Simulation of exact barotropic vorticity equation solutions using a spectral model



A numerical spectral model of the barotropic atmosphere is used to simulate well-known exact solutions of the vorticity equation for an ideal incompressible fluid on a rotating sphere. Primary emphasis is received to the behavior of the relative error between the exact and numerical solutions as well as to preserving the total kinetic energy, integral enstrophy, and geometric structure of the solutions (zonal flows, Rossby-Haurwitz waves, Wu-Verkley solutions, and Verkley's dipole modons). The 10-day integrations carried out with the model show that the classical exact solutions (RH waves) can be calculated to a good approximation. However, the instability of some exact generalized solutions with respect to initial errors and the errors associated with nonzero numerical model forcing can be a serious obstacle in simulating long-time behavior of such solutions. If it is the case then even highly truncated model with very small time step fails to resolve the problem, and the paths of the numerical and exact solutions diverge from each other with time. Nevertheless, the total energy and integral enstrophy of all the numerical solutions are conserved with a high degree of precision at least during first 10 days.

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