Lecture 20 Magnetic Interchange

← Previous Lecture Contents Next Lecture →

Lecture 20
Magnetic Interchange

Interchange modes are the cleanest place to see how equilibrium and stability talk to one another in MHD. They are the disturbances that try to exchange neighboring flux tubes while doing as little field-line bending as possible. That makes them the natural magnetic analogue of buoyant overturning, and it is why the gravitational interchange problem of the previous lecture is such a useful warm-up. In one language the instability is driven by bad curvature; in another it is governed by the geometry of the volume per unit flux, usually written as \(V'(\psi )\). Both points of view are useful, and both will reappear later in the ballooning lecture.

Overview

Big picture. Magnetic interchange is an energy-principle instability. The mode tries to move plasma across flux surfaces while minimizing magnetic-line bending, so the sign of \(\delta W\) is controlled primarily by curvature, pressure gradients, and the geometry of the volume available to a flux tube. In a paraxial mirror this produces the classic flute criterion; in a more global description it becomes the \(V'(\psi )\) criterion.

Historical Perspective

The classical mirror-interchange problem sits at the beginning of ideal-MHD stability theory. Kruskal and Schwarzschild first showed how a magnetized plasma can suffer a buoyancy-like instability even when the magnetic field is frozen into the fluid Kruskal and Schwarzschild (1954). Rosenbluth and Longmire then made the mirror connection explicit by interpreting bad magnetic curvature as an effective gravity and by showing why axisymmetric mirrors are generically vulnerable to flute modes Rosenbluth and Longmire (1957). Newcomb’s general stability formulation and the energy principle later placed these arguments on a rigorous variational footing Newcomb (1961). For mirrors this story remained central for decades, because any serious mirror design had to contend with interchange from the start: either by creating minimum-\(B\) geometry, by line-tying the ends, or by weighting the pressure toward good-curvature regions, as in the sloshing-ion idea of Hinton and Rosenbluth Hinton and Rosenbluth (1982). The modern axisymmetric treatment by Ryutov et al. is especially useful because it shows how the classical flute argument survives in a controlled paraxial expansion Ryutov et al. (2011).

Caution

Where this lecture stops. The interchange mode is the limit in which field-line bending is minimized. That is why the analysis can be so clean. Once parallel structure and bending energy become essential, one is already moving toward the ballooning problem. We therefore keep the focus here on flute-like perturbations and use the lecture as a bridge between equilibrium, the energy principle, and later ballooning theory.

20.1 Curvature as an Effective Gravity

Figure 20.1: Reminder that the gravitational interchange can be understood as charge build up and reinforced \(\vect {E}\times \vect {B}\) motion.

Single-particle picture. The basic intuition is already visible at the guiding-center level. In a slowly varying magnetic field the curvature and grad-\(B\) drifts are

\vect {v}_d = \frac {1}{qB^2}\, \B \times \left ( m v_\parallel ^2\,\vect {\kappa } + \mu \,\grad B \right ), \tag{20.1}

where

\vect {\kappa } \equiv (\vect {b}\cdot \grad )\vect {b}

is the magnetic curvature. For a low-\(\beta \) plasma in which pressure is neglible for equilibrium

\begin{aligned}\nabla \frac {B^2}{2 \mu _0} & = \frac {B^2}{\mu _0} \vect {\kappa } \\ \nabla B & = B \vect {\kappa }\end{aligned}

with approximately thermal perpendicular and parallel energies,

\mu B \sim \frac 12 m v_\perp ^2, \qquad m v_\parallel ^2 + \mu B \sim m v_\parallel ^2 + \frac 12 m v_\perp ^2,

so Eq. (20.1) suggests the effective acceleration

\vect {g}_{\rm eff} \sim - \left (v_\parallel ^2 + \frac 12 v_\perp ^2\right )\vect {\kappa } \sim c_s^2\,\vect {\kappa } \sim \frac {c_s^2}{R_{\rm curv}}\,\hat {\vect {R}}. \tag{20.6}

This is the magnetic counterpart of the gravitational acceleration in the previous lecture. Bad curvature means that the effective gravity points from high pressure toward low pressure, so a pressure gradient can drive a buoyant interchange exactly as in the Rayleigh–Taylor problem.

The fluid criterion. The single-particle picture is only a guide, but it points to the correct fluid quantity. When the plasma is pushed across the field, curvature enters the force balance through the same geometric combination that appeared in the anisotropic equilibrium lecture, namely \(\vect {\kappa }\cdot \grad p\). The local destabilizing tendency is therefore

\vect {\kappa }\cdot \grad p > 0. \tag{20.7}

This should be read in parallel with the gravitational criterion of Lecture 18: replace the ordinary gravity \(\vect g\) by the effective gravity associated with curvature, and the interchange analogy becomes immediate.

Orbit-averaged drift and pressure weighting. In a mirror one field line typically contains both good- and bad-curvature regions, so the orbit-averaged drift matters. For a single particle moving along a field line the net azimuthal displacement over one bounce is

\begin{aligned}\Delta \theta &= \oint \frac {v_{d,\theta }}{r}\,dt = \oint \frac {v_{d,\theta }}{r}\,\frac {d\ell }{v_\parallel } \nonumber \\ &= \oint \frac {\kappa }{qBr} \frac {mv_\parallel ^2 + \frac 12 m v_\perp ^2}{R_c} \frac {d\ell }{v_\parallel } \nonumber \\ &= \oint \sqrt {m} \frac {\kappa }{qBr} \frac { 2 \varepsilon - \mu B \ }{\sqrt {\varepsilon - \mu B}} d\ell\end{aligned} \tag{20.8}

Now lets consider the number \(N(\mu ,\varepsilon ) \) on a field line with given \(\mu \) and \(\varepsilon \). The average drift will be \(\langle \Delta \phi \rangle = \int d \mu d\varepsilon N(\mu , \varepsilon ) \Delta \phi (\mu , \varepsilon )\). The distribution function "density" at any point must vary as

f(\mu ,\varepsilon ,\ell ) = k N(\mu ,\varepsilon ) \frac {B}{v_\parallel }

where \(k\) is a constant. This follows directly from the real space divergence and the constancy of \(f v_\parallel dA = f \frac {v_\parallel }{B} d\Phi \) in a flux tube of flux \(d\Phi \).

