Formation of solitary zonal structures via the modulational instability of drift waves

Abstract

The dynamics of the radial envelope of a weak coherent drift wave is approximately governed by a nonlinear Schrödinger equation, which emerges as a limit of the modified Hasegawa-Mima equation. The nonlinear Schrödinger equation has well-known soliton solutions, and its modulational instability can naturally generate solitary structures. In this paper, we demonstrate that this simple model can adequately describe the formation of solitary zonal structures in the modified Hasegawa-Mima equation, but only when the amplitude of the coherent drift wave is relatively small. At larger amplitudes, the modulational instability produces stationary zonal structures instead. Furthermore, we find that incoherent drift waves with beam-like spectra can also be modulationally unstable to the formation of solitary or stationary zonal structures, depending on the beam intensity. Notably, we show that these drift waves can be modeled as quantumlike particles ("driftons") within a recently developed phase-space (Wigner-Moyal) formulation, which intuitively depicts the solitary zonal structures as quasi-monochromatic drifton condensates. Quantumlike effects, such as diffraction, are essential to these condensates; hence, the latter cannot be described by wave-kinetic models that are based on the ray approximation.

Theory of the Tertiary Instability and the Dimits Shift from Reduced Drift-Wave Models

“…in simulations of DW turbulence in a linear device [37].) Note that the propagating structures observed here are different from the DW-ZF solitons generated in the largeα limit [38]. They may be related to those seen in simulations of subcritical turbulence [39][40][41].…”

Nonlinear saturation and oscillations of collisionless zonal flows

“…This is understood from the fact that although a primary instability and dissipation would contribute additional terms in our WKE model (section 2.5), they would not affect the drifton phase-space trajectories that we discuss here. For example, in a recent study on the formation of solitary zonal structures [50], we demonstrate numerically that the structure formation is not significantly affected by external forcing and dissipation. Also, the trapping of driftons near phase-space equilibria, which will be discussed in section 4.1, can help explain the localization of DW activity reported recently within the Hasegawa-Wakatani model in [51] and may explain the results of some earlier studies of gyrokinetic plasmas [3, 52].…”

“…The striped structure in figure (b) is due to the interference between the shown trapped distribution and a similar distribution on the next spatial period (not shown). Such structures are discussed in further detail in [50]. (figure (a)) and the Φ 0 =0.35 (figure (b)) case in figure 6(d).…”

“…(Propagating zonal structures are studied in detail in our [50].) The assumption of spatially-monochromatic ZF holds approximately if the ZF's second and higher harmonics do not outpace the fundamental harmonic during the linear MI.…”

Structure formation in turbulence as an instability of effective quantum plasma

“…So far, it was successfully applied to manifestly quantumlike systems, such as those governed by the nonlinear Schrödinger equation [20][21][22][23][24][25][26] and the Klein-Gordon equation [27,28]. More recently, the same method was extended to drift-wave and Rossby-wave turbulence, where the wave function is governed by a Hamiltonian very different from a usual quantum particles, and a number of intriguing effects were identified as a result [29][30][31][32][33]. But this application is still limited to scalar waves, while the Wigner-Moyal approach could be useful also in more complex systems, where the wave function is a large-dimensional vector comprised of diverse fields (i.e., not just the electromagnetic field, as usual).…”