Lecture 25
Bussac: The Toroidal \(n=1\) Internal Kink
Overview
The ideal internal kink and the high-\(n\) ballooning mode are not separate uses of ideal
MHD. They are two different asymptotic reductions of the same axisymmetric tokamak
energy functional. In the ballooning appendix we started from the exact two-dimensional
toroidal functional, eliminated \(\xi _\parallel \) variationally, and then took a high-\(n\) field-line-localized
limit. Here we start from that same toroidal \(\delta W\), keep \(n=1\), and instead carry out an
inverse-aspect-ratio expansion. The result is the Bussac picture: toroidicity couples the
main \(m=1\) core shift to stable sidebands, and after those sidebands are eliminated the effective
\(n=1\) energy is no longer the cylindrical one.
The TAE chapter introduced toroidicity first in the cleanest possible language: neighboring
poloidal harmonics couple because the line-bending coefficient varies like \(1/R^2 \sim 1-2\epsilon \cos \theta +\cdots \). The Bussac calculation
uses that same harmonic bookkeeping, but inside a static energy principle rather than a wave
equation. So one may read this lecture as the \(n=1\), low-frequency counterpart of the TAE gap
calculation.
Historical Perspective
The cylindrical \(m=1\) internal kink was worked out by Rosenbluth, Dagazian, and Rutherford,
who showed that the minimizing displacement is essentially a rigid core shift cut off
at the \(q=1\) surface Rosenbluth et al. (1973). Bussac, Pellat, Edery, and Soulé then asked
what happens when the same mode is treated in toroidal geometry, where \(R=R_0+r\cos \theta \) and the
equilibrium can no longer be strictly cylindrical ?. Their answer is one of the classic
tokamak-stability results: the true \(n=1\) internal kink can be stable at low pressure because
sideband coupling produces a toroidal stabilizing term that is absent in the periodic
cylinder.
That is exactly why it is worth recasting the calculation in the language of the ballooning
appendix. One sees directly that Mercier, Bussac, and ballooning are not three unrelated
constructions. They are three different reductions of the same axisymmetric energy
principle.
Caution
Do not import the ballooning approximation into the internal-kink problem.
The first minimization over \(\xi _\parallel \) is exact and is shared by both problems. The second
ballooning step, the high-\(n\) elimination of \(U\), is not appropriate here. For the internal kink
one must keep the low-\(n\) poloidal sideband coupling generated by toroidicity. That is the
entire content of the Bussac correction.
25.1 Start from the same toroidal \(\delta W\)
The appendix derives the exact two-dimensional tokamak energy functional in the Connor–Hastie–Taylor
variables \(X\), \(U\), and \(Z\) and then carries out the first minimization over \(\xi _\parallel \) by eliminating \(Z\). Using the appendix
notation dictionary
\[B_\chi \rightarrow B_p, \qquad \chi \rightarrow \theta , \tag{28.1}\]
we therefore start from Appendix Eq. (I.44), namely \[\delta W = \pi \int \mathcal {J}\,d\Psi \,d\theta \, \Biggl [ \frac {B^2}{R^2B_p^2}|k_\parallel X|^2 + \frac {R^2}{\mathcal {J}^2}\left |\frac {\partial U}{\partial \theta }-I\frac {\partial }{\partial \Psi }\left (\frac {\mathcal {J}X}{R^2}\right )\right |^2 + B_p^2\left |inU+\frac {\partial X}{\partial \Psi }+\frac {j_\phi }{RB_p^2}X\right |^2 -2K|X|^2 \Biggr ]. \tag{28.2}\]
This is the exact low-frequency ideal-MHD functional after the parallel displacement has been minimized
away. The internal-kink calculation is therefore not a new principle; it is a different asymptotic use of the
same one.
Circular large-aspect-ratio geometry.
Take a circular tokamak with
\[R(r,\theta )=R_0+r\cos \theta , \qquad \epsilon \equiv \frac {r}{R_0}\ll 1, \qquad B_p\sim O(\epsilon B_0), \qquad \beta \sim O(\epsilon ^2). \tag{28.3}\]
With \(\chi =\theta \), the appendix metric gives \[d\Psi = RB_p\,dr, \qquad \mathcal {J}B_p=r, \qquad \mathcal {J}=\frac {r}{B_p}, \qquad dV = rR\,dr\,d\theta \,d\phi . \tag{28.4}\]
The appendix quantity \(\nu \) becomes the safety factor, \[\nu = \frac {I\mathcal {J}}{R^2} = \frac {rB_\phi }{RB_p}=q(r), \tag{28.5}\]
so for a single toroidal harmonic \(e^{in\phi }\), \[\B \cdot \nabla \longrightarrow \frac {1}{\mathcal {J}}\left (\frac {\partial }{\partial \theta }+inq\right ), \qquad ik_\parallel = \frac {1}{\mathcal {J}B}\left (\frac {\partial }{\partial \theta }+inq\right ). \tag{28.6}\]
Substituting Eqs. (??)–(??) into Eq. (??) gives \[\delta W = \pi \int _0^a dr\int _0^{2\pi } d\theta \, \Biggl [ \frac {1}{rR}\left |\left (\frac {\partial }{\partial \theta }+inq\right )X\right |^2 + \frac {R^3B_p^2}{r} \left |\frac {\partial U}{\partial \theta }-\frac {I}{RB_p}\frac {\partial }{\partial r}\left (\frac {rX}{R^2B_p}\right )\right |^2 \Biggr .\]
\[\Biggl . \qquad + rRB_p^2\left |inU+\frac {1}{RB_p}\frac {\partial X}{\partial r}+\frac {j_\phi }{RB_p^2}X\right |^2 -2rRK|X|^2 \Biggr ]. \tag{28.8}\]
Nothing has been approximated beyond axisymmetry and the circular-coordinate specialization. The low-\(n\)
calculation begins from Eq. (??).
