Vertical Stability Mathematics

Elongated and negative-triangularity plasmas are vertically unstable; the n=0 axisymmetric mode grows on the wall time and demands continuous feedback control.

Why elongation destabilizes

Shaping the plasma vertically (elongation kappa > 1) improves performance but makes the plasma column vertically unstable: a small vertical displacement grows because the external field gradient pushes it further. This is the axisymmetric n = 0 instability. The breeder's shaped, negative-triangularity cross-section must be actively balanced on the vertical axis at all times.

Vertical instability (rigid-displacement model):

  m * d2Z/dt2  =  F_dest * Z  +  F_coil(I_c)

  F_dest = I_p * dB_R_ext/dZ    (destabilizing field-index force)
  Field decay index n_idx = -(R/B_Z) dB_Z/dR ;  n_idx < 0 required

  Passive: wall eddy currents slow growth to tau_wall scale
  Growth rate gamma_z ~ sqrt(F_dest/m)  without wall

Passive plus active stabilization

The conducting vessel slows the growth to the resistive-wall timescale (milliseconds), inside which active control coils can act. The control problem is to hold Z near zero using vertical-field coils, given noisy position estimates. It is a fast, low-order loop that the shape controller and MPC must not fight.

# vertical position controller (2-state model)
# state x = [Z, Zdot]; input u = control-coil current
A = [[0, 1],[gamma_z**2, -damp]]     # unstable: gamma_z^2 > 0
B = [[0],[b_coil]]
K = lqr(A, B, Q, R)                  # stabilizing gain
# closed-loop eigenvalues moved into left half plane

Coupling with shape control

Vertical position and plasma shape share the same PF-coil set, so the vertical loop and the negative-triangularity shape loop are coupled multi-input multi-output control problems. The stack co-designs them: the fast vertical loop runs at high bandwidth, while MPC handles the slower shape targets, with an explicit allocation so coil authority is not double-booked. Negative-triangularity equilibria can sit closer to the vertical limit, tightening this coupling.

• Estimate Z and Zdot from magnetics faster than gamma_z.
• Reserve control-coil headroom for the vertical loop under all shapes.
• Report vertical stability margin as an envelope constraint to MPC.

Loss of vertical control is a primary disruption pathway, so the vertical margin is a first-class monitored quantity, and its degradation is a precursor the anomaly ensemble is trained to catch early.