Statistical behavior of filamentary plasmas
This work describes a study of plasmas with highly intermittent filamentary structures. A statistical model of two-dimensional magnetohydrodynamics is presented, based on a representation of the fluid as a collection of discrete current-vorticity concentrations. This approach is modeled after discrete vortex models of hydrodynamical turbulence. In a highly intermittent plasma, the induction force is small compared to the convective motion, and when this force is neglected, the plasma vortex system is described by a Hamiltonian. Canonical and micro-canonical statistical calculations show that both the vorticity and the current may exhibit large-scale structure, and the expected states revert to known hydrodynamical states as the magnetic field vanishes. These results show that when the filament calculation is expanded to include the inductive force, it approaches the Fourier equilibria in the low-temperature limit, and the previous Hamiltonian plasma vortex results in the high-temperature limit. A three-dimensional vortex model is also outlined. A statistical calculation in the canonical ensemble and numerical simulations show that a non-zero large-scale magnetic field is statistically favored, and that the preferred shape of this field is a long, thin tube of flux. In a tokamak, a stochastic magnetic field will give rise to strongly filamented current distributions. An external magnetic field possesses field lines described by a nonlinear map, while current fluctuations along these field lines have a toroidal dependence which takes the same form as the time dependence of a system of hydrodynamical vortices. Magnetic surfaces of a tokamak interior in steady state are given by the asymptotic limit of the behavior of the current filaments. Numerical simulations combining the effects of the internal and external fields show that ideal magnetic surfaces are always disrupted by current fluctuations along the field lines.
Hamiltonian structure of the Vlasov-Einstein system and the problem of stability for spherical relativistic star clusters
H.E. Kandrup, P.J. Morrison
The Hamiltonian formulation of the Vlasov-Einstein system, which is appropriate for collisionless, self-gravitating systems like clusters of stars that are so dense that gravity must be described by the Einstein equation, is presented. In particular, it is demonstrated explicitly in the context of a 3 + 1 splitting that, for spherically symmetric configurations, the Vlasov-Einstein system can be viewed as a Hamiltonian system, where the dynamics is generated by a noncanonical Poisson bracket, with the Hamiltonian generating the evolution of the distribution function ƒ (a noncanonical variable) being the conserved ADM mass-energy HADM. This facilitates a geometric understanding of the evolution of ƒ in an infinite-dimensional phase space, providing thereby a natural interpretation of the constraints associated with conservation of phase space. This geometric interpretation also facilitates the derivation of improved criteria for linear stability by focusing on dynamically accessible perturbations δƒ which satisfy all the constraints of phase space conservation. An explicit expression is derived for the energy δ(2)HADM associated with an arbitrary spherical phase space preserving perturbation of an arbitrary spherical equilibrium, and it is shown that the equilibrium must be linearly stable if δ(2)HADM is positive semi-definite. Insight into the Hamiltonian reformulation is provided by a description of general finite degree of freedom systems. Intuition derived from simple finite models clarifies several features of the Vlasov-Einstein system; for example, how, negative energy modes preclude necessary and sufficient conditions for stability and why, unlike the Newtonian case, the existence of negative energy perturbations for some static, isotropic equilibrium apparently signals the Onset of a linear instability. An Appendix exhibits the construction of a completely covariant bracket which generates the Vlasov-Einstein system for arbitrary configurations in a form independent of any assumed 3 + 1 splitting.
Vertical slot convection: A linear theory
A. McAllister, R. Steinolfson, T. Tajima
We investigate the linear stability properties of fluid convection in a vertical slot. We use a Fourier-Chebychev decomposition to set up the linear eigenvalue problems for the Vertical Slot Convection and Benard problems, and solved for eigenvalues, neutral stability surves, and critical point values of the Grashof number, G, and the wavenumber, α. Plots of the real and imaginary parts of the eigenvalues as functions of G and α are given for a wide rane of the Prandtl number, Pr, and special note is made of the complex mode that becomes linearly unstable above Pr ~ 12.5. A discussion comparing different special cases facilitates the physical understanding of the VSC equations, especially the interaction of the shear-flow and buoyancy induced physics. Making use of the real and imaginary eignevalues and the phas eproperties of the eigenmodes, we categorize the eigenmodes. We find that the mode structure becomes progressively simpler with increasing Pr, with the greatest complexity in the mid ranges where the terms in the heat equation are of roughly the same size.
