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Large-scale energy spectra in surface quasi-geostrophic turbulence. (English) Zbl 1065.76123
Summary: We study the large-scale energy spectrum in two-dimensional turbulence governed by the surface quasi-geostrophic equation \([\partial_t(-\Delta)^{1/2} \psi+J\big(\psi,(-\Delta)^{1/2}\psi\big)=\mu\Delta\psi+f]\). The nonlinear transfer of this system conserves 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 \(\langle\cdot\rangle\) denotes 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.

76F99 Turbulence
76U05 General theory of rotating fluids
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