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We could probably have a debate over the details of some of this. I don’t know what your background is, but I’m just an advanced layman who’s more than just a little bit literate on these topics. I agree that mathematically modeling this is hard. Very hard, in fact; but you can still do it well enough to extract valuable data from this. Just a note on such models ln how I see it.
If you look at global circulation models (GCM’s), they use physical data on the chemical composition of the ground, atmosphere, air and sea temperatures, wind velocities, rainfall, river flows, et al., the key variable in all these being greenhouse gases (‘GHG’s’ which we all know) because they absorb and re-radiate heat back to the Earth. The most important GHG is water vapor; but the problem is it’s inherently localized; making it the least controllable and hardest to obtain data on a global average; and it accounts for between 1/3rd or 2/3rd’s of the GHG effect. The next one is CO2, followed by methane and a few other industrial chemicals. If you compare things on a simple molecular basis, methane is more potent than CO2, but it doesn’t remain in the atmosphere for as long (about 5-6 years if I recall), compared to roughly a century for CO2. There’s much less methane in the atmosphere than CO2 but that could change if emissions from the methane clathrates in the sediment under the east Siberian shelf begin to accelerate.
This might be more to your point perhaps but yes it’s known that GCM’s have weaknesses. The first is that they assume vertical atmospheric thermal convection without wholly taking to account the horizontal component of circulation that arises from ocean currents and mixing. The other is that they divide the surface of the globe into icosahedrons but the minimum size of the grid area (with the exception of the poles) is way too large. You can’t simulate small scale behavior within just a few kilometers (which is nowhere near the 10’s of 100’s of meters necessary to predict local precipitation patterns or storm paths). That’s one of just a handful of problems which is why they also rely on comparative models, but the general trend of things has accelerated even faster than scientists forecast after factoring in the uncertainty of their models.
Allow me to clarify. I'm aware of some of the weaknesses in the models that are used purely for the physical systems, but I actually don't think they're that bad. They're actually pretty decent, AFAICT, where "pretty decent" is a sort of term of art to describe situations where you have an okay sense for the scale of your error analysis and can sort of understand when it might be a problem or not a problem.
Sigh. I really don't know that I have a good way of explaining this intuition. Maybe a story. Long ago, I paid a lot more attention to fairly straightforward aircraft control topics at conferences. One question that came up surprisingly often was, "...would you fly on an airplane that is being controlled by your controller?" There is just a qualitative sense that you develop for how bad the badness is. How the error is likely to be structured, how likely genuinely destructive failure modes are, magnitudes of expected error under various noise distributions, etc.
With that, I guess I'll just say again that I don't think the climate models are "that bad". They're not garbage. There is error, we know some of the sources of error (you mention a couple), and we have an intuitive sense for about how big it may be.
Conversely, what I have significant theoretical disputes with is specifically trying to model economic systems coupled with climate systems. We basically can't even get off the ground, theoretically speaking. The timescales are the wrong way round. When you talk about the apparent fastness of climate dynamics, they are still figuratively glacial in comparison to the dynamics of economic systems (and I sometimes tack on political systems, because it's amazing how many people try to make completely whacko claims about these, too). You did great to realize that attempts like using a static damage function and then proceeding with a simple amortization are bonkers. My point is that the underlying theoretical reason why they're bonkers is because that's just not how one does anything with timescale-separated coupled dynamical systems.
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