A Simple Truth; Computer Climate Models Cannot Work

Sir, your experience shows that climat change marketeers masquerading as grant snuffling scientists are not just laughably bad at science, what they do has major, potentially lethal, consequences.

Numerical solutions to partial, non-linear differential equations ALWAYS produce a tiny error and if you seek just the solution, numerically produced, to just one equation, you can construct the algorithm to make the error insignificant. (Actually, this is true also for linear – and far SIMPLER – differential equations as well).
But when the results of many of these numerical solutions are used as input parameters for the next set of equations, the “error” becomes magnified. Repeating this process thousands of times produces results with huge errors and the final results are simply WRONG.

This observation was originally made by Edward Lorentz in Deterministic Nonperiodic Flow in the 1960s. Lorenz concluded that very small variations could make immense changes in the state of deterministic systems over time. Since Lorentz’s work was in computational meteorology one would have thought the modelers would have considered the conclusions and at least qualified the discussion of model accuracy.

So how exactly do you think you know what the temperature is at any given time and any given place if you don’t have something measuring the temperature there?
You have a model that accurately replicates temperature in any given location entirely from first principles do you?

I’m trying to decide if this is another “I agree with Nick Stokes day”. On the one hand, you are absolutely correct when you say that weather stations do not contribute direct input to climate models. On the other hand, the climate models do have to be initialized, and because at least some of the subsystems that play a major role in the time evolution of the climate have very long characteristic times and because the climate is highly non-Markovian, one has to start them from initial conditions that are not horribly out of balance with respect to reality. This problem is somewhat exacerbated by the chaotic nature of the dynamics, of course.
Still, Tim Ball is incorrect to assert that it is the lack of station data per se that is the downfall of climate models. The downfall of climate models comes from the explicit assumption that the average of the averages of many climate models, each one producing an ensemble of chaotic trajectories from initial conditions that are explicitly assumed to be irrelevant in the long run, is a useful predictor of the one actual chaotic trajectory the Earth is following while self-integrating the actual physics at microscopic length scales instead of integrating equations that are called “physics” on an absurdly large spatiotemporal stepsize but that somebody basically made up.
Let’s see how sensible this is. Here is a typical Feigenbaum tree:
http://log.1am.me/2010/12/bifurcation-trees-and-fractals.html
Typical is good enough, since Feigenbaum basically showed that this tree has universal properties for iterated maps that lead to chaos, and the weather/climate system is of course the “iterated computational map” in which chaos was first discovered.
Even though the tree structure of the periods is universal, it is not insensitive to the underlying parameters used to generate it. Indeed, small changes in those parameters can lead to large shifts in precisely where the bifurcations occur, in the specific distribution of the strange attractors in whatever high dimensional space one evaluates the dynamics (in the case of weather/climate) but even in the simple few-dimensional systems typically used to generate graphs like this. We’ll stick to the few-dimensional case.
So imagine a set of figures like this, each one generated by a model, each model having a slightly different implementation of an iterated map (e.g. different parameters, slightly different functions being solved per step, different stepsize, and in all cases stepsizes so vastly larger than the stepsize needed to actually track the underlying nonlinear ordinary differential equations that pretending that the iterated map is somehow a “solution” to the ODEs becomes an exercise in the suspension of disbelief akin to that required by a typical space opera with FTL ships bopping all over the Universe in human-short times). Running each model with slightly different initial conditions in every case leads to completely different trajectories — it is in fact the opposite of the behavior of damped, driven linear oscillators, where initial conditions are indeed irrelevant, producing a transient behavior that damps away and leaves one with a nice, clean, periodic signal slaved to the periodic driver. In chaotic models this is in some sense inverted — even starting with almost the same initial condition, the differences grow until the correlation function between two initially almost identical trajectories decays pretty much to zero as the two models are, on average, completely decorrelated for the rest of eternity.
So let’s mentally average over a large set of these decorrelated trajectories, per model. This, of course, effectively linearizes them — the resulting trajectory is (gasp!) very close to the original non-chaotic linear-response trajectory that was split up into the tree by the nonlinearities in the appropriate regime. Do this for all of the distinct models, and one has a collection of nice, tame, linearized trajectories, all of them averages over chaos, all of them different (possibly even substantially different), and then let’s average the averages all together to produce a grand ensemble superaverage of the running averages of the individually produced chaotic trajectories evaluated by integrating a made-up dynamical system at distinct spatiotemporal length scales that completely erase the actual variation observed on all smaller length scales and that Nature seems to think are important to the ultimate dynamics of heat transport efficiency as it self-organizes them quite differently as conditions change on a must smaller length scale than the models track.
Now, claim that this super-averaged trajectory is a useful predictor of the future climate, even though the individual chaotic trajectories produced by each climate model have grossly incorrect features compared to the actual trajectory of the one chaotic nonlinear climate we live in.
