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 on a sphere at low to intermediate total kinetic energy levels where the Onsager temperature is positive: E(k)=Λ2/4πk[1+(4π/k)LJ1(kL)−2πexp(−k2/4)], 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)⩽C1k−2.5 for k⪡L1/2 where C1=Λ2L1/2 and we conjecture that C2k−3.5⩽E(k) for k⪡L1/2 from numerical evidence. For k>L1/2 we have E(k)=(Λ2/4π)k−1, and between k⪡L1/2 and k>L1/2, 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>L1/2 with k−5/3 behaviour.