Lecture 28 Toroidal Alfvén Eigenmodes

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Lecture 28
Toroidal Alfvén Eigenmodes

Overview

Toroidal Alfvén eigenmodes are a classic tokamak problem because they show, in unusually compact form, how geometry changes the spectrum of ideal MHD. In a cylinder the shear-Alfvén branch is a continuum of local field-line resonances. In a torus, neighboring poloidal harmonics couple, the continuum crossing becomes an avoided crossing, and a spectral gap appears. A discrete global mode can then live in that gap. That mathematical fact matters because such modes are easily driven by energetic particles and can redistribute or expel them.

Historical Perspective

The modern TAE story sits on top of several older layers. Alfvén’s original wave is already present in the local ideal-MHD system, but in a torus one must also confront the fact that each magnetic surface has its own parallel wave number and therefore its own local shear-Alfvén frequency. Newcomb’s one-dimensional variational formulation gave the clean language for radial singularities and continua Newcomb (1960). Kieras and Tataronis then gave an especially transparent formulation of the shear-Alfvén continuous spectrum in axisymmetric toroidal equilibria Kieras and Tataronis (1982). The large-aspect-ratio analyses of Cheng, Chen, and Chance made the key observation that the first toroidal correction couples neighboring poloidal harmonics and therefore opens a gap in the Alfvén continuum Cheng et al. (1985); Cheng and Chance (1987). Fu and Van Dam then pointed out the consequence that mattered for fusion: alpha particles and other energetic ions can resonate with a discrete gap mode and drive it unstable Fu and Van Dam (1989). The TFTR observations of beam-driven TAEs turned that point from an elegant theoretical possibility into an experimental reality Wong et al. (1991). The DIII-D beam-driven studies and the early global-mode comparison between theory and experiment made the same lesson concrete in a different experimental setting Heidbrink et al. (1991); Turnbull et al. (1993). Since then the field has broadened into a full spectroscopy of Alfvénic fluctuations—TAEs, EAEs, NAEs, RSAEs, BAEs—with continuum damping, kinetic corrections, and energetic-particle transport all becoming central parts of the subject Berk et al. (1991); Mett and Mahajan (1992); Fasoli et al. (2007); Lauber (2013); Chen and Zonca (2016); Heidbrink (2008); Heidbrink and White (2020).

This lecture is deliberately tied to the Rosenbluth–Bussac lecture. We use the same large-aspect-ratio notation, the same field-line mismatch

\Delta _m(r)\equiv n-\frac {m}{q(r)}, \tag{28.1}

and the same Newcomb radial displacement $\xi _m(r)$. The aim is to show how the static Newcomb functional becomes a wave functional, how the Alfvén continuum appears as a singular point of the radial equation, and how the first toroidal harmonic opens the TAE gap.

28.1 Physical picture

Start in a cylinder. For a Fourier harmonic

\xi _r(r,\theta ,\varphi ,t)=\xi _m(r)e^{i(m\theta -n\varphi )-i\omega t}, \tag{28.2}

the large-aspect-ratio parallel wave number is

k_{\parallel m}(r) = \frac {1}{R_0}\left (n-\frac {m}{q(r)}\right ) = \frac {\Delta _m(r)}{R_0}. \tag{28.3}

If $q$ varies with radius, then $k_{\parallel m}$ varies from one magnetic surface to the next. The local shear-Alfvén frequency also varies with radius,

\omega _{A,m}(r) = \left |k_{\parallel m}(r)\right |v_A(r) = \left |\Delta _m(r)\right |\frac {v_A(r)}{R_0}, \qquad v_A(r)=\frac {B_0}{\sqrt {\mu _0\rho (r)}}. \tag{28.4}

The cylinder therefore does not give one global shear-Alfvén frequency. It gives a family of local field-line resonances: the shear-Alfvén continuum.

Now add toroidicity. In a large-aspect-ratio tokamak the major radius varies around the poloidal angle,

R(r,\theta )=R_0+r\cos \theta , \qquad \epsilon (r)\equiv \frac {r}{R_0}\ll 1. \tag{28.5}

Because of the magnetic field variation within a flux surface, the $\cos \theta $ couples neighboring poloidal harmonics $m\leftrightarrow m\pm 1$, while preserving the same toroidal mode number $n$. Where two neighboring cylindrical continuum branches would have crossed, toroidicity converts the exact crossing into an avoided crossing. The interval between the split branches is the toroidicity-induced Alfvén gap.

Why that matters. A discrete mode can localize radially in that gap and thereby avoid the strongest ideal continuum resonances over the region where the mode lives. It is not undamped, but the leading continuum singularity has been pushed aside. At that point geometry has done its job; the stability question is handed to energetic particles, kinetic damping, and the detailed global mode structure.

Caution

A gap is not yet a global mode. The local gap is a statement about the spectrum on each surface. A true TAE still needs radial localization and boundary conditions. Only after the coupled radial problem is solved does one obtain a discrete global eigenmode.

28.2 Newcomb Lagrangian for waves

Tutorial

Step 1: start from the same ideal-MHD variational structure. For ideal MHD linearized about a static equilibrium,
$\rho _0\,\partial _t^2\boldsymbol {\xi } = \mathcal {F}(\boldsymbol {\xi }), \tag{28.6}$ where $\mathcal {F}$ is the self-adjoint linearized force operator. With $e^{-i\omega t}$ time dependence, $\mathcal {F}(\hat {\boldsymbol {\xi }})+\omega ^2\rho _0\hat {\boldsymbol {\xi }}=0. \tag{28.7}$ The corresponding Lagrangian is $\mathcal {L}[\hat {\boldsymbol {\xi }}] = \delta W[\hat {\boldsymbol {\xi }}]-\omega ^2K[\hat {\boldsymbol {\xi }}], \tag{28.8}$ where $\delta W = -\frac 12\int d^3x\,\hat {\boldsymbol {\xi }}^{\,*}\cdot \mathcal {F}(\hat {\boldsymbol {\xi }}), \qquad K=\frac 12\int d^3x\,\rho _0|\hat {\boldsymbol {\xi }}|^2. \tag{28.9}$ For an exact eigenfunction, $\omega ^2=\delta W/K$. But the more useful statement here is that $\delta \mathcal {L}=0$ gives the wave equation.

