Lecture 32 Magnetic Helicity, Relaxation, and Self-Organization

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Lecture 32
Magnetic Helicity, Relaxation, and Self-Organization

Overview

Magnetic helicity is the quantity that survives when ideal flux freezing is only weakly broken. That makes it the natural organizing principle for magnetic relaxation. Magnetic energy can be rapidly dissipated by reconnection and short-wavelength structure, but helicity is much harder to remove because it measures the large-scale linkage, twist, and writhe of the field.
For toroidal current-carrying plasmas this is not just a topological curiosity. Inductive drive tends to push the plasma toward a natural current profile set by Ohm’s law and the conductivity profile. That natural profile is often more peaked than the plasma can stably support. Tearing, quasi-interchange, and related MHD activity then act as a self-organization mechanism that broadens $\lambda \equiv \muo \,\mathbf J\!\cdot \!\mathbf B/B^2$ back toward marginal stability. In that sense, sawteeth, reversed-field-pinch relaxation, spheromak sustainment, flux pumping, and non-solenoidal startup methods all belong to one family of ideas: helicity is injected or approximately conserved, while the current profile reorganizes.

Historical Perspective

The modern story begins with Elsasser, who emphasized magnetic linkage in dynamo theory, and with Woltjer, who proved that magnetic helicity is conserved in ideal MHD and that the minimum-energy state at fixed helicity is force free Elsasser (1956); Woltjer (1958a,b). Kruskal and Kulsrud gave the related toroidal-surface formulation that later became standard in fusion applications Kruskal and Kulsrud (1958). Taylor’s crucial step was to argue that in a weakly resistive plasma reconnection can rapidly reduce magnetic energy while leaving the global helicity almost unchanged, so the plasma relaxes toward a constant-$\lambda $ state Taylor (1974). That point of view became central to reversed-field pinches, spheromaks, and compact toroids Prager (1990); Ortolani and Schnack (1993); Jarboe (1994).
The same logic later reappeared inside tokamaks in a less dramatic but equally important way. Jensen and Chu, and then Boozer, made explicit how helicity injection appears in a torus through the loop-voltage times toroidal-flux bookkeeping that is natural for transformer-driven plasmas Jensen and Chu (1984); Boozer (1986). Inductive current diffusion tends to peak the central current profile; MHD activity then broadens it again. Sawteeth are the classic example, but the same general idea underlies modern flux-pumping and hybrid-scenario physics, where a saturated $(m,n)=(1,1)$ structure can maintain $q_0\gtrsim 1$ without periodic crash-recovery cycles Jardin et al. (2015); Krebs et al. (2017); Jardin et al. (2020).

Caution

Two cautions matter here.
First, magnetic helicity is gauge dependent if magnetic flux crosses the boundary. For open systems one should really use relative helicity. In the derivations below we mostly assume a closed or perfectly conducting boundary with $\mathbf B\cdot \hat {\mathbf n}=0$, where ordinary helicity is well posed.
Second, $\lambda =\muo \,\mathbf J\!\cdot \!\mathbf B/B^2$ is a very useful proxy for tearing drive, but it is not a complete stability criterion. Tearing still depends on rational surfaces and matching indices such as $\Delta '$, while resistive-interchange activity also depends on pressure gradients and magnetic shear. The point of this lecture is not that $\lambda $ tells the whole story; it is that self-organization is easiest to understand when one first asks what profile inductive relaxation is trying to create.

32.1 Helicity as an ideal invariant

Definition and topological meaning. Magnetic helicity is defined by

K \equiv \int _V \mathbf A\cdot \mathbf B\,dV, \qquad \mathbf B = \nabla \times \mathbf A. \tag{32.1}

It is not a pointwise thermodynamic variable; it is a global measure of how magnetic flux is linked, twisted, and writhed. The reason it matters in MHD is already implicit in the frozen-in discussion of Lecture 4 ideal evolution preserves field-line connectivity [Eqs. (4.13)–(4.13)], so topological quantities built from that connectivity are especially robust.

Figure 32.1: A thin flux tube of flux $d\Phi $. Helicity may be interpreted as a measure of the linkage, twist, and writhe carried by such flux tubes.

Flux-tube interpretation. For a thin flux tube carrying flux $d\Phi $, one may write

K = \int _V \mathbf A\cdot d\boldsymbol {\ell }\; B\,dS = \Psi \,d\Phi , \tag{32.2}

where $\Psi $ is the poloidal or linking flux seen by that tube. Thus helicity counts flux weighted by linked flux. In axisymmetric toroidal geometry one may write, with the usual gauge choice $A_\phi = \psi /R$,

K = \int _V A_\phi B_\phi \,dV = \int _0^{\psi _{\rm wall}} \psi \, d\Phi _t = \int _0^{\psi _{\rm wall}} \psi \,q(\psi )\,d\psi , \tag{32.3}

where $\Phi _t$ is the toroidal flux and $q=d\Phi _t/d\psi $ in the standard axisymmetric notation. Equation (32.3) is one clean way to see that helicity couples current-profile physics to magnetic topology.

