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Lecture 40
Motion, Moving Walls, and Shear in MHD Stability

Overview

Motion changes MHD stability in three distinct ways. First, a bulk flow Doppler-shifts the spectrum, so every familiar wave and continuum is seen in the moving frame. Second, velocity shear introduces a loss of coherence: neighboring flux tubes no longer stay in phase, and a mode that would otherwise grow as a coherent global displacement can be smeared out before it completes an e-folding. Third, the flow itself is a reservoir of free energy. This is the dangerous side of the story: the same shear that can weaken interchange or kink modes can also drive Kelvin–Helmholtz-type instabilities.

This appendix is meant as a bridge lecture. It ties the static energy principle of Eq. (15.25) to moving boundaries, rotating plasmas, sheared-flow stabilization of mirrors and Z pinches, and the counterexample provided by Kelvin–Helmholtz instability. The common lesson is that “flow stabilization” is never a one-word theorem. One must always ask: which frame, which branch, and which free-energy source?

Historical Perspective

The clean static story belongs to Bernstein, Frieman, Kruskal, and Kulsrud and was recast for these notes in the energy-principle lecture. The corresponding theory for equilibria with flow was put into systematic form by Frieman and Rotenberg, who showed that the linearized problem is no longer governed by a purely self-adjoint force operator Frieman and Rotenberg (1960). At almost the same time, Chandrasekhar organized the hydrodynamic and hydromagnetic stability problems into a single intellectual framework, including the Kelvin–Helmholtz and rotating-flow problems Chandrasekhar (1961). Later developments split along two laboratory lines.

One line emphasized rotating plasmas and moving walls. In toroidal confinement, the classical question became whether a nearby wall and plasma rotation could stabilize external modes that are otherwise unstable without a wall Bondeson and Ward (1994). In non-neutral plasma physics, the “rotating wall” became a literal control technique: a rotating external perturbation transfers torque to the plasma and compresses it Huang et al. (1997). The second line emphasized velocity shear inside the plasma. In mirror devices it was learned that biased end plates can drive azimuthal shear strong enough to suppress flute-like transport Beklemishev et al. (2010). In Z pinches, axial shear was developed into a stabilization strategy for kink-like modes, first analytically and then experimentally Shumlak and Hartman (1995); Shumlak et al. (2003).

Caution

Three cautions are worth keeping in mind.

First, rigid motion is not the same as shear. Uniform rotation often does little more than produce a Doppler shift, whereas differential rotation can change the underlying operator and the continuum.

Second, a moving wall is best thought of as a boundary condition in the wall frame. The wall does not magically stabilize the plasma; it changes which field lines are frozen to which material points.

Third, shear is not automatically stabilizing. Smooth shear can decorrelate a flute or kink mode, but a sharp velocity jump is itself a classical free-energy source and can excite the Kelvin–Helmholtz branch.

40.1 Why motion changes the stability problem

For a static equilibrium, the linearized ideal-MHD system can be written in the familiar form

\[-\rho _0\omega ^2\,\vect {\xi }=\mathcal {F}_0[\vect {\xi }], \tag{F.1}\]
and the energy principle says that stability is determined by the sign of \(\delta W[\vect {\xi }]\), Eq. (15.25). The reason this is so powerful is that \(\mathcal {F}_0\) is self-adjoint. The eigenvalue appears only as \(\omega ^2\), the spectrum is real, and one can understand stability by minimizing a quadratic form.

The flowing problem. For a stationary equilibrium with background flow \(\vect {V}_0\), the corresponding displacement equation takes the schematic Frieman–Rotenberg form

\[\rho _0\,\partial _t^2\vect {\xi } +2\rho _0\,\vect {V}_0\!\cdot \!\grad \,\partial _t\vect {\xi } +\mathcal {F}_{V}[\vect {\xi }]=0, \tag{F.2}\]
where \(\mathcal {F}_V\) reduces to \(\mathcal {F}_0\) when \(\vect {V}_0=0\) Frieman and Rotenberg (1960). The exact form of \(\mathcal {F}_V\) contains the equilibrium-flow shear and centrifugal terms, but for present purposes the main point is simpler: the linear problem is no longer purely variational in the static sense.

