Coherent drift wave structures in sheared magnetic fields

For the problem of calculating the coherent drift wave structures in sheared magnetic fields, the authors have found it useful to derive the governing nonlinear pde from a variational principle. The variational principle is based on the free energy functional F[var-phi] = ∫V F(var-phi, ∇ var-phi, x)dx dy. The method is applied to the vortex with speed u derived in Su et al., given by ∇2 var-phi = (1 - vd/u) var-phi - Sm2/u2 (x - var-phi/u) (x - var-phi/2u) var-phi where space is measured in units of ρs, var-phi = (eΦ/Te)(Ln/ρs) and the magnetic shear parameter is Sm. While the linearized problem (var-phi much-lt ux) describes the usual shear induced damping, nonlinear solutions with trapped flow (var-phi > ur0) form nonlinear self-bound states, which are maxima of the free energy F. The authors discuss the analytic properties and the numerical procedures for solving these types of nonlinear pde's