Large-scale energy spectra in surface quasi-geostrophic turbulence

Abstract

The large-scale energy spectrum in two-dimensional turbulence governed by the surface quasi-geostrophic (SQG) equation $$\partial_t(-Δ)^{1/2}ψ+J(ψ,(-Δ)^{1/2}ψ) =μΔψ+f$$ is studied. The nonlinear transfer of this system conserves the two quadratic quantities $Ψ_1=<[(-Δ)^{1/4}ψ]^2>/2$ and $Ψ_2=<[(-Δ)^{1/2}ψ]^2>/2$ (kinetic energy), where $<\cdot>$ denotes a spatial average. The energy density $Ψ_2$ is bounded and its spectrum $Ψ_2(k)$ is shallower than $k^{-1}$ in the inverse-transfer range. For bounded turbulence, $Ψ_2(k)$ in the low-wavenumber region can be bounded by $Ck$ where $C$ is a constant independent of $k$ but dependent on the domain size. Results from numerical simulations confirming the theoretical predictions are presented.