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