Numerical study of rotational instabilities and beam ion effects in FRC using the HYM code

Numerical study of FRC spin-up and the effects of end-biasing on FRC stability has been performed using a 3D nonlinear hybrid version of the HYM code for TAE FRC experimental parameters. The n$=$1 tilt mode is found to be weakly unstable in S*$=$9 FRC, and it saturates nonlinearly at small amplitude. Simulations including the particle loss and periodic BCs show all low-n modes stable for large resistivity. End-shorting results in faster spin-up and instability of n$=$1 tilt and subsequent growth of the n$=$2 rotational mode. Depending on value of applied end-biasing electric field, the n$=$1 wobble or n$=$2 rotational mode becomes unstable. Hybrid simulations with non-symmetric BCs with/without end-shorting show strongly unstable n$=$1 radial shift (wobble) mode. The effects of energetic beam ions on FRC stability properties have been also investigated numerically using a hybrid version of the HYM code. Stability properties of co-interchange (kink) modes with different toroidal mode numbers n$=$1-4 and the beam driven instabilities have been studied for realistic TAE equilibrium and slowing down distribution for the beam ions.