A microscopic derivation of the equilibrium energy density spectrum for barotropic turbulence on a sphere

Abstract

We derive the equilibrium energy density spectrum E( k) for 2d Euler flows on a sphere at low to intermediate total kinetic energy levels where the Onsager temperature is positive: E(k)=Λ 2/4 πk[1+(4 π/k)LJ 1(kL)−2 π exp (−k 2/4)] , where L⪢1 is a large positive integer, and Λ is the total circulation. The proof is based on work of Wigner, Dyson and Ginibre on random matrices. Using this closed-form expression, we give a rigorous upper bound for the equilibrium energy density spectrum of Euler flows on the surface of a sphere: E( k)⩽ C 1 k −2.5 for k⪡ L 1/2 where C 1= Λ 2 L 1/2 and we conjecture that C 2 k −3.5⩽ E( k) for k⪡ L 1/2 from numerical evidence. For k> L 1/2 we have E( k)=( Λ 2/4π) k −1, and between k⪡ L 1/2 and k> L 1/2, the envelope of the graph of E( k) changes smoothly from a k −2.5 slope to a k −1 slope. Thus, for a punctured sphere with a hole over the south pole whose diameter determines L, such as the case of simple barotropic models for a global atmosphere with a mountainous southern continent or a ozone hole over the south pole, our calculations predict that there is a regime of wavenumbers k> L 1/2 with k −5/3 behaviour.