A distribution function \(f(v_\parallel ,\mu )\) on the other hand, weights different parts of the field line differently, so the collective drift of a thin flux tube depends on how the pressure is distributed along the orbit. For a small flux tube of flux \(d\Phi \) the distribution-function average is

\begin{aligned}N\,\langle \Delta \theta \rangle &= d \Phi \frac {\sqrt {m}}{q} \int _1^2 d\ell \int d^3v\; f(v_\parallel ,\mu ) \frac {\kappa }{rB^2} \left (mv_\parallel ^2 + \frac 12 m v_\perp ^2\right ) \nonumber \\ &= d \Phi \frac {\sqrt {m}}{q} \int _1^2 d\ell \; \frac {\kappa }{rB^2} \left (p_\parallel + p_\perp \right ).\end{aligned} \tag{20.10}

where the limits of integration might span the maximum field in the throat of the mirror where \(\kappa = 0\). This is the first hint that the combination \(p_\parallel +p_\perp \) and its weighting along the field line will control the interchange drive. It also explains why highly anisotropic distributions can sometimes stabilize a nominally bad-curvature mirror by placing pressure in the good-curvature regions Hinton and Rosenbluth (1982). For isotropic plasmas the stability is governed by the integral

\int _1^2 d\ell \; \frac {\kappa }{rB^2}, \tag{20.11}

we will return to this later.

Figure 20.2: A flute or interchange displacement exchanges neighboring flux tubes while minimizing field-line bending. That is why curvature and pressure gradients dominate its energetics.

20.2 Paraxial Axisymmetric Mirror Geometry

Flux coordinates in a thin mirror. We now specialize to a paraxial axisymmetric mirror with radial scale \(r\) much smaller than the axial variation scale \(L\),

\epsilon \equiv \frac {r}{L} \ll 1.

To lowest order the field is approximately axial,

\B \simeq B(z)\,\hat {\vect z}, \qquad B=B(z),

and the usual axisymmetric flux function satisfies

2\pi \psi \equiv \int _S \B \cdot d\vect S.

For a circle of radius \(r\) this gives

2\pi \psi \simeq \pi r^2 B(z) \qquad \Longrightarrow \qquad \psi \simeq \frac {B(z)r^2}{2}, \tag{20.15}

so the radius of the flux surface \(\psi ={\rm const}\) is

r(z;\psi ) = \sqrt {\frac {2\psi }{B(z)}}. \tag{20.16}

Equation (20.15) is the same paraxial relation used earlier in the anisotropic-equilibrium lecture.

Curvature and normal derivatives. The field-line curvature is

\vect {\kappa } \equiv (\vect b\cdot \grad )\vect b, \qquad \kappa \equiv \vect {\kappa }\cdot \hat {\vect n},

where \(\hat {\vect n}\) points outward across the flux surfaces. For a low-\(\beta \) vacuum field,

\kappa \simeq \frac {1}{B}\pp {B}{n}, \tag{20.18}

which is the paraxial form of the more general curvature relations developed in Eq. (11.6). Because Eq. (20.15) implies

\pp {}{n} = \pp {\psi }{n}\pp {}{\psi } = rB\pp {}{\psi }, \tag{20.19}

Eq. (20.18) may also be written as

\kappa = -rB^2\pp {}{\psi }\left (\frac {1}{B}\right ). \tag{20.20}

Equilibrium identities. The gyrotropic equilibrium relations derived in Lecture 11 will be used repeatedly below. Parallel balance is

\pp {p_\parallel }{\ell } = -\frac {p_\parallel -p_\perp }{B}\pp {B}{\ell }, \tag{20.21}

which is Eq. (11.11), and perpendicular balance is

\pp {}{n}\left (p_\perp + \frac {B^2}{2\muo }\right ) + \kappa \left (p_\parallel -p_\perp -\frac {B^2}{2\muo }\right ) = 0, \tag{20.22}

which is the paraxial form of Eq. (11.13). These relations tell us that even before doing any linear stability analysis, curvature has already entered the equilibrium force balance. The instability problem asks whether a neighboring flux tube can lower the energy further by exploiting that same curvature.

20.3 Flute Ordering and the Field-Line-Integrated Equation

Why flute modes are special. Interchange modes are characterized by short perpendicular structure and very weak parallel variation. Write the displacement as

\vect \xi (\vect x,t) = \vect \xi (\vect x)\,e^{-i\omega t + i m\theta },

with large azimuthal mode number \(m\) and negligible field-line bending to leading order. In a low-\(\beta \) vacuum equilibrium, \(\J _0 = \curl \B /\muo \simeq 0\), so the linearized momentum equation becomes

-\omega ^2 \rho \,\vect \xi = \delta \vect f + \delta \J \times \B , \tag{20.24}

where \(\delta \vect f \equiv -\divergence \delta \tens P\). This is the same force-balance operator already encountered in the energy-principle lecture, but now specialized to the flute limit where magnetic bending is deliberately minimized.

