Resistive Wall Mode Mathematics

A perfectly conducting wall stabilizes the external kink, but a real resistive wall lets it grow slowly; feedback control of that growth is a precise linear problem.

Wall stabilization and its limit

The external kink mode (n = 1) can be stabilized by image currents in a nearby conducting wall - but only if the wall is perfect. A real, finite-conductivity wall lets flux diffuse through on its resistive time, so the mode reappears as a slowly growing resistive wall mode (RWM) with a growth rate set by the wall time, not the fast Alfven time. This slow timescale is exactly what makes feedback control feasible on the breeder.

RWM dispersion (thin-wall model, schematic):

  gamma * tau_wall  =  ( delta-W_infinity ) / ( -delta-W_b )

  delta-W_infinity : energy with wall at infinity (no wall)
  delta-W_b        : energy with ideal wall at wall radius
  tau_wall = mu0 * sigma_wall * d * b   (wall resistive time)

  Stable-with-ideal-wall but unstable-without -> RWM window
  Growth rate ~ 1/tau_wall  (slow, milliseconds)

Feedback as pole placement

Active coils sense the growing radial field and drive currents to cancel it. Modeled as a low-order linear system, the RWM plus feedback is a pole-placement problem: choose controller gains so the closed-loop growth rate becomes negative. Because the open-loop growth is slow, modest gains suffice, but sensor noise and coil dynamics set the achievable margin.

# RWM feedback as a scalar state-space loop
# x_dot = gamma*x + b*u  ; y = x + noise ; u = -K*y
# closed loop: x_dot = (gamma - b*K) x - b*K*noise
def closed_loop_rate(gamma, b, K):
    return gamma - b*K          # want < 0 for stability
# margin shrinks as measurement noise or actuator lag grow

Where it fits the stack

The RWM controller is a fast inner loop; the AI supervises it rather than replacing it. The twin estimates delta-W margins and the mode amplitude from magnetics, and the anomaly ensemble flags a growing n = 1 signature as a disruption precursor. If the estimated growth outpaces the controller's authority, L3 requests a scenario change before the protection failsafe would trigger.

• Sensing: saddle loops and Mirnov arrays resolve the n = 1 radial field.
• Actuation: dedicated in-vessel or ex-vessel control coils.
• AI role: margin estimation, precursor detection, envelope enforcement.

The RWM model is a design-and-simulation construct; wall time and mode couplings will be measured and the model updated after first plasma.