Diminishing inverse transfer and non-cascading dynamics in surface quasi-geostrophic turbulence

The inverse transfer in the forced-dissipative surface quasi-geostrophic equation is studied, when the natural dissipation operator μ(−Δ)1/2μ(−Δ)1/2 is employed. The nonlinear transfer of this system conserves the two quadratic quantities Ψ1=〈|(−Δ)1/4ψ|2〉/2Ψ1=〈|(−Δ)1/4ψ|2〉/2 and Ψ2=〈|(−Δ)1/2ψ|2〉/2Ψ2=〈|(−Δ)1/2ψ|2〉/2 (kinetic energy), where ψψ is the stream function and 〈⋅〉〈⋅〉 denotes a spatial average. In the limit of infinite domain, the kinetic energy density Ψ2Ψ2 remains bounded, for the natural dissipation operator. For the power-law inverse-transfer region, the inverse flux of Ψ1Ψ1 diminishes as it proceeds toward sufficiently low wavenumbers, whenever the kinetic energy Ψ2Ψ2 remains bounded. This implies that no persistent (non-dissipative) inverse cascade of Ψ1Ψ1 to ever-lower wavenumbers is sustainable, as long as the dissipation parameter μμ is held fixed. This result does not rule out the possibility that for sufficiently small μμ, a finite inverse flux would reach a certain low wavenumber. Numerical results supporting the theoretical predictions are presented.