On the dynamics of the screw pinch

The coupling of the external circuits to the inward motion of a plasma in a screw pinch is considered by solving a set of non-linear partial differential equations for the plasma simultaneously with a set of non-linear ordinary differential equations for the circuits. The motion of the plasma couples the theta -circuit to the z-circuit, so that the simple correspondence between theta -circuit and Bz and between z-circuit and Btheta is lost. As a consequence, the plasma is seen by the circuits as having a self-inductance which is completely different from the self-inductance calculated in the usual way. A resonant condition between the circuit elements is derived which guarantees the generation of approximately constant pitch of the field lines, thereby nicely confirming the model of a constant-pitch force-free field for the outer region of the screw pinch. In the general case, the solution of the full set of non-linear equations is obtained as an expansion in a parameter which measures the deviation from the resonant condition and the amount of shear created.