Ideal induction and stream function. The frozen-in condition from Eq. (1.13) gives

\delta \B = \curl (\vect \xi \times \B ). \tag{20.25}

For the leading flute motion we set \(\delta \B \simeq 0\). Then

\curl (\vect \xi \times \B )=\vect 0 \qquad \Longrightarrow \qquad \vect \xi \times \B = \grad \varphi , \tag{20.26}

for a scalar stream function \(\varphi \). Taking the component along \(\B \) shows that

\vect b\cdot \grad \varphi = 0, \tag{20.27}

so \(\varphi \) is constant along each field line. The perpendicular displacement is therefore

\vect \xi _\perp = \frac {\B \times \grad \varphi }{B^2}. \tag{20.28}

In paraxial cylindrical geometry,

\xi _n = -\frac {1}{rB}\pp {\varphi }{\theta } = -\frac {i m}{rB}\,\varphi . \tag{20.29}

This is the basic kinematic ingredient behind the entire interchange reduction.

Perpendicular current. Cross Eq. (20.24) with \(\B /B^2\). The parallel part of the pressure force drops out, and one obtains the perpendicular perturbed current

\delta \J _\perp = \omega ^2\rho \,\frac {\B \times \vect \xi }{B^2} + \frac {\B \times \grad \delta p_\perp }{B^2} + (\delta p_\parallel -\delta p_\perp ) \frac {\B \times \vect \kappa }{B^2}. \tag{20.30}

Using Eq. (20.26),

\B \times \vect \xi = \grad \varphi ,

so the inertial term is simply proportional to \(\grad \varphi \). Equation (20.30) already exhibits the three pieces of flute physics: polarization current, diamagnetic response, and curvature coupling.

Current continuity integrated along the field line. Because charge is conserved,

\divergence \delta \J = 0.

Write

\delta \J = \delta J_\parallel \vect b + \delta \J _\perp .

Then

\divergence (\delta J_\parallel \vect b) = B\pp {}{\ell }\left (\frac {\delta J_\parallel }{B}\right ),

so integrating along the field line gives

\left (\frac {\delta J_\parallel }{B}\right )_2 - \left (\frac {\delta J_\parallel }{B}\right )_1 = \int _1^2 \frac {d\ell }{B}\,\divergence \delta \J _\perp . \tag{20.35}

This equation is important conceptually. For open or insulating ends the left-hand side vanishes; for line-tied or conducting ends it is the boundary channel through which the end condition modifies the interchange spectrum.

Divergence of the inertial term. Using Eq. (20.30) together with Eq. (20.26),

\divergence \delta \J _\perp = \omega ^2\divergence \left (\frac {\rho }{B^2}\grad \varphi \right ) + \divergence \left (\frac {\B }{B^2}\times \grad \delta p_\perp \right ) + \divergence \left [ (\delta p_\parallel -\delta p_\perp ) \frac {\B \times \vect \kappa }{B^2} \right ]. \tag{20.36}

The inertial term is the easiest to reduce. In paraxial cylindrical coordinates,

\begin{aligned}\divergence \left (\frac {\rho }{B^2}\grad \varphi \right ) &\simeq \frac {1}{r}\pp {}{r} \left (\frac {r\rho }{B^2}\pp {\varphi }{r}\right ) + \frac {1}{r^2}\pp {}{\theta } \left (\frac {\rho }{B^2}\pp {\varphi }{\theta }\right ) \nonumber \\ &= \frac {1}{r}\pp {}{r} \left (\frac {r\rho }{B^2}\pp {\varphi }{r}\right ) - \frac {m^2\rho }{r^2B^2}\,\varphi .\end{aligned} \tag{20.37}

Now use Eq. (20.15), namely \(r\pp {}{r}=2\psi \pp {}{\psi }\). Then

\begin{aligned}\frac {1}{r}\pp {}{r} \left (\frac {r\rho }{B^2}\pp {\varphi }{r}\right ) &= \frac {1}{r}\pp {}{r} \left (\frac {r\rho }{B^2}\,rB\pp {\varphi }{\psi }\right ) \nonumber \\ &= \frac {1}{r}\pp {}{r} \left (\frac {r^2\rho }{B}\pp {\varphi }{\psi }\right ) = \frac {1}{r}\pp {}{r} \left (\frac {2\rho \psi }{B^2}\pp {\varphi }{\psi }\right ) \nonumber \\ &= \frac {2}{B}\pp {}{\psi } \left (\rho \psi \pp {\varphi }{\psi }\right ).\end{aligned}

Likewise,

\frac {m^2\rho }{r^2B^2}\,\varphi = \frac {m^2\rho }{2\psi B}\,\varphi .

Therefore

\divergence \left (\frac {\rho }{B^2}\grad \varphi \right ) = \frac {2}{B} \left [ \pp {}{\psi }\left (\rho \psi \pp {\varphi }{\psi }\right ) - \frac {m^2\rho }{4\psi }\,\varphi \right ]. \tag{20.40}

Pressure contribution. The pressure terms can also be reduced explicitly. Using

\divergence (\vect A\times \vect C) = \vect C\cdot \curl \vect A - \vect A\cdot \curl \vect C,

with \(\vect A = \B /B^2\) and \(\vect C = \grad \delta p_\perp \) gives

\divergence \left (\frac {\B }{B^2}\times \grad \delta p_\perp \right ) = \curl \left (\frac {\B }{B^2}\right )\cdot \grad \delta p_\perp ,

since \(\curl \grad \delta p_\perp = \vect 0\). For a vacuum field,

\curl \left (\frac {\B }{B^2}\right ) = \grad \left (\frac {1}{B^2}\right )\times \B = \frac {2\B \times \grad B}{B^3},

so in paraxial geometry

\divergence \left (\frac {\B }{B^2}\times \grad \delta p_\perp \right ) = \frac {2}{rB^2}\pp {B}{n}\pp {\delta p_\perp }{\theta }. \tag{20.44}

Similarly,

\begin{aligned}\divergence \left [ (\delta p_\parallel -\delta p_\perp ) \frac {\B \times \vect \kappa }{B^2} \right ] &= \frac {1}{rB^2}\pp {B}{n}\pp {}{\theta } \left (\delta p_\parallel -\delta p_\perp \right ).\end{aligned} \tag{20.45}

Adding Eqs. (20.44) and (20.45) gives the compact result

\divergence \left (\frac {\B }{B^2}\times \grad \delta p_\perp \right ) + \divergence \left [ (\delta p_\parallel -\delta p_\perp ) \frac {\B \times \vect \kappa }{B^2} \right ] = \frac {1}{rB^2}\pp {B}{n}\pp {}{\theta } \left (\delta p_\parallel +\delta p_\perp \right ). \tag{20.46}

The interchange mode therefore depends only on the combination

P_\Sigma \equiv p_\parallel + p_\perp .

