Comparisons of theoretically predicted transport from ion temperature gradient instabilities to L-mode tokamak experiments
M. Kotschenreuther, V. Wong, P. Lyster, H.L. Berk, R. Denton, W.M. Miner, P. Valanju
Abstract
The theoretical transport from kinetic micro-instabilities driven by ion temperature gradients is a sheared slab is compared to experimentally inferred transport in L-mode tokamaks. Low noise gyrokinetic simulation techniques are used to obtain the ion thermal transport coefficient X. This X is much smaller than in experiments, and so cannot explain L-mode confinement. Previous predictions based on fluid models gave much greater X than experiments. Linear and nonlinear comparisons with the fluid model show that it greatly overestimates transport for experimental parameters. In addition, disagreements among previous analytic and simulation calculations of X in the fluid model are reconciled.
Existence and damping of toroidicity-induced Alfven eigenmodes
S.M. Mahajan, R.R. Mett
Abstract
A new method of analyzing the toroidicity-induced Alfven eigenmode (TAE) from kinetic theory is presented. The analysis includes electron parallel dynamics non-perturbatively, an effect which is found to strongly influence the character and damping of the TAE -- contrary to previous theoretical predictions. The normal electron Landau damping of the TAE is found to be higher than previously expected, and may explain recent experimental measurements of the TAE damping coefficient.
Nonlinear evolution of resistive tearing mode instability with shear flow and viscosity
L. Ofman, P.J. Morrison, R. Steinolfson
Abstract
The nonlinear evolution of the tearing mode instability with equilibrium shear flow is investigated via numerical solutions of the resistive magnetohydrodynamic (MHD) equations. The two-dimensional simulations are in slab geometry, are periodic in the x direction, and are initiated with solutions of the linearized MHD equations. The magnetic Reynolds number S was varied from 102 to 105, a parameter V that measures the strength of the flow in units of the average Alfvén speed was varied from 0 to 0.5, and the viscosity as measured by the Reynolds number Sν satisfied Sν≥103. When the shear flow is small (V≤0.3) the tearing mode saturates within one resistive time, while for larger flows the nonlinear saturation develops on a longer time scale. The two-dimensional spatial structure of both the flux function and the streamfunction distort in the direction of the equilibrium flow. The magnetic energy release decreases and the saturation time increases with V for both small and large resistivity. Shear flow decreases the saturated magnetic island width, and generates currents far from the tearing layer. The validity of the numerical solutions was tested by verifying that the total energy and the magnetic helicity are conserved. The results of the present study suggest that equilibrium shear flow may improve the confinment of tokamak plasma.
Anomalous ion thermal diffusion from eta-i modes
A. Beklemishev, W. Horton
Abstract
Models of ion temperature gradient-driven turbulence are reexamined in terms of the structure of the turbulent spectrum for radially localized modes to explain the significant difference between the radial profiles of ion heat conductivity inferred from local turbulence models and those observed in experiments. The strong radial inhomogeneity of the effective density of the turbulence spectrum in k is shown to produce a significant increase of the fluctuation level and ion thermal diffusion toward the plasma edge as compared with the local models under certain conditions. The conditions required are stated and should be tested with numerical simulations.
Percolation, statistical topography, and transport in random media
M.B. Isichenko
Abstract
A review of classical percolation theory is presented, with an emphasis on novel applications to statistical topography, turbulent diffusion, and heterogeneous media. Statistical topography involves the geometrical properties of the isosets (contour lines or surfaces) of a random potential ψ(x). For rapidly decaying correlations of ψ, the isopotentials fall into the same universality class as the perimeters of percolation clusters. The topography of long-range correlated potentials involves many length scales and is associated either with the correlated percolation problem or with Mandelbrot's fractional Brownian reliefs. In all cases, the concept of fractal dimension is particularly fruitful in characterizing the geometry of random fields. The physical applications of statistical topography include diffusion in random velocity fields, heat and particle transport in turbulent plasmas, quantum Hall effect, magnetoresistance in inhomogeneous conductors with the classical Hall effect, and many others where random isopotentials are relevant. A geometrical approach to studying transport in random media, which captures essential qualitative features of the described phenomena, is advocated.
