On Axial Current Induction and Stability of a Pinch

We present an approach for the induction of the plasma axial current in a pinching device, that consists in the use of apertures in the external tubular conductor that surrounds the plasma itself. These apertures are spaced at regular intervals along the axis of the device and a radial parallel plate waveguide is coupled to each of them. To the end of each waveguide, a time-varying power source is connected and operated synchronously with all the others. We analyze the stability of the plasma motion in this device by using a model, the ideal device, in which the actual electromagnetic field is approximated by that generated by a time-varying azimuthal current distributed on the internal surface of the external conductor and a time-varying axial current flowing on the surface of the plasma. The desired or reference plasma motion inside the ideal device is a sequence of cylindrical equilibrium configurations coaxial with the external conductor that extends indefinitely. We analyze the stability of a reference trajectory, along which the plasma is compressed and expanded under the action of a screw magnetic field, by using a simplified set of linear equations of motion, based on the quasi 1-D flow of a gas, with electromagnetic forces complying with the quasi static approximation. Under the hypothesis that the azimuthal and axial currents are not perturbed, we show that an axial magnetic flux frozen in the plasma and an external conductor are necessary for stability. We also show that an axial magnetic field external to the plasma is not necessary for stability, so that, in a real device, its role could be limited to that of replenishing the embedded one, while the plasma is in contact or nearly in contact with the wall of its container. As opposed to previous theoretical investigations, we do not use the energy principle to prove stability/instability of a steady-state pinch without computing the eigenvalues of the linear equations of motion. On the contrary, for the cylindrical configuration analyzed, after decomposing the plasma motion in its main components, that is, massless, fluid, and string modes (the sausage and kink modes in previous different models), we compute the generalized masses and stiffnesses of the fluid and string modes. In the steady-state case, this allows us to compute their eigenvalues as a function of the geometric and magnetic parameters and then ascertain stability or instability of a specific pinch. By using these expressions, we also formulate necessary and/or sufficient conditions for stability, on the basis of which we are able to show the consistency of our results with the ones obtained in previous investigations. In particular, we derive a sufficient stability condition which is similar to the Kruskal-Shafronov necessary stability condition. We also formulate stability conditions in the time-varying case, which seems to be a novelty, except for what was done by this author in a recently published paper.