Step 2: separate the static Newcomb coefficients from inertia. The coefficients in the Rosenbluth–Bussac lecture are potential-energy coefficients. In this lecture we should therefore write them first as $f_m^{(W)}$ and $g_m^{(W)}$. The wave coefficients are obtained only after subtracting $\omega ^2K$:
$\mathcal {L}_m(\omega ) = \delta W_m-\omega ^2K_m. \tag{28.10}$ For the radial displacement variable $\xi _m$, the leading cylindrical kinetic energy is not just an edge term. Use the Rosenbluth–Bussac normalization $\mathcal {N}\equiv \frac {2\pi ^2R_0}{\mu _0}. \tag{28.11}$ The small but important point is that an incompressible shear-Alfvén perturbation is not a purely radial displacement. Once $\xi _r=\xi _m$ is specified, incompressibility also fixes the binormal component $\xi _{\eta m}$. On a cylindrical magnetic surface define $\vect {e}_{\eta ,m} = (m/r)\vect {e}_{\theta } -(n/R_0)\vect {e}_\varphi {\left (m^2/r^2+n^2/R_0^2\right )^{1/2}}, \qquad k_{\eta m}^2\equiv \frac {m^2}{r^2}+\frac {n^2}{R_0^2} =\frac {M_m}{r^2}, \tag{28.12}$ where $M_m\equiv m^2+\frac {n^2r^2}{R_0^2}. \tag{28.13}$ Thus, to the accuracy needed here, the physical displacement is $\vect {\xi }_m = \xi _m\vect {e}_r+\xi _{\eta m}\vect {e}_{\eta ,m}, \tag{28.14}$ with any remaining transverse component chosen zero because it would add inertia without helping satisfy incompressibility. The divergence constraint is then $\nabla \cdot \vect {\xi }_m = \frac {1}{r}(r\xi _m)' + i k_{\eta m}\xi _{\eta m} =0, \tag{28.15}$ so $\xi _{\eta m} = \frac {i}{\sqrt {M_m}}(r\xi _m)', \qquad |\xi _{\eta m}|^2 = \frac {|(r\xi _m)'|^2}{M_m}. \tag{28.16}$ This is the whole origin of the derivative inertia. The kinetic energy contains both $|\xi _r|^2$ and $|\xi _\eta |^2$: $\frac {\omega ^2K_m}{\mathcal {N}} = \frac 12\int _0^a \mu _0\rho \omega ^2 r \left [ |\xi _m|^2+|\xi _{\eta m}|^2 \right ]dr = \frac 12\int _0^a \mu _0\rho \omega ^2 \left [ r|\xi _m|^2+ \frac {r}{M_m}\left |(r\xi _m)'\right |^2 \right ]dr . \tag{28.17}$ So the derivative term is not a boundary correction and it is not an extra assumption. It is simply the inertia of the binormal displacement required by $\nabla \cdot \boldsymbol {\xi }=0$.
To put (28.17) into Newcomb’s standard $f,g$ form, expand the second term:
$\begin{aligned}\frac {r}{M_m}|(r\xi _m)'|^2 &= \frac {r}{M_m}|r\xi _m'+\xi _m|^2 \nonumber \\ &= \frac {r^3}{M_m}|\xi _m'|^2 + \frac {r}{M_m}|\xi _m|^2 + \frac {r^2}{M_m}\frac {d}{dr}|\xi _m|^2.\end{aligned} \tag{28.18}$
The last term is the only part that becomes a surface contribution. With a nonuniform density,
$\begin{aligned}\int _0^a \mu _0\rho \omega ^2 \frac {r^2}{M_m}\frac {d}{dr}|\xi _m|^2dr &= \left [ \mu _0\rho \omega ^2\frac {r^2}{M_m}|\xi _m|^2 \right ]_0^a \nonumber \\ &\quad - \int _0^a \mu _0\omega ^2 \left (\frac {\rho r^2}{M_m}\right )' |\xi _m|^2dr .\end{aligned} \tag{28.19}$
Therefore the bulk kinetic coefficients are
$\begin{aligned}f_m^{(K)}(r) &= \mu _0\rho \frac {r^3}{M_m}, \\ g_m^{(K)}(r) &= \mu _0 \left [ \rho \left (r+\frac {r}{M_m}\right ) - \left (\frac {\rho r^2}{M_m}\right )' \right ].\end{aligned} \tag{28.20}$
Since the wave functional is $\delta W_m-\omega ^2K_m$, the full Newcomb wave coefficients are
$\begin{aligned}f_m(r,\omega ) &= f_m^{(W)}(r)-\omega ^2 f_m^{(K)}(r) = f_m^{(W)}(r)-\mu _0\rho \omega ^2\frac {r^3}{M_m}, \\ g_m(r,\omega ) &= g_m^{(W)}(r)-\omega ^2 g_m^{(K)}(r) \nonumber \\ &= g_m^{(W)}(r) - \mu _0\omega ^2 \left [ \rho \left (r+\frac {r}{M_m}\right ) - \left (\frac {\rho r^2}{M_m}\right )' \right ].\end{aligned} \tag{28.22}$
The integration by parts also changes the natural edge term by
$\mathcal {B}(\omega ) = \mathcal {B}^{(W)}- \mu _0\rho (a)\omega ^2\frac {a^2}{M_m(a)} \tag{28.24}$ for a free boundary. For the fixed-boundary internal problem this edge term is irrelevant, but the bulk terms in (28.22) and (28.23) remain. This is where the $\rho \omega ^2$ terms enter Newcomb’s $f_m$ and $g_m$.

Step 3: write the one-dimensional wave functional with f and g. After this subtraction has been made, a single harmonic has the form
$\mathcal {L}_m[\xi _m;\omega ] = \frac 12\int _0^a \left [ f_m(r,\omega )|\xi _m'|^2+g_m(r,\omega )|\xi _m|^2 \right ]dr +\frac 12\mathcal {B}(\omega )|\xi _m(a)|^2. \tag{28.25}$

Step 4: vary it once, carefully. Let $\xi _m\rightarrow \xi _m+\varepsilon \chi $. The term linear in $\varepsilon $ is
$\delta \mathcal {L}_m = \Re \left \{ \int _0^a \left [ f_m\xi _m'\chi ^{*\prime }+g_m\xi _m\chi ^* \right ]dr + \mathcal {B}\xi _m(a)\chi ^*(a) \right \}. \tag{28.26}$ Integrating the derivative term by parts, $\int _0^a f_m\xi _m'\chi ^{*\prime }dr = \left [f_m\xi _m'\chi ^*\right ]_0^a - \int _0^a (f_m\xi _m')'\chi ^*dr. \tag{28.27}$ Since $\chi $ is arbitrary in the interior, the Euler–Lagrange equation is $-\frac {d}{dr}\left (f_m\frac {d\xi _m}{dr}\right )+g_m\xi _m=0. \tag{28.28}$ If the edge is free rather than fixed, the natural boundary condition is $f_m(a)\xi _m'(a)+\mathcal {B}\xi _m(a)=0. \tag{28.29}$ For a fixed boundary one instead imposes $\xi _m(a)=0$.

