R. L. Dewar

Ballooning mode spectrum in general toroidal systems

A WKB formalism for constructing normal modes of short‐wavelength ideal hydromagnetic, pressure‐driven instabilities (ballooning modes) in general toroidal magnetic containment devices with sheared magnetic fields is developed. No incompressibility approximation is made. A dispersion relation is obtained from the eigenvalues of a fourth‐order system of ordinary differential equations to be solved by integrating along a line of force. Higher‐order calculations are performed to find the amplitude equation and the phase change at a caustic. These conform to typical WKB results. In axisymmetric systems, the ray equations are integrable, and semiclassical quantization leads to a growth rate spectrum consisting of an infinity of discrete eigenvalues, bounded above by an accumulation point. However, each eigenvalue is infinitely degenerate. In the nonaxisymmetric case, the rays are unbounded in a four‐dimensional phase space, and semiclassical quantization breaks down, leading to broadening of the discrete eigenvalues and the accumulation point of the axisymmetric unstable spectrum into continuum bands. Analysis of a model problem indicates that the broadening of the discrete eigenvalues is numerically very small, the dominant effect being broadening of the accumulation point.

Interaction between Hydromagnetic Waves and a Time‐Dependent, Inhomogeneous Medium

Consider a system made up of a hydromagnetic wave and the slowly varying background fluid in which it propagates. It is shown that both the effect of the background on the wave and that of the wave on the background may be derived from Hamiltons principle using the averaged hydromagnetic Lagrangian density. The waves propagate adiabatically, conserving the wave action, and act on the background via a wave pressure term. Total momentum, angular momentum, and energy are conserved. When many waves are superimposed, as in weak turbulence, the wave kinetic equation replaces the adiabatic conservation equation. The accuracy of the averaging approximation is examined, and it is shown that it may be extended to all orders in the inhomogeneity. Also, Eulerian and Lagrangian averaging are discussed.

Ideal MHD stability calculations in axisymmetric toroidal coordinate systems

Abstract A scalar form of the ideal MHD energy principle is shown to provide a more accurate and efficient numerical method for determining the stability of an axisymmetric toroidal equilibrium than the usual vector form. Additional improvement is obtained by employing a class of straight magnetic field line flux coordinates which allow for an optimal choice of the poloidal angle in the minor cross section of the torus. The usefulness of these techniques is illustrated by a study (using a new code, PEST 2) of the convergence properties of the finite element Galerkin representation in tokamak and spheromak geometries, and by the accurate determination of critical β values for ballooning modes.

FREQUENCY SHIFT DUE TO TRAPPED PARTICLES.

The time asymptotic distribution functions corresponding to adiabatic and sudden excitation of an electrostatic wave are calculated. These distributions are compared and used to calculate the nonlinear response of the plasma, and Poissons equation is used to find a nonlinear dispersion relation.

Energy principle with global invariants

A variational principle is proposed for constructing equilibria with minimum energy in a toroidal plasma. The total energy is minimized subject to global invariants which act as constraints during relaxation of the plasma. These global integrals of motion are preserved exactly for all ideal motions and approximately for a wide class of resistive motions. We assume, specifically, that relaxation of the plasma is dominated by a tearing mode of single helicity. Equilibria with realistic current density and pressure profiles may be constructed in this theory, which is also used here to study current penetration in tokamaks. The second variation of the free energy functional is computed. It is shown that if the second variation of any equilibrium constructed in this theory is positive, the equilibrium satisfies the necessary and sufficient conditions for ideal stability.

