Magnetic Islands

Source lecture(s): pc368_lec16_reconnection

Intuition

Magnetic islands form when reconnecting magnetic field lines create closed loops of flux. They are the “beaded” droplets in the water-beading analogy: the plasma minimizes magnetic energy by breaking a long thin current sheet into discrete islands.

Formal Definition

A magnetic island is a region of closed magnetic flux bounded by separatrices \(\\psi = \\text{const}\) in a reconnecting current sheet. The island width \(w\) is measured by the \(x\)-distance between the separatrices.

Mathematical Formulation

In the Sweet–Parker or Petschek picture, the island grows according to

\[\\frac{dw}{dt} \\sim \\Delta' v_A\]

where \(\\Delta'\) is the stability parameter of the tearing mode and \(v_A\) is the Alfvén speed.

Derivation

Start from a Harris sheet with \(\\psi(x)\) having an inflection point.
Linearize the tearing-mode eigenvalue problem in resistive MHD.
The inner region gives the matching condition \(\\Delta' \\gamma \\sim \\eta\).
Nonlinear evolution widens the island at the rate above.

Worked Example

For a tokamak sawtooth crash with \(B = 3\) T, \(\\rho_{\\text{pol}} = 1\) m, and \(\\eta = 10^{-6}\\,\\text{m}^2\)/s, the growth time is \(\\tau \\sim \\mu_0 \\sigma \\rho_{\\text{pol}}^2 \\sim 0.1\)–\(1\) s.

Common Mistakes

Equating islands with “bad” physics. They are natural, energetically favorable states.
Ignoring the separatrix. The topology inside and outside the island is fundamentally different.

Quiz Questions

Why does beading into islands lower the magnetic energy?
What determines whether a tearing mode grows or is stable?