Toroidal kinetic eta-i mode study in high temperature plasmas
J.Q. Dong, W.C. Horton, J.Y. Kim
Abstract
A new kinetic integral equation for the study of the ion-temperature-gradient-driven mode in toroidal geometry is developed that includes the ion toroidal (curvature and magnetic gradient) drift motion ωD, the mode coupling from finite k|| due to the toroidal feature of the sheared magnetic configuration. The integral equation allows the stability study for arbitrary k||v/(ω − ωD) and k⊥ ρi. A systematic parameter study is carried out for the low β circular flux surface equilibrium. Possible correlations between the unstable mode characteristics and some experimental results such as the fluctuation spectrum and the anomalous ion thermal transport measurements are discussed.
Edge turbulence scaling with shear flow
Y.Z. Zhang, S.M. Mahajan
Abstract
A formula relating turbulence levels with arbitrary shear flow is derived. When the diffusion coefficient is made a functional of the corresponding turbulence level, it is found that the scaling laws governing turbulence suppression are considerably modified. The results are compared with known formulas in various limiting cases, indicating that turbulence suppression mainly pertains in the moderate shear flow regime. The results also show that a flattened (steep) radial equilibrium gradient tends to enhance (eliminate) turbulence suppression due to the shear flow.
Self-consistent radial sheath in ignited plasmas
H. Xiao, R. Carrera, R.D. Hazeltine
Abstract
The radial sheath in ignited plasmas is studied based on the self-consistent effects of alpha particle's guiding-center orbits and E × B-induced toroidal rotation. The slowing-down alpha particle distribution determines the structure of the radial sheath through finite Larmor radius effects.
Neoclassical diffusion in a turbulent plasma
P. Yushmanov, A.I. Smolyakov
Abstract
The authors study a new diffusion mechanism which is observed when electrostatic fluctuations and toroidal drift are taken into account simultaneously. The diffusion is caused by transitions of particles from trajectories localized in the convective cells of electromagnetic fluctuations to trajectories aligned in the direction of toroidal drift. The spatial transport coefficient is found to be comparable to or higher than nonclassical diffusion in the plateau regime, regardless of the collision frequency, and is only weakly dependent on the parameters of the fluctuation spectrum for υ ≥ ud.
Drift wave vortices in nonuniform plasmas and sheared magnetic fields
X.N. Su, W. Horton, P.J. Morrison
Abstract
Nonlinear coherent structures governed by the coupled drift wave–ion-acoustic mode equations in nonuniform plasmas with sheared magnetic fields 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 analytic and numerical studies show that for a plasma in a sheared magnetic field, even without the temperature and drift velocity gradients, solitary vortex solutions are possible; however, these solutions are not exponentially localized due to the presence of a nonstructurally stable perturbative tail that connects to the core of the vortex. The new coherent vortex structures are dipolelike in their symmetry, but are not the modons of Larichev and Reznik. In the presence of a small temperature or drift velocity gradient, the new shear-induced dipole cannot survive and will separate into monopoles, like the case of the modon in a sheared drift velocity as studied in Su et al. [Phys. Fluids B 3, 921 (1991)]. The solitary solutions are found from the nonlinear eigenvalue problem for the effective potential in a quasi-one-dimensional approximation. The numerical simulations are performed in two dimensions with the coupled vorticity and parallel mass flow equations.
Particle simulation of non-circular toroidal plasmas in non-orthogonal curvilinear coordinates
K. Umegaki, M.J. Lebrun, T. Tajima
Abstract
A numerical algorithm has been developed for electrostatic particle simulation of arbitrarily shaped toroidal plasmas such as axisymmetric tokamaks with non-circular cross-section. Two-dimensional non-orthogonal flux surface coordinates are generated from the numerical solution of the Grad-Shafranov equation. Using a coordinate transformation technique, interpolation and smoothing of charge densities and electric fields are made in the transformed nonorthogonal coordinates. A fast Poisson solver has been developed in which a finite difference scheme for the radial direction and an FFT for toroidal and poloidal directions are employed. A time-decentered Lorentz pusher is used to integrate the momentum equation of ions where decentering is effective to eliminate the ion cyclotron limitation on time step. The Lorentz pushing is carried out in transformed local orthogonal coordinates where to of these coordinates correspond to poloidal and toroidal directions. The ion motion thus contains all electric and magnetic drifts including the polarization drift. Electrons are also pushed in the transformed local orthogonal coordinates using the drift equations. Algorithm is being incorporated into the Toroidal Particle Code.
