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Lecture 39
The Reversed-Field Pinch

Overview

The reversed-field pinch is one of the cleanest examples of a configuration that looks rather respectable in ideal MHD and rather unforgiving in resistive MHD. In cylindrical theory one can construct reversed-field diffuse pinches that are ideally stable at substantial \(\beta \), provided the pitch profile is shaped carefully, the conducting wall is close, and the total axial flux does not reverse. The practical difficulty is tearing. Because \(q(r)\) becomes small and then changes sign near the edge, the RFP places many low-\(m\) rational surfaces into a narrow radial interval, and the edge current roll-off that makes the configuration realistic also reintroduces the tearing drive.

Historical Perspective

Robinson’s 1971 paper gives the clean ideal-MHD statement of the diffuse-pinch problem. It shows that if the pitch profile develops a minimum, the configuration is generically unstable, and it explains why a stable high-\(\beta \) diffuse pinch with a vacuum region is naturally of reversed-field type Robinson (1971). Robinson’s 1978 paper then adds the resistive correction. A Taylor-like or Bessel-like core is helpful, but not sufficient; the outer current profile must also be shaped so that both the \(m=0\) and \(m=1\) tearing branches are stabilized by the edge structure and the nearby conducting wall Robinson (1978). The modern MST and simulation literature then shows what happens when that resistive stability is not achieved: a broad tearing spectrum, nonlinear \(m=0\) participation, a self-consistent dynamo, and stochastic transport Bodin and Newton (1980); Sovinec (1995). A useful modern bridge from those classic papers to the present experimental picture is the review by Marrelli et al., which ties together quiet periods, profile control, helical states, and the confinement consequences of tearing in the contemporary RFP Marrelli et al. (2021).

39.1 Why the RFP is a special MHD problem

Pitch, safety factor, and parallel current. For a cylindrical equilibrium one may characterize the field either by the pitch

\[P(r) \equiv \frac {r B_z}{B_\theta } = q(r)R, \tag{E.1}\]
or, in force-free language, by the field-aligned current parameter
\[\lambda (r) \equiv \mu _0\frac {\J \cdot \B }{B^2}. \tag{E.2}\]
When pressure is negligible and \(\J \times \B =0\), these are equivalent ways of specifying the equilibrium, since
\[\frac {dB_z}{dr} = -\lambda B_\theta , \qquad \frac {1}{r}\frac {d}{dr}\left (r B_\theta \right )=\lambda B_z. \tag{E.3}\]
A flat-core \(\lambda \) profile therefore generates the familiar Taylor or Bessel-function core, while the edge roll-off in \(\lambda \) is what makes the diffuse pinch realistic.

Why rational surfaces crowd the outer plasma. For a helical perturbation \(\propto e^{i(m\theta +kz)}\), the resonance condition is

\[F(r) \equiv \frac {m B_\theta }{r}+k B_z = 0. \tag{E.4}\]
Equivalently,
\[q(r_s) = -\frac {m}{n}, \qquad k=-\frac {n}{R},\]
up to the sign convention for the Fourier phase. What matters geometrically is that in an RFP the safety factor decreases to small magnitude and then changes sign near the reversal surface. The result is an accumulation of low-order \(m=1\) rational surfaces in the outer plasma. In cylindrical language, one gets a dense ladder of resonant surfaces approaching the reversal region. That is why tearing is not a small correction in the RFP; it is the natural resistive-MHD problem of the configuration Bodin and Newton (1980); Sovinec (1995).

A useful one-sentence summary. The tokamak has a few low-order rational surfaces that must be managed carefully. The RFP has a dense set of low-order rational surfaces that are difficult to avoid and even harder to ignore.

39.2 Ideal MHD: how well can the RFP do?

A minimum in pitch is bad news. Robinson’s ideal diffuse-pinch analysis is still the cleanest way to organize the problem. The qualitative result is simple and powerful: if the radial variation of the pitch develops a minimum, the configuration is unstable to a current-driven mode Robinson (1971). In RFP language this explains why a stable vacuum-bounded diffuse pinch is naturally of reversed-field type. Outside the plasma one has approximately vacuum behavior, so \(B_\theta \sim 1/r\) while the axial field changes slowly. The corresponding pitch tends to rise like \(P\sim r^2\). A non-reversed diffuse pinch whose pitch was already falling in the plasma would therefore have to turn around and create a minimum before joining the vacuum. Reversal of \(B_z\) is the natural way to avoid that minimum.

