Volume 20, Issue 4, April 2013

In this work, the development of modelbased feedback control that stabilizes an unstable equilibrium is obtained for the Modified HasegawaWakatani (MHW) equations, a classic model in plasma turbulence. First, a balanced truncation (a model reduction technique that has proven successful in flow control design problems) is applied to obtain a low dimensional model of the linearized MHW equation. Then, a modelbased feedback controller is designed for the reduced order model using linear quadratic regulators. Finally, a linear quadratic Gaussian controller which is more resistant to disturbances is deduced. The controller is applied on the nonreduced, nonlinear MHW equations to stabilize the equilibrium and suppress the transition to driftwave induced turbulence.
 LETTERS


On the linear stability of collisionless microtearing modes
View Description Hide DescriptionMicrotearing modes are an important drive of turbulent heat transport in presentday fusion plasmas. We investigate their linear stability under verylow collisionality regimes, expected for the next generations of devices, using gyrokinetic and driftkinetic approaches. At odds with current opinion, we show that collisionless microtearing instabilities may occur in certain experimental conditions, particularly relevant for such devices as reversed field pinches and spherical tokamaks.

On the toroidal current density flowing across a poloidalmagneticfield null in an axisymmetric plasma
View Description Hide DescriptionThe axisymmetry condition and two of Maxwell's equations are used to show that, in general, there are no nested magnetic surfaces around a poloidalmagneticfield null for a sufficiently small value of the toroidal current density flowing there. Hence, the toroidal current density at the axis of a magnetic configuration with extreme shear reversal cannot continuously approach zero unless nested surfaces are first broken or particular values are assigned to boundary conditions and other plasma parameters. The threshold of the toroidal currentdensity at which the topology changes is shown to be set by such parameters, and some examples of the predicted topology transition are presented using analytical solutions of the GradShafranov equation.
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 ARTICLES

 Basic Plasma Phenomena, Waves, Instabilities

Nonmodal analysis of the diocotron instability: Cylindrical geometry
View Description Hide DescriptionThe temporal evolution of the linear diocotron instability of the cylindrical annular plasma column is investigated by employing the extension of the shearing modes methodology to the cylindrical geometry. It was obtained that the spatial timedependent distortion of the electron density initial perturbations by shear flows leads to the nonmodal evolution of the potential, which was referred to as the manifestation of the continuous spectrum. The evolution process leads toward the convergence to the phaselocking configuration of the mutually growing normal modes.

Collisionless shock formation, spontaneous electromagnetic fluctuations, and streaming instabilities
View Description Hide DescriptionCollisionless shocks are ubiquitous in astrophysics and in the lab. Recent numerical simulations and experiments have shown how they can arise from the encounter of two collisionless plasma shells. When the shells interpenetrate, the overlapping region turns unstable, triggering the shock formation. As a first step towards a microscopic understanding of the process, we analyze here in detail the initial instability phase. On the one hand, 2D relativistic ParticleInCell simulations are performed where two symmetric initially cold pair plasmas collide. On the other hand, the instabilities at work are analyzed, as well as the field at saturation and the seed field which gets amplified. For mildly relativistic motions and onward, Weibel modes govern the linear phase. We derive an expression for the duration of the linear phase in good agreement with the simulations. This saturation time constitutes indeed a lowerbound for the shock formation time.

Gradient instabilities of electromagnetic waves in Hall thruster plasma
View Description Hide DescriptionThis paper presents a linear analysis of gradient plasma instabilities in Hall thrusters. The study obtains and analyzes the dispersion equation of highfrequency electromagnetic waves based on the twofluid model of a cold plasma. The regions of parameters corresponding to unstable high frequency modes are determined and the dependence of the increments and intrinsic frequencies on plasma parameters is obtained. The obtained results agree with those of previously published studies.

Viscous potential flow analysis of electrified miscible finitely conducting fluid through porous media
View Description Hide DescriptionIn this work, a viscous potential flow analysis is used to investigate capillary surface waves between two horizontal finite fluid layers. The two layers have finite conductivities and admit mass and heat transfer. A general dispersion relation is derived. The presence of finite conductivities together with the dielectric permeabilities makes the horizontal electric field play a dual role in the stability criterion. The phenomenon of negative viscosity is observed. A new growth rate parameter, depending on the kinematical viscosity of the lower fluid layer, is found and has a stabilizing effect on the unstable modes. The growth rates and neutral stability curve are given and applied to airwater interface. The effects of various parameters are discussed for the KelvinHelmholtz and the RayleighTaylor instabilities.