25.2 Low-\(n\) harmonic bookkeeping
For the internal kink we take \(n=1\) and expand in poloidal harmonics using the same \(e^{in\phi }\) convention as the
appendix,
\[X(r,\theta ,\phi ) = e^{i\phi }\sum _m X_m(r)e^{-im\theta }, \qquad U(r,\theta ,\phi ) = e^{i\phi }\sum _m U_m(r)e^{-im\theta }. \tag{28.9}\]
Define the field-line-pitch mismatch \[D_m(r)\equiv q(r)-m. \tag{28.10}\]
Then the parallel operator acts diagonally on each cylindrical harmonic, \[\left (\frac {\partial }{\partial \theta }+iq\right ) \left [e^{i\phi }X_m(r)e^{-im\theta }\right ] = iD_m(r) \,e^{i\phi }X_m(r)e^{-im\theta }. \tag{28.11}\]
Thus the dominant internal-kink harmonic is the \(m=1\) piece, \[X_1(r)e^{i(\phi -\theta )}, \qquad U_1(r)e^{i(\phi -\theta )}. \tag{28.12}\]
Why toroidicity generates only neighboring sidebands at first order.
Expand the toroidal coefficients in Eq. (??),
\[\begin{aligned}\frac {1}{R} &= \frac {1}{R_0}\left (1-\epsilon \cos \theta +\epsilon ^2\cos ^2\theta +\cdots \right ), \\ \frac {1}{R^2} &= \frac {1}{R_0^2}\left (1-2\epsilon \cos \theta +3\epsilon ^2\cos ^2\theta +\cdots \right ), \\ K(r,\theta ) &= K_0(r)+\epsilon K_1(r)\cos \theta +O(\epsilon ^2).\end{aligned} \tag{28.13}\]
Because
\[\cos \theta \,e^{-im\theta } = \frac 12 e^{-i(m-1)\theta }+\frac 12 e^{-i(m+1)\theta }, \tag{28.16}\]
all \(O(\epsilon )\) toroidal corrections acting on the \(m=1\) harmonic produce only \(m=0\) and \(m=2\) sidebands. The consistent low-\(\epsilon \) ansatz
is therefore \[X = e^{i\phi }\Bigl [X_1(r)e^{-i\theta }+\epsilon X_0(r)+\epsilon X_2(r)e^{-2i\theta }+O(\epsilon ^2)\Bigr ], \tag{28.17}\]
\[U = e^{i\phi }\Bigl [U_1(r)e^{-i\theta }+\epsilon U_0(r)+\epsilon U_2(r)e^{-2i\theta }+O(\epsilon ^2)\Bigr ]. \tag{28.18}\]
This is the low-\(n\) analogue of the ballooning transform. The main distinction is that the internal-kink
problem does not localize along one field line; instead it mixes a small set of neighboring poloidal
harmonics.
Carry out the bookkeeping explicitly.
To see where the sideband terms in \(\delta W\) actually come from, separate the three operator combinations
appearing in Eq. (??). First, the parallel line-bending operator gives
\[\left (\frac {\partial }{\partial \theta }+iq\right )X = i\,e^{i\phi } \Bigl [ D_1X_1e^{-i\theta } \epsilon D_0X_0 \epsilon D_2X_2e^{-2i\theta } O(\epsilon ^2) \Bigr ],\]
with \[D_0=q, \qquad D_1=q-1, \qquad D_2=q-2.\]
Only the \(m=1\) amplitude is nearly resonant at the \(q=1\) surface; the \(m=0\) and \(m=2\) neighbors remain off-resonant
there.
For the other two squared terms in Eq. (??), it is useful to define the cylindrical-order amplitudes
\[\mathscr {G}_m \equiv -im\,U_m -\frac {I}{R_0B_p}\frac {d}{dr}\!\left (\frac {rX_m}{R_0^2B_p}\right ), \qquad \mathscr {H}_m \equiv iU_m \frac {1}{R_0B_p}\frac {dX_m}{dr} \frac {j_\phi }{R_0B_p^2}X_m.\]
Then \[\frac {\partial U}{\partial \theta } -\frac {I}{RB_p}\frac {\partial }{\partial r}\!\left (\frac {rX}{R^2B_p}\right ) = e^{i\phi } \Bigl [ \mathscr {G}_1e^{-i\theta } \epsilon \mathscr {G}_0 \epsilon \mathscr {G}_2e^{-2i\theta } O(\epsilon ^2) \Bigr ],\]
\[iU+\frac {1}{RB_p}\frac {\partial X}{\partial r}+\frac {j_\phi }{RB_p^2}X = e^{i\phi } \Bigl [ \mathscr {H}_1e^{-i\theta } \epsilon \mathscr {H}_0 \epsilon \mathscr {H}_2e^{-2i\theta } O(\epsilon ^2) \Bigr ].\]
The important point is that the harmonic content has the same pattern in all three operator blocks: a
dominant \(m=1\) piece plus \(O(\epsilon )\) sidebands with \(m=0\) and \(m=2\).
