High mode number stability of an axisymmetric toroidal plasma

In the investigation of stability of a plasma confined by magnetic fields some of the most important modes of oscillation are those with long wavelength parallel to the magnetic field and short wavelength perpendicular to it. However, these characteristics conflict with the requirement of periodicity in a toroidal magnetic field with shear. This conflict can be resolved by transforming the calculation to one in an infinite domain without periodicity constraints. This transformation is the starting point for a full investigation of the magnetohydrodynamic stability of an axisymmetric plasma at large toroidal wave number n. (Small values of n can be studied by direct numerical computation but this fails when n is large.) For n > 1 there are two distinct length scales in the problem and a systematic approximation is developed around an eikonal representation, formally as an expansion in 1/n. In lowest order the oscillations of each magnetic surface are decoupled and a local eigenvalue is obtained. However, the mode structure is not fully determined in this lowest order. In higher orders a second eigenvalue equation is obtained which completes the determination of the structure of the mode and relates the local eigenvalue of the lower order theory to the true eigenvalue for the problem. This higher order theory shows that unstable modes are localized in the vicinity of the surface with the smallest local eigenvalue, that the true eigenvalue is close to the lowest local eigenvalue and that the most unstable high n modes occur for n-> oo. Hence the local theory, which involves no more than the solution of an ordinary differential equation, is normally adequate for the determination of stability of any axisymmetric plasma to high mode number oscillations.