Ideal Magnetohydrodynamic Model Research Articles

Inductive inhibition of cold-plasma stabilization of curvature-driven modes in finite-length plasmas

A modified quasistatic theory that incorporates inductive effects in earlier electrostatic models connects the conventional electrostatic and magnetohydrodynamic (MHD) pictures of line tying by cold plasma. The modified theory predicts that curvature-driven flute modes in mirror-confined plasmas can be stabilized by moderate concentrations of cold plasma if the beta of the hot, mirror-confined plasma is less than a critical value. The maximum stable beta for an idealized stratified model of a hot-ion plasma, separated from conducting end walls by cold plasma, is given approximately by βcrit≂(4RpRc/LhLc) ×[ω2pe(cold)/ (k2⊥c2+ω2pe(cold))]. For ω2pe(cold) ≪k2⊥ c2, this result is identical with the predictions of electrostatic models. In the opposite limit, the result resembles the predictions of ideal MHD models, namely, that unstable curvature-driven modes can occur even in the presence of dense cold plasmas if the hot-plasma beta exceeds a limiting value estimated to be π2RpRc/L2.

A mechanism for a class of solar coronal disturbances

In this paper, a new ideal magnetohydrodynamic (MHD) model is used to examine the dynamical response of the upper solar atmosphere to injection of cold mass from the photosphere akin to a surge. A significant new physical phenomenon is revealed: the formation of an almost stationary loop prominence in the atmosphere as a consequence of the ejected material. Simultaneously with the formation of this new loop, the simulation exhibits MHD waves that propagate outward (i.e., away from the loop) to excite coronal material. It is conjectured that these waves may trigger a class of coronal disturbances.

Finite β stellarators

A fully three-dimensional computer code based on an ideal magnetohydrodynamic model has been used to find stellarator configurations with finite critical values of the plasma parameter β. It is established that the WISTOR-U torsatron designed at the University of Wisconsin and the Heliotron E experiment constructed at Kyoto University have average β limits near 5 percent.

Stability Study of High-βFlux-Conserving Equilibria

The flux-conserving tokamak model suggested that rapid heating would yield equilibria with high relative energy density ($\ensuremath{\beta}=\frac{2p}{{B}^{2}}$) while nonetheless allowing control over $q$, the so-called safety factor for instability within the ideal magnetohydrodynamic plasma model. In this study, we show that this is adequate to provide stability to $\ensuremath{\beta}$ values of 10%, if there is a superconducting metal shell in the vicinity of the plasma.

High-beta tokamaks surrounded by force-free fields

The sharp-boundary model of ideal magnetohydrodynamics is presented in a form readily applicable to axisymmetric, high-β plasmas of arbitrary cross section. The usual vacuum region outside the plasma is replaced here by a layer of force-free currents, and the influence of force-free currents and cross-sectional shaping on the critical β is studied. Specific results are given for circular, elliptical, and racetrack-like cross sections. A new parameter q* is introduced in place of the Mercier q, and it is shown that q*≈1 is a reasonable estimate of the toroidal current limitation due to external kink modes in high-β plasmas with noncircular cross sections.

Experimental and Numerical Studies of Magnetohydrodynamic Stability Properties of a Rectangular-Cross-Section Finite-βToroidal Plasma

The dependence of the growth rate of the $n=1$ ($\ensuremath{\lambda}=2\ensuremath{\pi}{R}_{0}$) kink instability on plasma $\ensuremath{\beta}$ has been investigated experimentally in a rectangular-cross-section high-toroidal-$\ensuremath{\beta}$ [${\ensuremath{\beta}}_{t} (\mathrm{axis})\ensuremath{\gtrsim}10%$] device and compared with theoretical predictions. The ${\ensuremath{\beta}}_{t}$ for the onset of instability as well as the estimated values of the growth rates observed experimentally are in good agreement with predictions from a numerical ideal magnetohydrodynamic model.