With a normal mode

\[\vect {\xi }(\vect {x},t)=\hat {\vect {\xi }}(\vect {x})e^{-i\omega t},\]
Eq. (F.2) becomes
\[-\rho _0\omega ^2\hat {\vect {\xi }} -2i\omega \rho _0\,\vect {V}_0\!\cdot \!\grad \hat {\vect {\xi }} +\mathcal {F}_{V}[\hat {\vect {\xi }}]=0. \tag{F.4}\]
The middle term is anti-self-adjoint. It is the linear imprint of advection and it is why one no longer gets a simple “minimize \(\delta W\) and read off \(\omega ^2\)” rule.

The simplest limit: pure Doppler shift. If the equilibrium flow is locally uniform, so that its gradients may be ignored over one radial wavelength, then perturbations of the form \(e^{i\vect {k}\cdot \vect {x}-i\omega t}\) are governed by the Doppler-shifted frequency

\[\omega _D \equiv \omega -\vect {k}\cdot \vect {V}_0. \tag{F.5}\]
In this limit every familiar dispersion relation is obtained from the static one by replacing \(\omega \) with \(\omega _D\). Thus the local Alfvén relation becomes
\[\omega _D^2 = k_\parallel ^2 v_A^2, \tag{F.6}\]
and the Alfvén continuum is simply shifted in frequency. Rigid motion therefore changes the spectrum without necessarily changing the sign of the restoring forces.

Tutorial

Take a perturbation of the usual cylindrical form

\[\xi _r(r,\theta ,z,t)=\hat \xi (r)\,e^{im\theta +ikz-i\omega t}.\]
If the equilibrium has azimuthal and axial flow,
\[\vect {V}_0 = r\Omega (r)\,\etheta + V_z(r)\,\ez ,\]
then the local Doppler-shifted frequency is
\[\omega _D(r)=\omega -m\Omega (r)-kV_z(r). \tag{F.9}\]
Across a radial layer of width \(\Delta r\), different annuli see different phase velocities, so the spread in Doppler shift is
\[\Delta \omega _D \simeq \left |m\Omega '(r)+kV_z'(r)\right |\Delta r. \tag{F.10}\]
If the mode would grow without flow at a rate \(\gamma _0\), then a very crude but very useful stabilization criterion is
\[\Delta \omega _D \gtrsim \gamma _0. \tag{F.11}\]
This is not a theorem, and it does not replace a full eigenvalue calculation. But it captures the core mechanism of many “shear stabilization” results: the mode fails not because the local drive is gone, but because radial locations can no longer remain phase locked long enough to build a single coherent global eigenfunction.

Where the static energy principle still helps. Even though the flowing problem is not purely self-adjoint, the static energy principle still tells us what would happen in the absence of flow. That is very useful in practice. One first identifies the static branch — interchange, external kink, flute, ballooning, and so on — using Eq. (15.25) or one of its reductions such as the local Suydam criterion, Eq. (25.17). One then asks how advection, wall motion, or shear alter that branch. That is the viewpoint taken in the remaining sections.

40.2 Moving conducting walls and the wall frame

The cleanest way to think about a moving wall is to start with the ideal Ohm law in the plasma, Eq. (4.8), and the perfect-conductor condition in the wall frame,

\[\hat {\vect {n}}\times \left (\E + \vect {U}_w\times \B \right )=0, \tag{F.12}\]
where \(\vect {U}_w\) is the wall velocity. On the plasma side,
\[\E + \vect {V}\times \B =0.\]
Subtracting these gives
\[\hat {\vect {n}}\times \left [(\vect {V}-\vect {U}_w)\times \B \right ]=0. \tag{F.14}\]
Thus field lines that intersect a perfectly conducting wall are frozen into the wall material points. If the equilibrium magnetic field is tangent to the wall, \(\B \cdot \hat {\vect {n}}=0\), then Eq. (F.14) implies the kinematic condition
\[(\vect {V}-\vect {U}_w)\cdot \hat {\vect {n}}=0. \tag{F.15}\]
A fixed wall is simply the special case \(\vect {U}_w=0\).

