The modulational instability in the extended Hasegawa-Mima equation with a finite Larmor radius

Abstract

The effects of the finite Larmor radius on the generation of zonal flows by the four-wave modulational instability are investigated using an extended form of the Hasegawa-Mima equation. Growth rates of the zonal mode are quantified using analytical predictions from a four-mode truncated model, as well as from direct numerical simulation of the nonlinear extended Hasegawa-Mima equation. We not only consider purely zonal flows but also examine the generic oblique case and show that, for small Larmor radii, off-axis modes may become dominant. We find a key parameter Mρ which characterises the behaviour of the system due to changes in the Larmor radius. We find that, similarly to previous results obtained by changing the driving wave amplitude, two separate dynamical regimes can be accessed. These correspond to oscillatory energy transfer between zonal flows and a driving wave and the fully saturated zonal flow.