Sure, that works. If I apply it to a damped, driven rigid oscillator, I can prove that on average we expect to see it at rest. Or, if I’m slightly more sophisticated, I can show that it is still oscillating periodically. Or, if I’m nefarious, I can tweak the underlying model parameterization and “prove” lots of stuff, none of which has the slightest predictive value for the one trajectory that is actually observed, the single strange attractor of the underlying dynamics.
Of course nobody would be that stupid if they were studying a low dimensional chaotic system, or merely demonstrating chaos for a class. On the contrary, they would be pointing out to the class that this sort of thing is nearly pointless, because this:
http://video.mit.edu/watch/double-pendulum-6392/
is what one observes, completely differently every time, even without actually driving the e.g. double rigid pendulum with a noisy not-quite periodic force so that it never manages to come close to damping down to “simple” linearized small-oscillation behavior.
Here is one place where Tim Ball’s observations above are quite apropos. I think you missed this, but one of the points of the inadequacy of the measurement grid relative to the absurdly inadequate integration grid is that the latter by its nature requires assigning “average” numbers to entire grid cells. Those average numbers, in turn have to in some sense be physically relevant to the actual numbers within each cell. For example, would you say that the models make the implicit assumption, when they assign a temperature of 291.6719345 K and air pressure of .993221573 bar and water vapor content of 0.08112498 and air speed of 2.11558324 m/sec in some single vector direction to a volume of 10,000 cubic kilometers of air (100x100x1 km cell) is supposed to represent that actual average of the relevant measured quantities within that cell? Or are they simply toy parameters, variables created within a toy model that bear little to no necessary resemblance to the variables they are named after?
Damned either way, of course. If they are indeed supposed to represent coarse grained averages, the Ball’s observation that when he personally made actual observation of temperature distributions in just one small portion of the ground, he found that most of the assumptions of homogeneity required to assign an average value to even much smaller cells are simply false. He observed, for example, numerous inversions in the first few hundred meters of just one point in one cell, where the well-established dogma is that the lapse rate is (almost always) monotonic. (I would point out that a pattern of inversions is completely consistent with the topological folding process of lateral turbulent airflow across the warmed ground surface and could probably have been predicted — one can certainly observe them in everything from the original rotational mixing experiments to the patterns made by rising smoke from a cigarette or stick of incense.) On the other hand, if one asserts that they are not supposed to represent the actual averages of the physically relevant quantities in the grid cells, if one asserts that they are some sort of renormalized parameters that are only somehow monotonically connected to the actual averages by some sort of map, well, you’ve just acknowledged that in that case we have no good reason to think that they averages thus produced in 50 years represent the global averages that will be observed at that time! If they aren’t even an accurate representation of the average temperature(s) per cell now, but are the result of an unknown transformation of the average temperatures of the cell into the space where the coarse-grained dynamics is being evaluated as if the cells were “microscopic” in size, using effective interactions between the renormalized variables, how do you expect to be able to map the results of the computation back to actual temperatures then?
That’s really the problem, isn’t it? One of the appealing things about GCMs is that they produce something that really looks like actual weather. With enough tweaking (and a bit of brute force renormalization of energy per timestep to eliminate drift that would otherwise cause them to fail so badly that nobody could miss it) they produce a simulated world on which storms happen, rain falls, droughts occur, all with lots of beautiful chaos, and if one works very hard, one can actually keep the models from having egregious instabilities that take almost all initial states and (for example) collapse to the iceball Earth strange attractor when they are applied to initial conditions corresponding to (say) the middle of the Wisconsin glaciation, with runaway cooling from albedo-driven negative feedback, especially at times in the Wisconsin when atmospheric CO_2 apparently dropped to under 200 ppm and nearly caused mass extinction of plant life.
But the weather they produce isn’t the real weather. It doesn’t even qualitatively correspond to the real weather. And while the weather they produce can certainly be averaged, and while one can certainly assert that this average is “the climate”, the values in each cell of the system aren’t even in a simple one-to-one mapping with the average temperatures one would obtain in the cells with a simple rescaling of the integration grid, let alone the 30 orders of magnitude rescaling needed to contemplate resolving those little whorls of turbulent folding produced by every warmed leaf in the sun.
The conflict here is as old as physics itself. We cannot solve the actual problem of weather or climate and we know it. To be perfectly blunt, we will never be able to solve the actual problem of weather prediction or climate prediction, at least not in any sense that is formally defensible. We therefore tell ourselves that most of the things that prevent us from being able to do so do not matter, that we can average them away. We appeal to damping to erase the otherwise intractable detail that keeps us from being able to proceed. Since this work has to be funded, we make the infinitely adjustable assertion that the next doubling of computational power, the next halving of computational spatiotemporal step size, will at the same time reveal more of the missing detail and make the models more accurate, and yet that the unrevealed detail still remaining isn’t important to the long term predictions. We want to have our cake — models that already work well enough at the level we can afford to compute — and eat it too, the need to build models that work better as soon as we can afford to compute better.
