Charney Hasegawa Mima Equation Research Articles

Turbulence-driven vortex-flow around a magnetic island

We predict a new state of plasma self-organization, in the presence of magnetic islands, namely turbulence-driven island vortex-flows, which are non-axisymmetric flows. The interaction between an E × B sheared vortex-flow and the drift-wave (DW) turbulence driving it, is derived in the presence of a coherent static magnetic island. The turbulence is driven by a 3D density profile due to quasi-linear island-induced profile flattening, and the DWs thus follow the local electron diamagnetic drift along the island. The metric tensor is introduced, making the analysis easier in island geometry. An extended Charney–Hasegawa–Mima equation describes the DW turbulence—flow interaction, from which a wave-kinetic equation is obtained in island geometry. This yields a turbulence-vortex-flow predator–prey model which predicts a nonlinear threshold for island vortex-flow formation. The vortex-flow threshold decreases with increasing island-width as , which shows that wider islands may more easily drive vortex-flows.

Dissipative brackets for the Fokker–Planck equation in Hamiltonian systems and characterization of metriplectic manifolds

It is shown that the Fokker–Planck equation describing diffusion processes in noncanonical Hamiltonian systems exhibits a metriplectic structure, i.e. an algebraic bracket formalism that generates the equation in consistency with the thermodynamic principles of energy conservation and entropy growth. First, a microscopic metriplectic bracket is derived for the stochastic equations of motion that characterize the random walk of the elements constituting the statistical ensemble. Such bracket is fully determined by the Poisson operator generating the Hamiltonian dynamics of an isolated (unperturbed) particle. Then, the macroscopic metriplectic bracket associated with the evolution of the distribution function of the ensemble is induced from the microscopic metriplectic bracket. Similarly, the macroscopic Casimir invariants are inherited from microscopic dynamics. The theory is applied to construct the Fokker–Planck equation of an infinite dimensional Hamiltonian system, the Charney–Hasegawa–Mima equation. Finally, the canonical form of the symmetric (dissipative) part of the metriplectic bracket is identified in terms of a ‘canonical metric tensor’ corresponding to an Euclidean metric tensor on the symplectic leaves foliated by the Casimir invariants.

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

We review some recent results that have been obtained in the investigation of zonal flow emergence, by means of a gyrokinetic trapped ion model, in the regime of ion temperature gradient instabilities for tokamak plasmas. We show that an analogous formulation of the zonal flow dynamics in terms of the Reynolds tensor applies in the fluid and kinetic regimes, where polarization effects play a major role. The kinetic regime leads to the emergence of a resonant mode at a frequency close to the drift frequency. With the objective of modeling both separate fluid and kinetic regimes of zonal flows, we used in this paper a methodology for deriving both Charney–Hasegawa–Mima (CHM) and Hasegawa–Wakatani models. This methodology is based on the trapped ion model and is analogous to the hierarchy leading from the Vlasov equation to the macroscopic fluid equations. The nature of zonal flows in the hierarchy of the Mima, Hasegawa and Wakatani models is investigated and discussed through comparisons with global kinetic simulations. Applications to the CHM equation are discussed, which applies to a broad variety of hydrodynamical systems, ranging from large-scale processes met in magnetically confined plasma to the so-called zonostrophy turbulence emerging in the case of small-scale forced, two-dimensional barotropic turbulence (Sukoriansky et al. Phys. Rev. Letters, 101, 178501, 2008).

Large-scale drift and Rossby wave turbulence

We study drift/Rossby wave turbulence described by the large-scale limit of the Charney–Hasegawa–Mima equation. We define the zonal and meridional regions as and respectively, where is in a plane perpendicular to the magnetic field such that kx is along the isopycnals and ky is along the plasma density gradient. We prove that the only types of resonant triads allowed are and . Therefore, if the spectrum of weak large-scale drift/Rossby turbulence is initially in Z it will remain in Z indefinitely. We present a generalised Fjørtoft’s argument to find transfer directions for the quadratic invariants in the two-dimensional -space. Using direct numerical simulations, we test and confirm our theoretical predictions for weak large-scale drift/Rossby turbulence, and establish qualitative differences with cases when turbulence is strong. We demonstrate that the qualitative features of the large-scale limit survive when the typical turbulent scale is only moderately greater than the Larmor/Rossby radius.