Step 5: why continua appear in this same equation. The highest-derivative coefficient is $f_m(r,\omega )$. If $f_m$ is smooth and nonzero on $0<r<a$, the radial problem is an ordinary regular eigenvalue problem. If $f_m(r_A,\omega )=0$ at an interior surface, the differential operator is singular. That singular Newcomb solution is the ideal-MHD continuum.

28.3 The leading large-aspect-ratio wave functional

Use the same definitions as Lecture 26:

\Delta _m(r)=n-\frac {m}{q(r)}, \qquad \Sigma _m(r)=n+\frac {m}{q(r)}, \qquad F_m=-\frac {B_0}{R_0}\Delta _m. \tag{28.30}

The leading large-aspect-ratio Newcomb coefficients are

\begin{aligned}f_{2m}^{(W)}(r) &= \frac {B_0^2r^3}{m^2R_0^2}\Delta _m^2, \\ g_{2m}^{(W)}(r) &= \frac {B_0^2r}{m^2R_0^2}(m^2-1)\Delta _m^2 + \frac {2n^2r^2}{m^2R_0^2}\mu _0p'(r).\end{aligned} \tag{28.31}

The superscript $(W)$ reminds us that these are the potential-energy pieces; the inertial pieces still have to be added before this becomes a wave functional.

Add the inertial term explicitly. With this normalization, the leading kinetic contribution from (28.17) reduces, for $n^2r^2/R_0^2\ll m^2$ and slowly varying density, to

\frac {\omega ^2K_{2m}}{\mathcal {N}} = \frac 12\int _0^a \mu _0\rho \omega ^2 \left [ \frac {r^3}{m^2}|\xi _m'|^2 + \frac {m^2-1}{m^2}r|\xi _m|^2 \right ]dr +\hbox {surface term}. \tag{28.33}

The surface term is zero for a fixed boundary and regular axis, but the two bulk terms are not optional. In the same leading ordering, the Rosenbluth–Bussac potential-energy coefficients are

\begin{aligned}f_{2m}^{(W)} &= \frac {r^3}{m^2}\frac {B_0^2}{R_0^2}\Delta _m^2, \\ g_{2m}^{(W)} &= \frac {m^2-1}{m^2}r\frac {B_0^2}{R_0^2}\Delta _m^2 + \frac {2n^2r^2}{m^2R_0^2}\mu _0p'.\end{aligned}

The kinetic coefficients from (28.33) are

f_{2m}^{(K)}=\mu _0\rho \frac {r^3}{m^2}, \qquad g_{2m}^{(K)}=\mu _0\rho \frac {m^2-1}{m^2}r . \tag{28.36}

Subtracting $\omega ^2K_{2m}$ from the static Rosenbluth–Bussac coefficients therefore gives

\begin{aligned}f_{2m}(r,\omega ) &= f_{2m}^{(W)}-\omega ^2 f_{2m}^{(K)} \nonumber \\ &= \frac {r^3}{m^2} \left [ \frac {B_0^2}{R_0^2}\Delta _m^2(r)-\mu _0\rho (r)\omega ^2 \right ], \\ g_{2m}(r,\omega ) &= g_{2m}^{(W)}-\omega ^2 g_{2m}^{(K)} \nonumber \\ &= \frac {m^2-1}{m^2}r \left [ \frac {B_0^2}{R_0^2}\Delta _m^2(r)-\mu _0\rho (r)\omega ^2 \right ] + \frac {2n^2r^2}{m^2R_0^2}\mu _0p'(r).\end{aligned} \tag{28.37}

Equivalently,

f_{2m}=\frac {r^3}{m^2}\mathcal {A}_m, \qquad g_{2m}=\frac {m^2-1}{m^2}r\mathcal {A}_m + \frac {2n^2r^2}{m^2R_0^2}\mu _0p', \tag{28.39}

where

\mathcal {A}_m(r,\omega ) \equiv \frac {B_0^2}{R_0^2}\Delta _m^2(r)-\mu _0\rho (r)\omega ^2 = \mu _0\rho (r)\left [\omega _{A,m}^2(r)-\omega ^2\right ]. \tag{28.40}

This is the answer to the apparent mystery in the derivative term: the coefficient multiplying $\xi _m'$ is the line-bending coefficient $r^3B_0^2\Delta _m^2/(m^2R_0^2)$ minus the derivative part of the kinetic metric, $\mu _0\rho \omega ^2r^3/m^2$. Factoring out $r^3/m^2$ leaves $\mathcal {A}_m$. Thus the leading one-harmonic wave functional is

\frac {\mathcal {L}_{2m}}{\mathcal {N}} = \frac 12\int _0^a \left [ f_{2m}(r,\omega )|\xi _m'|^2+ g_{2m}(r,\omega )|\xi _m|^2 \right ]dr . \tag{28.41}

For the elementary TAE derivation below we set the pressure-gradient term aside. That is the low-$\beta $, shear-Alfvén limit. Pressure and compressibility are essential for BAEs and beta-induced gaps, but they are not needed to see the first toroidal Alfvén gap.

The useful rule is therefore not “use the static $f$ and $g$ unchanged.” It is

f_m=f_m^{(W)}-\omega ^2 f_m^{(K)}, \qquad g_m=g_m^{(W)}-\omega ^2 g_m^{(K)}. \tag{28.42}

At leading order, this rule is equivalent to replacing

\frac {B_0^2}{R_0^2}\Delta _m^2 \quad \longrightarrow \quad \frac {B_0^2}{R_0^2}\Delta _m^2-\mu _0\rho \omega ^2 \tag{28.43}

inside the leading line-bending pieces of both $f_m$ and $g_m$.

