We study the onset of dynamo action of the Riga and Karlsruhe experiments with the addition of an external wall, the electro-magnetic properties of which being different from those of the fluid in motion. We consider a wall of different thickness, conductivity and permeability. We also consider the case of a ferro-fluid in motion.

We present a feasible homopolar dynamo design consisting of a flat, multi-arm spiral coil, which is placed above a fast-spinning metal ring and connected to the latter by sliding liquid-metal electrical contacts. Using a simple, analytically solvable axisymmetric model, we determine the optimal design of such a setup. For small contact resistance, the lowest magnetic Reynolds number, Rm ≈ 34.6, at which the dynamo can work, is attained at the optimal ratio of the outer and inner radii of the rings R i /R o ≈ 0.36 and the spiral pitch angle 54.7• . In a setup of two copper rings with the thickness of 3 cm, R i = 10 cm and R o = 30 cm, self-excitation of the magnetic field is expected at a critical rotation frequency around 10 Hz.

To study the onset of a stationary dynamo in the presence of inner or outer walls of various electromagnetic properties, we propose a simple 1D-model in which the flow is replaced by an alpha effect. The equation of dispersion of the problem is derived analytically. It is solved numerically for walls of different thicknesses and of electric conductivity and magnetic permeability different from those of the fluid in motion. We also consider walls in the limit of infinite conductivity or permeability.PACS. 47.65.+a Magnetohydrodynamics and electrohydrodynamics -91.25.Cw Origins and models of the magnetic field; dynamo theories

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Rossby waves drifting in the azimuthal direction are a common feature at the onset of thermal convective instability in a rapidly rotating spherical shell. They can also result from the destabilization of a Stewartson shear layer produced by differential rotation as expected in the liquid sodium experiment (DTS) working in Grenoble, France.A usual way to explain why Rossby waves can participate to the dynamo process goes back to Busse (1975). In his picture, the flow geometry is a cylindrical array of parallel rolls aligned with the rotation axis. The axial flow component (the component parallel to the rotation axis) is (i) maximum in the middle of each roll and changes its sign from one roll to the next. It is produced by the Ekman pumping at the fluid containing shell boundary. The corresponding dynamo mechanism can be explained in terms of an α-tensor with non-zero coefficients on the diagonal. It corresponds to the heuristic picture given by Busse (1975).In rapidly rotating objects like the Earth's core (or in a fast rotating experiment), Rossby waves occur in the limit of small Ekman number (≈ 10 −15 ). In that case, the main source of the axial flow component is not the Ekman pumping but rather the "geometrical slope effect" due to the spherical shape of the fluid containing shell. This implies that the axial flow component is (ii) maximum at the borders of the rolls and not at the centers. If assumed to be stationary, such rolls would lead to zero coefficients on the diagonal of the α-tensor, making the dynamo probably less efficient if possible at all. Actually, the rolls are drifting as a wave, and we show that this drift implies non-zero coefficients on the diagonal of the α-tensor. These new coefficients are in essence very different from the ones obtained in case (i) and cannot be interpreted in terms of the heuristic picture of Busse (1975). They were interpreted as higher-order effects in Busse (1975). In addition we considered rolls not only drifting but also having an arbitrary radial phase shift as expected in real objects.