Semicollisional Regime Research Articles

Drag exerted by drift-acoustic radiation on a magnetic island

The Island Equilibrium and Transport (IslET) model is extended to allow for the effects of ion compressibility albeit for a constant electron temperature. The model describes fully relaxed, saturated, steady-state magnetic islands arising from the evolution of a tearing mode caused either by an instability or an externally imposed resonant perturbation. It assumes that the collision frequency lies in the semi-collisional regime. A salient new feature of the extended model is an equation describing the transport of parallel momentum. The model is applied to evaluate the drag exerted on a thin island by the emission of drift-acoustic waves.

Plasmoid instability in the semi-collisional regime

We investigate analytically and numerically the semi-collisional regime of the plasmoid instability, defined by the inequality $\unicode[STIX]{x1D6FF}_{\text{SP}}\gg \unicode[STIX]{x1D70C}_{s}\gg \unicode[STIX]{x1D6FF}_{\text{in}}$, where $\unicode[STIX]{x1D6FF}_{\text{SP}}$ is the width of a Sweet–Parker current sheet, $\unicode[STIX]{x1D70C}_{s}$ is the ion sound Larmor radius and $\unicode[STIX]{x1D6FF}_{\text{in}}$ is the width of the boundary layer that arises in the plasmoid instability analysis. Theoretically, this regime is predicted to exist if the Lundquist number $S$ and the length of the current sheet $L$ are such that $(L/\unicode[STIX]{x1D70C}_{s})^{14/9}<S<(L/\unicode[STIX]{x1D70C}_{s})^{2}$ (for a sinusoidal-like magnetic configuration; for a Harris-type sheet the lower bound is replaced with $(L/\unicode[STIX]{x1D70C}_{s})^{8/5}$). These bounds are validated numerically by means of simulations using a reduced gyrokinetic model (Zocco & Schekochihin, Phys. Plasmas, vol. 18 (10), 2011, 102309) conducted with the code Viriato. Importantly, this regime is conjectured to allow for plasmoid formation at relatively low, experimentally accessible, values of the Lundquist number. Our simulations obtain plasmoid instability at values of $S$ as low as ${\sim}250$. The simulations do not prescribe a Sweet–Parker sheet; rather, one is formed self-consistently during the nonlinear evolution of the initial tearing mode configuration. This proves that this regime of the plasmoid instability is realizable, at least at the relatively low values of the Lundquist number that are accessible to current dedicated experiments.

Finite Larmor radius effects on the (m = 2, n = 1) cylindrical tearing mode

New field solvers are developed in the gyrokinetic code GEM [Chen and Parker, J. Comput. Phys. 220, 839 (2007)] to simulate low-n modes. A novel discretization is developed for the ion polarization term in the gyrokinetic vorticity equation. An eigenmode analysis with finite Larmor radius effects is developed to study the linear resistive tearing mode. The mode growth rate is shown to scale with resistivity as γ ∼ η1∕3, the same as the semi-collisional regime in previous kinetic treatments [Drake and Lee, Phys. Fluids 20, 1341 (1977)]. Tearing mode simulations with gyrokinetic ions are verified with the eigenmode calculation.

The linear tearing instability in three dimensional, toroidal gyro-kinetic simulations

Linear gyro-kinetic simulations of the classical tearing mode in three-dimensional toroidal geometry were performed using the global gyro-kinetic turbulence code, GKW. The results were benchmarked against a cylindrical ideal MHD and analytical theory calculations. The stability, growth rate, and frequency of the mode were investigated by varying the current profile, collisionality, and the pressure gradients. Both collisionless and semi-collisional tearing modes were found with a smooth transition between the two. A residual, finite, rotation frequency of the mode even in the absence of a pressure gradient is observed, which is attributed to toroidal finite Larmor-radius effects. When a pressure gradient is present at low collisionality, the mode rotates at the expected electron diamagnetic frequency. However, the island rotation reverses direction at high collisionality. The growth rate is found to follow a η1∕7 scaling with collisional resistivity in the semi-collisional regime, closely following the semi-collisional scaling found by Fitzpatrick. The stability of the mode closely follows the stability analysis as performed by Hastie et al. using the same current and safety factor profiles but for cylindrical geometry, however, here a modification due to toroidal coupling and pressure effects is seen.