The explicit one-dimensional wave equation. Varying (28.41) gives the Newcomb wave equation

-\frac {d}{dr} \left [ f_{2m}(r,\omega )\frac {d\xi _m}{dr} \right ] + g_{2m}(r,\omega )\xi _m =0. \tag{28.44}

Using (28.37)–(28.38), this is

-\frac {d}{dr} \left [ \frac {r^3}{m^2}\mathcal {A}_m(r,\omega )\frac {d\xi _m}{dr} \right ] + \left [ \frac {m^2-1}{m^2}r\mathcal {A}_m(r,\omega ) + \frac {2n^2r^2}{m^2R_0^2}\mu _0p'(r) \right ]\xi _m =0. \tag{28.45}

In the low-$\beta $ TAE limit one drops the pressure-gradient term. The fixed-boundary version uses regularity at the magnetic axis and $\xi _m(a)=0$. The free-boundary version would use the corresponding natural condition obtained from the surface term.

Where the singularity is. The coefficient of the highest radial derivative vanishes when

\mathcal {A}_m(r_A,\omega )=0, \tag{28.46}

or equivalently

D_m(r_A,\omega ) \equiv \omega ^2-\omega _{A,m}^2(r_A) = \omega ^2-\omega _A^2(r_A)\Delta _m^2(r_A) =0, \tag{28.47}

where $\omega _A(r)=v_A(r)/R_0$. The two signs are useful for different purposes:

\mathcal {A}_m=-\mu _0\rho D_m. \tag{28.48}

Newcomb’s singular-surface language is naturally phrased as $\mathcal {A}_m=0$, because $\mathcal {A}_m$ multiplies the highest derivative. The continuum language is naturally phrased as $D_m=0$, because $D_m$ is the local oscillator denominator.

Local form of the singular solution. Assume $r_A$ is a simple resonance, so

\mathcal {A}_m(r,\omega ) \simeq \mathcal {A}'_{mA}(r-r_A), \qquad \mathcal {A}'_{mA}\equiv \left .\frac {\partial \mathcal {A}_m}{\partial r}\right |_{r_A}. \tag{28.49}

The leading singular part of (28.45) is then

-\frac {d}{dr} \left [ (r-r_A)\frac {d\xi _m}{dr} \right ] \simeq 0, \tag{28.50}

where smooth nonzero factors have been divided out. Integrating twice,

(r-r_A)\xi _m'=C_1, \qquad \xi _m(r)=C_0+C_1\ln |r-r_A|. \tag{28.51}

Thus the ideal-MHD continuum is not merely the algebraic formula $\omega =\omega _{A,m}(r)$. In the full radial Newcomb problem it is a logarithmic singularity caused by the vanishing of the highest-derivative coefficient.

Caution

Do not confuse two different uses of “resonance.” The rational surface $\Delta _m=0$ is the place where $k_{\parallel m}=0$, so the local shear-Alfvén frequency of that harmonic is zero. A finite-frequency Alfvén continuum resonance satisfies $D_m=0$, i.e. $\omega ^2=\omega _A^2\Delta _m^2$. For a fixed nonzero $\omega $, the resonant surface is generally not the rational surface.

28.4 Concrete crossing example: m = 2 and m = 3

Choose a simple monotone safety-factor profile. Take

q(r)=q_0+(q_a-q_0)\left (\frac {r}{a}\right )^2, \qquad q_0\simeq 1, \qquad q_a\simeq 4. \tag{28.52}

For arithmetic let $q_0=1$ and $q_a=4$. Then

q(r)=1+3\frac {r^2}{a^2}. \tag{28.53}

We will be searching for eigen functions that depend on the same $n$ since toroidal axisymmetry conserves the toroidal mode number. The basic TAE pair is therefore the pair

(m,n)=(2,1), \qquad (m+1,n)=(3,1), \tag{28.54}

At the crossing however the $\Delta _m$, hence $k_\parallel $ are of opposite signs.

Locate the uncoupled crossing. The two cylindrical continuum branches cross when their local Alfvén frequencies are equal,

\omega _A^2\Delta _m^2=\omega _A^2\Delta _{m+1}^2. \tag{28.55}

The crossing relevant to a TAE has opposite parallel wave numbers,

\Delta _m=-\Delta _{m+1}. \tag{28.56}

Using $\Delta _m=n-m/q$,

n-\frac {m}{q_c} = -\left (n-\frac {m+1}{q_c}\right ),

q_c=\frac {m+\tfrac 12}{n}. \tag{28.58}

For $m=2$ and $n=1$,

q_c=\frac 52. \tag{28.59}

With (28.53),

\frac {r_c}{a} = \left (\frac {q_c-q_0}{q_a-q_0}\right )^{1/2} = \left (\frac {2.5-1}{4-1}\right )^{1/2} = \frac {1}{\sqrt {2}} \simeq 0.707. \tag{28.60}

At this radius,

\Delta _2(r_c)=1-\frac {2}{2.5}=+\frac 15, \qquad \Delta _3(r_c)=1-\frac {3}{2.5}=-\frac 15. \tag{28.61}

Thus both branches satisfy the same continuum-resonance condition at the same radius when

\omega _0 = \omega _A(r_c)|\Delta _2(r_c)| = \omega _A(r_c)|\Delta _3(r_c)| = \frac {v_A(r_c)}{5R_0}. \tag{28.62}

This is the surface where toroidicity will open the gap.

The rational surfaces are nearby but different. For the same profile,

\Delta _2=0 \quad \Longleftrightarrow \quad q=2 \quad \Longrightarrow \quad \frac {r}{a}=\sqrt {\frac 13}\simeq 0.577, \tag{28.63}

and

\Delta _3=0 \quad \Longleftrightarrow \quad q=3 \quad \Longrightarrow \quad \frac {r}{a}=\sqrt {\frac 23}\simeq 0.816. \tag{28.64}

Those are the zero-frequency continuum points. The TAE crossing at $r_c/a\simeq 0.707$ lies between them, where $\Delta _2$ and $\Delta _3$ have opposite signs but equal magnitude.

28.5 The next-order functional and toroidal coupling

Separate the diagonal and off-diagonal next-order pieces. For this lecture it is useful to write the wave functional schematically as

\mathcal {L} = \mathcal {L}_2+ \mathcal {L}_{4,{\rm diag}}+ \mathcal {L}_{4,{\rm tor}}+ \cdots . \tag{28.65}

The diagonal part is obtained directly from the expanded $f$ and $g$ in (26.11). The toroidal part is the piece one sees only if the $\theta $-dependent geometry is kept before projecting onto poloidal Fourier harmonics. The first one shifts the branches; the second one opens the gap.