Now also expand the \(\theta \)-dependent coefficients multiplying the squared blocks,
\[\frac {1}{rR}=a_0(r)+\epsilon a_1(r)\cos \theta +O(\epsilon ^2), \qquad \frac {R^3B_p^2}{r}=b_0(r)+\epsilon b_1(r)\cos \theta +O(\epsilon ^2),\]
\[rRB_p^2=c_0(r)+\epsilon c_1(r)\cos \theta +O(\epsilon ^2), \qquad 2rRK=d_0(r)+\epsilon d_1(r)\cos \theta +O(\epsilon ^2).\]
When these are inserted into Eq. (??), the \(\theta \)-integral keeps only the combinations with zero net phase. The
diagonal pieces come from the coefficient \(1\), for example \[\int _0^{2\pi } d\theta \,|D_1X_1e^{-i\theta }|^2 = 2\pi |D_1X_1|^2, \qquad \int _0^{2\pi } d\theta \,|\mathscr {G}_1e^{-i\theta }|^2 = 2\pi |\mathscr {G}_1|^2,\]
and similarly for \(\mathscr {H}_1\) and \(X_1\) in the other terms.
The off-diagonal sideband couplings come from one explicit factor of \(\cos \theta \). Using \(\cos \theta =(e^{i\theta }+e^{-i\theta })/2\), one gets
\[\int _0^{2\pi } d\theta \, \cos \theta \, \Bigl (\mathscr {D}_1^*e^{i\theta }\Bigr )\mathscr {D}_0 = \pi \,\mathscr {D}_1^*\mathscr {D}_0, \qquad \int _0^{2\pi } d\theta \, \cos \theta \, \Bigl (\mathscr {D}_1^*e^{i\theta }\Bigr )\mathscr {D}_2e^{-2i\theta } = \pi \,\mathscr {D}_1^*\mathscr {D}_2,\]
where \(\mathscr {D}_m\equiv iD_mX_m\). Exactly the same algebra applies to \(\mathscr {G}_m\), \(\mathscr {H}_m\), and \(X_m\) itself in the curvature term. Thus the \(m=0\leftrightarrow 1\) source
generated by toroidicity is \[\delta W_{01} = \pi \epsilon \int _0^a dr\, \Bigl [ a_1\,\mathscr {D}_0^*\mathscr {D}_1 + b_1\,\mathscr {G}_0^*\mathscr {G}_1 + c_1\,\mathscr {H}_0^*\mathscr {H}_1 - d_1\,X_0^*X_1 \Bigr ] \;+\;\text {c.c.},\]
and the \(m=2\leftrightarrow 1\) source is \[\delta W_{21} = \pi \epsilon \int _0^a dr\, \Bigl [ a_1\,\mathscr {D}_2^*\mathscr {D}_1 + b_1\,\mathscr {G}_2^*\mathscr {G}_1 + c_1\,\mathscr {H}_2^*\mathscr {H}_1 - d_1\,X_2^*X_1 \Bigr ] \;+\;\text {c.c.}\]
No \(m=3\) or higher harmonic appears at this order because one factor of \(\cos \theta \) can only shift the dominant \(m=1\)
component by one unit. Likewise there is no direct \(m=0\leftrightarrow 2\) coupling at \(O(\epsilon )\): that would require two powers of \(\cos \theta \), hence
\(O(\epsilon ^2)\).
Block structure of the quadratic form.
If one inserts Eqs. (??)–(??) into Eq. (??), expands all coefficients through \(O(\epsilon ^2)\), and integrates over \(\theta \), the
energy takes the schematic form
\[\delta W = \delta W_{11}^{(0)} + \epsilon ^2 \Bigl [ \delta W_{11}^{(2)} + \delta W_{00}^{(0)} + \delta W_{22}^{(0)} + 2\Re \langle Y_0,\mathcal {M}_{01}Y_1\rangle + 2\Re \langle Y_2,\mathcal {M}_{21}Y_1\rangle \Bigr ] +O(\epsilon ^3), \tag{28.19}\]
where \[Y_m(r)\equiv \begin {pmatrix}X_m(r)\\ U_m(r)\end {pmatrix}. \tag{28.20}\]
Equation (??) already contains the whole Bussac logic. The same \(\cos \theta \) bookkeeping that opened
the TAE gap is operating here too: the \(m=1\) core shift forces stable \(m=0\) and \(m=2\) neighbors through the
first toroidal sideband couplings. The leading cylindrical \(m=1\) piece is degenerate, while the first
nontrivial toroidal correction is quadratic in the stable sidebands. After the non-radial variables are
minimized away, the \(m=0\) contribution can be absorbed into the same effective radial operator that
governs the \(m=2\) sideband, so the final correction is conventionally packaged as a single sideband
problem.
