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

064505-2
Rath, Onofri, and Barnes
Phys. Plasmas 23, 064505 (2016) Whether the plasma position is stable or unstable is then
determined by the relative magnitudes of the driving force from the equilibrium magnetic field and the restoring force from wall currents. If and only if the sum of driving and restoring force is positive when projected along the displace- ment, the plasma is unstable.
In the presence of a superconducting wall, the plasma can be stable to both axial and transverse displacements even if the wall is axisymmetric. This is because the magnetic field generated by the wall in response to transverse plasma displacements breaks the axisymmetry of the system, and thus Equation (11) is no longer valid.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 stabilize a plasma to rigid displacements.
The presence of a resistive wall in the vicinity of the plasma does not affect whether the plasma is stable or unsta- ble. This is because 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 cur- rents will dissipate some energy by ohmic heating and thereby prevent the plasma from ever returning to the (maxi- mum energy) equilibrium position.
While plasma stability (in the sense of unstable or sta- ble) does not differ between insulating and resistive walls, the growth rates of any instabilities are fundamentally differ- ent. There are three cases to consider for each potential dis- placement axis:
(1) If the driving force is negative when projected along a displacement, the plasma is stable to that displacement (no matter the wall configuration).
(2) If the sum of projected driving force and restoring force is positive for some displacement, the plasma is Alfvenically unstable, i.e., the growth rate is determined by plasma inertia and the magnitude of Fzc þ Fzw. This is because there is a net-force on the plasma and acceler- ation is only limited by plasma inertia.
(3) If the projected driving force is positive, but the pro- jected net force negative, the plasma is “wall-stabilized.” This means that for a superconducting wall the plasma is stable, but for a resistive wall the plasma is unstable.
As shown earlier, the first case is never realized for all potential displacements at the same time. The second case results in fast loss of confinement because the instability is generally too fast for active control, so the third case is most important in practice. In this case, the growth time of the instability is proportional to the resistivity of the wall and the magnitude of the driving force. Plasma inertia becomes negligible. The proportionality can be shown rigorously by constructing a linearized model that includes the electromag- netic interaction between plasma and wall,5 but also follows from considering the limiting cases of very large resistivity (for which the wall can be treated as insulating, so the growth-rate is Alfvenic) and very small resistivity (for which the wall can be treated as superconducting, so the growth rate has to go to zero). With decreasing growth rate, the iner- tial term in the plasma momentum equation becomes smaller. The driving force, however, is determined only by the plasma position which implies that driving and restoring
0 Fxc 0 0 1 ~~@A~~
FðnÞ1⁄4 0 Fxc 0 n F n: (10) 0 0 Fzc
If Fzc > 0, the plasma is unstable to axial displacements.
If Fxc > 0, the plasma is unstable to transverse displacements.
From r B~ 1⁄4 0, we find that in an axisymmetric v
system
r@2w @wþr@2w1⁄40; (11) @z2 @r @r2
which, when plugged into Equation (5), gives
Fzc 1⁄4 2Fxc: (12)
Thus, the plasma is always unstable to either axial or trans- verse displacements, but never to both.
The above analysis has assumed that there are no conduc- tors in the vicinity of the plasma. In the presence of a conduct- ing wall, movement of the plasma will excite wall currents that keep the flux through the wall constant. The magnetic field gen- erated by these currents exerts a restoring force on the plasma.
For the simple example of an axial displacement inside an axisymmetric, thin wall (in the sense that the variation of the magnetic field within the wall thickness is negligible) the restoring force can be calculated analytically. The toroidal current line density IwðzÞ in a wall of radius rwðzÞ that is induced if the plasma moves along the axis is given by
IwðzÞ 1⁄4
1 Lðrw ðzÞÞ
ð @jðr0;z0Þ
plasma
(13) h M r0;rwðzÞ;z z0 dV0;
@z0
where M is the mutual inductance between two current rings with radius rw and r0 and separated by an axial distance of z z0, and L(r) is the self-inductance of a current loop with radius r. The unit of Iw is current per wall length per dis- placement distance.
The axial force on the plasma resulting from the wall currents is
ðð
Fzw 1⁄4 dV0jh plasma
Bwðr0; z0; zÞ dz;
(14)
where Bwðr0; z0; zÞ is the radial magnetic field at ðr0; z0Þ pro- duced by a current Iw at ðrwðz0Þ; z0Þ.
The calculations for more complicated, three-dimensional walls and displacements can be performed in similar fashion:
(1) Calculate ð~n rÞ~jp to determine the change in plasma current from a displacement ~n.
(2) Calculate the flux through the wall from the magnetic field generated by the perturbed plasma current, ð~n rÞ~j p .
(3) Calculate the wall currents that conserve the flux, cancel- ling the contribution from the displaced plasma.
(4) Calculate the magnetic field excited by the flux- conserving currents in the wall.
(5) Calculate the resulting Lorentz force on the plasma.
wall
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