Impeded inverse energy transfer in the Charney–Hasegawa–Mima model of quasi-geostrophic flows

Abstract

The behaviour of turbulent flows within the single-layer quasi-geostrophic (Charney–Hasegawa–Mima) model is shown to be strongly dependent on the Rossby deformation wavenumber $\lambda$ (or free-surface elasticity). Herein, we derive a bound on the inverse energy transfer, specifically on the growth rate ${\rm d}\ell/{\rm d} t$ of the characteristic length scale $\ell$ representing the energy centroid. It is found that ${\rm d}\ell/{\rm d} t\le2\|q\|_\infty/(\ell_s\lambda^2)$, where $\|q\|_\infty$ is the supremum of the potential vorticity and $\ell_s$ represents the potential enstrophy centroid of the reservoir, both invariant. This result implies that in the potential-energy-dominated regime ($\ell\ge\ell_s\,{\gg}\,\lambda^{-1}$), the inverse energy transfer is strongly impeded, in the sense that under the usual time scale no significant transfer of energy to larger scales occurs. The physical implication is that the elasticity of the free surface impedes turbulent energy transfer in wavenumber space, effectively rendering large-scale vortices long-lived and inactive. Results from numerical simulations of forced-dissipative turbulence confirm this prediction.