Control of ideal and resistive magnetohydrodynamic modes in reversed field pinches with a resistive wall

Abstract

Numerical studies of magnetohydrodynamic (MHD) instabilities with feedback control in reversed field pinches (RFPs) are presented. Specifically, investigations are performed of the stability of m=1 modes in RFPs with control based on sensing the normal and tangential magnetic fields at the resistive wall and applying two-parameter feedback proportional to these fields. The control scheme is based on that of [J. M. Finn, Phys. Plasmas 13, 082504 (2006)], which is here modified to use a more realistic plasma model. The plasma model now uses full resistive MHD rather than reduced MHD, and it uses three realistic classes of equilibrium parallel current density profiles appropriate to RFPs. Results with these modifications are in qualitative agreement with [J. M. Finn, Phys. Plasmas 13, 082504 (2006)]: the feedback can stabilize tearing modes (with resistive or ideal-wall) and resistive wall ideal modes. The limit for stabilization is again found to be near the threshold for ideal modes with an ideal-wall. In addition to confirming these predictions, the nature of the instabilities limiting the range of feedback stabilization near the ideal-wall ideal-plasma threshold are studied, and the effects of viscosity, resistive wall time, and plasma resistivity are reported.

Control of linear modes in cylindrical resistive magnetohydrodynamics with a resistive wall, plasma rotation, and complex gain

“…These four values of k 0 are, respectively, the current limits for resistive plasma, resistive wall; resistive plasma, ideal wall; ideal plasma, resistive wall; and ideal plasma, ideal wall. The alternate possible order k rp,rw < k ip,rw < k rp,iw < k ip,iw , i.e., interchange of the two middle values, is in principle possible, but the inner inequality k rp,iw < k ip,rw was observed to hold 39,40 for all RFP equilibria investigated. The other inequalities must always hold.…”

“…15, which used a very simple qualitative model based on reduced resistive MHD 41 for the plasma dynamics. More recent investigations in full visco-resistive MHD in a cylindrical model for RFPs 39,40 have shown this ability to stabilize close to the ideal plasma-ideal wall limit, depending on the plasma viscosity and resistivity. In Ref.…”

“…This is because rotation of the plasma relative to the wall suppresses the resistive wall mode by preventing the flux from diffusing through the wall. For b 0 > b rp,iw , rotation relative to the wall is found to be destabilizing: in this regime, the resistive plasma mode is unstable even with an ideal wall, and for the feedback to succeed, the flux needs to diffuse through the wall in order to form a virtual wall 4 inside the actual wall, 15,39,40 i.e., a radius r with r < r w where the perturbed flux is zero. The stabilizing effect of rotation or G i for b 0 < b rp,iw and the destabilizing effect for b 0 > b rp,iw is similar to the dependence on the wall time observed in Ref.…”

“…1,2,5, 7,8,[11][12][13]17,18,20,23,26,28 Studies have also been performed for reversed field pinches (RFPs). 4, 39,40 Earlier studies in tokamak geometry 3, 6,9,10,14,30,38 investigated sensing either the radial or the poloidal component of the magnetic field, concluding that it is better to sense the poloidal component, and that the latter measurement is of more use if it is inside the wall. 14,38 Results in Refs.…”

“…40, the work in Ref. 39 was extended to a model measuring the radial component and two tangential components of the magnetic field, again in RFP geometry. This work included the presence of two walls, the (inner) vacuum vessel, and a better conducting external copper shell, with sensors between the two walls, as suggested by the configuration of the RFXmod facility.…”