On spectral scaling laws for averaged turbulence on the sphere

Spectral analysis for a class of Lagrangian-averaged Navier--Stokes (LANS) equations on the sphere is carried out. The equations arise from the Navier--Stokes equations by applying a Helmholtz filter of width $\alpha$ to the advecting velocity $\beta$ times. Power laws for the energy spectrum are derived and indicate a $\beta$-dependent scaling at wave numbers $l$ with $\alpha l\gg 1$. The energy and enstrophy transfer rates distinctly depend on the averaging, allowing control over the energy flux and the enstrophy flux separately through the choice of averaging operator. A necessary condition on the averaging operator is derived for the existence of the inverse cascade in two-dimensional turbulence. Numerical experiments with a structure-preserving integrator confirm the expected energy spectrum scalings and the robustness of the double cascade under choices of the averaging operator.