← Previous LectureContentsNext Lecture →

Lecture 26
Rosenbluth, Dagazian and Rutherford: The cylindrical \(m=1\) Internal Kink

Overview

After the local pressure-driven tests have been passed, the next question is the global fixed-boundary problem. In a large-aspect-ratio tokamak the \(m\ge 2\) families are relatively benign at low \(\beta \), but the \(m=1\) family is exceptional: its leading restoring term vanishes, a rigid core shift becomes possible, and the classic Rosenbluth and Bussac analyses determine the sign of the internal-kink energy.

Historical Perspective

Large-aspect-ratio tokamak theory translated Newcomb’s general screw-pinch language into the experimentally meaningful language of the safety factor \(q(r)\) Shafranov (1966); Newcomb (1960). The \(m=1\) internal kink later became central to sawtooth theory, and the Rosenbluth–Dagazian–Rutherford and Bussac calculations remain the classic route from variational formalism to a concrete instability threshold Rosenbluth et al. (1973); Bussac et al. (1975).

26.1 Large-aspect-ratio reduction

Ordering. Take a tokamak of major radius \(R_0\) and minor radius \(a\) with

\[\epsilon \equiv \frac {r}{R_0}\ll 1, \qquad \frac {B_\theta }{B_z}\sim \epsilon , \qquad \beta \sim \epsilon ^2,\]
so that \(B_z\simeq B_0\) to leading order. Write the axial wavenumber as
\[k=-\frac {n}{R_0}\]
and use the cylindrical safety factor
\[q(r)=\frac {rB_z}{R_0 B_\theta }. \tag{26.3}\]
Then
\[B_\theta (r)\simeq \frac {rB_0}{R_0 q(r)}.\]

The field-line factors. With (26.3),

\[F(r) = \frac {mB_\theta }{r}+kB_z = \frac {B_0}{R_0}\left (\frac {m}{q(r)}-n\right ), \tag{26.5}\]
\[F^\dagger (r) = kB_z-\frac {mB_\theta }{r} = -\frac {B_0}{R_0}\left (n+\frac {m}{q(r)}\right ). \tag{26.6}\]
Also,
\[k_0^2r^2=m^2+\frac {n^2r^2}{R_0^2}=m^2+O(\epsilon ^2).\]

Large-aspect-ratio forms of \(f\) and \(g\). To leading order Eqs. (23.83) and (23.91) become

\[\begin{aligned}f(r) &= \frac {rF^2}{k_0^2} \approx \frac {B_0^2 r^3}{m^2R_0^2}\left (n-\frac {m}{q(r)}\right )^2, \\ g(r) &\approx \frac {B_0^2 r}{m^2R_0^2}(m^2-1) \left (n-\frac {m}{q(r)}\right )^2 + \frac {2n^2r^2}{m^2R_0^2}\muo p'(r).\end{aligned} \tag{26.8}\]

The first term in (26.9) is line bending and geometry; the second is the pressure drive already encoded locally by Suydam.

Large-aspect-ratio forms of \(f\) and \(g\) to next order. Let \[ \Delta (r)\equiv n-\frac {m}{q(r)}, \qquad \Sigma (r)\equiv n+\frac {m}{q(r)}, \qquad \delta (r)\equiv \frac {n^2r^2}{m^2R_0^2}\ll 1. \] Using \[ F=-\frac {B_0}{R_0}\Delta , \qquad F^\dagger =-\frac {B_0}{R_0}\Sigma , \qquad k_0^2=\frac {m^2}{r^2}\left (1+\delta \right ), \] one finds

\[\begin{aligned}f(r) &= \frac {B_0^2r^3}{m^2R_0^2}\Delta ^2 \left [ 1-\frac {n^2r^2}{m^2R_0^2} +O\!\left (\frac {r^4}{R_0^4}\right ) \right ], \\ g(r) &= \frac {B_0^2r}{m^2R_0^2}(m^2-1)\Delta ^2 +\frac {2n^2r^2}{m^2R_0^2}\muo p'(r) +\frac {n^2B_0^2r^3}{m^4R_0^4} \left ( 3n^2-\frac {2nm}{q(r)}-\frac {m^2}{q(r)^2} \right ) +O\!\left (\frac {r^4}{R_0^4}\muo p',\frac {r^5B_0^2}{R_0^6}\right ).\end{aligned} \tag{26.11}\]

For the internal kink with \(m=n=1\), this reduces to

\[\begin{aligned}f_{11}(r) &= \frac {B_0^2r^3}{R_0^2} \left (1-\frac {1}{q}\right )^2 \left [ 1-\frac {r^2}{R_0^2} +O\!\left (\frac {r^4}{R_0^4}\right ) \right ], \\ g_{11}(r) &= \frac {2r^2}{R_0^2}\muo p'(r) + \frac {B_0^2r^3}{R_0^4} \left (1-\frac {1}{q(r)}\right ) \left (3+\frac {1}{q(r)}\right ) +O\!\left (\frac {r^4}{R_0^4}\muo p',\frac {r^5B_0^2}{R_0^6}\right ).\end{aligned} \tag{26.13}\]

