On the Helical Dynamo of Lortz as a Model for the Steady Main Geomagnetic Field

Abstract

Summary An attempt is made to evaluate the infinite cylindrical dynamo of Lortz as a basis for a model of the steady geomagnetic field. Provided it can be deformed from cylindrical to toroidal geometry, when then embedded near the equatorial plane of the liquid outer core, such a dynamo appears to be geophysically relevant. 1. Introduction and scaling Dynamo theory is widely believed to be the fundamental tool required to understand the fluid dynamics of the Earth’s liquid outer core, particularly the manner in which forced fluid motions sustain the main geomagnetic field in the face of inevitable ohmic decay. In the relatively short period since 1958, when Backus and Herzenberg provided rigorous existence proofs of homogeneous spherical dynamo action, a host of more plausible dynamos have appeared (excellent reviews are given by Roberts 1971; Weiss 1971; Gubbins 1974). It very much remains to sort out this array and apply the results within the geophysical context. This paper describes a preliminary effort to utilize dynamo theory on behalf of geomagnetism. The steady cylindrical helical dynamo of Lortz (1968) holds a unique place in this subject. Not only is it ‘ the simplest case of a self-exciting homogeneous dynamo known ’ (as remarked by P. H. Roberts 1971), but it is also the only one presently available for which exact closed form analytic solutions of the induction equation can be extracted (but, surprisingly, none have yet been displayed or studied). Solely on these grounds it deserves further development because it thereby avoids the notoriously troublesome truncation-convergence difficulties inherent in most other dynamos. The present work is motivated by a desire to learn whether or not the Lortz dynamo can also satisfy the particular restrictions presented by steady geomagnetism. We first attempt to elucidate some aspects of how this dynamo functions in general (Section 2). An explicit example of a Lortz dynamo is presented in Section 3. Finally, in Section 4, we address the question of its geophysical relevance. It is customary to presume that a proper theory for Earth’s main magnetic field rests on finding simultaneous solutions of the equations of fluid dynamics, with Lorentz forces and a driving body force included, together with the induction equation of magnetohydrodynamics. For a Boussinesq liquid of nearly uniform density po executing incompressible motions, the relevant system can be cast into the form: av - + (v V)v = - Vn- 2Ct0 x v+ (V x a) x a- VV x (V x v) + f,