The diagonal L4 pieces from the earlier lecture. Equation (26.11) gives the potential-energy expansion

f_m^{(W)}=f_{2m}^{(W)}+f_{4m}^{(W)}+\cdots , \qquad g_m^{(W)}=g_{2m}^{(W)}+g_{4m}^{(W)}+\cdots , \tag{28.66}

with

f_{4m}^{(W)} = -\frac {B_0^2n^2r^5}{m^4R_0^4}\Delta _m^2, \tag{28.67}

and, in the low-$\beta $ shear-Alfvén limit,

g_{4m}^{(W)} = \frac {n^2B_0^2r^3}{m^4R_0^4} \left ( 3n^2-\frac {2nm}{q(r)}-\frac {m^2}{q^2(r)} \right ). \tag{28.68}

These are the static line-bending pieces. The wave problem requires the same-order expansion of the kinetic metric.

Derive the fourth-order kinetic terms from xi-eta. The important point is that no new displacement component is being introduced at fourth order. The fourth-order kinetic terms are the next term in the same binormal inertia already used in (28.17). Start again from

\frac {\omega ^2K_m}{\mathcal {N}} = \frac 12\int _0^a \mu _0\rho \omega ^2 \left [ r|\xi _m|^2+ \frac {r}{M_m}\left |(r\xi _m)'\right |^2 \right ]dr, \qquad M_m=m^2+\frac {n^2r^2}{R_0^2}. \tag{28.69}

The leading calculation replaced $M_m$ by $m^2$. To retain the next piece, write

M_m=m^2(1+\lambda _m), \qquad \lambda _m(r)\equiv \frac {n^2r^2}{m^2R_0^2}=O(\epsilon ^2), \tag{28.70}

so that

\frac {1}{M_m} = \frac {1}{m^2}\left (1-\lambda _m+O(\lambda _m^2)\right ). \tag{28.71}

The coefficient of $|\xi _m'|^2$ in the kinetic energy is therefore

\begin{aligned}f_m^{(K,{\rm raw})} &= \mu _0\rho \frac {r^3}{M_m} \nonumber \\ &= \mu _0\rho \frac {r^3}{m^2} - \mu _0\rho \frac {n^2r^5}{m^4R_0^2} +O\!\left (\mu _0\rho \frac {r^7}{R_0^4}\right ).\end{aligned} \tag{28.72}

The word “raw” means that this is still the coefficient inside $K$, before multiplication by $-\omega ^2$ in $\mathcal {L}=\delta W-\omega ^2K$. Hence the fourth-order inertial contribution that appears in the wave functional is

f_{4m}^{({\rm inert})} \equiv -\omega ^2\left [-\mu _0\rho \frac {n^2r^5}{m^4R_0^2}\right ] = +\mu _0\rho \omega ^2\frac {n^2r^5}{m^4R_0^2}. \tag{28.73}

This plus sign is often where mistakes creep in: the next term in $1/M_m$ is negative, but the kinetic energy is subtracted from $\delta W$.

Now do the same algebra for the $|\xi _m|^2$ coefficient. For locally constant or slowly varying density, use the bulk coefficient from (28.21),

g_m^{(K,{\rm raw})} = \mu _0\rho \left [ r+\frac {r}{M_m}-\left (\frac {r^2}{M_m}\right )' \right ]. \tag{28.74}

Expanding each piece gives

\begin{aligned}\frac {r}{M_m} &= \frac {r}{m^2} - \frac {n^2r^3}{m^4R_0^2} +O\!\left (\frac {r^5}{R_0^4}\right ), \\ \left (\frac {r^2}{M_m}\right )' &= \frac {2r}{m^2} - \frac {4n^2r^3}{m^4R_0^2} +O\!\left (\frac {r^5}{R_0^4}\right ).\end{aligned} \tag{28.75}

Therefore

\begin{aligned}g_m^{(K,{\rm raw})} &= \mu _0\rho \left [ r+\left (\frac {r}{m^2}-\frac {n^2r^3}{m^4R_0^2}\right ) -\left (\frac {2r}{m^2}-\frac {4n^2r^3}{m^4R_0^2}\right ) \right ] +\cdots \nonumber \\ &= \mu _0\rho \left [ \frac {m^2-1}{m^2}r + \frac {3n^2r^3}{m^4R_0^2} \right ] +\cdots .\end{aligned} \tag{28.77}

Thus the fourth-order inertial contribution to the $g$ coefficient in the wave functional is

g_{4m}^{({\rm inert})} \equiv -\omega ^2\left [\mu _0\rho \frac {3n^2r^3}{m^4R_0^2}\right ] = -\mu _0\rho \omega ^2\frac {3n^2r^3}{m^4R_0^2}. \tag{28.78}

If the density varies appreciably across the radial scale of interest, the more general expression (28.21) should be expanded instead; the extra terms are proportional to derivatives of $\rho $. For the local TAE gap calculation below, $\rho $ is frozen at the crossing surface, so (28.73) and (28.78) are the relevant kinetic corrections.

Combining the static and inertial pieces gives the diagonal fourth-order functional

\frac {\mathcal {L}_{4,{\rm diag}}}{\mathcal {N}} = \frac 12\sum _m\int _0^a \left [ \left (f_{4m}^{(W)}+f_{4m}^{({\rm inert})}\right )|\xi _m'|^2+ \left (g_{4m}^{(W)}+g_{4m}^{({\rm inert})}\right )|\xi _m|^2 \right ]dr . \tag{28.79}

For the elementary gap calculation, these diagonal terms are less important than the off-diagonal toroidal term because they do not couple $m$ to $m+1$. They shift the uncoupled continuum branches; by themselves they cannot produce an avoided crossing.

Where the coupling comes from: project the cosine term explicitly. The off-diagonal coupling is not contained in the already-projected one-harmonic coefficients $f_m$ and $g_m$. It appears only if the poloidal dependence of the toroidal metric is kept before the Fourier projection. For circular flux surfaces,

\frac {R_0^2}{R^2(r,\theta )} = 1-2\epsilon (r)\cos \theta +O(\epsilon ^2), \qquad \epsilon (r)=\frac {r}{R_0}. \tag{28.80}

The selection rule is just the Fourier identity

\cos \theta \,e^{i(m\theta -n\varphi )} = \frac 12e^{i[(m+1)\theta -n\varphi ]} + \frac 12e^{i[(m-1)\theta -n\varphi ]}. \tag{28.81}

To see the matrix element, write the local line-bending amplitude as

S(r,\theta ) \equiv \sum _m \Delta _m(r)\xi _m(r)e^{im\theta }, \tag{28.82}

where the common factor $e^{-in\varphi }$ has been suppressed. The minimal local shear-Alfvén Lagrangian before angular projection is

