Lecture 30 Furth, Killeen and Rosenbluth: Tearing Modes

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Lecture 30
Furth, Killeen and Rosenbluth: Tearing Modes

Overview

Tearing modes are the first place in these notes where the ideal-MHD stability machinery must be used with a small but essential non-ideal correction. Outside a very narrow resistive layer, the plasma is still described by the ideal equations and by the same Newcomb-style outer matching used in the pinch lectures. At the rational surface, however, the ideal solution becomes singular. Finite resistivity regularizes that singularity, allows magnetic reconnection, and introduces the outer matching parameter $ \Delta ' = \bigl [\psi '/\psi \bigr ]_{r_s^-}^{r_s^+}. $ In the classical constant-$\psi $ regime the mode is unstable when $\Delta '>0$, and the growth rate scales like $\gamma \propto \eta ^{3/5}(\Delta ')^{4/5}$.
What makes tearing so important is not merely that it is unstable, but that it changes magnetic topology. In toroidal confinement the same basic mechanism underlies sawtooth reconnection, low-order $3/2$ and $2/1$ islands, neoclassical tearing modes, locked modes, and ultimately the stochastic transport produced by island overlap. Tearing is therefore one of the main bridges between linear stability theory, nonlinear self-organization, and disruption physics.

Historical Perspective

The modern tearing-mode story begins with the paper of Furth, Killeen, and Rosenbluth, who showed that a plasma that is almost ideal everywhere can still reconnect if a thin resistive boundary layer forms at a resonant surface Furth et al. (1963). That was the real conceptual step: the instability is not an ordinary ideal-MHD mode, but neither is it a uniformly resistive diffusion problem. It is a matched asymptotic problem, with an ideal outer region and a non-ideal inner layer.
Coppi, Greene, and Johnson soon carried this logic into cylindrical pinch geometry, showing how the same layer-matching procedure works for screw-pinch equilibria Coppi et al. (1966). Rutherford then showed that once an island is larger than the linear resistive layer, the growth becomes algebraic rather than exponential, giving the weakly nonlinear evolution law now called the Rutherford equation Rutherford (1973). In toroidal confinement this physics entered the mainstream through the interpretation of the tokamak sawtooth as a current-profile relaxation driven by the $m=n=1$ internal tearing mode Kadomtsev (1975). By the 1990s and 2000s the same island physics had reappeared in a new form: low-order $3/2$ and $2/1$ islands driven nonlinearly by the bootstrap current, and wall-locked modes that often mark the last stage before disruption Hegna (1998); La Haye (2006); La Haye et al. (1992).

Caution

Two cautions are worth stating at the start.
First, $\Delta '$ is an outer-region quantity. It tells us how the ideal solution leans into the resonant layer, not the growth rate by itself. The inner-layer physics is still needed to convert $\Delta '$ into a dispersion relation.
Second, the classical FKR result assumes the constant-$\psi $ ordering. Large-$\Delta '$, viscous, two-fluid, or collisionless layers modify both the layer width and the dispersion relation. The virtue of the classical problem is not that it contains everything, but that it teaches the matching idea once and for all.

30.1 Why tearing is different from ideal MHD

The starting point is the induction equation introduced in the opening lecture, Eq. (1.13). In ideal MHD the resistive term vanishes and magnetic flux is frozen into the fluid. The ideal perturbation theory developed in the diffuse-pinch lecture then leads to the Newcomb equation, and its singular points occur where the helical field-line bending term vanishes. In cylindrical language that singularity sits at the rational surface

F(r) \equiv \frac {m B_\theta (r)}{r} + k B_z(r) = \frac {m B_\theta (r)}{r}\left (1-\frac {n q(r)}{m}\right )=0, \tag{30.1}

with $k=-n/R$ and $q=r B_z/(R B_\theta )$. This is the same resonant surface that appeared in the ideal calculation, compare Eq. (25.5). In ideal MHD one must demand a regular solution through that surface. In tearing theory we do something more subtle: we keep the ideal solution everywhere except in a thin neighborhood of $r=r_s$, and we let resistivity regularize the singularity there.

The physical picture is simple. If the outer ideal solution arrives at the resonant surface with a positive logarithmic derivative jump,

\Delta ' \equiv \left .\frac {\psi '}{\psi }\right |_{r=r_s^-}^{r=r_s^+}, \tag{30.2}

then the field lines on the two sides of the resonant surface are trying to reconnect in a way that lowers the magnetic energy. The inner resistive layer provides the actual reconnection, while the outer region supplies the free-energy bookkeeping.

30.2 Linear cylindrical equations

Take a screw-pinch equilibrium

\B _0(r)=B_\theta (r)\,\etheta + B_z(r)\,\ez , \qquad \J _0(r)=J_z(r)\,\ez ,

and perturb it by a single helical harmonic

\sim e^{\gamma t + i m \theta + i k z}, \qquad k=-\frac {n}{R}.

