3 year ~ $200k/year

(1) to provide direct support to the MRX-CT proposal by providing design and optimization calculations and an assessment of particle orbits,

(2) to advance our computational capability for extended MHD stability analysis of FRC’s by developing both linear and non-linear full-particle-orbit MHD codes for large ion-orbit MHD including the effects of strongly sheared rotation, and

(3) to interpret the results of these calculations in terms of modern relaxation theory.

An important associated theoretical activity, not directly requiring funding from this proposal, will be an attempt to extend the gyro-kinetic simulation model to FRC’s, thus enabling more efficient calculations than that provided by the full-particle-orbit simulation models.

There is a general difficulty in connecting an initial equilibrium to a relaxed state. The choice of initial equilibrium is probably ill informed and leads to exceedingly complicated (ugly) intermediate dynamics.

There is a technical problem in the existence of difficult (i.e. small) length scales. In single fluid MHD the ion self-helicity is not preserved in small scale (k > 1/l_c) for single fluid model. Two-fluid (e.g. hall-MHD) has the usual two-fluid complications where small length scales need to be resolved.

The two-fluid relaxed states have several weaknesses. The Natural two fluid length scale ignores finite orbit effects. In FRCs, the proper scale is the ion gyroradius, which is comparable to l_c for beta ~ 1

First year: Method for finding 2D(r, z) relaxed equilibrium. Probably not a Grad-Shafranov (GS) like solver. Instead, develop an algorithm based on an "artificial" minimization principal and compare with experiment. First year: globally relaxed state with system boundary at separatrix

FRC-spheromak bifurcation, finding intermediate relaxed states (ideal). Find evolution of "ideal" states caused by weak dissipation (visco-resistive). The goal is to explain the nondurability of intermediate states and subsequent bifurcation into either FRC or spheromak (as obsrved on TS-3).

Later, develop a method for finding 2D (r,z) relaxed equilibria by extending relaxed equilibrium algorithm to allow partially relaxed states. For example relaxed core/ unrelaxed edge or incomplete relaxation of flows. Try and develop "free energy" concept for equilibria with incomplete relaxation.

Work to date has been on Ion Ring Configuration at Cornell. Existing code is Hybrid PIC with reduced EM equations, massless electron fluid response, PIC/fluid ions, charge quasi-neutrality, slowing-down collisions, and cylindrical coordinates. They contain low-frequency whistler and Alfven modes. The FLAME code is fully 3D and uses a direct B-formulation. It is object-oriented, parallel, and optimized. The 3D simulations so far have focused on beam injection. A stable 3D configuration has been computed. There will be FLAME upgrades for this proposal. We will add finite electron pressure (1-3 months), plasma-neutral coupling/ionization (6-8 months), background ion-fluid description (?), a model for energetic electrons (?), and ion-ion PIC viscosity (?). The focus will be on magnetic compression (static/liner), transport of rings, FRC stability (tilt and precession), and hybrid FRC formation/stability.

The M3D code is a comprehensive code that can be run with many physics options. It can have single or 2 fluid, gyrokinetic hot particles and/or full kinetic ions and fluid electrons. In the medium term we will stay with fluid electrons and kinetic ions. Need to change geometry and change from gyro-to full kinetic ion.

2D toroidally symmetric MHD simulation of spheromak merging. With co-helicity it forms a spheromak. With counter helicity, it leads to a FRC. Have also done linear and non-linear MHD stability analysis of the 2D equilibrium formed by merging. Future plans are to find a configuration stable to gross and localized MHD modes. We will study the effects of passive stabilizers and flux-core coils

Plans for NOVA calculations of FRC. Numerical equilibrium, plasma flow, finite Larmor radii, full orbit width, wave-particle interaction. We also have the non-perturbative approach of treating particle kinetic effects for sawtooth stabilization and fishbone stabilization.

The plan is to develop a FRC equilibrium code with plasma flow. Then use the non-perturbative approach for FRC stability calculation. We will next include plasma flow effect in the MHD version of NOVA code. We will then include ion diamagnetic drift effect. Next, include FLR effects with both diamagnetic drift and Bessel function modifications. Next include wave-particle resonance.

(Note that there may be a linear code at Cornell by Friedman, et al that can be used for some of these studies)

Will look at single particle orbits. See paper by Wang and Wiley a while back. See Paul Bellan in Phys Rev Lett. See Norman Rostocker in Science magazine.

Simulation of tilting in FRC. Relativistic equations of motion plus full electromagnetic equations. Concentrate on the difference between cycling and non-cycling ions. Needs beta at the separatrix of 0.2 to stabilize internal tilt mode. Beam ions are also stabilizing. Define an effective S characteristic of neutral beam.