J. Fluid Mech. (1978), zyxwvutsr uol. 87, part 2, pp. 305-319 Printed in Great Britain 305 Two-dimensional turbulence on the surface of a sphere By CHA-ME1 TANG? AND STEVEN A. ORSZAG Department of Mathematics, Massachusetts Institute of Technology, Cambridge (Received 25 August 1977) zyxwv Large-scale atmospheric flow shares certain attributes with two-dimensional turbu- lence. In this paper, we study the effect of spherical geometry on two-dimensional turbulence. Energy transfer is multi-component in spherical geometry in contrast to energy transfer among triads of wave vectors in Cartesian geometry. It follows that energy transfer is more local in spherical than in Cartesian geometry. Enstrophy transfer to higher wavenumbers in spherical geometry is less than enstrophy transfer to higher wavenumbers in Cartesian geometry. Since both energy and enstrophy are inviscid constants of motion, the back transfer of energy is also less in spherical than in Cartesian geometry. Therefore, with a finite viscosity, enstrophy decays more slowly in spherical geometry than in Cartesian geometry. Here these conjectures are tested numerically by spectral methods. The numerical results agree well with the conjec- tures. 1. Introduction I n this paper, we compare two-dimensional turbulent flows on the surface of a sphere with two-dimensional turbulent flows in Cartesian geomet,ry. Two-dimensional turbu- lent flow in Cartesian geometry has been the subject of much study (Onsager 1949; Lee 1951; Fjartoft 1953; Kraichnan 1967; Batchelor 1969; Lilly 1971; Leith 1971; Herring et zyxwvutsrqp al. 1974). While two-dimensional turbulence is not yet realizable in the laboratory, it is thought that these flows are relevant to atmospheric dynamics. Three- dimensional quasi-geostrophic flow in the atmosphere away from the equator was found by Charney (1971) to exhibit two scalar invariants, the total kinetic energy and mean-square zyxwvut ' pseudo-potential vorticity ', similar to the quadratic invariants of inviscid two-dimensional turbulence. In $ 2, we review the important concepts of two-dimensional turbulence in Cartesian geometry. In $3, we formulate the problem in spherical geometry. We shall show that the appropriate two-dimensional wavenumber for flows on the sphere is the degree zy n, not the order zyxwvut m of the surface harmonic YE. Energy and enstrophy transfer will be shown to be more local in wave space for two-dimensional turbulence on a sphere than for planar turbulence. The consequences of this locality of transfer on spheres will be discussed and substantiated by numerical results in zyxw $4. t Present address : Applied Physics Laboratory, The Johns Hopkins University, Laurel, Maryland. I1 FLM 87