Kinematic dynamo action with helical symmetry in an unbounded fluid conductor: Part 1. Formulation and survey of low order cases

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Abstract
Kinematic dynamo action in an unbounded homogeneous, fluid conductor is studied from an exact, analytical viewpoint, by exploiting the mathematical tractibility of “helical symmetry,” uncovered by Lortz [1968a). The principal motivation is to elucidate how helical dynamos operate, to clarify their energetics, to extend Lortz's work where possible and to place helical dynamos within the more general context of dynamo theory. Lortz's novel helical coordinate system is first motivated and derived rationally. The velocity and magnetic fields are then projected onto helical coordinates and represented in terms of helical defining scalar functions dependent on only two helical coordinates. The unsteady dynamo equations, in helical variables, are next derived, and their invariance properties deduced. Attention is then focused on the steady state. Low order finite Fourier expansions for the dependence on the primary helical coordinate are introduced. Three different possible low order dynamo classes are studied: (A) dynamos with magnetic field lines lying on circular cylinders (as in Lortz's prototype case); (B) dynamos with velocity streamlines lying on. circular cylinders; (C) dynamos with fully three-dimensional motions and fields. The dynamo mechanism is discussed, and the character of the mathematical problem to be solved for extraction of details is considered for each class of helical dynamo. Part 2 elaborates and refines a previously published explicit, exact, closed-form solution in Class A. The eigenvalue character of the problem is now incorporated directly, the field-line and streamline patterns are portrayed, and the dynamo energetics are clarified.

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