Evolution equation. Starting from

\mathbf E = -\pp {\mathbf A}{t} - \nabla \phi , \qquad \pp {\mathbf B}{t} = -\nabla \times \mathbf E, \tag{32.4}

one finds

\begin{aligned}\pp {}{t}(\mathbf A\cdot \mathbf B) &= \pp {\mathbf A}{t}\cdot \mathbf B + \mathbf A\cdot \pp {\mathbf B}{t} \nonumber \\ &= (-\mathbf E-\nabla \phi )\cdot \mathbf B - \mathbf A\cdot (\nabla \times \mathbf E) \nonumber \\ &= -\mathbf E\cdot \mathbf B - \nabla \phi \cdot \mathbf B -\left [\mathbf E\cdot (\nabla \times \mathbf A)-\nabla \cdot (\mathbf A\times \mathbf E)\right ] \nonumber \\ &= -2\,\mathbf E\cdot \mathbf B - \nabla \cdot (\phi \mathbf B) + \nabla \cdot (\mathbf A\times \mathbf E).\end{aligned}

Therefore

\pp {}{t}(\mathbf A\cdot \mathbf B) = -2\,\mathbf E\cdot \mathbf B + \nabla \cdot \left (\mathbf A\times \mathbf E-\phi \mathbf B\right ). \tag{32.6}

Integrating over the volume gives

\dv {K}{t} = -2\int _V \mathbf E\cdot \mathbf B\,dV + \oint _S\left (\mathbf A\times \mathbf E-\phi \mathbf B\right )\cdot d\mathbf S. \tag{32.7}

For a closed ideal system with $\mathbf B\cdot \hat {\mathbf n}=0$ and a fixed-boundary gauge choice, the surface term vanishes. Then ideal MHD gives $\mathbf E\cdot \mathbf B=0$ from Eq. (1.9), so

\dv {K}{t}=0. \tag{32.8}

This is the global statement behind the frozen-flux arguments of Lecture 4.

Resistive decay of energy and helicity. If the only non-ideal effect retained in Ohm’s law is scalar resistivity,

\mathbf E + \mathbf v\times \mathbf B = \eta \mathbf J,

then $\mathbf E\cdot \mathbf B = \eta \,\mathbf J\cdot \mathbf B$. For a closed system, Eq. (32.7) becomes

\dv {K}{t} = -2\eta \int _V \mathbf J\cdot \mathbf B\,dV. \tag{32.10}

Likewise the magnetic energy

W = \int _V \frac {B^2}{2\muo }\,dV \tag{32.11}

obeys

\dv {W}{t} = -\eta \int _V J^2\,dV \tag{32.12}

when the boundary is closed and the flow does no net work on it.

A useful spectral cartoon is obtained by writing $\tilde B = \sum _{\mathbf k} B_k e^{i\mathbf k\cdot \mathbf x}$. Then schematically

W \sim \sum _k B_k^2, \qquad K \sim \sum _k \frac {B_k^2}{k},

so short-wavelength structure carries relatively more energy than helicity. Under resistive decay,

\dv {W}{t}\sim -\eta \sum _k k^2 B_k^2, \qquad \dv {K}{t}\sim -2\eta \sum _k k B_k^2. \tag{32.14}

Thus small scales dissipate energy very efficiently while helicity is weighted more toward the large scales. This is the heuristic reason that Taylor relaxation works at all.

32.2 Woltjer–Taylor relaxation

Force-free equilibria. A force-free equilibrium satisfies

\mathbf J\times \mathbf B = 0,

so current flows parallel to the magnetic field:

\nabla \times \mathbf B = \muo \mathbf J = \lambda (\mathbf x)\,\mathbf B. \tag{32.16}

In a general equilibrium $\lambda $ may vary from surface to surface. Taylor’s claim is stronger: a weakly resistive plasma that relaxes rapidly while approximately conserving $K$ should approach a state with constant $\lambda $.

Constrained minimization. We therefore minimize the magnetic energy,

W = \int _V \frac {B^2}{2\muo }\,dV,

subject to fixed helicity,

K = \int _V \mathbf A\cdot \mathbf B\,dV = \text {const}.