Reversed field, but not reversed total axial flux. The subtle point in Robinson’s 1971 paper is that one wants the field to reverse near the edge, not the total axial flux. The stable examples keep

\[\Phi _z \propto \int _0^a B_z(r)\,r\,dr > 0, \tag{E.6}\]
while still allowing \(B_z(a)<0\) near the boundary Robinson (1971). This is a good ideal-MHD formulation of reversal: the outer layer may reverse to avoid the destabilizing pitch minimum, but the global axial flux should remain of one sign.

Local and global criteria both matter. At finite pressure the RFP must satisfy the local Suydam criterion as well as the global Newcomb problem. The local statement controls pressure-gradient drive, while the global pitch variation controls the large-scale current-driven instability Suydam (1958); Newcomb (1960). Robinson’s 1971 paper is best read as the combination of those two ideas: the pressure gradient must be tolerable locally, and the pitch profile must avoid the global geometry that makes an \(m=1\) mode possible Robinson (1971).

Newcomb’s variables in the cylindrical RFP problem. It is worth laying out the reduced variables explicitly, because they make the logic of the ideal section much clearer. For a helical perturbation \(\propto e^{i(m\theta +kz)}\) one defines

\[F(r)\equiv \frac {mB_\theta }{r}+kB_z, \qquad k_0^2(r)\equiv \frac {m^2}{r^2}+k^2, \tag{E.7}\]
and uses the radial displacement amplitude \(\xi (r)\) as the dynamical variable. After the algebraic variables are eliminated, Newcomb’s ideal-MHD reduction gives
\[\delta W = \int _0^a \left [f(r)\,|\xi '|^2 + g(r)\,|\xi |^2\right ]dr, \qquad f(r)=\frac {rF^2}{k_0^2}, \tag{E.8}\]
with Euler–Lagrange equation
\[\frac {d}{dr}\!\left (f\,\frac {d\xi }{dr}\right )-g\,\xi =0. \tag{E.9}\]
The rational surface is the radius \(r_s\) where \(F(r_s)=0\). Because \(f\propto F^2\), field-line bending vanishes at \(r_s\), which is exactly why both the global Newcomb problem and the local pressure-driven problem focus so sharply on rational surfaces.

Why reversed plasmas are often good from the Suydam point of view. In pitch language, with

\[P(r)=\frac {rB_z}{B_\theta }=q(r)R, \tag{E.10}\]
the cylindrical Suydam condition may be written schematically as
\[\frac {rB_z^2}{8\mu _0}\left (\frac {P'}{P}\right )^2 + \frac {dp}{dr} > 0, \tag{E.11}\]
up to conventional normalization choices Suydam (1958); Robinson (1971). Since \(dp/dr<0\) in an ordinary plasma column, magnetic shear is the stabilizing term. This is one reason the RFP is more subtle than the caricature “low-\(q\) therefore bad.” Near reversal, \(q\) is small and varies rapidly, so the shear term is often large. In that limited but important sense, reversed plasmas are frequently good from the Suydam point of view even while they remain vulnerable to global or resistive tearing.

Suydam, yes; Mercier mostly not in Robinson’s diffuse-pinch papers. If one asks specifically about the Robinson diffuse-pinch papers, the natural pairing is Suydam plus the global Newcomb problem, not Suydam versus Mercier. Of course the Mercier correction is of order \(1-q_a^2\) and so when \(q_a\ll 1 \) it becomes negligible. The 1978 paper’s finite-\(\beta \) extension explicitly compares the Suydam and Newcomb limits for the same cylindrical reversed-field diffuse pinch Robinson (1978). Mercier is the toroidal completion of the same singular local problem once curvature and magnetic-well terms are retained, so it enters more naturally in later toroidal high-\(\beta \) discussions than in Robinson’s cylindrical construction Marrelli et al. (2021). So for this appendix the clean statement is: the RFP’s large shear makes the cylindrical Suydam test surprisingly favorable, whereas Mercier is better treated as the toroidal afterword rather than the main actor.