\frac {\mathcal {L}_{\rm loc}}{\mathcal {N}} = \frac 12\int dr\,\mu _0\rho r \left \{ \omega _A^2 \left \langle \left [1-2\epsilon \cos \theta \right ]|S(r,\theta )|^2 \right \rangle _\theta - \omega ^2 \left \langle \left |\sum _m\xi _m e^{im\theta }\right |^2 \right \rangle _\theta \right \}, \tag{28.83}

with

\left \langle A\right \rangle _\theta \equiv \frac {1}{2\pi }\int _0^{2\pi }A(\theta )\,d\theta . \tag{28.84}

The diagonal part follows from ordinary orthogonality,

\left \langle e^{i(m-\ell )\theta }\right \rangle _\theta =\delta _{m\ell }, \qquad \left \langle |S|^2\right \rangle _\theta = \sum _m\Delta _m^2|\xi _m|^2. \tag{28.85}

The off-diagonal part follows from the same identity with one factor of $\cos \theta $:

\left \langle \cos \theta \,e^{i(m-\ell )\theta } \right \rangle _\theta = \frac 12\left (\delta _{\ell ,m+1}+\delta _{\ell ,m-1}\right ). \tag{28.86}

Using (28.86) in $|S|^2$ gives

\begin{aligned}\left \langle \cos \theta \,|S|^2\right \rangle _\theta &= \frac 12\sum _m \Delta _m\Delta _{m+1} \left ( \xi _m^*\xi _{m+1}+\xi _{m+1}^*\xi _m \right ).\end{aligned} \tag{28.87}

This equation is the crux. The factor $1/2$ comes from the cosine selection rule, while the metric perturbation carries the factor $-2\epsilon \cos \theta $. Their product is $-\epsilon $. Thus the projected local Lagrangian is

\begin{aligned}\frac {\mathcal {L}_{\rm loc}}{\mathcal {N}} &= \frac 12\int dr\,\mu _0\rho (r)r \Bigg \{ \sum _m \left [\omega _A^2(r)\Delta _m^2(r)-\omega ^2\right ]|\xi _m|^2 \nonumber \\ &\hspace {7em} - \epsilon (r)\omega _A^2(r) \sum _m\Delta _m(r)\Delta _{m+1}(r) \left ( \xi _m^*\xi _{m+1}+\xi _{m+1}^*\xi _m \right ) \Bigg \}.\end{aligned} \tag{28.88}

For the concrete $(m,n)=(2,1)$, $(3,1)$ pair, the same statement is

\left \langle e^{-i3\theta }\cos \theta \,e^{i2\theta }\right \rangle _\theta =\frac 12, \qquad -2\epsilon \times \frac 12=-\epsilon , \tag{28.89}

so the line-bending part contains the cross term

-\frac 12\int dr\,\mu _0\rho r\, \epsilon \omega _A^2\Delta _2\Delta _3 \left (\xi _2^*\xi _3+\xi _3^*\xi _2\right ). \tag{28.90}

The full radial version replaces the diagonal algebraic factors in (28.88) by the Newcomb differential operators in (28.45) and keeps the corresponding nearest-neighbor operator couplings. For locating the local gap edges, (28.88) is the useful minimal form.

Stationarity gives the tridiagonal continuum matrix. Now vary (28.88) with respect to $\xi _m^*$. The diagonal term contributes

\frac {\partial }{\partial \xi _m^*} \left [ \left (\omega _A^2\Delta _m^2-\omega ^2\right )|\xi _m|^2 \right ] = \left (\omega _A^2\Delta _m^2-\omega ^2\right )\xi _m . \tag{28.91}

The off-diagonal sum contains two terms involving $\xi _m^*$: the pair $(m,m+1)$ and the pair $(m-1,m)$. Therefore

\begin{aligned}\frac {\partial }{\partial \xi _m^*} \sum _\ell \Delta _\ell \Delta _{\ell +1} \left (\xi _\ell ^*\xi _{\ell +1}+\xi _{\ell +1}^*\xi _\ell \right ) &= \Delta _m\Delta _{m+1}\xi _{m+1} + \Delta _{m-1}\Delta _m\xi _{m-1}.\end{aligned} \tag{28.92}

Stationarity gives

\left [\omega _A^2\Delta _m^2-\omega ^2\right ]\xi _m - \epsilon \omega _A^2 \left [ \Delta _m\Delta _{m+1}\xi _{m+1} + \Delta _m\Delta _{m-1}\xi _{m-1} \right ] =0. \tag{28.93}

Multiplying by $-1$ puts it in the usual continuum-denominator form,

\left [\omega ^2-\omega _A^2\Delta _m^2\right ]\xi _m + \epsilon \omega _A^2\Delta _m\Delta _{m+1}\xi _{m+1} + \epsilon \omega _A^2\Delta _m\Delta _{m-1}\xi _{m-1} =0. \tag{28.94}

This is the cleanest algebraic statement of toroidicity-induced coupling in the same $\Delta _m$ notation used above.

28.6 The toroidicity-induced gap

Keep only the two harmonics that cross. Near the crossing of the $m$ and $m+1$ branches, keep only those two amplitudes. Equation (28.94) becomes

\begin {pmatrix} \omega ^2-\omega _A^2\Delta _m^2 & \epsilon \omega _A^2\Delta _m\Delta _{m+1} \\[4pt] \epsilon \omega _A^2\Delta _m\Delta _{m+1} & \omega ^2-\omega _A^2\Delta _{m+1}^2 \end {pmatrix} \begin {pmatrix} \xi _m\\ \xi _{m+1} \end {pmatrix} =0. \tag{28.95}

A nontrivial solution requires

\left (\omega ^2-\omega _A^2\Delta _m^2\right ) \left (\omega ^2-\omega _A^2\Delta _{m+1}^2\right ) - \epsilon ^2\omega _A^4\Delta _m^2\Delta _{m+1}^2 =0. \tag{28.96}

Solving for $\omega ^2$,

\omega _\pm ^2 = \frac {\omega _A^2}{2}\left (\Delta _m^2+\Delta _{m+1}^2\right ) \pm \omega _A^2 \left [ \left (\frac {\Delta _m^2-\Delta _{m+1}^2}{2}\right )^2 + \epsilon ^2\Delta _m^2\Delta _{m+1}^2 \right ]^{1/2}. \tag{28.97}

This is the avoided-crossing formula.

