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

Abstract

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.

17 pages, 2 figures, submitted to Physica D

References in corpus (2)

• Nonlinear transfer and spectral distribution of energy in $\alpha$ turbulence
• Extensivity of two-dimensional turbulence