Effect of alpha particles on toroidal Alfven eigenmodes
An overview is given of the analytic structure for linear theory of the Toroidal Alfven Eigenmode (TAE), where multiple gap structures occur. A discussion is given of the alpha particle drive and the various dissipation mechanisms that can stabilize the system. A self-cocnsistent calculation of the TAE mode, for a low-beta high-aspect-ratio plasma, indicates that though the alpha particle drive is comparable to the dissipation mechanisms, overall stability is still achieved for ignited ITER-like plasma. A brief discussion is given of the nonlinear theory for the TAE mode and how nonlinear alpha particle dynamics can be treated by mapping methods.
Scenarios for the nonlinear evolution of beam-driven instability with a weak source
H.L. Berk, B.N. Breizman
If the field amplitude reaches the level where orbit stochasticity occurs, the particle diffusion leads to a further conversion of the distribution's free energy to wave energy. This leads to a rapid quaslinear relaxation (a phase space explosion) of the distribution function. Hence the overall response of the system is characterized by a relatively long time interval where the source needs to build up the distribution to its unstable shape as well as provide a sufficient amount of free energy for the instability to grow to the stochastic threshold of particle motion. The particle distribution is then flattened by the quasilinear diffusion in a relatively short time interval to regenerate the cycle.
Linked mirror neutron source
V.P. Pastukhov, H.L. Berk
A toroidally linked mirror system is proposed as a neutron source. The mirror core has the favourable feature of a high plasma beta in a relatively small volume. A low beta highly elliptical toroidal linkage prevents rapid electron thermal loss and enhances the power efficiency of previous designs of mirror machine based neutron sources to 30-40%. Specific problems related to the following are discussed: low beta toroidal MHD equilibrium, finite beta distortion of the plasma column, radial electric field, particle drift orbits, rotational stability, beam ion kinetics and cross-field transport.
Drift wave coherent vortex structures in inhomogeneous plasmas
Nonlinear drift wave vortex structures in magnetized plasmas are studied theoretically and numerically in the various physical environments. The effects of density and temperature gradients on drift wave vortex dynamics are analyzed using a fully nonlinear model with the Boltzmann density distribution. The equation, based on the full Boltzmann relation, possess no localized monopole solution in the short wavelength ([approximately][rho][sub s]) region, while in the longer wavelength (~(ρsrn)1/2) region the density profile governs the existence of monopole-like solution. In the longer wavelength regime, however, the monopoles cannot be localized sufficiently to avoid coupling to propagating drift waves due to the inhomogeneity of the plasma. Thus, the monopole vortex is a long lived coherent structure, but it is not precisely a stationary structure since the coupling results in a "flapping" tail. The tail causes energy of the vortex to leak out, but the effect of the temperature gradient is to reduce the leaking of this energy. Nonlinear coherent structures governing by the coupled drift wave-ion acoustic mode equations in sheared magnetic field are studied analytically and numerically. A solitary vortex equation that includes the effects of density and temperature gradients and magnetic shear is derived and analyzed. The results show that for a plasma in a sheared magnetic field, there exist the solitary vortex solutions. The new vortex structures are dipole-like in their symmetry, but not the modon type of dipoles. The numerical simulations are performed in 2-D with the coupled vorticity and parallel mass flow equations. The vortex structures in an unstable drift wave system driven by parallel shear flow are studied. The nonlinear solitary vortex solutions are given and the formation of the vortices from a turbulent state is observed from the numerical simulations.
Beam-Beam interaction effects on particle dynamics
The effects of the beam-beam interaction on particle dynamics in a synchrotron collider are investigated. The main highlight of this work is the investigation of collective effects of the beam-beam interaction in a self-consistent approach that naturally incorporates the correct single-particle dynamics. The most important target of this simulation is to understand and predict the long-time (108-109 rotations) behavior of the beam luminosity and lifetime. For this task a series of computer codes in one spatial dimension has been developed in increasing order of sophistication. They are: the single-particle dynamics tracking code, the strong-strong particle-in-cell (PIC) code, and the particle code based on the δf algorithm. The latter two include the single-particle dynamics of the first. The third approach is used to understand beam lifetime by trying to improve the numerical noise problem in the second. Scans in tune ν0 and tune shift Δν0 show regions of stability and instability that correspond to the regions predicted by a lineax theory. Strong resonance beam blowup is observed just above ν0 = 1/2 and ν0 = 1/4, where the rate of beam blowup drops with the order of the resonance. In both the strong-strong code and δf code using the reference parameters of the Superconducting Super Collider, oscillations in the tune shift, Δν, are observed. The odd moments of the beam are increasing in oscillation amplitude with rotation number, while the amplitudes of the even moments either decrease or remain constant. The "flip-flop" effect is observed in the strong-strong code simulations and is found to be sensitive to the initial conditions.