For the magnetic perturbation it is convenient to introduce the flux function $\psi (r)$ by

\B _1 = -\curl \bigl (\psi \,\ez \bigr ) = -\frac {i m}{r}\psi \,\er + \psi '\,\etheta , \qquad \psi = -i\frac {r}{m} b_r. \tag{30.5}

The dominant perturbed current is then the axial component

\muo j_{1z} = \frac {1}{r}(r\psi ')' - \frac {m^2}{r^2}\psi \equiv \nabla _\perp ^2 \psi . \tag{30.6}

For incompressible motion we may similarly introduce a streamfunction $\phi $:

\uvec _1 = \curl (\phi \,\ez ), \qquad v_r = -\frac {i m}{r}\phi , \qquad v_\theta = \phi '. \tag{30.7}

Tutorial

From the full linearized equations to the reduced tearing pair. The cleanest way to see what is being kept and what is being discarded is to start from the full linearized resistive-MHD system for a single helical harmonic. Write
$\rho _0\gamma \,\uvec _1 = \J _1\times \B _0 + \J _0\times \B _1 - \nabla p_1, \tag{30.8}$ $\gamma \B _1 = \nabla \times (\uvec _1\times \B _0) + \frac {\eta }{\muo }\nabla ^2\B _1,$ $\gamma p_1 + v_r p_0' + \Gamma p_0\nabla \cdot \uvec _1 = 0, \qquad \nabla \cdot \B _1 = 0.$ We keep $p_0(r)$ for the moment because this is exactly the same starting point that will be needed again in the resistive-interchange lecture.
Now take the $\hat z$ component of the curl of (30.8). Since $\nabla \times \nabla p_1=0$, the pressure term is annihilated by this step:
$\bigl (\nabla \times \rho _0\gamma \uvec _1\bigr )_z = \rho _0\gamma \,\nabla _\perp ^2\phi .$ On a single helical harmonic, $\B _0\cdot \nabla \to iF(r)$, while $b_r=-im\psi /r$ and $\muo j_{1z}=\nabla _\perp ^2\psi $. Therefore $\begin{aligned}\bigl (\nabla \times (\J _1\times \B _0 + \J _0\times \B _1)\bigr )_z &= \bigl (\B _0\cdot \nabla \,j_{1z}\bigr ) + \bigl (\B _1\cdot \nabla \,J_{0z}\bigr ) \\ &= \frac {iF(r)}{\muo }\,\nabla _\perp ^2\psi - \frac {i m}{r}J_z'(r)\,\psi .\end{aligned}$
This gives the reduced vorticity equation
$\boxed { \rho \gamma \,\nabla _\perp ^2\phi = \frac {iF(r)}{\muo }\,\nabla _\perp ^2\psi - \frac {i m}{r}J_z'(r)\,\psi . } \tag{30.14}$
The induction equation is just as direct. Its radial component is
$\gamma b_r = iF(r) v_r + \frac {\eta }{\muo }\nabla _\perp ^2 b_r.$ Using $b_r=-im\psi /r$ and $v_r=-im\phi /r$ then yields $\boxed { \gamma \psi + iF(r)\phi = \frac {\eta }{\muo }\,\nabla _\perp ^2\psi . } \tag{30.16}$
So pressure has not been “forgotten”; it disappears from the leading tearing equations because we projected the dynamics onto the $z$-vorticity and induction sectors. The resistive- interchange lecture will return to the same linearized starting point but will keep the perpendicular force-balance / total-pressure sector instead of annihilating it.

Equations (30.14) and (30.16) already display the layered structure of the problem. The factor $F(r)$ vanishes at the resonant surface, so the ideal relation between $\phi $ and $\psi $ breaks down there. Away from $r_s$, however, both resistivity and inertia are small, so the outer problem is nearly ideal.

The outer equation. In the outer region we drop the inertial term in (30.14) and the resistive term in (30.16). The second equation then gives $ \phi = i\gamma \psi /F. $ Substituting into the first and neglecting the small factor $\gamma ^2$ produces

\frac {1}{r}(r\psi ')' - \frac {m^2}{r^2}\psi - \frac {\muo m J_z'(r)}{r F(r)}\,\psi = 0. \tag{30.17}

This is the cylindrical tearing analogue of the Newcomb equation: it is an ideal-MHD problem everywhere except at the resonant surface, where $F(r_s)=0$.

Figure 30.1: The top hat current profile used in text. Both m/n=2/1 and 3/2 are unstable for the choice of c/a

What $\Delta '$ measures. Once (30.17) is solved from the magnetic axis outward and from the wall inward, the only information the inner layer needs from the outer ideal region is the jump (30.2). The parameter $\Delta '$ is therefore the cylindrical tearing counterpart of the ideal-MHD boundary data that appeared in the energy-principle lectures. When $\Delta '>0$, the two outer solutions arrive at the resonant layer with a kink in $\psi '$ that can be healed by reconnection; when $\Delta '<0$, the outer solution resists such healing.

30.3 A fully analytic worked example for $\Delta '$

The most useful first example is not the smoothest equilibrium but the one for which the algebra can be carried all the way to the end. We therefore take the reduced-MHD current profile

J_z(r)= \begin {cases} J_0, & 0<r<c,\\[4pt] 0, & c<r<a, \end {cases} \tag{30.18}

with a perfectly conducting wall at $r=a$. The equilibrium poloidal field follows from Ampère’s law,

B_\theta (r)= \begin {cases} \dfrac {\muo J_0 r}{2}, & 0<r<c,\\[10pt] \dfrac {\muo J_0 c^2}{2r}, & c<r<a. \end {cases} \tag{30.19}

If $B_z$ is taken constant, then the safety factor is

q(r)=\frac {rB_z}{R B_\theta }= \begin {cases} q_c, & 0<r<c,\\[4pt] q_c\,\dfrac {r^2}{c^2}, & c<r<a, \end {cases} \qquad q_c \equiv \frac {2B_z}{\muo J_0 R}. \tag{30.20}

A rational surface in the current-free annulus therefore satisfies

q(r_s)=\frac {m}{n} \qquad \Longrightarrow \qquad r_s = c\sqrt {\frac {m}{n q_c}}. \tag{30.21}

We shall work out the case $m=2$. The reason for not choosing $m=1$ is structural: for $m=1$ the tearing problem sits much closer to the ideal kink problem, whereas $m=2$ cleanly exhibits the sign change of $\Delta '$.

