Suydam Criterion Research Articles

Analytic Models of High-ß Reversed Field Pinches

A class of simple analytic models for equilibria neighbouring Taylor's force-free states has been examined. It is shown that some members of this class are high-P equilibria which satisfy Suydam's criterion and Robinson's condition for magnetohydrodynamic stability.

A Criterion for Stability of High-m Quasikink Modes and Its Growth Rates in Plasma in an External Gravitational Field

In the presence of an external gravitational force, high-m quasikink modes of plasma are investigated. The modes become unstable for the mode number m below a certain critical value. A stability criterion is obtained, which is more stringent than Suydam's criterion. A dispersion relation for the modes quadratic in ω 2 is also derived. For a constant-pitch field, the growth rates of the modes are calculated and their contour map is presented. It is shown that the stability of the modes does not depend on β=2 P / B 2 but B θ 2 / B 2 and the maximum growth rates exist only for k · B >0.

Effect of finite Larmor radius on the interchange instability in a cylindrical plasma

The eigenvalue equation for the interchange instability is derived from guiding-centre equations including finite-Larmor-radius corrections. Analytic solutions are found which are valid near the k⃗⋅B⃗ = 0 surface and away from this surface. Matching these two solutions over their common range of validity determines the eigenvalue and hence the mode frequency. The growth rate and radial profile of the mode may readily be evaluated explicitly. The analysis is applicable to the high-β pinch, since cylindrical effects dominate over toroidal. For typical parameters, FLR stabilization is found to increase the marginally stable pressure gradient by a factor two over that given by the Suydam criterion.

Spectra and growth rates of a generalized screw pinch

The theory of Weitzner for the bumpy θ pinch is extended to the case of a generalized screw pinch in which the equilibrium magnetic field is B = (εBr, εBθ, Bz). ε represents the slow periodic z variation of the axisymmetric equilibrium and the stability relative to their r variation. Perturbation expansions in ε and in the bumpiness of the flux surfaces are performed on the linearized equations of ideal magnetohydrodynamics. From the theory of Grad there should be four stable continua, two of which are found for modes with ω2 = O (ε2), while for the transverse modes a generalized Suydam criterion for local instability is derived. The (global) growth rates of the most unstable transverse modes are computed numerically for various Bθ and the results are compared to those of the bumpy θ pinch (where Bθ ≡ 0)) as well as to the ordinary screw pinch (where Br ≡ 0).

New approach to magnetohydrodynamic stability: I. A practical stability concept

An equilibrium is called σ stable if growth faster than exp (σt) does not occur. On the basis of this definition a modified energy principle is obtained by means of which the stability of plasma confinement systems can be tested for times of thermonuclear interest, instead of the infinitely long times which are pertinent to the usual stability analysis. The theory is applied to the diffuse linear pinch, a theorem for σ stability is derived, and the connection with the normal-mode analysis is shown to be given with the Sturmian property, which holds for the unstable side of the spectrum, whereas the stable side consists of Sturmian and anti-Sturmian point spectra separated by continuous spectra. Growth rates and eigen-functions of Suydam modes are numerically calculated, and it is shown that violation of Suydam's criterion in the high and intermediate shear case leads to nonlocalized rapidly growing m = 1 modes. Consequently, this criterion obtains new relevance in the σ-stability analysis for this regime.

Generalization of Suydam's criterion

For low-shear systems Suydam's criterion is generalized to include stability against localized quasikink instabilities. In the obtained criterion, containing pressure gradient, shear, and a term responsible for the quasikink, the shear enters in a completely different way than in an earlier derived criterion by Shafranov. The reason for the discrepancy is pointed out. The present criterion is more restrictive than Suydam's criterion if m is below a certain critical value depending on the shear. The special case of a force-free field having a pitch extremum is separately treated. Several limitations of the use of local criteria are indicated.

Flute instability of a current-carrying curved plasma column

Suydam's criterion for an arbitrarily curved plasma column is generalized. Mercier's well-known results concerning the local stability of a plasma in the vicinity of the magnetic axis are a special case arising from this criterion. In a curved plasma column, in addition to the stabilizing effect of shear, the stabilizing role of the “magnetic well” assumes importance. The magnetic well is produced by a displacement of the magnetic axis into the region where the toroidal magnetic field is weaker. The contribution to the magnetic well made by that part of the displacement which is associated with ballooning of the plasma column as a whole exactly cancels the ballooning mode. The part of the displacement due to electrodynamic stretching of the current column creates a “vacuum” magnetic well that is independent of plasma pressure. In stability criterion, given in this paper the stabilizing role of the vacuum magnetic well manifests itself in the coefficient F, which is dependent both on the mean curvature of the plasma column and on its harmonics. The condition for stabilization by the magnetic well, F > 1, imposes a limitation on the “stability safety factor” q = 2πr Bζ (r)/LBθ. The author considers regions where there are variations in the values q ensuring stability for systems of the racetrack and spatial figure-eight types.

A necessary condition for hydromagnetic stability of plasma with axial symmetry

In this work a necessary condition is derived for the stability of a plasma with axial symmetry. This condition corresponds to displacements which are localized in the neighborhood of those surfaces of constant pressure on which the lines of force are closed. This criterion reduces to Suydam's criterion in the case of cylindrical symmetry.