Introduce the functional

\mathcal F[\mathbf A] = \int _V \frac {B^2}{2\muo }\,dV - \frac {\lambda }{2\muo }\int _V \mathbf A\cdot \mathbf B\,dV. \tag{32.19}

Its variation is

\begin{aligned}\delta \mathcal F &= \frac {1}{\muo }\int _V \mathbf B\cdot (\nabla \times \delta \mathbf A)\,dV - \frac {\lambda }{2\muo }\,\delta \int _V \mathbf A\cdot \mathbf B\,dV.\end{aligned}

Using

\mathbf B\cdot (\nabla \times \delta \mathbf A) = \nabla \cdot (\mathbf B\times \delta \mathbf A) + \delta \mathbf A\cdot (\nabla \times \mathbf B)

and discarding the boundary term for a fixed conducting shell gives

\delta W = \frac {1}{\muo }\int _V \delta \mathbf A\cdot (\nabla \times \mathbf B)\,dV + \frac {1}{\mu _0}\int _S (\delta \vect {A} \times \vect {B} ) \cdot \hat {\vect {n}} dS

Similarly,

\begin{aligned}\delta K &= \int _V (\nabla \times \delta \mathbf A)\cdot \mathbf A\,dV + \int _V \mathbf B\cdot \delta \mathbf A\,dV \nonumber \\ &= \int _V \nabla \cdot (\mathbf A\times \delta \mathbf A)\,dV + 2\int _V \mathbf B\cdot \delta \mathbf A\,dV + \int _S (\vect {A}\times \delta \vect {A}) \cdot \hat {\vect {n}} dS \nonumber \\ &= 2\int _V \mathbf B\cdot \delta \mathbf A\,dV, + \frac {1}{\mu _0}\int _S \left ( \hat {\vect {n}} \times \delta \vect {A} \right ) \cdot \left ( \vect {B} + \lambda \vect {A} \right ) dS\end{aligned}

We can see the boundary term vanishes by noting that $\hat {\vect {n}} \times \delta \vect {E} =- i \omega \hat {\vect {n}} \times \delta \vect {A}=0$ on a flux conserving boundary. Therefore

\delta \mathcal F = \frac {1}{\muo }\int _V \delta \mathbf A\cdot \left (\nabla \times \mathbf B - \lambda \mathbf B\right )\,dV.

Since this must vanish for arbitrary $\delta \mathbf A$, the Euler–Lagrange condition is

\nabla \times \mathbf B = \lambda \mathbf B, \tag{32.25}

with $\lambda $ a spatial constant. This is the Taylor relaxed state.

Cylindrical Bessel family. Inside a cylindrical conducting shell the constant-$\lambda $ state has the familiar form

\begin{aligned}B_z(r) &= B_0 J_0(\lambda r), \\ B_\theta (r) &= B_0 J_1(\lambda r), \\ B_r &= 0.\end{aligned} \tag{32.26}

Writing $x=r/a$, $\xi = \lambda a$, and $A=R_0/a$, the safety factor becomes

q(x) = \frac {rB_z}{R_0 B_\theta } = \frac {x}{A}\,\frac {J_0(\xi x)}{J_1(\xi x)}, \qquad q_a = \frac {J_0(\xi )}{A J_1(\xi )}. \tag{32.29}

For $\xi $ below the first zero of $J_0$, $q_a>0$ and the profiles are tokamak-like. When $\xi $ passes that zero, the edge toroidal field reverses and $q_a$ becomes RFP-like. This continuous mapping is one reason the Bessel family forms such a useful bridge between tokamak-like and RFP-like profiles.

Figure 32.2: Mapping from the Bessel parameter $\xi =\lambda a$ to edge safety factor $q_a$ in the cylindrical Taylor family. Positive $q_a$ corresponds to tokamak-like profiles; crossing the first zero of $J_0$ leads to RFP-like reversal.

Figure 32.3: Representative $q(r)$ profiles in the cylindrical Bessel/Taylor family. The window $0.5\lesssim q_a\lesssim 2$ corresponds to broad, relatively low-shear tokamak-like profiles, whereas $q_a\approx 0$ and $q_a<0$ are progressively more RFP-like.

The same parameter $\xi =\lambda a$ that labels the Taylor family also labels the shear. Broad tokamak-like states with $0.5\lesssim q_a\lesssim 2$ have comparatively weak shear across much of the minor radius, while RFP-like states and strongly peaked-current tokamak states have much larger shear. That distinction matters below when we ask whether the natural inductive profile already resembles a flat-$\lambda $ state or whether MHD must reorganize it.

32.3 Helicity content in a tokamak

Why the loop voltage can appear. Equation (32.7) looks as though a closed conducting boundary would leave only the volume term $-2\int \mathbf E\cdot \mathbf B\,dV$. That is correct for a simply connected volume. A tokamak is subtler because the plasma region is toroidal and therefore multiply connected: the vector potential has nontrivial toroidal and poloidal loop integrals that cannot be removed by a single-valued gauge. In that case the gauge-invariant quantity is not the naive $K$ but the helicity content of Boozer Boozer (1986); Jensen and Chu (1984),

K_0 \equiv \int _V \mathbf A\cdot \mathbf B\,dV - \Psi \,\Psi _p, \tag{32.30}

where $\Psi $ is the toroidal flux enclosed by the wall and $\Psi _p$ is the linked poloidal flux. More generally one writes the subtraction in terms of the poloidal and toroidal loop integrals of $\mathbf A$, but Eq. (32.30) is the clean axisymmetric version.

