The Ballooning Equation

The eigenvalue equation for pressure-driven MHD modes that localize on the outboard, bad-curvature side.

Pressure-driven instability

On the outboard side of a torus, the magnetic field curves away from the plasma (bad curvature), so a pressure gradient can drive an interchange-like instability. Ballooning modes have short wavelength across the field and localize where curvature and pressure gradient conspire. The ballooning equation is the eigenvalue problem that determines their stability.

d/dtheta [ (1 + Lambda^2) dX/dtheta ] + alpha (cos theta + ...) X = -gamma^2 (1+Lambda^2) X

This is a one-dimensional equation along the field line (the extended poloidal angle theta), with the pressure-gradient parameter alpha and the magnetic shear s as the controlling parameters. The growth rate gamma^2 comes out as the eigenvalue.

The s-alpha diagram

Plotting stability boundaries in the shear-alpha plane gives the classic s-alpha diagram, with a first stable region at low pressure gradient, an unstable band, and, at high shear, a second stable region reachable by strong shaping. Access to second stability is a design lever for high-beta operation.

How it is solved numerically

• Integrate the ballooning ODE along the field line for each flux surface
• Scan the ballooning phase angle to find the most unstable mode
• Map marginal alpha versus radius to build the stability boundary

Ballooning stability, combined with kink stability, sets the beta limit summarized by the Troyon coefficient.

Shaping and the edge

Strong shaping, negative triangularity, and high shear change the local curvature and can raise the ballooning limit. The Hyperion breeder uses negative triangularity (-0.30), which alters edge stability and is analyzed with ballooning and peeling-ballooning theory to characterize the pressure the plasma can hold.