Modeling of Feedback and Rotation Stabilization of the Resistive Wall Mode in Tokamaks

This paper describes the modeling of the feedback control and rotational stabilization of the resistive wall mode (RWM) in tokamaks. A normal mode theory for the feedback stabilization of the RWM has been developed for an ideal plasma with no toroidal rotation. This theory has been numerically implemented for general tokamak geometry and applied to the DIII-D tokamak. It is found that feedback with poloidal field sensors is superior to feedback with radial field sensors. The strength of the RWM that can be stabilized for a series of DIII-D equilibria are quantified. A general formulation is further developed for the feedback stabilization of tokamak with toroidal rotation and plasma dissipation. It has been used to understand the role of the external resonant field in affecting the plasma stability and compared with the resonant field amplification phenomenon observed in DIII-D. The effectiveness of a differentially rotating resistive wall in stabilizing the RWM has also been studied numerically. It is found that the maximum flow speed required is quite modest for a resistive wall with a long resistive wall time constant. It is orders of magnitude smaller than the required speed of plasma rotation. For a noncircular tokamak, a wide range of flow patterns have all found to be effective. The structure of the resistive wall mode predicted from ideal MHD theory has been compared with signals from various diagnostics. Simulation of the stabilization of the RWM in ITER-FEAT has been studied by using the MARS code coupled with the ONETWO transport code. It is also projected that 33 MW of negative neutral beam injection will be able to sustain plasma rotation sufficient to stabilize the RWM without relying on feedback. 1. Normal Mode Approach [1] to Feedback Stabilization A practical tokamak fusion reactor must operate at high beta normal and high current [2]. This requires steady-state operation of the tokamak above the no wall βN limit with sustained stabilization of the resistive wall mode (RWM) [3]. Plasmas in future reactors are expected to rotate with negligible rotation speed. Feedback stabilization of plasmas with no or negligible rotation is therefore of particular interest. In this case, the plasma dynamics is determined by a set of normal modes. The behavior of the feedback can then be prescribed completely from the properties of the normal modes. Central to this approach is the consideration of the quadratic energy functional of the perturbed plasma displacement ξ in the plasma and the perturbed magnetic fields δB in the outside “vacuum” region: δWp + δWv + Dw + δEc = 0 . (1) In Eq. (1), δWp is the perturbed plasma potential energy, δWv the perturbed vacuum energy, Dw the dissipation energy in the resistive wall, and δEc the energy exchange between the feedback coil and the plasma resistive wall system. During the open loop operation δEc = 0, Eq. (1) is self-adjoint and determines a set of normal modes, {ξi, δBi}, with growth rates {γi} and with Dw being the norm. This is an energy principle extended from that of the usual ideal MHD energy principle using Dw as norm replacing the plasma kinetic energy. Only one of these normal modes (the RWM) has been found to be unstable. The rest are stable MODELING OF FEEDBACK AND ROTATIONAL STABILIZATION OF M.S. Chu, et al. THE RESISTIVE WALL MODE IN TOKAMAKS GENERAL ATOMICS REPORT GA–A24132 2 (damped) modes in which the resistive wall provides the dissipation. During the closed loop operation, the feedback currents and δEc are non-zero and the requirement of Eq. (1) then determines that the amplitude of the normal modes {αi} are determined by ∂αi ∂t − γ iαi =Ei Ic . (2) In Eq. (2), Ei c is the excitation matrix which describes the excitation of the eigenmode ξi, δBi by the feedback current Ic. The circuit equations for the currents Ic incorporate the feedback logic ∂Ic ∂t + 1 τc ′ c I ′ c = Gcl Fli αi ( ) . (3) In Eq. (3), τc ′ c is determined by the self and mutual inductances of the coils, Fli is the sensor matrix which measures the magnetic fluxes induced by the eigenmodes in the sensor loop l, and Gcl is the gain matrix. The stability of the feedback is completely described by Eq. (2) and (3) above and the closed loop feedback problem is reduced to a small set of coupled lumped circuit equations. This set of equations is, in general, non-self-adjoint. For feedback with a single array of sensors and a single array of feedback coils the stability may be studied by using the method of Nyquist diagram [4], and for multiple sensor arrays and multiple feedback coils the characteristic equations of Eqs. (2) and (3) have to be solved. This approach has been applied to the DIII-D geometry shown in Fig. 1, with up to three bands (central midplane, upper, and lower ) of external feedback coils. Shown in Fig. 2(a) is the Nyquist diagram for the stability of a set of equilibria with different βN. For this set of equilibria, the plasma (which has not been optimized for its β value) is stable with no external wall at βN = 2.0 and stable with the DIII-D wall infinitely conducting at βN = 2.6. The curves are symmetric with respect to the horizontal axis. Only the upper half of each curve is shown. The Nyquist theorem demands that for plasma stability, the locus of the stability curve has to encircle the point (–1, 0). It is observed that the curves for βN = 2.06 and 2.13 encircle (–1, 0), Resistive Wall Sensor Loop