Mima Equation Research Articles

An extended Hasegawa–Mima equation for nonlinear drift wave turbulence in general magnetic configurations

We propose an extended Hasegawa–Mima equation describing the evolution of nonlinear drift wave turbulence in general magnetic configurations. Such HMGM equation can be derived within the kinetic framework of guiding center motion or from a two-fluid model of an ion-electron plasma by application of a drift wave turbulence ordering that does not involve conditions on spatial derivatives of magnetic field and plasma density. The HMGM equation is therefore appropriate to describe the evolution of drift wave turbulence in strongly inhomogeneous magnetized plasmas, such as magnetospheric and stellarator plasmas, involving complex magnetic field geometries and non-uniform plasma density distributions. We find conservation laws (mass, energy, and generalized enstrophy) of the HMGM equation, study its algebraic (Hamiltonian) structure, and prove a nonlinear stability criterion for steady solutions through the energy-Casimir method. We then apply these results to describe drift waves and infer the existence of stable toroidal zonal flows with radial shear in dipole magnetic fields.

Fast ion effects on zonal flow generation: A simple model

Fast ions' effects on turbulence-driven zonal flow generation are investigated in the context of a simple reduced model based on the Hasegawa–Mima equation. Fast ions' much higher characteristic frequency of parallel motion in comparison with the drift wave's phase velocity along the magnetic field facilitates a derivation of the reduced model equations. Nonlinear mode coupling analyses show that the threshold amplitude of drift wave required for the zonal flow modulational instability is significantly reduced, making its generation easier. This occurs as both a down-shift of the drift wave's frequency and a reduction of dispersion in the presence of the fast ions cause a decrease in the mismatch between the primary drift wave frequency and the zonal flow modulated sideband drift wave's characteristic frequency. This finding could be a common nonlinear physics mechanism behind numerous recent results on tokamak plasma confinement enhancement caused by the fast ions.

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.

Zonal flow generation by small-scale drift-ion-acoustic waves in electron–positron–ion plasmas

The generation of zonal flows by small-scale coupled drift-ion-acoustic waves is investigated. The problem is analyzed in magnetized electron–positron–ion plasmas by the system of the generalized Hasegawa–Mima equation and the equation of parallel motion of ions. It is concluded that the inclusion of positrons enhances zonal flow growth rates.

A generalized Hasegawa–Mima equation in curved magnetic fields

We derive a model equation describing electrostatic plasma turbulence in general (inhomogeneous and curved) magnetic fields by analysing the effect of curved geometry on the ion fluid polarization drift velocity. The derived nonlinear equation generalizes the Hasegawa–Mima equation governing drift wave turbulence in a straight homogeneous magnetic field, and may serve as a toy model for the description of turbulent plasmas. The equation is most appropriate for configurations with a small$\boldsymbol {E}\times \boldsymbol {B}$drift velocity divergence, or a mild spatial change in$\boldsymbol {E}\times \boldsymbol {B}$drift velocity. We identify the conserved energy of the system, and obtain conditions on magnetic field topology for conservation of generalized enstrophy. Through numerical examples, we further show how the curvature of the magnetic field reshapes self-organized steady turbulent states.

Equilibria and condensates in Rossby and drift wave turbulence

We study the thermodynamic equilibrium spectra of the Charney–Hasegawa–Mima (CHM) equation in its weakly nonlinear limit. In this limit, the equation has three adiabatic invariants, in contrast to the two invariants of the 2D Euler or Gross–Pitaevskii equations, which are examples for comparison. We explore how the third invariant considerably enriches the variety of equilibrium spectra that the CHM system can access. In particular we characterise the singular limits of these spectra in which condensates occur, i.e. a single Fourier mode (or pair of modes) accumulate(s) a macroscopic fraction of the total invariants. We show that these equilibrium condensates provide a simple explanation for the characteristic structures observed in CHM systems of finite size: highly anisotropic zonal flows, large-scale isotropic vortices, and vortices at small scale. We show how these condensates are associated with combinations of negative thermodynamic potentials (e.g. temperature).

Phase transition to blob-hole coherent structure in the Hasegawa–Mima model for plasmas

An equilibrium statistical mechanics theory for the Hasegawa–Mima equations of toroidal plasmas, with canonical constraint on energy and microcanonical constraint on potential enstrophy, is solved exactly as a spherical model. The use of a canonical energy constraint instead of a fixed-energy microcanonical approach is justified by the preference for viewing real plasmas as an open system. A significant consequence of the results obtained from the partition function, free energy and critical temperature, is the condensation into a ground state exhibiting a blob-hole-like structure observed in real plasmas.