Piecewise outer solutions. Away from the current jump at $r=c$ and away from the resonant layer at $r=r_s$, the current gradient vanishes, so (30.17) reduces to Laplace’s equation,

\frac {1}{r}(r\psi ')' - \frac {m^2}{r^2}\psi = 0, \tag{30.22}

with solutions $\psi = A r^m + B r^{-m}$. The three relevant pieces are therefore

\begin{aligned}\psi _I(r) &= A\,r^m, \qquad 0<r<c, \\ \psi _{II}(r) &= B\,r^m + C\,r^{-m}, \qquad c<r<r_s, \\ \psi _{III}(r) &= D\left (r^m-a^{2m}r^{-m}\right ), \qquad r_s<r<a,\end{aligned} \tag{30.23}

where regularity at the axis killed the $r^{-m}$ term in region I and the wall condition $\psi (a)=0$ fixed the form in region III.

Jump condition at the current step. Because $J_z$ jumps discontinuously at $r=c$, the derivative of $\psi $ also jumps there. Integrating (30.17) across $r=c$ gives

\bigl [r\psi '\bigr ]_{c^-}^{c^+} = \int _{c^-}^{c^+} \frac {\muo m J_z'(r)}{F(r)}\,\psi \,dr. \tag{30.26}

For the step profile (30.18), $ J_z'=-J_0\delta (r-c), $ and using (30.20)–(30.21) one finds

F(c)=\frac {mB_\theta (c)}{c}\left (1-\frac {n q(c)}{m}\right ) = \frac {m\muo J_0}{2}\left (1-\frac {c^2}{r_s^2}\right ).

Equation (30.26) therefore becomes

\bigl [r\psi '\bigr ]_{c^-}^{c^+} = -\frac {2}{1-c^2/r_s^2}\,\psi (c) = -\frac {2r_s^2}{r_s^2-c^2}\,\psi (c). \tag{30.28}

Together with continuity of $\psi $, this determines the coefficients $B$ and $C$ in terms of $A$:

\begin{aligned}B &= A\,\frac {m(c^2-r_s^2)+r_s^2}{m(c^2-r_s^2)}, \\ C &= -A\,\frac {c^{2m}r_s^2}{m(c^2-r_s^2)}.\end{aligned} \tag{30.29}

At this point the only remaining task is to evaluate the logarithmic derivatives at the resonant surface.

The right-hand derivative. From (30.25) we obtain immediately

\left .\frac {\psi '}{\psi }\right |_{r_s^+} = \frac {m}{r_s}\,\frac {1+(a/r_s)^{2m}}{1-(a/r_s)^{2m}} = -\frac {m}{r_s}\,\frac {1+y^{2m}}{y^{2m}-1}, \tag{30.31}

where for compactness we introduced

x \equiv \frac {c}{r_s}, \qquad y \equiv \frac {a}{r_s}. \tag{30.32}

Figure 30.2: Left: A unstable current profile and Right: stable current profile. The kink in the $\psi $ profile determines whether flux increases or decreases in the layer. Concave up $\Delta '>0$ leads to growth.

The left-hand derivative. Using (30.24), (30.29), and (30.30) gives

\left .\frac {\psi '}{\psi }\right |_{r_s^-} = \frac {m}{r_s} \frac {x^{2m}+m x^2-m+1}{m x^2-m-x^{2m}+1}. \tag{30.33}

Subtracting (30.33) from (30.31) yields the desired matching parameter. For $m=2$ the result is especially transparent:

\Delta ' = \frac {4}{r_s} \frac {2x^2 y^4 - x^4 - y^4}{(1-x^2)^2(y^4-1)}. \tag{30.34}

This is the main reason for choosing the step profile as the worked example: the sign of $\Delta '$ can be seen directly.

In the large-wall limit $a\gg r_s$ (so $y\to \infty $), Eq. (30.34) reduces to

\Delta ' \longrightarrow \frac {4}{r_s}\,\frac {2x^2-1}{(1-x^2)^2}. \tag{30.35}

Hence the same equilibrium can be either tearing-stable or tearing-unstable depending on where the resonant surface sits:

\Delta '>0 \quad \Longleftrightarrow \quad x>\frac {1}{\sqrt {2}} \quad \Longleftrightarrow \quad r_s<\sqrt {2}\,c. \tag{30.36}

Every step is analytic, and yet the mode does not have a predetermined sign.

The same profile can host $3/2$ and $2/1$ surfaces. Because the annular safety-factor profile (30.20) is monotone,

q_c < \frac {m}{n} < q_a \qquad \Longrightarrow \qquad r_{m/n}=c\sqrt {\frac {m}{n q_c}}.

In particular,

r_{3/2}=c\sqrt {\frac {3}{2q_c}}, \qquad r_{2/1}=c\sqrt {\frac {2}{q_c}}. \tag{30.38}

So the step profile supports both a $3/2$ and a $2/1$ tearing surface whenever

q_c<\frac {3}{2}, \qquad q_a>2. \tag{30.39}

This is the regime one would naturally use for a field-line-tracing exercise aimed at island overlap.

