Hamiltonian formulation of reduced magnetohydrodynamics

Abstract

Reduced magnetohydrodynamics (RMHD) is a principal tool for understanding nonlinear processes, including disruptions, in tokamak plasmas. Although analytical studies of RMHD turbulence are useful, the model’s impressive ability to simulate tokamak fluid behavior has been revealed primarily by numerical solution. A new analytical approach, not restricted to turbulent regimes, based on Hamiltonian field theory is described. It is shown that the nonlinear (ideal) RMHD system, in both its high‐beta and low‐beta versions, can be expressed in Hamiltonian form. Thus a Poisson bracket, { , }, is constructed such that each RMHD field quantity ξ_i evolves according to ξ̇_i ={ξ_i,H}, where H is the total field energy. The new formulation makes RMHD accessible to the methodology of Hamiltonian mechanics; it has lead, in particular, to the recognition of new RMHD invariants and even exact, nonlinear RMHD solutions. A canonical version of the Poisson bracket, which requires the introduction of additional fields, leads to a nonlinear variational principle for time‐dependent RMHD.