A generalization of the Taylor constraint to the advection of axisymmetric poloidal magnetic fields

Abstract

The range of ▿ ∧ (u ∧ B) is determined for an axisymmetric poloidal incompressible B and incompressible u, by solving ▿ ∧ (u ∧ B) = ▿ ∧ v for u, assumed to be confined to within a surface of revolution. Two constraints on V are shown to be necessary for the existence of solutions, viz that the integral of rç/|B| must vanish on each flux surface, and that the integral of v1 around any dosed field line must vanish. The construction of the general solution proves that the constraints are sufficient conditions, providing also that the second derivative, with respect to the poloidal flux function, of the volume contained by flux surfaces does not vanish. The general solution is stated for the homogeneous case, v = 0. For the particular case of poloidal v1 an integral property of the solution u is established.