It is also useful to record the large-wall result for general $m$. Taking $y=a/r_s\to \infty $ in the algebra above gives

\Delta '_m \longrightarrow \frac {2m\bigl [mx^2-(m-1)\bigr ]} {r_s(1-x^2)^2\displaystyle \sum _{\ell =0}^{m-2}(m-1-\ell )x^{2\ell }}, \qquad x\equiv \frac {c}{r_s}. \tag{30.40}

Since the denominator is positive for $0<x<1$, the sign is set by the numerator:

\Delta '_m>0 \qquad \Longleftrightarrow \qquad x^2>\frac {m-1}{m}. \tag{30.41}

Using $x^2 = q_c/(m/n)=n q_c/m$, this becomes

\Delta '_m>0 \qquad \Longleftrightarrow \qquad n q_c > m-1. \tag{30.42}

Hence this simple profile makes a neat point: both the $2/1$ and $3/2$ branches are tearing-unstable in the large-wall limit when $q_c>1$, marginal when $q_c=1$, and tearing-stable when $q_c<1$ (provided the corresponding rational surfaces still exist).

Caution

Why use the step profile? It is not the most realistic current profile, but it is probably the best first $\Delta '$ example because the outer equation reduces to power laws in every region. A smooth profile such as $J_z=J_0(1-r^2/a^2)^\nu $ is a very good next step.

Interactive Tearing-Mode Explorer

Open a browser companion to the lecture’s smooth-profile extension. The app uses $J_z(r)=\hat J(1-r^2/a^2)^\nu$, reconstructs $q(r)$ and $F(r)$, solves the outer tearing equation on both sides of the resonant layer, and reports $\Delta\!\!\prime$ for selectable $m/n$ branches.

Open the tearing-mode explorer

For the timescale ingredients that enter the FKR and Lundquist-number estimates, jump to the Braginskii formulary calculator.

30.4 The resistive inner layer and the FKR scaling

Near the resonant surface we set

s=r-r_s, \qquad F(r)\simeq F'(r_s)s = -\frac {m B_{\theta s}}{r_s}\left (\frac {q_s'}{q_s}\right )s \equiv -\frac {mB_{\theta s}}{r_s L_s}s, \tag{30.43}

where

L_s \equiv \frac {q_s}{|q_s'|}

is the shear length and subscript $s$ means evaluation at $r=r_s$. In the constant-$\psi $ regime the perturbed flux is nearly constant across the layer, so we keep $\psi \simeq \psi _s$ whenever it is not differentiated and where is differentiated we will assume "slab" like behavior

\nabla _\perp ^2 \approx \frac {\partial ^2}{\partial s^2};

locally most variation is normal the the surface (the $\theta $ dependence is much smaller).

With the streamfunction definition (30.7), the leading inner equations become¹

\begin{aligned}\gamma \phi '' &= -\frac {m B_{\theta s}}{\muo \rho L_s}\,s\,\psi ''- \frac {m^2}{\rho r^2} \frac {d J_\phi }{d r} \psi , \\ \gamma \psi - \frac {m B_{\theta s}}{r_sL_s}\,s\,\phi &= \frac {\eta }{\muo }\,\psi ''.\end{aligned} \tag{30.46}

The equations are greatly simplified if we assume $b_r\sim \psi $ (not $\psi ''$) is constant over the inner region. We can later computer the solution directly and check. In this case we can substitute $\psi ''$ from Eq. (30.47) into Eq. (30.46) to derive

\begin{aligned}\gamma \phi '' & = -\frac {m B_{\theta s}}{\muo \rho r_s L_s}\,s\, \frac {\mu _0}{\eta } ( \gamma \psi _s - \frac {m B_{\theta s}}{r_sL_s}\,s\,\phi ) \\ \gamma \phi '' - s^2 \frac {m^2 B_\theta ^2}{\eta \rho r_s^2 L_s^2} \phi & = \underbrace {s \frac {B_{\theta s}\psi _s }{\eta r_s \rho L_s}}_{odd} - \underbrace {\frac { m^2}{\rho r^2} J_z'(r_s)\,\psi _s}_{even}.\end{aligned}

The characteristic layer width is obtained by balancing the two terms on the right-hand side of (30.47) against the inertial response implied by (30.46):

d^4 \sim \frac {\eta }{\muo }\,\frac {\gamma L_s^2}{k_\perp ^2 V_{A\theta }^2}, \qquad k_\perp \equiv \frac {m}{r_s}, \qquad V_{A\theta }\equiv \frac {B_{\theta s}}{\sqrt {\muo \rho }}. \tag{30.50}

Tutorial

Putting back the numerical factor in the FKR matching. To go beyond the scaling argument, introduce the dimensionless variables
$x \equiv \frac {s}{d}, \qquad y(x) \equiv -\frac {\muo }{\eta }\frac {d}{r_s }\frac { m B_{\theta s}}{L_s}\frac {\phi }{\psi _s}. \tag{30.51}$ Focusing on the odd part of Eqs. (30.46) and (30.47) the inner equation of motion reduces to $\begin{aligned}\left (\frac {d^2 y}{dx^2}-x^2 y\right )& =-x,\\ \frac {d^2 y}{dx^2} & = -x (1 - x y)\end{aligned} \tag{30.53}$
which can be solved in different ways include simple numerical integration.
The constant-$\psi $ form of Ohm’s law (Eq. (30.16)) becomes
$\psi _{xx} = \frac {\muo \gamma d^2}{\eta }\,\psi _s\,[1+x y(x)]. \tag{30.54}$ Once the function $y$ is known, the jump in $\psi '$ from one side to the other of the inner region can be calculated by extending the limit of integration to infinity: $\begin{aligned}\Delta ' &= \frac {1}{\psi _s d}\int _{-\infty }^{\infty }\psi _{xx}\,dx \\ &= \frac {\muo \gamma d}{\eta } \int _{-\infty }^{\infty }[1+x y(x)]\,dx.\end{aligned} \tag{30.56}$
This problem is solved analytically in Appendix H and numerical solutions are showing in Fig. 30.3
$\int _{-\infty }^{\infty }[1+x y(x)]\,dx = C_\Delta , \qquad C_\Delta = \frac {\Gamma (1/4)}{2\pi \Gamma (3/4)} \approx 2.12 \tag{30.57}$ Hence $\Delta ' = C_\Delta \,\frac {\muo \gamma d}{\eta }. \tag{30.58}$ Substituting (30.50) then yields the classical constant-$\psi $ dispersion relation with its familiar numerical coefficient, $\gamma \simeq 0.55 \left [ \frac {\eta ^3}{\muo ^3} \frac {k_\perp ^2 V_{A\theta }^2}{L_s^2} \left (\Delta '\right )^4 \right ]^{1/5}. \tag{30.59}$