Kinetic theory of toroidicity and ellipticity-induced Alfven eigenmodes
R.R. Mett, S.M. Mahajan
Toroidicity-induced Alfven eigenmodes (TAE) and ellipticity-induced Alfven eigenmodes (EAE) are currently of great interest because they may destroy the confinement of fast ions in a burning tokamak plasma. The present study focuses on kinetic effects, extending the non-perturbative kinetic analysis of the TAE to the EAE. One finds that the parameter which measures the kinetic character of the EAE is significantly smaller than it is for the TAE for elongated plasmas like DIII-D. The parameter is rather small for the lower mode numbers but attains values of order unity or larger for the higher mode numbers, since the parameter scales as the square of the mode number. Consequently, one expects the lower mode number EAE's to have a strongly magnetohydrodynamic (MHD) character, and to suffer only perturbative damping that depends linearly on the dissipative mechanisms. However, while the former is true, the latter is not necessarily the case. This work examines these kinetic T/EAE(KT/EAE) modes in further detail.
Magnetic reconnection and current-sheet formation at X-type neutral points
R.S. Steinolfson, L. Ofman, P.J. Morrison
Numerical solutions of the nonlinear, resistive magnetohydrodynamic (MHD) equations are used to study the evolution of a perturbed or stressed x-type neutral point. By performing individual simulations for both compressible and incompressible plasmas, we are able to demonstrate that the important physics for this problem involves just the interaction between the plasma flow velocity and the magnetic field and that the thermodynamics has a relatively passive effect. We have also done separate simulations for both solid, conducting wall boundary conditions at a fixed distance from the x-point and for open boundary conditions that adjust as required by the evolving solution within the boundaries. With solid, conducting wall boundary conditions, our solutions for azimuthally symmetric disturbances agree with those obtained in previous analytic linear studies. In this case the stressed x-point relaxes back to the unstressed state on a time scale somewhat shorter than the time scale for the linear resistive tearing mode. Perturbations that are not azimuthally symmetric can relax even faster than the symmetric modes. When the conditions at the boundary are free to adjust, the disturbances grow in amplitude on an Alfven time scale with the eventual formation of a current sheet separating two y-points. This rapid growing behavior is, of course, in sharp contrast to the relatively slow decaying solutions obtained with closed boundaries. The growing solutions qualitatively agree with previous analytic x-point solutions that have been suggested as an explanation for the rapid energy conversion in flares and substorms.
Magnetic reconnection at stressed x-type neutral points
L. Ofman, P.J. Morrison, R.S. Steinolfson
The reconnection and relaxation of two-dimensional stressed (non-potential) x-type neutral point magnetic fields are studied via solution of the nonlinear resistive 2-D MHD equations and by analytical solution of the linear eigenvalue problem. The linear dispersion relation was generalized for azimuthally nonsymmetric perturbations, and have found that for modes with azimuthal mode numbers m > 0, the relaxation can occur at a rate faster than that for n = m = 0, where n is the radial quantum'' number. One finds that for nearly azimuthally symmetric magnetic perturbations that are zero at the boundary; i.e. the frozen-in'' (sometimes called fine-tied'') boundary conditions, the fields relax incompressibly and nonlinearly to the unstressed x-type neutral point at a rate close to that predicted by linear theory. Also, fully compressible nonlinear MHD simulations have been performed, which show that the interaction between the plasma flow velocity and the magnetic field is the important physical effect, while the inclusion of thermodynamics does not affect the evolution considerably. A Lyapunov functional for the nonlinear incompressible 2-D resistive MHD equations is derived to show that the current-free x-point configuration is a global equilibrium to which general initial conditions relax.