26.2 Fixed-boundary internal modes

Why \(\xi _a=0\) is imposed here. Throughout this lecture we stay in Newcomb’s fixed-boundary sector,

\[\xi _a\equiv \xi (a)=0.\]
So every instability discussed below is internal by construction. The edge is held fixed while the plasma tries to reorganize inside. If the minimizing mode requires \(\xi _a\neq 0\), that mode belongs to the external problem treated in the next lecture, not to the present one.

Low-\(\beta \) internal stability for \(m\ge 2\). If the pressure term is small enough that the local Suydam test has already been passed, then

\[\frac {\delta W}{2\pi ^2 R_0/\muo } \approx \frac {B_0^2}{R_0^2} \int _0^a \left (n-\frac {m}{q}\right )^2 \left [ \frac {r^3}{m^2}|\xi '|^2+\frac {m^2-1}{m^2}r|\xi |^2 \right ]dr. \tag{26.15}\]
For \(m\ge 2\) the integrand is manifestly positive. Thus the leading-order large-aspect-ratio tokamak is internally stable to those families.

The \(m=1\) internal kink. For \(m=1\) the explicit \(g\) term vanishes at leading order and a nearly rigid core displacement becomes possible. This is the exceptional current-driven family isolated by Rosenbluth, Dagazian, and Rutherford. If

\[q_0<1<q_a,\]
there is a resonant surface \(r_s<a\) satisfying \(q(r_s)=1\). A highly effective outer trial function is
\[\xi (r)\approx \begin {cases} \xi _0, & 0<r<r_s,\\[0.3em] 0, & r>r_s, \end {cases}\]
with a narrow transition layer. Because it vanishes for \(r>r_s\), it already satisfies the fixed-edge condition \(\xi _a=0\). This top-hat structure is the cylindrical Rosenbluth–Dagazian–Rutherford result: in their linear theory the minimizing singular solution is constant for \(r<r_s\) and zero for \(r>r_s\), and the exact resolved inner-layer profile tends to the same step in the ideal limit Rosenbluth et al. (1973). Since the leading \(O(\epsilon ^2)\) functional is degenerate, the next-order contribution sets the sign of the energy:
\[\delta W_{4,cyl} = \frac {2 \pi ^2 B_z^2}{\mu _0 R_0} \xi _0^2 n^2 \int _0^{r_s} \left [ r\left (\dd {\beta }{r}\right ) + \frac {r^2}{R_0^2} \left (1-\frac {1}{q}\right )\left (3+\frac {1}{q}\right ) \right ]r\,dr, \tag{26.18}\]
Equation (26.18) is the large-aspect-ratio cylindrical contribution obtained by Rosenbluth, Dagazian, and Rutherford Rosenbluth et al. (1973). Toroidal corrections, evaluated in the next lecture) from the \(R=R_0(1 + \frac {r}{R_0}\cos \theta )\) variation of the equilibrium field enter at the same order and were evaluated by Bussac et al. Bussac et al. (1975) in a fashion similar to Mercier:
\[\delta W_{4,tor} = \frac {3 n^2 r_1^4}{R_0^2} \frac {2 \pi ^2 B_z^2}{\mu _0 R_0} \xi _0^2 (1 - q_0) \left ( \frac {13}{144} - \hat {\beta }_p^2 \right )\]
where \(r_1\equiv r_s\) is the \(q=1\) radius and \(\hat {\beta }_p\) is the poloidal \(\beta \) inside that surface.

As shown in the next lecture, after including toroidal geometry the \(1/R\) variation of \(B_\phi \) couples the rigid \(m=1\) shift to \(m=0\) and \(m=2\) sidebands, and Bussac et al. showed that after those sidebands are eliminated variationally the first surviving contribution is

\[\delta W = \left (1 - \frac {1}{n^2}\right ) \delta W_{4,cyl} + \delta W_{4,tor}\]
For the true internal kink, \(n=1\), the cylindrical prefactor \(\left (1-\frac {1}{n^2}\right )\) vanishes, so the overall stability is set by the toroidal term.

Tutorial

Two classic papers are being compressed into one formula. Rosenbluth, Dagazian, and Rutherford analyze the cylindrical \(m=1\) mode and explain why the minimizing displacement is a rigid core shift with a narrow singular layer at \(q=1\) Rosenbluth et al. (1973). Bussac, Pellat, Edery, and Soule then redo the problem in toroidal geometry, expand in inverse aspect ratio, track the sidebands generated by the \(1/R\) variation of the toroidal field, and derive the toroidal correction that survives for the true \(n=1\) internal kink Bussac et al. (1975). The logic below follows that sequence.

