The Ballooning Mode Equation and s-alpha Space

High-toroidal-number pressure-driven modes reduce to a 1D ODE along the field line; its stability chart in shear-alpha space bounds the breeder's pressure gradient.

Ballooning transformation

At high toroidal mode number n the two-dimensional stability problem separates: the mode localizes on a flux surface and varies rapidly across it but slowly along the field line. The ballooning transformation converts the eigenproblem into a single ordinary differential equation along the extended poloidal angle. This is the workhorse limit for the breeder's edge stability, where negative triangularity does its work.

Ballooning equation (along field-line coordinate theta):

  d/dtheta [ (1 + Lambda^2) dF/dtheta ]
      + alpha [ cos theta + Lambda sin theta ] F  =  0

  Lambda = s*theta - alpha*sin(theta)
  s     = (r/q) dq/dr             (magnetic shear)
  alpha = -(2 mu0 R q^2 / B^2) dp/dr   (normalized pressure gradient)

  Marginal stability -> boundary curve in the (s, alpha) plane

The s-alpha stability chart

Solving the ballooning ODE for marginal stability over a grid of shear s and normalized pressure gradient alpha produces the classic s-alpha diagram: a stable region, a first unstable region, and at high shear a second stable region. Where the breeder's edge sits on this chart determines whether the pressure pedestal is ballooning-limited.

# marginal ballooning boundary by shooting method
def ballooning_growth(s, alpha):
    def rhs(theta, y):
        F, dF = y
        Lam = s*theta - alpha*sin(theta)
        coef = 1 + Lam**2
        d2F = -(alpha*(cos(theta)+Lam*sin(theta))*F
                + dLam_terms(...)*dF) / coef
        return [dF, d2F]
    y = integrate(rhs, span=(-Nturns*pi, Nturns*pi), y0=[1,0])
    return decay_rate(y)     # bounded -> stable ; growing -> unstable

Negative triangularity in this chart

Negative triangularity (delta -0.30) modifies the local magnetic geometry entering the coefficients, moving the edge operating point relative to the ballooning boundary and enabling good edge confinement without a steep, ELM-prone pedestal. The stack encodes the marginal curve as a differentiable constraint so scenario optimization keeps the edge pressure gradient inside the stable region by construction.

• alpha too high at low s: first-region ballooning instability.
• Access to second stability: high shear can re-stabilize steep gradients.
• Negative-delta geometry reshapes access to these regions at the edge.

These boundaries are computed offline in simulation and used as design constraints; they will be re-validated against edge diagnostics once the breeder operates.