Control of magnetohydrodynamic modes in reversed field pinches with normal and tangential magnetic field sensing and two resistive walls

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Numerical results are presented on control of magnetohydrodynamic (MHD) modes in reversed field pinches (RFPs) for a geometry with two resistive walls. We use measurements of the normal component of the magnetic field and introduce the use of both tangential components. In Richardson et al (2010 Phys. Plasmas 17 112511), RFP control studies were performed sensing the radial (normal) component of the magnetic field and a single tangential component just inside the wall, showing that it is possible to stabilize the MHD modes in an RFP for current up to the ideal plasma–ideal wall limit in that configuration. Here, we extend our modeling by including two resistive walls, in a configuration relevant to experiments such as RFX-mod, and measuring all three magnetic field components, i.e. including a second tangential component, as an exploratory effort. We present our study incrementally, starting with a single resistive wall, and conclude that with the first tangential sensor located inside the wall, the plasma can be stabilized up to the ideal plasma–ideal wall limit, as in Richardson et al. With the first tangential sensor outside the wall, stabilization is possible only up to the ideal wall–resistive plasma (tearing) limit. We then show that for experimentally relevant parameters the thin-wall approximation is indeed valid for the MHD modes of interest but invalid for the high-frequency magnetosonic mode (Richardson et al) driven by the (first) tangential component feedback. In fact, when a thick-wall formulation with realistic parameters is considered, the high-frequency magnetosonic mode is found to be destabilized only for a very high gain parameter, and we conclude that this mode can be ignored for an experimentally relevant analysis. Consequently, the plasma can be stabilized in a much larger region of feedback gain parameter space than found in Richardson et al. In the presence of two walls, with the first tangential component measured just outside the inner wall and with RFX-mod relevant time constants, we show that feedback control can stabilize the plasma at currents much larger than the ideal wall–resistive plasma limit. The current limit is still less, however, than the ideal plasma–ideal wall limit. Use of the second tangential component appears in all cases to lead to significantly different but not necessarily improved feedback stabilization. These results may lead to better understanding and improved stability properties in current-day RFP experiments through robust access to quasi-single-helicity states.