This browser-side companion to the ballooning lecture solves the local ŝ–α equation on an extended field line, d/dθ[(1+Λ²)X′] + α(cosθ + Λ sinθ)X = λ(1+Λ²)X, while tying the local parameters back to a chosen q(r) profile and a reference surface q_ref. The same calculation then reconstructs the sideband packet centered on m₀ ≈ n q_ref.
Different profile families change the local magnetic shear felt at the reference surface.
Sets the core safety factor of the chosen profile family.
Sets the edge value used to build the global q profile.
The local ballooning packet is built around the surface where q(r) = q_ref.
Multiplies the local shear inferred from the q profile, so you can compare profile-driven and hand-adjusted shear.
This is the standard local ballooning drive parameter from the lecture.
Used only to reconstruct the sideband packet around m₀ ≈ n q_ref.
α = 1.20, q_ref = 3, and a moderate positive local shear.
The large panel shows the dominant even field-line envelope X(θ). The smaller panels connect that local mode back to the global q(r) profile and to the neighboring poloidal sidebands that make up a finite-n ballooning packet.
The local magnetic shear is read from the chosen q(r) family at the surface where q = q_ref.
The ballooning transform reconstructs a narrow packet around the centroid m₀ ≈ n q_ref.
Model summary. The app solves the lecture’s local equation with
Λ(θ) = ŝ θ - α sinθ, A(θ) = 1 + Λ², V(θ) = α(cosθ + Λ sinθ), and (A X′)′ + V X = λ A X.