- Ray Pierrehumbert
As I said on that thread, I do indeed think we can do much better. Here is a link to a presentation by Paul Williams that shows a small example of how we can do better.http://www.newton.ac.uk/files/seminar/20101208153016101-152652.pdfWhat is interesting about it I think is the demonstration that changes to the time step in a GCM can affect the long time averages of the solution.Perhaps at some point, MT, I will send you a longer exposition of recent progress in computational fluid dynamics. There has been some interesting recent work showing that in fact the glosses about current simulation practice need revision. Unfortunately, the new work shows that things are not nearly as reliable or accurate as many have thought.
I will make a couple of preliminary comments here because they are really very obvious and easy to verify. They probably were drowned out at ATTP by irrelevancies. There is a truly vast literature on better ways to stabilize numerical schemes through unwinding and also a vast literature on turbulence modeling. If, as I hope, some climate modeler is reading this, please lets follow up on it. I have references for all this.I believe that at least some GCMs use high order centered spatial discretizations stabilized by hyper viscosity. There is a vast literature on unwinding as a better way to stabilize these things. There are also modern finite element methods such as SUPG developed by Tom Hughes at Austin. Leszek Demkowicz has done the best work I know of in rigorous higher order methods and adaptive use of those methods. Whether or not it will change things dramatically I don't think anyone can say as there appears to be NO work on this other than Williams very small first step. But regardless, there are huge benefits to working on it if for no other reason than that computer costs can come down dramatically. Artificial viscosity methods are really very outdated.On this topic the leapfrog scheme Williams improved is really a very bad method and this is really not even debated as everyone knows it very well. When I was a graduate student we studied this method and even then the textbook stated that the method was prone to nonlinear instabilities. There has been 40 years of progress here and modern methods are dramatically better both in their dissipation but also in their accuracy and error control. Some are even adaptive. Even what Williams did ( which is still very far from optimal) had a very significant effect on weather skill for the model worked on.Turbulence is I believe very important in the atmosphere. We know from balloon and flight testing that it is also quite strong at altitude. However, there are also regions where the flow is largely laminar. This issue of transition from laminar to turbulent flow is an unsolved problem, but once again there has been tremendous progress since the 1970's. In the early days algebraic models were all that could be afforded. Basically, the eddy viscosity is computed just from the local properties of the fluid. A tremendous step forward was PDE based methods where the eddy viscosity satisfies a global differential equation that is solved simultaneously with the Navier-Stokes equations. Well, at least in theory, but that's another story. Most RANS codes actually don't really solve it very well. This seems to me to be an area where in principle there could be very large gains in accuracy. Nick Stokes, who perhaps will comment here, gave me a reference and quite frankly it seemed like an old method and there were no references to the recent literature. It's certainly worth a little time to look into as modeling methods have gotten a lot better even in the last decade.Sorry to not provide references here. I'll do that in due time. Googling SUPG is a way to start. Demkowicz has a web site at the ICES institute as does Hughes.
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