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

We derive the equilibrium energy density spectrum E(k) for 2d Euler flows o n a sphere at low to intermediate total kinetic energy levels where the Ons ager temperature is positive: E(k) = Lambda (2)/4 pik[1 + (4 pi /k)LJ(1)(kL ) - 2 pi exp(-k(2)/4)], where L much greater than1 is a large positive inte ger, and Lambda is the total circulation The proof is based on work of Wign er, Dyson and Ginibre on random matrices. Using this closed-form expression , we give a rigorous upper bound for the equilibrium energy density spectru m of Euler flows on the surface of a sphere: E(k)less than or equal toC(1)k (-2.5) for k much less thanL(1/2) where C-1 = Lambda L-2(1/2) and we conjec ture that C(2)k(-3.5) less than or equal toE(k) for k much less thanL(1/2) from numerical evidence. For k > L-1/2 we have E(k) = (Lambda (2)/4 pi )k(- 1), and between k much less thanL(1/2) and k > L-1/2, the envelope of the g raph 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 deter mines L, such as the case of simple barotropic models for a global atmosphe re with a mountainous southern continent or a ozone hole over the south pol e, our calculations predict that there is a regime of wavenumbers k > L-1/2 with k(-5/3) behaviour. (C) 2001 Elsevier Science B.V. All rights reserved .