Drift waves enstrophy, zonal flow, and nonlinear evolution of the modulational instability

The interaction of the drift wave (DW) turbulence and zonal flow (ZF) is investigated with the modified Hasegawa–Mima equation taking into account the backreaction of ZF velocity on DW turbulence. It is shown that the y-averaged enstrophy of DW turbulence and the velocity of ZF are intrinsically related. By utilizing this feature, a nonlinear stage of DW modulational instability is considered within the framework of the wave kinetic equation. It is shown that in this approximation, the nonlinear stage of the modulational instability results in the collapsing solutions, accompanied by the “wave breaking” phenomenon. Numerical simulations based on the Hasegawa–Mima equation show that for a weak DW turbulence, Φ̃=(eφ̃/Te) (Ln/ρs)⪝1, the collapsing-like features on both ZF and y-averaged enstrophy of DW turbulence decay in time and then re-emerge again at different locations. For the case of a strong DW turbulence, Φ̃>1, where nonlinear interactions of DW harmonics dominate, stable spatial structures of ZF and y-averaged enstrophy of DW turbulence emerge.

Enstrophy non-conservation and the forward cascade of energy in two-dimensional electrostatic magnetized plasma turbulence

A fluid system is derived to describe electrostatic magnetized plasma turbulence at scales somewhat larger than the Larmor radius of a given species. It is related to the Hasegawa–Mima equation, but does not conserve enstrophy, and, as a result, exhibits a forward cascade of energy, to small scales. The inertial-range energy spectrum is argued to be shallower than a $-11/3$ power law, as compared to the $-5$ law of the Hasegawa–Mima enstrophy cascade. This property, confirmed here by direct numerical simulations of the fluid system, may help explain the fluctuation spectrum observed in gyrokinetic simulations of streamer-dominated electron-temperature-gradient driven turbulence (Plunk et al., Phys. Rev. Lett., vol. 122, 2019, 035002), and also possibly some cases of ion-temperature-gradient driven turbulence where zonal flows are suppressed (Plunk et al., Phys. Rev. Lett., vol. 118, 2017, 105002).

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.

Solitary zonal structures in subcritical drift waves: a minimum model

Solitary zonal structures have recently been identified in gyrokinetic simulations of subcritical drift-wave (DW) turbulence with background shear flows. However, the nature of these structures has not been fully understood yet. Here, we show that similar structures can be obtained within a reduced model, which complements the modified Hasegawa–Mima equation with a generic primary instability and a background shear flow. We also find that these structures can be qualitatively reproduced in the modified Hasegawa–Wakatani equation, which subsumes the reduced model as a limit. In particular, we illustrate that in both cases, the solitary zonal structures approximately satisfy the same ‘equation of state’, which is a local relation connecting the DW envelope with the zonal-flow velocity. Due to this generality, our reduced model can be considered as a minimum model for solitary zonal structures in subcritical DWs.

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

The dynamics of the radial envelope of a weak coherent drift-wave (DW) is approximately governed by a nonlinear Schrödinger equation (NLSE), which emerges as a limit of the modified Hasegawa–Mima equation (mHME). The NLSE has well-known soliton solutions, and its modulational instability (MI) 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 mHME, but only when the amplitude of the coherent DW is relatively small. At larger amplitudes, the MI produces stationary zonal structures instead. Furthermore, we find that incoherent DWs 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 DWs can be modeled as quantum-like particles (‘driftons’) within a recently developed phase-space (Wigner–Moyal) formulation, which intuitively depicts the solitary zonal structures as quasi-monochromatic drifton condensates. Quantum-like 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.

Turbulence characteristics, energy equipartition, and zonal flow generation in coupled drift wave-parallel velocity gradient driven turbulence

We perform a computational study of characteristics of coupled drift wave (DW)-parallel velocity gradient (PVG) driven turbulence. The three-dimensional Hasegawa–Mima equation combined with ion parallel flow dynamics is used for this study. Energetic analyses, both analytic and computational, show the energy equipartition of the parallel Reynolds power (i.e. the free energy due to a gradient in equilibrium parallel velocity) in parallel and perpendicular directions. A considerable portion of the perpendicular energy that is transferred through the parallel coupling generates the perpendicular zonal flow (ZF), implying the increase of the ZF level when an equilibrium PVG is present. A careful analysis of parallel-perpendicular coupling dynamics shows that the vortex stretching-like term in this system plays a significant role in the ZF generation. This is in contrast to the neutral fluid turbulence where it hinders the inverse cascade process. However, the ZF generation from PVG turbulence is found to be less effective in comparison with that of DW turbulence.

