Negative-Triangularity Shaping Mathematics

The breeder runs at negative triangularity delta -0.30; parameterizing and controlling that inverted-D shape is a specific geometric problem layered on Grad-Shafranov.

Parameterizing the boundary

The plasma boundary is described by a small set of shape parameters that the shape controller and twin share. Triangularity delta measures how the plasma cross-section deviates from an ellipse toward a D (positive) or an inverted D (negative). The breeder is canonically delta -0.30: the crescent points inward, a choice tied to edge stability and heat-flux handling.

Miller-style boundary parameterization:

  R(theta) = R0 + a * cos(theta + arcsin(delta) * sin(theta))
  Z(theta) = kappa * a * sin(theta)

  R0     : major radius        a      : minor radius
  kappa  : elongation          delta  : triangularity

  Breeder canonical shape: delta = -0.30  (negative)
  positive delta -> D pointing out ; negative -> D pointing in

Why negative delta changes the physics

Negative triangularity shifts the magnetic geometry so that the plasma edge accesses improved confinement without the strong edge pressure pedestal that drives edge-localized modes (ELMs). For the breeder this is a route to an ELM-free-like edge, reducing pulsed heat loads on the divertor. The trade is that negative-delta equilibria are harder to hold and interact differently with vertical stability, which the control math must respect.

# geometric moments the shape controller regulates
def shape_descriptors(R, Z):
    R0 = 0.5*(R.max()+R.min()); a = 0.5*(R.max()-R.min())
    kappa = 0.5*(Z.max()-Z.min())/a
    R_upper = R[argmax(Z)]; R_lower = R[argmin(Z)]
    delta = ((R0 - R_upper) + (R0 - R_lower)) / (2*a)
    return R0, a, kappa, delta   # target delta approx -0.30

Coupling to control

The shape descriptors R0, a, kappa, delta are the controlled outputs; PF-coil currents are the inputs. The free-boundary GS solve provides the sensitivity matrix (Jacobian) from coil currents to descriptors, which MPC uses to hold delta near -0.30 against disturbances. Because negative-triangularity equilibria can be closer to vertical-instability limits, the shape and vertical-stability loops are co-designed rather than treated independently.

This shape target is a design-and-simulation choice validated in equilibrium and stability codes; it is not yet a measured operating point. The same parameterization will initialize reconstruction once FOAK hardware exists ~2030.