Evolution with solenoidal drive. Let $V_s$ denote the toroidal loop voltage produced by the central solenoid. Differentiating Eq. (32.30) and using Eq. (32.7) gives

\dv {K_0}{t} = 2V_s\Psi - 2\int _V \mathbf E\cdot \mathbf B\,dV - 2\oint _w \Phi _w\,\mathbf B\cdot d\mathbf a, \tag{32.31}

where $\Phi _w$ is the electric potential on the wall. For an equipotential conducting wall with no open-field-line voltage drop, the last term vanishes. The important point is that the solenoidal loop voltage appears explicitly as a helicity-injection term $2V_s\Psi $.

From $\mathbf E\cdot \mathbf B$ to the plasma loop voltage. On a set of closed magnetic surfaces one may write

\mathbf E = -\nabla \Phi + \frac {V_\ell (\psi )}{2\pi }\nabla \phi , \tag{32.32}

with $V_\ell (\psi )$ chosen so that $\Phi $ is single valued on each surface. Dotting with $\mathbf B$ gives

\mathbf E\cdot \mathbf B = -\mathbf B\cdot \nabla \Phi + \frac {V_\ell (\psi )}{2\pi }\,\mathbf B\cdot \nabla \phi .

The first term vanishes on a flux-surface average, so

\begin{aligned}\int _V \mathbf E\cdot \mathbf B\,dV &= \int _V \frac {V_\ell (\psi )}{2\pi }\,\mathbf B\cdot \nabla \phi \,dV \nonumber \\ &= \int _0^\Psi d\psi \int \frac {d\theta \,d\phi }{(2\pi )^2}\,V_\ell (\psi ) \nonumber \\ &= \int _0^\Psi V_\ell (\psi )\,d\psi ,\end{aligned} \tag{32.34}

because $(\mathbf B\cdot \nabla \phi /2\pi )\,dV=d\psi \,d\theta \,d\phi /(2\pi )^2$. Therefore

\dv {K_0}{t} = 2\int _0^\Psi \left [V_s - V_\ell (\psi )\right ]\,d\psi - 2\oint _w \Phi _w\,\mathbf B\cdot d\mathbf a. \tag{32.35}

If the plasma loop voltage is nearly constant across surfaces and the wall term vanishes, this reduces to

\dv {K_0}{t}\approx 2\Psi \left (V_s - V_\ell \right ). \tag{32.36}

This is the tokamak form that often appears in helicity-conservation arguments.

No contradiction with ideal conservation. There is no conflict between Eqs. (32.8) and (32.35). Equation (32.8) refers to the ordinary helicity $K$ in a simply connected closed system with no externally linked transformer flux. Equation (32.35) refers to the gauge-invariant helicity content $K_0$ of a multiply connected torus, for which changing the linked solenoidal flux injects helicity through the term $2V_s\Psi $. What looked like a pure boundary/gauge subtlety in Eq. (32.7) becomes an explicit loop-voltage term once the toroidal topology is treated correctly.

32.4 Inductive drive and the natural current profile

Surface-averaged Ohm’s law. Reuse Eq. (32.32) for a stationary axisymmetric discharge. Dotting Eq. (1.9) with $\mathbf B$ gives

\eta \,\mathbf J\cdot \mathbf B = \mathbf E\cdot \mathbf B.

Now flux-surface average. The electrostatic term disappears because $\langle \mathbf B\cdot \nabla \Phi \rangle = 0$ on a closed surface, so one obtains the standard stationary-state condition

\eta (\psi )\,\langle \mathbf J\cdot \mathbf B\rangle = \frac {V_\ell (\psi )}{2\pi }\,\langle \mathbf B\cdot \nabla \phi \rangle . \tag{32.38}

Equivalently,

\langle \mathbf J\cdot \mathbf B\rangle = \sigma (\psi )\,\frac {V_\ell (\psi )}{2\pi }\,\langle \mathbf B\cdot \nabla \phi \rangle , \qquad \sigma \equiv \eta ^{-1}. \tag{32.39}

This is the current profile that inductive diffusion tries to build if no instability intervenes.

Definition of the natural profile. Motivated by Taylor’s constant-$\lambda $ state, define the surface-averaged proxy

\lambda _{\rm nat}(\psi ) \equiv \muo \,\frac {\langle \mathbf J\cdot \mathbf B\rangle }{\langle B^2\rangle } = \muo \,\sigma (\psi )\,\frac {V_\ell (\psi )}{2\pi }\, \frac {\langle \mathbf B\cdot \nabla \phi \rangle }{\langle B^2\rangle }. \tag{32.40}

At large aspect ratio one may read this schematically as

\lambda _{\rm nat} \propto \sigma \,\frac {E_\phi B_\phi }{B^2}. \tag{32.41}

Equation (32.40) is the central quantity of this lecture. It is the inductively relaxed current profile: the profile the plasma would approach if current diffusion operated without tearing, quasi-interchange, or sawtooth-like reorganization.

