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In geophysical and plasma contexts, zonal flows (ZFs) are well known to arise out of turbulence. We elucidate the transition from homogeneous turbulence without ZFs to inhomogeneous turbulence with steady ZFs. Starting from the equation for barotropic flow on a β plane, we employ both the quasilinear approximation and a statistical average, which retains a great deal of the qualitative behavior of the full system. Within the resulting framework known as CE2, we extend recent understanding of the symmetry-breaking zonostrophic instability and show that it is an example of a Type I s instability within the pattern formation literature. The broken symmetry is statistical homogeneity. Near the bifurcation point, the slow dynamics of CE2 are governed by a wellknown amplitude equation. The important features of this amplitude equation, and therefore of the CE2 system, are multiple. First, the ZF wavelength is not unique. In an idealized, infinite system, there is a continuous band of ZF wavelengths that allow a nonlinear equilibrium. Second, of these wavelengths, only those within a smaller subband are stable. Unstable wavelengths must evolve to reach a stable wavelength; this process manifests as merging jets. These behaviors are shown numerically to hold in the CE2 system. We also conclude that the stability of the equilibria near the bifurcation point, which is governed by the Eckhaus instability, is independent of the Rayleigh-Kuo criterion. Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. New J. Phys. 16 (2014) 035006 J B Parker and J A Krommes New J. Phys. 16 (2014) 035006 J B Parker and J A Krommesdeformation radius, ξ is white-noise forcing, μ is the constant friction, and ν is the viscosity with hyperviscosity factor h. The ZF behavior in numerical simulations of (2) is shown in figure 1(b).Here and the rest of the paper, unless otherwise specified, the parameters take the values β = 1, New J. Phys. 16 (2014) 035006 J B Parker and J A Krommes 4 New J. Phys. 16 (2014) 035006 J B Parker and J A Krommes 5 New J. Phys. 16 (2014) 035006 J B Parker and J A Krommes 6 New J. Phys. 16 (2014) 035006 J B Parker and J A Krommes New J. Phys. 16 (2014) 035006 J B Parker and J A Krommes 13 W U and = z z z { , } W U , we have the unsymmetrized version N un , New J. Phys. 16 (2014) 035006 J B Parker and J A Krommes

The wave kinetic equation (WKE) describing drift-wave (DW) turbulence is widely used in studies of zonal flows (ZFs) emerging from DW turbulence. However, this formulation neglects the exchange of enstrophy between DWs and ZFs and also ignores effects beyond the geometrical-optics limit. We derive a modified theory that takes both of these effects into account, while still treating DW quanta ("driftons") as particles in phase space. The drifton dynamics is described by an equation of the Wigner-Moyal type, which is commonly known in the phase-space formulation of quantum mechanics. In the geometrical-optics limit, this formulation features additional terms missing in the traditional WKE that ensure exact conservation of the total enstrophy of the system, in addition to the total energy, which is the only conserved invariant in previous theories based on the WKE. Numerical simulations are presented to illustrate the importance of these additional terms. The proposed formulation can be considered as a phase-space representation of the second-order cumulant expansion, or CE2.Comment: 14 pages, 4 figure

Zonal flows are well known to arise spontaneously out of turbulence. We show that for statistically averaged equations of the stochastically forced generalized Hasegawa-Mima model, steady-state zonal flows and inhomogeneous turbulence fit into the framework of pattern formation. There are many implications. First, the wavelength of the zonal flows is not unique. Indeed, in an idealized, infinite system, any wavelength within a certain continuous band corresponds to a solution. Second, of these wavelengths, only those within a smaller subband are linearly stable. Unstable wavelengths must evolve to reach a stable wavelength; this process manifests as merging jets.Zonal flows (ZFs) -azimuthally symmetric, generally banded, shear flows -are spontaneously generated from turbulence and have been reported in atmospheric 1 and laboratory plasma 2 contexts. Recently, they have also been observed in astrophysical simulations. 3 In magnetically confined plasmas, ZFs are thought to play a crucial role in regulation of turbulence and turbulent transport. 4,5 A greater understanding of ZF behavior is valuable for untangling a host of nonlinear processes in plasmas, including details of transitions between modes of low and high confinement.Zonal flows remain incompletely understood, even regarding the basic question of the jet width (wavelength). In the plasma literature, one finds modulational or secondary instability calculations of ZF generation, 5,6 but these cannot provide information on a saturated state. Other theories typically make an assumption of longwavelength ZFs and leave the ZF scale as an undetermined parameter. 7 Within geophysical contexts, various authors have attempted to relate the jet width or spacing to length scales that emerge from the vorticity equation by heuristically balancing the magnitudes of the Rossby wave term and the nonlinear advection. Those scales include the Rhines scale and other, similar scales. 8-10 A Rhines-like length scale is also obtained from arguments based on potential vorticity staircases. 11,12 However, neither the heuristic Rhines estimates nor the paradigm of potential vorticity inversion and mixing generalize to more complex situations involving realistic plasma models. We are therefore motivated to seek a more systematic approach to determining the ZF width that may offer such a generalization.A related topic is the merging of jets. Coalescence of two or more jets is ubiquitous in numerical simulations. 13,14 The merging process occurs during the initial transient period before a statistically steady state is reached. It is clear that the merging is part of a dynamical process through which the ZF reaches its preferred length scale, but the merging phenomeon has not been understood thus far.Our present work addresses these questions in the context of the stochastically forced generalized Hasegawa-Mima (GHM) equation, 15,16 a model of magnetized plasma turbulence in the presence of a background density gradient. This model is mathematically similar to the barotropic vortici...