Vertical Stability of Elongated Plasmas
The physics behind why shaped plasmas are vertically unstable and how conducting structures and feedback set the stability limit.
The instability drive
To elongate a plasma, the external field must have a decay index that makes the vertical restoring force negative: displace the plasma up, and the field configuration pulls it further up. The same field curvature that gives elongation gives vertical instability. The stronger the elongation, the stronger the drive.
Growth rate and the wall
Without conductors the mode grows on the Alfven timescale - microseconds, effectively uncontrollable. Nearby conducting walls carry induced eddy currents that oppose the motion, slowing growth to the wall's resistive L/R time, typically milliseconds. The instability becomes a resistive-timescale mode that feedback can catch.
The stability margin
A common figure of merit is the stability margin m_s, related to the ratio of stabilizing to destabilizing field indices. m_s must be positive with margin for the plasma to be stabilizable by a realistic active system. Designers trade elongation against m_s: more elongation improves performance but shrinks the margin and stresses the vertical loop.
What sets the practical limit
- Passive conductor placement and time constant
- Fast-coil geometry and available voltage
- Power-supply bandwidth and current limits
- Control-loop delay and sensor noise
- Required robustness to disturbances like edge-localized modes
Spherical tokamak context
Spherical tokamaks like the Kronos breeder Hyperion naturally run at high elongation, which sharpens the vertical-stability problem and makes fast vertical control central to the scenario. In design and simulation this constrains the achievable shape; it is a modeling constraint, not a measured hardware result.