Kinetic waterbag model of global collisional drift waves and ion temperature gradient instabilities in cylindrical geometry
View Description Hide DescriptionCollisional drift waves and ion temperature gradient (ITG) instabilities are studied using a linear waterbag kinetic model [P. Morel et al., Phys. Plasmas 14, 112109 (2007)]. An efficient spectral method, already validated in the case of drift waves instabilities [E. Gravier et al., Eur. Phys. J. D 67, 7 (2013)], allows a fast solving of the global linear problem in cylindrical geometry. The comparison between the linear ITG instability properties thus computed and the ones given by the COLUMBIA experiment [R. G. Greaves et al., Plasma Phys. Controlled Fusion 34, 1253 (1992)] shows a qualitative agreement. Moreover, the transition between collisional drift waves and ITG instabilities is studied theoretically as a function of the ion temperature profile.

Modeling of long range frequency sweeping for energetic particle modes
View Description Hide DescriptionLong range frequency sweeping events are simulated numerically within a onedimensional, electrostatic bumpontail model with fast particle sources and collisions. The numerical solution accounts for fast particle trapping and detrapping in an evolving wave field with a fixed wavelength, and it includes three distinct collisions operators: Drag (dynamical friction on the background electrons), Krooktype collisions, and velocity space diffusion. The effects of particle trapping and diffusion on the evolution of holes and clumps are investigated, and the occurrence of nonmonotonic (hooked) frequency sweeping and asymptotically steady holes is discussed. The presented solution constitutes a step towards predictive modeling of frequency sweeping events in more realistic geometries.

An improved model to determine the inception of positive upward leader–streamer system considering the leader propagation during dark period
View Description Hide DescriptionStem–leader transition and frontstreamer inception are two essential conditions for the inception of positive upward leader–streamer system (LSS). Previous models have not considered the initialleader propagation during dark period and have not been verified systematically. In this paper, a series of positive upward discharge simulation experiments was designed and carried out. Characteristic parameters of the discharge process related to the inception of positive upward LSS, namely, the firstcorona inception voltage, the firstcorona charge, the dark period, and the LSS inception voltage, were obtained. By comparing these experiment results with simulation results calculated using previous models, it was found that it is improper to assume that the length of the initial leader is a fixed value. Finally, an improved inception model of positive upward LSS considering the leader propagation during dark period was developed and verified with experiment results.

Ion response in a weakly ionized plasma with ion flow
View Description Hide DescriptionWe study the ion response to an initial perturbation in a weakly ionized plasma with ion flow driven by a dc electric field. The analysis is made by extending the classical Landau work [J. Phys. (USSR) 10, 25 (1946)] to the ion kinetic equation including ionneutral collisions and a dc electric field. We show, in particular, that the complex frequencies of ion waves can be directly found from a known expression for the ion susceptibility [A. V. Ivlev et al., Phys. Rev. E 71, 016405 (2005); V. A. Schweigert, Phys. Rep. 27, 997 (2001)]; this is not obvious from its original derivation, because it only aims to describe the ion response for real frequencies.

Stability of compressible reduced magnetohydrodynamic equilibria—Analogy with magnetorotational instability
View Description Hide DescriptionStability analyses for equilibria of the compressible reduced magnetohydrodynamics (CRMHD) model are carried out by means of the EnergyCasimir (EC) method. Stability results are compared with those obtained for ideal magnetohydrodynamics (MHD) from the classical criterion. An identification of the terms in the second variation of the free energy functional for CRMHD with those of is made: two destabilizing effects present for CRMHD turn out to correspond to the kink and interchange instabilities in usual MHD, while the stabilizing roles of field line bending and compressibility are also identified in the reduced model. Also, using the EC method, stability conditions in the presence of toroidal flow are obtained. A formal analogy between CRMHD and a reduced incompressible model for magnetized rotating disks, due to Julien and Knobloch [EAS Pub. Series, 21, 81 (2006)], is discovered. In light of this analogy, energy stability analysis shows that the condition for magnetorotational instability (MRI) for the latter model corresponds to the condition for interchange instability in CRMHD, with the Coriolis term and shear velocity playing the roles of the curvature term and pressure gradient, respectively. Using the EC method, stability conditions for the rotating disk model, for a large class of equilibria with possible nonuniform magnetic fields, are obtained. In particular, this shows it is possible for the MRI system to undergo, in addition to the MRI, another instability that is analogous to the kink instability. For vanishing magnetic field, the Rayleigh hydrodynamical stability condition is recovered.