Monopole vortices in inhomogeneous plasmas
W. Horton, X. Su, P.J. Morrison
Abstract
Drift wave turbulence in weakly driven or decaying states possesses strong correlations requiring the concept of a weakly correlated vortex gas. Recent progress on the effects of inhomogeneities on the structure, stability and life-time of the vortices is reviewed. In particular, two cases (i) of a finite temperature gradient, and (ii) of a shearing of the magnetic field across the vortex structure are analyzed. A new formulation of drift wave turbulence proposed by Zakharov (1991) in terms of the separation of short and long scales is applied.
Neo-ballooning theory via spontaneous symmetry breaking
Y.Z. Zhang
Abstract
The ballooning symmetry, referred to a translational invariance in an axisymmetric toroidal plasma pinch, is shown to be spontaneously broken for non-ideal systems, i.e., the lowest order mode amplitude varies (exponentially) in radical direction. The ballooning equation has thus to be modified according to the solution of the solvability condition in higher order ballooning theory. Derived in this letter is a new set of equations suitable for non-ideal systems. It may yield significant modifications to plasma stability described by the conventional ballooning theory for such systems.
Preliminary abstracts - US-Japan workshop on nuclear fusion in dense plasmas
T. Tajima, S. Ichimaru
Continuum damping of high-mode-number toroidal Alfven waves
M.N. Rosenbluth, H.L. Berk, D. Lindberg, J.W. Van Dam
Abstract
An asymptotic theory is developed to determine the continuum damping of short-wavelength toroidal Alfvén eigenmodes, which is essential for ascertaining thresholds for alpha-particle-driven instability in ignited tokamaks.
Enhancement of current diffusion in the presence of a kink mode or an Alfven wave
A.D. Beklemishev
Abstract
Many characteristic features of Alfven waves and related instabilities are strongly dependent on the inhomogeneity of the background density and the magnetic field. On the other hand, these waves also have an influence on the inhomogeneity, which is caused by the enhancement of the cross-field transport through wave- distortion of flux surfaces. This problem is addressed here within the framework of the single-fluid reduced MHD model and generalized Lagrangian representation of motion. The new effect of transport enhancement is identified as a consequence of the local squeezing of adjacent flux surfaces, which results in increased radial gradients and cross-field fluxes. This effect is found to be proportional to the second power of the ratio of the magnetic field perturbation to the normal field component. The result is applied to several problems related to m = 1 equilibrium relaxation and Alfven resonance broadening.
Stabilization of the M=1 tearing mode by resonance detuning
A. Thyagaraja, R.D. Hazeltine, A.Y. Aydemir
Abstract
The possibility of stabilizing the m = 1 tearing mode by rapidly oscillating the resonance point rs(t) about its mean position is examined. The calculations are carried out for both externally controlled, coherent oscillations of rs(t), as well as those resulting from turbulent plasma motions. Complete stabilization is possible in the coherent case, while turbulent fluctuations may yield substantial reduction in the growth rate. The technique seems to apply to any linear mode that depends upon local properties around a resonant surface.
Effects of a stratified atmosphere on the production of x-ray and particle energy spectra in solar flames
K. Holcomb, T. Tajima, B. Meerson
Abstract
In a stratified solar corona in which the Alfvén speed increases with height, compressional Alfvén waves can accelerate particles to high energies by the mechanism of phase-locked trapping. We demonstrate by numerical simulations that this mechanism can produce power-law spectra of electron and photon energy, both of which show strong similarities, in quantities such as the exponent indices, to the spectra actually observed in X-ray flares and solar energetic particle (SEP) events. Isotopic effects of 4He and 3He ions are also studied in order to explore a possible mechanism for heavy-ion enrichment of such flares.
From invariance in convective-diffusive systems with applications to impurity transport
S.M. Mahajan, P.M. Valanju, W. Rowan
Abstract
Simple, closed from Green's functions are derived for convective- diffusive systems in slab and cylindrical geometries. In slab geometry, they are Gaussian, while in cylindrical geometry, they are of the form e-ar2I0(βr), where I0 is the modified Bessel function, and α and β depend on time via e-t. Time dependent profiles resulting from various pulsed and oscillatory sources can be obtained using these Green's functions. These solutions can be used to model a variety of time dependent transport experiments such as pulsed impurity transport, oscillating gas puff, perturbative RF heating, and sawtooth heat pulse propagation. A successful comparison of our profiles with pulsed impurity injection experiments on TEXT is presented.
