Finite Ion Larmor Radius Effects Research Articles

Self-generation of hollow current profile and tilt instability in field-reversed configuration

Two-dimensional electromagnetic particle simulation is performed to investigate the profile relaxation from a magnetohydrodynamic (MHD) equilibrium to a kinetic one and the physical property of the kinetic equilibrium in the field-reversed configuration. The radial oscillation is excited in order to relax an excess energy in the MHD equilibrium. After this profile oscillation, the system spontaneously relaxes toward a kinetic equilibrium, in which the electron current profile becomes hollow as a result of the combined effects of the gradient-B drift near the field-null line and the E×B drift generated by the ion finite Larmor radius effect near the magnetic separatrix. On the other hand, the ion current profile becomes peaked due to the effect of the ion meandering orbit near the field-null line. The stability of the obtained kinetic equilibrium against the tilt mode is also studied by means of three-dimensional full electromagnetic particle simulation. It is found that the growth rate of the tilt instability in the case of the hollow current profile and high separatrix beta value is smaller than that in the case of the peaked current profile.

Violating Suydam criterion produces feeble instabilities

Eigenmode analysis of a magnetic shear localized ideal interchange mode indicates a marginal stability condition consistent with the Suydam (and Mercier) criterion. However, the linear eigenmode analysis shows that it is not a very restrictive condition because the mode growth rate is exponentially small just above the marginal stability condition and the modes are easily stabilized by finite ion Larmor radius (FLR) effects. This suggests that for practical applications to toroidal confinement experiments, the Suydam–Mercier criterion does not introduce a restrictive condition for operational limits. The criterion for robust growth appears to be almost a factor of two greater than the Suydam criterion. It is also shown that even the presence of usual nonideal effects (FLR, electron diamagnetic flow and resistivity) does not alter this higher criterion for robust growth.

The origins of electrical resistivity in magnetic reconnection: Studies by 2D and 3D macro particle simulations

This article argues the roles of electrical resistivity in magnetic reconnection, and also presents recent 3D particle simulations of coalescing magnetized flux bundles. Anomalous resistivity of the lower-hybrid-drift (LHD) instability, and collisionless effects of electron inertia and/or off-diagonal terms of electron pressure tensor are thought to break the frozen-in state that prohibits magnetic reconnection. Studies show that, while well-known stabilization of the LHD instability in high-beta plasma condition makes anomalous resistivity less likely, the electron inertia and/or the off-diagonal electron pressure tensor terms make adequate contributions to break the frozen-in state, depending on strength of the toroidal magnetic field. Large time and space scale particle simulations show that reconnection in magnetized plasmas proceeds by means of electron inertia effect, and that electron acceleration results instead of Joule heating of the MHD picture. Ion inertia contributes positively to reconnection, but ion finite Larmor radius effect does negatively because of charge separation of ions and magnetized electrons. The collisionless processes of the 2D and 3D simulations are similar in essence, and support the mediative role of electron inertia in magnetic reconnection of magnetized plasmas.

Numerical study of tilt stability of prolate field-reversed configurations

Global stability of the field-reversed configuration (FRC) has been investigated numerically using both three-dimensional magnetohydrodynamic and hybrid (fluid electron and δf particle ion) simulations. The stabilizing effects of velocity shear and finite ion Larmor radius (FLR) on the n=1 internal tilt mode in the prolate FRCs have been studied. Sheared rotation is found to reduce the growth rate, however a large rotation rate with Mach number of M≳1 is required in order for significant reduction in the instability growth rate to occur. Kinetic effects associated with large thermal ion orbits have been studied for different kinetic equilibria. The simulations show that there is a reduction in the tilt mode growth rate due to FLR effects, but complete linear stability has not been found, even when the thermal ion gyroradius is comparable to the distance between the field null and the separatrix. The instability existing beyond the FLR theory threshold could be due to the resonant interaction of the wave with ions whose Doppler shifted frequency matches the betatron frequency.

Critical magnetic shear for the tokamak ideal internal kink mode in the ion kinetic regime

It is shown that for the ideal internal kink mode of tokamak plasma, there exists a critical magnetic shear s0c for its excitation when the finite ion Larmor radius effect is considered in the mode singular layer. This can be related to the sudden onset of the sawtooth and also to the stabilization of the sawtooth by the trapped hot ions observed experimentally. The alpha particle effect on this mode is revisited and some new results are reported.