Advection of the pressure sum. Because flute motion moves an entire thin flux tube sideways with little parallel rearrangement, the leading perturbation of \(P_\Sigma \) is simply the sideways advection of the equilibrium profile:

\delta P_\Sigma \simeq -\xi _n\pp {P_\Sigma }{n}. \tag{20.48}

Using Eq. (20.29) and then Eq. (20.19),

\begin{aligned}\delta P_\Sigma &= -\left (-\frac {i m}{rB}\varphi \right )\pp {P_\Sigma }{n} \nonumber \\ &= \frac {i m}{rB}\varphi \,\pp {P_\Sigma }{n} = i m\varphi \,\pp {P_\Sigma }{\psi }.\end{aligned} \tag{20.49}

Substituting Eq. (20.49) into Eq. (20.46) and using \(\pp {B}{n}=\kappa B\) gives

\text {pressure terms} = - m^2\,\frac {\kappa }{rB}\,\pp {P_\Sigma }{\psi }\,\varphi . \tag{20.50}

This is the local curvature-drive term. It has exactly the sign one expects: outward-decreasing pressure together with bad curvature is destabilizing.

Field-line-integrated eigenvalue equation. Insert Eqs. (20.40) and (20.50) into Eq. (20.35). Because \(\varphi \) is constant along the field line, all \(z\)-dependence sits inside the line integrals. Define

\begin{aligned}I(\psi ) &\equiv \int _1^2 \frac {\rho (\psi ,\ell )}{2B^2(\psi ,\ell )}\,d\ell , \\[0.5em] D(\psi ) &\equiv -\int _1^2 \frac {\kappa (\psi ,\ell )}{r(\psi ,\ell )B^2(\psi ,\ell )} \pp {P_\Sigma }{\psi }(\psi ,\ell ) \,d\ell .\end{aligned} \tag{20.51}

Then the flute equation becomes

i\omega \left [ \left (\frac {\delta J_\parallel }{B}\right )_2 - \left (\frac {\delta J_\parallel }{B}\right )_1 \right ] = \omega ^2 \left [ 4\pp {}{\psi }\left (I\psi \pp {\varphi }{\psi }\right ) - \frac {m^2 I}{\psi }\,\varphi \right ] - m^2 D\,\varphi . \tag{20.53}

Equation (20.53) is the paraxial mirror interchange equation. It is the mirror analogue of the energy-principle form encountered in the previous two lectures: an inertial denominator weighted by \(I(\psi )\) and a drive weighted by the curvature integral \(D(\psi )\). With the sign convention of Eq. (20.52), bad curvature and an outward-decreasing \(P_\Sigma \) give \(D>0\).

20.4 Variational Form, Sharp-Boundary Mirrors, and Pressure Weighting

Open ends and the variational principle. For insulating or effectively open ends,

\left (\frac {\delta J_\parallel }{B}\right )_1 = \left (\frac {\delta J_\parallel }{B}\right )_2 = 0,

so the left-hand side of Eq. (20.53) vanishes. Multiply the remaining equation by \(\varphi ^*(\psi )\) and integrate from the axis to the plasma edge. Assuming the boundary terms vanish,

\begin{aligned}0 &= \omega ^2 \int d\psi \; \varphi ^* \left [ 4\pp {}{\psi }\left (I\psi \pp {\varphi }{\psi }\right ) - \frac {m^2 I}{\psi }\,\varphi \right ] - m^2\int d\psi \;D|\varphi |^2 \nonumber \\ &= -\omega ^2 \int d\psi \; \left [ 4 I\psi \left |\pp {\varphi }{\psi }\right |^2 + \frac {m^2 I}{\psi }|\varphi |^2 \right ] - m^2\int d\psi \;D|\varphi |^2.\end{aligned}

Therefore

\omega ^2 = - m^2 \frac {\displaystyle \int d\psi \;D(\psi )|\varphi |^2} {\displaystyle \int d\psi \;\left [ 4I(\psi )\psi \left |\pp {\varphi }{\psi }\right |^2 + \frac {m^2 I(\psi )}{\psi }|\varphi |^2 \right ]}. \tag{20.56}

This is an energy principle in the same spirit as Eq. (15.18). The denominator is positive definite, so instability requires

\int d\psi \;D(\psi )|\varphi |^2 > 0.

The flute mode is therefore unstable whenever the weighted bad-curvature drive dominates.

Local high-\(m\) form. If the mode is radially localized, the derivative term in Eq. (20.56) is subdominant and the local dispersion relation becomes

\begin{aligned}\omega ^2 & \simeq -\frac {\psi D(\psi )}{I(\psi )} \nonumber \\ & \sim \frac {B r^2 \frac {\kappa }{r\cancel {B^2} } \frac {\partial p}{\partial r } \frac {\partial r}{\partial \psi } \cancel {L}}{\frac {\rho }{2 \cancel {B^2} \cancel {L} }} \nonumber \\ & \sim \frac {\frac { p}{L_p}}{R_{curv} \rho }\nonumber \\ & \sim \frac {v_{Ti}^2}{r R_{curv}} \nonumber \\ &\sim \frac {g}{L_p}\end{aligned}

This is the simplest statement of the flute instability. A positive \(D\) gives \(\omega ^2<0\), hence pure exponential growth. Recall Brunt Vasaila frequency \(N^2 \sim g / L_p \). The tendency of the fastest ideal modes to run to short perpendicular wavelength is one reason finite-Larmor-radius physics later becomes important. We will return to that regularization in the Roberts–Taylor and anisotropic-wave lectures.