A self-consistent turbulence generated scenario for L-H transition
Y.Z. Zhang, S.M. Mahajan
The turbulence-induced ion banana polarization current associated with steep ion temperature gradients is explored as a possible mechanism for generating poloidal momentum at the tokamak edge. In the light of a recently developed two-dimensional turbulence theory, one can obtain a simple closed expression relating this current (determined by turbulence levels) to the derivatives of the poloidal rotation speed. A self-consistent system, then, emerges, if we balance the turbulence-induced poloidal momentum with that dissipated by viscosity. Under suitable conditions this system may show a bifurcation controlled by a parameter dependent on temperature gradients. Both the bifurcation point, and the shear layer width are predicted for a prescribed flow in terms of a scale characterizing the nonlinearity of viscosity. The crucial relevance of the flow parity with the turbulence scenario is analyzed.
Two-dimensional analytic solution for toroidal Alfven eigenmodes
Z. Sedlacek, H. Ye, S.M. Mahajan
A two-dimensional analysis of the toroidal Alfven eigenmodes (TAE) is presented, based on an integrodifferential equation describing the shear Alfven perturbation of a toroidal plasma equilibrium in terms of coupling among the toroidal Alfven continua with the usual gap structure. Using a method similar to the Van Kampen-Case analysis for the Vlasov equation, we derive exact analytic expressions for the dispersion function and the two-dimensional eigenmode structure. The dispersion function is expressed in terms of Cauchy-type integrals, which explicitly expresses the global character of TAE modes and facilitates the calculation of their damping. The continuum-damped TAE modes are shown to be, in general, not true eigenmodes of the toroidal plasma equilibrium, but rather resonances corresponding to zeros of the analytic continuation of the dispersion function into unphysical sheets of its Riemann surface. Approximate but explicit expressions for the dispersion relation and the eigenfunction are also obtained in the limit of vanishing inverse aspect ratio.
Multiple-gap theory of toroidal Alfven waves with kinetic effects
X.D. Zhang, Y.Z. Zhang, S.M. Mahajan
The stability of kinetic toroidal Alfven waves with multi-gap coupling is analyzed by using the two-dimensional ballooning transform. An alternate convergence scheme, based on the smallness of the inverse aspect ratio, is devised. The resulting wave functions are oscillatory and do not balloon in contrast to the wave functions of conventional ballooning theory. It is shown that the single-gap theory is a special, weak shear (s → 0) limit of the formalism. Analytical and numerical results for the two fundamental branches, the ideal toroidal Alfven eigenmode (TAE), and the kinetic toroidal Alfven eigenmode (KTAE) are presented and discussed.
Kinetic Toroidal ion temperature gradient instability in the presence of sheared flows
J.Q. Dong, W. Horton
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 fluxsurface. Magnetic gradient and curvature drifts of the ions as well as finite ion Larmor radious effects are induced. 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 alos to the shrinking of the mode width. Using the mixing length formulas for the thermal transport associated with theunstable modes we show 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
Low frequency fluctuations of plasma magnetic fields
It is shown that even a nonmagnetized plasma with temperature T sustains zero-frequency magnetic fluctuations in thermal equilibrium. Fluctuations in electric and magnetic fields, as well as in densities, are computed. Four cases are studied: a cold, gaseous, isotropic, nonmagnetized plasma; a cold, gaseous plasma in a uniform magnetic field; a warm, gaseous plasma described by kinetic theory; and a degenerate electron plasma. For the simple gaseous plasma, the fluctuation strength of the magnetic field as a function of frequency and wave number is calculated with the aid of the fluctuation-dissipation theorem. This calculation is done for both collisional and collisionless plasmas. The magnetic-field fluctuation spectrum of each plasma has a large zero-frequency peak. The peak is a Dirac δ function in the collisionless plasma; it is broadened into a Lorentzian curve in the collisional plasma. The plasma causes a low-frequency cutoff in the typical blackbody radiation spectrum, and the energy under the discovered peak approximates the energy lost in this cutoff. When the imposed magnetic field is weak, the magnetic-field wave-vector fluctuation spectra of the two lowest modes are independent of the strength of the imposed field. Further, these modes contain finite energy even when the imposed field is zero. It is the energy of these modes that forms the zero-frequency peak of the nonmagnetized plasma. In deriving these results, a simple relationship between the dispersion relation and the fluctuation power spectrum of electromagnetic waves is found. The warm plasma is shown, by kinetic theory, to exhibit a zero-frequency peak in its magnetic-field fluctuation spectrum as well. For the degenerate plasma, we find that electric-field fluctuations and number-density fluctuations vanish at zero frequency; however, the magnetic-field power spectrum diverges at zero frequency.