The past, present, and future of Z pinches

The Z pinch is enjoying a renaissance as the world’s most powerful yet efficient soft x-ray source which can energize large volume hohlraums for indirectly driven inertial confinement fusion. It has the advantages of being efficient and having high energy and power density. Its early history will be traced from the 18th century to the present day. The most notable feature of the Z pinch is its instability. The various regimes of stability analysis will be reviewed, including resistive and finite ion Larmor radius effects. Work in the last 10 years on single fibres, especially of cryogenic deuterium, gave neutrons that were of the same origin, namely, beam–plasma interactions, as reported by Kurchatov. The renaissance has come about through the implosion of arrays of fine wires. Research at Sandia National Laboratory has shown that by using more and finer wires, the x-ray radiation emitted at stagnation increased in power and decreased in pulse width. The understanding of these results has been advanced considerably by theory, simulation and smaller-scale, well diagnosed experiments showing the early uncorrelated m=0 instabilities on each wire, the inward jetting of plasma to the axis, the global Rayleigh–Taylor instability and the mitigating effect of nested arrays.

Regularized magnetic islands. II. The role of polarization current

The role of polarization current in the problem of rotating magnetic islands in the absence of drift effects is discussed for the case of nonregularized and regularized velocity profiles governed by the perpendicular viscosity. The polarization current is calculated taking into account the finite ion Larmor radius effects. It is shown that, in correspondence with the general rule by Waelbroeck and Fitzpatrick [Phys. Rev. Lett. 78, 1703 (1997)], this current is destabilizing. The destabilizing contribution of the polarization current into the generalized Rutherford equation for the magnetic island width does not depend on whether the profile function is regularized or not. It is shown that the Larmor corrections to the current consist of the sum of two oppositely directed currents and do not contribute into the generalized Rutherford equation.

A kinetic‐fluid model

A nonlinear kinetic‐fluid model for high β plasmas with multiple ion species which can be applied to multiscale phenomena is presented. The model embeds important kinetic effects due to finite ion Larmor radius (FLR), wave‐particle resonances, magnetic particle trapping, etc., in the framework of simple fluid descriptions. When further restricted to low‐frequency phenomena, with frequencies less than the ion cyclotron frequency, the kinetic‐fluid model takes a simpler form in which the fluid equations of multiple ion species collapse into one‐fluid density and momentum equations and a low‐frequency generalized Ohm's law. The kinetic effects are introduced via plasma pressure tensors for ions and electrons which are computed from particle distribution functions that are governed by the Vlasov equation or simplified plasma dynamics equations such as the gyrokinetic equation. The ion FLR effects provide a finite parallel electric field, a perpendicular velocity that modifies the E × B drift, and a gyroviscosity tensor, all of which are neglected in the usual one‐fluid MHD description. Eigenmode equations are derived, which include magnetosphere‐ionosphere coupling effects for low‐frequency waves (e.g., kinetic/inertial Alfvén waves and ballooning‐mirror instabilities).

Generalized MHD for numerical stability analysis of high-performance plasmas in tokamaks

A set of generalized magnetohydrodynamic (MHD) equations is formulated to accommodate the effects associated with high ion and electron temperatures in high-performance plasmas in tokamaks. The effects of neoclassical bootstrap current, neoclassical ion viscosity, the ion finite Larmor radius effect and electron and ion drift effects are taken into account in two-fluid MHD equations together with gyroviscosity, parallel viscosity, electron parallel inertia and collisionless ion heat flux. The ion velocity is identified as the plasma velocity, while the electron velocity is expressed in terms of the plasma velocity and electric current. Ion and electron momentum equations are combined to give the plasma momentum equation. The perpendicular (with respect to the equilibrium magnetic field) ion momentum equation is used as perpendicular Ohm's law and the parallel electron momentum equation - as parallel Ohm's law. Perpendicular Ohm's law allows for the Hall and ion drift effects. Parallel Ohm's law includes the electron drift effect, collisionless skin effect and bootstrap current. In addition, both perpendicular and parallel Ohm's laws contain the resistivity. Due to the quasineutrality condition, the ions and electrons are characterized by the same number density which is described by the ion continuity equation. On the other hand, the ion and electron temperatures are allowed to be different. The ion temperature is described by the ion energy equation allowing for the oblique heat flux, in addition to the perpendicular ion heat flux. The electron temperature is determined by the condition of high parallel electron heat conductivity. The ion and electron parallel viscosities are represented in a form valid for all the collisionality regimes (Pfirsch-Schluter, plateau, and banana). An optimized form of the generalized MHD equations is then represented in terms of the toroidal coordinate system used in the JET equilibrium and stability codes. The derived equations provide a basis for development of generalized MHD codes for numerical stability analysis of high-performance plasmas in tokamaks.

A magnetohydrodynamic model of kinetic Alfvén waves with finite ion gyroradius

Low-frequency magnetohydrodynamic equations including the effects of finite ion Larmor radius are derived. These equations contain all the physical terms necessary to describe kinetic Alfvén waves. Among other applications, these waves are used to interpret ultra-low-frequency wave phenomena in the Earth’s magnetosphere. The finite ion Larmor radius effects are usually disregarded; however, under magnetospheric conditions, the ion temperature cannot be considered negligible since Ti≥Te. The finite ion gyroradius is accounted for by the ion stress tensor calculated in this Brief Communication. The nonlinear one-fluid system of equations is shown to adequately describe linear kinetic Alfvén waves in a low-β(β≪1) homogeneous plasma. It is demonstrated that energy of the plasma system is conserved.