Lattice Boltzmann model for collisionless electrostatic drift wave turbulence obeying Charney–Hasegawa–Mima dynamics

A lattice Boltzmann method (LBM) approach to the Charney–Hasegawa–Mima (CHM) model for adiabatic drift wave turbulence in magnetised plasmas is implemented. The CHM-LBM model contains a barotropic equation of state for the potential, a force term including a cross-product analogous to the Coriolis force in quasigeostrophic models, and a density gradient source term. Expansion of the resulting lattice Boltzmann model equations leads to cold-ion fluid continuity and momentum equations, which resemble CHM dynamics under drift ordering. The resulting numerical solutions of standard test cases (monopole propagation, stable drift modes and decaying turbulence) are compared to results obtained by a conventional finite difference scheme that directly discretizes the CHM equation. The LB scheme resembles characteristic CHM dynamics apart from an additional shear in the density gradient direction. The occurring shear reduces with the drift ratio and is ascribed to the compressible limit of the underlying LBM.

Percolation transition in the kinematics of nonlinear resonance broadening in Charney–Hasegawa–Mima model of Rossby wave turbulence

We study the kinematics of nonlinear resonance broadening of interacting Rossby waves as modelled by the Charney–Hasegawa–Mima equation on a biperiodic domain. We focus on the set of wave modes which can interact quasi-resonantly at a particular level of resonance broadening and aim to characterize how the structure of this set changes as the level of resonance broadening is varied. The commonly held view that resonance broadening can be thought of as a thickening of the resonant manifold is misleading. We show that in fact the set of modes corresponding to a single quasi-resonant triad has a non-trivial structure and that its area in fact diverges for a finite degree of broadening. We also study the connectivity of the network of modes which is generated when quasi-resonant triads share common modes. This network has been argued to form the backbone for energy transfer in Rossby wave turbulence. We show that this network undergoes a percolation transition when the level of resonance broadening exceeds a critical value. Below this critical value, the largest connected component of the quasi-resonant network contains a negligible fraction of the total number of modes in the system whereas above this critical value a finite fraction of the total number of modes in the system are contained in the largest connected component. We argue that this percolation transition should correspond to the transition to turbulence in the system.

Two-dimensional gyrokinetic turbulence

Two-dimensional gyrokinetics is a simple paradigm for the study of kinetic magnetised plasma turbulence. In this paper, we present a comprehensive theoretical framework for this turbulence. We study both the inverse and direct cascades (the ‘dual cascade’), driven by a homogeneous and isotropic random forcing. The key characteristic length of gyrokinetics, the Larmor radius, divides scales into two physically distinct ranges. For scales larger than the Larmor radius, we derive the familiar Charney–Hasegawa–Mima equation from the gyrokinetic system, and explain its relationship to gyrokinetics. At scales smaller than the Larmor radius, a dual cascade occurs in phase space (two dimensions in position space plus one dimension in velocity space) via a nonlinear phase-mixing process. We show that at these sub-Larmor scales, the turbulence is self-similar and exhibits power-law spectra in position and velocity space. We propose a Hankel-transform formalism to characterise velocity-space spectra. We derive the exact relations for third-order structure functions, analogous to Kolmogorov's four-fifths and Yaglom's four-thirds laws and valid at both long and short wavelengths. We show how the general gyrokinetic invariants are related to the particular invariants that control the dual cascade in the long- and short-wavelength limits. We describe the full range of cascades from the fluid to the fully kinetic range.

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

We study the modulational instability of geophysical Rossby and plasma drift waves within the Charney–Hasegawa–Mima (CHM) 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 (Phys. Fluids, vol. 6, 1994, p. 1158). We find that, for strong primary waves, these narrow zonal jets further roll up into Kármán-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).