Figure 30.3: Matched odd solution of the inner tearing-mode equation $y''=-x(1-xy)$ with $y(0)=0$ and $xy\to 1$ as $x\to \infty $. Left: the odd profile $y(x)$ together with the outer-match guide $y\sim 1/x$. Right: the even integrand $1-xy$, whose dominant positive core and weak negative tails combine to give the finite matching integral $2\int _0^\infty (1-xy)\,dx = 2.12364827298\ldots $. The zero crossing occurs at $x\simeq \pm 2.367$.

If one only wants the exponents, the algebra can be read off directly from (30.50) and (30.58):

\begin{aligned}\Delta ' &\sim \frac {\muo \gamma }{\eta } \left (\frac {\eta }{\muo }\frac {\gamma L_s^2}{k_\perp ^2V_{A\theta }^2}\right )^{1/4} \\ &= \left (\frac {\muo }{\eta }\right )^{3/4} \gamma ^{5/4} \left (\frac {L_s^2}{k_\perp ^2V_{A\theta }^2}\right )^{1/4},\end{aligned}

\gamma ^{5/4} =\frac {1}{2.12} \left (\frac {\eta }{\muo }\right )^{3/4} \left (\frac {k_\perp ^2V_{A\theta }^2}{L_s^2}\right )^{1/4} \Delta '.

which reduces to

\gamma = 0.55 \left (\frac {\eta }{\muo a^2}\right )^{3/5} \left (\frac { V_{A\theta }}{a}\right )^{2/5} \left (\frac {k_\perp a^2 }{L_s }\right )^{2/5} (a\Delta ')^{4/5} = \frac {0.55}{\tau _R^{3/5}\tau _A^{2/5}} (k_\perp a)^{2/5} \left (\frac {a}{L_s}\right )^{2/5}(a\Delta ')^{4/5}.

Raising the last relation to the $4/5$ power reproduces the FKR exponents, while the preceding tutorial box fixes the numerical coefficient. The corresponding layer width is

d \sim \left (\frac {\eta L_s^2}{\muo \gamma }\right )^{1/4} \propto \eta ^{2/5}(\Delta ')^{-1/5}. \tag{30.64}

30.5 Magnetic islands, Rutherford evolution, and why tearing matters

Figure 30.4: The tearing perturbation changes the topology of the magentic field surrounding the rational surface.

Once reconnection has begun, the helical flux function may be written locally as

\Psi (r,\theta ,z) = \Psi _0(r) + \tilde \psi \cos (m\theta +kz), \qquad k=-\frac {n}{R}.

Near the resonant surface, $ \Psi _0(r)\simeq \Psi _0(r_s)+\tfrac 12\Psi _0''(r_s)(r-r_s)^2, $ so the island half-width follows from balancing the perturbation against the quadratic variation of the equilibrium flux:

\frac 12\,|\Psi _0''(r_s)|\left (\frac {w}{2}\right )^2 \sim |\tilde \psi |.

Therefore

w \sim 4\sqrt {\frac {|\tilde \psi |}{|\Psi _0''(r_s)|}}. \tag{30.67}

When the island width becomes larger than the linear resistive layer, $w\gg d$, inertia falls out of the inner problem and the growth turns algebraic. In that Rutherford regime Rutherford (1973),

\frac {dw}{dt} \sim \frac {\eta }{\muo }\,\Delta ', \tag{30.68}

up to a geometry-dependent factor of order $r_s^{-1}$. The important physical point is that the instability now evolves on the resistive time scale and gradually flattens the current profile across the island.

Why this matters for confinement. Tearing is not just another eigenmode in a spectrum. Ideal modes can displace a flux surface and then return it. A tearing mode changes the connectivity of field lines, flattens current and pressure across a rational surface, and leaves behind a magnetic island chain. In tokamaks the $m=n=1$ internal tearing mode explains the sawtooth crash, while the low-order $3/2$ and $2/1$ surfaces are the classic seats of neoclassical tearing modes. In reversed-field pinches a broad spectrum of tearing modes can overlap and stochasticize a large fraction of the plasma column. So the real importance of tearing is that it is simultaneously a stability problem, a transport problem, and a self-organization problem Hegna (1998); La Haye (2006).