Fast particle destabilization of toroidal Alfven eigenmodes

High-n, toroidal Alfven eigenmodes (TAEs) are studied on the basis of a kinetic model that includes full thermal ion finite Larmor radius effects, trapped electron collisions and fast particle instability drive. Lower kinetic TAEs (KTAEs) are shown to be non-existent in the nearly collisionless limit. Like TAEs, upper KTAEs are shown to exist because of thermal ion finite Larmor radius effects in the dissipationless limit. Dissipation effects on the stability of both TAEs and upper KTAEs can be treated perturbatively. However, owing to their extended mode structure in the ballooning space, upper KTAEs usually remain stable or weakly unstable even with large fast particle free energy. On the other hand, TAEs can be strongly destabilized. A new resonant TAE (RTAE) can be excited when the fast particle drive is significantly large. The RTAE is a beam-like mode, with its frequency determined mainly by the wave-particle resonance condition. The frequency of the RTAE can be much less than the TAE gap frequency and may be interpreted as the BAE observed in DIII-D experiments. As plasma beta increases, the TAE, RTAE and kinetic ballooning modes strongly couple; the TAE changes into the RTAE and eventually connects to the kinetic ballooning mode. Numerical results and analytical analysis on the stability of the RTAEs and KTAEs will be presented and compared with the TAE stability

Alpha particle effects on the internal kink and fishbone modes

The effects of alpha particles on the internal kink and fishbone modes are studied analytically. The nonadiabatic contribution from untrapped alpha particles is negligible. Finite inverse aspect ratio, plasma β, and plasma shaping effects can significantly enhance the trapped particle drift reversal domain in the pitch angle space and reduce the bounce-averaged magnetic drift frequency. The decrease of the drift magnitude and drift reversal effects on the ideal kink mode is small, but the βα threshold for the fishbone mode can be much lower than previously predicted [B. Coppi, S. Migliuolo, F. Pegoraro, and F. Porcelli, Phys. Fluids B 2, 927 (1990)]. Moreover, the fishbone mode could be excited by alpha particles, even when the plasma is stable in the ideal magnetohydrodynamic (MHD) limit. In addition, the ion diamagnetic drift frequency (finite ion Larmor radius effect) has a strong destabilizing effect on the fishbone mode when it is comparable with the trapped alpha-averaged precessional drift frequency, even though it stabilizes the plasma in the ideal MHD limit.

Experimental study of curvature-driven flute instability in the gas-dynamic trap*

A curvature-driven flute instability will be excited in the magnetized plasmas if the magnetic field lines curve toward the entire plasma boundary. Conditions under which it can be effectively stabilized in axisymmetric geometry have been experimentally studied in a gas-dynamic trap (GDT) at Novosibirsk. Flexible design of the experimental device and the availability of neutral beams and ion cyclotron heating enabled the pressure-weighted curvature to be varied over a wide range. The stability limits were thus measured and compared with those predicted by the modified Rosenbluth–Longmire criterion. Characteristics of unstable curvature-driven flute modes were also measured and found to conform to a theory including finite ion Larmor radius (FLR) effects. Stable operation during neutral beam injection was achieved with a cusp end cell, resulting in an increase in Te to 45 eV, limited by end losses rather than anomalous power losses.

Larmor radius effect on Rayleigh-Taylor instability of a partially ionized plasma in composite medium

The simultaneous effects of finite ion Larmor radius (FLR) and collisions with neutral atoms are investigated on the stability of an infinitely conducting plasma of variable density in the presence of a horizontal magnetic field. The perturbations propagating along the ambient magnetic field are considered. It is established that, real part ofn is negative, wheren is the growth rate of disturbance, so that instability does not arise in the form of increasing amplitude, i.e. overstability. A variational principle is used to obtain an approximate solution of the problem. Further the case of two semi-infinitely extending plasmas of constant densities separated by a horizontal interface is considered. It is found that the system is stable (for some wave numbers) for potentially stable configuration and unstable (for other wave numbers) for potentially unstable configuration even if there are collisions with dust particles. Also criteria determining stability and instability are independent of FLR effects.

Cascading process in the flute‐mode turbulence of a plasma

The cascades of ideal invariants in the flute‐mode turbulence are analyzed by considering a statistics based on an elementary three‐mode coupling process. The statistical dynamics of the system is investigated on the basis of the existence of the physically most important (PMI) triad. When finite ion Larmor radius effects are considered, the PMI triad describes the formation of zonal flows.

