This browser-side companion now compares five outer-region tearing models. The tokamak-like branch uses the smooth current profile Jz(r) = Ĵ (1 - r²/a²)ν. One RFP-like branch uses a Robinson-style force-free equilibrium with a constant-λ core and an edge roll-off to zero at the wall. A second tokamak-like branch uses the lecture’s top-hat current profile, with a uniform current core and a current-free annulus where the rational surface sits analytically. A second RFP-like branch uses the phenomenological alpha-profile λ(r)=λ0[1-(r/a)α], which is a simple beyond-Taylor-state model. A third RFP-like branch uses the Sprott-Shen modified polynomial function model as a non-Taylor-state profile with finite-pressure structure. In every case the explorer reconstructs the equilibrium, solves the outer tearing equation on both sides of the resonant surface, and reports the logarithmic-derivative jump Δ′.
Switches between two tokamak-like branches and three RFP-like current-profile models.
Sets the long axial pitch scale through k = -n / R0.
Physical size in meters. This also sets the conducting-wall radius in the lecture model.
Treated as constant, exactly as in the large-aspect-ratio smooth-profile extension of the lecture.
Together with q(a), this fixes the smooth current peaking exponent.
Sets the total current through the cylindrical edge relation used in the equilibrium lecture.
The explorer evaluates the outer tearing equation for a single selected helical branch.
Modes are valid only when a resonant surface exists inside the plasma, that is, when q(rs) = m/n.
Applies the lecture’s outer vacuum matching at the plasma boundary. The no-wall limit gives ψ′(a)=-(m/a)ψ(a).
q(0) ≈ 1, q(a) = 3, and the 2/1 branch selected.
The large panel shows the left and right outer solutions, each normalized to unity as they approach the resonant layer. The small panels show the underlying safety-factor and current profiles, so the resonance can be read directly from the equilibrium rather than treated as an abstract input.
This thinner panel shows the normalized radial slope a ψ′(r) on each side of the rational surface. The left and right endpoints need not meet, and that mismatch is the visual signature behind Δ′.
The resonant condition q(rs) = m/n is read directly from the smooth profile implied by the current model.
The tearing drive depends on the current gradient and on the helical bending term F(r), which changes sign at the resonance.
Model summary. The explorer integrates the cylindrical outer equation
ψ'' + (1/r)ψ' - m²ψ/r² - μ0 m J'z(r)ψ/[r F(r)] = 0, with F(r) = m Bθ(r)/r - n Bz/R0, and then evaluates Δ′ = (ψ'/ψ)+ - (ψ'/ψ)- just outside the resistive layer.