Page 4 - Positional stability of field-reversed-configurations in the presence of resistive walls

we treat the displacement of the plasma as a linear pertur- bation of an axisymmetric equilibrium, caused by currents in the vessel wall. This description may at first sound back- wards (because it’s the displacement that induces the cur- rents), but is justified by the slow evolution of the instability: since the evolution is slow, plasma inertia can be neglected. Therefore, the plasma must be in approximate force bal- ance and the force from the equilibrium field that drives the instability must be approximately canceled by the restor- ing force from the induced wall currents. Thus the magni- tude of the wall currents uniquely defines the position of the plasma. With this in mind, the plasma reluctance for rigid plasma displacements can be calculated as follows:
For an arbitrary perturbation ⃗Φx to the external field the resulting force on the equilibrium plasma is given by Maxwell’s equations. Since this relation is linear, the force F⃗w (⃗Φx ) may be written as matrix Fw (where the subscript in- dicates that we expect the force to be due to eddy currents in the wall):
F⃗w (⃗Φx ) = Fw .⃗Φx (19)
Fw is a mapping from the infinite vector space of surface functions to the physical, 3-D space of forces on the plasma. For a given rigid displacement ⃗ξ, the resulting force is given by equation (10). On the time scale of wall current evolution, plasma mass can be neglected and the plasma is assumed to be in force balance. This means that the plasma position is determined by the condition that driving and
restoring forces balance each other:
⃗⃗
Fv.ξ+Fw.Φx =0 (20)
This equation can be inverted to give plasma position ⃗ξ as a function of external perturbation:
(Lw+Mwp.R.Mpw)d⃗Iw =−Rw.⃗Iw (29) vwx dt
B.
Wall Coupling
With the plasma response captured in the reluctance ma- trix, the evolution of the coupled plasma-wall system is driven by the evolution of the wall currents. If the variation of the magnetic field within the wall thickness is small, wall currents may be considered surface currents and can be de- scribed by their expansion coefficients ⃗Iw in some set of sur- face functions {g(⃗x)} defined on the wall. The wall currents can be shown to obey a circuit equation6:
d⃗Φw =−Rw.⃗Iw (25) dt
where ⃗Φw are the expansion coefficients of the normal mag- netic field at the wall, and Rw is a matrix of resistances. The changing flux through the wall has components from the wall currents themselves, and components from the mov- ing plasma:
d⃗Φw =Lw.d⃗Iw +Mwp.d⃗Ip (26) dt dt dt
where Lw is the self-inductance of the wall, and Mwp is the mutual inductance between the control surface and the wall. The currents ⃗Ip can be calculated from knowing the plasma reluctance matrix, so that
d⃗Φw =L .d⃗Iw +M .R.d⃗Φp (27) dt wdt wp dt
d⃗Iw =(Lw+Mwp.R.Mpw) dt (28)
we thus obtain a closed system of linear ODEs describing the evolution of the coupled plasma-wall system:
4

⃗ξ=−F−1.F .⃗Φ (21)
If the plasma is displaced by ⃗ξ, the resulting change in the plasma generated magnetic field may be expanded as in equation (17):
(p)1 ⃗⃗ ⃗
C. Numerical Implementation
Equations (29) and (24) form the mathematical model of rigid FRC evolution in the presence of an arbitrary shaped, resistive wall. To solve these equations, we have adapted the VALEN code3, which was developed to simulate the evolu- tion of resistive wall modes in Tokamaks.
VALEN represents the wall as a collection of finite tri- angular and quadrilateral elements with circulating branch currents and varying resistivity. The finite element repre- sentation provides a natural basis for the orthogonal surface functions {gi } and allows calculation of finite dimensional inductance and resistivity matrices Lw and Rw. To deter- mine the plasma reluctance matrix R, VALEN relies on the DCON or IPEC13 codes, which perform a diagonalization of the force operator for a given axisymmetric equilibrium. Ei- ther code produces both the surface functions {fi } and the plasma reluctance matrix for the outermost closed flux sur- face. Using the provided {fi } VALEN then proceeds to cal- culate the mutual inductances Mpw and Mwp. The resulting
fi(⃗x) (ξ·∇)Bp(⃗x) ·dA
=−1 f(⃗x) (F−1.F.⃗Φ)·∇)B⃗(⃗x) ·d⃗A (23)
Ii = μ 0
(22)
μivwxp 0
Comparing the above expression with equation (18), the rows of the reluctance matrix are thus given by
−1 −1 ⃗ ⃗ Ri = μ fi(⃗x) (Fv .Fw)·∇)Bp(⃗x) ·dA
0
(24)
In this expression, inner products are to be taken over the three spatial dimensions. The definition of the reluctance matrix for rigid displacements depends on the equilibrium plasma field B⃗p, the driving force matrix Fv and the restoring force matrix Fw. The driving force matrix in turn depends on equilibrium plasma current ⃗jp, and the restoring force matrix depends on ⃗jp and the wall geometry.