<h2>Chaos, CFD and GCMs.</h2>

There has been a flurry of skeptic blogging (and commentary from me) on chaos and climate models. It's generally along the lines that chaos renders GCMs unworkable because of small changes magnifying or some such, with words like coupled and non-linear. Kip Hansen has a series at WUWT, finishing here. Like many such, it shows the Lorenz trajectories produced by a set of three slightly non-linear equations. I'll develop that with a gadget to explore these curves and their attractor in a future post. Tomas Milanovic has one of an intermittent series of posts (latest "Determinism and predictability") at Climate Etc, of which the general theme is the unsolvability of Navier-Stokes equations due to some effect of non-linearity negating proof of existence and uniqueness, or some such.

My standard response to all this is, look at Computational Fluid Dynamics (CFD, which has been my professional activity for the last thirty years). It is a major established engineering tool based on numerically solving the Navier Stokes equations, and has dealt with the chaos (turbulence) from the beginning. And the climate models are just large scale CFD. There are certainly difficulties with the solution, mainly to do with the necessary sub-grid modelling (in both CFD and GCMs). But they aren't to do with the fact that the solutions don't relate to initial conditions. In fact, that is a benefit, since initial conditions are hardly ever known accurately.

And the theoretical issues of existence and uniqueness etc don't impinge on practice. Algorithms are used which generate solutions on a gridded or meshed space with time stepping. These solutions satisfy on that scale the conservation laws of momentum, mass and energy, which are also expressed by the N-S equations. If you find such a solution, it doesn't matter whether it's existence could be proved in advance. As for uniqueness, the solution procedure itself will generally indicate whether different solution pathways are possible. One CFD scientist, David Young, has been objecting that some recent work, in which he has a part, does show non-unique solutions. But as far as I see, this is in situations like near-stall on a wing, where reality itself is far from predictable.

The CE post had an odd answer to this - yes, CFD works, but only on a scale of up to a few metres. This is of course unphysical - there is no such restriction on the physical laws, nor in the discretised algorithms is any physical scale limitation built in. And of course, GCM's are just Numerical Weather Prediction (NWP) programs, run for longer periods. Most sensible people concede that these work quite well, despite the many km scale.

What people who like to show fancy chaos pictures rarely dwell on is the nature of attractors. These are what distinguish chaos from randomness. And they are typically the results that are sought from CFD analysis. In CFD, initial conditions are usually just a nuisance (because you rarly have good data, and when you try and specify them, there is usually something that will generate unintended disturbances). The standard remedy is to run the program for a while to let these settle out. This takes advantage of the fact that initial conditions are swept away in chaos. GCM's do the same. They typically "wind back" to start at some time well before the period of interest. This would be bad if initial conditions mattered, because data back then is less reliable. But it isn't bad, because they don't. Again, it is better to let artefacts settle before the solutions are needed.

This lack of concern with initial conditions in a search for attractors, relates to the frequent criticism of GCMs as predictors. GCM's find out about climate (attractor), but don't predict the trajectories that converge to them (weather). That relates to the initial condition issue - models can only generate trajectories that are possible in the circumstances, not ones that will reproducibly happen.

When trying to explain why GCMs do really work, and attractors are the key, I often post this GFDL video of modelled ocean SST over seasons. I say that it shows many transient effect, from various eddies to longer term events like ENSO. None of these are predictions for Earth. The actual eddies won't happen, nor will the ENSO events, at least not at the stated times. But this solution which just came from specifying bottom topography and various long term forcings (energy input) comes up with familiar patterns like Gulf Stream and other major ocean currents. The wiggles vary, but the current is there. There is underlying physics which determines the transfer of heat from the Caribbean to the North Atlantic. And GCMs can tell you how that effect of physics will relate to changes in forcing. Anyway, here is the video:



In my next post, I'll develop the notion of an attractor using the simple Lorenz system of differential equations. These show two important things. Trajectories follow a path with a pattern that is, after some convergence from the initial point, similar for all cases. This is the attractor, and in contrast to the hypersensitive dependence on initial conditions, the dependence of that trajectory on the three parameters of the system is gradual, although over the full range allows many very different shapes. To do this, I'll show a Javascript/WebGL gadget that allows you to vary initial conditions and parameters, and visualise the trajectories in 3D.