Quantum-mechanics as a generalization of Nambu dynamics to the Weyl-Wigner formalism
I. Bialynicki-Birula, P.J. Morrison
Abstract
It is shown that Nambu dynamics can be generalized to any number of dimensions by replacing the O(3) algebra, a prominent feature of Nambu's formulation, by an arbitrary Lie algebra. For the infinite dimensional algebra of rotations in phase space one obtains quantum mechanics in the Weyl-Wigner representation from the generalized Nambu dynamics. Also, this formulation can be cast into a canonical Hamiltonian form by a natural choice of canonically conjugate variables.
Vlasov-Maxwell system: 2-D Equilibria, reversed field pinches
S.M. Mahajan, W.Q. Li, A. Sen
Abstract
Exact solutions to the Vlasov-Maxwell system in the two dimensional circular cylindrical model are presented. The magnetic surfaces are shifted circles for the m = 1 case, where the shift is determined by a parameter ε; ε = 0 gives concentric circles and represents 1-D solutions. For m greater than or equal to 2 cases, the solutions are singular at the origin and the magnetic surfaces contain islands and separatrices. An improved one dimensional model with currents in both the axial and azimuthal directions is also presented. It is shown that this simple finite pressure model can yield field reversed equilibria in the presence of appropriate boundary constraints.
Nonlinear interactions of tearing modes in the presence of shear flow
X.L. Chen, P.J. Morrison
Abstract
The interaction of two near-marginal tearing modes in the presence of shear flow is studied. To find the time asymptotic states, the resistive magnetohydrodynamic (MHD) equations are reduced to four amplitude equations, using center manifold reduction. These amplitude equations are subject to the constraints due to the symmetries of the physical problem. For the case without flow, the model that is adopted has translation and reflection symmetries. Presence of flow breaks the reflection symmetry, while the translation symmetry is preserved, and hence flow allows the coefficients of the amplitude equations to be complex. Bifurcation analysis is employed to find various possible time asymptotic states. In particular, the oscillating magnetic island states discovered numerically by Persson and Bondeson [Phys. Fluids 29, 2997 (1986)] are discussed. It is found that the flow-introduced parameters (imaginary part of the coefficients) play an important role in driving these oscillating islands.
Asymptotic spectra in two-dimensional drift wave turbulence
Y.Z. Zhang
Abstract
A hybrid field Ψ, is defined as the linear combination of the vorticity and the logarithm of electron density, is a constant of motion along the perturbed orbit in two-dimensional electrostatic turbulences. As a result, the spatial correlation of the fluctuating hybrid field is logarithmically divergent at small distance, which suggests a large wave number asymptote of the correlation proportional to k-2. For the Hasegawa-Mima turbulence it follows that the asymptotic energy spectrum Ek ∼ k-4.
Geometric phase, rotational transforms and adiabatic invariants in toroidal magnetic fields
A. Bhattacharjee, G.M. Schreiber, J.B. Taylor
Abstract
The rotational transform associated with the magnetic surfaces of a toroidal magnetic field with a nonplanar axis is an example of the angle anholonomy which recently has been much discussed in quantum and classical dynamics (the Berry phase and Hannay angle). The same anholonomic angle appears in the phase of a charged particle spiraling around its guiding center in a strong magnetic field. This accounts for a contribution to the longitudinal invariant, associated with the guiding-center motion, which is different for guiding-center orbits that circulate in opposite directions and is absent for orbits that are reflected between mirrors.
Kinetic theory of RF current drive and helicity injection
R. Mett
Abstract
Current drive and helicity injection by plasma waves are examined with the use of kinetic theory. The Vlasov equation yields a general current drive formula which contains resonant and nonresonant contributions. Standard quasilinear current drive is described by the former, while helicity current drive may be contained in the latter. Since direct analytical comparison of the sizes of the two terms is in general difficult, a new approach is taken. Solution of the drift-kinetic equation shows that the standard Landau damping/transit time magnetic pumping quasilinear diffusion coefficient is the only contribution to steady-state current drive to leading order in ε=ρL/Ll, where ρL is the Larmor radius and L is the inhomogeneity scale length. All nonresonant contributions, including the helicity, appear at higher order, after averages are taken over a flux surface, over azimuth, and over time. Consequently, at wave frequencies well below the electron cyclotron frequency, a wave helicity flux perpendicular to the magnetic field does not influence the parallel motion of electrons to leading order and therefore will not drive a significant current. Any current associated with a wave helicity flux is then either ion current or electron current stemming from effects not included in the drift-kinetic treatment, such as cyclotron, collisional, or nonlinear.