Evaluate it at the crossing. At the uncoupled crossing,

\Delta _m=-\Delta _{m+1}, \qquad |\Delta _m|=|\Delta _{m+1}|=\frac {1}{2q_c}, \tag{28.98}

\omega _0 = \frac {v_A}{2q_cR_0}. \tag{28.99}

Equation (28.97) reduces to

\omega _\pm ^2 = \omega _0^2(1\pm \epsilon _c), \qquad \epsilon _c\equiv \frac {r_c}{R_0}. \tag{28.100}

Therefore

\Delta (\omega ^2)=\omega _+^2-\omega _-^2=2\epsilon _c\omega _0^2, \qquad \Delta \omega \equiv \omega _+-\omega _- \simeq \epsilon _c\omega _0 \quad (\epsilon _c\ll 1). \tag{28.101}

The cylinder would have given an exact crossing. The torus gives a gap whose fractional width is of order inverse aspect ratio.

Return to the m = 2, 3 and n = 1 example. For $q_c=5/2$,

\omega _0=\frac {v_A(r_c)}{5R_0}, \qquad \omega _\pm ^2=\omega _0^2(1\pm \epsilon _c), \qquad \epsilon _c=\frac {r_c}{R_0}=\frac {a}{\sqrt {2}R_0}. \tag{28.102}

Thus the local TAE gap opens around $v_A/(5R_0)$ at the radius where $q=2.5$. This is the simplest concrete example of the continuum crossing becoming an avoided crossing.

What is being plotted in the continuum figures. For each surface one first draws the uncoupled cylindrical continua

\omega _{A,m}(r)=\left |\Delta _m(r)\right |\frac {v_A(r)}{R_0}, \qquad \omega _{A,m+1}(r)=\left |\Delta _{m+1}(r)\right |\frac {v_A(r)}{R_0}. \tag{28.103}

Then one retains only the crossing pair $m$ and $m+1$, solves the local two-by-two system (28.95) on that surface, and plots its two eigenvalues $\omega _\pm (r)$ from (28.97). The solid split branches in the figures below are therefore the local toroidal continuum obtained by diagonalizing the coupled continuum matrix on each surface. They are not yet the discrete global TAE eigenfunction; the radial envelope problem still comes afterward.

Figure 28.1: Concrete $n=1$ crossing example for the pair $(m,n)=(2,1)$ and $(3,1)$. Panel (a) shows the profile $q(r)=1+3r^2/a^2$ together with the surfaces $q=2$, $q=2.5$, and $q=3$, corresponding to the $m=2$ rational surface, the TAE crossing surface, and the $m=3$ rational surface. Panel (b) shows the illustrative Alfvén-speed profile used in the plot, $ \rho (r)=\rho _0[1-0.55(r/a)^2]$ and notes explicitly that $ v_A(r)/v_{A0}=[1-0.55(r/a)^2]^{-1/2}$. Panel (c) overlays the cylindrical continua $\omega _{A,2}(r)=|\Delta _2|v_A/R_0$ and $\omega _{A,3}(r)=|\Delta _3|v_A/R_0$ with the toroidally split local branches $\omega _\pm (r)$ obtained from the two-harmonic coupling matrix (28.95). At $q=2.5$ the cylindrical crossing is replaced by an avoided crossing, and the local TAE gap opens around $\omega _0=v_A(r_c)/(5R_0)$.

A schematic in the style of Heidbrink’s review. Figure 28.2 presents the same avoided crossing in the more schematic style used in Heidbrink’s review Heidbrink (2008). The upper panel isolates the pair of counterpropagating cylindrical branches whose equal-magnitude parallel wave numbers define the center of the gap. The lower panel overlays the positive cylindrical continua and the toroidally split continua for the $n=4$, $(m,m+1)=(4,5)$ pair, together with the next pair $(6,7)$ that would cross farther out. In this presentation the cylindrical branches are plotted as positive frequencies, so the crossing represents two waves with opposite signs of $k_\parallel $ but equal $|k_\parallel |$.

Figure 28.2: TAE continuum plotted in the style of Fig. 3 of Heidbrink Heidbrink (2008). The upper panel highlights the two counterpropagating cylindrical branches and the frequency gap that appears once toroidicity is retained. The lower panel overlays dashed cylindrical continua with solid toroidal continua for the $n=4$ pairs $(m,m+1)=(4,5)$ and $(6,7)$. The toroidal branches are again the local eigenvalues $\omega _\pm (r)$ of the coupled two-harmonic matrix (28.95), evaluated surface by surface.

Takeaways

The leading cylindrical Newcomb equation gives singular continuum surfaces where $D_m=0$. Toroidicity does not remove the continuum everywhere; it couples neighboring singular branches. At the branch crossing, that coupling splits the two local continuum frequencies and opens a gap. A TAE is a global mode that can live in that gap.

28.7 From the local gap to a global eigenmode

The radial problem is still necessary. The local gap calculation gives the split continuum branches on each surface, but a global eigenmode requires a radial envelope. After the two-mode system is diagonalized locally, the envelope satisfies a bound-state problem of the schematic form

\frac {d^2A}{dr^2}+k_r^2(r,\omega )A=0. \tag{28.104}

If $k_r^2>0$ in some interval and $k_r^2<0$ outside, the mode oscillates in the allowed interval and decays outside it. The TAE is then trapped in a radial cavity.

Quantization of the cavity. In a WKB treatment the bound-state condition is

\int _{r_1}^{r_2}k_r(r,\omega )dr \simeq \pi \left (\ell +\frac 12\right ), \qquad \ell =0,1,2,\ldots . \tag{28.105}

This is the step that converts the local gap into a discrete ladder of global eigenvalues.

Why continuum damping is reduced but not banished. If the eigenfrequency lies inside the gap over the region where the mode is localized, then the dominant resonant condition $D_m(r,\omega )=0$ is avoided in the core of the mode. The logarithmic singularity does not appear there. But damping can still occur if the radial tails reach the continuum, if additional couplings reconnect the mode to another branch, or if kinetic Alfvén-wave conversion, ion Landau damping, electron Landau damping, or radiative damping is strong enough Rosenbluth et al. (1992); Zonca and Chen (1993); Chen and Zonca (2016). A gap mode is best thought of as a mode that has evaded the leading-order continuum resonance, not as a mode that has escaped all dissipation.

The same physics as phase mixing. The continuum can also be understood in initial-value language. If neighboring surfaces oscillate at slightly different local frequencies,

\xi (r,t)\sim \xi _0(r)e^{-i\omega _A(r)t},

then

\partial _r\xi \sim -it\,\omega _A'(r)\xi . \tag{28.107}

The radial gradient grows linearly in time, the perpendicular scale collapses, and the energy cascades toward small scales until non-ideal physics dissipates or phase-mixes it away.

