Axially symmetrical unsteady motion of plasma in a non uniform magnetic channel

A successive approximation method is proposed. The method is small β expansion. The plasma is assumed to be an infinitely conductive ideal gas. The hydromagnetic equations are rewritten in a time-dependent curvilinear coordinate system, the one family & coordinate lanes of which attaches the field lines frozen with the plasma. The resultant equations are separable into two systems of equations. One of the systems is essentially representing a one dimensional problem along each magnetic line of force. This problem is solvable in itself by numerical calculations when the time variation of the field is already known. The other system of equations describes that the Lorentz' force j×B must balance the other forces perpendicular to the lines of force; pressure gradient, inertia, centrifugal force and Coriolis force. These circumstances enable us to do iteration when a zeroth order magnetic field, for example, vacuum field, is known and β is sufficiently small compared to unity. The proposed method should be carefully examined in relation to the singularity appearing in the case of small Alfven number, however, that is now under way. (author)