Axisymmetric dynamo action is possible with anisotropic conductivity

Abstract

A milestone of dynamo theory is Cowling's theorem, known in its modern form as the impossibility for an axisymmetric velocity field to generate an axisymmetric magnetic field by dynamo action. Using anisotropic electrical conductivity, we show that an axisymmetric dynamo is in fact possible with a motion as simple as solid-body rotation. On top of that, the instability analysis can be conducted entirely analytically, leading to an explicit expression of the dynamo threshold, which is the only example in dynamo theory.

Axisymmetric dynamo action produced by differential rotation, with anisotropic electrical conductivity and anisotropic magnetic permeability

“…For , corresponding to isotropy of both conductivity and permeability, (3.12) and (3.13) are diffusion equations, leading to a free decaying solution (no dynamo action). For an isotropic permeability, , (3.12) and (3.13) are identical to the equations derived in Plunian & Alboussière (2020).…”

“…We check that for an isotropic permeability, , is the same as that given in Plunian & Alboussière (2020). In figure 4 the curves of the dynamo threshold versus are plotted for (isotropic magnetic permeability), and different values of .…”

“…The role in the dynamo effect of an anisotropic electrical conductivity has been studied for different geometries: Cartesian (Ruderman & Ruzmaikin 1984; Alboussière, Drif & Plunian 2020), cylindrical (Plunian & Alboussière 2020) and toroidal (Lortz 1989). Although at first glance it is difficult to imagine such an anisotropic electrical conductivity in natural objects, it is far from impossible.…”

“…From a theoretical point of view, an interesting consequence of considering an anisotropic conductivity is the possibility of obtaining an axisymmetric dynamo effect (Plunian & Alboussière 2020), allowing one to bypass the well-known Cowling antidynamo theorem (Cowling 1934; Kaiser & Tilgner 2014). Indeed, if it is anisotropic, then the electrical conductivity becomes a tensor, instead of being a scalar, which defeats Cowling's neutral point argument.…”

“…In addition, the use of an anisotropic conductivity leads to dynamo action for a motion as simple as shear (Ruderman & Ruzmaikin 1984; Alboussière et al. 2020; Plunian & Alboussière 2020), which is otherwise impossible to achieve.…”