Volume 12, Issue 12, December 2005
 ARTICLES

 Basic Plasma Phenomena, Waves, Instabilities

Magnetic dynamics of simple collective modes in a twosphere plasma model
View Description Hide DescriptionA plasma blob is modeled as consisting of two homogeneous spheres of equal radius and equal but opposite charge densities that can move relative to each other. Relative translational and rotational motion are considered separately. Magnetic effects from the current density caused by the relative motion are included. Magnetic interaction is seen to cause an inductive inertia. In the relative translation case the Coulomb attraction, approximately a linear force for small amplitudes, causes an oscillation. For a large number of particles, the corresponding oscillation frequency will not be the Langmuir plasma frequency, because of the large inductive inertia. For rotation an external magnetic field is included and the energy and diamagnetism of the plasma in the model is calculated. Finally, it is noted how the neglect of resistivity is motivated by the results.

Confinement of Coulomb balls
View Description Hide DescriptionA model for the confinement of the recently discovered Coulomb balls is proposed. These spherical threedimensional plasma crystals are trapped inside a rf discharge under gravity conditions and show an unusual structural order in complex plasmas. Measurements of the thermophoretic force acting on the trapped dust particles and simulations of the plasma properties of the discharge are presented. The proposed model of confinement considers thermophoretic, iondrag, and electric field forces, and shows excellent agreement with the observations. The findings suggest that selfconfinement does not significantly contribute to the structural properties of Coulomb balls.
 Nonlinear Phenomena, Turbulence, Transport

Vortex formation in an electron plasma with a sheared flow
View Description Hide DescriptionThe formation of vortex structures in an electron plasma with a sheared flow is investigated. The electron fluid is drifting in a selfelectric field generated by an unshielded electron population. This setting is linearly unstable and an instability of diocotron (slippingstream) type occurs. The time scale of the dynamics is assumed to be much shorter than the ion plasma and ion gyroperiods. Consequently, the ions do not respond to the wave potential and serve only as a neutralizing background. An equation determining the nonlinear evolution of the electrostatic potential in a plane perpendicular to an external magnetic field is derived within the drift approximation. The governing equation is then analyzed for the case with a localized shear in the electron fluid velocity. Possible final states of the diocotron instability are investigated analytically and solutions in the form of a tripolar vortex, a zonal flow, and a vortex street are found. The nonlinear time evolution of the diocotron instability is investigated by solving the governing equation numerically. In particular, the dynamics of nonlinearly saturated states and the formation of such states are discussed. Numerical solutions show a vortex street structure in a saturated state. The relevance of our investigation for space and laboratory plasmas is discussed.

Bifurcation theory of the transition to collisionless iontemperaturegradientdriven plasma turbulence
View Description Hide DescriptionThe collisionless limit of the transition to iontemperaturegradientdriven plasma turbulence is considered with a dynamicalsystems approach. The importance of systematic analysis for understanding the differences in the bifurcations and dynamics of linearly damped and undamped systems is emphasized. A model with ten degrees of freedom is studied as a concrete example. A fourdimensional center manifold (CM) is analyzed, and fixed points of its dynamics are identified and used to predict a “Dimits shift” of the threshold for turbulence due to the excitation of zonal flows. The exact value of that shift in terms of physical parameters is established for the model; the effects of higherorder truncations on the dynamics are noted. Multiplescale analysis of the CM equations is used to discuss possible effects of modulational instability on scenarios for the transition to turbulence in both collisional and collisionless cases.

Effects of varying the step particle distribution on a probabilistic transport model
View Description Hide DescriptionThe consequences of varying the step particle distribution on a probabilistic transportmodel, which captures the basic features of transport in plasmas and was recently introduced in Ref. 1 [B. Ph. van Milligen et al., Phys. Plasmas11, 2272 (2004)], are studied. Different superdiffusive transport mechanisms generated by a family of distributions with algebraic decays (Tsallis distributions) are considered. It is observed that the possibility of changing the superdiffusive transport mechanism improves the flexibility of the model for describing different situations. The use of the model to describe the low and high confinement modes is also analyzed.

Spectral transfers and zonal flow dynamics in the generalized CharneyHasegawaMima model
View Description Hide DescriptionThe mechanism of four nonlinearly interacting drift or Rossby waves is used as the basic process underlying the turbulent evolution of both the CharneyHasegawaMimaequation (CHME) and its generalized modification (GCHME). Hasegawa and Kodama’s concept of equivalent action (or quanta) is applied to the fourwave 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 finiteamplitude drift wave (the pump wave), and all the modulationally unstable modes are present at the same low level ( times the pump amplitude). The simulation shows that at first the fastestgrowing modulationally unstable modes dominate but reveals that at a later time, before pump depletion occurs, long and shortwavelength modes, driven by pairs of fastgrowing modes, grow at . The numerical simulation illustrates the development of a spectrum of turbulent modes from a finiteamplitude pump wave.