A schematic generalized Rutherford equation. Once pressure flattening, bootstrap current, curvature, and applied current drive are included, the weakly nonlinear evolution is no longer described by the single term (30.68). There is no unique universal normal form, because different authors package the transport thresholds and small-island physics in slightly different ways, but a very useful schematic version is

I_1\tau _R\frac {a}{r_s}\frac {d\hat w}{dt} = \Delta ' - f_{\rm sat}\hat w + f_{\rm bs}\frac {\hat w}{\hat w^2+\hat w_d^2} - \frac {f_{\rm curv}}{\sqrt {\hat w^2+\hat w_d^2}} - \frac {f_{\rm pol}}{(\hat w^2+\hat w_d^2)^{3/2}} - f_{\rm CD}(\hat w). \tag{30.69}

where $\hat w\equiv w/a$ is the normalized island width and $\hat w_d\equiv w_d/a$ is the small-island scale below which transport does not fully flatten the pressure inside the separatrix. The first term is the classical tearing drive from the outer ideal solution, the second is Rutherford saturation, the third is the destabilizing bootstrap-current loss, the fourth and fifth represent stabilizing curvature and polarization effects, and the last term represents active control by driven current or heating Hegna (1998); Hegna and Callen (1997); La Haye (2006).

The really important lesson is that a neoclassical tearing mode is a nonlinear instability. A surface can be linearly stable in the classical sense, $\Delta '<0$, and still grow once a seed island becomes wide enough that the bootstrap term dominates the classical stabilizing term. In tokamak language this is why sawteeth, edge-localized events, error fields, or nonlinear mode coupling can seed a $3/2$ or $2/1$ island which then grows on its own. The low-order island is not merely a passenger on the equilibrium; it becomes a new current-transport channel.

Mode locking and the emergence of external character. As long as an island rotates relative to the wall, plasma response and wall eddy currents partially shield the external resonant field. A simple toroidal torque balance may be written schematically as

I_{\rm eff}\frac {d\Omega }{dt} = \underbrace {T_{\rm EM}}_{\text {error field + wall}} +\underbrace {T_{\rm visc}}_{\text {viscous / neoclassical drag}} +\underbrace {T_{\rm src}}_{\text {beam or rf torque}}. \tag{30.70}

When electromagnetic braking wins and $\Omega \to 0$, the island locks to the wall. At that point the perturbation ceases to behave like a nearly internal rotating chain and acquires a more external character: the vacuum region and the wall response participate directly, the edge magnetic perturbation becomes larger, shielding is lost, and disruption probability rises sharply La Haye et al. (1992); Lazzaro et al. (2002); Fitzpatrick (1993). That is why in real devices the topics “tearing mode,” “error field,” and “locked mode” are not separate stories but successive stages of one island problem.

A single tearing mode gives an island chain. A spectrum of tearing modes gives many island chains. Once neighboring separatrices overlap,

\mathcal {S} \equiv \frac {w_1+w_2}{\Delta r} \gtrsim 1, \tag{30.71}

field lines wander stochastically and transport rises sharply. In this sense the tearing mode is the simplest topological route from a nested set of flux surfaces to a stochastic magnetic web. The field-line diffusion and electron-heat-transport estimates of Rechester–Stix and Rechester–Rosenbluth are the classical asymptotic descriptions of that stochastic regime Rechester and Stix (1976); Rechester and Rosenbluth (1978).

Figure 30.5: Poincaré sections for two resonant tearing perturbations in the cylindrical step-profile equilibrium, showing the transition from intact flux surfaces to overlap-driven stochasticity. Left: weak $2/1$ and $3/2$ perturbations produce two distinct island chains with nominal single-island widths $w_{2/1}/a \approx 0.013$ and $w_{3/2}/a \approx 0.010$, separated by $\Delta r/a \approx 0.078$, so nested flux surfaces remain between the resonances. Right: stronger perturbations increase the nominal widths to $w_{2/1}/a \approx 0.048$ and $w_{3/2}/a \approx 0.041$, while the resonant-surface separation is unchanged, and the puncture points begin to wander through the inter-island region, indicating the onset of ergodicity.

30.6 Rechester–Rosenbluth transport in stochastic fields

Once neighboring island chains overlap strongly enough that $\mathcal {S}\gtrsim 1$, field lines cease to remain on smooth flux surfaces and instead execute a radial random walk. The stochastic-field picture itself goes back to Rechester and Stix, who analyzed magnetic braiding due to weak asymmetry, while the associated electron heat-transport estimate most often used in confinement theory is due to Rechester and Rosenbluth Rechester and Stix (1976); Rechester and Rosenbluth (1978). A standard way to characterize the stochastic field is through the magnetic-field-line diffusion coefficient,

D_M \equiv \lim _{L\to \infty }\frac {\left \langle \left [r(L)-r(0)\right ]^2\right \rangle }{2L},

where $L$ is the distance measured along the field. In the quasilinear island-overlap picture one writes

D_M \sim L_c\left (\frac {\delta B_\perp }{B}\right )^2,

with $L_c$ (or $L_{ac}$) the field-line autocorrelation length. Because the electron parallel thermal conductivity is so large, the stochastic radial heat diffusivity is obtained by combining the parallel heat flow with the field-line random walk,

\chi _{e,\perp } \sim \frac {\chi _{\parallel e}}{\lambda _{\mathrm {mfp},e}}\,D_M \sim v_{th,e} D_M,

in the collisionless Rechester–Rosenbluth limit $\lambda _{\mathrm {mfp},e}\gg L_c$. A useful interpolation that makes contact with experimental analysis is

\chi _{RR} \sim v_{th,e}L_{\rm eff}\left (\frac {\delta B_\perp }{B}\right )^2, \qquad L_{\rm eff}^{-1}=L_{ac}^{-1}+\lambda _{\mathrm {mfp},e}^{-1}.

This should be understood as an asymptotic transport law for a strongly stochastic region: near the overlap threshold, partial barriers, surviving island remnants, and finite-correlation effects can all modify the simple diffusive picture.