Finite orbit energetic particle linear response to toroidal Alfven eigenmodes
H.L. Berk, B. Breizman, H. Ye
Abstract
The linear response of energetic particles to the TAE modes is calculated taking into account their finite orbit excursion from the flux surfaces. The general expression reproduces the previously derived theory for small banana width: when the banana width Δb is much larger than the mode thickness Δm, we obtain a new compact expression for the linear power transfer. When Δm/Δb << 1, the banana orbit effect reduces the power transfer by a factor of Δm/Δb from that predicted by the narrow orbit theory. A comparison is made of the contribution to the TAE growth rate of energetic particles with a slowing-down distribution arising from an isotropic source, and a balance-injected beam source when the source speed is close to the Alfven speed. For the same stored energy density, the contribution from the principal resonances (|ν||| = νA) is substantially enhanced in the beam case compared to the isotropic case, while the contribution at the higher sidebands (|ν||| = νA/(2l - 1) with l ≥ 2) is substantially reduced.
Magnetohydrodynamic studies of ideal and resistive tearing modes with equilibrium shear flow
X.L. Chen
Abstract
Two magnetohydrodynamic (MHD) instabilities are studied. A simple sufficient condition is given for the linear ideal instability of plane parallel equilibria with antisymmetric shear flow and symmetric or antisymmetric magnetic field. Application of this condition demonstrates the destabilizing effect of the magnetic field on shear flow driven Kelvin-Helmholz instabilities. For the resistive tearing instability, the effect of equilibrium shear flow is systematically studied, using the boundary layer approach. Both the constant-psi tearing mode and the nonconstant-psi tearing mode are analyzed in the presence of flow. It is found that the shear flow has a significant influence on both the external ideal region and the internal resistive region. In the external ideal region, the shear flow can dramatically change the value of the matching quantity δ. In the internal resistive region, the tearing mode scalings are sensitive to the flow shear at the magnetic null plane. When the flow shear is larger than the magnetic field shear at the magnetic null plane, both tearing modes are stabilized. Also, the transition to ideal instability was traced. Furthermore, the influence of small viscosity on the constant-ψ tearing mode in the presence of shear flow is considered. It is found that the influence of viscosity depends upon the parameter, v0(O)B0(O), where V0(O) and B0(O) denote the flow shear and magnetic field shear at the magnetic null plane, respectively. Viscosity basically tends to suppress the tearing mode. Finally, the nonlinear interaction of two near-marginal tearing modes in the presence of shear flow is studied. To find the time asymptotic states, the resistive MHD equations are reduced to four amplitude equations, using center manifold reduction. These amplitude equations are subject to the constraint of translational symmetry of the physical problem.
Shock formation in a poloidally rotating tokamak plasma
K.C. Shaing, R.D. Hazeltine, H. Sanuki
Abstract
When the Mach number Mp of the poloidal rotation in a tokamak approaches unity, the poloidal variations of plasma density and potential appear to have the characteristics of a shock whose front lies on a plane (ribbon) of a fixed poloidal angle η0. The shock first appears, when 1−Mp≤(ε)1/2 (ε is the inverse aspect ratio), on the inside of the torus at a shock angle η0≥π if the plasma rotates counterclockwise poloidally. As Mp increases, η0 moves in the direction of the poloidal rotation. At Mp=1, η0=2π. When Mp −1≤(ε)1/2, the shock angle is at η0≤π. The parallel viscosity associated with the shock is collisionality independent, in contrast to the conventional neoclassical viscosity. The viscosity reaches its maximum at Mp=1, which is the barrier that must be overcome to have a poloidal supersonic flow. Strong up–down asymmetric components of poloidal variations of plasma density and potential develop at Mp ≈1. In the edge region, the convective poloidal momentum transport weakens the parallel viscosity and facilitates the L–H transition.
