Nonlinear, three-dimensional magnetohydrodynamics of noncircular tokamaks

TLDR

The equations describe magnetohydrodynamics in the low‑β, incompressible, large‑aspect‑ratio limit. The authors generalized Rosenbluth’s nonlinear tokamak equations to three dimensions, derived conservation laws and the energy principle, and numerically solved the resulting equations to study fixed‑boundary kink modes in rectangular tokamaks, using linearized two‑dimensional results as initial conditions for the nonlinear three‑dimensional simulations. The study found that fixed‑boundary kink modes in rectangular tokamaks, which are marginally stable in circular geometry, become highly unstable with growth rates comparable to free‑boundary modes, yet the nonlinear evolution perturbs the magnetic field only slightly while inducing strong plasma convection from the chamber center to the walls.

Abstract

Rosenbluth’s nonlinear, approximate tokamak equations of motion are generalized to three dimensions. The equations describe magnetohydrodynamics in the low β, incompressible, large aspect ratio limit. Conservation laws are derived and a well-known form of the energy principle is recovered from the linearized equations. The equations are solved numerically to study kink modes in tokamaks with rectangular cross section. Fixed-boundary kink modes, for which the plasma completely fills the conducting chamber, are considered. These modes, which are marginally stable to lowest order in circular tokamaks, become unstable with large growth rates, comparable to the growth rates of free boundary kink modes. The unstable modes are found using linearized, two-dimensional equations. The linear results are used as initial values in the nonlinear, three-dimensional computations. The nonlinear results show that the magnetic field is perturbed only slightly, while a large amount of plasma convection takes place carrying plasma from the center of the chamber to the walls.