The Ballooning-Mode Equation

The high-mode-number pressure-driven instability that localizes on the bad-curvature outboard side.

The Instability

Ballooning modes are pressure-driven ideal-MHD instabilities that balloon outward on the low-field, outboard side of a torus where field-line curvature is unfavorable, meaning the curvature points in the same direction as the pressure gradient. There the pressure gradient does positive work on a displacement, driving growth, while magnetic tension and shear resist. They have high toroidal mode number and short perpendicular wavelength.

The Ballooning Transform

High-n modes are hard to treat because they vary rapidly across flux surfaces. The ballooning transformation maps the problem onto a single field line parametrized by the extended poloidal angle, converting the two-dimensional eigenvalue problem into a one-dimensional ordinary differential equation along the line. The resulting ballooning equation balances field-line bending (stabilizing) against the curvature-pressure drive (destabilizing) as a function of the shear and the normalized pressure gradient alpha.

s-alpha Diagram

Stability is summarized in the s-alpha diagram, with magnetic shear s on one axis and the normalized pressure-gradient parameter alpha on the other. A first stability region exists at low alpha. Remarkably, at high alpha a second stability region reopens, accessible when the Shafranov shift deepens the local magnetic well enough to restabilize the mode, a route advanced high-beta scenarios exploit.

Relevance

Ballooning stability sets an important part of the pressure-gradient limit, particularly in the steep-gradient edge pedestal, where it combines with peeling modes to trigger edge-localized modes. For the Hyperion breeder concept, ballooning analysis across the profile and access to second stability are design-stage considerations for a simulated machine; the negative triangularity of -0.30 alters the curvature and pedestal behavior in this analysis.