Large-scale energy spectra in surface quasi-geostrophic turbulence

The large-scale energy spectrum in two-dimensional turbulence governed by the surface quasi-geostrophic (SQG) equation

partial derivative(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 <center dot > 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.