Sharp-boundary mirror. The sign of \(D\) is especially transparent when the pressure falls sharply to zero at the last closed surface. Let the plasma edge be \(r=a(z)\) on the flux surface \(\psi =\psi _a\). Since Eq. (20.15) gives \(B(z)a^2(z)=2\psi _a\), one has

\frac {1}{aB^2} = \frac {a^3}{4\psi _a^2}.

For a vacuum paraxial field the boundary curvature is approximately

\kappa \simeq -\dd {^2 a}{z^2}.

If \(P_\Sigma \) is constant on the field line and drops to zero across a thin edge layer, then Eq. (20.52) gives

D_{\rm edge} \propto -\int _1^2 a^3(z)\dd {^2 a}{z^2}\,dz\;[P_\Sigma ], \tag{20.61}

where \([P_\Sigma ]\) is the pressure jump across the edge. Integrating by parts,

\begin{aligned}-\int _1^2 a^3\dd {^2 a}{z^2}\,dz &= -\left [a^3\dd {a}{z}\right ]_1^2 + 3\int _1^2 a^2\left (\dd {a}{z}\right )^2dz.\end{aligned}

At the turning points of a symmetric mirror \(da/dz=0\), so the boundary term vanishes and

-\int _1^2 a^3\dd {^2 a}{z^2}\,dz = 3\int _1^2 a^2\left (\dd {a}{z}\right )^2dz >0. \tag{20.63}

Thus an isotropic sharp-boundary axisymmetric mirror is generically interchange unstable. This is one of the classical reasons simple mirrors are so unforgiving.

Pressure weighting and sloshing ions. The same calculation also shows how anisotropy or energetic trapped populations can help. Suppose the pressure weighting along the field line is \(P_\Sigma =P_\Sigma (a)\) rather than a constant. Then

D \propto -\int _1^2 P_\Sigma (a)\,a^3\dd {^2 a}{z^2}\,dz.

Integrating by parts again gives

\begin{aligned}D &= -\left [P_\Sigma a^3 \dd {a}{z}\right ]_1^2 + \int _1^2 \dd {a}{z}\,\dd {}{z}\left (P_\Sigma a^3\right )dz \nonumber \\ &= \int _1^2 \left (\dd {a}{z}\right )^2 \dd {}{a}\left (P_\Sigma a^3\right )dz.\end{aligned} \tag{20.65}

For ordinary isotropic pressure, \(\dd {}{a}(P_\Sigma a^3)>0\) and the mode remains unstable. Stability demands instead that

\dd {}{a}\left (P_\Sigma a^3\right ) < 0, \tag{20.66}

which means that the pressure must be concentrated strongly enough in the good-curvature regions. This is the logic behind sloshing-ion stabilization Hinton and Rosenbluth (1982).

Figure 20.3: Pressure weighting along the field line can convert the same mirror geometry from interchange-unstable to interchange-stable if enough pressure is concentrated in the good-curvature regions Hinton and Rosenbluth (1982).

End conditions and partial line-tying. Equation (20.35) also shows how end conditions modify the instability. For a high-\(m\) localized mode we may drop the radial-derivative term in Eq. (20.53), obtaining

i\omega \left [ \left (\frac {\delta J_\parallel }{B}\right )_2 - \left (\frac {\delta J_\parallel }{B}\right )_1 \right ] = -\omega ^2\frac {m^2 I}{\psi }\,\varphi - m^2 D\,\varphi . \tag{20.67}

If each end is connected to a sheath or limiter with small-signal impedance \(Z_{\rm sh}\), then the linearized end response can be modeled as

\delta J_{\parallel ,2} = +\frac {\varphi }{Z_{\rm sh}}, \qquad \delta J_{\parallel ,1} = -\frac {\varphi }{Z_{\rm sh}}. \tag{20.68}

For a symmetric mirror with end field \(B_{\rm end}\),

\left (\frac {\delta J_\parallel }{B}\right )_2 - \left (\frac {\delta J_\parallel }{B}\right )_1 = \frac {2\varphi }{B_{\rm end}Z_{\rm sh}}. \tag{20.69}

Substituting into Eq. (20.67) yields

\omega ^2 + i\Gamma _{\rm LT}\omega + \Gamma _0^2 = 0, \tag{20.70}

with

\Gamma _0^2 \equiv \frac {\psi D}{I}, \qquad \Gamma _{\rm LT} \equiv \frac {2\psi }{m^2 I B_{\rm end} Z_{\rm sh}}. \tag{20.71}

For open ends \(\Gamma _{\rm LT}=0\) and one recovers \(\omega ^2=-\Gamma _0^2\). For strong line-tying, \(\Gamma _{\rm LT}\gg \Gamma _0\), the growth rate becomes

\gamma \simeq \frac {\Gamma _0^2}{\Gamma _{\rm LT}}, \tag{20.72}

so line-tying does not change the ideal drive itself but can make the mode arbitrarily slow by providing the parallel current needed to short out the flute polarization.

20.5 A Second Energy Principle: the \(V'(\psi )\) Criterion

Why a second formulation is useful. The mirror calculation above emphasizes local curvature and current continuity. There is, however, a more geometric way to express the same interchange physics. If the displacement really does exchange neighboring flux tubes with very little field-line bending, then the central quantity is the volume available to a tube of given flux. This is the origin of the \(V'(\psi )\) criterion. It is especially useful because it makes the instability look transparently like a buoyancy problem in a nontrivial magnetic geometry.