30.7 Experimental tests of Rechester-Rosenbluth in the RFP

The reversed-field pinch provides a particularly clean laboratory test of stochastic magnetic transport because its standard state contains a broad spectrum of internally resonant tearing modes driven by the current-density gradient, and the resulting islands overlap across a substantial radial interval. In the Madison Symmetric Torus, Biewer, Forest, and collaborators used power balance, equilibrium reconstruction, and modeled internal fluctuations to compare the measured electron thermal diffusivity to stochastic-field transport. They found that the electron heat flux was primarily conductive rather than convective, that the measured $\chi _e$ agreed well with both field-line tracing and Rechester– Rosenbluth-like estimates in the overlap region, and that transport fell near the reversal surface where magnetic diffusion is small Biewer et al. (2003). Follow-on experiments documenting the behavior of fast electrons showed that the thermal diffusivity depended strongly on $v_\parallel $, further supporting the Rechester-Rosenbluth picture O‚ O’Connell et al. (2003).

A sharper test was later carried out through the full sawtooth cycle by Reusch, Forest, and collaborators using high-time-resolution Thomson scattering together with nonlinear resistive-MHD simulations at the experimental Lundquist number. If one estimates the transport directly from field-line wandering,

\chi _{MD}=v_{th,e}D_{\rm mag},

the result tends to overpredict the measured diffusion. Agreement with experiment was recovered when the parallel transport was reduced by the circulating-particle fraction,

\chi _e \simeq f_c\,v_{th,e}D_{\rm mag},

which makes explicit that trapped particles do not freely stream along the stochastic web Reusch et al. (2011). This is closely analogous to the suppression of conduction in tangled astrophysical magnetic fields discussed by Chandran, Cowley, and collaborators for galaxy-cluster plasmas Chandran and Cowley (1998); Chandran et al. (1999).

30.8 Closing bridge: unified $m=1$ theory and the two tearing branches

The cylindrical tearing calculation above deliberately emphasized the ordinary $m>1$ tearing mode because it gives a clean first example of $\Delta '$. The historical tokamak problem, however, is the $m=n=1$ internal mode. This case is special because the same displacement of the core can appear as an ideal internal kink, a resistive internal kink, or an $m=1$ tearing mode, depending on how close the equilibrium is to ideal marginal stability. Coppi, Galvão, Pellat, Rosenbluth, and Rutherford showed that these are not three unrelated instabilities: they are different limits of a single inner–outer matching problem Coppi et al. (1976). The later review by Ara and collaborators and the tokamak-focused review by Migliuolo are useful road maps through this literature Ara et al. (1978); Migliuolo (1993).

For the $m=1$ internal mode the ideal outer eigenfunction is nearly a rigid translation of the plasma inside the $q=1$ surface, and is small outside it:

\xi (r) \simeq \xi _0 \text { for } r<r_1, \qquad \xi (r) \simeq 0 \text { for } r>r_1, \qquad q(r_1)=1. \tag{30.78}

The thin layer at $r=r_1$ must decide how that discontinuous ideal displacement is healed. Two local times are useful:

\tau _H \equiv \left (\frac {\sqrt {\mu _0\rho }}{q'B_\theta }\right )_{r_1}, \qquad \tau _R \equiv \left (\frac {\mu _0 r^2}{\eta }\right )_{r_1}. \tag{30.79}

Here $\tau _H$ is the Alfvénic time based on the magnetic shear at the $q=1$ surface, while $\tau _R$ is the resistive diffusion time across the local radius. Near ideal marginality the outer-region potential energy controls the matching. In the ideal-stable, positive-$\Delta '$ limit one finds, schematically,

\Delta ' = \frac {\pi ^2 R\left (r q'^2 B_\theta ^2/\mu _0\right )_{r_1}} {\delta W/\xi _0^2}, \tag{30.80}

so a small stabilizing ideal energy denominator produces a large tearing drive. This is the cleanest way to state the $m=1$ peculiarity: the mode is ideal-stable in energy-principle language, but the resistive layer can still reconnect the displaced core.

The three limits are then easy to remember. The ideal internal kink is obtained when the resistive layer is irrelevant. The resistive internal kink is obtained near $\delta W\simeq 0$, where the layer itself sets the rate,

\gamma _{\eta k}\sim \tau _H^{-2/3}\tau _R^{-1/3}. \tag{30.81}

The $m=1$ tearing limit is obtained on the ideal-stable side, where Eq. (30.80) may be inserted into the constant-$\psi $ tearing law. With the normalization of Eq. (30.79), this gives

\gamma \simeq 0.55 \left [ \frac {\pi ^2 R\left (r q'^2B_\theta ^2/\mu _0\right )_{r_1}} {\delta W/\xi _0^2} \right ]^{4/5} \tau _H^{-2/5}\tau _R^{-3/5}. \tag{30.82}

The same calculation can be written as a single Coppi dispersion relation. With the energy-principle sign convention used here, $\delta W<0$ is ideal-kink unstable, so I write the interpolation with an explicit minus sign. Defining

\delta W_H \equiv \frac {\delta W/\xi _0^2} {2R\left (rB_\theta q'\right )_{r_1}^2/\mu _0}, \tag{30.83}

one convenient form is

\gamma \tau _H = -\delta W_H \left \{ \frac {1}{8}\left (\frac {\gamma }{\gamma _{\eta k}}\right )^{9/4} \frac { \Gamma \!\left [\left ((\gamma /\gamma _{\eta k})^{3/2}-1\right )/4\right ]} {\Gamma \!\left [\left ((\gamma /\gamma _{\eta k})^{3/2}+5\right )/4\right ]} \right \}. \tag{30.84}

One should not get lost in the gamma functions. Their purpose is to make the interpolation exact: large $\gamma /\gamma _{\eta k}$ gives the ideal internal-kink limit $\gamma \tau _H\simeq -\delta W_H$, $\delta W_H\rightarrow 0$ gives the resistive-kink rate (30.81), and $\gamma \ll \gamma _{\eta k}$ on the ideal-stable side gives the tearing law (30.82).