The stochastic field transport associated with the slab ITG modes

Many models for anomalous transport consider the turbulent E × B transport arising from electrostatic micro-instabilities. In this paper we investigate whether the perturbed magnetic field that is associated with such instabilities at small but finite values of β can lead to significant stochastic magnetic field transport. Using the tearing parity, long wavelength ion temperature gradient (ITG) modes in a plasma slab with magnetic shear as an example, we calculate the amplitude of the perturbed magnetic field at the resonant surface that results. The plasma model consists of a Braginskii description for the electrons with the dissipation at the resonant surface evaluated for the semi-collisional regime, while a finite ion Larmor radius kinetic model is invoked for the ions. The resulting stochastic field transport is estimated and also compared with an estimate for the E × B transport due to the ITG mode.

Mechanisms of Spontaneous Reconnection: From Magnetospheric to Fusion Plasma

Very often space plasma is treated as collisionless. We check the validity of this paradigm considering various regimes of tearing mode (spontaneous reconnection) including effects of particle collisions and shear of magnetic field. We briefly describe Pitaevskii’s effect of effective modification of collision frequency due to the finite particle Larmor radius in the presence of magnetic field. This effect results in a significant increase of the role of collisionality, especially in a weakly magnetized systems. Another popular paradigm is related with application of MHD description to collisionless or weakly collisional systems. We show, that for current sheets observed in the Earth magnetotail and magnetopause as well as for current sheets formed in Solar corona and in laboratory devices most appropriate is the kinetic semi-collisional tearing regime. Role of “collisions” could play usual Coulomb pair collisions of electrons and ions (e.g. in Solar corona) or effective collisions (scattering) of electrons with the microturbulence wave modes. Transition to real MHD modes requires either very large collisions frequencies and/or very large amplitudes of the magnetic field shear. The largest domain in the parameter space is occupied by the kinetic regimes of tearing mode growth where dissipation is provided either by Landau damping or by real (or effective) collisions.

Numerical Kinetic Model of Axial Confinement in a Mirror Trap

Numerical kinetic model of axial confinement is developed. Simulations show that in semi-collisional regime axial losses are determined by narrow “beam” of cold ions. Analytical treatment is proposed for the case of square-well magnetic field. Simulation of the ambipolar trapping of axial losses in the intermediate collision frequency range shows that essential suppression of axial losses is possible even when ambipolar barrier is of the order of ion temperature.

Magnetic reconnection in weakly collisional highly magnetized electron-ion plasmas

A reduced three-field model of two-dimensional magnetic reconnection in a weakly collisional, highly magnetized plasma consisting of isothermal electrons and cold ions is derived from a set of Braginskii-like fluid equations. The model is then used to calculate the linear growth rate of the reconnecting instability in collisionless and semicollisional parameter regimes.

Model of a source-driven plasma interacting with a wall in an oblique magnetic field

A fluid model of a magnetized source-driven plasma is discussed for regimes with (Debye length)⪡(ion Larmor radius)⪡(plasma size and collisional mean-free path). Plasma collection by the wall is determined in terms of angle of incidence, magnetic strength, and plasma collisionality. For nonparallel incidence, a three-scale asymptotic analysis reveals a three-region matched structure consisting of a magnetically aligned bulk region, the Chodura layer, and the Debye sheath. Sonic Chodura and Bohm conditions define the singular region transitions. For near-parallel incidence, a separate analysis demonstrates the presence of a diffusive-collisional bulk region followed by a thin collisionless layer, which differs partially from the Chodura layer. A parametric analysis unveils the presence of four regimes depending on plasma collisionality: (1) a collisionless regime, with the magnetically channeled bulk region governed by plasma production; (2) a resistive semicollisional regime, where collisions retard the plasma transport in the bulk region; (3) a diffusive semicollisional regime, where the E×B drift dominates the ion flux in the bulk region; and (4) a collisional regime, where collisions cancel out magnetic effects. At grazing incidence, plasma collection is found to vary nonmonotonically with plasma collisionality. Nonzero Debye-length effects are discussed briefly.

