Triple Cascade Behavior in Quasigeostrophic and Drift Turbulence and Generation of Zonal Jets

Abstract

We study quasi-geostrophic turbulence and plasma drift turbulence within the Charney-Hasegawa-Mima (CHM) model. 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 [1,2] in two special cases -large-scale turbulence and anisotropic turbulence. Papers [1, 2] 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 Fjørtoft argument of [1, 2] 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 Fjørtoft 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.

Hasegawa–Wakatani and Modified Hasegawa–Wakatani Turbulence Induced by Ion-Temperature-Gradient Instabilities

“…The intermittence is characterized by the generation of a turbulence burst at time tω d0 25, as the result of the emergence of the resonant TIM with toroidal number n = 1 (note that the initial perturbed mode n = 5 has disappeared at that time). Diagrams in Figure 10 reveal the occurrence of (nonlinear) streamers with a broad spectrum (with a toroidal number of n ∼ [16][17][18][19][20][21][22][23][24], as can be seen in the plot shown at time tω d0 = 25.6. At time tω d0 = 26.8, an inverse cascade takes place leading to a dominant mode on the toroidal number n = 9, as indicated in the bottom panel in Figure 9.…”

“…However, this term disappears when we consider its average over α. From (19) (20) which clearly indicates that the nature of the emerged ZF is driven only by the Reynolds tensor. The similarity between the reduced Hamiltonian model for trapped ions and the CHM equation had been already pointed out in Ref.…”

“…Even in its simplified barotropic version (with infinite Rossby deformation radius), the commingling of strong nonlinearity, strong anisotropy and Rossby waves give rise to complicated dynamics (see for instance Refs. [19,20]). …”

On a Quasi-Invariant Associated with the Emergence of Anisotropy in Two-Dimensional Turbulence on a Rotating Sphere

“…This quasi-invariant was discovered by Balk (1991) and Balk (2005) demonstrated that its conservation leads to the spectral anisotropy. The quasi-invariant was termed as zonostrophy by Nazarenko and Quinn (2009). The conservation of zonostrophy is related to the concept of resonant triad interactions.…”

“…In such cases, zonostrophy is conserved approximately, in other words, it is conserved relatively well compared to other non-conserved quantities. Nazarenko and Quinn (2009) conducted numerical time-integrations of two-dimensional vorticity equation on a β-plane and showed that zonostrophy is conserved relatively well when the nonlinearity is sufficiently weak and that its conservation corresponds to the anisotropic upscale energy transfer. For the spherical system, quasi-invariants such as zonostrophy have not been discovered.…”