Page 2 - Positional stability of field-reversed-configurations in the presence of resistive walls

PHYSICS OF PLASMAS 23, 064505 (2016)
Positional stability of field-reversed-configurations in the presence
of resistive walls
N. Rath,a) M. Onofri, and D. C. Barnes
Tri Alpha Energy, P.O. Box 7010, Rancho Santa Margarita, California 92688-7010, USA
(Received 14 March 2016; accepted 19 May 2016; published online 24 June 2016)
We show that in a field-reversed-configuration, the plasma is unstable to either transverse or axial rigid displacement, but never to both. Driving forces are found to be parallel to the direction of displacement with no orthogonal components. Furthermore, we demonstrate that the properties of a resistive wall (geometry and resistivity) in the vicinity of the plasma do not affect whether the plasma is stable or unstable, but in the case of an unstable system determine the instability growth rate. Depending on the properties of the wall, the instability growth is dominated by plasma inertia (and not affected by wall resistivity) or dominated by ohmic dissipation of wall eddy currents (and thus proportional to the wall resistivity). Published by AIP Publishing. [http://dx.doi.org/10.1063/1.4953417]
A field-reversed configuration (“FRC”) is an axisym- metric toroidal confinement scheme.6 In contrast to the Tokamak, there is no central column, almost no toroidal field, and almost no poloidal plasma current. External coils and plasma current create axial magnetic fields in opposite directions.
In this brief communication, we elucidate the stability
of FRCs to rigid displacement of the plasma. In FRCs, rigid
displacements are of particular interest because they are less
affected by the large ion gyradii that are hypothesized to sta-
bilize many non-rigid instabilities.1,3,6 Furthermore, recent
experiments with beam-driven FRCs have achieved plasma
2
Rigid FRC translation and rotation have been looked at before in the context of deriving stability limits for the Solovev equilibrium.4 However, the relation between axial and transverse displacement and the effects of a resistive wall have not yet been documented in the literature.
We use a cylindrical ðr; h; zÞ coordinate system and assume a current distribution of the form ~jpðr;zÞ 1⁄4 jhðr; jzjÞ ^h. The vacuum magnetic field is expressed in terms of the poloidal flux w and toroidal angle h as
B~ ðr; zÞ 1⁄4 rwðr; jzjÞ rh: (1) v
Generally, MHD stability of a plasma can be determined by analyzing the linearized force operator
~f 1⁄2~nð~xÞ 1⁄4 rp B~ ðr Q~Þ þ~j Q~; (2) where p is the perturbed pressure and Q~ is the perturbed
magnetic field. However, if we restrict our attention to rigid displacements of the entire plasma, the resulting force is more easily calculated in the frame where the plasma remains at the same position, but the vacuum magnetic field
a)Electronic mail: nrath@trialphanenergy.com
is displaced. In this case, the only force resulting from the
displacement is the Lorentz force between the plasma current
~j and the change in the vacuum magnetic field B~ produced pv
by the equilibrium coils. For small, rigid/constant displace- ments ~n, we thus have
~fð~nÞ1⁄4 ~j 1⁄2ð~n rÞB~ : (3) pv
The same result can be obtained by evaluating Equation (2) for a rigid ~nð~xÞ 1⁄4 ~n.
The net force on the plasma is
ð
~~ ~~
FðnÞ1⁄4 fðnÞdV: (4)
After integrating over the toroidal angle, and setting ~n 1⁄4 fnx; ny; nzg this becomes
lifetimes exceeding 10 ms, a time-scale on which the vac- uum vessel stops to act as a superconductor. The interaction of the positional instability with a resistive wall has therefore become important.
ð 2 ~~ @w 1@w
Fn x^1⁄4pnx jhðr;zÞ@r2 r@rdrdz; (5) ð @2w1@w
F~~n y^1⁄4pny jhðr;zÞ@r2 r@rdrdz; (6) ð @2w
F~~n z^1⁄42pnz jhðr;zÞ@z2 drdz: (7) In other words, for a displacement in any transverse or axial
direction, the resulting force will be parallel or antiparallel to the displacement, without any orthogonal components.
The stability of a rigid plasma is thus described by two stability parameters which we call Fzc and Fxc (we have introduced the c subscript to indicate that these quantities are computed from the field produced by the equilibrium coils)
With this convention, calculation of the force for a given dis- placement ~n becomes a linear operation, so that Equation (4) can be written as
~~
Fzc 1⁄4z^ Fð^zÞ1⁄4kFðz^Þk (8) ~~
Fxc 1⁄4 x^ Fðx^Þ 1⁄4 kFðx^Þk: (9)
1070-664X/2016/23(6)/064505/3/$30.00 23, 064505-1 Published by AIP Publishing.
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