Large-scale energy spectra in surface quasi-geostrophic turbulence

Details Large-scale energy spectra in surface quasi-geostrophic turbulence Large-scale energy spectra in surface quasi-geostrophic turbulence

2004-12-10

arxiv.org/abs/nlin/0412024v1

Nonlinear Sciences

Chaotic Dynamics

11 pages, 4 figures, to appear in JFM

Scientific paper

10.1017/S0022112004002848

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.

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