The reason this belongs at the end of the tearing lecture is that it prepares the reader for the plasmoid calculation in Lecture 33, especially the high-Lundquist-number current-sheet analysis of Loureiro, Schekochihin, and Cowley Loureiro et al. (2007). In the plasmoid problem the control parameter is not $\delta W_H$ but the wavelength along a long current sheet. Take a local sheet half-thickness $a$, local Alfvén time $\tau _{A,a}=a/V_A$, magnetic diffusivity $\eta _m=\eta /\mu _0$, and $S_a=aV_A/\eta _m$. For a Harris-like sheet, the outer matching parameter is

\Delta 'a = 2\left (\frac {1}{ka}-ka\right ). \tag{30.85}

Thus the unstable long-wave side has $ka<1$, but within that unstable interval there are still two asymptotic branches. The constant-$\psi $, or FKR, branch is

\gamma \tau _{A,a} \sim S_a^{-3/5}(\Delta 'a)^{4/5}(ka)^{2/5}, \qquad \Delta '\delta _{\rm in}\ll 1. \tag{30.86}

For the Harris sheet at $ka\ll 1$, this becomes

\gamma \tau _{A,a} \sim S_a^{-3/5}(ka)^{-2/5}, \qquad ka\,S_a^{1/4}\gg 1, \tag{30.87}

which is the high-$k$ tearing branch used later in Eq. (33.31). Here “high-$k$” means high relative to the crossover scale $S_a^{-1/4}/a$, not $ka>1$, since $ka>1$ is already stable for this simple sheet.

At smaller $k$, the FKR approximation predicts an inner layer so wide that $\Delta '\delta _{\rm in}$ is no longer small. The perturbation is then nonconstant across the layer, and the Coppi branch replaces the FKR branch:

\gamma \tau _{A,a} \sim S_a^{-1/3}(ka)^{2/3}, \qquad ka\,S_a^{1/4}\ll 1. \tag{30.88}

This is the low-$k$ branch used later in Eq. (33.33). The fastest plasmoid mode sits where the two branches meet,

ka\,S_a^{1/4}\sim 1, \qquad k_{\,max}a\sim S_a^{-1/4}, \qquad \gamma _{\max }\tau _{A,a}\sim S_a^{-1/2}. \tag{30.89}

This is the same lesson as the unified $m=1$ theory, translated from a tokamak core to an elongated reconnection layer: a single resistive inner-layer problem has several outer limits, and the observed branch is selected by the matching parameter.

Takeaways

Tearing modes are matched-asymptotic instabilities: ideal outside, resistive inside.

The outer ideal problem determines the single number $\Delta '$, defined in Eq. (30.2).

In the constant-$\psi $ regime the classical growth law is $\gamma \propto \eta ^{3/5}(\Delta ')^{4/5}$.

The two-region current profile (30.18) is an especially useful worked example because $\Delta '$ can be computed analytically, changes sign, and can host both $3/2$ and $2/1$ resonant surfaces.

The generalized Rutherford equation explains why neoclassical tearing modes are subcritical: even a classically stable surface can grow if a seed island is large enough.

Once an island locks to the wall, the perturbation takes on a more external character and can become a direct disruption precursor.

The unified $m=1$ theory organizes the ideal kink, resistive kink, and $m=1$ tearing mode as limits of one matching problem.

The same matching logic produces the high-$k$ FKR and low-$k$ Coppi branches that underlie the plasmoid instability in Lecture 33.

In the nonlinear phase, multiple islands can overlap and make the magnetic field stochastic.

Bibliography

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Problems

Problem 30.1.: Starting from (30.17), rederive the jump condition (30.28) for the step profile (30.18).
Problem 30.2.: For the $m=2$ example, verify Eq. (30.34) explicitly and show that the large-wall limit gives Eq. (30.35).
Problem 30.3.: Starting from Eqs. (30.31) and (30.33), derive the general large-wall result (30.40) and the sign criterion (30.42). Use it to discuss whether the step profile supports tearing-unstable $3/2$ and $2/1$ surfaces for a given choice of $q_c$ and $q_a$.
Problem 30.4.: Using the schematic generalized Rutherford equation (30.69), explain in words why a neoclassical tearing mode is a nonlinear instability even when $\Delta '<0$. Which terms provide the seed threshold and which terms provide saturation?
Problem 30.5.: Repeat the outer calculation for a smooth current profile such as $J_z=J_0(1-r^2/a^2)$ and compare the sign of $\Delta '$ with the step-profile result.
Problem 30.6.: Starting from the tutorial box around Eqs. (30.56)–(30.59), show how the exact matching coefficient modifies the pure scaling argument for the FKR law.
Problem 30.7.: Using Eqs. (30.87) and (30.88), show that the fastest tearing mode of a local Harris sheet satisfies Eq. (30.89). Then use the Sweet–Parker relation $a/L\sim S^{-1/2}$ to recover $k_{\max }L\sim S^{3/8}$ and $\gamma _{\max }\tau _{A,L}\sim S^{1/4}$, as in Lecture 33.