Brenda Quinn

Modulational instability of Rossby and drift waves and generation of zonal jets

We study the modulational instability of geophysical Rossby and plasma drill waves within the Charney-Hasegawa-Mima (CH M) model both theoretically, using truncated (four-mode and three-mode) models, and numerically, using direct simulations of CHM equation in the Fourier space. We review the linear theory of Gill (Geophys. Fluid Dyn., vol. 6, 1974, p. 29) and extend it to show that for strong primary waves the most unstable modes are perpendicular to the primary wave, which correspond to generation of a zonal flow if the primary wave is purely meridional. For weak waves, the maximum growth occurs for off-zonal inclined modulations that are close to being in three-wave resonance with the primary wave. Our numerical simulations confirm the theoretical predictions of the linear theory as well as the nonlinear jet pinching predicted by Manin & Nazarenko (Pit vs. Fluids, vol. 6, 1994, p. 1158). We find that, for strong primary waves, these narrow zonal jets further roll up into Karman-like vortex streets, and at this moment the truncated models fail. For weak primary waves, the growth of the unstable mode reverses and the system oscillates between a dominant jet and a dominate primary wave, so that the truncated description holds for longer. The two-dimensional vortex streets appear to be more stable than purely one-dimensional zonal jets, and their zonal-averaged speed can reach amplitudes much stronger than is allowed by the Rayleigh-Kuo instability criterion for the one-dimensional case. In the long term, the system transitions to turbulence helped by the vortex-pairing instability (for strong waves) and the resonant wave wave interactions (for weak waves).

Triple cascade behavior in quasigeostrophic and drift turbulence and generation of zonal jets

We study quasigeostrophic (QG) and plasma drift turbulence within the Charney-Hasegawa-Mima (CHM) model. We focus on the zonostrophy, an extra invariant in the CHM model, and on its role in the formation of zonal jets. We use a generalized Fjørtoft argument for the energy, enstrophy, and zonostrophy and show that they cascade anisotropically into nonintersecting sectors in k space with the energy cascading towards large zonal scales. Using direct numerical simulations of the CHM equation, we show that zonostrophy is well conserved, and the three invariants cascade as predicted by the Fjørtoft argument.

Feedback of zonal flows on wave turbulence driven by small-scale instability in the Charney-Hasegawa-Mima model

We demonstrate theoretically and numerically the zonal-flow/drift-wave feedback mechanism in an idealised 2-dimensional model of plasma turbulence driven by a small-scale instability. Zonal flows are generated by a secondary modulational instability of the modes which are directly driven by the primary instability. The zonal flows then suppress the small scales thereby arresting the energy injection into the system, a process which can be described using nonlocal (in scale) wave turbulence theory. Finally, the arrest of the energy input results in saturation of the zonal flows at a level which can be estimated from the theory and the system reaches stationarity without large-scale dissipation.

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

A detailed study of the Charney-Hasegawa-Mima model and its extensions is presented. These simple nonlinear partial differential equations suggested for both Rossby waves in the atmosphere and also drift waves in a magnetically-confined plasma exhibit 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. A numerical proof is provided for the extra invariant in Rossby and drift wave turbulence -zonostrophy and the invariant cascades are shown to be characterised by the zonostrophy pushing the energy to the zonal scales. A small scale instability forcing applied to the model demonstrates the well-known drift wave - zonal flow feedback loop in which the turbulence which initially leads to the zonal flow creation, 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.

Triple cascade behaviour in QG and drift turbulence and generation of zonal jets

We study quasi-geostrophic turbulence and plasma drift turbulence within the Charney-Hasegawa-Mima (CHM) model extending the work reported in (Nazarenko and Quinn, 2009). We focus, theoretically and using numerical simulations, on conservation of zonostrophy and on its role in the formation of the zonal jets. The zonostrophy invariant was first predicted in (Balk et al., 1990, 1991) in two special cases – large-scale turbulence and anisotropic turbulence. Papers (Balk et al., 1990, 1991) also predicted that the three invariants, energy, enstrophy and zonostrophy, will cascade anisotropically into non-intersecting sectors in the k-space, so that the energy cascade is “pushed” into the large-scale zonal scales. In the present paper, we consider the scales much less than the Rossby deformation radius and generalise the Fjortoft argument of (Balk et al., 1990, 1991) to find the directions of the three cascades in this case. For the first time, we demonstrate numerically that zonostrophy is well conserved by the CHM model, and that the energy, enstrophy and zonostrophy cascade as prescribed by the Fjortoft argument if the nonlinearity is sufficiently weak. Moreover, numerically we observe that zonostrophy is conserved surprisingly well at late times and the triple-cascade picture is rather accurate even if the initial nonlinearity is strong.