Rotating boundary condition. For a cylindrical perturbation

\[\propto e^{im\theta +ikz-i\omega t},\]
a wall rotating with angular velocity \(\Omega _w\) and translating at speed \(U_{wz}\) sees the mode frequency
\[\omega _w = \omega - m\Omega _w - kU_{wz}. \tag{F.17}\]
This simple replacement is the boundary analogue of the Doppler shift in the plasma. It is why so many “moving wall” results can be interpreted as ordinary wall physics evaluated in a different frame.

Why a close wall helps. The static wall effect was already made concrete in the diffuse-pinch and Kruskal–Shafranov lectures. A nearby ideal wall alters the vacuum contribution to \(\delta W\) by changing the allowable external solution, as seen for example in Eqs. (24.5) and (24.10). If the wall is no longer fixed, its response carries the shifted frequency \(\omega _w\). In practice, this means that either a physically moving wall or a rotating plasma can change how quickly wall currents are induced and therefore how effectively the wall can mimic an ideal conductor.

Two laboratory realizations. Tokamak resistive-wall-mode theory is the obvious magnetic-confinement realization: plasma rotation can make a resistive wall behave more like an ideal one over the mode timescale Bondeson and Ward (1994). A very different but conceptually related realization appears in non-neutral plasma physics, where a rotating external perturbation literally exerts torque on the plasma column and compresses it; there the phrase “rotating wall” is not an analogy but a control technique Huang et al. (1997).

Caution

Rigid rotation is often only a frame change.

If \(\Omega \) is constant across the plasma, then many branches are simply shifted by \(\omega \rightarrow \omega -m\Omega \). The interesting physics begins when either the wall or the plasma introduces a gradient in the Doppler shift, because then different radii or different boundaries no longer remain phase locked.

40.3 Shear as stabilization by loss of coherence

The broadest lesson of flow stabilization is that a global mode needs global phase coherence. A flute mode, an external kink, or a low-order current-driven mode grows by keeping a large radial region moving in roughly the same phase. Velocity shear attacks exactly that coherence.

A generic estimate. Suppose a branch is unstable without flow with growth rate \(\gamma _0\) and radial width \(\Delta r\). Then Eq. (F.10) suggests the characteristic shearing rate

\[\gamma _{\rm sh} \sim \left |m\Omega '(r)+kV_z'(r)\right |\Delta r. \tag{F.18}\]
When
\[\gamma _{\rm sh}\gtrsim \gamma _0,\]
one expects appreciable weakening of the coherent global eigenmode. This estimate is the common backbone of several otherwise rather different stabilization stories.

Why this is different from line bending. Line bending raises \(\delta W\) by adding a genuine restoring force. Shear stabilization is more subtle. The local free-energy source may still be present; what changes is that the unstable layer no longer communicates efficiently with neighboring radii. In that sense shear stabilization is closer to phase mixing than to the creation of a new positive-definite energy term.

Why low-shear equilibria are special. This is also why low magnetic shear and low velocity shear often appear together in discussions of core MHD. A low magnetic shear weakens the field-line restoring force, as in the Suydam and resistive-interchange lectures. If the flow shear is also low, nothing prevents a broad core mode from organizing itself coherently. If the flow shear is high enough, the same pressure-driven or current-driven branch may be blurred out before it can form a global structure.

40.4 Mirror traps, interchange, and sheared \(E\times B\) flow

The mirror and magnetic-interchange lectures already showed that flute-like displacements are the magnetic analogue of buoyant interchange. In the absence of flow one writes schematically

\[\partial _t^2\xi _r = \gamma _0^2\,\xi _r, \tag{F.20}\]
with \(\gamma _0^2>0\) when the integrated curvature/pressure drive outweighs line tying and field line bending.