The \(\beta \) lesson. The ideal calculation also gives a useful bound: the beta measured against the field produced by the plasma current must remain below unity, and the practical optimum from Robinson’s numerical study is of order one-half Robinson (1971). More optimistically, the same paper shows that stable reversed-field diffuse pinches can be ideally stable at central \(\beta \) values up to about \(17\%\), corresponding to average \(\beta \) of order \(35\%\) Robinson (1971). So the fairest ideal-MHD verdict is not “the RFP is hopeless,” but rather “the RFP can be an honest ideal-MHD configuration if the pitch profile, pressure gradient, and wall are all used correctly.”

39.3 Resistive tearing in \(\lambda (r)\) language

Outer region: the same logic as the tearing lecture. The resistive tearing calculation follows exactly the structure used in the tearing lecture. Outside the thin resistive layer one solves the ideal outer problem and summarizes its behavior by

\[\Delta ' \equiv \left [\frac {\psi '}{\psi }\right ]_{r_s^-}^{r_s^+}. \tag{E.12}\]
In the constant-\(\psi \) regime the layer calculation then yields the familiar FKR statement that instability requires \(\Delta '>0\), with growth rate scaling like \(\gamma \propto \eta ^{3/5}(\Delta ')^{4/5}\) Furth et al. (1963); Coppi et al. (1966).

Robinson’s outer equation. For a pressureless force-free equilibrium,

\[\J = \frac {\lambda (r)}{\mu _0}\,\B ,\]
and a helical perturbation \(\propto e^{i(m\theta +kz)}\), Robinson introduces the scaled radial-flux function
\[\psi (r) \equiv \frac {r^{3/2} b_r}{\sqrt {m^2+k^2 r^2}}, \tag{E.14}\]
so that the outer ideal equation becomes
\[\frac {d^2\psi }{dr^2} + A(r)\psi = 0, \tag{E.15}\]
with
\[\begin{aligned}-A(r) ={}& \frac {m^2+k^2 r^2}{r^2} - \lambda ^2 - \frac {2mk\lambda }{m^2+k^2 r^2} - \frac {m^4 + 10m^2 k^2 r^2 - 3k^4 r^4}{4r^2\left (m^2+k^2 r^2\right )^2} \notag \\ &\quad + \lambda '(r) \frac {m B_z - k r B_\theta }{m B_\theta + k r B_z}.\end{aligned} \tag{E.16}\]

The last term is the decisive one. Its denominator is just \(rF(r)\) with \(F\) defined in Eq. (E.4), so the singular current-gradient drive is proportional to \(\lambda '(r_s)/F\). A constant-\(\lambda \) core removes that drive completely, while any edge roll-off in \(\lambda \) reintroduces it.

Tutorial

Robinson’s comparison-equation trick, written in the language of the tearing lecture.

The outer equation (E.15) is singular at the rational surface \(r=r_s\), exactly because the current-gradient term contains \(1/F\). Robinson’s idea is to expand the coefficient near resonance by writing

\[A(r) = \frac {G}{x} + G_1 + \cdots , \qquad x\equiv r-r_s,\]
and then solve a comparison equation
\[u'' + Q(x)u = 0, \qquad Q(x)=\frac {G}{x}+G_1+R(x),\]
whose two linearly independent solutions are known explicitly. In the notation used by Robinson one may take
\[u_1(x)=\frac {x}{1+ax+bx^2}, \qquad u_2(x)=u_1(x)\left [-\frac {1}{x}+2a\ln |x|+(a^2+2b)x+abx^2+\frac {b^2x^3}{3}\right ],\]
with
\[a=\frac {G}{2}, \qquad b=\frac {G^2+G_1}{6}.\]
The outer solution is then written as
\[\psi (x)=\alpha (x)u_1(x)+\beta (x)u_2(x),\]
and one imposes the auxiliary condition
\[\alpha ' u_1 + \beta ' u_2 = 0.\]
This converts the second-order singular equation into the regular first-order system
\[\alpha ' = \frac {u_2\,(Q-A)\psi }{W_0}, \qquad \beta ' = -\frac {u_1\,(Q-A)\psi }{W_0}, \tag{E.23}\]
where \(W_0=u_1u_2'-u_2u_1'\) is the Wronskian. Because \(Q-A=O(x)\), the coefficients \(\alpha \) and \(\beta \) remain regular through the singularity.