Correlation theory of a two-dimensional plasma turbulence with shear flow
Y.Z. Zhang, S.M. Mahajan
When the ion sound effect is neglected, a wide class of electrostatic plasma turbulence can be modeled by a two-dimensional equation for the generalized enstrophy Ψ, an inviscid constant of motion along the turbulent orbits. Under the assumption of a Gaussian stochastic electrostatic potential, an averaged Green's function method is used to rigorously derive equations for the N-particle correlation functions for a dissipative and sheared flow. This approach is equivalent to the cumulant expansion method [T. H. Dupree, Phys. Fluids 15, 334 (1972); 21, 783 (1978)] used to study the Vlasov–Poisson system. For various cases of interest, appropriate equations are solved to obtain the absolute level as well as the detailed structure of the two-point correlation function C(r), and its Fourier transform, the enstrophy spectral function I(k). Uniformly valid analytical expressions are derived for the dissipative but shearless case resulting in a "fluctuation–dissipation" theorem relating the total spectral intensity to classical viscosity. These self-consistent results show a strong logarithmic modification of the mixing length estimates for the turbulence levels. For the extremely important and interesting problem of a sheared flow, the suppression of turbulence is demonstrated by using asymptotic analytical techniques in the inviscid range, and uniformly valid numerical methods for the dissipative system. The current asymptotic methods reproduce the results obtained in the orbit picture [Y. Z. Zhang and S. M. Mahajan, Phys. Fluids B 4, 1385 (1992)], but provide much clearer physical perspective and a better definition of crucial parameters like the decorrelation time. The uniformly valid numerical approach allows the determination of the change in spectral shape and intensity due to the presence of shear. It is found that the suppression is more effective for longer wavelengths as compared to the shorter wavelengths. This and other relevant issues, concerning the role of flows with shear (including its radial variation) in the understanding of the L–H transitions in tokamaks, are discussed.
The shear-Alfven harmonic chain vortex of current-carrying low-β plasma cylinder
J. Lui, Y. Chen, W. Horton
Nonlinear shear-Alfven mode is low-β (β << me/mi) current-carrying plasma cylinder is studied by using two-fluid model. A new type of solitary vortex solution is given. This type of solution consists of monopole and multipole parts, along the azimuthal direction it forms a chain of interowven cyclones and anticyclones, while the structure globally propagates in the azimuthal direction of plasma with constant angular velocity. Comparing with related structures previously obtained for other nonlinear modes, the equilibrium current of plasma affects the vortex structure. The fact that the radial size of the vortex is comparable to the radius of plasma cylinder may provide some convenience for experimental observation of this coherent structure.
Electron-temperature-gradient-induced instability in tokamak scrape-off layers
H.L. Berk, R.H. Cohen, D.D. Ryutov, Yu.A. Tsidulko, X.Q. Xu
An electron temperature instability driven by the Kunkel-Guillory sheath impedance has been applied to the scrape-off layer of tokamaks. The formalism has been generalized in order to more fully account for parallel wavelength dynamics, to differentiate between electromagnetic and electrostatic perturbations and to account for particle recycling effects. It is conjectured that this conducting wall instability leads to edge fluctuations in tokamaks that produce scrape-off layers tens of ion gyroradii thick. The predicted instability characteristics have similarities to experimental edge fluctuation data, and the scrape-off layer width in the DIII-D experiment agrees with theoretical estimates that can be derived from mixing length theory.
Particle simulations in toroidal geometry
A computational took to be used in kinetic aimulations of toroidal plasmas is being developed. The initial goal of the project is to develop an electrostatic gyrokinetic model for studying transport and stability problems in tokamaks. In this brief report, preliminary results from the early stages of this effort are presented.
Map model for nonlinear alpha particle interaction with Alfven waves
H.L. Berk, B.N. Breizman, H. Ye
A map model has been developed for studying the nonlinear interaction of alpha particles with the toroidal Alfvén eigenmodes. The map is constructed by assuming a linear interaction during a single poloidal transit, which allows the study of the nonlinear interaction over many transits. By using this map, analytic expressions are obtained for the particle nonlinear bounce frequency, and the wave amplitude threshold for the onset of particle orbit stochasticity. The map model can also facilitate self-consistent simulations which incorporate the time variation of the waves.
