Page 1 - Positional stability of field-reversed-configurations in the presence of resistive walls

Modeling feedback control of unstable separatrix location in beam-driven field-reversed configurations
N. Rath,1, a) M. Onofri,1 and D. Barnes1
Tri Alpha Energy, P.O. Box 7010, Rancho Santa Margarita, CA 92688-7010
We present a linear model for rigid displacement of a toroidally symmetric plasma. We find that FRC plasmas are unstable to either radial or axial displacement. When feedback control is feasible, plasma inertia can be ne- glected and the instability instability growth rate is proportional to the wall resisitivity. We benchmark the linear model against non-linear, hybrid, axisymmetric simulations of an axially unstable system to fix the one free pa- rameter of the model. The resulting parameter-free model is again benchmarked in linear and non-linear closed- loop simulations with active feedback control. The feedback algorithm is designed in Simulink and integrated into the non-linear code without replicating the control logic by compiling it into a shared object and making use of the stop/restart functionality of the plasma simulation code. In closed loop simulations, the predictions of the parameter-free linear model agree satisfactory with the non-linear results. Implications for feedback control of unstable FRC location in experiments are discussed.
I. INTRODUCTION
A field-reversed configuration (“FRC”) is an axisymmet- ric toroidal confinement scheme. In contrast to the Toka- mak, there is no central column (so the vacuum chamber can be cylindrical) and almost no toroidal field. The config- uration is called field-reversed, because the total magnetic field on the central machine axis points in the direction op- posite of the vacuum field. This reversal is generated by strong toroidal plasma currents and results in formation of a separatrix that separates a compact toroid with closed field lines from the open-field line scrape-off layer. The main at- tractions of using FRCs for controlled nuclear fusion are the relative engineering simplicity (no center stack, no toroidal field coils, and no limit on divertor plate area) and the ex- tremely high beta (field becomes zero at X-points and O- points). An excellent review of FRCs can be found in Stein- hauer 15 .
FRCs are predicted to be MHD unstable14, and early FRCs never achieved lifetimes of more than a millisecond16 However, recent experiments with beam-driven FRCs4 have achieved lifetimes exceeding 10 ms, i.e. much longer than all the relevant dynamic plasma timescales. The exact stabi- lization mechanisms are not fully understood yet, but large ion gyroradii (in the order of the machine radius) and radial electric potentials are known to play important roles1,8,15. Generally, predicting and modelling the stability of an FRC experiment is challenging, and requires time-dependent, non-linear, 3-D, ion-kinetic codes2.
However, there is one kind of instability that may not be significantly affected by physics beyond basic MHD: the po- sitional instability9. If the FRC can be moved “as a whole” to a position of lower energy, it can be expected to move into this direction regardless of the complexity of its internal dy- namics. In most of the past FRC experiments, the positional instability has been stabilized by a conducting wall which (on the timescale of the experiments) was acting as a per- fect flux conserver. However, with the recent advances in
a) Electronic mail: nrath@trialphanenergy.com
beam-driven FRCs, the positional instability is expected to require feedback stabilization.
This paper describes modeling and simulation tech- niques for the positional instability that are suitable for the development of feedback control systems. The plasma po- sition will be defined as the (r,z) centroid of the separatrix at fixed toroidal angle. We assume that the FRC is in an un- stable, axisymmetric equilibrium with no toroidal magnetic fields. We also assume that in the absence of perturbations there are no wall currents, i.e. the vacuum magnetic field is solely produced by equilibrium coils. This assumption is reasonable because feedback control of the positional insta- bility only becomes necessary when the plasma lifetime ex- ceeds the characteristic eddy current decay time, so at that point any eddy currents that may have been generated by FRC formation will have decayed away.
The paper is structured as follows. We first provide a brief analysis of positional stability for an FRC in insulating, su- perconducting, and resistive chambers (section II). We then proceed with the derivation of a linear model for the in- teresting case of a resistive wall (section III). In section IV, we compare the predictions of the linear model with non- linear simulations computed by Q2D (a hybrid FRC simula- tion code) and use the non-linear results to fix the free pa- rameters of the linear model. After introducing a method for convenient integration of Simulink-designed control al- gorithms into both linear and non-linear closed-loop sim- ulations (section V), we evaluate the suitability of the linear model for feedback control design (section VI D) and dis- cuss the implications for position control experiments (sec- tion VI E). A summary of the paper is given in section VII.
II. A.
POSITIONAL STABILITY Insulating Walls
Generally, MHD stability of a plasma can be determined by analyzing the linearized force operator,
⃗f ⃗ξ(⃗x ) = −∇π − B⃗ × (∇ × Q⃗ ) + ⃗j × Q⃗ (1)