Electrostatic Drift Research Articles

Study of toroidal ion temperature gradient electrostatic drift waves

The toroidal ion temperature gradient electrostatic drift waves are investigated including the effect of ion dynamics parallel to the ambient magnetic fields. By using the transport equations of Braginskii, the mode equations are derived. The local as well as nonlocal dispersion relations are obtained and various limiting cases are discussed. Also addressed is the issue of the anomalous ion energy transport that is caused by the nonthermal ion temperature gradient drift waves. A possible stationary solution of the nonlinear mode coupling equations is presented. The relevance of our investigation to the ion energy confinement in tokamaks is pointed out.

Almost two-dimensional treatment of drift wave turbulence

The approximation of two-dimensionality is studied and extended for electrostatic drift wave turbulence in a three-dimensional, magnetized plasma. It is argued on the basis of the direct interaction approximation that in the absence of parallel viscosity, purely 2-D solutions exist for which only modes with k∥=0 are excited, but that the 2-D spectrum is unstable to perturbations at nonzero k∥. A 1-D equation for the parallel profile gk⊥(k∥) of the saturated spectrum at steady state is derived and solved, allowing for parallel viscosity; the spectrum has finite width in k∥, and hence finite parallel correlation length, as a result of nonlinear coupling. The enhanced energy dissipation rate, a 3-D effect, may be incorporated in the 2-D approximation by a suitable renormalization of the linear dissipation term. An algorithm is presented that reduces the 3-D problem to coupled 1- and 2-D problems. Numerical results from a 2-D spectral direct simulation, thus modified, are compared with the results from the corresponding 3-D (unmodified) simulation for a specific model of drift wave excitation. Damping at high k∥ is included. It is verified that the 1-D solution for gk⊥(k∥) accurately describes the shape and width of the 3-D spectrum, and that the modified 2-D simulation gives a good estimate of the 3-D energy saturation level and distribution E(k⊥).

The structure and dynamics of electrostatic and magnetostatic drift holes

Localized, nonwavelike phase-space density fluctuations with self-trapping potentials, called drift holes, are studied for systems with density gradients. The three-dimensional structure of maximally probable, isolated electrostatic drift holes is determined and shown to satisfy a condition for self-trapping. The dynamics of such structures is examined. It is shown that under a broad set of conditions the potential amplitude, hole depth, and velocity-space width grow nonlinearly while maintaining the self-trapping structure. The nonlinear instability responsible for growth requires dissipative coupling between the trapped particles and the passing particles of the opposite species. Two saturation mechanisms for this instability are described and saturated amplitude estimates are given. It is shown that when the hole velocity is small compared to the speed of light, self-consistent coupling of stationary magnetic perturbations to the electrostatic potential is possible and produces structures called magnetostatic drift holes. The role of electrostatic and magnetostatic drift holes as a constituent of intermittent plasma turbulence is discussed.

Three-dimensional numerical simulations of simple electron drift wave turbulence in a sheared and curved magnetic field

A three-dimensional code is used to study the scaling of an electrostatic electron drift wave turbulence with respect to shear and curvature of the magnetic field as well as the driving strength. The electrons are modeled with a simple formula for the deviation from a near Boltzmann response and the ions are modeled as a nonlinear fluid. For sheared slab geometry suppressing curvature (or toroidal) effects, the turbulence level and transport scale according to the mixing length rules with the mixing length given by the linear normal mode widths and the poloidal wavelengths scaled to the ion gyroradius. Toroidal coupling results in a more complex behavior. Some evidence for nonlocal or convective turbulence effects is found.

Nonlinear electrostatic drift waves: Higher-order structures

In this paper we will show that the Hasegawa-Mima equation (which governs the electrostatic drift waves in an inhomogeneous plasma) can possess generalized solitary-wave type solutions even when both the quadratic and cubic nonlinearities are included. Further, the Hasegawa-Mima equation is also shown to possess a double-layer type solution when both the quadratic and cubic nonlinearities are included provided a certain condition is satisfied.

Modulational instability of electrostatic drift waves in an inhomogeneous plasma.