Bessel-model evaluation. In the cylindrical Taylor family of Eqs. (32.26)–(32.27), $\mathbf B\cdot \nabla \phi $ is proportional to $B_z$, so Eq. (32.40) becomes

\lambda _{\rm nat}(x) \propto \sigma (x) \frac {J_0(\xi x)}{J_0^2(\xi x)+J_1^2(\xi x)}. \tag{32.42}

To compare shapes rather than absolute normalization, it is convenient to define

\hat \lambda _{\rm nat}(x) \equiv \frac {\lambda _{\rm nat}(x)}{\langle \lambda _{\rm nat}\rangle }, \tag{32.43}

where the average is weighted by cylindrical area. A perfectly aligned Taylor-like state would have $\hat \lambda _{\rm nat}=1$.

For the conductivity we may consider two simple models. A flat electron-temperature profile gives

\sigma _{\rm flat}(x)=\sigma _0, \tag{32.44}

whereas a peaked profile may be represented by

T_e(x)=T_{e,\rm edge} + \left (T_{e0}-T_{e,\rm edge}\right )(1-x^2)^2, \qquad \sigma (x)\propto T_e^{3/2}(x). \tag{32.45}

These two choices are enough to show the basic point.

Figure 32.4: Normalized natural current-profile proxy $\hat \lambda _{\rm nat}$ for the Bessel/Taylor family. Left: flat conductivity, corresponding to a flat $T_e$ profile. Right: peaked conductivity, $\sigma \propto T_e^{3/2}$, using Eq. (32.45). The horizontal dashed line is the perfectly aligned constant-$\lambda $ limit.

A particularly convenient scalar measure of misalignment is

\mathcal A \equiv \left \langle \left (\hat \lambda _{\rm nat}-1\right )^2\right \rangle ^{1/2}, \tag{32.46}

again with cylindrical area weighting. In the Bessel model one finds a remarkably clean result: for flat $\sigma $ and roughly

0.5 \lesssim q_a \lesssim 2, \tag{32.47}

$\hat \lambda _{\rm nat}$ is nearly flat. In other words, the natural inductive profile already lies close to the Taylor-like reference and there is little need for violent magnetic relaxation. By contrast, if $T_e$ is strongly peaked or if the discharge is RFP-like with $q_a\approx 0$ or negative, Eq. (32.42) becomes strongly peaked and can even turn negative near the edge. Then the plasma is being driven toward a current profile that is generically hostile to tearing stability, so self-organization is practically inevitable.

Why this matters for tearing and interchange. The cylindrical tearing lecture showed that peaking of the parallel-current profile is exactly what provides free energy for $\Delta '$-driven island formation. The resistive-interchange lecture then showed that if pressure is also peaked and magnetic shear is low, pressure-driven parity can join the story as well. The present lecture provides the missing zeroth-order picture: before any instability appears, inductive relaxation is already pushing the system toward $\lambda _{\rm nat}(\psi )$. Magnetic self-organization is what happens when the plasma refuses to stay on that natural profile.

This viewpoint is especially useful for the tokamak sawtooth problem. If a peaked $T_e$ profile forces $\lambda _{\rm nat}$ to peak strongly, the central safety factor is driven downward, and $(m,n)=(1,1)$ activity becomes unavoidable. Depending on the pressure profile and central shear, the nonlinear saturating structure may look more tearing-like, more resistive-interchange-like, or more quasi-interchange-like; in practice the distinction is often one of emphasis rather than of separate worlds. Conversely, when $T_e$ is flat enough that Eq. (32.42) is already close to constant $\lambda $, the plasma can live in a broad-current, low-sawtooth state.

Low shear versus strong shear. The same comparison also clarifies why tearing and interchange often trade places rather than coexisting with equal strength. In the alignment window $0.5\lesssim q_a\lesssim 2$, the current profile is broad and the shear is relatively flat; if the electron-temperature profile is also broad, the natural inductive state is already close to constant $\lambda $ and there is little reason for a large sawtooth-like relaxation. Those same low-shear states are precisely the ones that are most vulnerable to resistive-interchange or quasi-interchange behavior if the pressure gradient is allowed to sharpen. By contrast, RFP-like states and tokamaks with strongly peaked current profiles generally have much stronger shear. That strong shear helps dramatically against interchange, but it comes with a more peaked $\lambda _{\rm nat}$, so the price is a greater drive for tearing and magnetic relaxation.

