Statistical Dynamics of Two-Dimensional Inviscid Flow on a Sphere

We derive the statistical mechanical equilibrium properties of two-dimensional flow on a sphere, described by the truncated inviscid nondivergent barotrapic model. It is found that probability distribution functions and expectation values are considerably different from those on a rotating beta-plane. The approach of numerical simulations to equilibrium is established by integrations for 208 days starting with observed meteorological global fields. It is found that the smaller scales of the observed initial field with a rhomboidal truncation wavenumber of 15 are indistinguishable from those of a realization of the equilibrium ensemble, while by days 70–80 this applies to all scales including the slowly changing zonal flow contributions. The effects of changing the resolution in the numerical simulations are examined and it is shown that the growth of differences between high-and low-resolution integrations may also be explained in terms of the nonuniforin relaxation of different scales to equilibrium. We find that differences between such simulations started with identical fields are under-estimated for the first few days, compared with using increased initial resolution in the high-resolution integration, while subsequently differences at the larger scales tend to be overestimated as the smaller scales equilibrate. For a given number of degrees of freedom, the equilibrium spectra are found to be relatively insensitive to the truncation scheme used and we propose the use of a more efficient parauelogrammic truncation scheme for numerical spectral models.