The \(O(\epsilon ^2)\) cylindrical functional has no \(|\xi |^2\) cost. Setting \(m=n=1\) in (26.15) gives

\[\frac {\delta W_2}{2\pi ^2 R_0/\muo } \approx \frac {B_0^2}{R_0^2} \int _0^a \left (1-\frac {1}{q}\right )^2 r^3 |\xi '|^2\,dr. \tag{26.21}\]
The explicit \(|\xi |^2\) term vanishes because \(m^2-1=0\). Thus only the thin matching region near \(q=1\) costs energy. This is the cylindrical degeneracy behind the Rosenbluth–Dagazian–Rutherford step function Rosenbluth et al. (1973).

A simple minimizing sequence. Take \(q(r_s)=1\) and define

\[\xi _\delta (r)= \begin {cases} \xi _0, & r<r_s-\delta /2,\\[0.3em] \xi _0\left (\dfrac {r_s+\delta /2-r}{\delta }\right ), & |r-r_s|<\delta /2,\\[0.8em] 0, & r>r_s+\delta /2. \end {cases} \tag{26.22}\]
Near \(r_s\),
\[1-\frac {1}{q(r)} \approx \left .\dd {}{r}\left (1-\frac {1}{q}\right )\right |_{r_s}(r-r_s) = \frac {s_1}{r_s}(r-r_s), \qquad s_1\equiv r_s q'(r_s),\]
so in the layer,
\[\begin{aligned}\frac {\delta W_2}{2\pi ^2 R_0/\muo } &\approx \frac {B_0^2}{R_0^2} \int _{-\delta /2}^{\delta /2} \left (\frac {s_1 x}{r_s}\right )^2 r_s^3\left (\frac {\xi _0}{\delta }\right )^2 dx \nonumber \\ &= \frac {B_0^2}{R_0^2} \frac {s_1^2 r_s\xi _0^2}{12}\,\delta \longrightarrow 0 \qquad (\delta \to 0).\end{aligned} \tag{26.24}\]

So the top-hat profile is not merely heuristic: it is the minimizing sequence of the \(O(\epsilon ^2)\) problem, which is why Rosenbluth et al. can take the outer solution to be constant inside \(r_s\) and zero outside Rosenbluth et al. (1973).

The exact resolved jump from the singular layer. A triangular ramp is enough to prove (26.24), but the ideal problem by itself has only an infimum, not a smooth minimizer, so one must retain the small inertial term that resolves the singular layer. Rosenbluth–Dagazian–Rutherford then obtain the exact layer profile. Let

\[x\equiv r-r_s, \qquad k\cdot B \approx (k\cdot B)'_s x,\]
so their layer equation reduces to
\[\frac {d}{dx} \left [ \left ( 4\pi \rho _s\gamma ^2+(k\cdot B)_s'^2 x^2 \right ) \frac {d\xi }{dx} \right ] =0. \tag{26.26}\]
Integrating once gives
\[\frac {d\xi }{dx} = \frac {C}{4\pi \rho _s\gamma ^2+(k\cdot B)_s'^2 x^2},\]
and imposing
\[\xi \to \xi _s \quad (x\to -\infty ), \qquad \xi \to 0 \quad (x\to +\infty ),\]
fixes the constants. The result is
\[\xi (x) = \frac {\xi _s}{2} \left [ 1- \frac {2}{\pi } \tan ^{-1} \left ( \frac {(k\cdot B)'_s x}{\gamma (4\pi \rho _s)^{1/2}} \right ) \right ]. \tag{26.29}\]
Equivalently, with
\[\Delta \equiv \frac {\gamma (4\pi \rho _s)^{1/2}}{(k\cdot B)'_s},\]
Eq. (26.29) is the arctangent-smoothed top hat. As \(\Delta \to 0\) it reduces to the same rigid core shift Rosenbluth et al. (1973).

Takeaways

The cylindrical \(m=1\) internal kink is the cleanest example of an ideal-MHD mode whose minimizing sequence is almost discontinuous. The outer-region functional drives the core toward a rigid displacement inside the \(q=1\) surface and near-zero displacement outside. The singular layer does not change that geometric picture; it resolves it. Once inertia is retained, the sharp top hat is replaced by the arctangent profile (26.29), which smooths the jump across a layer whose width shrinks with the growth rate.

Bibliography

    V. D. Shafranov. Plasma equilibrium in a magnetic field. In M. A. Leontovich, editor, Reviews of Plasma Physics, volume 2, pages 103–151. Consultants Bureau, New York, 1966. Classic long review; includes the large-aspect-ratio treatment used for what is commonly called the Shafranov shift.

    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.

    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.