A useful viewpoint. A very compact way to say all of this is the following. Inductive systems are almost never trying to reach the marginally stable current profile. They are trying to reach the natural resistive profile imposed by Ohm’s law and conductivity. Tearing, sawteeth, and flux pumping are the mechanisms that stop them.

32.5 Experimental and modern perspective

RFPs and spheromaks. Reversed-field pinches and spheromaks are the cleanest laboratories for Taylor relaxation. There magnetic self-organization is not a small correction to an otherwise quiescent equilibrium; it is the main event. The Bessel family is therefore not just a classroom model but a genuine first approximation to the observed states, with relaxation repeatedly clamping the current profile near the edge of tearing marginality Prager (1990); Ortolani and Schnack (1993); Jarboe (1994). In that sense spheromaks are closely connected to the present discussion: they are systems in which helicity injection and magnetic relaxation are not side issues but the confinement principle itself.

32.6 Energy and Helicity Dissipation

In resistive MHD its easily shown that

\begin{aligned}\dv {W}{t} &= -\eta \int _V J^2\,dV,\\ \dv {K}{t} &= -2\eta \int \bm {J}\cdot \bm {B}\,dV.\end{aligned}

Considering the characteristic decay times of the large scale field gives similar times scales for resistive decay of both magnetic energy and helicity

\tau _W \sim \frac {\mu _0 a^2}{\eta }, \qquad \tau _K \sim \frac {\mu _0 a}{\eta \lambda },

so that

\frac {\tau _W}{\tau _K}\sim \frac {1}{a\lambda } \sim 1

for typical RFP and spheromak parameters.

During relaxation, however, there is a broad spacial spectrum of modes generated. In the RFP, for example, the mode spectrum consists primarily of $m=0,1$ modes and a broad range of $n$ numbers. Figuratively,

\tilde {B} = \sum _{\bf k} B_k \exp {i \vect {k}\cdot \vect {x}}

When considering the decay of the spectrum of modes

\begin{aligned}\dv {W}{t} &= -\eta \sum _k \int _V k^2 B_k^2 \,dV,\\ \dv {K}{t} &= -2\eta \sum _k \int _V k B_k^2 \\\end{aligned}

When the spectrum consists of considerable short wavelength "turbulence" (large $k$ modes) Helicity wins out as the energy quickly dissipates. Thus magnetic energy dissipates much faster than helicity, justifying Taylor’s hypothesis.

32.7 Beyond Taylor.

Real plasmas do not land on an exact global Taylor state. The relaxed profile is instead set by a competition between inductive peaking, finite conductivity gradients, and the requirement that the dominant tearing spectrum sit near marginality. In an RFP this means that the plasma is repeatedly peaked by Ohmic drive and then relaxed by reconnecting activity. The outcome is not $\lambda (r)=\text {const}$ everywhere, but a broadened profile with residual structure, especially near the edge where $T_e$ and $\sigma $ fall rapidly.

The measured fluctuation-induced emf. A useful way to say this is to write the mean parallel Ohm’s law as

E_\parallel + \mathcal E_\parallel = \eta J_\parallel , \qquad \mathcal E_\parallel \equiv \left \langle \tilde {\mathbf v}\times \tilde {\mathbf b}\right \rangle \cdot \frac {\mathbf B}{B}. \tag{32.56}

Classical MST measurements—first with edge probe correlations and later with direct spectroscopic measurements of the flow fluctuations—showed that $\mathcal E_\parallel $ accounts for the “missing” term in the parallel Ohm’s law during relaxation events and, in appropriate regimes, between them as well Ji et al. (1994); Ji and Prager (2002). In other words, the correlated fluctuations do the profile-broadening work that a purely resistive equilibrium cannot do.

In mean-field language one would write

\mathcal E_\parallel \simeq \alpha _{\rm dyn} B - \beta \,\muo J_\parallel + \cdots , \tag{32.57}

so that $\alpha _{\rm dyn}$ is the coefficient multiplying the mean field in the same way that the usual $\alpha $ effect is written in solar and astrophysical mean-field theory. This is mathematically the same object that appears in the solar $\alpha \Omega $ dynamo: the $\Omega $ effect shears poloidal field into toroidal field, while the $\alpha $ effect closes the loop by regenerating the complementary mean-field component Cameron et al. (2017). The comparison to the present lecture is especially clean because

\frac {E_\parallel }{B} + \alpha _{\rm dyn} \;\simeq \; \eta \,\frac {J_\parallel }{B} = \frac {\eta }{\muo }\,\lambda . \tag{32.58}

Thus $\alpha _{\rm dyn}$ enters the mean-field closure in essentially the same algebraic slot that $\lambda $ occupies in the relaxation problem: $\lambda $ measures the required field-aligned current per unit field, while $\alpha _{\rm dyn}$ measures the fluctuation-induced emf per unit field.

