On the Problem of Dynamic Stabilization

The basic equations used in this work are the ideal magnetohydrodynamic equations. A general theory is developed for investigating the stability of a plasma with an oscillatory motion. The oscillatory motion is assumed to be forced on the system and controlled from outside. It takes place around some equilibrium of the system. This equilibrium may be 'weakly' unstable or only marginally stable. 'Weakly' unstable in this context means that the characteristic frequency of any instability present in the static system must be much less than the frequency of the forced oscillations. The aim of the investigation is to determine whether the dynamic effect of such oscillations can improve the stability of the system. First a vector equation is derived for the Lagrangian displacement vector of an arbitrary fluid element. This equation determines the stability of a hydromagnetic fluid in an arbitrary motion. Then the assumption of a small-amplitude periodic motion about some equilibrium is made. To deal with the explicit time dependence of the problem the multiple-time-scale method is used. An expansion is made in terms of a small parameter which is associated with the applied frequency and the amplitude of the oscillations. The first relevant equations obtained result in stability conditions related to the energy principle for the static case. The results are not restricted to any particular geometry. The special case of a plasma separated from the vacuum region by a discontinuity surface is worked out. Up to now the general theory has been applied to axially symmetric plasma columns in the limit β = 1 (that is, no internal magnetic field). The oscillatory motion is produced and determined by the external magnetic field and the system is found to be stable for sufficiently high applied frequency. The possibility of resonances is briefly discussed. (author)