I used to see this all the time in field theory computations presented at conferences — so much so that I named it the “fundamental theorem of diagrammatic quantum field theory”. Speaker after speaker would present the results of their computations in some problem in QFT, and explain the Feynman diagrams that they included in their computation. This was typically a fairly small set of all Feynman diagrams for the problem, because these problems are also basically uncomputable as there are an infinite set of Feynman diagrams and little a priori reason to think that any given diagram omitted from a given computation is, in fact, unimportant. Everybody knows this. We can’t even prove that sums of specific sub-classes of diagrams necessarily converge, even as the formal theory is derivably exact if only one could compute and sum over an infinite number of diagrams.
Each talk would thus without fail begin with a short introduction, explaining the diagrams that were included (usually because they were the diagrams they could algebraically derive formulas for and that they could afford to add up with whatever their computing resources of that particular year were) and then they would invoke the fundamental theorem: “All of the diagrams we omitted from this computation do not significantly contribute.”
Amazingly, the very same people would return two years later, where, armed with new computers and more time they would have redone the entire computation and included several new diagrams omitted two years earlier. The results they obtained would (unsurprisingly) have changed. And yet there it was, at the beginning of the new talk, instead of saying “Hey dudes, we learned that you just can’t trust this diagrammatic perturbation theory thing to work because we included a couple of new diagrams that we thought were unimportant a couple of years ago and they turned out to be important after all, maybe we should all try a completely different approach to solving the problem.” they opened, as before with “All of the diagrams we omitted…”
And this was still better than climate science, because in at least a few cases the computations could be directly compared to measured quantities known to fairly high precision. So each talk would not only compare results of computations carried out to different order in diagrammatic perturbation theory with sums over all ladders, or single loops, or whatever (while still omitting, note well, countless diagrams even at low order as there are a lot of diagrams for even fairly simple systems), they would compare the results to the numbers they were trying to compute. Sometimes adding diagrams made the results better. Sometimes it made them worse. As I said, no theorem of uniform convergence, no real theorem of asymptotic convergence in sums over diagrams of a specific type.
But as a jobs program for physicists, it was and remains today marvelous. And from that point of view or the point of view of pure science, the work is by no means without merit! Some very famous physicists who have made real contributions have done their time doing “wax on, wax off” computations of this sort, or “swinging the chain, swinging the chain” (two movie references, in case this is confusing, google them:-).
At least they knew better than to take the results of forty distinct diagrammatic perturbation theory computations done by forty different groups to forty different orders with forty different precisions and forty different total expenditures of computational resources and with forty different sources of graduate-student-induced computational/method error, carried out over forty years and then do the flat average over all of the results and present the result to the world as a better evaluation of the number in question than the direct experimental measurement of the actual number in nature!
I think that is Tim’s real assertion here. In actual fact, we haven’t got any particularly good idea what the global average temperature actually is. In a sense it is worse than our knowledge of the output of climate models that are supposed to simulate it. We have data on an irregular grid that samples some regions finely and most regions enormously coarsely, even worse than the better GCMs (that at least use a regular grid, even if the usual lat/long decomposition in degrees on a sphere is dumb, dumb, dumb). It is probably excessive to claim that we know the actual global average surface temperature to within one whole degree absolute — if we could agree on some way to define global average surface temperature, if the global average surface temperature we agreed on could be somehow mapped in a sensible way into the average implemented in a renormalized way at the grid resolution we can afford to compute this year.
In summary, let’s say that today I half agree with Nick. Yes, the measurement grid is sadly irrelevant to starting climate models as much as it is in fact relevant to weather prediction with pretty much the same sort of models. The models do not use any actual measurements to initialize, which is at least partly due to the fact that we don’t have any set of actual measurements they could use to initialize that wasn’t egregiously in error over most of the globe. So they start up any old way (hopefully close enough to some actual state of the planet at the past times one starts the models that the model building process remains on the same attractor, oops, hmmm, maybe the correspondence of initial state and reality matters after all!) and hope that after a decade or so the specific behavior associated with the transient goes away and leaves them with, um, “something” predictive of the actual future climate.
It is highly relevant to the model building process itself, as well as the model validation process. Specifically, how can we know that the detail omitted in the climate models is unimportant, especially when it is important in every other chaotic system we’ve ever studied, including many that are far simpler? And, how can we demonstrate that these enormously complex models are actually working when we don’t have a very good idea of the actual state of the planet that can be compared to their predictions? When we don’t know the actual mean surface temperature to within a whole degree (or even how to define “mean surface temperature”), and when our ignorance of that temperature however you define it increases significantly as one moves backwards in time, how can we even resolve model failure?
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