Mean-field model of the von Kármán sodium dynamo experiment using soft iron impellers

It has been observed that dynamo action occurs in the von-Kármán-Sodium (VKS) experiment only when the rotating disks and the blades are made of soft iron. The purpose of this paper is to numerically investigate the role of soft iron in the VKS dynamo scenario. This is done by using a mean-field model based on an axisymmetric mean flow, a localized permeability distribution, and a localized $\alpha$ effect modeling the action of the small velocity scales between the blades. The action of the rotating blades is modeled by an axisymmetric effective permeability field. Key properties of the flow giving to the numerical magnetic field a geometric structure similar to that observed experimentally are identified. Depending on the permeability of the disks and the effective permeability of the blades, the dynamo that is obtained is either oscillatory or stationary. Our numerical results confirm the leading role played by the ferromagnetic impellers. A scenario for the VKS dynamo is proposed.

Figure 1

Bottom half of the meridian section of the VKS numerical model.

Figure 2

Heuristics for the $\alpha$ model. The jet between the blades (top and bottom left) expels the fluid outward (thin arrow and outward symbol for ${\mathbf{u}}^{\prime }$) and deforms the azimuthal component of the mean magnetic induction $\mathbf{B}$ (thick arrow), resulting in small scale perturbations ${\mathbf{b}}^{\prime }$ (top and bottom center panels) which, in turn, generate an electromotive force ${\mathbf{u}}^{\prime }\times {\mathbf{b}}^{\prime }$ (top and bottom right). Note that ${\mathbf{u}}^{\prime }\times {\mathbf{b}}^{\prime }$ is opposite to $\mathbf{B}$. The same argument holds for a vertical mean magnetic induction. This argument shows also that the radial component of $\mathbf{B}$ yields an electromotive force that is zero on average, thereby implying that ${\mathbb{a}}_{rr}$ can be neglected.

Figure 3

Time evolution of the toroidal, poloidal, and total magnetic energy in induction runs with the MND velocity field with ${\mu }_d=60$, ${\mu }_b=10$ using $h=0.9$ (a) and $h=0.7$ (b).

Figure 4

Induction runs with different MND flows. Profiles as a function of $z$ at $r=0.3$ of the steady magnetic field with different MND flows: (a) $h=0.9$; (b) $h=0.7$.

Figure 5

Largest growth rates of axisymmetric modes for $\alpha \in [-0.03,0.05]$, $R_{\mathrm{m}}=30$, ${\mu }_b=10$, and ${\mu }_d=60$ (dipolar and quadrupolar eigenvectors).

Figure 6

Largest growth rates of axisymmetric modes for $\alpha \in [-0.03,0.03]$ $R_{\mathrm{m}}=30$, ${\mu }_b=10$, and ${\mu }_d=\infty$ (dipolar and quadrupolar eigenvectors).

Figure 7

Dipolar eigenmode for ${\mu }_d=\infty$ (a) and ${\mu }_d=60$ (b), with ${\mu }_b=10$, $R_{\mathrm{m}}=30$, and $\alpha =-0.03$ in both cases. Shown in panels (a) and (b) are (left) a meridional section, where colors represent the out-of-plane magnetic field component and the arrows materialize the poloidal magnetic field; and (right) an isosurface of $20\%$ of the maximum of the magnetic energy and magnetic field lines colored by $H_z$ [from blue (near axis) $H_z=-0.01$ to red (outside axis) $H_z=0]$.

Figure 8

Dipolar eigenmode for ${\mu }_d=\infty$ (top row) and ${\mu }_d=60$ (bottom row), with ${\mu }_b=10$, $R_{\mathrm{m}}=30$, and $\alpha =-0.03$. Radial profiles of $H_r$ (a),(d), $H_{\theta }$ (b),(e), and $H_z$ (c),(f) at $z=0,\pm 0.51$, as indicated.

Figure 9

Dynamo runs with the toroidal MND flow using ${\mu }_b=10$, ${\mu }_d=60$, $R_{\mathrm{m}}=30$. (a) Comparison of dipolar growth rate versus $\alpha$ for the toroidal MND field (labeled Dip-noPol) and the full MND velocity field (labeled Dip as in Fig. 5). (b) Dipolar eigenmode for $\alpha =-0.03$ using toroidal MND field; (c) Dipolar eigenmode for $\alpha =-0.03$ using full MND field [as in Fig. 7]. Colors represent the out-of-plane component of the magnetic field and the arrows materialize the poloidal component.

Figure 10

Dynamo runs with the toroidal MND velocity field for ${\mu }_b=10$, ${\mu }_d=60$, $R_{\mathrm{m}}=30$, and $\alpha =-0.03$. Radial profiles of $H_r$ (a), $H_{\theta }$ (b), and $H_z$ (c) at $z=0,\pm 0.51$ as indicated.

Figure 11

Largest growth rate of the axisymmetric dipolar mode with ${\mu }_d=60$ for $\alpha \in [-0.03,0.05]$, using either ${\mu }_b=10$ (as in Fig. 5, solid line labeled 10-60) or ${\mu }_b=60$ (dashed line labeled 60-60).

Figure 12

Growth rates of Fourier modes $m=0$ (solid line labeled $m=0$) and $m=1$ (dashed line labeled $m=1$) with $\alpha =-0.03$ for $R_{\mathrm{m}}\in [30,430]$: (a) with ${\mu }_b=1$, ${\mu }_d=60$; (b) with ${\mu }_b=1$, ${\mu }_d=1$.