Adding azimuthal flow. Now let the equilibrium have an \(E\times B\) azimuthal velocity

\[\vect {V}_E = r\Omega _E(r)\,\etheta .\]
For a flute perturbation \(\propto e^{im\theta -i\omega t}\), the time derivative is convected to
\[\partial _t \rightarrow \partial _t + i m\Omega _E(r),\]
so the local equation becomes schematically
\[\left (\partial _t + i m\Omega _E\right )^2\xi _r = \gamma _0^2\,\xi _r + \cdots . \tag{F.23}\]
The dots stand for the same line-bending and compressibility terms that determined \(\gamma _0\) in the static problem. The new ingredient is the radial variation of \(\Omega _E\).

Stabilization criterion. If the unstable eigenfunction spans a radial width \(\Delta r\), then neighboring radii drift apart in phase at the rate

\[\Delta \omega _E \sim m\,\Omega _E'\,\Delta r.\]
The resulting stabilization estimate is
\[m\,|\Omega _E'|\,\Delta r \gtrsim \gamma _0. \tag{F.25}\]
This is the mirror-trap counterpart of Eq. (F.11). It says that what matters is not the absolute rotation rate but the radial shear in that rotation.

Why rigid rotation is not enough for the rigid flute mode. The classical \(m=1\) flute in an open system is almost a rigid displacement of the whole column. A uniform azimuthal drift simply convects that displacement. It does not tear the eigenfunction apart. This is why the experimentally successful mirror-trap strategies use end-plate biasing to create differential rotation rather than mere rigid spin.

The GDT lesson. In the gas-dynamic trap and related mirror experiments, biased edge potentials drive strong edge rotation and thereby suppress flute-dominated transport. The famous “vortex confinement” regime is not a tiny linear correction; it is a global nonlinear operating point. But its physical seed is already visible in Eq. (F.25): the unstable interchange branch is denied the radial phase coherence it needs Beklemishev et al. (2010).

40.5 Z pinches and axial sheared flow

For the Z pinch, the corresponding story is usually told with axial rather than azimuthal flow. Without flow, one has the current-driven branches discussed in the Kruskal–Shafranov and diffuse- pinch lectures. In the simplest cylindrical picture, a kink-like mode behaves schematically as

\[\partial _t^2\xi _r = \gamma _0^2\,\xi _r,\]
where \(\gamma _0\) is set by current drive, line bending, and wall stabilization.

Adding axial flow. If

\[\vect {V}_0 = V_z(r)\,\ez ,\]
then a perturbation \(\propto e^{ikz-i\omega t}\) sees the Doppler shift
\[\omega _D(r)=\omega -kV_z(r).\]
Across a layer of width \(\Delta r\), the phase spread is therefore
\[\Delta \omega _D \sim k\,V_z'(r)\,\Delta r.\]
The crude stabilization estimate becomes
\[k\,|V_z'|\,\Delta r \gtrsim \gamma _0. \tag{F.30}\]
This is the axial-flow counterpart of Eq. (F.25).

Why the wall still matters. Equation (F.30) should not be read as if the magnetic geometry has disappeared. The wall location and vacuum solution still determine the static branch that is being sheared apart. In that sense, the axial-flow-stabilized Z pinch is not a new category of mode. It is the ordinary kink or sausage branch viewed in a plasma whose different radii are no longer locked together.

Connection to experiments. This was the line followed by Shumlak and Hartman, who showed that a sheared axial flow can stabilize the \(m=1\) kink branch in Z-pinch geometry Shumlak and Hartman (1995). Later ZaP experiments made the point concrete by directly diagnosing the flow profile and correlating reduced MHD activity with sufficient axial shear Shumlak et al. (2003). The basic picture is exactly the one behind Eq. (F.30): the unstable kink cannot remain radially coherent.

40.6 Kelvin–Helmholtz as the counterexample

The previous sections might tempt one to summarize everything by saying that “shear is good.” Kelvin–Helmholtz instability is the reason that slogan is false.

Planar interface. Consider two incompressible plasmas separated by \(y=0\), with equal densities \(\rho _1=\rho _2=\rho \), equal magnetic fields \(\vect {B}_1=\vect {B}_2=B_0\,\bx \) parallel to the interface, and flows

\[\vect {V}_1=+U\,\bx , \qquad \vect {V}_2=-U\,\bx .\]
Let the interface be displaced by
\[\eta (x,t)=\hat \eta \,e^{ikx-i\omega t}.\]
The perturbation decays away from the interface as \(e^{-|k||y|}\).