Discrete particle noise in particleincell simulations of plasma microturbulence
View Description Hide DescriptionRecent gyrokinetic simulations of electron temperature gradient (ETG) turbulence with the global particleincell(PIC) code GTC [Z. Lin et al., Proceedings of the 20th Fusion Energy Conference, Vilamoura, Portugal, 2004 (IAEA, Vienna, 2005)] yielded different results from earlier fluxtube continuum code simulations [F. Jenko and W. Dorland, Phys. Rev. Lett.89, 225001 (2002)] despite similar plasma parameters. Differences between the simulation results were attributed to insufficient phasespace resolution and novel physics associated with global simulation models. The results of the global PIC code are reproduced here using the fluxtube PIC code PG3EQ [A. M. Dimits et al., Phys. Rev. Lett.77, 71 (1996)], thereby eliminating global effects as the cause of the discrepancy. The latetime decay of the ETG turbulence and the steadystate heat transport observed in these PIC simulations are shown to result from discrete particle noise. Discrete particle noise is a numerical artifact, so both these PG3EQ simulations and, by inference, the GTC simulations that they reproduced have little to say about steadystate ETG turbulence and the associated anomalous heat transport. In the course of this work several diagnostics are developed to retrospectively test whether a particular PIC simulation is dominated by discrete particle noise.

Turbulence spreading, anomalous transport, and pinch effect
View Description Hide DescriptionA paradigmatic model describing transport and turbulence spreading as coupled processes is proposed, trying to unify the approaches of penetrative overshoot and overshoot. As a natural consequence asymmetric radial spreading of the turbulence, upgradient transport of the particle density, and front propagation are observed. The model accounts for the interaction between the microscale of the turbulence and the meso, respectively, system scale on which profile modifications occur. Comparison with direct numerical simulations of twodimensional interchange turbulence shows qualitatively good agreement with the proposed transportmodel. Key transport features are reproduced even in the presence of coherent bloblike structures. Features of density pulse dynamics are also investigated.

Nonlinear instability and saturation of linearly stable currentcarrying pair plasmas
View Description Hide DescriptionThe nonlinear instability of currentcarrying pair plasmas is investigated with a Vlasov–Poisson model for the twoparticle species. It is shown that linearly stable configurations are unstable against small incoherent perturbations of the particle distribution functions. The instability gives rise to a selfacceleration and growth of phasespace holes. After the growth of the phasespace holes, the instability reaches a chaotic saturation where the finiteamplitude holes interact and merge, and after a long time, the system attains a stable equilibrium state with a smaller drift and a larger temperature than the initial one, and where a few stable phasespace holes are present.

Electromagnetic fluid drift turbulence in static ergodic magnetic fields
View Description Hide DescriptionNumerical simulations of threedimensional nonlinear electromagnetic fluid drift turbulence in a tokamak plasma with externally applied stochastic magneticfield perturbations are presented. The contributions to the radial particle transport due to nonlinearities arising from advection and magnetic flutter are investigated for perturbation fields of varying strengths in the cases of low and high collisionalities. The perturbation strength is varied to study the physics for Chirikov parameters above 1. In all the cases considered a significant increase of transport is found. A static contribution in the density and velocity perturbations contributes significantly to the total radial transport. For low collisionality, the external perturbation leads to enhanced density and velocity fluctuations over a broad range in the toroidal wavenumber spectrum, resulting in an enhanced turbulent flux. For high collisionality, the density fluctuations stay roughly the same and the velocity fluctuations are increased in an intermediate range of the toroidal wave number spectrum, separated from the maximum of the density fluctuations, thus leaving the turbulent flux almost unchanged.

