Rossby and drift wave turbulence and zonal flows: The Charney–Hasegawa–Mima model and its extensions

Abstract

A detailed study of the Charney-Hasegawa-Mima model and its extensions is presented. These simplest nonlinear partial differential equations suggested for both Rossby waves in the atmosphere and also drift waves in a magnetically-confined plasma exhibits some remarkable and nontrivial properties, which in their qualitative form survive in more realistic and complicated models, and as such form a conceptual basis for understanding the turbulence and zonal flow dynamics in real plasma and geophysical systems. Two idealised scenarios of generation of zonal flows by small-scale turbulence are explored: a modulational instability and turbulent cascades. A detailed study of the generation of zonal flows by the modulational instability reveals that the dynamics of this zonal flow generation mechanism differ widely depending on the initial degree of nonlinearity. The jets in the strongly nonlinear case further roll up into Kármán-like vortex streets and saturate, while for the weaker nonlinearities, the growth of the unstable mode reverses and the system oscillates between a dominant jet, which is slightly inclined to the zonal direction, and a dominant primary wave. A numerical proof is provided for an the extra invariant in Rossby and drift wave turbulence -zonostrophy. While the theoretical derivations of this invariant stem from the wave kinetic equation which assumes weak wave amplitudes, it is shown to be relatively well-conserved for higher nonlinearities also. Together with the energy and enstrophy, these three invariants cascade into anisotropic sectors in the k-space as predicted by the Fjørtoft argument. The cascades are characterised by the zonostrophy pushing the energy to the zonal scales. A small scale instability forcing applied to the model has demonstrated the well-known drift wavezonal flow feedback loop. The drift wave turbulence is generated from this primary instability. The zonal flows are then excited by either one of the generation mechanisms, extracting energy from the drift waves as they grow. Eventually the turbulence is completely suppressed and the zonal flows saturate. The turbulence spectrum is shown to diffuse in a manner which has been mathematically predicted. The insights gained from this simple model could provide a basis for equivalent studies in more sophisticated plasma and geophysical fluid dynamics models in an effort to fully understand the zonal flow generation, the turbulent transport suppression and the zonal flow saturation processes in both the plasma and geophysical contexts as well as other wave and turbulence systems where order evolves from chaos.

Three-dimensional slow Rossby waves in rotating spherical density-stratified convection

“…Rossby waves [1,2] which are caused by the combined effect of rotation and curvature of the surface, exist in various hydrodynamic flows (see [3][4][5], and references therein). Rossby waves have been found in different geophysical phenomena (see [3,4,[6][7][8][9], and references therein), e.g., they are observed in the atmosphere as the large meanders of the mid-latitude jet stream that are responsible for the prevailing seasonal weather patterns and their day-to-day variations (see [3,6,7], and references therein).…”

“…Zonal flows are coherent structures with a jet-like velocity profile in anisotropic flows due to the large aspect ratio and planetary rotation. Zonal flows are produced near the tropopause in the atmosphere (the polar and subtropical jet streams) and in the oceans (the Antarctic circumpolar current) [5,19]. The generation of zonal flows are caused by generation of small-scale meridional Rossby waves by a modulational instability and nonlinear interaction of the Rossby waves [5].…”

Large-scale drift and Rossby wave turbulence

“…Note that pure zonal flows with = k 0 y will never appear in a system if they are not present initially in the CHM model. For other basic models of the CHM family, including models such as the Hasegawa-Wakatani equations, see Connaughton et al (2015).…”

Wave turbulence in the two-layer ocean model

“…Drawing intuition from the one-layer case, it would be natural to assume that in our two-layer system the inverse energy transfer to the barotropic mode is also nonlocal. This is what we will now consider using a similar scale separation technique to that used for the one-layer model (Balk et al, 1990;Connaughton et al, 2010), but now for dominant {− − +} triads.…”

“…Similar to work done in the one-layer case in (Connaughton et al, 2010) the kinetic equation for the small scales n − k can be written as the following anisotropic diffusion equation in k-space: ∂n…”