The properties of photonic band gaps for threedimensional plasma photonic crystals in a diamond structure
View Description Hide DescriptionIn this paper, the properties of photonic band gaps (PBGs) for two types of threedimensional plasma photonic crystals (PPCs) composed of isotropic dielectric and unmagnetized plasma with diamond lattices are theoretically investigated for electromagnetic waves based on a modified plane wave expansion method. The equations for type1 structure are theoretically deduced, which depend on the diamond lattices realization (dielectric spheres immersed in plasma background). The influences of dielectric constant of dielectric, plasma collision frequency, filling factor, and plasma frequency on PBGs are investigated, respectively, and some corresponding physical explanations and the possible methods to realize the threedimensional PPCs in experiments are also given. From the numerical results, it has been shown that not only the locations but also the gap/midgap ratios of the PBGs for two types of PPCs can be tuned by plasma frequency, filling factor, and the relative dielectric constant, respectively. However, the plasma collision frequency has no effect on the frequency ranges and gap/midgap ratios of the PBGs for two types of PPCs.

Parametric study of nonrelativistic electrostatic shocks and the structure of their transition layer
View Description Hide DescriptionNonrelativistic electrostatic unmagnetized shocks are frequently observed in laboratory plasmas and they are likely to exist in astrophysical plasmas. Their maximum speed, expressed in units of the ion acoustic speed far upstream of the shock, depends only on the electrontoion temperature ratio if binary collisions are absent. The formation and evolution of such shocks is examined here for a wide range of shock speeds with particleincell simulations. The initial temperatures of the electrons and the 400 times heavier ions are equal. Shocks form on electron time scales at Mach numbers between 1.7 and 2.2. Shocks with Mach numbers up to 2.5 form after tens of inverse ion plasma frequencies. The density of the shockreflected ion beam increases and the number of ions crossing the shock thus decreases with an increasing Mach number, causing a slower expansion of the downstream region in its rest frame. The interval occupied by this ion beam is on a positive potential relative to the far upstream. This potential preheats the electrons ahead of the shock even in the absence of beam instabilities and decouples the electron temperature in the foreshock ahead of the shock from the one in the far upstream plasma. The effective Mach number of the shock is reduced by this electron heating. This effect can potentially stabilize nonrelativistic electrostatic shocks moving as fast as supernova remnant shocks.

Nonlinear modulation of short wavelength compressional Alfvén eigenmodes
View Description Hide DescriptionMost Alfvénic activity in the frequency range between toroidal Alfvén eigenmodes and roughly one half of the ion cyclotron frequency on National Spherical Torus eXperiment [Ono et al., Nucl. Fusion 40, 557 (2000)], that is, approximately 0.3 MHz up to ≈1.2 MHz, are modes propagating counter to the neutral beam ions. These have been modeled as Compressional and Global Alfvén Eigenmodes (CAE and GAE) and are excited through a Dopplershifted cyclotron resonance with the beam ions. There is also a class of copropagating modes at higher frequency than the counterpropagating CAE and GAE. These modes have been identified as CAE, and are seen mostly in the company of a low frequency, n = 1 kinklike mode. In this paper, we present measurements of the spectrum of these high frequency CAE (hfCAE) and their mode structure. We compare those measurements to a simple model of CAE and present a predatorprey type model of the curious nonlinear coupling of the hfCAE and the low frequency kinklike mode.

Visible imaging and spectroscopy of disruption runaway electrons in DIIID
View Description Hide DescriptionThe first visible light images of synchrotron emission from disruption runaway electrons are presented. The forwarddetected continuum radiation from runaways is identified as synchrotron emission by comparing two survey spectrometers and two visible fast cameras viewing in opposite toroidal directions. Analysis of the elongation of 2D synchrotron images of ovalshaped runaway beams indicates that the velocity pitch angle ranges from 0.1 to 0.2 for the detected electrons, with energies above 25 MeV. Analysis of synchrotron intensity from a camera indicates that the tail of the runaway energy distribution reaches energies up to 60 MeV, which agrees with 0D modeling of electron acceleration in the toroidal electric field generated during the current quench. A visible spectrometer provides an independent estimate of the upper limit of runaway electron energy which is roughly consistent with energy determined from camera data. Synchrotron spectra reveal that approximately 1% of the total postthermal quench plasma current is carried by the detected highenergy runaway population with energies in the range of 25–60 MeV; the bulk of the plasma current thus appears to be carried by relativistic electrons with energy less than 25 MeV. In addition to stable oval shapes, runaway beams with other shapes and internal structure are sometimes observed.