Tutorial
In the ballooning appendix the second reduction is driven by the ordering \(\partial /\partial \Psi \sim n\) and \(k_\parallel =O(1)\), so \(U\) is
slaved to \(X\) and the problem collapses to a one-dimensional field-line functional.
For the internal kink, \(n=1\), there is no such large-\(n\) ordering. The small parameter is instead the
inverse aspect ratio \(\epsilon =r/R_0\). Toroidicity then appears as a weak \(\cos \theta \) modulation of the coefficients
in Eq. (??). That weak modulation couples the dominant \(m=1\) harmonic to neighboring
sidebands. So the second reduction is now \[ \text {keep }m=1\text { and its }m\pm 1\text { sidebands,}\] \[ \text {solve the stable sideband equation,} \]
\[ \text {substitute back into }\delta W. \] That is the low-\(n\) Bussac reduction of the same toroidal energy principle.
25.3 Recover the cylindrical Rosenbluth problem from the same functional
If one drops all \(O(\epsilon )\) toroidal pieces in Eq. (??), the Fourier harmonics decouple. After variation with respect
to the in-surface component \(U_m\), the diagonal \(m\) sector reduces to the usual Newcomb functional,
\[\delta W_m^{\rm cyl} = \frac {2\pi ^2R_0}{\mu _0} \int _0^a dr\, \Bigl [f_m(r)|\xi _m'|^2+g_m(r)|\xi _m|^2\Bigr ], \tag{28.21}\]
with \[\begin{aligned}f_m(r) &= \frac {B_0^2r^3}{m^2R_0^2}\left (1-\frac {m}{q(r)}\right )^2, \\ g_m(r) &= \frac {B_0^2r}{m^2R_0^2}(m^2-1)\left (1-\frac {m}{q(r)}\right )^2 + \frac {2r^2}{m^2R_0^2}\mu _0p'(r).\end{aligned} \tag{28.22}\]
Here \(\xi _m\) is the radial displacement harmonic defined by
\[X_m = R_0B_p\,\xi _m + O(\epsilon ). \tag{28.24}\]
For \(m=1\), the explicit line-bending part of \(g_m\) vanishes, \[g_1(r)=\frac {2r^2}{R_0^2}\mu _0p'(r)+O(\epsilon ^4), \tag{28.25}\]
so at the leading cylindrical order the energy depends only on the narrow region where \(\xi _1\) changes across the \(q=1\)
surface. Thus the minimizing sequence is the rigid core shift \[\xi _1(r)\longrightarrow \xi _0 H(r_1-r), \qquad q(r_1)=1, \tag{28.26}\]
which is exactly the Rosenbluth–Dagazian–Rutherford result.
Why the cylindrical contribution becomes singularly weak.
For the top-hat family,
\[\delta W_1^{\rm cyl} \propto \int _{r_1-\delta /2}^{r_1+\delta /2} \left (1-\frac {1}{q}\right )^2 r^3\left |\frac {d\xi _1}{dr}\right |^2dr \sim O(\delta ) \rightarrow 0, \tag{28.27}\]
so the \(O(\epsilon ^2)\) cylindrical functional has an infimum rather than a smooth minimizer. This is the degeneracy that
toroidicity resolves.
The resolved cylindrical layer.
To turn the minimizing sequence into an actual eigenfunction, one retains the small inertial term that
resolves the singular layer at \(q=1\). Let
\[x\equiv r-r_1, \qquad \mathbf {k}\cdot \mathbf {B}\approx (\mathbf {k}\cdot \mathbf {B})'_1 x, \tag{28.28}\]
so the local ideal layer equation becomes \[\frac {d}{dx} \left [ \left (4\pi \rho _1\gamma ^2+(\mathbf {k}\cdot \mathbf {B})_1'^2x^2\right ) \frac {d\xi }{dx} \right ] =0. \tag{28.29}\]
Integrating once and imposing \[\xi \to \xi _0 \quad (x\to -\infty ), \qquad \xi \to 0 \quad (x\to +\infty ), \tag{28.30}\]
leads to the arctangent-smoothed top hat \[\xi (x) = \frac {\xi _0}{2} \left [ 1- \frac {2}{\pi } \tan ^{-1} \left ( \frac {(\mathbf {k}\cdot \mathbf {B})'_1x}{\gamma (4\pi \rho _1)^{1/2}} \right ) \right ]. \tag{28.31}\]
As the inertial width tends to zero, Eq. (??) reduces back to Eq. (??). So the cylindrical Rosenbluth
solution is already contained in the same toroidal functional; one simply suppresses the toroidal
sidebands.
