(For a test case involving the primitive equations in spherical geometry
see Polvani, Scott,
and Thomas (2004).)
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Baroclinic instability in the "inviscid" shallow water equations at t=144 hours, computed at spectral resolutions T170, T341, and T682. Initial conditions are as specified in Galewsky, Scott, and Polvani (2004). Click on the image to see a higher resolution version (500 kbytes) |
The figure shows the development of barotropic instability in the shallow water equations. A small amount of diffusion is needed at each resolution to control the build-up of enstrophy at small scales. The higher the resolution, the weaker the diffusion needed. Decreasing the diffusion as resolution is increased is an attempt to converge to the inviscid equations.
From the figure, it appears that numerical convergence in this case will be attained somewhere between T1365 and T2730. Note that the "picture norm" definition of convergence is being used here, in which convergence can be said to have been achieved when doubling the resolution results in no perceptible change in the picture. This is actually a much more stringent test of convergence than many traditional norms. For example, the usual L2 norm involves a global integral that averages many of the errors at lower resolution, and indicates convergence in this case already at around T341.
Because the inviscid equations require such high resolution for convergence (in any dynamically interesting case), they are unsuitable as test case solutions for model development. A more suitable suitable test case is obtained by using the diffusive equations with a fixed diffusion coefficient, i.e. a diffusion coefficient that is held constant as resolution is increased. See Galewsky, Scott, and Polvani (2004) for further details.