Effect of twotemperature trapped electrons to nonlinear dustionacoustic solitons
View Description Hide DescriptionPropagation of threedimensional dustionacoustic solitons is investigated in a dusty plasma consisting of positive ions, negatively variablecharged dust particles, and twotemperature trapped electrons. We use the reductive perturbation theory to reduce the basic set of fluid equations to one evolution equation called damped modified KadontsevPetviashivili equation. Exact solution of this equation is not possible, so we obtain the time evolution solitary wave form approximate solution. It is found that only compressive soliton can propagate in this system. We develop a theoretical estimate condition under which the solitons can propagate. It is found that this condition is satisfied for Saturn’s F ring. It is found also that low electrontemperature has a role on the behavior of the soliton width, i.e., for lower (higher) range of low electrontemperature the soliton width decreases (increases). However, high electrontemperature decreases the width. The trapped electrons have no effect on the soliton width. The ratio of free low (high) to trapped low (high) electrontemperatures increases the soliton amplitude. Also, the amplitude increases with free low and free high electrontemperatures. To investigate the stabilty of the waves, we used a method based on energy consideration to obtain a condition for stable solitons. It is found that this condition depends on dust charge variation, streaming velocity, directional cosine of the wave vector along the axis, and temperatures of dust particles, ions, and free electrons.

Driven dissipative whistler wave turbulence
View Description Hide DescriptionDriven dissipative whistler waveturbulence in twodimensional electron magnetohydrodynamics is investigated using very high resolution nonlinear fluid simulations. It is shown that a dual cascade phenomenon of mean magnetic potential and energy invariants is in agreement with predictions based on statistical ensemble and Kolmogorov’s theories. Turbulent length scales larger than the electron skin depth (whistler wave regime) exhibit a spectral break in the vicinity of the forcing wave number that separates the inverse and forward cascade regimes. On the other hand, length scales smaller than the electron skin depth behave like hydrodynamiceddies in which both small and large scale regimes exhibit identical turbulent spectra. In both cases, however, turbulent fluctuations follow an exact Kolmogorovtype spectra. While waveeffects are strong in the whistler wave regime, they are absent entirely in the hydrodynamics regime of the driven electron magnetohydrodynamicturbulence.

Drift wave driven zonal flows in plasmas
View Description Hide DescriptionThe generation of largescale zonal flows by smallscale electrostaticdrift waves in a plasma is considered. The generation mechanism is based on the parametric excitation of convective cells by finite amplitude drift waves. To describe this process a generalized Hasegawa–Mima equation containing both vector and scalar nonlinearities is used. The drift waves are supposed to have arbitrary wavelengths (as compared with the ion Larmor radius). A set of coupled equations describing the nonlinear interaction of drift waves and zonal flows is deduced. The generation of zonal flows turns out to be due to Reynolds stresses produced by finite amplitude drift waves. It is found that the wave vector of the fastest growing mode is perpendicular to that of the drift pump wave. Explicit expressions for the maximum growth rate as well as for the optimal spatial dimensions of the zonal flows are obtained. A comparison with previous results is carried out. The present theory can be used for interpretations of drift wave observations in laboratory plasmas.

The scaling of forced collisionless reconnection
View Description Hide DescriptionWe present twofluid simulations of forced magnetic reconnection with finite electron inertia in a collisionless twodimensional slab geometry. Reconnection in this system is driven by a spatially localized forcing function that is added to the ion momentum equation inside the computational domain. The resulting forced reconnection process is studied as a function of the temporal and spatial structure of the forcing function, the plasma , and strength of the outofplane guide magnetic field component, and the electron to ion mass ratio. Consistent with previous results found in unforced, large systems, for sufficiently strong forcing the reconnection process is found to become Alfvénic, i.e., the inflow velocity scales roughly like some small fraction of the Alfvén speed based on the reconnecting component of the magnetic field just upstream of the dissipation region. The magnitude of this field and thus the rate of reconnection is controlled by the behavior of the forcing function. When the forcing strength is below a certain level, fast reconnection is not observed.
 Magnetically Confined Plasmas, Heating, Confinement

Kinetic properties of shear Alfvén eigenmodes in tokamak plasmas
View Description Hide DescriptionThis work reports on numerical calculations concerning the kinetic properties of low, lowtoroidal Alfvén eigenmodes(TAEs) in tokamakplasmas for fusion relevant parameters. The selfconsistent and nonperturbative code LIGKA [Ph. Lauber, Ph.D. thesis, TU München (2003)] is employed. It is based on a linear gyrokinetic model consisting of the quasineutrality equation and the moment equation for the perturbed current. It is shown that in a certain limit the underlying equations of LIGKA can be simplified to the equations known as the “reduced kinetic model.” An antennalike version of LIGKA allows one to systematically find all shearAlfvéntype modes in a given frequency interval, such as kinetic TAEs (KTAEs) and kinetically modified TAEs. The coupling to the kinetic Alfvén wave (KAW) is found in the form of continuum damping and radiative damping. For the cases examined here, no mode conversion in the centre is found. In the case of a large nonideal parameter, damping rates around 0.5%–1% are found, close to experimental measurements.

