Homogeneous and Isotropic Turbulence on the Sphere

Abstract

The assumption that the streamfunction for two-dimensional nondivergent flow on the sphere is a homogeneous and isotropic random field is used to obtain a variety of results for the study of large-scale atmospheric turbulence. These results differ somewhat from those for Cartesian geometry. It is shown that the necessary and sufficient condition that the turbulent flow be homogeneous and isotropic is the statistical independence of the spherical harmonic expansion coefficients of the streamfunction and the dependence of the variance of the expansion coefficients only an the wavenumber n. The homogeneity and isotropy of the vorticity field and of the velocities along and perpendicular to the geodesic connecting points on the sphere follows. The usual zonal and meridional velocity components on the sphere are an-isotropic although, at a point, these velocity components are uncorrelated and have equal amounts of kinetic energy. Equations for the covariance function and the spectrum for flow on the sphere am obtained from the barotropic vorticity equation.