This is the Robinson version of the same outer matching used in the tearing lecture. One constructs the regular solution from the axis, the “small” solution on the outer interval, and then evaluates the logarithmic-derivative jump. In Sovinec’s notation, with the wall normalized to \(r=1\) and the small solution normalized to unit slope at \(r_s\), the result is

\[\Delta ' = -\frac {y_0(1)}{y_1(1)\,y_0(r_s)}, \tag{E.24}\]
which is just the RFP realization of the same \(\Delta '\) logic used throughout the tearing lecture Robinson (1978); Sovinec (1995).

What constant \(\lambda \) buys you. Equation (E.16) explains why the Taylor core is so attractive. If \(\lambda '=0\), then the singular current-gradient term disappears from the outer problem. The cylindrical Bessel or Taylor core is therefore naturally resistant to tearing Taylor (1974); Robinson (1978). But a laboratory pinch cannot keep constant \(\lambda \) all the way to the wall. The edge current must decrease, so \(\lambda '\) returns in the outer plasma, and with it the tearing drive. The actual problem is therefore not whether the Taylor core is useful; it is how gently one can connect that core to a low-current edge.

39.4 A Robinson-style stable diffuse pinch in modern notation

Core plus edge roll-off. A simple modern restatement of Robinson’s logic is to take

\[\lambda (r) \simeq \begin {cases} \lambda _0, & 0<r<c,\\[4pt] \text {monotone roll-off}, & c<r<a, \end {cases} \tag{E.25}\]
with a close conducting wall at or just outside \(r=a\). In the core, Eq. (E.3) then gives the Bessel-function form
\[B_z \propto J_0(\lambda _0 r), \qquad B_\theta \propto J_1(\lambda _0 r), \tag{E.26}\]
which is exactly the Taylor-like core Robinson emphasizes. The edge roll-off then reduces the current density and makes the diffuse pinch physically plausible.

How the tearing calculation is actually done. Once the equilibrium is fixed, the calculation proceeds exactly as in the tearing lecture.

\[\text {equilibrium } \Longrightarrow \text {outer equation } \Longrightarrow \Delta ' \Longrightarrow \text {inner-layer dispersion relation.}\]
More explicitly:

1.
Integrate the regular outer solution from the axis to the rational surface.
2.
Integrate the wall solution inward, imposing \(\psi (a)=0\) at the conducting wall.
3.
Use Robinson’s comparison-equation construction to pass cleanly through the singularity.
4.
Form \(\Delta '\) and infer stability from its sign in the constant-\(\psi \) regime.

The physical interpretation is just as simple. A steep outer roll-off in \(\lambda \) tends to make \(\Delta '>0\). A broader roll-off, together with a close wall, can make the same branch stable.

What Robinson 1978 actually showed. The important fact is that this program succeeds. Robinson showed that tearing-mode-stable reverse-field diffuse pinches exist at \(\beta =0\), with a small vacuum region between plasma and wall, with pinch parameter \(\Theta \) as large as about three, and with the wall playing an essential role in suppressing both \(m=0\) and \(m=1\) branches Robinson (1978). The key point is that the tearing problem is hard, but not trivial in the sense of being unsalvageable: one can stabilize it in cylindrical theory, though only with deliberate current-profile shaping.

Caution

Ideal stability and tearing stability are adjacent questions, not identical ones. Robinson’s diffuse-pinch work is valuable precisely because it shows that a profile that looks acceptable to the ideal-MHD eye may still fail once the thin resistive layer is allowed to reconnect it.

39.5 External kink, resistive walls, and smart shells

Why this is the next question once tearing is reduced. To connect directly back to the main external-kink lecture, it is useful to keep the same free-boundary language. Internal modes were tested in the restricted class \(\xi _a=0\). External modes reopen the variational problem and allow \(\xi _a\neq 0\) at the plasma edge. In the RFP this matters as soon as the interior has been made more Taylor-like: once the broad internal tearing spectrum is weakened, the remaining free-boundary \(m=1\) branch is the next ideal-MHD obstacle. Because the edge safety factor is small, the cylindrical RFP admits a broader set of unstable toroidal harmonics than the tokamak case that is usually first taught Marrelli et al. (2021).

