The nonlinear dynamics of the modulational instability of drift waves and the associated zonal flows

Abstract

The linear and nonlinear dynamics of zonal flows and their interactions with drift wave turbulence is considered in the simple but illuminating generalized Charney–Hasegawa–Mima model due to Smolyakov et al. [Phys. Plasmas 7, 1349 (2000)]. Two positive definite, exact, integral invariants associated with the full generalized Charney–Hasegawa–Mima system are derived. For an initial monochromatic drift wave pump with small but finite amplitude, a modulational instability can occur, characterized by growing zonal flow and sideband perturbations (i.e., a four-wave interaction). The pump threshold for instability is readily satisfied, depending on the zonal flow wave number. The fully nonlinear Charney–Hasegawa–Mima equations are solved with a numerical scheme which is validated by demonstrating the conservation of the two exact invariants. The simulations show that the validity of the four-wave model is limited to approximately three instability growth times. The radial structure of the zonal flow can be “jet-like” or highly oscillatory depending upon the ratio of the system size to the density scale length and initial conditions. It is found that zonal flows can be dramatically reduced if the most unstable zonal flow wave number does not fit into the system.

Improved model of quasi-particle turbulence (with applications to Alfvén and drift wave turbulence)

“…In recent years, considerable progress has been made in the wave kinetic approach to plasma turbulence. [1][2][3] Several aspects of strong nonlinear plasma behavior were successfully described by this approach, such as photon Landau damping of relativistic electron plasma waves, 4,5 photon acceleration by laser wakefields 6,7 or by relativistic ionization fronts, 8,9 formation of soliton like structures 10 and modulational instabilities of photon beams, 11 drift wave and ion temperature gradient wave excitation of zonal flows, [12][13][14][15] excitation of ion acoustic waves by electron plasma turbulence, [16][17][18] or even the excitation of dust crystal oscillations by ion acoustic turbulence. 19 With some exceptions (see for instance Ref.…”

Photon acceleration in laser wakefield accelerators

“…The numerical model is based on a particle-in-cell treatment of quasi-particles, the "drift waves," while we describe the plasma as a fluid. In our work we can easily treat broadband drift mode turbulence whereas other numerical treatments deal with a monochromatic drift mode pump wave [42,43,44]. Earlier work on the broadband nature of the turbulence was restricted to linear theory [41].…”

“…The drift wave problem has been tackled by a number of authors. The work of Smolyakov et al [41] considered the linear theory of broadband drift modes coupled to zonal flows while Lashmore-Davies et al [42,43] considered a monochromatic treatment with a drift mode pump wave coupling through the zonal flow to a series of discrete Stokes and anti-Stokes sidebands. There is not only a conceptual difference in the treatment of the problem: the two approaches solve different aspects of the nonlinear coupling problem.…”

Bursting events in zonal flow-drift wave turbulence

“…4,5 Several theoretical models, based on the HasegawaMima equation, 6 have been proposed in order to explain the emergence of ZFs in tokamak plasmas. [7][8][9][10][11][12] The simplest model involves the modulational instability of a monochromatic drift wave ͑the pump͒, generating two sidebands and a ZF that finally saturate by depletion of the pump wave. [9][10][11][12] However, models involving only a small number of waves cannot describe correctly the broad-band turbulence occurring in tokamak experiments and simulations.…”

“…[7][8][9][10][11][12] The simplest model involves the modulational instability of a monochromatic drift wave ͑the pump͒, generating two sidebands and a ZF that finally saturate by depletion of the pump wave. [9][10][11][12] However, models involving only a small number of waves cannot describe correctly the broad-band turbulence occurring in tokamak experiments and simulations. In particular, a distinct feature recently observed in several large-scale numerical simulations is the appearance of ''bursts,'' i.e., punctuated events during which radial transport is considerably enhanced.…”