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 $${\partial }_t(-\Delta {)}^{1/2}\psi +J(\psi ,(-\Delta {)}^{1/2}\psi )=\mu \Delta \psi +f$$ is studied. The nonlinear transfer of this system conserves the two quadratic quantities ${\Psi }_1=<[(-\Delta {)}^{1/4}\psi {]}^2>/2$ and ${\Psi }_2=<[(-\Delta {)}^{1/2}\psi {]}^2>/2$ (kinetic energy), where $<\cdot >$ 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.

arXiv comment11 pages, 4 figures, to appear in JFM

Cited by in corpus (2)

• Effective degrees of nonlinearity in a family of generalized models of two-dimensional turbulence
• Thermalization and free decay in Surface Quasi-Geostrophic flows