Closure and transport theory for highcollisionality electronion plasmas
View Description Hide DescriptionSystems of algebraic equations for a highcollisionality electronion plasma are constructed from the general moment equations with linearized collision operators [J.Y. Ji and E. D. Held, Phys. Plasmas 13, 102103 (2006) and J.Y. Ji and E. D. Held, Phys. Plasmas 15, 102101 (2008)]. A systematic geometric method is invented and applied to solve the system of equations to find closure and transport relations. It is known that some closure coefficients of Braginskii [S. I. Braginskii, Reviews of Plasma Physics (Consultants Bureau, New York, 1965), Vol. 1] are in error up to 65% for some finite values of x (cyclotron frequency × electronion collision time) and have significant error in the largex limit [E. M. Epperlein and M. G. Haines, Phys. Fluids 29, 1029 (1986)]. In this work, fitting formulas for electron coefficients are obtained from the 160 moment (Laguerre polynomial) solution, which converges with increasing moments for and from the asymptotic solution for large xvalues. The new fitting formulas are practically exact (less than 1% error) for arbitrary x and Z (the ion charge number, checked up to Z = 100). The ion coefficients for equal electron and ion temperatures are moderately modified by including the ionelectron collision operator. When the ion temperature is higher than the electron temperature, the ionelectron collision and the temperature change terms in the moment equations must be kept. The ion coefficient formulas from 3 moment (Laguerre polynomial) calculations, precise to less than 0.4% error from the convergent values, are explicitly written.

Electromagnetic envelope solitary waves with transverse perturbation in a plasma
View Description Hide DescriptionThe system of fluidMaxwell equations governing the twodimensional dynamics of electromagnetic waves in a plasma is analyzed by means of multiple scale perturbation method. It is shown that the evolution of the amplitude of wave field is governed by a twodimensional nonlinear Schrödinger equation. The stability of bright envelope solitons is studied using the variational method. It is found that the development of transverse periodic perturbations on bright solitons is faster for a plasma with near critical density. Dynamics of electromagnetic bright solitons is investigated in the longwave approximation. Our model predicts the appearance of collapse of electromagnetic waves in plasmas and describes the collapse dynamics at initial stages.

Guiding center equations for ideal magnetohydrodynamic modes
View Description Hide DescriptionGuiding center simulations are routinely used for the discovery of modeparticle resonances in tokamaks, for both resistive and ideal instabilities and to find modifications of particle distributions caused by a given spectrum of modes, including large scale avalanches during events with a number of large amplitude modes. One of the most fundamental properties of ideal magnetohydrodynamics is the condition that plasma motion cannot change magnetic topology. The conventional representation of ideal magnetohydrodynamic modes by perturbing a toroidal equilibrium field through , however, perturbs the magnetic topology, introducing extraneous magnetic islands in the field. A proper treatment of an ideal perturbation involves a full Lagrangian displacement of the field due to the perturbation and conserves magnetic topology as it should. In order to examine the effect of ideal magnetohydrodynamic modes on particle trajectories, the guiding center equations should include a correct Lagrangian treatment. Guiding center equations for an ideal displacement are derived which preserve the magnetic topology and are used to examine mode particle resonances in toroidal confinement devices. These simulations are compared to others which are identical in all respects except that they use the linear representation for the field. Unlike the case for the magnetic field, the use of the linear field perturbation in the guiding center equations does not result in extraneous mode particle resonances.

Pitch angle scattering of an energetic magnetized particle by a circularly polarized electromagnetic wave
View Description Hide DescriptionThe interaction between a circularly polarized wave and an energetic gyrating particle is described using a relativistic pseudopotential that is a function of the frequency mismatch. Analysis of the pseudopotential provides a means for interpreting numerical results. The pseudopotential profile depends on the initial mismatch, the normalized wave amplitude, and the initial angle between the wave magnetic field and the particle perpendicular velocity. For zero initial mismatch, the pseudopotential consists of only one valley, but for finite mismatch, there can be two valleys separated by a hill. A large pitch angle scattering of the energetic electron can occur in the twovalley situation but fast scattering can also occur in a single valley. Examples relevant to magnetospheric whistler waves show that the energetic electron pitch angle can be deflected 5°towards the loss cone when transiting a 10 ms long coherent wave packet having realistic parameters.

Electron random walk and collisional crossover in a gas in presence of electromagnetic waves and magnetostatic fields
View Description Hide DescriptionThis paper deals with random walk of electrons and collisional crossover in a gas evolving toward a plasma, in presence of electromagnetic (EM) waves and magnetostatic (B) fields, a fundamental subject of importance in areas requiring generation and confinement of wave assisted plasmas. In presence of EM waves and B fields, the number of collisions N suffered by an electron with neutral gas atoms while diffusing out of the volume during the walk is significantly modified when compared to the conventional field free square law diffusion; where Λ is the characteristic diffusion length and λ is the mean free path. There is a distinct crossover and a time scale associated with the transition from the elastic to inelastic collisions dominated regime, which can accurately predict the breakdown time (τ_{c} ) and the threshold electric field (E_{BD} ) for plasma initiation. The essential features of cyclotron resonance manifested as a sharp drop in τ_{c} , lowering of E_{BD} and enhanced electron energy gain is well reproduced in the constrained random walk.