In this Brief Report, we investigate the modulational instability of electrostatic drift waves in an inhomogeneous plasma with respect to quasi-one-dimensional modulations that primarily propagate along the drift direction (like the drift solitons given by Lakhin, Mikhailovskii, and Onishchenko [Phys. Lett. A 119, 348 (1987); Plasma Phys. Contr. Fusion 30, 457 (1988)]).

Nonlinear fluid equations for fully toroidal electromagnetic waves in nonuniform magnetized plasmas.

By means of the two-fluid model of Braginskii [Reviews of Plasma Physics, edited by M. A. Leontovich (Consultants Bureau, New York, 1965), Vol. I, p. 205], this paper derives a set of nonlinear differential equations governing the dynamics of fully toroidal low-frequency electromagnetic waves in an inhomogeneous electron-ion plasma immersed in a nonuniform magnetic field. The effects of equilibrium electron and ion pressure gradients, magnetic drifts caused by the nonuniform magnetic field, as well as the contribution of heat fluxes in the particle-energy transport equations are incorporated. In the linear limit, expressions for the ion and electron number-density perturbations together with a new dispersion relation for fully toroidal electromagnetic waves are obtained. A new long-wavelength electromagnetic drift wave instability is shown to exist in a nonuniform magne- tized electron plasma. The formation of dipolar vortices in the two-dimensional ion-temperature-gradient-driven electrostatic drift wave turbulence has been demonstrated.

Coupling between Varma, modified-convective-cell and Alfvén modes in an inhomogeneous plasma

The effect of parallel electron dynamics is incorporated in the study of the Varma mode, which is a flute-like electrostatic ion drift perturbation in an inhomogeneous magnetic field. Starting from the electron and ion continuity equations, an equation for the conservation of the ion magnetic moment, the generalized Ohm's law, and Faraday's and Ampère's laws, we derive a set of four nonlinearly coupled differential equations. The nonlinearities arising from the E × B0 convection of the density and magnetic-moment variations, nonlinear ion polarization drift and Lorentz force, and the coupling of the parallel electron flow to the perturbed magnetic field are included. In the linear limit we find that the Varma mode becomes coupled with the modified-convective-cell and Alfvén modes. The instability of the coupled system is analysed and an expression for the growth rate is obtained. In the nonlinear regime we present the localized dipole vortex solution of the coupled Varma and convective-cell modes as well as the coupled Varma and Alfvén modes. It is found that for the former case the nonlinear structure is a solution of a second-order partial differential equation, whereas for the coupled Varma and inertial Alfvén waves the dipole vortex belongs to a family of fourth-order differential equations. Our results can be useful in understanding the electrostatic and electromagnetic fluctuations in mirror reactors, tokamaks and astrophysical plasmas.

New electron-temperature-gradient drift wave instabilities and anomalous thermal transport in magnetized electron plasmas

New electron-temperature-gradient-driven electrostatic drift wave instabilities are shown to arise in magnetized electron plasmas. The quasilinear electron energy transport caused by such nonthermal drift waves is also obtained.

Ion diffusion coefficients from resonant interactions with broadband turbulence in the magnetotail

A potentially important source of hot ions within the central plasma sheet (CPS) is warm plasma sheet boundary layer (PSBL) ions which interact with broadband electrostatic waves and electric field drift toward the CPS. In the PSBL, the local interaction between the waves and particles can be assumed to be unmagnetized. The effect of the magnetic field is to organize wave and particle distributions in velocity space. Under these conditions, particle diffusion is, in general, two‐dimensional and is similar to magnetized diffusion. The general quasi‐linear equations describing such diffusion are presented assuming that a spectrum of waves is excited similar to the waves observed in the boundary layer. In order to apply the general quasi‐linear diffusion coefficients, two particle distribution models are assumed based on PSBL observations. One is for the outer edge and one is for the inner edge of the boundary. An expression for the unmagnetized dielectric function is given and evaluated for wave frequency and growth rate for the assumed particle distribution models. It is found that slow and fast mode ion‐sound waves can be unstable for the range of plasma parameters considered. The diffusion coefficients are then evaluated for resonant warm PSBL ion interactions with ion‐sound waves. The results illustrate how resonant ion diffusion rates vary with pitch angle and speed, and how the diffusion rates depend upon the distribution of wave energy in k space. For the model of the outer edge of the PSBL, pitch angle diffusion is found to dominate energy diffusion, whereas both types of diffusion can be important for the model of the inner edge of the PSBL. It is found that the characteristic time for resonant warm boundary layer ions to diffuse in velocity space is ∼10 min (for a wave electric field of 1 mV/m), which is approximately an ion bounce period. In addition, significant pitch angle and energy diffusion should occur resulting in isotropization of the warm PSBL ion beams to form the hot isotropic ion component in the CPS.