Strong turbulence, self-organization and plasma confinement

This paper elucidates the close connections between hydrodynamic models of two-dimensional fluids and reduced models of plasma dynamics in the presence of a strong magnetic field. The key element is the similarity of the Coriolis force to the Lorentz force. The reduced plasma model, the Hasegawa–Mima equation, is equivalent to the two-dimensional ion vortex equation. The paper discusses the history of the Hasegawa–Mima model and that of a related reduced system called the Hasegawa–Wakatani model. The 2D fluid ↔ magnetized plasma analogy is exploited to argue that magnetized plasma turbulence exhibits a dual cascade, including an inverse cascade of energy. Generation of ordered mesoscopic flows in plasmas (akin to zonal jets) is also explained. The paper concludes with a brief explanation of the relevance of the quasi-2D dynamics to aspects of plasma confinement physics.

Formation of Nonlinear Solitary Vortical Structures by Coupled Electrostatic Drift and Ion-Acoustic Waves

Propagation of coupled electrostatic drift and ion-acoustic waves (DIAWs) is presented. It is shown that nonlinear solitary vortical structures can be formed by low-frequency coupled electrostatic DIAWs. Primary waves of distinct (small, intermediate and large) scales are considered. Appropriate set of 3D equations consisting of the generalized Hasegawa–Mima equation for the electrostatic potential (involving both vector and scalar nonlinearities) and the equation of motion of ions parallel to magnetic field are obtained. According to experiments of laboratory plasma mainly focused to large scale DIAWs, the possibility of self-organization of DIAWs into the nonlinear solitary vortical structures is shown analytically. Peculiarities of scalar nonlinearities in the formation of solitary vortical structures are widely discussed.

A characterization of the inertial range in forced-damped Hasegawa-Mima turbulence

From the nonlinear simulations of the two-dimensional forced-damped Hasegawa–Mima equation, we show that a Reynolds number-like parameter, Rew, can represent a power law exponent of the energy spectrum in Hasegawa–Mima turbulence. Rew is defined as the ratio of nonlinear convection to dissipation. For a same value of Rew, the power law exponent in the inertial range is shown to be uniquely determined regardless of the forcing and dissipation conditions. At high Rew, the power law exponent asymptotically converges to −7.2, consistent with a recent theoretical prediction based on the shell model [Gürcan et al., Plasma Phys. Controlled Fusion 52, 045002 (2010)].

The Arithmetic Geometry of Resonant Rossby Wave Triads

Linear wave solutions to the Charney--Hasegawa--Mima equation with periodic boundary conditions have two physical interpretations: Rossby (atmospheric) waves, and drift (plasma) waves in a tokamak. These waves display resonance in triads. In the case of infinite Rossby deformation radius, the set of resonant triads may be described as the set of integer solutions to a particular homogeneous Diophantine equation, or as the set of rational points on a projective surface $X$. The set of all resonant triads was found by Bustamante and Hayat (2013) via mapping to quadratic forms. Our work independently finds all resonant triads via a rational parametrization of $X$. We provide a fiberwise description of $X$ as a rational singular elliptic surface, yielding many new results about the set of wavevectors belonging to resonant triads. In particular, we show there is an infinite number of resonant triads (with relatively prime wavevectors) containing a wavevector $(a,b)$ with $a/b=r$, where $r$ is any given rationa...

Poloidal rotation driven by nonlinear momentum transport in strong electrostatic turbulence

Virtually, all existing theoretical works on turbulent poloidal momentum transport are based on quasilinear theory. Nonlinear poloidal momentum flux— is universally neglected. However, in the strong turbulence regime where relative fluctuation amplitude is no longer small, quasilinear theory is invalid. This is true at the all-important plasma edge. In this work, nonlinear poloidal momentum flux in strong electrostatic turbulence is calculated using the Hasegawa–Mima equation, and is compared with quasilinear poloidal Reynolds stress. A novel property is that symmetry breaking in fluctuation spectrum is not necessary for a nonlinear poloidal momentum flux. This is fundamentally different from the quasilinear Reynold stress. Furthermore, the comparison implies that the poloidal rotation drive from the radial gradient of nonlinear momentum flux is comparable to that from the quasilinear Reynolds force. Nonlinear poloidal momentum transport in strong electrostatic turbulence is thus not negligible for poloidal rotation drive, and so may be significant to transport barrier formation.

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.

Rescaling effects on a low-frequency drift wave in dusty plasmas

The effect of dust on the low-frequency drift wave in inhomogeneous magnetized dusty plasmas is investigated. It is shown that a low-frequency drift wave can be modeled by the Hasegawa–Mima equation (HME) both in mobile and immobile dusty plasmas, which are dust-modified HME and HME in dusty background, respectively. The former is rescaled significantly by the presence of the dust and the space-time scale greatly increases with the increasing density and mass of dust, while the latter is not rescaled, but an additional driving force appears to drive the drift waves.