Masafumi Kito

Wall stabilization of a sharp boundary toroidal plasma

The stability of a toroidally symmetric sharp boundary plasma is studied in a wall whose cross section is an arbitrary shape. The wall stabilization has an effect on a fat plasma. The kink mode of the plasma has two stable regions in which toroidal fields and poloidal fields are, respectively, dominant. The second harmonic plays an important role for the stability. For a fixed aspect ratio of the plasma, the stable regions increase accordingly as the plasma shifts outward. If circular walls, elliptical walls, and oval‐shaped walls have the same cross‐sectional area, the circular wall is the worst case for the stability.

Magnetohydrodynamic Stability of a Sharp Boundary Toroidal Plasma

In the toroidal coordinates, a stability analysis is presented for a toroidally symmetrical, sharp boundary plasma. The beta ratio β and aspect ratio, R p / r p , are assumed to be arbitrary. In the equilibrium, the maximum poloidal beta ratio β p is evaluated as Max β p =1/2 ( R p / r p +1). For the kink mode, Curves of β vs a safety factor q and curves of β p vs β in the marginal stability are described for the several aspect rations. Both curves show that the stable region for a fat plasma is wider than for a slender one. The curve of β vs q depends on the aspect ratio even at β=0, and the plasma is unstable for all values of q , if a value of β exceeds a critical value.

Critical β of a D‐shaped toroidal plasma with sharp boundary

Under the cap‐cyclide coordinate system, the marginal stability of a sharp boundary toroidal plasma whose cross section is D shaped is examined using the energy principle. The analytical results are valid for arbitrary β, arbitrary aspect ratio, and arbitrary elongation of the plasma. There exists a maximum elongation for a given aspect ratio. The critical beta β c is computed for several aspect ratios and elongations. For given aspect ratio, β c increases as long as the elongation increases.

A Stability Condition of Plasma for Quasi-Interchange Modes in External Gravitational Potential

In the presence of an external gravitational potential, the necessary and sufficient condition for stability of interchanges in a cylindrical plasma with a constant pitch of the magnetic field is obtained. Both the equation of motion and the energy integral are used to derive the stability condition. Two limiting cases for interchanges, k · B →0 and k · B =0, are considered. The stability condition with k · B →0 differs from the condition with k · B =0. For small k · B ≠0 (quasi-interchanges), a dispersion relation is obtained to explain the differences. A calculation of the growth rates shows that the discontinuity results from the existence of two types of the quasi-interchanges, i.e., for k · B =0 one transformed into pure-interchanges and the other into pure-translations. Suydams condition with a centrifugal potential and the constant-pitch field is suitable to the stability condition for quasi-interchanges.

Magnetohydrodynamic Stability of an Axisymmetric Toroidal Sharp-Boundary Plasma with Flattened Cross Section

A stability of an axisymmetric toroidal sharp-boundary plasma whose cross section is horizontally elongated is presented in the flat-ring cyclide coordinate system. The solution of Laplaces equation referred to as the Wangerin function is used and the energy principle is reduced to the quadratic form without using the tokamak ordering. The marginal stability of the plasma is numerically studied. Numerical results of the kink mode show that an increase of the elongation ratio for a fixed aspect ratio leads to decreases of both the maximum plasma beta and poloidal beta, and an increase of the minimum safety factor. It is also shown that the stable region for a plasma with a circular cross section is wider than with a flattened cross section.

MHD Stability Analysis of Axisymmetric Surface Current Model Tokamaks Close to the Spheromak Regime

In the toroidal coordinates, a stability analysis is presented for very low-aspect-ratio tokamaks with circular cross section which is described by a surface current model (SCM) of axisymmetric equilibria. The energy principle determining the stability of plasma is treated without any expansion of aspect ratio. Numerical results show that, owing to the occurrence of the non-axisymmetric ( n =1) unstable modes, there exists no MHD-stable ideal SCM spheromak characterized by zero external toroidal vacuum field. Instead, a stable spheromak-type plasma which comes to the ideal SCM spheromak is provided by the configuration with a very weak external toroidal field. Close to the spheromak regime (1.0<aspect ratio\( \lesssim \)1.1), the minimum safety factor and the critical β-values increase monotonically with aspect ratio decreasing from a large value, and curves of β p versus β in the marginal stability approach to an ideal SCM spheromak line β p =β.

A Criterion for Stability of Surface-Preserving Modes in Axisymmetric Toroidal Plasmas

A stability criterion for surface-preserving modes in closed-line systems in axisymmetric toroidal plasmas is derived from an energy principle. The criterion is compared with a criterion for localized modes in a plasma with closed-line field. It is found that there exists a certain discontinuity between those criteria. The stability criterion for the surface-preserving modes is diagrammed by elliptic quadratic curves for toroidal plasmas with a large aspect ratio.

A Equilibrium Magnetic Field for a Toroidal Plasma

In the toroidal coodinates, a equilibrium analysis is presented for a toroidally symmetrical, sharp boundary plasma. The plasma β and aspect ratio, Rp/rp, are assumed arbitrarily. In the equilibrium, the maximum plasma βp is evaluated as Max βp=1/4 (Rp/rp+rp/Rp) +1/2. When corrents flow on te plasmasurface and external conductors, a equilibrium magnetic iiela and the currents is analitically solved. In the case in which there exist the external conductors infinitely far from the plasma, a stagnation point and separatrix can be found near the plasma surface. Furthermore, the magnetic surfaces similar to the external conductors form can be found outside such the separatrix. The current density flowing on coils placed outside magnetic surfaces is much higer than that on inside them. The stagnation points exist inside and outside the coiles, or on the ones.

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 Suydams 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.

Numerical Solution of Dispersion Equation for a Bi‐Maxwellian Plasma

In the presence of a uniform magnetic field, longitudinal oscillations of a plasma with a bi‐Maxwellian distribution function are studied numerically. A dispersion equation which involves a function Z(ζ) = π−1/2 ∫ −ωωdxexp(−x2)(x − ζ), where ζ is complex, is derived. A root of the equation is obtained by using a formula which combines the asymptotic expansion of Z(ζ) with the power series of Z(ζ). The real part of roots gives a mode of cold plasma oscillations and modes of multiple oscillations of the cyclotron frequency. These modes are coupled and hybrid modes exist. Instabilities arise along long strips which are along modes of multiple oscillations. If an anisotropy of the bi‐Maxwellian distribution is constant, then instabilities are growing while the temperature of the plasma is increasing.