This browser-side companion to the TAE lecture shows exactly what the continuum-gap figures are plotting. The dashed curves are the uncoupled cylindrical continua Ωm = |n - m/q| vA/vA0 and Ωm+1 = |n - (m+1)/q| vA/vA0. The solid curves are the local toroidal eigenvalues Ω± obtained by diagonalizing the same two-harmonic continuum matrix that appears in the lecture. This is not yet the discrete TAE eigenmode; it is the local avoided crossing and the resulting gap.
The lecture’s basic example uses q(0)=1.
For q(r)=1+3r²/a², the edge value is q(a)=4.
The paired branch is always m+1, as in the simplest TAE coupling.
Toroidal axisymmetry keeps n fixed while toroidicity couples neighboring m.
This sets the local two-harmonic coupling strength and therefore the gap width.
Uses the lecture’s illustrative profile ρ(r)=ρ0[1-cρ(r/a)²], so vA/vA0 = [1-cρ(r/a)²]-1/2.
q(r)=1+3r²/a² with the (m,m+1,n)=(2,3,1) crossing near q=2.5.
The large panel overlays the dashed cylindrical continua with the solid toroidally split branches. The two smaller panels show the underlying q(r) geometry and the local avoided crossing in a Heidbrink-style frequency view. This is the local continuum picture on each surface, before the later radial eigenmode problem is imposed.
The rational surfaces q = m/n and q = (m+1)/n bracket the TAE crossing surface q = (2m+1)/(2n).
The upper-style view isolates the two counterpropagating cylindrical branches and shows the avoided crossing once toroidicity is retained.
The explorer uses the two-harmonic continuum matrix
(Ω²-Δm²) ξm + ε ΔmΔm+1 ξm+1 = 0,
with the partner equation for m+1, where
Δm = n - m/q(r) and Ω = ωR0/vA0.
The solid branches are the positive-frequency eigenvalues of that local matrix.
At the crossing surface Δm = -Δm+1, the cylindrical crossing is replaced
by an avoided crossing, and the gap edges are split by toroidicity. A true TAE would still need
the later radial localization problem to turn this local gap into a global mode.