Sheared flow effects on ballooning instabilities in threedimensional equilibria
View Description Hide DescriptionThe stability of ideal magnetohydrodynamicballooning modes in the presence of sheared flow is investigated for threedimensional equilibria. Application of ballooning formalism reduces the problem to a partial differential equation in three dimensions that can be solved in the limit of small flow. Analytic calculations demonstrate the stabilizing effect of shear flow. The derived stability criterion generalizes prior work related to axisymmetric equilibrium with sheared toroidal flow.

Gyro center invariant and associated diamagnetic current
View Description Hide DescriptionThe gyro center radial Clebsch coordinate is an exact invariant in confining fields where the gyro center is restricted to move on a magnetic flux surface, and could also be expected to be a useful approximating invariant in other confining magnetic fields. A radial drift invariant generalizes the invariance of if there are oscillatory gyro center radial displacements off the magnetic surface. Expressions for and are obtained for gyrating particles in the drift ordering. An exact energy integral is proven to exist for the firstorder drift motion of the gyro center. The gyro center parallel motion is periodic with respect to a certain curve parameter (the “proper time” for the parallel motion) that deviates slightly, due to the slow perpendicular drift, from the ordinary time. A modification of the parallel invariant is derived which leads to an exact (not only adiabatic) invariant to first order. By using in solutions of the Vlasov equation, it is demonstrated that the approximating gyro center invariant determines the perpendicular plasma diamagnetic current. It is also shown that a fourth stationary motional invariant is required to calculate the parallel plasma current. Several systems with four time independent invariants are identified, and the general solution for straight cylindrical Vlasov equilibria with adiabatic particle motion is determined. A set of four invariants is proposed for adiabatic equilibria in general geometry, including systems where single valued flux surfaces may not exist.

Analytic description of high poloidal beta equilibrium with a natural inboard poloidal field null
View Description Hide DescriptionAnalytical high poloidal beta equilibria for toroidally axisymmetric plasmas with arbitrary aspect ratio and elongation are described. These equilibria that can describe a transition from nondivertor to divertor configuration are exact solutions of the GradShafranov equation when the toroidalcurrent density is quasiuniform. Generally, these are high poloidal beta equilibria, limited by the appearance of a natural inboard poloidal field null. Some of their properties, including the nonuniformity of the poloidal magnetic field in the poloidal direction, the safety factor profile and the magnetic shear profile near the separatrix, the parameter dependence of the poloidal beta and , as well as the toroidal beta on the aspect ratio and the elongation of the magnetic surface, are discussed. Applications to experiments of the Tokamak Fusion Test Reactor (TFTR) [Sabbagh et al., Phys. FluidsB3, 2277 (1991)] are particularly analyzed.

Runaway electron generation in a cooling plasma
View Description Hide DescriptionThe usual calculation of Dreicer [Phys. Rev.115, 238 (1959); 117, 329 (1960)] generation of runaway electrons assumes that the plasma is in a steady state. In a tokamak disruption this is not necessarily true since the plasma cools down quickly and the collision time for electrons at the runaway threshold energy can be comparable to the cooling time. The electrondistribution function then acquires a highenergy tail which can easily be converted to a burst of runaways by the rising electric field. This process is investigated and simple criteria for its importance are derived. If no rapid losses of fast electrons occur, this can be a more important source of runaway electrons than ordinary Dreicer generation in tokamak disruptions.

Ion heating during magnetic relaxation in the helicity injected torusII experiment
View Description Hide DescriptionIon doppler spectroscopy (IDS) is applied to the helicity injected torus (HITII) spherical torus to measure impurity ion temperature and flows. [A. J. Redd et al., Phys. Plasmas9, 2006 (2002)] The IDS instrument employs a 16channel photomultiplier and can track temperature and velocity continuously through a discharge. Data for the coaxial helicity injection (CHI), transformer, and combined current drive configurations are presented. Ion temperatures for transformerdriven discharges are typically equal to or somewhat lower than electron temperaturesmeasured by Thomson scattering. Internal reconnection events in transformerdriven discharges cause rapid ion heating. The CHI discharges exhibit anomalously high ion temperatures, which are an order of magnitude higher than Thomson measurements, indicating ion heating through magnetic relaxation. The CHI discharges that exhibit current and poloidal flux buildup after bubble burst show sustained ion heating during current drive.