Numerical study of an axisymmetric anisotropic conductivity dynamo

May 4, 2020, Revision: 1.12

Abstract

Plunian & Alboussière (2020) have recently presented an analytic solution to an axisymmetric dynamo for an anisotropically conducting medium. The velocity field corresponds to that of a rigidly rotating cylinder embedded in a conductor at rest. Here we present numerical simulations for smooth angular velocity profiles. We also consider the case where the anisotropy is confined to be within the rotating cylinder only.

1 Introduction

A major obstacle in dynamo theory has been the fact that the magnetic field must always be nonaxisymmetric and therefore three-dimensional. But this is not strictly true if an allows for the conductivity to be anisotropic. For that case, Plunian & Alboussière (2020) found recently a solution which is extremely remarkable in many respects. (i) Using cylindrical coordinates, (r, ϕ, z), the three components of the magnetic field depend only on radius r and height z, but not on the azimuthal angle ϕ. (ii) The velocity field is very simple, corresponding to a rigidly rotating cylinder embedded in a conducting solid at rest. (iii) The growth rate increases with increasing conductivity without limit and does not go to zero as for slow dynamos. (iv) The solution is given in closed form.

Understanding the properties of such a dynamo could be of interest not only for the realization of laboratory dynamos, but also for dynamos in nearly collisionless plasmas, where the vastness of the full six-dimensional phase space constitutes a significant technical challenge. It should be kept in mind, however, that the experimental realization of such an anisotropically conducting medium may, in practice, involve a nonaxisymmetric design. This was already discussed by Plunian & Alboussière (2020), who suggested a composite of conducting layers arranged in a logarithmic spiral interlaced with poorly contacting or insulating layers.

The connection between dynamos in a medium of anisotropic conductivity and the homopolar disk dynamo was already pointed out by Ruderman & Ruzmaikin (1984). They considered a twodimensional dynamo in Cartesian coordinates, for which Alboussière et al. (2020) found an analytic solution. A feasible design of the homopolar disk dynamo was presented by Priede & Avalos-Zúñiga (2020).

Plunian & Alboussière (2020) emphasize that the rotation must be retrograde. Dynamo action requires then α > 0. For α < 0, dynamo action occurs for prograde rotation.

2 The model

We adopt rigid rotation with angular velocity Ω_{0} in the inner cylinder of radius R, with the material at rest outside, i.e.,

Ω(r) = Ω_{0} f(r)  (1)

where Ω_{0} is a constant and f(r) is a radial profile modelling the change from finite to vanishing angular velocity in a continuous fashion, given by

f(r) = 1 − Θ(r, d).  (2)

We use the same profile function to also allow for a radial truncation of the anisotropy using

η_{ij} = η_{0} δ_{ij} + η_{1}(r) q_{i}q_{j}  (3)

where η_{1} = η_{10}f(r). We solve the uncurled induction equation,

\frac{∂A_{i}}{∂t} = (u × B)_{i} − η_{ij}J_{j}  (4)

We define a magnetic Reynolds number as

R_{Ω} = Ω_{0}R^{2}/η_{0},  (5)

and the degree of anisotropy as

s = η_{10}/η_{0}.  (6)

Plunian & Alboussière (2020) found dynamo action for |R_{Ω}| > 14.61 for s → ∞ using the optimal angle α = 0.16π ≈ 29^{∘} and wavenumber kR = 1.1, corresponding to a minimal height of L_{z}/R = 5.7.

Figure 1: psav_64x32r1 Anisotropic conductor both in the interior and the exterior. Here, η_{0} = 0.0440, L_{r} = 3, and L_{z} = 3.9

3 Results

In Figure 1 we show the result for the case where η_{1} = const. The critical maximal magnetic diffusivity is η_{0} = 0.0440, corresponding to R_{Ω} = 22.7, and the vertical wavenumber is kR = 1.6. In Figure 2 we show the case where η_{1} → 0 for r > R. The critical maximal magnetic diffusivity is η_{0} = 0.0218, corresponding to R_{Ω} = 45.9, and the vertical wavenumber is kR = 1.7.

References

Alboussière, T., Drif, K., & Plunian, F., Dynamo action in sliding plates of anisotropic electrcal conductivity. Phys. Rev. E 2020, 101, 033107.

Figure 2: psav_I64x32k4 Isotropic conductor in the exterior, outside the cylinder. Here, η_{0} = 0.0218, L_{r} = 3, and L_{z} = 3.6

Figure 3: p Growth rate and k.

Plunian, F., & Alboussière, T., Axisymmetric dynamo action is possible with anisotropic conductivity. Phys. Rev. Research 2020, 2, 013321.

Priede, J., & Avalos-Zúñiga, R., Feasible homopolar dynamo with sliding liquid-metal contacts. Phys.

Table 1: Models with η_{1}/η_{0} = 5, α = 28.8^{∘}, and kR = 1.1.