28.8 What other Alfvénic families teach us

TAEs are only the first gap family. The TAE is the cleanest example because it is opened by the first toroidal harmonic and therefore by $m\leftrightarrow m+1$ coupling. But once one sees that logic, other families are almost inevitable. Ellipticity couples $m\leftrightarrow m+2$ and opens the EAE gap; higher-order shaping opens still higher gaps such as the NAE; reversed shear can localize modes near $q_{\min }$, producing RSAEs; and finite compressibility together with geodesic curvature opens the low-frequency BAE gap Betti and Freidberg (1991); Heidbrink et al. (1993, 1999); Kramer et al. (1998); Edlund et al. (2010); Gorelenkov et al. (2007); Chen and Zonca (2016).

RSAEs as a simple example of current-profile sensitivity. Near a minimum of the safety factor profile,

q(r)=q_{\min }+\frac 12 q_0''(r-r_0)^2+\cdots , \tag{28.108}

k_{\parallel m}(r) = \frac {1}{R_0}\left (n-\frac {m}{q(r)}\right ) \simeq k_{\parallel 0}+\alpha (r-r_0)^2+\cdots . \tag{28.109}

If $nq_{\min }$ is near an integer $m$, then $k_{\parallel 0}$ is small and the mode can localize near $r_0$. As the current profile evolves and $q_{\min }$ changes, the frequency chirps. This is why RSAEs became such a useful spectroscopic signature of current-profile evolution in advanced tokamak scenarios Edlund et al. (2009); Kramer et al. (2004); Edlund et al. (2010); Heidbrink et al. (2013).

28.9 Energetic-particle drive and experimental importance

Why energetic particles couple so easily. The simplest passing-particle resonance is

\omega \simeq k_\parallel v_\parallel , \tag{28.110}

which already explains why neutral-beam ions and fusion alphas are such effective drivers of TAEs: the phase velocity of the mode is of order $v_A$, and fast ions often satisfy $v_\parallel \sim v_A$. In a more faithful toroidal description one resolves transit, bounce, and drift frequencies, leading schematically to resonances of the form

\omega -n\omega _\phi -\ell \omega _b\simeq 0, \tag{28.111}

where $\ell $ labels the relevant orbital harmonic.

Drive versus damping. A convenient bookkeeping formula is

\gamma =\frac {P_h-P_d}{2W}, \tag{28.112}

where $W$ is the mode energy, $P_h$ is the power transferred from energetic particles to the wave, and $P_d$ is the sum of damping channels. This formula is schematic, but it captures the physics. Geometry determines whether a weakly damped gap mode can exist at all; the fast-ion distribution and the non-ideal damping mechanisms determine whether that mode is actually unstable.

What experiments actually care about. In experiments, AE physics matters because it affects fast-ion confinement. The earliest observations of beam-driven TAEs on TFTR made that point clearly Wong et al. (1991). DIII-D then provided a particularly important experimental thread: beam-driven Alfvén instabilities were characterized in detail, beta-induced Alfvén eigenmodes were identified, and reversed-shear cases showed that Alfvénic activity can flatten the fast-ion profile Heidbrink et al. (1991, 1993, 2008). Since then the subject has broadened from “Can one see the mode?” to “How much does it redistribute fast ions, flatten pressure profiles, or cause losses?” In present-day devices that question is tied directly to heating efficiency, current drive, and burning-plasma performance Fasoli et al. (2007); Heidbrink (2008); Lauber (2013); Heidbrink and White (2020).

Takeaways

The shear-Alfvén continuum and the TAE gap are two outputs of the same variational ideal-MHD framework. In the cylindrical Newcomb problem, the wave functional contains the coefficient $\mathcal {A}_m=\mu _0\rho (\omega _{A,m}^2-\omega ^2)$. Where this coefficient vanishes, the highest radial derivative vanishes and the ideal solution becomes logarithmically singular. In a torus, the $\cos \theta $ variation of the metric couples neighboring poloidal harmonics with the same $n$. At the place where $\Delta _m=-\Delta _{m+1}$, the two continuum branches would cross in a cylinder; toroidicity splits them and opens the TAE gap.
The lecture can be condensed into six statements.
1.
The local shear-Alfvén branch is $\omega ^2=k_\parallel ^2v_A^2$, with $k_{\parallel m}=\Delta _m/R_0$.
2.
The Newcomb wave equation is singular where $\mathcal {A}_m=0$, equivalently where $D_m=\omega ^2-\omega _A^2\Delta _m^2=0$.
3.
Near a simple resonant surface, $\xi _m=C_0+C_1\ln |r-r_A|$.
4.
The basic TAE pair has the same toroidal mode number $n$ and neighboring poloidal mode numbers $m$ and $m+1$.
5.
The crossing condition is $q_c=(m+1/2)/n$, or $q_c=5/2$ for the $m=2,3$, $n=1$ example.
6.
Toroidicity converts that crossing into the split branches $\omega _\pm ^2=\omega _0^2(1\pm \epsilon _c)$ at leading order.

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Problems

Problem 28.1.: Starting from $\mathcal {L}_m=\frac 12\int (f_m|\xi _m'|^2+g_m|\xi _m|^2)dr$, rederive (28.28). Keep the boundary term and identify the fixed-boundary and natural-boundary choices.
Problem 28.2.: Starting from (28.41), derive the wave equation (28.45). Identify exactly which coefficient vanishes at the continuum resonance.
Problem 28.3.: Let $\mathcal {A}_m\simeq \mathcal {A}'_{mA}(r-r_A)$. Derive the logarithmic singularity (28.51) and explain why the $|\xi _m|^2$ term is subdominant in the local singular balance.
Problem 28.4.: For $q(r)=1+3r^2/a^2$, compute the $m=2$ rational surface, the $m=3$ rational surface, and the $m=2,3$, $n=1$ TAE crossing radius. Explain why these are three different radii.
Problem 28.5.: Starting from (28.95), verify the determinant condition (28.96) and the avoided-crossing formula (28.97).
Problem 28.6.: For the $m=2,3$, $n=1$ example, assume $a/R_0=1/3$. Estimate $\epsilon _c$, $\omega _0$, and the leading gap width $\Delta \omega $ in terms of $v_A/R_0$.
Problem 28.7.: Using (28.112), discuss qualitatively how increasing beam power, increasing electron temperature, and increasing magnetic shear should affect the observability of a TAE.