J. L. Johnson

Resistive instabilities in general toroidal plasma configurations

Previous work by Johnson and Greene on resistive instabilities is extended to finite‐pressure configurations. The Mercier criterion for the stability of the ideal magnetohydrodynamic interchange mode is rederived, the generalization of the earlier stability criterion for the resistive interchange mode is obtained, and a relation between the two is noted. Conditions for tearing mode instability are recovered with the growth rate scaling with the resistivity in a more complicated manner than η3/5. Nyquist techniques are used to show that favorable average curvature can convert the tearing mode into an overstable mode and can often stabilize it.

Geodesic Acoustic Waves in Hydromagnetic Systems

In toroidal systems with geodesic curvature an electrostatic acoustic mode occurs with plasma motion in the magnetic surfaces, perpendicular to the field. In typical stellarators this mode should dominate ordinary sound waves associated with motion along the field.

Determination of Hydromagnetic Equilibria

Techniques for calculating hydromagnetic equilibria in toroidal systems which differ little from a uniform field are developed. The zeroth‐order magnetic surfaces in these systems differ appreciably from concentric circular toroids. Care is taken to match onto reasonable external fields at the plasma boundary. Expressions for various equilibrium properties including the rotational transform, the net current on each surface, and the magnetic lines of force are obtained. As an illustration of the theory it is shown that the application of a particular field perpendicular to the plane of the torus reduces the distortion associated with the introduction of pressure into the system.

Some Stable Hydromagnetic Equilibria

Hydromagnetic equilibria are obtained for a variety of situations which differ little from that of a zero pressure uniform axial magnetic field. Criteria for ascertaining the stability of these equilibria are found by means of an energy principle. In particular, if helically invariant fields are present, stable equilibria with nonzero pressure and net axial current can be found.

Hydromagnetic Instability in a Stellarator

Stability diagrams are calculated for hydromagnetic kink instabilities in a zero‐pressure plasma confined in a stellarator geometry with arbitrary mixtures of l = 2 and l = 3 multipolar helical fields and with various radial distributions of the current parallel to the confining magnetic field. The introduction of small pressure gradients increases the size of the unstable regions. Modes with small azimuthal wavenumber grow more rapidly than those with large m. Experimental data, obtained with the model C Stellarator, show that the plasma is macroscopically stable except for certain intervals of rotational transform. These intervals agree qualitatively with those in which the theory predicts the plasma should be unstable against the kink instability.

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.

Interchange instabilities in ideal hydromagnetic theory

Interchange instabilities are examined using an ideal hydromagnetic model. The nature of the energy sources that can drive the instability is clarified.

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.

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.

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.