η_{0}  η_{1}  λ k
0.0445 0.0035
0.04607 -0.0004
0.0467 -0.0014

Table 2: Models with η_{1}/η_{0} = 5, α = 30^{∘}, and kR = 1.1.

\begin{matrix}η_{0} & L_{z} & λ \\ 1 × 10^{−2} & 1 & 0.3221 \\ 1 × 10^{−3} & 0.2 & 1.816 \\ 2 × 10^{−4} & 0.06 & 2.9404 \\ 5 × 10^{−5} & 0.03 & 3.6706 \\ 1 × 10^{−5} & 0.03 & 3.6706 \\ 1e-5 & 0.010 & 3.3899 \\ 1e-5 & 0.012 & 3.9734 \\ 1e-5 & 0.014 & 4.155 \\ 1e-5 & 0.016 & 4.1450 \\ 1e-5 & 0.018 & 4.0539 \\ 5e-6 & 0.010 & 4.130 \\ 5e-6 & 0.008 & 3.883 \end{matrix}

Lett. A 2013, 377, 2093–2096.

Ruderman, M. S., & Ruzmaikin, A. A., Magnetic field generation in an anisotropically conducting fluid. Geophys. Astrophys. Fluid Dynam. 1984, 28, 77–88.

Table 3: Models with η_{1} = 0 for r ≫ 1 and η_{10}/η_{0} = 5.

\begin{matrix}η_{0} & L_{r} & L_{z} & λ \\ 0.010 & 2 & 2.0 & 0.0886 \\ 0.014 & 2 & 2.0 & 0.0441 \\ 0.016 & 2 & 2.0 & 0.0208 \\ 0.018 & 2 & 2.0 & −0.0030 \\ 0.018 & 2 & 2.1 & +0.0024 \\ 0.018 & 2 & 2.2 & +0.0066 \\ 0.018 & 2 & 2.4 & +0.0123 \\ 0.018 & 2 & 2.6 & +0.0155 \\ 0.018 & 2 & 2.8 & +0.0171 \\ 0.018 & 2 & 3.0 & +0.0177 \\ 0.018 & 2 & 3.2 & +0.0176 \\ 0.020 & 2 & 3.2 & +0.0071 \\ 0.022 & 2 & 3.2 & −0.0035 \\ 0.022 & 2 & 3.0 & −0.0059 \\ 0.022 & 2 & 3.3 & −0.0027 \\ 0.022 & 2 & 3.4 & −0.0021 \\ 0.022 & 2 & 3.5 & −0.0016 \\ 0.022 & 2 & 3.6 & −0.0012 \\ 0.022 & 2 & 3.7 & −0.0009 \\ 0.022 & 2 & 3.8 & −0.0007 \\ 0.022 & 2 & 3.9 & −0.0006 \\ 0.022 & 2 & 4.0 & −0.0006 \\ 0.022 & 3 & 4.0 & −0.0002 \\ 0.022 & 4 & 4.0 & −0.0011 \\ 0.022 & 3 & 4.2 & −0.0007 \\ 0.022 & 3 & 3.8 & −0.0002 \\ 0.0216 & 3 & 3.8 & +0.0014 \\ 0.0216 & 3 & 3.7 & +0.0014 \\ 0.0216 & 3 & 3.6 & +0.0013 \\ 0.0218 & 3 & 3.6 & +0.0001 \end{matrix}

Table 4: Models with η_{1} = 0 for r ≫ 1 and η_{10}/η_{0} = 5.

\begin{matrix}d_{Ω} & d_{η} & η_{0} & L_{r} & L_{z} & λ \\ 0.02 & 0.02 & 0.0218 & 3 & 3.6 & +0.0001 \\ 0.05 & 0.02 & 0.0218 & 3 & 3.6 & −0.0013 \\ 0.05 & 0.05 & 0.0218 & 3 & 3.6 & +0.0034 \end{matrix}

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Table 5: Models with anisotropy throughout, i.e., η_{1} = η_{10} = 5 η.

\begin{matrix}η_{0} & L_{r} & L_{z} & λ \\ 0.0218 & 3 & 1.8 & +0.1010 \\ 0.0300 & 3 & 1.8 & −0.0082 \\ 0.0300 & 3 & 2.0 & +0.0051 \\ 0.0300 & 3 & 2.2 & +0.0292 \\ 0.0294 & 3 & 1.8 & −0.0080 \\ 0.0294 & 3 & 2.2 & +0.0360 \\ 0.0306 & 3 & 2.2 & +0.0220 \\ 0.0312 & 3 & 2.2 & +0.0146 \\ 0.0324 & 3 & 2.2 & +0.0018 \\ 0.0324 & 3 & 2.4 & +0.0197 \\ 0.0340 & 3 & 2.6 & +0.0197 \\ 0.0360 & 3 & 2.8 & +0.0148 \\ 0.0380 & 3 & 3.0 & +0.0095 \\ 0.0400 & 3 & 3.2 & +0.0042 \\ 0.0410 & 3 & 3.2 & −0.0018 \\ 0.0410 & 3 & 3.3 & +0.0017 \\ 0.0410 & 3 & 3.4 & +0.0046 \\ 0.0420 & 3 & 3.5 & +0.0018 \\ 0.0425 & 3 & 3.6 & +0.0016 \\ 0.0430 & 3 & 3.7 & +0.0013 \\ 0.0435 & 3 & 3.8 & +0.0008 \\ 0.0440 & 3 & 3.9 & +0.0002 \end{matrix}