Keep the same surface form of \(\delta W\). Using the notation of the external-kink lecture, the free-boundary energy for a single \((m,n)\) harmonic can be written as

\[\frac {\delta W}{2\pi ^2 R_0/\mu _0} = \left [ \left (\frac {F^2}{k_0^2}\frac {r\xi '}{\xi }\right )_{r=a} + \left (\frac {F F^\dagger }{k_0^2}\right )_{r=a} + a^2 F_a^2\,\Lambda _m \right ]|\xi _a|^2, \tag{E.28}\]
where
\[F_a\equiv F(a)=\frac {B_{za}}{R_0}\left (\frac {m}{q_a}-n\right ), \qquad F_a^\dagger \equiv F^\dagger (a)=-\frac {B_{za}}{R_0}\left (n+\frac {m}{q_a}\right ), \tag{E.29}\]
and the vacuum-wall response factor is
\[\Lambda _m=\frac {1+\alpha _m}{1-\alpha _m}, \qquad \alpha _m\equiv \left (\frac {a}{b}\right )^{2m}. \tag{E.30}\]
Here \(b\) is the shell radius and \(q_a\) is the safety factor evaluated on the plasma side of the shell. Equation (E.28) is just Eq. (24.16) of the main external-kink lecture rewritten in RFP notation; the only question is what the RFP interior contributes through the boundary ratio \((r\xi '/\xi )_a\).

Constant-\(\lambda \) Taylor equilibrium at the edge. To make the surface form completely explicit, take the simplest Taylor profile,

\[B_z(r)=B_0 J_0(\lambda r), \qquad B_\theta (r)=B_0 J_1(\lambda r), \tag{E.31}\]
which is just Eq. (E.26) with the normalization restored. Then
\[q(r)=\frac {r B_z(r)}{R_0 B_\theta (r)} = \frac {r}{R_0}\frac {J_0(\lambda r)}{J_1(\lambda r)}, \qquad q_a=\frac {a}{R_0}\frac {J_0(x_a)}{J_1(x_a)}, \qquad x_a\equiv \lambda a. \tag{E.32}\]
The low-\(q\) RFP regime corresponds to \(0<q_a<1\) on the plasma side of the shell. If one pushes the plasma boundary into the already reversed zone so that \(q_a\le 0\), the unfactorized edge form (E.28) is the safer starting point.

The three terms in \(\delta W\) for the rigid \(m=1\) branch. For a Taylor or Bessel-like core, the lowest long-pitch \(m=1\) external displacement is approximately rigid in the interior, so one may use

\[\frac {a\,\xi '(a)}{\xi _a}\simeq 0. \tag{E.33}\]
Equation (E.33) is the RFP analogue of the rigid \(m=1\) approximation used in Eq. (24.17). In that ordering the interior contribution in Eq. (E.28) is
\[\frac {\delta W_{\rm int}}{2\pi ^2 R_0/\mu _0} = \left (\frac {F_a^2}{k_0^2}\frac {a\xi '(a)}{\xi _a}\right )|\xi _a|^2 \simeq 0, \tag{E.34}\]
while the surface and vacuum pieces become
\[\begin{aligned}\frac {\delta W_{\rm surf}}{2\pi ^2 R_0/\mu _0} &= \left (\frac {F_a F_a^\dagger }{k_0^2}\right )|\xi _a|^2 \simeq -\frac {a^2 B_{za}^2}{R_0^2 q_a^2}(1-n^2 q_a^2)|\xi _a|^2, \\ \frac {\delta W_{\rm vac}(b)}{2\pi ^2 R_0/\mu _0} &= a^2 F_a^2 \Lambda _1 |\xi _a|^2 \simeq \frac {a^2 B_{za}^2}{R_0^2 q_a^2}\Lambda _1(1-nq_a)^2|\xi _a|^2,\end{aligned} \tag{E.35}\]

