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Pinch Equilibria

Source lecture(s): PC368 Lec 14

Intuition

Run a current through a plasma and it squeezes itself. Parallel currents attract, so a column of current contracts until magnetic tension balances the outward plasma pressure. It is the simplest confinement scheme imaginable, it was the first one tried, and its failure modes taught the field almost everything it knows about MHD stability.

The three pinches

Current Field Confining stress
Z-pinch axial \(J_z\) azimuthal \(B_\theta\) tension in curved \(B_\theta\)
θ-pinch azimuthal \(J_\theta\) axial \(B_z\) magnetic pressure of \(B_z\)
Screw pinch both both both — this is a tokamak

The Z-pinch and the Bennett relation

Radial force balance is

\[\frac{dp}{dr} = -J_zB_\theta,\qquad B_\theta(r) = \frac{\mu_0 I(r)}{2\pi r}\]

Multiply by \(r^2\), integrate across the column, and integrate the pressure term by parts. All the profile detail cancels, leaving a statement about totals only:

Bennett relation

$\(\frac{\mu_0 I^2}{8\pi} = N k_B(T_e + T_i)\)$ where \(N\) is the number of ions per unit length.

Beautifully profile-independent: whatever the density and temperature shapes, a given current confines a given line-density at a given temperature. It is the cheapest useful equilibrium result in plasma physics.

Worked example

A Z-pinch carries \(I = 1\) MA with line density \(N = 10^{19}\) m⁻¹. What temperature does it confine?

\[k_B(T_e+T_i) = \frac{\mu_0I^2}{8\pi N} = \frac{4\pi\times10^{-7}\times10^{12}}{8\pi\times10^{19}} = 5\times10^{-15}\ \text{J} \approx 31\ \text{keV}\]

So \(T_e = T_i \approx 16\) keV — comfortably fusion-relevant, from a single megamp and no external field at all. This is why the Z-pinch looked so promising in the 1950s, and why its instability was such a bitter disappointment.

Why it does not work

The Z-pinch is violently unstable, in two named ways:

  • \(m=0\) sausage. Squeeze the column locally: \(B_\theta = \mu_0I/2\pi r\) rises where \(r\) falls, so the pinching force grows, so the neck squeezes harder. Runaway. The column necks off in microseconds.
  • \(m=1\) kink. Bend the column: field lines bunch on the inside of the bend, where the pressure is now higher, pushing the bend further out. Also runaway.

An axial \(B_z\) inside the plasma cures both — it resists both compression (pressure) and bending (tension), which converts the Z-pinch into a screw pinch and, with the right safety-factor profile, into a tokamak. See kink instability and the Kruskal–Shafranov limit.

The modern afterlife

  • Z machine (Sandia): wire-array Z-pinches producing the most intense laboratory X-ray source on Earth, used to drive inertial-fusion capsules.
  • Dense plasma focus: a compact pulsed Z-pinch used as a neutron and X-ray source.
  • Sheared-flow-stabilised Z-pinch: velocity shear suppresses the sausage and kink modes; a live line of fusion research, if a contested one.
  • Astrophysical jets are widely modelled as current-carrying columns and show exactly the kink morphology — the wiggles in radio jets are \(m=1\) modes on a cosmic scale.

Common mistakes

  • Reading the Bennett relation as an equilibrium guarantee. It says an equilibrium exists; it says nothing about whether it survives a perturbation. It does not.
  • Confusing θ-pinch and Z-pinch. The subscript names the current direction; the field is perpendicular to it.
  • Thinking a θ-pinch is stable and therefore useful. It is far more stable — but it is straight, so it loses everything out the ends. Closing it into a torus reintroduces the problems.

Knowledge graph position

Prerequisites: MHD, magnetic stress tensor. Leads to: kink instability, energy principle, tokamak equilibrium.

Quiz

Q1 (conceptual). Why is the Bennett relation independent of the pressure profile?

Answer

Integrating the force balance against \(r^2\,dr\) and using integration by parts converts the pressure gradient into the total pressure integral, while \(B_\theta \propto I(r)/r\) turns the magnetic term into the total current. Profile shape drops out; only the integrals survive.

Q2 (computational). Doubling the current in a Z-pinch at fixed line density does what to temperature?

Answer

\(T \propto I^2/N\), so temperature quadruples. The steep scaling is what made pinches attractive — until stability was accounted for.

Q3 (MCQ). The sausage instability grows because a local constriction:

  • (a) lowers the local \(B_\theta\)
  • (b) raises the local \(B_\theta\) since \(B_\theta \propto 1/r\), increasing the pinch force further
  • (c) increases the plasma pressure enough to push back
  • (d) is stabilised by the axial current
Answer

(b). Positive feedback: narrower radius → stronger azimuthal field → stronger inward force → narrower still. An internal \(B_z\) breaks the loop, because compressing it costs magnetic pressure.