Volume per unit flux. For an axisymmetric flux surface labeled by \(\psi \), let \(V(\psi )\) be the volume enclosed by that surface. A thin shell between \(\psi \) and \(\psi +d\psi \) has volume

dV = 2\pi \,d\psi \oint \frac {d\ell }{B},

V'(\psi ) \equiv \frac {dV}{d\psi } = 2\pi \oint \frac {d\ell }{B}. \tag{20.74}

Up to the conventional factor \(2\pi \), the same geometric content is carried by

U(\psi ) \equiv \oint \frac {d\ell }{B}. \tag{20.75}

The prime notation is traditional in equilibrium theory, so in this lecture it is natural to emphasize \(V'(\psi )\).

Flux-Tube Derivation of the Marginal Pressure Profile The flux-tube argument used in the main text is worth writing out in full. Take a tube of flux \(d\psi \) with volume \(d V = V'(\psi )d\psi \). Adiabatic transport implies

p(\psi )\,[V'(\psi )]^\gamma = p_\star (\psi +d\psi )\,[V'(\psi +d\psi )]^\gamma .

Therefore

p_\star (\psi +d\psi ) = p(\psi ) \left [ \frac {V'(\psi )}{V'(\psi +d\psi )} \right ]^\gamma .

Expanding

V'(\psi +d\psi )=V'(\psi )+\frac {d V'}{d\psi }d\psi ,

we have

\begin{aligned}\left [\frac {V'(\psi )}{V'(\psi +d\psi )}\right ]^\gamma &= \left [ 1+\frac {1}{V'}\frac {d V'}{d\psi }d\psi \right ]^{-\gamma } \nonumber \\ &\simeq 1-\gamma \frac {1}{V'}\frac {d V'}{d\psi }d\psi .\end{aligned}

Hence

p_\star (\psi +d\psi ) \simeq p(\psi ) \left [ 1-\gamma \frac {1}{V'}\frac {d V'}{d\psi }d\psi \right ].

The ambient pressure at the new location is

p(\psi +d\psi ) = p(\psi )+\frac {d p}{d\psi }d\psi .

Marginal stability means \(p_\star (\psi +d\psi )=p(\psi +d\psi )\), so

\frac {d p}{d\psi } = -\gamma \frac {p}{V'}\frac {d V'}{d\psi }.

Dividing by \(p\) gives

\frac {d\ln p}{d\psi } = -\gamma \frac {d\ln V'}{d\psi },

p(\psi )\,[V'(\psi )]^\gamma = \mathrm {const}.

In terms of a midplane radius label \(r_0\),

L_{p,\mathrm {crit}} \equiv -\frac {p}{\frac {dp}{d r_0}} = \frac {V'}{\gamma \, \frac {dV'}{d r_0}}.

Interchanging two neighboring tubes. Consider two thin neighboring flux tubes carrying the same infinitesimal flux \(d\Phi \) and centered at \(\psi \) and \(\psi +d\psi \). Their volumes are

V_1 = U(\psi )\,d\Phi , \qquad V_2 = \left [U(\psi )+\pp {U}{\psi }d\psi \right ]d\Phi =\left [U(\psi )+\delta U \right ]d\Phi ,

and their pressures are

p_1 = p(\psi ) = p, \qquad p_2 = p(\psi )+\pp {p}{\psi }d\psi = p + \delta p

Now interchange the two tubes. Each tube preserves its entropy, so the pressure of tube 1 after it occupies the volume \(V_2\) is

p_1' = p_1\left (\frac {V_1}{V_2}\right )^\gamma ,

and similarly

p_2' = p_2\left (\frac {V_2}{V_1}\right )^\gamma .

The thermal energy before the interchange is

W_i = \frac {p_1V_1}{\gamma -1} + \frac {p_2V_2}{\gamma -1},

and after the interchange it is

W_f = \frac {p_1'V_2}{\gamma -1} + \frac {p_2'V_1}{\gamma -1}.

Expanding to second order in \(d\psi \) gives

\begin{aligned}\Delta W \equiv W_f-W_i & = \delta p \delta U + \gamma p \frac {(\delta U)^2}{U} \\& = (d\psi )^2 \frac {d}{d\psi }\ln \left [p\,U^\gamma \right ].\end{aligned} \tag{20.92}

Since \(V'(\psi )=2\pi U(\psi )\) differs only by a constant factor, this may be written equally well as

\Delta W \propto \frac {d}{d\psi }\ln \left [p\,\bigl (V'(\psi )\bigr )^\gamma \right ]. \tag{20.94}

This is the geometric interchange energy principle. The role of \(V'(\psi )\) is completely analogous to the role of density stratification in ordinary buoyancy. Since the second term in Eq. (20.92) is manifestly positive, a sufficient condition for stability is that

\pp {p}{\psi } \pp {V'}{\psi } > 0

which in the general case in which \(p\) decreases outward requires the \(V'\) also decrease outward.

We can come to this same conclusion by returning to the particle drift picture. Combining the stability integral Eq. (20.11) and the expression for \(\kappa \) from Eq. (20.20), the stability integral requires

\int _1^2 \frac {\kappa }{rB^2} d\ell = \int _1^2 d\ell \frac {\partial }{\partial \psi } \frac {1}{B} = \frac {\partial }{\partial \psi } \int _1^2 \frac {d\ell }{B} > 0.

For the plasma to gain outward directed energy, the volume must expand.

