Neighboring equilibria and integrity of elongated FRC configurations

In the TAE C-2W experiment, confinement and stability of Field

Reversed Configurations (FRC) 

reliy on the presence of energetic ions from neutral beam injection

and their relatively large Larmor radius. The radial and axial sizes

of FRC are constrained by various practical limits, along with a few

stability requirements.

Here, the integrity  of an elongated FRC configuration against splitting

to internal FRC-like structures is studied computationally using 

perturbed Grad-Shafranov equation for isotropic equilibrium 

$$

\Delta^*\bar{\Psi}=-rP(\bar{\Psi})

,\quad

\bar{\Psi}=\bar{\Psi}_0+\psi

,\quad

\Delta^*\psi+r\frac{dP}{d\bar{\Psi}}\psi

=

0

,$$

where $P$ is the derivative of plasma pressure $d\mu_0p/d\bar{\Psi}$.

The existence of solutions for $\psi$ indicates the possibility of

perturbations of magnetic configuration, including the change of

topology. Ssuming the Soloviev  FRC model, it is shown that an FRC

with the elongation of 4 has 4 instabilities 

along the axis of symmetry and 3 other, similar to tearing modes,

symmetric to the middle plane. If observed experimentally these might

require multi-point feedback stabilization. 

At the same time, the presence of a Scrape off Layer (SoL) around FRC

provides significant stabilization. Thus, for the reference case

$P(\bar{\Psi})=$const, considered so far, and the pressure drop inside

the SoL

equal to pressure 

drop inside FRC all axisymmetric instabilities are eliminated up to at

least the elongations of 5. At present, a new numerical code for

perturbed equilibria with arbitrary function $P(\bar{\Psi})$ is under

development.