Kinematic condition. The normal velocity on each side must match the material motion of the interface, so

\[\left (\partial _t + U_j\partial _x\right )\eta = v_{y,j} \qquad (j=1,2). \tag{F.33}\]

Dynamic condition. For incompressible ideal MHD, continuity of the total pressure across the interface gives the classic dispersion relation

\[\rho _1\left (\omega -kU_1\right )^2+ \rho _2\left (\omega -kU_2\right )^2 = \frac {k^2(B_1^2+B_2^2)}{\muo }. \tag{F.34}\]
For the symmetric case above this reduces to
\[\omega ^2 = k^2\left (V_A^2-U^2\right ), \qquad V_A\equiv \frac {B_0}{\sqrt {\muo \rho }}. \tag{F.35}\]
Equivalently,
\[\omega = k\bar U \pm k\sqrt {V_A^2-\frac {\Delta U^2}{4}}, \qquad \bar U\equiv \frac {U_1+U_2}{2}, \quad \Delta U \equiv U_1-U_2. \tag{F.36}\]
Thus instability requires
\[|\Delta U| > 2V_A. \tag{F.37}\]

Tutorial

Equation (F.37) is the cleanest possible warning sign. Shear can stabilize one branch and drive another.

In the mirror and Z-pinch examples, the shear was smooth enough that its main effect was to make neighboring radii drift out of phase. The velocity profile did not itself create a strong new surface free energy. In the Kelvin–Helmholtz problem, by contrast, the shear is the free energy. The magnetic field then plays the role of line tension. If it is strong enough, \(|\Delta U|<2V_A\), the two fluids are tied together. If it is too weak, the interface rolls up.

That is why one should not ask merely whether a plasma is “sheared.” One must ask whether the flow profile is acting like a smooth phase mixer or like a sharp free-shear layer.

Connection to earlier lectures. The form of Eq. (F.37) is the interface analogue of the firehose logic in Eq. (11.22): motion weakens magnetic tension when it becomes too large along the field. Likewise, the rotating-cylinder story connects back to the MRI lecture, where the role of differential rotation is organized by the epicyclic frequency in Eq. (19.46). The appendix point is therefore not that Kelvin–Helmholtz, firehose, and MRI are the same instability. It is that all three teach the same structural lesson: flow changes stability by modifying the balance between driving and tension.

40.7 Experimental perspective

A literal moving wall is rare in hot-plasma MHD. More often the wall motion is implemented indirectly: either the plasma rotates with respect to a fixed wall, or an external rotating field makes a fixed structure behave as if the boundary were moving. This is why the language of wall motion and plasma rotation should be regarded as complementary rather than competing.

Tokamaks and resistive walls. In toroidal confinement, the main practical question is whether plasma rotation can keep a resistive wall from “looking resistive” on the timescale of an external mode. That is the conceptual bridge from the static no-wall / ideal-wall story to resistive-wall stabilization by rotation Bondeson and Ward (1994).

Non-neutral rotating walls. The most literal rotating-wall experiments are found in non-neutral plasmas, where an externally applied rotating perturbation spins up and compresses the plasma column Huang et al. (1997). Although the magnetic geometry is different from fusion MHD, the lesson is beautiful: one can use a boundary condition in a chosen rotating frame as an active control tool rather than merely as a passive stabilizer.

Mirrors and GDT. In mirror devices, biased end plates create differential \(E\times B\) rotation and thereby suppress flute-dominated transport. The GDT vortex-confinement program is the best-known modern example Beklemishev et al. (2010). It is a reminder that interchange stabilization in an open system is not only a matter of line tying; it is also a matter of whether the unstable displacement can stay globally coherent.