Stability criterion. The equilibrium is stable to such an interchange if \(\Delta W>0\). Thus a sufficient instability condition is

\frac {d}{d\psi }\ln \left [p\,\bigl (V'(\psi )\bigr )^\gamma \right ] < 0 \qquad \text {when } V''(\psi )>0. \tag{20.97}

Equivalently,

\frac {1}{p}\pp {p}{\psi } + \gamma \,\frac {1}{V'(\psi )}\pp {V'(\psi )}{\psi } < 0. \tag{20.98}

This is exactly the same kind of statement as the gravitational energy principle: a sufficiently steep pressure gradient can overturn the magnetic buoyancy supplied by the geometry.

Straight-current geometry. For a straight current channel the field is approximately

B_\theta = \frac {\muo I}{2\pi r}.

A magnetic field line is a circle of circumference \(2\pi r\), so

U(r) = \oint \frac {d\ell }{B} = \frac {2\pi r}{B_\theta } \propto r^2,

and therefore

V'(r) \propto r^2. \tag{20.101}

Then Eq. (20.98) becomes

\frac {1}{p}\dd {p}{r} + \frac {2\gamma }{r} < 0.

Defining the pressure scale length

L_p \equiv -\frac {p}{dp/dr},

one obtains the familiar estimate

L_p > \frac {r}{2\gamma } \tag{20.104}

for marginal interchange stability. This is the same z-pinch intuition encountered from the local curvature point of view, but expressed in the cleaner geometric language of \(V'(\psi )\). This will be revisited in Lecture 25.

Dipole geometry and self-organization. The \(V'(\psi )\) language is especially attractive in a dipole, where the flux geometry strongly expands outward. Dipoles can come in an open-field line geometry like for planets but also in closed-field line geometries like levitated dipoles. Assuming for now an axisymmetric point dipole moment \(m\) (perhaps suitable for magnetospheres),

\psi (r,\theta ) = r A_\phi = \frac {\muo m\sin ^2\theta }{4\pi r},

with spherical coordinates \(r,\theta ,\phi \) A straightforward volume integral gives

V(\psi ) \propto R^3, \qquad V'(\psi ) \propto R^4, \tag{20.106}

where \(R\) is the equatorial crossing radius of each field line. Equation (20.98) then implies that the real space gradient of pressure in the equatorial plane is governed by

\frac {1}{p}\dd {p}{R} + \frac {4\gamma }{R} < 0.

Marginality therefore gives

p(R) \propto R^{-4\gamma }. \tag{20.108}

For a monatomic gas, \(\gamma =5/3\), so

p(R) \propto R^{-20/3}.

This is a strong profile, but it carries an important physical message: interchange mixing can self-organize a plasma toward a nearly marginal state, just as ordinary convection tends to drive a stratified atmosphere toward \(N^2\approx 0\).

Entropy mixing and marginality. The same dipole example makes another point. If interchange turbulence mixes flux tubes efficiently, it is natural to expect not only pressure but particle content per unit flux to relax. If

\rho V' = \text {const},

then

\frac {1}{\rho }\dd {\rho }{R} + \frac {1}{V'}\dd {V'}{R} =0.

Combining this with the marginal pressure profile yields

\dd {}{R}\left (\frac {p}{\rho ^\gamma }\right )=0,

so the entropy becomes approximately flat in radius. That is precisely the magnetic analogue of the way ordinary convection mixes an atmosphere toward constant entropy.

How the two formulations fit together. The local \(D(\psi )\) formulation and the global \(V'(\psi )\) formulation are not competitors. They are two views of the same ideal-MHD interchange physics. The \(D(\psi )\) form is better for paraxial mirrors, end conditions, and pressure weighting along the field line. The \(V'(\psi )\) form is better for seeing the underlying geometry and for making contact with the energy principle. In later lectures the ballooning problem will inherit both viewpoints: a local curvature drive and a global statement about how neighboring flux tubes gain or lose accessible volume.

Takeaways

Takeaway. Interchange modes are ideal-MHD buoyancy modes. They minimize field-line bending, so their stability is decided by curvature, pressure gradients, and flux-tube geometry. In a paraxial mirror the drive is encoded in the field-line integral \(D(\psi )\); in a more geometric language it is encoded in \(V'(\psi )\). Simple axisymmetric mirrors are therefore generically unstable, and classical mirror stabilization strategies—minimum-\(B\) design, pressure weighting, and line-tying—can all be understood as ways of changing the sign or effectiveness of the same interchange energy principle.

20.6 Bonus: Non-Paraxial, Interchange-Stable Axisymmetric Mirrors

From paraxial intuition to the full flux geometry. The paraxial mirror analysis is useful for building intuition, but the WHAM geometry is better described by the exact flux-tube volume per unit flux,

V'(\psi )=2\pi \oint \frac {d\ell }{B},

evaluated on the axisymmetric vacuum flux surfaces generated by the HTS mirror coils. In this non-paraxial treatment the outermost relevant surface is not chosen from a small-radius expansion, but from the separatrix itself. For the present WHAM coil set the midplane null occurs at

R_X \simeq 1.412~\mathrm {m},

and we take the separatrix flux \(\psi _{\mathrm {sep}}=\psi (R_X,0)\) as the natural limiting surface.

Interchange criterion. The geometric interchange condition may be written as

\frac {1}{p}\pp {p}{\psi } + \gamma \frac {1}{V'}\pp {V'}{\psi } < 0,

or, in terms of the midplane radius label \(r_0\),

L_{p,\mathrm {crit}}(r_0) = \frac {V'}{\gamma \, \dd {V'}{r_0}}.

The important point is that the exact WHAM flux geometry produces a very rapid increase in \(V'(\psi )\) as the separatrix is approached. That geometric expansion strongly relaxes the pressure-gradient constraint compared with the paraxial-core estimate.