where \(B_{za}\equiv B_z(a)\) and we have kept the same long-pitch ordering \(ka\ll 1\) used in the main external-kink lecture, so \(k_0^2 a^2\simeq 1\) and \(\Lambda _1=(1+\alpha _1)/(1-\alpha _1)\). Summing (E.34)–(E.36) gives

\[\frac {\delta W_{m=1}^{\rm RFP}}{2\pi ^2 R_0/\mu _0} \simeq \frac {2 a^2 B_{za}^2}{R_0^2 q_a^2(1-\alpha _1)} \,(n q_a-1)(n q_a-\alpha _1)\,|\xi _a|^2 = \frac {2 B_{\theta a}^2}{1-\alpha _1} \,(n q_a-1)(n q_a-\alpha _1)\,|\xi _a|^2, \tag{E.37}\]
with \(B_{\theta a}\equiv B_\theta (a)\). The second form uses \(q_a=aB_{za}/(R_0 B_{\theta a})\) and makes it clear that the \(q_a\to 0^+\) limit is not singular.

Why the RFP has a broad unstable band when \(q_a<1\). Equation (E.37) shows that the ideal external kink is unstable in the interval

\[\alpha _1 < n q_a < 1. \tag{E.38}\]
In the no-wall limit, \(\alpha _1\to 0\), so every positive toroidal harmonic with
\[0 < n q_a < 1 \tag{E.39}\]
is unstable. Thus when \(0<q_a<1\), one does not merely inherit the tokamak’s single \(n=1\) warning; rather, the same long-pitch ordering admits a family of unstable harmonics, roughly
\[n=1,2,\dots ,\left \lfloor \frac {1}{q_a}\right \rfloor , \tag{E.40}\]
subject to the wall-shifted lower bound in Eq. (E.38). For the highest \(n\) at very small \(q_a\), one should replace the \(ka\ll 1\) vacuum reduction by the exact modified-Bessel solution, but (E.37) is the right organizing formula for the low-\(n\) external branch. For a rigid core with uniform density,
\[\mathcal K \simeq \pi ^2 \rho R_0 a^2 |\xi _a|^2,\]
so the corresponding ideal growth rate is
\[\left (\frac {\gamma a}{v_{A\theta a}}\right )^2 \simeq \frac {-4 (nq_a-1)(nq_a-\alpha _1)}{1-\alpha _1}, \qquad v_{A\theta a}^2\equiv \frac {B_{\theta a}^2}{\mu _0 \rho }. \tag{E.42}\]
A plot of (E.42) versus \(q_a\) at fixed \(\alpha _1\) therefore gives the same banded structure as the tokamak constant-\(q\) example in the main lecture, but now several low-\(n\) branches can fit into the interval \(0<q_a<1\).

No-wall and wall pieces. Exactly as in Eqs. (24.31) and (24.32) of the main lecture, it is useful to write

\[\delta W_{\rm ideal}^{\rm RFP}(b)=\delta W_\infty ^{\rm RFP}+\delta W_b^{\rm RFP}. \tag{E.43}\]
The no-wall and wall terms are
\[\begin{aligned}\frac {\delta W_\infty ^{\rm RFP}}{2\pi ^2 R_0/\mu _0} &= \frac {2 a^2 B_{za}^2}{R_0^2 q_a^2} \,n q_a(n q_a-1)\,|\xi _a|^2 = 2B_{\theta a}^2\,n q_a(nq_a-1)\,|\xi _a|^2, \\ \frac {\delta W_b^{\rm RFP}}{2\pi ^2 R_0/\mu _0} &= \frac {2 a^2 B_{za}^2}{R_0^2 q_a^2} \frac {\alpha _1}{1-\alpha _1}(1-n q_a)^2\,|\xi _a|^2 = 2B_{\theta a}^2\frac {\alpha _1}{1-\alpha _1}(1-n q_a)^2\,|\xi _a|^2,\end{aligned} \tag{E.44}\]

with \(\delta W_b^{\rm RFP}>0\). The ideal shell therefore stabilizes the lower part of the no-wall interval, so the RWM strip is

\[0<n q_a<\alpha _1, \tag{E.46}\]
where \(\delta W_\infty ^{\rm RFP}<0\) but \(\delta W_{\rm ideal}^{\rm RFP}(b)>0\).