Renormalized perturbation theory: Vlasov–Poisson system, weak turbulence limit, and gyrokinetics

The self-consistency of the renormalized perturbation theory of Zhang and Mahajan [Phys. Rev. 132, 1759 (1985)] is demonstrated by applying it to the Vlasov–Poisson system and showing that the theory has the correct weak turbulence limit. Energy conservation is proved to arbitrary high order for the electrostatic drift waves. The theory is applied to derive renormalized equations for a low-beta gyrokinetic system. Comparison of this theory with other current theories is presented.

Effects of an Ambipolar Field on Stability of Electrostatic Drift Waves in Sheet Plasmas

The effects of an ambipolar field on the stability of electrostatic drift waves in sheet plasmas are investigated numerically. The analysis is based on an integral equation in the wave number space derived from the profiles of Gaussian density and parabolic ambipolar potential. The ambipolar field dependence of the eigenfrequency and eigenfunction of the drift waves is obtained. It is found that the drift waves are stabilized for a strong ambipolar potential of both the hill-type and well-type. The stabilization is due to the ion Landau damping caused by the velocity shear effect of the E × B drift. Purely growing (zero real frequency) drift modes are also found in the case of a hill-type ambipolar potential.

Current driven turbulence and microturbulent spectra in the TORTUR Tokamak

In the TORTUR Tokamak (R=0.46 m, a-0.085 m, BT=2.9 T, Ip<or=55 kA, Te approximately=Ti<or=1 keV, n<or=1020 m-3, Zeff(imp)<2) plasmas are heated by current driven turbulence in the build-up phase to high beta values and electron temperatures a factor of two above values retrieved from scaling law studies reported in literature for ohmically heated Tokamaks. A stationary state of current driven turbulence with the same beta value can be maintained during the plateau state by virtue of relatively high loop voltages (2-4 times the classical value) giving proportionally enhanced resistive dissipation. The turbulent fluctuations of the plateau state have been observed at r/a=0.7 by collective scattering of microwaves in a wide spectral region for Ip approximately=30 kA corresponding with plasmas in the safe q range: q(a)=7. Three frequency domains of turbulence have distinguished with rather arbitrary separation, interpreted as follows: (a) A low-frequency region (0-0.7 MHz) containing the frequently reported electrostatic drift modes and low-frequency electromagnetic modes which may account for a considerable fraction of the electron heat diffusivity chi e(r); (b) Unstable magnetic microtearing modes driven by a supercritical electron temperature gradient characterized by real frequencies in the interval 0.7-3 MHz. Indirect evidence for the presence of this magnetic tearing exists by the observation of distinct satellites in the Thomson scattering spectrum; (c) A magnetic fluctuation spectrum in the high-frequency region (5-55 MHz) which can be ascribed to current driven modes from non-thermal electron tails around 50 keV. All spectral domains exhibit slow time periodicities synchronous with the development of high-energy tails of the ions and with the Mirnov-type small sawteeth found on the loop voltage, electron cyclotron emission signals and soft X-rays.

Electrostatic drift waves in a two-electron-temperature plasma.

A new type of electrostatic drift waves, which can exist in a plasma whose electrons can be grouped into distinct hot and cold components, is discussed. These waves involve electron dynamics only and can be in a higher-frequency range than that of the ordinary drift waves for which ion motion is crucial. Nonlinear states associated with these drift waves are also investigated.