Linear tearing mode stability equations for a low collisionality toroidal plasma

Tearing mode stability is normally analysed using MHD or two-fluid Braginskii plasma models. However for present, or future, large hot tokamaks like JET or ITER the collisionality is such as to place them in the banana regime. Here we develop a linear stability theory for the resonant layer physics appropriate to such a regime. The outcome is a set of ‘fluid’ equations whose coefficients encapsulate all neoclassical physics: the neoclassical Ohm's law, enhanced ion inertia, cross-field transport of particles, heat and momentum all play a role. While earlier treatments have also addressed this type of neoclassical physics we differ in incorporating the more physically relevant ‘semi-collisional fluid’ regime previously considered in cylindrical geometry; semi-collisional effects tend to screen the resonant surface from the perturbed magnetic field, preventing reconnection. Furthermore we also include thermal physics, which may modify the results. While this electron description is of wide relevance and validity, the fluid treatment of the ions requires the ion banana orbit width to be less than the semi-collisional electron layer. This limits the application of the present theory to low magnetic shear—however, this is highly relevant to the sawtooth instability—or to colder ions. The outcome of the calculation is a set of one-dimensional radial differential equations of rather high order. However, various simplifications that reduce the computational task of solving these are discussed. In the collisional regime, when the set reduces to a single second-order differential equation, the theory extends previous work by Hahm et al (1988 Phys. Fluids 31 3709) to include diamagnetic-type effects arising from plasma gradients, both in Ohm's law and the ion inertia term of the vorticity equation. The more relevant semi-collisional regime pertaining to JET or ITER, is described by a pair of second-order differential equations, extending the cylindrical equations of Drake et al (1983 Phys. Fluids 26 2509) to toroidal geometry.

Nonlinear tearing mode in inhomogeneous plasma: I. Unmagnetized islands

A theory of the nonlinear growth and propagation of magnetic islands in the semi-collisional regime is presented. The theory includes the effects of finite electron temperature gradients and uses a fluid model with cold ions in slab geometry to describe islands that are unmagnetized in the sense that their width is less than ρs, the ion Larmor radius calculated with the electron temperature. The polarization integral and the natural phase velocity are both calculated. It is found that increasing the electron temperature gradient reduces the natural phase velocity below the electron diamagnetic frequency and thus causes the polarization current to become stabilizing.

The Effect of Semicollisional Accretion on Planetary Spins

Planetesimal accretion during planet formation is usually treated as collisionless. Such accretion from a uniform and dynamically cold disk predicts protoplanets with slow retrograde rotation. However, if the building blocks of protoplanets, planetesimals, are small, of the order of a meter in size, then they are likely to collide within the protoplanet's sphere of gravitational influence, creating a prograde accretion disk around the protoplanet. The accretion of such a disk results in the formation of protoplanets spinning in the prograde sense with the maximal spin rate allowed before centrifugal forces break them apart. As a result of semicollisional accretion, the final spin of a planet after giant impacts is not completely random, but is biased toward prograde rotation. The eventual accretion of the remaining planetesimals in the post-giant-impact phase might again be in the semicollisional regime and delivers a significant amount of additional prograde angular momentum to the terrestrial planets. We suggest that in our solar system, semicollisional accretion gave rise to the preference for prograde rotation observed in the terrestrial planets and perhaps the largest asteroids.

Testing a model for triggering sawtooth oscillations in tokamaks

Simulations of JET and TFTR discharges with the BALDUR integrated modeling code are used to test a sawtooth model that consists of sawtooth triggering mechanisms [Porcelli et al., Plasma Phys. Contolled Fusion 38, 2163 (1996)] together with a modified version of the Kadomtsev sawtooth reconnection model [Kadomtsev, Sov. J. Plasma Phys. 1, 389 (1975)]. In simulations of 12 TFTR low confinement (L-mode) and 5 JET high confinement (H-mode) discharges, sawtooth triggering conditions are examined in order to determine which physical mechanisms are responsible for the crashes. It is found that most sawtooth crashes in the simulations are triggered by the m=1 resistive internal kink instability in plasmas that are in the semicollisional regime. However, in some discharges, some of the sawtooth crashes are triggered as a consequence of the driving force for the internal kink instability overcoming the fast ion stabilization. In rare instances, a sawtooth crash is triggered when the driving force for the internal kink instability overcomes the stabilization produced by the diamagnetic rotation of thermal ions. Generally, the median sawtooth period is found to increase as the magnetic reconnection fraction is increased. Optimal agreement with experimental data for the discharges considered is obtained with a magnetic reconnection fraction of approximately 37% although there is considerable scatter in the sawtooth periods observed in each discharge, both in the experimental data and in the simulations.

Spherical Accretion

We compare different examples of spherical accretion onto a gravitating mass. Limiting cases include the accretion of a collisionally dominated fluid and the accretion of collisionless particles. We derive expressions for the accretion rate and density profile for semi-collisional accretion which bridges the gap between these limiting cases. Particle crossing of the Hill sphere during the formation of the outer planets is likely to have taken place in the semi-collisional regime.

