Ponomarenko dynamo sustained by a free swirling jet

Abstract

We present numerical results on dynamo action in a flow driven by an azimuthal body force localized near the end of an elongated cylindrical container. The analysis focuses on the central region of the cylinder, where axial variations in the flow are relatively weak, allowing the magnetic field to be represented as a helically traveling wave. Four magnetic impeller configurations and multiple forcing intensities are examined. In all cases, the velocity profiles in the central region display a similar \propto r^{-2} dependence across a wide range of Reynolds numbers and forcing region widths. The magnetic field is found to start growing under conditions similar to those of the Riga dynamo. However, the growing modes exhibit a substantial nonzero group velocity, indicating that the associated instability is convective: the flow can amplify an externally applied magnetic field but cannot sustain it autonomously. We outline several approaches for overcoming this limitation in order to realize a working laboratory dynamo based on an internally unconstrained swirling jet-type flow.

Figures (6)

• Figure 1: Schematic of the four numerically simulated configurations (a) and the corresponding radial profiles of the azimuthal body force at the magnet-facing surface (b). The flow in the liquid cylinder (C) is driven either by a rotating permanent magnet (PM) or by a rotating magnetic dipole (D). The cylinder has a length-to-diameter ratio of 2.5 in cases 1--3 and 3.0 in case 4 (shown with dashed lines). The permanent magnet has a diameter and length equal to 0.25 and 0.35 cylinder diameters in cases 1 and 2, respectively, and 0.3 cylinder diameters in case 4. The gap between the PM and the liquid is 0.05 cylinder diameters. In case 3, the dipole is positioned 0.25 cylinder diameters away from the liquid.
• Figure 2: Axial (a) and angular (b) velocity distributions for case 1. Colors are superimposed in the central half of the cylinder, over which the profiles in Fig. \ref{['Vp:fig']} are averaged. The isoline interval is 0.2 times the corresponding maximum absolute value in the averaging domain.
• Figure 3: Axial (a) and angular (b) velocity profiles averaged over $z$ in the central half of the cylinder. The numbers in the legend correspond to the cases shown in Fig. \ref{['scheme:fig']}. The velocity profile fits defined by Eq. (\ref{['Vprofiles']}) are shown as curves, and the corresponding fit parameters are listed in Table \ref{['fit:table']}.
• Figure 4: Magnetic Reynolds number (a) and oscillation frequency (b) of marginal modes of the magnetic field versus axial wavenumber $k$ and $m=1$ for the four velocity profiles corresponding to the configurations shown in Fig. \ref{['scheme:fig']}.
• Figure 5: The marginal $Rm$ as a function of marginal frequency $\omega$ for case 1 with a uniform back-flow $v$.