What a resistive shell changes. An ideal shell does not add a new drive; it adds the stabilizing increment \(\delta W_b^{\rm RFP}\). Once the shell conductivity is finite, only a fraction of that increment is restored. The natural shell timescale is

\[\tau _w \sim \mu _0 a\,\delta \,\sigma , \tag{E.47}\]
up to geometry factors, or more specifically for a thin cylindrical wall,
\[\tau _{w,m}\sim \frac {\mu _0 r_w d\,\sigma }{2m}. \tag{E.48}\]
Using the thin-wall interpolation and dispersion relation from the Kruskal–Shafranov lecture, Eqs. (22.107) and (22.110), the RFP resistive-wall branch is
\[\gamma \tau _w = -\frac {2b^2}{b^2-a^2} \frac {\delta W_\infty ^{\rm RFP}}{\delta W_{\rm ideal}^{\rm RFP}(b)}. \tag{E.49}\]
In the explicit constant-\(\lambda \) \(m=1\) model this becomes
\[\gamma \tau _w = \frac {2b^2}{b^2-a^2} \frac {nq_a(1-\alpha _1)}{\alpha _1-nq_a}, \qquad 0<nq_a<\alpha _1, \tag{E.50}\]
which is the RFP counterpart of Eq. (24.42) in the main lecture. Very close to \(nq_a=\alpha _1\) the separation between the slow RWM branch and the Alfvénic ideal branch breaks down, but away from that edge the mode satisfies
\[\gamma \tau _w = O(1), \qquad \gamma \tau _A \ll 1. \tag{E.51}\]
That is precisely the regime explored in the thin-shell RFP experiments and in the associated linear and nonlinear modelling Ho (1991); Paccagnella et al. (2002); Marrelli et al. (2021). In fact, Marrelli et al. summarize both the early thin-shell experiments and the nonlinear DEBS result that external kinks remain unstable whenever the surrounding wall is not ideal Marrelli et al. (2021).

The smart-shell idea. The natural response to this problem was the intelligent or “smart” shell. Bishop’s idea was to use matched sensor and saddle-coil arrays so that the powered coils imitate the eddy currents of an ideal conducting shell Bishop (1989). In its simplest local form one writes

\[V^{\rm coil}_{ij} = -G_{ij}\,\frac {d\Phi ^{\rm sensor}_{ij}}{dt}, \tag{E.52}\]
so the controller attempts to keep the linked magnetic flux frozen. In Fourier language this becomes a mode-control law of the form
\[I_{m,n}=G_{m,n}\, b_{m,n}, \tag{E.53}\]
which is the natural extension when several unstable harmonics coexist.

Why coil number matters. Because the RFP can have several unstable external harmonics at once, a smart shell must also contend with toroidal sidebands. The key criterion emerging from the linear theory is that the number of toroidal control locations should satisfy roughly

\[N_c \gtrsim \Delta n, \tag{E.54}\]
where \(\Delta n\) is the span of simultaneously unstable toroidal mode numbers. Otherwise the sideband \(n+hN_c\) generated by the discrete actuator array can land on another unstable branch and spoil the control strategy Fitzpatrick and Yu (1999); Marrelli et al. (2021). This is a specifically RFP-flavored problem: the low-\(q\) edge makes a broad spectrum possible, so the shell and controller must be thought of spectrally, not mode by mode in isolation.

What was demonstrated experimentally. The EXTRAP-T2R and RFX-mod programs turned that idea into a practical MHD control system. The review by Marrelli et al. summarizes the arc nicely: thin resistive shells exposed the RWMs, and then active feedback with intelligent- shell or mode-control algorithms stabilized multiple RWMs over many wall times Brunsell et al. (2004, 2005, 2006); Marrelli et al. (2021). That history is worth calling out in an appendix because it shows that the RFP external-kink problem was not merely diagnosed; it became one of the clearest laboratories for multi-mode resistive-wall control.

Takeaways

Once the internal tearing spectrum has been reduced, the next question is the free-boundary \(m=1\) branch. In a low-\(q\) RFP the no-wall interval is \(0<nq_a<1\), the ideal-wall external-kink band is \(\alpha _1<n q_a<1\), and the lower strip \(0<nq_a<\alpha _1\) becomes the RWM branch once the shell is resistive. A smart shell tries to recover the missing ideal-wall increment actively.