A hot-ion edge state. To illustrate a non-paraxial WHAM case with a controlled core beta, we choose an edge state just inside the separatrix with

n_{\mathrm {edge}} = 1.0\times 10^{17}~\mathrm {m^{-3}}, \qquad T_{e,\mathrm {edge}} = 12.3~\mathrm {eV}, \qquad T_{i,\mathrm {edge}} = 123.0~\mathrm {eV},

so that

T_e:T_i = 1:10, \qquad p_{\mathrm {edge}} \simeq 2.17~\mathrm {Pa}.

We then integrate the marginal profile inward from the separatrix-limited edge using

\dd {}{r_0}\ln p = -\frac {1}{L_{p,\mathrm {crit}}}.

To close the thermodynamics we additionally assume constant entropy,

\frac {p}{n^\gamma }=\mathrm {const}, \qquad \gamma =\frac {5}{3},

and preserve the electron-to-ion temperature ratio throughout the profile.

What the profiles show. The resulting pressure profile remains modest in the core even though the geometric expansion near the separatrix is very strong. At the first sampled core point, \(r_0=0.02~\mathrm {m}\), the profile gives

\beta (0)\approx 0.20, \qquad \beta (r_0\approx 1~\mathrm {m})\approx 0.57.

Thus the core and much of the mid-radius region remain in a regime where an MHD discussion based on the vacuum flux geometry is at least qualitatively sensible. The same run gives

n(0.02~\mathrm {m}) \simeq 1.12\times 10^{19}~\mathrm {m^{-3}},

with hot-ion temperatures

T_e(0.02~\mathrm {m}) \simeq 2.87\times 10^2~\mathrm {eV}, \qquad T_i(0.02~\mathrm {m}) \simeq 2.87\times 10^3~\mathrm {eV}.

Meanwhile the line-wise mirror ratio grows from about

R_m \simeq 64

on axis to

R_m \gtrsim 8\times 10^4

just inside the separatrix, reflecting the collapse of the midplane field near the null.

Interpretation. This non-paraxial construction shows that the exact \(V'(\psi )\) geometry of WHAM can support a much larger pressure buildup than one would infer from a purely paraxial estimate, while still allowing the core beta to remain below unity for a suitably chosen edge state. That is encouraging from the standpoint of interchange stability. At the same time, the very large values of \(V'\), mirror ratio, and \(\beta \) as the separatrix is approached are a warning sign: the outermost flux surfaces lie in a region where the vacuum field is becoming singular and the magnetic geometry is highly sensitive to finite-pressure corrections.

What comes next. For that reason this section should be viewed as a geometric and thermodynamic guide, not as the final equilibrium solution. The next step is to compute the full high-\(\beta \) MHD equilibrium and revisit the flux geometry self-consistently. That will determine how much of the large non-paraxial stabilization survives once the pressure modifies the field rather than simply filling a fixed vacuum geometry.

Figure 20.4: Non-paraxial WHAM separatrix geometry, the exact \(V'(\psi )\) profile, the critical pressure-gradient scale length, and the marginal pressure profile obtained by integrating inward from the separatrix-limited edge state.

Figure 20.5: Midplane beta profile for the hot-ion separatrix-limited case. Here so the core remains below unity while \(\beta \) rises rapidly near the separatrix as the midplane field collapses.

Figure 20.6: Density and temperature profiles obtained from the marginal pressure profile under the constant-entropy closure with fixed \(T_e:T_i=1:10\).

Figure 20.7: Mirror ratio along the sampled WHAM flux tubes. The mirror ratio becomes extremely large as the separatrix is approached because the midplane field tends to zero at the null.

Takeaways

The mirror-interchange lecture has two complementary morals. In the paraxial problem the drive is encoded in the orbit-weighted curvature integral \(D(\psi )\), which makes clear why simple axisymmetric mirrors are generically interchange unstable and why line-tying or pressure weighting can help. In the non-paraxial WHAM geometry the same physics is reorganized by the exact flux-tube volume \(V'(\psi )\): strong separatrix expansion can greatly relax the marginal pressure-gradient bound, but only until finite-\(\beta \) corrections to the magnetic geometry become unavoidable.

Bibliography

M. D. Kruskal and M. Schwarzschild. Some instabilities of a completely ionized plasma. Proceedings of the Royal Society of London A, 223:348–360, 1954. doi:10.1098/rspa.1954.0120.

M.N Rosenbluth and C.L Longmire. Stability of plasmas confined by magnetic fields. Annals of Physics, 1(2):120–140, 1957. doi:10.1016/0003-4916(57)90055-6.

W. A. Newcomb. Convective instability induced by gravity in a plasma with a frozen-in magnetic field. Physics of Fluids, 4:391–396, 1961. doi:10.1063/1.1706342.

F L Hinton and M N Rosenbluth. Stabilization of axisymmetric mirror plasmas by energetic ion injection. Nuclear Fusion, 22(12):1547–1557, 1982. doi:10.1088/0029-5515/22/12/001.

D D Ryutov, H L Berk, B I Cohen, A W Molvik, and T C Simonen. Magneto-hydrodynamically stable axisymmetric mirrors. Physics of Plasmas, 18 (9):092301, 2011. doi:10.1063/1.3624763.

Problems

Problem 20.1.: Starting from Eq. (20.30), re-derive Eq. (20.53) carefully and check every sign in the definition of \(D(\psi )\).
Problem 20.2.: Show explicitly that the sharp-boundary isotropic mirror gives \(D>0\) by carrying out the integration by parts in Eqs. (20.61)–(20.63).
Problem 20.3.: Starting from the interchange of two neighboring flux tubes, derive Eq. (20.93) without skipping the second-order expansion.
Problem 20.4.: Evaluate \(V'(\psi )\) for a large-aspect-ratio tokamak with circular surfaces and discuss why pure interchange is harder to realize once line bending is included.
Problem 20.5.: In the local line-tying model, solve Eq. (20.70) exactly and recover the slowed-growth limit Eq. (20.72) for \(\Gamma _{\rm LT}\gg \Gamma _0\).