Flow-stabilized Z pinches. In Z pinches, the axial-shear program culminating in the ZaP family of experiments made it possible to test, in a particularly direct way, the idea that sufficiently strong internal flow shear can weaken current-driven global modes Shumlak and Hartman (1995); Shumlak et al. (2003). Whatever one thinks of the eventual reactor prospects, the value of these experiments is enormous: they are among the cleanest laboratory demonstrations that stability is not solely a property of a magnetic equilibrium but of the equilibrium together with its motion.

Takeaways
  • The static energy principle remains the natural starting point for identifying the branch that would be unstable without motion, but flowing equilibria are not governed by the same purely self-adjoint minimization problem.
  • Rigid motion mainly Doppler-shifts the spectrum. Differential motion changes stability by creating a spread in Doppler shift across the eigenfunction.
  • A moving conducting wall is best understood through the wall-frame condition \(\hat {\vect {n}}\times (\E +\vect {U}_w\times \B )=0\): boundary anchoring is evaluated in the wall frame.
  • Sheared \(E\times B\) flow in mirrors and sheared axial flow in Z pinches can suppress interchange- and kink-like modes when the shear-induced phase spread exceeds the no-flow growth rate.
  • Kelvin–Helmholtz instability is the cautionary counterexample: a velocity profile can be a stabilizer for one branch and the free-energy source for another.

Bibliography

    E. Frieman and Manuel Rotenberg. On hydromagnetic stability of stationary equilibria. Reviews of Modern Physics, 32(4):898–902, 1960. doi:10.1103/RevModPhys.32.898.

    S. Chandrasekhar. Hydrodynamic and Hydromagnetic Stability. Clarendon Press, Oxford, 1961.

    A. Bondeson and D. J. Ward. Stabilization of external modes in tokamaks by resistive walls and plasma rotation. Physical Review Letters, 72(17):2709–2712, 1994. doi:10.1103/PhysRevLett.72.2709.

    X.-P. Huang, F. Anderegg, E. M. Hollmann, C. F. Driscoll, and T. M. O’Neil. Steady-state confinement of non-neutral plasmas by rotating electric fields. Physical Review Letters, 78(5): 875–878, 1997. doi:10.1103/PhysRevLett.78.875.

    Alexei D. Beklemishev, Peter A. Bagryansky, Maxim S. Chaschin, and Elena I. Soldatkina. Vortex confinement of plasmas in symmetric mirror traps. Fusion Science and Technology, 57(4): 351–360, 2010. doi:10.13182/FST10-A9497.

    U. Shumlak and C. W. Hartman. Sheared flow stabilization of the \(m = 1\) kink mode in z pinches. Physical Review Letters, 75(18):3285–3288, 1995. doi:10.1103/PhysRevLett.75.3285.

    U. Shumlak, B. A. Nelson, R. P. Golingo, S. L. Jackson, E. A. Crawford, and D. J. Den Hartog. Sheared flow stabilization experiments in the zap flow z pinch. Physics of Plasmas, 10(5):1683–1690, 2003. doi:10.1063/1.1558294.

Problems

Problem 40.1.
Show directly from Eqs. (F.12) and (F.14) that if \(\B \cdot \hat {\vect {n}}=0\) on a perfectly conducting wall then the plasma normal velocity at the boundary must equal the wall normal velocity.
Problem 40.2.
Consider a mode with \((m,k)\) and radial width \(\Delta r\) in a plasma with \(\Omega (r)=\Omega _0+\Omega _1 r\) and \(V_z(r)=V_0+V_1 r\). Derive the phase-spread estimate (F.10) and determine the condition under which shear beats a no-flow growth rate \(\gamma _0\).
Problem 40.3.
Starting from Eq. (F.34), derive the symmetric result (F.35) and the instability threshold (F.37). Repeat the derivation for unequal densities and determine how the threshold changes.
Problem 40.4.
Use Eq. (F.25) to estimate the edge rotation shear needed to stabilize a flute mode with \(m=1\), width \(\Delta r\), and no-flow growth rate \(\gamma _0\) in a mirror trap. Compare with the corresponding Z-pinch estimate, Eq. (F.30), and explain physically why the mode number enters one case while the axial wavenumber enters the other.