Caution

An opinion on terminology. The formal analogy with mean-field dynamo theory is real, and the RFP literature noticed it very early—apparently going back at least to Gimblett and Watkins Gimblett and Watkins (1975); Ji and Prager (2002). But I would still use the word dynamo cautiously here. In the solar $\alpha \Omega $ problem the large-scale magnetic field energy is regenerated from mechanical free energy associated with convection and differential rotation. In the RFP the external circuit has already supplied the helicity and current drive; the measured $\langle \tilde {\mathbf v}\times \tilde {\mathbf b}\rangle $ mainly redistributes current and flux so that the plasma can remain near a marginally stable state. In fact, the magnetic energy explicitly decreases. Calling this term a fluctuation-induced emf, or a relaxation emf, is therefore cleaner than implying a self-excited astrophysical dynamo in the strict energetic sense. By the same token, “flux pumping” in tokamaks is best viewed as closely related jargon for the same kind of mean emf, not as a wholly separate mechanism.

Profile models beyond the exact Taylor state. Several phenomenological models are used once one gives up on a globally constant $\lambda $:

Modified Bessel-function models,
Polynomial expansions for $B(r)$,
so-called “$\alpha $-models” with $\lambda (r)=\lambda _0(1-r^\alpha ),$ allowing finite pressure and edge gradients.

Here the profile-shape exponent $\alpha $ is completely unrelated to the mean-field dynamo coefficient $\alpha _{\rm dyn}$ in Eq. (32.57).

These models reproduce observed MST/RFP profiles remarkably well, with typical shape exponents $\alpha \simeq 4$. The plasma appears to self-organize toward states near tearing marginality, maintaining residual free energy that sustains turbulence and transport.

What survives from Taylor’s picture. Taylor relaxation therefore remains the organizing principle, but no longer as a literal prediction of a globally constant-$\lambda $ final state. What survives is the hierarchy of ideas:

Helicity is a comparatively robust invariant.
Magnetic energy is preferentially dissipated.
The plasma relaxes toward force-free or nearly force-free states.
Real devices hover near, rather than exactly on, the Taylor manifold.
The fluctuation-induced emf is the mechanism by which the system broadens a current profile that inductive drive alone would otherwise overpeak.

Although idealized, this framework explains many of the universal features of self-organized, toroidal plasmas.

It is also worth noting that helicity conservation and its scale dependence reappear in MHD turbulence and in the mean-field dynamo problem addressed in Lecture 10. Turbulence is beyond our scope here; the point for present purposes is simply that the same $\langle \tilde {\mathbf v}\times \tilde {\mathbf b}\rangle $ measured in RFPs is the laboratory cousin of the emf parameterized by the $\alpha $ effect in astrophysical dynamo theory Ji and Prager (2002); Cameron et al. (2017).

32.8 Helicity Injection and Tokamak Startup

In toroidal current-carrying devices, helicity injection is most naturally viewed as the injection of linkage between toroidal and poloidal magnetic flux. In fact, the conventional tokamak transformer already does this: a loop voltage $V_{\rm loop}$ acting on a plasma with toroidal flux $\Phi _t$ injects helicity at a rate of order

\dot K \sim 2 V_{\rm loop}\Phi _t,

up to the usual relative-helicity and boundary-flux conventions. From this viewpoint, non-solenoidal start-up methods are not different in kind from conventional inductive current drive; they are alternative ways of applying helicity flux to the torus Jensen and Chu (1984); Boozer (1986); Yoshida (1990).

The earliest tokamak demonstrations of dc helicity injection were carried out by Ono, Darrow, Forest, Park, Stix, and collaborators on ACT-1/CDX, where a low-energy electron beam from a biased cathode was launched along open helical field lines to an anode or limiter at vessel potential, producing and sustaining a tokamak discharge without transformer drive Ono et al. (1987). Subsequent CDX and CCT studies clarified the basic mechanism: when a dc voltage $V_{\rm inj}$ is applied between electrodes linked by an injector flux $\psi _{\rm inj}$, current follows the magnetic connection between the electrodes and injects helicity at a rate $\dot K \sim 2V_{\rm inj}\psi _{\rm inj}$ Jensen and Chu (1984); Darrow et al. (1990). In point-source implementations the cathode is a localized edge emitter and the return is a limiter or vessel component; in coaxial implementations the two electrodes are typically inner and outer divertor or vessel surfaces separated by insulators, with open field lines linking them Raman and Shevchenko (2014). The common requirement is therefore geometric rather than technological: the biased electrodes must be pierced by the same magnetic flux so that the imposed voltage appears along field lines that can communicate with the toroidal current channel.