Derivation of reduced two-dimensional fluid models via Dirac’s theory of constrained Hamiltonian systems

We present a Hamiltonian derivation of a class of reduced plasma two-dimensional fluid models, an example being the Charney–Hasegawa–Mima equation. These models are obtained from the same parent Hamiltonian model, which consists of the ion momentum equation coupled to the continuity equation, by imposing dynamical constraints. It is shown that the Poisson bracket associated with these reduced models is the Dirac bracket obtained from the Poisson bracket of the parent model.

Zonal flow generation by the stretching of Rossby modons

The numerical study is presented of the zonal flow generation in shallow rotating fluids and in magnetized plasmas, in a strongly nonlinear regime described by the Charney–Hasegawa–Mima equation. It is demonstrated that coherent vortices, often regarded as the building blocks of the strong turbulence, are unstable in the presence of inhomogeneities. While the monopolar vortices, both cyclones and anticyclones, are rapidly dispersed by a finite Rossby velocity, the dipolar vortices (or modons) undergo a qualitative modification by the action of the scalar nonlinearity arising from the β effect. The westward propagating modons rapidly topple, disintegrating into two monopoles that propagate independently and rapidly disperse. Conversely, for the eastward propagating modons, the β-effect produces the change of the direction of the propagation, followed by the stretching in the east–west direction. On a long time scale, such modons expand to a length equal to the size of the computational box, and essentially an one-dimensional zonal flow is created, whose transverse (north–south) scale is determined by the initial size of the modon.

Hamiltonian derivation of the Charney–Hasegawa–Mima equation

The Charney–Hasegawa–Mima equation is an infinite-dimensional Hamiltonian system with dynamics generated by a noncanonical Poisson bracket. Here a first principle Hamiltonian derivation of this system, beginning with the ion fluid dynamics and its known Hamiltonian form, is given.

Hamiltonian structure for the Charney–Hasegawa–Mima equation in the asymptotic model regime

We show that the Charney–Hasegawa–Mima (CHM) equation in the asymptotic model (CHM-AM) regime possesses the non-canonical Hamiltonian structure. The CHM-AM corresponds to the CHM equation in the asymptotic limit of length scales large compared to the Rossby deformation radius. It is shown that the Hamiltonian structure of the CHM-AM cannot be derived directly from that of the CHM equation by taking a simple limit of a length scale. Both the two-dimensional (2-D) Euler equation and the CHM-AM are regarded as special cases of a generalized 2-D fluid system, the so-called α -turbulence system. The existence of the Hamiltonian structure of the CHM-AM obtained in this study and that of the 2-D Euler equation implies the existence of the Hamiltonian structure of the α -turbulence system.

Spectral transfers and zonal flow dynamics in the generalized Charney-Hasegawa-Mima model

The mechanism of four nonlinearly interacting drift or Rossby waves is used as the basic process underlying the turbulent evolution of both the Charney-Hasegawa-Mima-equation (CHME) and its generalized modification (GCHME). Hasegawa and Kodama’s concept of equivalent action (or quanta) is applied to the four-wave system and shown to control the distribution of energy and enstrophy between the modes. A numerical study of the GCHME is described in which the initial state contains a single finite-amplitude drift wave (the pump wave), and all the modulationally unstable modes are present at the same low level (10−6 times the pump amplitude). The simulation shows that at first the fastest-growing modulationally unstable modes dominate but reveals that at a later time, before pump depletion occurs, long- and short-wavelength modes, driven by pairs of fast-growing modes, grow at 2γmax. The numerical simulation illustrates the development of a spectrum of turbulent modes from a finite-amplitude pump wave.

Infrared Dynamics of Decaying Two-Dimensional Turbulence Governed by the Charney-Hasegawa-Mima Equation

The low wave number range of decaying turbulence governed by the Charney-Hasegawa-Mima (CHM) equation is examined theoretically and by direct numerical simulation. Here, the low wave number range is defined as values of the wave number k below the wave number kE corresponding to the peak of the energy spectrum, or alternatively the centroid wave number of the energy spectrum. The energy spectrum in the low wave number range in the infrared regime (k →0) is theoretically derived to be E(k) ∼k5, using a quasinormal Markovianized model of the CHM equation. This result is verified by direct numerical simulation of the CHM equation. The wave number triads (k,p,q) responsible for the formation of the low wave number spectrum are also examined. It is found that the energy flux Π(k) for k k. Furthermore, the contribution of nonlocal triad interactions to the energy flux is found to be predominant in the range log (k/kE)<-0.5, where the nonlocal interactions are defined to be those triad interactions for which the ratio of the largest leg of the triad to the smallest leg is larger than four. ©2001 The Physical Society of Japan

