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 \[\partial_t(-\Delta)^{1/2}\psi+J\big(\psi,(-\Delta)^{1/2}\psi\big)=\mu\Delta\psi+f\] is studied. The nonlinear transfer of this system conserves the two quadratic quantities $\Psi_1\,{=}\,\langle[(-\Delta)^{1/4}\psi]^2\rangle/2$ and $\Psi_2\,{=}\,\langle[(-\Delta)^{1/2}\psi]^2\rangle/2$ (kinetic energy), where $\langle{\bm \cdot}\rangle$ denotes a spatial average. The energy density $\Psi_2$ is bounded and its spectrum $\Psi_2(k)$ is shallower than $k^{-1}$ in the inverse-transfer range. For bounded turbulence, $\Psi_2(k)$ 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.