25.4 Toroidal sideband elimination in the TAE language
The practical low-\(\epsilon \) calculation is obtained by substituting the sideband ansatz, Eqs. (??)–(??), into
Eq. (??), expanding through \(O(\epsilon ^2)\), and varying with respect to the sideband amplitudes. After the
non-radial pieces are eliminated, the remaining radial sideband problem can be written in terms of
\[\iota (r)\equiv \frac {1}{q(r)} \tag{28.32}\]
and the stable cylindrical operators \[L_m[\xi ] \equiv \frac {d}{dr}\left [r^3\bigl (m\iota -1\bigr )^2\frac {d\xi }{dr}\right ] - r(m^2-1)\bigl (m\iota -1\bigr )^2\xi . \tag{28.33}\]
The toroidally forced \(m=2\) sideband obeys \[L_2[\xi _2] + \frac {d}{dr}\left [r^3T_1(r)\frac {d\xi _1}{dr}\right ] + r^2T_2(r)\frac {d\xi _1}{dr} =0. \tag{28.34}\]
The coefficients \(T_1\) and \(T_2\) are the explicit \(O(\epsilon )\) sideband couplings, \[\begin{aligned}T_1(r) &= \bigl (-6\iota ^2+6\iota -1\bigr )\Delta _S'(r) + \left (-4\iota ^2+10\iota -\frac {11}{2}\right )\frac {r}{R_0}, \\ T_2(r) &= \bigl (6\iota ^2-6\iota +3\bigr )\Delta _S'(r) + \left (2\iota ^2-4\iota +\frac {7}{2}\right )\frac {r}{R_0},\end{aligned} \tag{28.35}\]
with the Shafranov-shift derivative written in the standard circular form
\[\Delta _S'(r) = -\frac {r}{R_0}\left [\beta _p(r)+\frac {l_i(r)}{2}\right ], \tag{28.37}\]
where \[\beta _p(r) \equiv \frac {4\mu _0}{r^2B_p^2(r)}\int _0^r p(r')\,r'\,dr', \qquad l_i(r) \equiv \frac {2}{r^2B_p^2(r)}\int _0^r B_p^2(r')\,r'\,dr'. \tag{28.38}\]
Equation (??) is the low-\(n\) analogue of the ballooning \(U\)-equation: it tells us how the stable neighboring
harmonic is slaved to the main \(m=1\) displacement.
Show the minimization before reducing to \(\xi _2\).
At the level of the full sideband amplitudes, the \(m=2\) contribution to the expanded energy has the generic
quadratic structure
\[\delta W_{(2)}[Y_1,Y_2] = \delta W_{11}^{(0)}[Y_1] + \epsilon ^2 \left [ \langle Y_2,\mathcal {L}_{22}Y_2\rangle + \langle Y_2,\mathcal {S}_{21}Y_1\rangle + \langle \mathcal {S}_{21}Y_1,Y_2\rangle \right ],\]
where \[Y_2(r)\equiv \begin {pmatrix} X_2(r)\\ U_2(r) \end {pmatrix}, \qquad Y_1(r)\equiv \begin {pmatrix} X_1(r)\\ U_1(r) \end {pmatrix}.\]
Here \(\mathcal {L}_{22}\) is the self-adjoint cylindrical \(m=2\) operator coming from the diagonal \(m=2\) block of Eq. (??), while \(\mathcal {S}_{21}\) is the
explicit \(O(\epsilon )\) source constructed from the \(m=1\leftrightarrow 2\) couplings in Eqs. (??). Varying with respect to \(Y_2^\dagger \) therefore gives the
vector sideband equation \[\mathcal {L}_{22}Y_2+\mathcal {S}_{21}Y_1=0.\]
This is the exact low-\(n\) analogue of the TAE step in which one varies the coupled two-harmonic functional
with respect to the stable neighboring harmonic.
Reduce the vector equation to one radial amplitude.
The in-surface sideband variable \(U_2\) enters Eq. (??) without singular behavior, so it may be eliminated
algebraically in the same way that \(U_m\) was eliminated in the cylindrical Rosenbluth reduction. After that step
one is left with a single radial sideband amplitude \(\xi _2\), defined through \(X_2=R_0B_p\,\xi _2+O(\epsilon )\), and Eq. (??) collapses to the scalar
form (??). In other words, \(L_2\) in Eq. (??) is just the reduced one-dimensional version of the
diagonal block \(\mathcal {L}_{22}\), and the forcing term involving \(T_1\) and \(T_2\) is the reduced one-dimensional version of
\(\mathcal {S}_{21}Y_1\).
The quadratic sideband form.
The \(m=2\) part of the expanded energy can be written as
\[\delta W_{\rm sb}[\xi _1,\xi _2] = \mathcal {N} \int _0^a dr\, \Bigl [ \xi _2^*L_2\xi _2 + \xi _2^*\mathcal {F}[\xi _1] + \mathcal {F}[\xi _1]^*\xi _2 \Bigr ], \tag{28.39}\]
where \(\mathcal {N}>0\) is an overall normalization and \[\mathcal {F}[\xi _1] \equiv \frac {d}{dr}\left [r^3T_1(r)\frac {d\xi _1}{dr}\right ] + r^2T_2(r)\frac {d\xi _1}{dr}. \tag{28.40}\]
Varying Eq. (??) with respect to \(\xi _2^*\) gives Eq. (??). Because the \(m=2\) cylindrical sector is stable, \(L_2\) is positive on
the relevant domain and the minimizing sideband is therefore \[\xi _2 = - L_2^{-1}\mathcal {F}[\xi _1]. \tag{28.41}\]
Substituting this back into Eq. (??) gives the negative-definite correction \[\delta W_{\rm sb,min}[\xi _1] = -\mathcal {N} \int _0^a dr\,\mathcal {F}[\xi _1]^*L_2^{-1}\mathcal {F}[\xi _1]. \tag{28.42}\]
Equation (??) is the precise variational meaning of the phrase “toroidal sideband stabilization”: the
toroidal \(m=1\leftrightarrow 2\) coupling produces a source \(\mathcal {F}\), the stable sideband responds through \(L_2^{-1}\), and substituting that
response back gives the effective toroidal correction to the main harmonic.
