Page 2 - Demo

042504-2 Rath et al.
feasibility of a beam-driven fusion reactor by investigating
the scaling of C-2U results to higher magnetic fields and temperatures. A thin inconel vessel enables ramp-up of the magnetic field during the plasma pulse. The expected plasma lifetimes of up to 30 ms will be an order of magnitude longer than the L/R time of the vessel. The positional instability is thus no longer passively stabilized, and active control is required to prevent the FRC from becoming unstable within a few ms after formation.
This paper describes the modeling and simulation tech- niques for the positional instability that have been developed to guide the design of the feedback control hardware for C- 2W. The objective was to derive the conservative upper bounds on the C-2W hardware requirements that give a high confidence of successful control in experiments. This was achieved by running simulations with a wide-variety of plasma equilibria and potential feedback hardware configura- tions. We describe the physical model and its validation, as well as the most important findings on the design of the hard- ware. The assumed capabilities of the feedback hardware reflect the final choice for the systems that are going to be deployed in the C-2W device.
The hardware that will be available for feedback control in C-2W includes 8 axisymmetric “trim” coils (located out- side of the resistive wall and distributed approximately uni- formly along the machine axis), 16 saddle coils (sitting directly on the resistive wall and arranged in a 4 4 layout), 192 magnetic sensors (distributed on the interior of the resis- tive wall), 20 toroidal flux loops, and 20 bolometers (distrib- uted along the z axis with radial views). In this paper, we focus on the requirements for using the axisymmetric trim coils to control the axial plasma position.
The plasma position will be defined as the (r, z) centroid of the separatrix at a fixed toroidal angle. We assume that the FRC is in an unstable, axisymmetric equilibrium with no toroidal magnetic fields. We also assume that in the absence of perturbations, there are no wall currents.
This paper is structured as follows: We first provide a brief review of positional stability for an FRC (Section II). We then proceed with the derivation of a linear model for the interesting 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 simulation code) and use the non-linear results to fix the free parameters of the linear model. We evaluate the suit- ability of the linear model for feedback control design (Section VI A) and its advantages in evaluating the effects of non-axisymmetric walls (Section VIB) and discuss the implications for experimental position control hardware (Section VI C).
II. POSITIONAL STABILITY
A derivation of the positional stability of an FRC can be found in Rath, Onofri, and Dan.18 In short, the stability of a rigid FRC in the presence of a resistive wall is described by two stability parameters. The first one, Fzc, is defined by the equation
Phys. Plasmas 24, 042504 (2017) 0 Fzc=2 0 01
F~~n 1⁄4@
0 Fzc=2 0 A:~n; (1) 0 0 Fzc

where ~n is a displacement vector and F~ ~n the resulting line- arized force on the plasma. This implies that the plasma is always unstable to either axial or transverse displacement but never to both. Fusion-relevant FRC plasmas generally have Fzc > 0 (i.e., axial instability and radial stability) because of the need for a midplane-peaked vacuum field to maintain sufficient elongation to avoid the tilt instability. (The “zc” subscript indicates that the parameter quantifies the force per unit displacement in the axial direction (“z”) that is exerted by the equilibrium coils (“c”) on the plasma. For the special case discussed here, the force for displace- ments along the other directions can be calculated from Fzc). Fzc can be computed from the plasma current distribution and the vacuum magnetic field.
The second stability parameter is the force due to the eddy currents that would be induced by the displacement if the wall was superconducting rather than resistive. If Fzc > 0, the only relevant component is the axial force, Fzw (with “w” indicating that this parameter is related to the con- ducting wall rather than the equilibrium coils). There are two possible cases:
1. If Fzc þ Fzw > 0, the plasma is Alfvenically unstable, i.e., the growth rate is determined by plasma inertia and the magnitude of Fzc þ Fzw.
2. If Fzc þ Fzw < 0, the plasma is wall-stabilized, i.e., the plasma is unstable but the growth rate is determined by the wall resistivity and the magnitude of Fzc.
In the first case, the instability is generally too strong to make the feedback control feasible. To avoid this situation in experiments, strategic installation of conductors can be used to increase the magnitude of Fzw (which is always negative). This paper focuses on the second case.
III. LINEAR MODEL
In the wall-stabilized situation, the evolution of the plasma displacement is strongly coupled to the electromag- netic properties of the resistive wall. The resulting instability shares several characteristics with the resistive wall mode (RWM) of the Tokamak which has attracted considerable research interest.19 To derive a linear model for the posi- tional stability of an FRC plasma in the presence of resistive walls, we use several theoretical concepts that have been developed for the analysis of RWMs.
A. Plasma reluctance
Plasma evolution that develops on a time scale much slower than the Alfven time of the plasma may be considered as a continuous sequence of MHD equilibria.20 For suffi- ciently short intervals, this sequence can be modeled as a lin- ear perturbation to the initial equilibrium. If the evolution is driven by changes in the external magnetic field (as opposed to, e.g., internal energy dissipation or loss of particles), the perturbed equilibrium at any point in time is fully defined by