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Magnetic Confinement

Source lecture(s): PC368 Lec 1

Intuition

A charged particle cannot cross a magnetic field line easily — it spirals around it with a Larmor radius of a millimetre or so in a fusion field. Perpendicular confinement is therefore free. Parallel confinement is the entire problem: along the field line the particle is unimpeded, and it leaves at thermal speed.

The history of magnetic confinement is a sequence of increasingly clever answers to "where does the field line go?"

Attempt 1: the straight solenoid

Uniform \(B\hat{z}\). Particles spiral tightly, and stream straight out the ends at \(\sim10^6\) m/s. Confinement time: microseconds. Fails.

Attempt 2: the magnetic mirror

Make the field stronger at both ends. Conservation of the magnetic moment \(\mu = mv_\perp^2/2B\) converts parallel motion into perpendicular as a particle enters the strong-field region, and if \(v_\perp\) uses up all the energy the particle turns around. This genuinely works — for some particles. Those with too little \(v_\perp\) escape through the loss cone, and because collisions continuously refill the loss cone, the plasma drains. Fails, but instructively — the same physics traps the Van Allen belts.

Attempt 3: bend it into a torus

No ends, no end losses. Except a purely toroidal field is non-uniform by geometry — \(B \propto 1/R\) — so it has both a gradient and curvature, and both drive charge-dependent drifts:

\[\mathbf{v}_{\nabla B + \text{curv}} = \frac{m\left(v_\parallel^2 + \tfrac12 v_\perp^2\right)}{qB^3}\,\mathbf{B}\times\nabla B\]

Ions drift up, electrons drift down. The resulting vertical electric field drives an \(\mathbf{E}\times\mathbf{B}\) drift that is outward for both species, and the entire plasma walks into the outer wall in milliseconds. Fails — and you can watch it fail in the drift orbit lab, curvature-drift mode.

Attempt 4: twist the field — tokamaks and stellarators

The cure is to make each field line spiral, so it spends half its time on top and half on the bottom. A particle following it samples both signs of the vertical drift, which then averages away. The twist is measured by the safety factor

\[q(r) = \frac{r B_\phi}{R B_\theta}\]

— the number of toroidal turns per poloidal turn. Two ways to get the poloidal field:

  • Tokamak. Drive a large toroidal current through the plasma itself; that current makes \(B_\theta\). Simple, axisymmetric, excellent confinement — but the current is a free energy source for kink instabilities, needs \(q > 1\) at the edge (Kruskal–Shafranov), and is fundamentally pulsed unless driven non-inductively. Disruptions — sudden loss of the whole current — are the defining engineering hazard.
  • Stellarator. Build the twist into external coils instead. No plasma current, so no disruptions and inherently steady-state; the price is a fiendish 3-D coil geometry and historically worse confinement. Wendelstein 7-X showed that modern optimisation largely closes the gap.

The two devices worth knowing

ITER (Saint-Paul-lès-Durance, France) — the largest tokamak ever built, designed for \(Q \ge 10\): 500 MW of fusion power from 50 MW of plasma heating, the first burning plasma where alpha self-heating dominates. Originally budgeted near $4B, now above $25B; a seven-member international collaboration with additional participants including Thailand.

TT-1 (Thailand Tokamak 1) — the first tokamak in Thailand and in ASEAN, a refurbished HT-6M donated by ASIPP, China, installed at TINT in Ongkharak, Nakhon Nayok.

Milestone Date
Commissioning at ASIPP, China May 2022
Disassembly and transport August 2022
Assembly at TINT November 2022
Commissioning at TINT January 2023
Ownership transfer April 2023

Its stated development path is three steps: a foundation phase reaching \(10^6\) K and building domestic engineering capability; a superconducting device reaching fusion-relevant \(10^8\) K within ten years; and, at thirty years, a Thai-designed power plant.

Common mistakes

  • "The magnetic field holds the plasma like a container." It restricts motion across field lines only. Everything hard about confinement is about the parallel direction and about drifts that carry particles across lines anyway.
  • Forgetting that the toroidal field is non-uniform by necessity. \(B \propto 1/R\) is forced by Ampère's law around the central column — you cannot design it away, only compensate for it.
  • Thinking \(q\) is about safety margins in the engineering sense. It is a geometric winding number; the name is historical, though staying above \(q = 2\) really does keep you safe from the most violent kinks.

Knowledge graph position

Prerequisites: Drift motions, Larmor radius, adiabatic invariants. Leads to: MHD equilibrium, MHD stability, pinch equilibria.

Quiz

Q1 (conceptual). Why does bending a solenoid into a torus not, by itself, confine a plasma?

Answer

It removes end losses but introduces \(B \propto 1/R\), hence ∇B and curvature drifts. These are charge-dependent, so ions and electrons separate vertically; the resulting \(\mathbf{E}\) drives an \(\mathbf{E}\times\mathbf{B}\) drift that is radially outward for both species. The plasma is lost in milliseconds.

Q2 (computational). A tokamak has \(R = 6\) m, \(a = 2\) m, \(B_\phi = 5\) T and a plasma current giving \(B_\theta = 0.6\) T at the edge. What is the edge safety factor?

Answer

\(q = rB_\phi/(RB_\theta) = (2)(5)/((6)(0.6)) = 2.8\) — comfortably above the Kruskal–Shafranov limit of \(q = 1\), and above the \(q = 2\) working threshold.

Q3 (MCQ). The essential difference between a tokamak and a stellarator is:

  • (a) the tokamak is toroidal and the stellarator is linear
  • (b) the poloidal field comes from plasma current in one, external coils in the other
  • (c) only the stellarator confines particles magnetically
  • (d) the stellarator does not need a toroidal field
Answer

(b). Both are toroidal and both need helical twist; they differ only in where the poloidal component comes from — and every other difference (disruptions, steady-state operation, coil complexity) follows from that one choice.