Kaufmann and Stern contained a reference to the provocatively titled Govindan et al. , Global Climate Models Violate Scaling of the Observed Atmospheric Variability, Phys. Rev. Lett. 89, available here . I’ll comment at some time on the scaling issues, but it contained the following concise description of GCMs which I liked:
The models [coupled atmosphere-ocean general circulation models (AOGCMs) ] provide numerical solutions of the Navier Stokes equations devised for simulating mesoscale to large-scale atmospheric and oceanic dynamics. In addition to the explicitly resolved scales of motions, the models also contain parametrization schemes representing the so-called subgrid-scale processes, such as radiative transfer, turbulent mixing, boundary layer processes, cumulus convection, precipitation, and gravity wave drag. A radiative transfer scheme, for example, is necessary for simulating the role of various greenhouse gases such as CO2 and the effect of aerosol particles. The differences among the models usually lie in the selection of the numerical methods employed, the choice of the spatial resolution, and the subgrid-scale.
Note carefully the focus on Navier-Stokes equations, which are notoriously intractable with very difficult mathematics. In fact, they are one of the Clay Insitute’s seven "Hilbert" problems for the 21st century – a $1 million prize is attached to any progress on the mathematics of the Navier-Stokes equations. So one should not assume that brute force numerical methods necessarily evade difficult and subtle mathematical problems. The Clay Institute’s snapshot of the problem is:
Mathematicians and physicists believe that an explanation for and the prediction of both the breeze and the turbulence can be found through an understanding of solutions to the Navier-Stokes equations. Although these equations were written down in the 19th Century, our understanding of them remains minimal. The challenge is to make substantial progress toward a mathematical theory which will unlock the secrets hidden in the Navier-Stokes equations.
Charles Fefferman writes in the Clay Institute problem statement paper, Existence and Smoothness of the Navier-Stokes equation here :
Fluids are important and hard to understand. There are many fascinating problems and conjectures about the behaviour of solutions of the Navier-Stokes equations. Since we don’t even know whether these solutions exist, our understanding is at a very primitive level. Standard methods from PDE appear inadequate to settle the problem. Instead, we probably need some deep, new ideas.
In May 2002, the Clay Mathematics Institute (CMI) of Cambridge, Massachusetts, in an initiative to further the study of mathematics, allocated a $7m prize fund for the solution of seven Millennium Problems, ‘focusing on important classic questions that have resisted solution over the years’. One of the $1m problems stands out for its massive practical importance: the solution of the Navier-Stokes equations (NSEs) for fluid flow.
Although there are many named variants and special cases, the fundamental equations are the incompressible Navier-Stokes for Newtonian fluids. In their most compact form, they comprise a pair of vector partial differential equations (PDEs): one expresses the forces acting (pressure, viscosity and body forces); the other is the continuity equation, which says that divergence of the velocity field is zero for an incompressible fluid (that is, ‘what comes in, goes out’).
The NSEs are among the most-studied partial differential systems, the subject of around 15-20 published papers a week. Nevertheless, they’re among the least understood at a theoretical level.
Figure from Girvan article.
In short, A GCM "control run", is essentially one numerical run from a hugely complicated Navier-Stokes equation, the deep mathematical properties of which mathematicians say they know very little. Climatologists on the other hand appear to know the results to high degres of certainty – remarkable.
Reference: R. B. Govindan, Dmitry Vyushin, Armin Bunde, Stephen Brenner,Shlomo Havlin,1 and Hans-Joachim Schellnhuber, Global Climate Models Violate Scaling of the Observed Atmospheric Variability. 2002, Phys Rev Letters, 89 http://www.atmosp.physics.utoronto.ca/people/vyushin/Papers/Govindan_Vyushin_PRL_2002.pdf