Page 3 - Positional stability of field-reversed-configurations in the presence of resistive walls

The stability of the plasma position is then determined by the relative magnitudes of the driving force from the equi- librium magnetic field (Fzc for axial displacements) and the restoring force from wall currents (Fzw for axial displace- ments). If Fzc + Fzw < 0, the plasma is axially unstable.
In the presence of a superconducting wall, the plasma can be stable to both axial and radial displacements even if the wall is axisymmetric. This is because the magnetic field generated by the wall in response to radial plasma dis- placements breaks the axisymmetry of the system, and thus equation (11) is no longer valid. With careful placement of a superconducting wall, it is thus possible to completely sta- bilize a plasma to rigid displacements.
(RWM) of the Tokamak which has attracted considerable re- search interest7. To derive a linear model for the positional stability of an FRC plasma in the presence of resistive walls, we use several theoretical concepts that have been devel- oped for the analysis of RWMs.
C.
Resistive Walls
Plasma evolution that developes on a time scale much slower than the Alfven time of the plasma may be consid- ered as a continous sequence of MHD equilibria5. For suf- ficiently short intervals, this sequence can be modeled as a linear perturbation to the initial equilibrium. If the evolu- tion is driven by changes in the external magnetic field (as opposed to e.g. internal energy dissipation or loss of parti- cles), the perturbed equilibrium at any point in time is fully defined by the perturbation of the external magnetic field at that time. The relation between the perturbation of the ex- ternal magnetic field and the resulting plasma response is described by the plasma reluctance matrix R. The plasma reluctance matrix is defined by means of a control surface that separates the plasma from the sources of the external field.
On the control surface, any external magnetic field B⃗x may be expanded in some set of orthonormal surface basis functions{fi(⃗x)},
(x) ⃗ ⃗
Φi = fi (⃗x)Bx(⃗x)·dA (15)
where integration is to be performed over the control sur- face. The coefficient vector ⃗Φx fully defines the external field inside the control surface.
Furthermore, any magnetic field B⃗p due to the perturbed plasma currents may be described in terms of a current po- tential κ5,6 such that on the control surface
B⃗p(⃗x)·nˆ =μ0κ(⃗x) (16)
(κ(⃗x) is called a current potential because a surface cur- rent ∇κ × nˆ flowing on the control surface would give rise to the same normal field B⃗p that is produced by the plasma). The expansion coefficients ⃗Ip of κ(⃗x) in the basis functions {fi(⃗x)}are
(p) 1 ⃗ ⃗
Ii = fi (⃗x)Bp(⃗x)·dA (17)
μ0
and they fully define the magnetic field outside the control surface due to the plasma currents
The plasma reluctance matrix is defined as the mapping from external perturbations ⃗Φx to the resulting plasma re- sponse ⃗Ip5:
⃗Ip =R.⃗Φx (18)
For the analysis of resistive wall modes, the reluctance matrix is calculated by diagonalization of the force opera- tor (equation 1). To model positional stability of an FRC,
The stability of a plasma inside a resistive wall is the same as with insulating walls. This is easy to see: on one hand the wall currents will eventually decay and thus cannot stabilize the plasma in an off-equilibrium position permanently, but on the other hand the wall currents will dissipate some en- ergy by ohmic heating and thereby prevent the plasma from ever returning to the (maximum energy) equilibrium posi- tion.
While plasma stability does not differ between insulating and resistive walls, the growth rates of any instabilities are fundamentally different. There are three cases to consider:
1. If Fzc < 0 the plasma is stable (no matter the wall con- figuration).
2. If Fzc + Fzw > 0 the plasma is Alfvenically unstable, i.e. the growth rate is determined by plasma inertia and the magnitude of Fzc + Fzw.
3. If Fzc + Fzw < 0 the plasma is wall-stabilized, i.e. for a superconducting wall the plasma is stable, but for a resistive wall the plasma is unstable and (as will be shown) the growth rate is determined by the wall re- sistivity and the magnitude of Fzc.
In this paper, we are concerned with the last case. The first case doesn’t need feedback control, and in the second case the instability is generally too violent to make feed- back control feasible. Fusion-relevant FRC plasmas gener- ally have Fzc > 0 because of the need for a midplane-peaked vacuum field to maintain sufficient elongation to avoid the tilt instability. Conductors are then placed in such a way to ensure that Fzc + Fzw < 0.
A.
Plasma Reluctance
3
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 instabil- ity shares several characteristics with the resistive wall mode