Excitation of the modified Simon–Hoh instability in an electron beam produced plasma

An intermediate frequency (fci≤f≤fce) electrostatic instability has been observed in an electron beam produced, cylindrical plasma column. This instability has been identified as a new instability, the modified Simon–Hoh instability (MSHI), which has an instability mechanism similar to the Simon–Hoh instability (SHI). This instability can occur in a cylindrical collisionless plasma if a radial dc electric field exists and if this radial dc electric field and the radial density gradient are in the same direction. The origin of the dc electric field is found to be the difference between the ion and the electron radial density profiles. In such a plasma if the ions are essentially unmagnetized but the electrons are magnetized, a velocity difference in the θ direction can arise because of the finite ion Larmor radius effect. This leads to a space charge separation between the electron and ion density perturbations in the θ direction. The consequent perturbed azimuthal electric field Eθ1 and the enhancement of the density perturbation by the Eθ1×B0 velocity occur in the same manner as in the SHI. The instability frequency is decided by the ion azimuthal drift velocity. This new instability has been investigated through experiments and theory.

Kinetic quasitoroidal ion temperature gradient instability in the presence of sheared flows

The gyrokinetic integral equations for the study of the ion temperature gradient driven mode (ηi mode) in toroidal geometry, at low plasma pressure, are extended to include equilibrium ion parallel v0∥(r) and perpendicular vE(r) sheared flows, where r is the minor radius of the flux surface. Magnetic gradient and curvature drifts of the ions, as well as finite ion Larmor radius effects, are included. The parallel sheared flow is shown to be destabilizing. The perpendicular sheared flow is a stabilizing mechanism. Mixing length estimates show that the ηi-mode induced ion thermal transport increases with increasing parallel flow shear, and decreases with perpendicular flow shear. The decrease of the ion transport is due not only to the decrease of the mode growth rate, but also to the shrinking of the mode width. Using the mixing length formulas for the thermal transport associated with the unstable modes, it is shown that the results are consistent with the experimental observations concerning the improvement of confinement in H-mode discharges coincident with the increase of cross-field sheared flows.

Alfvén-wave heating of a high-beta plasma column

Alfvén-wave heating experiments are performed on the High Beta Q Machine (HBQM) [Phys. Fluids 25, 262 (1982)] straight plasma column. An investigation is made of Alfvén-wave energy absorption by the Alfvén-resonant layer at which the local shear Alfvén velocity is equal to the phase velocity of the driven plasma standing wave. The radial position of the resonance in the plasma is defined by the equilibrium magnetic field diamagnetic profile and nearly constant electron number density profile. The experiment uses a 250 MW ringing LC circuit in the 615–720 kHz frequency range with the capacitors driving the HBQM compression coil segments that are connected to form a 2.0 m long m=±1 antenna configuration with axial wave number kz=15.7 m−1. Experiments were conducted on both hydrogen and deuterium plasmas. In the hydrogen case the Alfvén resonance is predicted to lie inside the plasma. Internal magnetic field probes were used to measure both the rf and equilibrium magnetic fields. The rf heating produces a larger change in the hydrogen plasma axial magnetic field profile that is centered around r=6 cm, near the Alfvén-resonance location. The rf B̃θ signal shows a phase reversal at the same radius. Other measurements show a 25% increase in plasma pressure for hydrogen. The antenna resistance increases from 0.06 Ω vacuum value to approximately 0.2 Ω in the presence of hydrogen plasma with a 31% absorption efficiency. The rf-induced effects in the deuterium plasma are 20%–50% smaller, agreeing with estimates based on collisional viscous damping, or collisonless model with finite ion cyclotron and Larmor radius effects. In the hydrogen case both these models predict insufficient energy absorption to explain observed heating. However, a magnetofluid calculation taking account of the singularity at the Alfvén-resonant layer reproduces the value of the antenna resistance for the hydrogen case.

On finite ion Larmor radius MHD equations

The finite ion Larmor radius effects in magnetohydrodynamic equations are formulated without imposing plasma incompressibility. The result is in agreement with the gyrokinetic formulation.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Rayleigh-Taylor instability of a plasma with finite ion Larmor radius effects

The stability of an infinitely conducting plasma of variable density has been investigated taking into account the finiteness of the ion Larmor radius. The perturbations propagating along the ambient magnetic field are considered. It is established that, in general,n2 is necessarily real, wheren is the growth rate of disturbance, thus ruling out the possibility of overstability or damped oscillations. The solution is shown to be characterized by a variational principle, which provides the basis for obtaining an approximate solution of the problem. Two density distributions are considered: (i) a continuously stratified plasma layer and (ii) two semi-infinitely extending plasmas of constant densities separated by a horizontal interface. In both cases it has been shown that for the said disturbances the stability criterion remains unaffected by the inclusion of finite Larmor radius effects, though the amplified motion is strongly inhibited due to their inclusion.