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This is an IPCC chart. It greatly overstated the competence of the models, both in the past and now.

They aren't even close to being able to accurately include the greatest heat reservoir of the Earth, oceans, in their modeling in a gross manner, not to say anything about specific situations as ocean currents. The effect of aerosols is very poorly understood. The effect of clouds is poorly understood. They don't even try to consider the most highly likely main driver of climate, the Sun and the ways it can affect temperature except in the most gross manner, and definitely have undercalculated its effect. Cosmic rays, bacteria, etc... many other things are completely ignored.

Of those factors that are known and reasonably well understood, very few are known well enough to provide data inputs to the models with 2% or lower error bars. Only 30 such factors make would throw off the calculations so much as to make them useless, and there are many more than 30 factors involved in this problem. (I'm being so generous to the IPCC 'scientists' in this paragraph that I'm almost disgusted with myself)

As you say, also, there is a great problem with "dynamics". The "mesh" that the present models are using is ridiculously large (by necessity) and there is no foreseeable computational technique or equipment on the horizon to overcome that. Meanwhile, these GCMs can't even get close to reproducing the past without heroic and by-chance tweaking of the many "free parameters".

Once I saw a calculation of the number of moves in a Chess game - a problem whose complexity is child's play compared to the Earth's climate. The author did something along the lines of this: He calculated the "exponential explosion" for the number of possible chess games and came up with what I found to be a reasonable number (I don't recall what it was, but it was pretty darn big) He then postulated that a chess computer could be made of a single electron, and that each of those 'computers' could evaluate one game in a second. He gave a parallel computation of the number of electrons in the known universe. The long and short of it was that there are not enough electrons in the known universe to have examined every possible chess game in 15billion or so years.

Not Even Close.

The GCM's are examining something far more complex than a chess game, and far more chaotic, and with very few rules that are really known. It is no surprise to me that they are not competent, and I believe it very highly unlikely that they will become so in my lifetime.

As of now, one of the ways they decide if a model run with given parameters is worth "keeping" and continuing or not is whether shows the temperature to increase with increasing CO2 after a few iterations. That alone biases the results of all runs toward increasing temperature regardless of any other parameters, and taints all possible evidence from these models. It would be nice if someday they have to simply admit the major deficiencies.

In my view, the best way to express a concern about the very likely minuscule amount that humans are having on the Earth's climate is to continue observations of the Sun, CO2, methane, cosmic rays, clouds, bacteria(?), etc., and chart correlations of such gross inputs with the actual temperature records. We need to eschew these silly temperature 'models' in order to have a better idea of the chaotic earths long term climate, where the true believer environmentalists will tell you that even a butterfly flapping will have it's effect on the Europe's weather next week.

By the way, I work intimately with supercomputer modeling and the results of such computations. Believe me when I say that even very well understood physical models are subject to huge errors when it comes to comparison of the outputs of these runs with real world data and inputs. We're often stymied as to what is going on because our models didn't suggest things we actually see in or simple, well measured, lab situations.