus of knowledgeable nuclear engineers, they strongly disagreed, and claimed that anyone who understood the material had a conflict of interest, and was therefore disqualified from judging the situation. Only lawyers and judges (without technical training) could make those decisions. Now the shoe is on the other foot…
lucia April 11th, 2008 at 12:58 pm
rxc– jumping in is great! But I think, based on my response, why I sort of chuckled at that.
This equation IS an oversimplification. But, when I get time to write up the portion on the random equation, that’s where some discussion of features of the NS would come in. (The same response would go to James G.)
rxc April 11th, 2008 at 1:07 pm
Lucia,
One last thought. What is the parameter of interest that is being plotted linearly against the figure of merit? [CO2]? [H2O]? [CH4]? Solar irradiation? Haze/particulates? Cloud cover? Are these all independent of one another, dependent, or interlinked in some impossibly complex fashion?
From what I have read of the IPCC models, they take some serious computational fluid models and simplify them. When you do that, you invariably lose phenomena, and you have to evaluate which ones you can give up. I have seen no such assessment anywhere. The other problem with these simplified models is that they are only good for the range of interest in the underlying phenomenological models, and their extrapolation outside the range of applicability is very dangerous - there lie monsters…
lucia April 11th, 2008 at 1:40 pm
rxc–
I’ve been particularly interested in applying hypothesis tests to IPCC projections of GMST. Their most recent document (the AR4) includes projections showing nearly linear behavior for the underlying trend for early portions of this century. They provide an estimate for the slope of the line.
So, simple minded or not, that’s what they project, and that’s what I test. But, one of the question is: does linear make any physical sense under any circumstances. And of course as a simplification for a “one lump” earth, linear is meaningful under a range of restrictions. Presumably, if the IPCC is predicting that, their models think it applies (more or less), for the period in which the projections are near linear.
Ultimately, the models the IPCC uses for these projections do, indeed, appear to be greatly simplified compared to GCMs. They aren’t simplified as much as the equation I’m using to explain “why linear”, but they are simplified and them tuned to AOGCM’s.