This Princeton line of work continued in CDX-U. Forest, Hwang, Ono, and Darrow showed that once sufficient current is generated, the plasma self-field can exceed the imposed vacuum poloidal field locally and produce closed flux surfaces in a small-aspect-ratio tokamak geometry Forest et al. (1992). Magnetic reconstructions by Hwang, Forest, Darrow, Greene, and Ono then quantified the resulting noninductive current profiles Hwang et al. (1992). These papers are historically useful because they connect the original dc helicity-injection experiments to the later spherical-tokamak start-up, where, of course, bootstrap current is also creating helicity. Following CDX-U, a number of spherical tokamaks have continued to work on this reasearch: local helicity injection on Pegasus Battaglia et al. (2011), transient CHI on QUEST Kuroda et al. (2024), lower-hybrid/RF startup on TST-2 Shinya et al. (2015), and center-solenoid-free merging startup on UTST Inomoto et al. (2015). These are different experimental realizations of one common idea: inject linked flux and let the plasma relax into a current-carrying torus.

In ac helicity injection, the same topological idea is retained, but the boundary drive is oscillatory rather than steady. Instead of maintaining a fixed dc bias between two electrodes, one oscillates the applied voltage and/or the linking flux so that the cycle-averaged product of voltage and linked flux is nonzero. In the formulations of Jensen and Chu and of Bellan, the required phase relation between the oscillating fields rectifies into a net parallel current drive Jensen and Chu (1984); Bellan (1984, 1985). In practice the oscillating drive must still act on field lines that connect the driven boundary region to the toroidal current channel; the hardware may therefore be external coils, moving or oscillating flux surfaces, or oscillatory boundary voltages rather than fixed dc electrodes. Thus the distinction between dc and ac helicity injection is mainly whether the boundary drive is steady or time-periodic; in both cases the physics relies on magnetic connection, helicity transport, and subsequent relaxation of the toroidal plasma.

Tokamak-oriented oscillating-flux experiments were explored on DIII-D by modulating the linked toroidal flux through programmed plasma shaping Yamaguchi et al. (1995). AC helicity injection has also been demonstrated in the reversed field pinch context; partial current sustainment was demonstrated on ZT-40 Schoenberg et al. (1988) and similar results were also found on MST McCollam et al. (2005).

Flux pumping and hybrid scenarios. Modern flux-pumping work makes the self-organization picture especially sharp. In the nonlinear M3D-C$^1$ calculations of Jardin, Ferraro, and Krebs, a saturated central $(1,1)$ quasi-interchange mode generates a dynamo loop voltage that broadens the current profile and keeps $q_0\gtrsim 1$ in a stationary state Jardin et al. (2015); Krebs et al. (2017); Jardin et al. (2020). That outcome is exactly what this lecture suggests: the inductive system is trying to create a peaked $\lambda _{\rm nat}$, while the plasma self-organizes to clamp the actual profile near marginal stability. This is one reason the language of flux pumping is so natural in the ITER-hybrid context: one is trying to hold a broad current profile without repeated sawtooth reorganization. The language changes from RFP relaxation to hybrid-tokamak flux pumping, but the physics rhyme is unmistakable.

Takeaways

Magnetic helicity is the slowly changing invariant that survives magnetic relaxation. Taylor’s constant-$\lambda $ state is therefore the natural reference state, but the more revealing quantity for an inductively driven plasma is the natural profile $\lambda _{\rm nat}=\muo \langle \mathbf J\cdot \mathbf B\rangle /\langle B^2\rangle $, obtained from stationary Ohm’s law. If that natural profile already aligns with a broad, tearing-stable state, little magnetic reorganization is needed. If it is strongly peaked, the plasma must self-organize through tearing, sawteeth, quasi-interchange, or flux pumping. Helicity is the bookkeeping device; self-organization is the response.

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Problems

Problem 32.1.: Starting from Eq. (32.4), rederive Eq. (32.7) carefully, keeping all surface terms. Under what assumptions does it reduce to Eq. (32.10)?
Problem 32.2.: Starting from the functional (32.19), verify explicitly that the factor of $1/2$ in front of the helicity term is the one that leads to the Euler–Lagrange equation (32.25) without extra numerical coefficients.
Problem 32.3.: Starting from Eq. (32.30), derive the loop-voltage form (32.35). Show carefully where the term $2V_s\Psi $ enters and explain why this does not contradict the ideal-conservation statement (32.8).
Problem 32.4.: For the Bessel family, derive Eq. (32.29). Show that the first zero of $J_0$ marks the transition from tokamak-like $q_a>0$ to reversed-field-pinch-like $q_a<0$.
Problem 32.5.: Using Eq. (32.42), evaluate $\hat \lambda _{\rm nat}(x)$ for a flat conductivity profile and $q_a=1$. Show that it is nearly constant.
Problem 32.6.: Repeat the previous problem for a peaked temperature profile of the form Eq. (32.45). Discuss why the stronger peaking of $\hat \lambda _{\rm nat}$ suggests a greater tendency toward sawtooth-like or tearing-driven reorganization.