A mesoscopic linear accelerator driven by superintense subpicosecond laser pulses
B. Meerson, T. Tajima
Abstract
We reconsider the idea of a solid-state laser-driven linac in view of the latest development in the creation of super-intense subpicosecond laser pulses and studying their interaction with surfaces. We suggest that by guiding such pulses in a disposable mesoscopic (several hundred nm diameter) hollow metal waveguide it will be possible to avoid deleterious effects of the breakdown and provide very high acceleration gradients.
Power law energy spectrum and orbital stochasticity
K. Mima, W. Horton, T. Tajima, A. Hasegawa
Abstract
The power-law energy distributions are observed in space and fusion plasmas. The power law decay of the two-time velocity correlation function and the corresponding frequency spectrum of the correlation function are shown to be related to the power law distribution of the time interval of acceleration, which produces a power-law energy distribution. In particular, the two time correlation function, the distribution of acceleration duration, namely the distribution of the trapping time of the quasi-trapped orbits in the vicinity of the magnetic null such as the geomagnetic tail configurations are shown to produce a power law energy distribution function. The statistical property is applicable under conditions given here to the energy spectra of cosmic rays, electrons in laser-plasma interaction and the radio-frequency heated confined plasmas.
Excitation on nonlinear wakefield in a plasma for particle acceleration
B. Breizman, D.L. Fisher, P.Z. Chebotaev, T. Tajima
Abstract
Excitation of large amplitude wake fields in a plasma for acceleration of particles is theoretically and computationally considered. The wake electric field can be generated either by short laser pulses or charged particle (electrons) beam pulses. We treat both cases from a unified point of view and compare them. In two (or three) dimensional investigations, the wake causing agencies are treated as rigid, while in the one dimensional cases the feedback of the wake field on the driving pulse is accounted for fully kinetically and relativistically. We elucidate transverse and longitudinal wavebreaking effects, nonlinear wake field effects, pulse shaping, multiple pulses, the coherency length of wake fields and comparison of laser and electron beam pulses.
Ion-temperature-gradient-driven transport in a density modification experiment on the TFTR tokamak
W. Horton, D. Lindberg, J.Y. Kim, J.Q. Dong
Abstract
TFTR profiles from a supershot density-modification experiment are analyzed for their local and ballooning stability to toroidal ηi}-modes in order to understand the initially puzzling results showing no increase in Xi when a pellet is used to produce an abrupt and large increase in the ηi parameter. The local stability analysis assumes that k|| = 1/qR and ignores the effects of shear, but makes no assumption on the magnitude of k||vti/ω. The ballooning stability analysis determines a self-consistent linear spectrum of k||`s including the effect of shear and toroidicity, but it expands in k||vti/ω ≤ 1, which is a marginal assumption for this experiment. Nevertheless, the two approaches agree well and show that the mixing length estimate of the transport rate does not change appreciably during the density-modification and has a value close to or less than the observed Xi, in contrast to most previous theories which predicted Xi`s which were over an order-of-magnitude too large. However, we are still unable to explain the observed increase Xi(r) with minor radius by adding the effects of the finite beta drift - MHD mode coupling, the slab-like mode, or the trapped electron response. The experimental tracking 0.2 < Xe/Xi < 0.7 suggest that both grad Ti and trapped-electron driving mechanisms are operating.
Impurity and neutral effects on the dissipative drift wave in tokamak edge plasmas
Y.Z. Zhang, S.M. Mahajan
Abstract
Possible destabilizing mechanisms for the linear electrostatic dissipative drift waves (in tokamak edge plasmas) are investigated in slab geometry. The effects of processes such as ionization, charge exchange, radiation, and rippling are examined. In particular, the impurity condensation associated with radiation cooling is evaluated appropriately for the drift wave ordering, which is found to be an important driving mechanism in contrast to the results of earlier studies (R.J. Thayer and P. H. Diamond, Phys.Rev.Lett. 65, 2784 (1990)). It is also shown that the role of ionization is quite complicated, and depends strongly on the manner in which the equilibrium is achieved. The linear eigenmode equation is studied both analytically and numerically. For the range of parameters relevant to TEXT tokamak (K.W. Gentle, Nucl.Fusion Technol. 1, 479 (1981)), both the charge exchange and the rippling effect are found to be unimportant for instability.