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

The inverse transfer in two-dimensional turbulence governed by the surface quasi-geostrophic (SQG) equation 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 $\psi$ is the streamfunction and $<\cdot>$ denotes a spatial average. In the limit of infinite domain, the kinetic energy density $\Psi_2$ remains bounded. For power-law inverse-transfer region, the inverse flux of $\Psi_1$ diminishes as it proceeds toward sufficiently low wavenumbers, implying that no persistent inverse cascade of $\Psi_1$ is sustainable. The unrealizability of an inverse cascade of $\Psi_1$ implies that there is no direct cascade of $\Psi_2$. Hence, the dual-cascade picture which is widely believed to be realizable in two-dimensional Navier--Stokes turbulence does not apply to SQG turbulence. Numerical results supporting the theoretical predictions are presented.