Axisymmetric dynamo action is possible with anisotropic conductivity

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 an 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.


I. INTRODUCTION
Since the pioneering study of Cowling [1] there has been a constant effort to improve the demonstration of the so-called Cowling's (antidynamo) theorem. In its modern form this theorem states that an axisymmetric magnetic field cannot be generated by dynamo action under the assumption of axisymmetry of velocity field, electrical conductivity, magnetic permeability and shape of the conductor [2][3][4][5][6]. Cowling's theorem encompasses time-dependent flows [7,8], non-solenoidal flows and variable conductivity [9,10]. However nothing has yet been said about the effect of an anisotropic electrical conductivity and how in this case Cowling's theorem is overcome. A demonstration of dynamo action with shear and anisotropic conductivity has already been given [11], but for a different geometry and within asymptotic limits relevant to the fast dynamo problem.
Beyond its theoretical interest, this issue is relevant to at least three fields of physics. In astrophysics it is well known that, in the mean-field approximation, an anisotropic tensor of magnetic diffusivity may naturally occur from anisotropic gradients of magnetohydrodynamic turbulence [12]. In plasma physics, just like thermal conductivity [13], the electrical conductivity in the magnetic field direction is different from the electrical conductivity in the direction perpendicular to the magnetic field [14]. This usually occurs in a plasma which is already magnetized. Although this does not preclude dynamo action we will not examine this issue here, considering that there is no external magnetic field. Finally, as will be shown below, a dynamo experiment can be designed on the basis of our anisotropic conductivity model. The results show that such an experiment is feasible, which is welcome because experimental dynamo demonstrations are rather rare. * Franck.Plunian@univ-grenoble-alpes.fr † Thierry.Alboussiere@ens-lyon.fr The inner-cylinder of radius r0 rotates as a solidbody within an outer-cylinder at rest. The radial boundary r1 of the outer-cylinder is rejected at infinity. In the limit η1 = ∞ the electric currents follow logarithmic spiral trajectories. Right: The vector q, which makes a constant angle α with the radial direction, is perpendicular to the spiraling current.

II. ANISOTROPIC CONDUCTIVITY
Let us consider a material of electrical conductivity σ such that σ = σ 1 in a given direction q, and σ = σ 0 ≥ σ 1 in the directions perpendicular to q. We choose q as a unit vector in the horizontal plane, where (e r , e θ , e z ) is a cylindrical coordinate system and α a constant angle. In a companion paper another choice for q, within a cartesian frame, is studied [15]. In Fig. 1 the curved lines correspond to the directions of the large conductivity σ 0 . They are perpendicular to q and describe logaritmic spirals.
Writing Ohm's law j = σ 1 E in the direction of q, and j = σ 0 E in the directions perpendicular to q, leads to the following conductivity tensor We consider the solid body rotation u of a cylinder of radius r 0 embedded in an infinite medium at rest (Fig. 1), both regions having the same resistivity tensor R ij . The magnetic induction B satisfies the equation where [η] is the magnetic diffusivity tensor defined as η ij = R ij /µ 0 , µ 0 being the magnetic permeability of vacuum. Renormalizing the distance, magnetic diffusivity and time by respectively r 0 , (µ 0 σ 0 ) −1 and µ 0 σ 0 r 2 0 , the dimensionless form of the induction equation is identical to (4), but with and where Ω is the dimensionless angular velocity of the inner-cylinder.

III. RESOLUTION
Provided the velocity is stationary and z-independent, an axisymmetric magnetic induction can be searched in the form with (Ã,B) = (A, B) exp(γt + ikz) where γ is the instability growthrate, k the vertical wavenumber of the corresponding eigenmode, and where A and B depend only on the radial coordinate r. Thus the magnetic induction takes the form dynamo action corresponding to {γ} > 0. From (6) and (8) we find that ∇ × (u × B) = 0 in each region r < 1 and r > 1. Replacing (5) and (6) in the induction equation (4) leads to where D ν (X) = ν 2 X − ∂ r 1 r ∂ r (rX) , c = cos α and s = sin α.

IV. DYNAMO MECHANISM
The dynamo mechanism can be described as a two step process as illustrated in Fig. 4. The boundary condition (17) implies that B θ is generated from B r by differential rotation between the inner and outer cylinders. This leads to distorsion of magnetic field lines as shown in the right of Fig. 4. In return the first term on the right hand side of (9) corresponds to the generation of B r from B θ , provided η 1 csk = 0. This appears more clearly rewritting (9-10) as FIG. 4. Left: Three dimensional sketch of some trajectories of the current density j and the magnetic field B. Right: Magnetic field lines in the horizonthal plane z = 0. The magnetic field is distorded by the differential rotation while the current density is bent by the conductivity anisotropy.
In the left of Fig. 4 the horizonthal currents are represented to follow the direction of logaritmic spirals. To show it, the current density j = ∇ × B is written in the form From (9) taken at the threshold γ = 0, we find that corresponding to the equation of logaritmic spirals. In the limit η 1 → ∞ we find that j · q = 0, the currents following the trajectories given in Fig.1. Dynamo action thus occurs through differential rotation conjugated to anisotropic diffusion. For η 1 = 0 (isotropic diffusion) or cs = 0, in (20-21) B r and B θ are decoupled, canceling any hope of dynamo action in accordance with Cowling's theorem.
It is interesting to note that in (20) and (21), in each equation it is the first term on the right-hand side which helps for dynamo action. These terms correspond to the off-diagonal coefficients of the anisotropic diffusivity tensor (5). Therefore the diagonal and off-diagonal coefficients act respectively against and in favour of dynamo action.

V. CONCLUSIONS
The neutral point argument of Cowling relies on the impossibility, in an axisymmetric configuration, of maintaining a toroidal current density [1]. This argument falls as soon as the conductivity is a tensor because, in this case, the cross product of a toroidal velocity field with a poloidal magnetic field can actually produce a toroidal current density. In other words, the anistropic conductivity forces the current density to follow spiraling trajectories, with nonzero azimuthal components, thus overcoming Cowling's theorem.
Beyond the fact that with an anisotropic conductivity an axisymmetric dynamo can be operated from a simple solid-body rotation, it is interesting to put some numbers on the previous results. Considering an innercylinder of radius r 0 = 0.05m, taking the conductivity of copper µ 0 σ 0 ≈ 72.9s.m −2 , leads to a dynamo threshold f * = Ω * (2πµ 0 σ 0 r 2 0 ) −1 ≈ 12.8Hz. Provided the cylinder height and outer radius r 1 are sufficiently large, this is experimentally achievable. Such an anisotropic conductivity can be easily manufactured by alternating thin layers of two materials with different conductivities and a logarithmic spiral arrangement of these thin layers. Of course, the resulting conductivity is no longer homogeneous and, more importantly, it does not satisfy the axisymetry hypothesis of Cowling's theorem. However, provided the layers are thin enough, an anisotropic conductivity model is relevant to design such a dynamo experiment. Another dynamo experiment design with spiraling wires has been studied [16]. Though the geometry is different, the dynamo threshold is comparable to the present one.