STM-D-1069Paper1995Published and peer-reviewed

Rotating nonlinear magnetic islands in a tokamak plasma

Andrei I. Smolyakov · Akira Hirose · Enzo Lazzaro · G. B. Re · James D. Callen

Abstract and summary · read the original at the source

In one page

A tokamak holds its plasma inside nested magnetic surfaces. When one of those surfaces tears and reconnects it forms a magnetic island — a self-made bubble of field lines that spins with the plasma and lets heat leak out of the core. Smolyakov, Hirose, Lazzaro, Re and Callen write the full time-dependent equations for such an island, solving its width and its rotation frequency together in a two-fluid model that keeps the plasma’s own rotation and the viscosity that comes from the machine’s doughnut shape. Two results carry the paper. Because the toroidal geometry ties flow along the field to flow across it, the plasma in the island region behaves as if it were far heavier than a flat-slab calculation says, so a much stronger stray field is needed before an island locks and stops turning. And viscosity holds the island at an angle behind the applied field, raising that threshold again. The result is a control law: what to apply, how strong, and how fast to turn it.

Why it matters hereChapter 9 is about plasma that organises itself into large rotating magnetic structures and then holds that shape; the tokamak island is the version engineers can instrument, and this paper is its equation of motion. Chapter 12 needs it because a machine that cannot steer its own islands cannot hold a burning plasma long enough to be a power source.

What it claims

01Nonlinear evolution equations for the island width and for the toroidal rotation frequency are derived together inside one two-fluid magnetohydrodynamic model that keeps the plasma rotation and the neoclassical parallel viscosity, so that how fast an island grows and how fast it spins are solved as a single time-dependent system rather than as two separate problems.Abstract; Sections II and III, Eqs. 25 and 31
Published and peer-reviewed
02Toroidal geometry couples the plasma flow along the magnetic field to the flow across it, and the neoclassical damping of the poloidal rotation then renormalises the effective plasma inertia in the island region upward by the factor q squared times R squared divided by r squared compared with the flat-slab model — which raises the critical stray magnetic field required to force an island open.Section VI, Eq. 25 set against the slab Eq. 17; Section V B
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03The perpendicular, anomalous plasma viscosity holds the induced island at a finite phase angle behind the applied external perturbation at the point of bifurcation, and that phase lag raises the critical magnetic field again — the same kind of lag a resistive wall imposes.Abstract; Section VI, Eq. 111 with Eqs. 68 and 69
Published and peer-reviewed
04The ion finite-Larmor-radius correction can reverse the sign of the inertia term, so that inertia destabilises rather than stabilises a non-rotating island when the radial electric field profile is inverted such that the diamagnetic flow is very nearly cancelled by the E-cross-B flow. The authors offer this as a candidate explanation for the measured asymmetry in critical field between tokamak discharges heated by tangential and by perpendicular neutral beams.Section VI, discussion of the modified inertial term in Eq. 25
Published and peer-reviewed
05Even with no wall to push against, an island’s rotation slows to the toroidal rotation frequency of the bulk plasma through viscous transfer of toroidal momentum, so that frequency is the island’s natural one. For a large, naturally occurring island carrying constant momentum, the rotation frequency then falls inversely with the island width — the behaviour reported for islands in the TEXT tokamak.Section VI; Eqs. 88 and 89
Published and peer-reviewed
06Comparing the measured phase of a locked mode against the phase this model predicts would settle whether present tokamaks sit in the inertia-dominated or the viscosity-dominated regime, which existing data cannot yet decide; and applying an external helical field that rotates steadily at an amplitude slightly above the error-field level should keep induced islands small, because resistive-wall stabilisation still acts on a rotating island.Section VI, closing paragraphs, Eq. 111 and the rotating-field proposal
What to watch

Read it · abstract

Abstract

The nonlinear dynamics of rotating low m (poloidal mode number) tearing modes in a tokamak with external resonant magnetic perturbations is examined. Nonlinear evolution equations for the island width and the toroidal rotation frequency are derived within the two-fluid magnetohydrodynamic model, taking into account the plasma rotation and neoclassical parallel viscosity. The nonlinear stability of magnetic islands interacting with a static external magnetic perturbation is considered, and the critical magnetic field for the appearance of a locked mode is determined. It is shown that the coupling of the perpendicular and longitudinal plasma flow due to the neoclassical plasma viscosity enhances the amplitude of the critical magnetic field compared to the value obtained in a slab approximation. The perpendicular plasma viscosity causes a finite phase shift between the applied external field and the magnetic island, and further increases the value of the critical magnetic field required to induce a magnetic island.

A. I. Smolyakov, A. Hirose, E. Lazzaro, G. B. Re and J. D. Callen, Rotating nonlinear magnetic islands in a tokamak plasma, Physics of Plasmas 2, issue 5, 1581–1598 (1995).

(Abstract only — see the rights note above. On this site, Porcelli and colleagues’ modelling of macroscopic magnetic islands in tokamaks is at /library/stm-1990916bc3, the same lead author’s later work on how a shear flow profile changes island stability is at /library/stm-95d2311119, Lazzaro and Shchepetov on bifurcated tokamak equilibria with vanishing edge current density is at /library/stm-3d22ea9fd7, and flow generated by electrostatic turbulence in tokamaks is at /library/stm-ea244957db.)

The way in

https://doi.org/10.1063/1.871308Published as Physics of Plasmas 2, issue 5, pages 1581 to 1598 (1995), copyright 1995 American Institute of Physics, with no licence recorded in Crossref, so the sheet is abstract-only. An author-deposited copy sits in the University of Wisconsin-Madison institutional repository (handle 1793/8574), but it carries the publisher’s notice that the article may be downloaded for personal use only, so it is not an open licence and no text beyond the published abstract is reproduced here. That deposited copy was read in full on 2026-09-08 through the repository’s REST interface — the old bitstream URL now 301s to a single-page application — and every claim below is located to a section or equation of it. Pagination and the funding record come from the OSTI bibliographic record for this DOI (OSTI ID 46360; United States Department of Energy grant DE-FG02-86ER53218, with support also from the Natural Sciences and Engineering Research Council and National Fusion Program of Canada and from the Consiglio Nazionale delle Ricerche of Italy). Given names are expanded from the repository’s own author record for this article (Smolyakov, Andrei I.; Hirose, Akira; Lazzaro, Enzo; Callen, James D.); the fourth author is recorded there only as Re, G. B., so that name is left as the journal prints it.

How to cite it

Andrei I. Smolyakov, Akira Hirose, Enzo Lazzaro, G. B. Re, James D. Callen (1995) Rotating nonlinear magnetic islands in a tokamak plasma. doi:10.1063/1.871308

Where it sits in the curriculum

Plasmoids, charge clusters and the orbs Lattice confinement fusion