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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.

##### MSC:
 76F99 Turbulence 76U05 General theory of rotating fluids
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