Effect of transverse magnetic field on axial magnetic surfaces of a three-turn stellarator perturbed by an equilibrium plasma

The question of maintaining plasma equilibrium in a three-turn stellarator by means of an external transverse field H⊥ is investigated in an ideal magnetohydrodynamic model as a first approximation with respect to β, where β is the ratio of the gas-kinetic pressure of the plasma to the magnetic pressure. An “island” structure of the paraxial magnetic surfaces is observed under conditions of precise compensation and slight undercompensation of the homogeneous component of the plasma magnetic field by the field H⊥. The optimum conditions are found for controlling the position of the magnetic axis with an external transverse field. Relations are obtained for estimating the maximum equilibrium pressure of the plasma in a three turn stellarator in the presence of an external transverse magnetic field.

Stability of a high-β, l=3 stellarator

The stability of an infinitely long, high β, l=3 stellarator is investiated. The calculation is carried out by using the new scyllac expansion in the sharp boundary ideal magnetohydrodynamic model. It is found that for any given size l=3 field allowed by equilibrium considerations and mode number m, an infinite but discrete set of wavenumbers k exist for which the plasma is unstable to all β; that is, the critical β equals zero. These modes can be described as long wavelength interchanges. Thus, with regard to sharp boundary stability, l=3 is less desirable than l=1 for the basic scyllac magnetic field.

Recent developments in the computational aspects of MHD stability

In recent years variational methods have been applied successfully to the numerical determination of the spectra for simple linearized ideal magnetohydrodynamic models. Currently, several extensive efforts have been undertaken to implement these approaches to determine the normal modes for general axisymmetric toroidal equilibria, especially those applicable to the Tokamak experimental program. This paper reviews the motivation, the difficulties, and the various numerical approaches employed. Several simple illustrative results are given.

Bifurcated stability of a family of stellarator equilibria

The ideal magnetohydrodynamic free boundary model is employed to study the bifurcation of a family of helically symmetric equilibria. The equilibria correspond to an L=2 stellarator. The plasma column has finite length and is surrounded by a vacuum region enclosed by an outer conductor. The stability analysis is based on a determination of the sign of the second variation of the potential energy. The column length is chosen to avoid ’’kink instabilities’’ for some range of the angular wavenumber, m, for which the theory applies. At a critical value of beta, a marginally stable state of principally m=1 long wavelength modes is determined and serves as the starting point of bifurcation. Neighboring bifurcated equilibria are found to be stable when the conducting wall is within 1.15 plasma column radii of the plasma vacuum interface. This position was not sufficient to stabilize the original equilibria.

Motion of a plasma column in a perturbing magnetic field

The motion of a plasma column acted upon by a small external magnetic field is calculated using the sharp-boundary ideal magnetohydrodynamic model. The natural oscillation frequencies of the system are obtained. For the theta pinch, simple approximate formulas useful in considering feedback and dynamic stabilization schemes are derived.

Stability of a finite β, l =2 stellarator

The stability of an infinitely long, high β, l = 2 stellarator is investigated. The calculation is carried out by using the new scyllac expansion in the sharp boundary ideal magnetohydrodynamic model. It is found that for any given size l = 2 field and mode number m, an infinite but discrete set of wavenumbers k exist for which the plasma is unstable to all β; that is, the critical β equals zero. These modes can be described as long wavelength interchanges. Thus, with regard to sharp boundary stability, l = 2 is less desirable than l = 1 for the basic scyllac field, and in addition does not offer any particular advantage over l = 0 as a sideband field required for toroidal equilibrium in the l = 1 configuration. configuration.

Free Boundary Long Helical Wavelength Equilibria

A straight high beta helical equilibrium approximating a theta pinch is perturbed into a large aspect ratio torus by the addition of other helically symmetric fields in such a way that stability of the torus to most significant modes is unaffected. The ideal magnetohydrodynamic free boundary model is employed. The straight helical equilibrium is obtained by an expansion in the two parameters which are small but of the same order of magnitude, the ratio of helical magnetic field to theta pinch field and the product of helical wavenumber times plasma column radius. Neither beta nor the distortion of the plasma column from a true cylinder are taken small and the helical fields are principally L = 1. The equilibrium without net z current has desirable stability properties to m = 1 long wavelength modes, although higher m modes are unstable. The stability conditions are more readily satisfied than earlier ones where it was assumed that the plasma column deformation was small and helical wavenumber times plasma radius was not small.