Historical research guide · 1069 · 1995
Rotating nonlinear magnetic islands in a tokamak plasma
Andrei I. Smolyakov · Akira Hirose · Enzo Lazzaro · G. B. Re · James D. Callen
In one minute
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.
Connection
Chapter 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.
From the source
Nonlinear 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
Toroidal 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
The 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
The 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
Even 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
Comparing 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
The way in
https://doi.org/10.1063/1.871308How 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