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042504-3 Rath et al.
the perturbation of the external magnetic field at that time.
The relationship between the perturbation of the external 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 B~ x
may be expanded in some set of orthonormal surface basis functions ffið~xÞg,
U
Phys. Plasmas 24, 042504 (2017)
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 ~n, the resulting force is given by Equation (1). The plasma position is determined by force balance
Fv:~nþFw:U~x 1⁄40; (7) ~n 1⁄4 F 1:Fw:U~x: (8)
v
If the plasma is displaced by ~n, the resulting change in the plasma generated magnetic field may be expanded as in Equation (4)
ðxÞ ð
f ð~xÞ B~ ð~xÞ dA~; (2) iix
1⁄4
ðpÞ 1ð h i
~ ~~~
where integration is to be performed over the control surface.
The coefficient vector Ux fully defines the external field inside the control surface.
Furthermore, any magnetic field B~ due to the perturbed p
plasma currents may be described in terms of a current potential j20,21 such that on the control surface
Ii 1⁄4l fið~xÞ n r Bpð~xÞ dA; (9)
of the reluctance matrix are thus given by
1ð 1 ~ ~
Ri 1⁄4 l fið~xÞ Fv :Fw r Bpð~xÞ dA: (11) i0
0ðh i
1
1⁄4 l fið~xÞ Fv :Fw:Ux r Bpð~xÞ dA: (10)
1 ~ ~ ~ Comparing the above expression with Equation (5), the rows
B~ ð~xÞ n^ 1⁄4 l jð~xÞ: p0
(3) The expansion coefficients ~Ip of jð~xÞ in the basis functions
ðpÞ 1ð
I 1⁄4 f ð~xÞ B~ ð~xÞ dA~; (4)
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
0
ff ð~xÞg are
i
lip 0
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 B~ , the driving force matrix F , and the restor- pv
ing force matrix Fw (the subscripts standing for “vacuum” and “wall,” respectively). The driving force matrix in turn depends on the equilibrium plasma current ~jp, and the restor- ing force matrix depends on ~jp and the wall geometry.
B. Wall coupling
With the plasma response captured in the reluctance matrix, 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 as surface currents and can be described by their expansion coefficients ~Iw in some set of surface functions fgð~xÞg defined on the wall. The wall cur- rents can be shown to obey21
from external perturbations U~ x to the response ~Ip20
~Ip 1⁄4R:U~x:
resulting
plasma
(5)
For the analysis of resistive wall modes, the reluctance matrix is calculated by diagonalization of the force operator. To model the positional stability of an FRC, we treat the dis- placement of the plasma as a linear perturbation of an axi- symmetric equilibrium, which is caused by currents in the vessel wall. This description may at first sound backwards (because it is the displacement that induces the currents) 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 balance and the force from the equilibrium field that drives the insta- bility must be approximately canceled by the restoring force from the induced wall currents. Thus, the magnitude of the wall currents uniquely defines the position of the plasma. With this in mind, the plasma reluctance for rigid plasma dis- placements can be calculated as follows:
For an arbitrary perturbation ~Ux to the external field, the
resulting force on the equilibrium plasma is given by
dU~w dt
1⁄4 Rw:~Iw; (12)
Maxwell’s equations. Since this relation is linear, the force
dU~w1⁄4L:d~IwþM :d~Ip; (13) w wp
where ~Uw 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 moving plasma
F~ U~ may be written as matrix F (where the subscript wxw
dt dt dt
indicates that we expect the force to be due to eddy currents in the wall)

F~wU~x1⁄4Fw:U~x: (6)
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