Theory and simulation of rotational shear stabilization of turbulence

Numerical simulations of ion temperature gradient (ITG) mode transport with gyrofluid flux tube codes first lead to the rule that the turbulence is quenched when the critical E×B rotational shear rate γE−crit exceeds the maximum of ballooning mode growth rates γ0 without E×B shear [Waltz, Kerbel, and Milovich, Phys. Plasmas 1, 2229 (1994)]. The present work revisits the flux tube simulations reformulated in terms of Floquet ballooning modes which convect in the ballooning mode angle. This new formulation avoids linearly unstable “box modes” from discretizing in the ballooning angle and illustrates the true nonlinear nature of the stabilization in toroidal geometry. The linear eigenmodes can be linearly stable at small E×B shear rates, yet Floquet mode convective amplification allows turbulence to persist unless the critical shear rate is exceeded. The flux tube simulations and the γE−crit≈γ0 quench rule are valid only at vanishing relative gyroradius. Modifications and limits of validity on the quench rul...

Long‐wavelength kink instabilities in low‐pressure, uniform axial current, cylindrical plasmas with elliptic cross sections

The magnetohydrodynamic stability of a straight plasma column with elliptic cross section, carrying a uniform axial current, is investigated by extremizing the Lagrangian of the system using a natural coordinate system based on the magnetic field lines. Stability criteria are derived and growth rates are obtained analytically for systems with a uniform mass density inside the plasma. It is shown that the coupling between kink modes and Alfven waves produced by noncircularity is a destabilizing effect. A technique for solving the problem numerically is also discussed and used to demonstrate the effect of a spatially varying plasma density on the growth rate.

Coupled tearing modes in plasmas with differential rotation

The global asymptotic matching equations for multiple coupled resistive modes of arbitrary parity in a cylindrical plasma are derived. Three different variational principles are given for the outer region matching data, while the inner region analysis features a careful treatment of the symmetry‐breaking effect of a gradient in the equilibrium current for a zero‐β slab model. It is concluded that the usual constant‐ψ result remains valid and constrains the matrix matching formalism. The dispersion relation is compared with initial value calculations of a double tearing mode when there are small relative rotation velocities between the rational surfaces. In treating differential rotation within the asymptotic matching formalism, flow is ignored in the outer region and is assumed to affect the inner response solely through a Doppler shift. It is shown that the relative rotation can have a strong stabilizing effect by making all but one rational surface effectively ideal.

A Lagrangian theory for nonlinear wave packets in a collisionless plasma

A Lagrangian, for the slowly varying complex amplitude of an almost monochromatic electrostatic plasma wave in an unmagnetized plasma, is derived. The method is a variant of the averaged Lagrangian technique of Whitham, adapted for use in plasma physics by employing Lows Lagrangian, together with certain elimination procedures to simplify the Lagrangian to usable form. The expansion in powers of the wave amplitude is carried out to quartic terms, and non-adiabatic terms are also retained. Variations with respect to the amplitude lead to a nonlinear Schrodinger equation for the amplitude, while variations of the particle trajectories lead to a modified Vlasov equation, which includes the nonlinear reaction of the wave on the average trajectories. It is assumed that there are no resonant particles. These equations are coupled through the nonlinear frequency shift. It is found that the particle aspect of the plasma has a profound effect on the stability of the system, due to resonance of the wave envelope with particles moving at the group velocity. This effect, nonlinear Landau damping, is always destabilizing, leading to growth of modulations. In terms of the nonlinear Schrodinger equation, the effect changes the equation from a Hartree-Fock equation with a delta-function interaction to one with a non-momentum-conserving, non-local interaction. In the limit of fairly small wavelength, of the modulations, it is shown that the growth rate approaches that expected from plasma kinetic theory.

Oscillation center quasilinear theory

A new formulation of the quasilinear theory of weakly turbulent plasmas is presented, which explicitly separates resonant and nonresonant wave‐particle interactions from the outset. This is achieved by making a canonical transformation to “oscillation center variables” before attempting to solve the Vlasov equation. A systematic method of constructing the generating function to any order in the wave amplitude is presented, based on a variant of Hamilton‐Jacobi perturbation theory. Momentum and energy split naturally into a wave and a particle component. The results are generalized to apply to weakly inhomogeneous plasmas, and verified by demonstrating momentum and energy conservation.