Fusion, Magnetic confinement
Particle simulation algorithm with short range forces in MHD and fluid flow
S. Cable, T. Tajima, K. Umegaki
Attempts are made to develop numerical algorithms for handling fluid flows involving liquids and liquid-gas mixtures. In these types of systems, the short-range intermolecular interactions are important enough to significantly alter behavior predicted on the basis of standard fluid mechanics and magnetohydrodynamics alone. We have constructed a particle-in-cell (PIC) code for the purpose of studying the effects of these interactions. Of the algorithms considered, the one which has been successfully implemented is based on a MHD particle code developed by Brunel et al. In the version presented here, short range forces are included in particle motion by, first, calculating the forces between individual particles and then, to prevent aliasing, interpolating these forces to the computational grid points, then interpolating the forces back to the particles. The code has been used to model a simple two-fluid Rayleigh-Taylor instability. Limitations to the accuracy of the code exist at short wavelengths, where the effects of the short-range forces would be expected to be most pronounced.
Toroidal effects on drift wave turbulence
J.J. Lebrun, T. Tajima, M. Gray, G. Furnish, W. Horton
The universal drift instability and other drift instabilities driven by density and temperature gradients in a toroidal system are investigated in both linear and nonlinear regimes via particle simulation. Runs in toroidal and cylindrical geometry show dramatic differences in plasma behavior, primarily due to the toroidicity-induced coupling of rational surfaces through the poloidal mode number m. In the toroidal system studied, the eigenmodes are seen to possess (i) an elongated, nearly global radial extent, (ii) a higher growth rate than in the corresponding cylindrical system, (iii) an eigenfrequency nearly constant with radius, and (iv) a global temperature relaxation and enhancement of thermal heat conduction. Most importantly, the measured χi shows an increase with radius and an absolute value on the order of that observed in experiment. On the basis of the present observations, it is argued that the increase in χi with radius observed in experiment is caused by the global nature of heat convection in the presence of toroidicity-induced mode coupling.
Dynamics of the ion-ion acoustic instability in the thermalization of ion beams
J-H. Han, W. Horton, J-N. Leboeuf
Particle simulation using a nonlinear adiabatic electron response with two streaming ion species and nonlinear theory are used to study the collisionless thermalization of ion beams in a hot electron plasma. The slow beam or subsonic regime is investigated and the criterion for the transition from predominantly light ion to predominantly heavy ion heating is developed. Long-lived ion hole structures are observed in the final state.
The electrostatic wake of a superthermal test electron in a magnetized plasma
A.A. Ware, J.C. Wiley
The electrostatic potential is determined for a test electron with v|| >> vTe in a uniform magnetized plasma (ωce >> ωpe). In the frame of the test electron, part of the spatially oscillatory potential has spherical symmetry over the hemisphere to the rear of the electron and is zero ahead of the electron. A second part of different character, which makes the potential continuous at the plane containing the electron, is oscillatory in the radial direction but decreases almost monotonically in the axial direction.
Nonlinear studies of M=1 modes in high temperature plasmas
Nonlinear evolution of the m=1 mode is examined in high-temperature plasmas where the mode is in the semicollisional or collisionless regime. Unlike the finite −Δ(m>=2) tearing modes, the nonlinear evolution of which is collisional, both the semicollisional and collisionless m=1 modes exhibit nonlinearly enhanced growth rates that far exceed their linear values, thus making their nonlinear evolution collisionless; this accelerated growth of a collisionless m=1 mode may explain the fast sawtooth crashes observed in large tokamaks.
Fast magnetic field penetration into an intense neutralized ion beam
R. Armale, N. Rostoker
Experiments involving propagation of neutralized ion beams across a magnetic field indicate a magnetic field penetration time determined by the Hall resistivity rather than the Spitzer or Pedersen resistivity. In magnetohydrodynamics the Hall current is negligible because electrons and ions drift together in response to an electric field perpendicular to the magnetic field. For a propagating neutralized ion beam, the ion orbits are completely different from the electron orbits and the Hall current must be considered. There would be no effect unless there is a component of magnetic field normal to the surface which would usually be absent for a good conductor. It is necessary to consider electron inertia and the consequent penetration of the normal component to a depth c/ωp. In addition it is essential to consider a component of magnetic field parallel to the velocity of the beam which may be initially absent, but is generated by the Hall effect. The penetration time is determined by whistler waves rather than diffusion.