Toroidicity effects in electrostatic drift wave turbulence in Tokamaks. II. The incoherent response

For pt. I see ibid., vol.29, no.8, pp.993-1018 (1987). The concept of clumps is used in the Tokamak turbulence theory. Using the Misguich-Balescu stochastic orbit treatment (1985), the two-point fluctuation correlation equation is studied in detail. It is shown that the Hirshman-Molvig effect (1979) has a special influence on the clumps theory: the general quasilinear diffusion regime does not exist in Tokamaks. It is found that the clumps cannot be considered only as a destabilizing mechanism ('clumps-induced instability'), the presence of Delta omega (the frequency width at fixed wavenumber) stabilizes the plasma. It is also shown that Delta omega approximately omega c> omega as a result of the Hirshman-Molvig effect on the propagator. The saturation appears as a result of the balance between the clumps and the ion Compton scattering. The fluctuation spectrum is estimated, the large ktheta asymptotic form is ktheta -103/.

Toroidicity effects in electrostatic drift wave turbulence in Tokamaks. I. The coherent electron and ion response

Two methods (renormalization and stochastic orbit treatment) are used to discuss the ion and electron coherent responses in a turbulent Tokamak plasma. The ballooning representation for the non linear E*B term is questioned. Some coherent dissipation and saturation mechanisms are studied in detail. Specifically, it is found that the Hirshman-Molvig effect (1979) plays the dominant role in the determination of the diffusion characteristics in the Tokamak geometry. Also, using the averaged radial wave-number vector technique the ion Compton scattering is investigated analytically as an effective saturation mechanism.

On The Solitary Vortex Solutions of Nonlinear Drift Wave Equations

The Hasegawa-Mima equation for a quasi-two-dimensional electrostatic drift wave is generalized to include arbitrary distributions of the equilibrium electrostatic potential and the density and temperature of plasma. It is shown that the spatial dependence of the equilibrium density distribution plays a decisive role in determining the behavior of the steady vortex solution, that is, whether it is a single soliton or pair-solitons, etc. The relation between the drift wave equation and the barotropic equation for Rossby waves is discussed, and, in particular, a remark is made on Petviashvili's equation which has an additional nonliruear term proportional to the temperature gradient.

Vortices in an anisotropic plasma

Localized electrostatic nonlinear drift wave structures in a plasma with hot electrons and anisotropic ions are considered. The anisotropic energy distribution leads to a new nonlinearity in the evolution equation of finite amplitude drift waves. One- and two-dimensional solutions of physical interest are presented.

Instability and saturation of drift-convective modes in an inhomogeneous plasma

It is found that the inclusion of the electron inertia effect (parallel to an external magnetic field) can provide a linear coupling between the electrostatic drift and the convective modes in a non-uniform plasma. This coupling leads to new branches of rapidly growing modes, which are calculated in the kinetic as well as in the hydrodynamic regimes. To study the saturation of the linear unstable modes, we account for the mode coupling and derive a set of model nonlinear fluid equations. A perturbation technique is employed to obtain a nonlinear evolution equation. In the steady state, the latter yields the saturated electric potential. It is argued that the enhanced low-frequency fluctuations can cause anomalous particle transport in a magnetoplasma.

Generalized moments for low-frequency modes in tokamaks and their application to drift and ballooning instabilities

It is shown that the recently proposed electrostatic ballooning formulation in tokamak geometry absolutely destabilizes electrostatic drift waves (universal instability), and also further destabilizes the magnetohydrodynamic ballooning instability (finite ion Larmor-radius destabilization). The ion-density perturbation responsible for electrostatic ballooning effects have been derived by three independent methods: fluid approximation, the gyro-kinetic equation in a toroidal geometry, and the full kinetic equation in a plane-slab geometry with effective g (gravitational acceleration). All have yielded a consistent result. Toroidicity-induced frequency downshift is responsible for the universal mode, which smoothly reduces to the (stable) drift mode in slab geometry. For finite ion Larmor-radius destabilization of ballooning modes, toroidicity (due to magnetic drifts)-induced coupling with drift waves is also responsible.