Lecture 27 Bussac: The Toroidal \(n=1\) Internal Kink

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Lecture 27
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.

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 Bussac et al. (1975). 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.

27.1 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{27.1}

we therefore start from Appendix Eq. (I.44), namely

\delta W = \pi \int J\,d\Psi \,d\theta \, \Biggl [ \frac {B^2}{R^2B_p^2}|k_\parallel X|^2 + \frac {R^2}{J^2}\left |\frac {\partial U}{\partial \theta }-I\frac {\partial }{\partial \Psi }\left (\frac {JX}{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{27.2}

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{27.3}

With \(\chi =\theta \), the appendix metric gives

d\Psi = RB_p\,dr, \qquad JB_p=r, \qquad J=\frac {r}{B_p}, \qquad dV = rR\,dr\,d\theta \,d\phi . \tag{27.4}

The appendix quantity \(\nu \) becomes the safety factor,

\nu = \frac {IJ}{R^2} = \frac {rB_\phi }{RB_p}=q(r), \tag{27.5}

so for a single toroidal harmonic \(e^{in\phi }\),

\B \cdot \nabla \longrightarrow \frac {1}{J}\left (\frac {\partial }{\partial \theta }+inq\right ), \qquad ik_\parallel = \frac {1}{JB}\left (\frac {\partial }{\partial \theta }+inq\right ). \tag{27.6}

Substituting Eqs. (27.4)–(27.6) into Eq. (27.2) 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{27.8}

Nothing has been approximated beyond axisymmetry and the circular-coordinate specialization. The low-\(n\) calculation begins from Eq. (27.8).

27.2 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{27.9}

Define the field-line-pitch mismatch

D_m(r)\equiv q(r)-m. \tag{27.10}

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{27.11}

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{27.12}

Why toroidicity generates only neighboring sidebands at first order. Expand the toroidal coefficients in Eq. (27.8),

\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{27.13}

Because

\cos \theta \,e^{-im\theta } = \frac 12 e^{-i(m-1)\theta }+\frac 12 e^{-i(m+1)\theta }, \tag{27.16}

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{27.17}

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{27.18}

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.

Block structure of the quadratic form. If one inserts Eqs. (27.17)–(27.18) into Eq. (27.8), 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{27.19}

where

Y_m(r)\equiv \begin {pmatrix}X_m(r)\\ U_m(r)\end {pmatrix}. \tag{27.20}

Equation (27.19) already contains the whole Bussac logic. 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. (27.8). 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.

27.3 Recover the cylindrical Rosenbluth problem from the same functional

If one drops all \(O(\epsilon )\) toroidal pieces in Eq. (27.8), 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{27.21}

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{27.22}

Here \(\xi _m\) is the radial displacement harmonic defined by

X_m = R_0B_p\,\xi _m + O(\epsilon ). \tag{27.24}

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{27.25}

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{27.26}

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{27.27}

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{27.28}

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{27.29}

Integrating once and imposing

\xi \to \xi _0 \quad (x\to -\infty ), \qquad \xi \to 0 \quad (x\to +\infty ), \tag{27.30}

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{27.31}

As the inertial width tends to zero, Eq. (27.31) reduces back to Eq. (27.26). So the cylindrical Rosenbluth solution is already contained in the same toroidal functional; one simply suppresses the toroidal sidebands.

27.4 Toroidal sideband elimination

The practical low-\(\epsilon \) calculation is obtained by substituting the sideband ansatz, Eqs. (27.17)–(27.18), into Eq. (27.8), 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{27.32}

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{27.33}

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{27.34}

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{27.35}

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{27.37}

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{27.38}

Equation (27.34) 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.

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{27.39}

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{27.40}

Varying Eq. (27.39) with respect to \(\xi _2^*\) gives Eq. (27.34). 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{27.41}

Substituting this back into Eq. (27.39) 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{27.42}

Equation (27.42) 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{27.43}

Then

\xi _2(r) = -\int _0^a G_2(r,r')\,\mathcal {F}[\xi _1](r')\,dr', \tag{27.44}

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{27.45}

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. (27.45).
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 \(\delta W\).

27.5 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{27.46}

where \(\hat {\delta W}_c\) is the normalized cylindrical contribution and \(\hat {\delta W}_T\) is the sideband-generated toroidal correction. Equation (27.46) 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{27.47}

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.

27.6 Evaluation for the rigid core shift

For the ideal internal kink the dominant \(m=1\) eigenfunction is the rigid core shift, Eq. (27.26). In the sharp-layer limit,

\frac {d\xi _1}{dr} = -\xi _0\,\delta (r-r_1), \tag{27.48}

so the sideband source in Eq. (27.40) 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{27.49}

Therefore Eqs. (27.44)–(27.45) 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{27.50}

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{27.51}

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{27.52}

Equation (27.52) 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{27.53}

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{27.54}

Equation (27.53) 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. (27.53) implies instability only when

\hat \beta _{p1} > \frac {\sqrt {13}}{12}\approx 0.30. \tag{27.55}

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. (27.52) 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. (27.53) 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. (27.2);
2.
specialize to circular, large-aspect-ratio geometry, giving Eq. (27.8);
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. (27.34);
6.
substitute that minimizing sideband back into \(\delta W\), giving the Green-function form Eq. (27.52);
7.
for \(n=1\), retain only the toroidal correction, Eq. (27.47);
8.
evaluate it for the Bussac parabolic-profile model to obtain Eq. (27.53) and the threshold Eq. (27.55).

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

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.

M. N. Bussac, R. Pellat, D. Edery, and J. L. Soule. Internal kink modes in toroidal plasmas with circular cross sections. Physical Review Letters, 35(24):1638–1641, 1975. doi:10.1103/physrevlett.35.1638.