39.6 What this means for the operational RFP

Why the laboratory RFP is tearing dominated. The operational RFP does not ordinarily sit in Robinson’s optimized tearing-stable corner of profile space. Instead, it lives in a regime where many \(m=1\) tearing modes are resonant in the interior and couple to \(m=0\) activity near the reversal surface. That coupled spectrum provides the dynamo that redistributes current and helps sustain reversal, but it also makes the magnetic field stochastic and degrades confinement Schnack et al. (1985); Nebel et al. (1989); Kusano and Sato (1990); Ho and Craddock (1991).

Transport is the real penalty. Once the tearing spectrum is broad enough, the issue is no longer just the existence of islands. Field-line stochasticity turns the magnetic fluctuation spectrum into a transport problem. The MST-era measurements and simulations make this especially clear: magnetic-fluctuation-induced transport is directly observable, and transient current-profile control reduces both fluctuations and transport Fiksel et al. (1994); Stoneking et al. (1994); Sarff et al. (1994); Sovinec (1995).

Why current-profile control is central. This is the operational lesson of the RFP. Because the tearing drive is so tightly linked to the current profile, the natural control knob is the current profile itself. That is why PPCD, RF-current-drive ideas, and helicity-injection profile control all enter the RFP story so naturally. In the language of this appendix, the objective is always the same: make the core more Taylor-like, soften the edge roll-off enough that \(\Delta '\) becomes less dangerous, and keep the broad \(m=1\) spectrum from coupling strongly to the reversal-region \(m=0\) modes Ho (1991); Sovinec (1995).

39.7 From ZETA and “catching” to PPCD

Quiet periods were already the clue. The modern review by Marrelli et al., with Sarff among the coauthors, points back to the quiet periods in ZETA as some of the earliest evidence that an RFP-like discharge could evolve into a lower-fluctuation, more ordered state Butt et al. (1958); Robinson and King (1968); Marrelli et al. (2021). In hindsight that already contains the seed of the later current-profile story: the discharge is not uniformly turbulent, but moves through profiles that are more or less favorable to the tearing spectrum.

“Catching” as the ancestor of transient profile control. The same review also recalls an older Culham/HBTX idea, called “catching”, in which the loop-voltage program during current decay was shaped so as to hold the plasma on a more favorable resistive trajectory Watterson and Gimblett (1983); Marrelli et al. (2021). In present language this sits very naturally beside self-similar current decay and other transient profile-control ideas: one exploits the fact that the inductively relaxing RFP is already moving through profile space on the resistive timescale Nebel et al. (2002).

Why this matters for PPCD. That historical thread makes PPCD feel less like an isolated trick and more like the mature experimental version of an old RFP instinct: use the inductive evolution itself as a control knob. The later OPCD work in RFX and the PPCD work on MST sit naturally on that lineage, even though the exact waveforms and hardware differ Bartiromo et al. (1999); Sarff et al. (1994); Sovinec (1995). For a historical appendix, it is worth saying plainly that the later success of PPCD did not come from nowhere; it was prepared by earlier attempts to “catch” the plasma while its current profile was inductively relaxing.

So how does the RFP fare? The fairest answer is nuanced. As an ideal-MHD configuration, the RFP fares better than its reputation suggests: reversed-field diffuse pinches can be globally stable, and even finite-\(\beta \) stable, if the pitch profile is chosen correctly and the wall is close Robinson (1971); Antoni et al. (1986); Robinson (1978). As a resistive-MHD system, however, the RFP is severe. Its low-\(q\) geometry and required edge current roll-off make tearing modes easy to excite, numerous, and nonlinearly effective. So the operational verdict is simple enough: the RFP does reasonably well in ideal MHD, poorly in generic classical tearing stability, and best when one treats current-profile control as the central experimental task rather than as an optional refinement.

Takeaways

For the tokamak, tearing is an important defect superimposed on an otherwise good ideal-MHD equilibrium. For the RFP, tearing is much closer to the organizing principle of the discharge. That is precisely why it remains such a useful and classic problem in MHD but also why its confinement lacks for fusion purposes.

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