Natural Velocity of Magnetic Islands

The phase velocity of magnetic islands is calculated in the semicollisional regime with cold ions. Two solution branches arise, corresponding to islands propagating with the ions and with the electrons. For the ion branch the phase velocity and the polarization current are small. For the electron branch, the phase velocity depends on the ratio of W, the half-width of the island, and rho(s), the Larmor radius calculated with the electron temperature. For W >>rho(s) the phase velocity is larger than the electron drift velocity and the polarization current is destabilizing. For W << rho(s), the situation is reversed provided that the density and temperature gradients point in the same direction.

Gyrokinetic δf simulation of the collisionless and semicollisional tearing mode instability

The evolution of collisionless and semicollisional tearing mode instabilities is studied using an electromagnetic gyrokinetic δf particle-in-cell simulation model. Drift-kinetic electrons are used. Linear eigenmode analysis is presented for the case of fixed ions and there is excellent agreement with simulation. A double peaked eigenmode structure is seen indicative of a positive Δ′. Nonlinear evolution of a magnetic island is studied and the results compare well with existing theory in terms of saturation level and electron bounce oscillations. Electron-ion collisions are included to study the semicollisional regime. The algebraic growth stage is observed and compares favorably with theory. Nonlinear saturation following the algebraic stage is observed.

Streaming Instabilities in Plasmas with Strongly Coupled Dust Grains

Two-stream instabilities of electrostatic waves in a strongly coupled dusty plasma are investigated. The appropriate dielectric constants for strongly correlated dust grains and weakly coupled electrons and ions are used to derive dispersion relations. The latter admit a new class of streaming instability in a collision-dominated regime. Explicit expressions for the growth rates and thresholds are presented. Also examined are the hydrodynamic and kinetic instabilities in semi-collisional regimes. The relevance of our investigation to turbulence induced dust grain melting in dusty plasma discharges is discussed.

Neutral beam stabilization of sawtooth oscillations in JET

Recent experiments in the Joint European Torus (JET) have provided evidence of sawtooth stabilization by fast ions arising from deuterium neutral beam injection (NBI). A possible theoretical basis for the interpretation of the observed sawtooth period behaviour is investigated and predictions are compared with experimental results, using a sawtooth period model developed to predict the sawtooth period in the International Thermonuclear Experimental Reactor (ITER). Unlike the case of ion cyclotron resonance heating, a detailedcomparison between theory and experiment for NBI has not yet been made.In the model employed in this paper,a beam ion contribution to the internal kink potential energy has been incorporated, using a simple analytical expression valid in the limit of isotropic fast particles.This analytical expression has been found to compare well withdetailed calculations performed with a hybrid kinetic/MHD code NOVA-K, using fast particle distribution functions computed with a plasma analysis code TRANSP. The beam ion contribution has been implemented in a transport code PRETOR and a few representative JET discharges have been analysed and modelled. The beam ion term is found to be sufficiently stabilizing to produce simulated sawtooth periods in agreement with the experimental results. Sawtooth periods computed without taking this term into account are much shorter than the measured periods. The model indicates that sawteeth are triggered in these JET discharges by excitation of the internal kink in the semi-collisional ion-kinetic regime: this was found by previous authors to be the sawtooth trigger most likely to be relevant to ITER. The role of beam ions in determining the sawtooth period in JET is thus found to be similar to the predicted role of α-particles in ITER.

Effect of electron diffusion on m=2 tearing mode

The effect of electron diffusion due to background drift wave turbulence on the stability of the m=2 tearing mode, in the collisionless and the semicollisional regimes, has been studied. The coherent direct interaction approximation has been applied to the drift-kinetic equation to obtain the renormalized tearing mode equation in the turbulent background. It has been found from the present calculations that, in the collisionless regime, the electron diffusion does not affect the stability of the mode significantly, but in the semicollisional regime, it significantly stabilizes the mode.

Effect of a stochastic electron background on low-m tearing modes

The effect of a stochastic electron background on the stability of low-m tearing modes is studied. The coherent direct interaction approximation is applied to the drift-kinetic equation in order to include the effects of current density diffusion, electron density diffusion and magnetic field diffusion in the tearing mode equations. It is found that for the m=1 mode in the collisionless regime the magnetic field diffusion destabilizes the mode further, while in the collisional regime these diffusions may not significantly affect the mode; for the m[ges ]2 modes in the collisional and semicollisional regimes the electron density diffusion stabilizes the modes, while in the collisionless regime the effects of current density diffusion and electron density diffusion may balance each other without significantly affecting the modes.