Green-function form.
Let the Green function of the stable sideband operator satisfy
\[L_2\,G_2(r,r')=\delta (r-r'), \qquad G_2(0,r')\ \text {regular}, \qquad G_2(a,r')=0. \tag{28.43}\]
Then \[\xi _2(r) = -\int _0^a G_2(r,r')\,\mathcal {F}[\xi _1](r')\,dr', \tag{28.44}\]
and the minimizing sideband contribution becomes \[\delta W_{\rm sb,min}[\xi _1] = -\mathcal {N} \int _0^a\!dr\int _0^a\!dr'\, \mathcal {F}[\xi _1]^*(r) G_2(r,r') \mathcal {F}[\xi _1](r'). \tag{28.45}\]
This is the exact formal reduction. The remaining work is profile-specific: choose \(q(r)\) and \(p(r)\), solve the stable \(m=2\)
Green-function problem, and evaluate the double integral.
Takeaways
One toroidal functional, two very different second reductions.
For ballooning modes the second reduction is a high-\(n\) minimization over the in-surface
variable \(U\), leading to a local field-line ODE.
For the internal kink the second reduction is a low-\(\epsilon \) elimination of the stable sideband \(\xi _2\),
leading to the effective radial energy Eq. (??).
So the conceptual bridge is exact: Mercier, Bussac, and ballooning are all descendants
of Appendix I; they simply keep different pieces of the same toroidal \(\delta W\).
25.5 How Bussac’s decomposition emerges
After the sideband has been eliminated, the effective large-aspect-ratio energy can be written in the Bussac
form
\[\delta W = W_0\left [ \left (1-\frac {1}{n^2}\right )\hat {\delta W}_c + \frac {1}{n^2}\hat {\delta W}_T \right ], \tag{28.46}\]
where \(\hat {\delta W}_c\) is the normalized cylindrical contribution and \(\hat {\delta W}_T\) is the sideband-generated toroidal correction.
Equation (??) is the low-\(n\) counterpart of the high-\(n\) ballooning reduction: the same toroidal \(\delta W\) has been
reduced, but the surviving one-dimensional object is now a radial functional rather than a field-line
functional.
The key physical consequence is immediate. For the true internal kink, \(n=1\), so
\[\delta W_{n=1} = W_0\hat {\delta W}_T, \tag{28.47}\]
that is, the cylindrical \(m=1\) contribution cancels identically. This is the sharpest possible statement of why the
toroidal calculation matters: the actual ideal internal-kink threshold is determined entirely by the toroidal
sideband problem.
25.6 Evaluation for the rigid core shift
For the ideal internal kink the dominant \(m=1\) eigenfunction is the rigid core shift, Eq. (??). In the sharp-layer
limit,
\[\frac {d\xi _1}{dr} = -\xi _0\,\delta (r-r_1), \tag{28.48}\]
so the sideband source in Eq. (??) is localized at the \(q=1\) radius: \[\mathcal {F}[\xi _1] = -\xi _0 \left [ \frac {d}{dr}\Bigl (r^3T_1(r)\,\delta (r-r_1)\Bigr ) + r^2T_2(r)\,\delta (r-r_1) \right ]. \tag{28.49}\]
Therefore Eqs. (??)–(??) collapse to a localized response problem at the singular surface. In operator form
one may write \[\xi _2(r) = \xi _0\,\mathcal {J}_{r_1}G_2(r,r_1), \tag{28.50}\]
with the source operator \[\mathcal {J}_{r_1} \equiv -\left [\frac {d}{dr_1}\bigl (r_1^3T_1(r_1)\,\cdot \,\bigr )+r_1^2T_2(r_1)\right ], \tag{28.51}\]
and the toroidal correction becomes \[\delta W_{n=1} = -\mathcal {N}\xi _0^2\, \mathcal {J}_{r_1}^{\dagger }G_2(r_1,r_1)\mathcal {J}_{r_1}. \tag{28.52}\]
Equation (??) is the cleanest formal statement of the Bussac calculation. Everything after this point is
explicit but profile-dependent algebra.
Parabolic-profile model.