The effect of sheared diamagnetic flow on turbulent structures generated by the Charney–Hasegawa–Mima equation

The generation of electrostatic drift wave turbulence is modeled by the Charney–Hasegawa–Mima equation. The equilibrium density gradient n0=n0(x) is chosen so that dn0/dx is nonzero and spatially variable (i.e., v*e is sheared). It is shown that this sheared diamagnetic flow leads to localized turbulence which is concentrated at max(∇n0), with a large dv*e/dx inhibiting the spread of the turbulence in the x direction. Coherent structures form which propagate with the local v*e in the y direction. Movement in the x direction is accompanied by a change in their amplitudes. When the numerical code is initialized with a single wave, the plasma behavior is dominated by the initial mode and its harmonics.

Passive particle dynamics in a flow exhibiting transition to turbulence

The behavior of a passive particle in a flow that exhibits bifurcations in the transition to a turbulent regime is investigated. The flow considered is a variant of the Charney–Hasegawa–Mima equation. The scalar particle dynamics is considered for different regimes of the main flow. A regime of anomalous diffusion (hypodiffusion) is observed when the field has few harmonics whereas normal diffusion occurs in the strange attractor regime. The analysis of the singular orbit reveals the presence of traps and flights that control the transport.

On the most probable states of two-dimensional plasmas

The Charney-Hasegawa-Mima equation and two-dimensional magnetohydrodynamics (MHD) have ideal Lagrangian invariants that are used as the basis for most probable state analysis. An information-theoretic entropy defined in terms of the Lagrangian invariants leads to predictions for relationships between the field quantities. These relationships are tested through numerical solution of the equations through the turbulent relaxation of random initial conditions to coherent states. It is found that the predictions for the most probable state for the Charney—Hasegawa—Mima equation are essentially correct, whereas for two-dimensional MHD they are not. Questions and issues are raised pertaining to why a most probable state analysis based upon Lagrangian invariants seems to work in the case of the Charney—Hasegawa—Mima equation but not for two-dimensional MHIJ. A qualitatively correct entropy is proposed for two-dimensional MHD based upon field variables that are not local invariants.

Exact solutions for a system of nonlinear plasma fluid equations

A method is presented for constructing exact solutions to a system of nonlinear plasma fluid equations that combines the physics of reduced magnetohydrodynamics and the electrostatic drift-wave description of the Charney–Hasegawa–Mima equation. The system has nonlinearities that take the form of Poisson brackets involving the fluid field variables. The method relies on modifying a class of simple equilibrium solutions, but no approximations are made. A distinguishing feature is that the original nonlinear problem is reduced to the solution of two linear partial differential equations, one fourth order and the other first order. The first-order equation has Hamiltonian characteristics and is easily integrated, supplying information about the propagation of solutions.

A point vortex dipole model of an isolated modon

A point vortex dipole model for an isolated modon governed by the Charney–Hasegawa–Mima (CHM) equation is developed, building on the point vortex formulation of Zabusky and McWilliams [Phys. Fluids 25, 2175 (1982)]. The model dipole is compared to the exact modon solution in order to determine parameter values for which the model dipole matches the modon’s speed and far-field behavior. The model allows one to study nonuniform motions analytically. It predicts that right-moving modons in uniform motion should be stable in the sense that their paths exhibit small-amplitude oscillations in response to small perturbations of their initial orientation. It also predicts that left-moving modons in uniform motion should be unstable, being pushed into finite-amplitude motions by arbitrarily small perturbations. These predictions are confirmed by direct numerical simulation of modons evolving under the CHM equation. It is noted that although the distribution of vorticity within modons may be Lyapunov stable in nearly uniform motions, the paths of modons may be unstable to asymmetric perturbations.