z-Pinch Stabilization by a Frozen-In Axial Magnetic Flux

We consider a neutral plasma, within a tubular dielectric container, under the action of a screw pinch whose magnetic field has the axial component completely embedded in the plasma and the azimuthal one completely external to the plasma itself. The axial magnetic field is then generated by currents flowing only in the plasma, instead of flowing, as proposed so far, also in fixed coaxial coils external to the plasma. We theoretically argue that the stabilizing effect of the axial field on the sausage mode is in this way preserved, while, at the same time, the excitation of the kink mode is avoided. Indeed, in this case, the axial current flowing in the plasma, pinching the plasma itself, remains aligned with the axial magnetic field, as both move with the plasma. To provide quantitative indications, we develop a model based on the quasi-1-D flow of a gas. By using this model, we highlight the role played in plasma magnetic confinement by both the axial magnetic flux, ideally frozen into the plasma, and the axial current returning through a coaxial conductor closely fitting the tubular container. With reference to a cylindrical configuration that extends indefinitely, we show that none of the modes of the linearized equations of motion grow exponentially, faster the higher the mode number, provided that the axial magnetic flux is sufficiently strong. We analyze the case that the plasma is steadily kept compressed (steady operating modality), as well as the case that the plasma is alternately confined by the wall of its container or by the magnetic field (cyclic operating modality). In the first case, provided that the axial flux is sufficiently intense, the linearized equations of motion turn out to be a wave equation plus a simple harmonic oscillator. In the second case, the presence of time-varying coefficients prevents such a simple interpretation, and we resort to using the Liouville–Green approximation to show that, over a finite time interval, their solutions behave almost as in the time-invariant case, provided that the length of the container is sufficiently short. We conclude the article by illustrating a way of diffusing and maintaining over time the axial magnetic field in the plasma. We suggest, in particular, illuminating the plasma with an external axial magnetic field while it is in touch with the wall of its container, and kink instability cannot occur, and then quickly compress and expand the plasma itself, before the embedded axial magnetic field decays too much. This operating modality, however, raises the doubt that the time of confinement at high temperatures may be too short according to Lawson’s criterion. We then show that this drawback can be overcome by adopting a controlled expansion of the hot plasma that allows the energy stored in it to be recovered as electromagnetic energy.