For the standard Bussac equilibrium with circular cross section, parabolic current profile, and \(|q_0-1|\ll 1\), the
Green-function problem can be evaluated analytically. The result is
\[\delta W = \frac {2\pi ^2 B_0^2}{\mu _0 R_0}\,\xi _0^2\, \frac {3r_1^4}{R_0^2} (1-q_0) \left ( \frac {13}{144}-\hat \beta _{p1}^{\,2} \right ). \tag{28.53}\]
Here \(q_0\equiv q(0)\), \(r_1\) is the \(q=1\) radius, and \(\hat \beta _{p1}\) is the poloidal beta inside that surface, \[\hat \beta _{p1} \equiv \frac {4\mu _0}{r_1^2B_p^2(r_1)}\int _0^{r_1} p(r)\,r\,dr. \tag{28.54}\]
Equation (??) is the classic Bussac toroidal correction in large-aspect-ratio circular geometry.
Bussac’s criterion.
Because \((1-q_0)>0\) when the magnetic axis lies below unity, Eq. (??) implies instability only when
\[\hat \beta _{p1} > \frac {\sqrt {13}}{12}\approx 0.30. \tag{28.55}\]
So in a torus of circular cross section the ideal \(n=1\) internal kink is stable at sufficiently low pressure. This is
the opposite of the naive cylindrical intuition.
Caution
Where the derivation becomes profile specific. Up to Eq. (??) the reduction is
completely generic: start from the appendix toroidal functional, eliminate \(\xi _\parallel \), expand in
low inverse aspect ratio, solve the stable sideband problem, and substitute back.
The final numerical coefficient \(13/144\) in Eq. (??) is not universal. It is the outcome of
evaluating the Green-function expression for the specific circular, parabolic-profile
equilibrium used by Bussac et al. That is why the final criterion changes when one
changes aspect ratio, shear, or shape.
Takeaways
Minimal summary. Starting from the appendix tokamak functional,
-
1.
- eliminate \(\xi _\parallel \) exactly, giving Eq. (??);
-
2.
- specialize to circular, large-aspect-ratio geometry, giving Eq. (??);
-
3.
- expand in low-\(\epsilon \) sidebands about the dominant \(m=1\) harmonic;
-
4.
- recover the cylindrical Rosenbluth top-hat mode as the degenerate leading-order
problem;
-
5.
- solve the stable \(m=2\) sideband equation, Eq. (??);
-
6.
- substitute that minimizing sideband back into \(\delta W\), giving the Green-function form
Eq. (??);
-
7.
- for \(n=1\), retain only the toroidal correction, Eq. (??);
-
8.
- evaluate it for the Bussac parabolic-profile model to obtain Eq. (??) and the
threshold Eq. (??).
That is the clean one-to-one derivation of the Bussac internal-kink correction from the same
toroidal \(\delta W\) used in the ballooning appendix.
Bibliography
William A Newcomb. Hydromagnetic stability of a diffuse linear pinch. Annals of Physics, 10(2):232–267, 1960. doi:10.1016/0003-4916(60)90023-3.
C. E. Kieras and J. A. Tataronis. The shear Alfv\'en continuous spectrum of axisymmetric toroidal equilibria in the large aspect ratio limit. Journal of Plasma Physics, 28(3):395–414, 1982. doi:10.1017/S0022377800000386.
C. Z. Cheng, L. Chen, and M. S. Chance. High-n ideal and resistive shear Alfvén waves in tokamaks. Annals of Physics, 161:21–47, 1985. doi:10.1016/0003-4916(85)90335-5.
C. Z. Cheng and M. S. Chance. Low-n shear Alfvén spectra in axisymmetric toroidal plasmas. Journal of Computational Physics, 71:124–153, 1987. doi:10.1016/0021-9991(87)90023-4.
G. Y. Fu and J. W. Van Dam. Excitation of the toroidicity-induced shear Alfvén eigenmode by fusion alpha particles in an ignited tokamak. Physics of Fluids B, 1(10):1949–1952, 1989. doi:10.1063/1.859057.
K. L. Wong, R. J. Fonck, S. F. Paul, D. R. Roberts, E. D. Fredrickson, R. Nazikian, M. G. Bell, N. Bretz, C. E. Bush, Z. Chang, et al. Excitation of toroidal Alfvén eigenmodes in tftr. Physical Review Letters, 66:1874–1877, 1991. doi:10.1103/PhysRevLett.66.1874.
W. W. Heidbrink, E. J. Strait, E. Doyle, G. Sager, and R. T. Snider. An investigation of beam driven Alfv\'en instabilities in the DIII-D tokamak. Nuclear Fusion, 31(9):1635–1648, 1991. doi:10.1088/0029-5515/31/9/002.
A. D. Turnbull, E. J. Strait, W. W. Heidbrink, M. S. Chu, H. H. Duong, J. M. Greene, L. L. Lao, T. S. Taylor, and S. J. Thompson. Global Alfv\'en modes: Theory and experiment. Physics of Fluids B, 5(7):2546–2553, 1993. doi:10.1063/1.860742.
H. L. Berk, J. W. Van Dam, Z. Guo, and D. M. Lindberg. Continuum damping of low-n toroidicity-induced shear Alfvén eigenmodes. Technical report, Institute for Fusion Studies, The University of Texas at Austin, 1991.
R. R. Mett and S. M. Mahajan. Kinetic theory of toroidicity-induced Alfvén eigenmodes. Physics of Fluids B, 4(9):2885–2893, 1992. doi:10.1063/1.860459.
