R.C. Grimm

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.

Numerical determination of axisymmetric toroidal magnetohydrodynamic equilibria

Numerical schemes for the determination of stationary axisymmetric toroidal equilibria appropriate for modeling real experimental devices are given. Iterative schemes are used to solve the elliptic nonlinear partial differential equation for the poloidal flux function psi. The principal emphasis is on solving the free boundary (plasma-vacuum interface) equilibrium problem where external current-carrying toroidal coils support the plasma column, but fixed boundary (e.g., conducting shell) cases are also included. The toroidal current distribution is given by specifying the pressure and either the poloidal current or the safety factor profiles as functions of psi. Examples of the application of the codes to tokamak design at PPPL are given.

Dependence of ideal-MHD kink and ballooning modes on plasma shape and profiles in tokamaks

Extensive numerical studies of ideal-MHD instabilities have been carried out to gain insight into the parametric dependence of critical βs in tokamaks. The large number of interrelated equilibrium quantities involved in establishing a critical β has demanded a careful, systematic survey in order to isolate this dependence. The results of this survey establish the scaling with geometrical quantities including aspect ratio, elongation, and triangularity in the parameter regimes appropriate to both current and reactor-sized plasmas. A moderate dependence on the pressure profile and a strong variation with the current profile is found. The principal result is that, for aspect ratio R/a ≈ 3, critical βs are of the order of 2% for circular cross-sections and 5% for plasmas with elongation K ≈2; somewhat higher values could be achieved with more optimal shaping. Finally, sequences of equilibria have been analysed to compare critical β as a function of toroidal mode number n. It is concluded that the infinite-n analytic ballooning theory provides a sufficient condition for ideal-MHD internal-mode stability. Low-n free-boundary modes appear to set a lower limit.

Study of the MHD spectrum of an elliptic plasma column

The MHD spectrum associated with small perturbations about an analytic equilibrium is determined for a configuration with magnetic flux surfaces which are nested similar elliptic cylinders generated by a uniform axial current. Since the system is periodic, it models the essential features of a toroid. Both analytic and numerical techniques are used to investigate many properties of the modes, including the continuous shear Alfven and slow acoustic spectra as well as the discrete modes associated with fast magnetosonic waves, kinks, and instabilities where the interchange criterion is violated. In particular, the relationship between discrete and continuum modes is described both qualitatively and quantitatively, and it is shown how the unstable interchange modes emerge from the shear-Alfven branch of the spectrum.

Comparative Numerical Studies of Ideal Magnetohydrodynamic Instabilities

Stability properties associated with a specific analytic equilibrium have been calculated to compare the accuracy of three large computational programs that have been developed at Garching, Princeton, and Lausanne. All three use a Galerkin formulation of the variational principle for determining spectra. Good agreement is found, verifying the efficacy of all three codes.

Tilting and shifting modes in a spheromak

In the absence of a conducting wall, typical spheromak plasmas are unstable to tilting and/or shifting modes. The effects of the cross-sectional shape, aspect ratio, and the location of a conducting wall on the stability of these modes are investigated. A circular cross-section (b/a ~ 1) configuration with a flux hole δ = 0.5 (aspect ratio R/a = 2) will will be stabilized by an ellipsoidal wall of mean separation of 1.3 minor radii from the plasma.

MHD stability properties of bean-shaped tokamaks

A study of the MHD stability properties of bean-shaped tokamak plasmas is presented. For ballooning modes, while increased indentation gives larger stable-beta configurations, the existence and accessibility of the second stable region are sensitive to the pressure and safety factor profiles. The second stable region appears at lower beta values for large aspect ratio and moderately high q-values. Finite-Larmor-radius kinetic effects can significantly improve the stability properties. For low-q ( 1) operation, long-wavelength (n ~ 2, 3) internal-pressure-driven modes occur at modest βp values and accessibility to higher beta operation is unlikely. Indentation modifies the nature of the usually vertical axisymmetric instability, but the mode can be passively stabilized by placing highly conducting plates near the tips of the plasma bean. At constant q, indentation has a stabilizing effect on tearing modes.

The conductivity of a toroidal plasma

The effect of particle trapping on the conductivity of a toroidal plasma is investigated, taking account of like-particle collisions. The customary large-aspect-ratio approximation is not invoked, and results applicable to all aspect ratios are obtained. These agree with the previously known asymptotic results.

Anisotropic pressure tokamak equilibrium and stability considerations

Investigation of the effect of pressure anisotropy on tokamak equilibrium and stability is made with a magnetohydrodynamic model. Realistic perpendicular and parallel pressure distributions, P⊥(ψ,B) and P∥(ψ,B), are obtained by solving a one‐dimensional Fokker–Planck equation for neutral beam injection to find a distribution function f (E,v∥/v) at the position of minimum field on each magnetic surface and then using invariance of the magnetic moment to determine its value at each point on the surface. The shift of the surfaces of constant perpendicular and parallel pressure from the flux surfaces depends strongly on the angle of injection. This shift explains the observed increase or decrease in the stability conditions. Estimates of the stabilizing effect of hot trapped ions indicate that a large fraction must be nonresonant and thus decoupled from the bad curvature before it becomes important.

Dynamical grid method for time dependent simulations of axisymmetric instabilities in tokamaks

A natural nonorthogonal time-dependent coordinate transformation based on the magnetic field lines is utilized for the numerical integration of the two-dimensional axisymmetric time-dependent ideal MHD equations in tokamak geometry. The finite-difference grid is treated as a dynamical variable, and its equations of motion are integrated simultaneously with those for the fluid and magnetic field. The method is applicable to tokamak systems of arbitrary pressure and cross section. It is particularly useful for the nearly incompressible ideal MHD modes which are of interest in tokamak stability studies.