Thursday, May 29, 2008

Climate models: What went wrong

You shall not curse a deaf man, nor place a stumbling block before the blind ....
- Leviticus 19

Climate models of any kind, including GCMs, involve multiple layers of approximation. They are not the full theory. That is unsolvable and cannot even be properly stated in its full complexity, which is why climate models are used in its place. From a mathematical point of view, the key question is the nature of those approximations.

On the grid. The most obvious approximation is the discrete spacetime "grid" that replaces the spacetime continuum. A continuum and any field on that continuum (pressure, temperature, etc.) contain an infinite amount of information and cannot be represented by a finite list of numbers and thus on a computer. The first step is to replace the continuum with a grid of discrete points in space and instants in time.

A set of thermohydrodynamic variables (pressure, temperature, wind, water) is solved for within a grid cell. In place of variables that are functions of continuously varying time and space, we get discrete points and instants. The continuous integrodifferential equations of physics are replaced in the model approximation by discrete difference and sum equations. This approximation gives rise to discretization error, which is bounded and can be controlled by making the space and time cell size smaller. Hence the quest for ever more computer memory, to handle larger and larger numbers of smaller and smaller climate "cells."

However, this approximation is far from the only one inherent in climate modeling. The mistaken assumption that it is feeds the illusion that bigger computers are all that's needed to reduce the uncertainties. Given the immense complexity and chaotic nature of climate, it's also not the case that bigger computers are a practical solution even for just this modeling error. Attempts to forecast weather over weeks and months have consistently led to the conclusion that computers much larger than any built, calculating for times longer than the lifetime of the universe, would be needed to cope with weather chaos, which in climate manifests itself as atmospheric turbulence.

Poorly defined statistical averages. In place of dealing with chaos directly, climate models sample sets or ensembles of initial conditions, then average over the samples. The climate modeling fallacy arises from this averaging over undefined model spaces. No one understands the full climate theory well enough to enumerate possible climates and assign them probabilities. (Mathematically, there's no "measure on the space of models.") How do you average? How do you know you've got a representative sample of the space of possible "weathers"? No one knows. The workaround today is more ad hoc handwaving, making convenient simplified assumptions there's no way to check and which further butcher the theory.*

Forcing closure on the equations. Discretization of continuous spacetime gives rise to other, more technical modeling errors as well. These additional approximations fall into two broad classes, although these classes of errors interact with each other.

1. Inherent in the complete theory of climate are continuous symmetries of the laws of physics. These laws are independent of translation in space, rotational orientation, and what time it is. Each gives rise to a conserved flow: densities of momentum, angular momentum, and energy. When the theory is discretized to form the numerical approximation, these symmetries are broken and the conservation laws violated. These violated conservation laws (momentum, angular momentum, and energy appearing from and disappearing into nothing) have to be "fixed up" in some way, so as to not produce nonsensical results. These "fixing up" methods, which we'll meet in the next post, themselves introduce ad hoc and uncontrolled approximations.

2. The unchanging identity of a parcel of dry air and a parcel of water gives rise to further conservation laws, relating the flows and densities of water and air. The full theory of climate includes within it the hardest equation of physics, first discovered in the 19th century, the Navier-Stokes equation. It describes the dynamics of fluids (air and water, both in gaseous and liquid states - physicists use "fluid" for both). These equations cannot, even on their own (without the effect of radiation and of the phase transitions of water from ice to liquid to vapor), be solved or even be stated in complete form. Instead, fluid dynamicists in physics and engineering introduce simplified approximations ("forced closure") to covert the fluid dynamics into something that can at least be stated as a complete, self-consistent mathematical problem. Introducing the phase transitions of water and the radiation passing through, being reflected, absorbed, and re-radiated, makes the problem even more intractable. So further approximations (more "forced closures") are introduced.

From a mathematical point of view, these "forced closures" are not controlled approximations. There's no way to bound or estimate the error made in introducing them. In laboratory or engineering applications, we have an "out," namely, controlled experiments that provide an alternative source of insight into the behavior of fluids. We have no controlled experiments for the atmosphere, with its mix of air, discontinuously changing water, and radiation.

Known unknowns and unknown unknowns. Reliable knowledge in the sciences arises from controlled contexts: deduction from explicit assumptions, laboratory experiments, mathematical approximations with bounded errors. In such situations, even if we can't arrive at an exact answer, we know the right questions to ask and get a range of the numerical values we seek. In climate modeling, we are lost. Not much has been attempted in the way of rigorous deduction from the full climate theory, partly because it's so complex. We have no controlled laboratory experiments. And the leap from the full, unsolvable theory to any known model (including the GCMs) is made with uncontrolled approximations. We might know the right question to ask, but have a only vague idea of the numerical range we're aiming for - perhaps on the order of five or so degrees C. It's quantitatively too fuzzy to serve for the kinds of precise conclusions that people seek, temperature changes on the order of tenths of degree C, or even a full degree.

What is climate anyway? And there's a more basic problem: we don't know what "climate" means, unless it means the exact state of the whole atmosphere and oceans at one instant. That's far too vast to comprehend or measure, and it might not even be necessary to know all of it. What's lacking is a reduction of "climate state" that can serve as a simplified abstraction to track. Such a state would need to track something about the state and flow of the air, the heat, and the water. There's no "temperature of the Earth," in spite of the meaningless numbers bandied about. All such intensive thermodynamic measurements are local and vary in space and time. We need something that captures the spatially spread-out nature of climate and the fact that it's controlled by flows, not static reservoirs, of heat, air, and water.

Just as there are few controlled approximations in modeling climate dynamics, there's no controlled and well-defined "state" of climate even to talk about. These are open scientific questions. Unfortunately, they're almost always taken as somehow already answered or are never even asked. But they need to asked, and we need to face the fact that, at present, there are no good answers.

POSTSCRIPT: Chapter 3 of Lorenz's chaos lectures discusses the origins of GCMs from the point of view of someone who was there.
* Readers of this blog might remember this problem from over a year ago in a very different context, the failure of the "multiverse" or "landscape" picture of string theory. There was no way in that case to specify a list of universes and assign their probabilities either.

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