A. Fasoli, C. Gormenzano, H. L. Berk, B. N. Breizman, S. Briguglio, D. S. Darrow, et al. Progress in the iter physics basis—chapter 5: Physics of energetic ions. Nuclear Fusion, 47:S264–S284, 2007. doi:10.1088/0029-5515/47/6/S05.
P. Lauber. Super-thermal particles in hot plasmas—kinetic models, numerical solution strategies, and comparison to tokamak experiments. Physics Reports, 533:33–68, 2013. doi:10.1016/j.physrep.2013.07.001.
L. Chen and F. Zonca. Physics of alfv\'en waves and energetic particles in burning plasmas. Reviews of Modern Physics, 88:015008, 2016. doi:10.1103/RevModPhys.88.015008.
W. W. Heidbrink. Basic physics of Alfv\'en instabilities driven by energetic particles in toroidally confined plasmas. Physics of Plasmas, 15(5):055501, 2008. doi:10.1063/1.2838239.
W. W. Heidbrink and R. B. White. Mechanisms of energetic-particle transport in magnetically confined plasmas. Physics of Plasmas, 27(3):030901, 2020. doi:10.1063/1.5136237.
M. N. Rosenbluth, H. L. Berk, J. W. Van Dam, and D. M. Lindberg. Continuum damping of high-mode-number toroidal Alfv\'en waves. Physical Review Letters, 68(5):596–599, 1992. doi:10.1103/PhysRevLett.68.596.
Fulvio Zonca and Liu Chen. Theory of continuum damping of toroidal Alfv\'en eigenmodes in finite-$ $ tokamaks. Physics of Fluids B: Plasma Physics, 5(10):3668–3690, 1993. doi:10.1063/1.860839.
R. Betti and J. P. Freidberg. Ellipticity induced Alfvén eigenmodes. Physics of Fluids B, 3(8):1865–1871, 1991. doi:10.1063/1.859655.
W. W. Heidbrink, E. J. Strait, M. S. Chu, and A. D. Turnbull. Observation of beta-induced Alfv\'en eigenmodes in the DIII-D tokamak. Physical Review Letters, 71(6):855–858, 1993. doi:10.1103/PhysRevLett.71.855.
W. W. Heidbrink, E. Ruskov, E. M. Carolipio, J. Fang, M. A. Van Zeeland, and R. A. James. What is the ``beta-induced Alfv\'en eigenmode?''. Physics of Plasmas, 6(4):1147–1161, 1999. doi:10.1063/1.873359.
G. J. Kramer, M. Saigusa, T. Ozeki, L. Chen, C. Z. Cheng, Y. Kusama, K. Tobita, M. Nemoto, and the JT-60 Team. Noncircular triangularity and ellipticity-induced Alfvén eigenmodes observed in jt-60u. Physical Review Letters, 80:2594–2597, 1998. doi:10.1103/PhysRevLett.80.2594.
E. M. Edlund, M. Porkolab, G. J. Kramer, L. Lin, Y. Lin, N. Tsujii, and S. J. Wukitch. Experimental study of reversed shear Alfvén eigenmodes during the current ramp in the alcator c-mod tokamak. Technical report, Princeton Plasma Physics Laboratory, 2010. PPPL-4544.
N. N. Gorelenkov, H. L. Berk, E. D. Fredrickson, and S. E. Sharapov. Predictions and observations of low-shear beta-induced shear alfv\'en–acoustic eigenmodes in toroidal plasmas. Physics Letters A, 370(1):70–75, 2007. doi:10.1016/j.physleta.2007.05.113.
E. M. Edlund, M. Porkolab, G. J. Kramer, L. Lin, N. Tsujii, and S. J. Wukitch. Observation of reversed shear Alfvén eigenmodes during the sawtooth cycle in alcator c-mod. Physical Review Letters, 102:165003, 2009. doi:10.1103/PhysRevLett.102.165003.
G. J. Kramer, N. N. Gorelenkov, C. Z. Cheng, and R. Nazikian. Finite pressure effects on reversed shear Alfvén eigenmodes. Technical report, Princeton Plasma Physics Laboratory, 2004. PPPL-4002.
W. W. Heidbrink, M. E. Austin, D. A. Spong, B. J. Tobias, and M. A. Van Zeeland. Measurements of the eigenfunction of reversed shear Alfv\'en eigenmodes that sweep downward in frequency. Physics of Plasmas, 20(8):082504, 2013. doi:10.1063/1.4817950.
W. W. Heidbrink, M. A. Van Zeeland, M. E. Austin, K. H. Burrell, N. N. Gorelenkov, G. J. Kramer, Y. Luo, M. A. Makowski, G. R. McKee, C. Muscatello, R. Nazikian, E. Ruskov, W. M. Solomon, R. B. White, and Y. Zhu. Central flattening of the fast-ion profile in reversed-shear DIII-D discharges. Nuclear Fusion, 48(8):084001, 2008. doi:10.1088/0029-5515/48/8/084001.
Marshall N Rosenbluth, R Y Dagazian, and P H Rutherford. Nonlinear properties of the internal m = 1 kink instability in the cylindrical tokamak. The Physics of Fluids, 16(11):1894–1902, 1973. doi:10.1063/1.1694231.