Rossby waves andα-effect

Abstract

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.

Selection of axial dipole from a seed magnetic field in rapidly rotating dynamo models

“…For a frozen flow (saturated velocity field at a given time) we do not find any dynamo action in our low-E parameter regimes as classical kinematic theory deals with [9]. This suggests that Rossby wave plays an important role in magnetic field generation in the rapidly rotating spherical shell dynamo models [11]. The roles of drifting frequency of the columnar structure (for exmaple, Kumar-Roberts flow) on dynamo action is briefly studied in the framework of kinematic models [12].…”

Cylindrical anisotropic $α^{2}$ dynamos

“…Similar flow structures were recently discussed in the context of quasi-geostrophic dynamos (Schaeffer and Cardin 2006). A detailed description of such "rings of rolls" and the α tensor resulting from them has been derived by Avalos et al (2007) and is given in Appendix A. The mean e.m.f.…”

“…For some flow configurations, it has been shown (Avalos et al 2007) that the resulting matrix α κλ is of the form…”

From isotropic to rapidly rotating anisotropic alpha-squared dynamos

“…Further discussion is given in paper I. Moffatt's force balance is also ageostrophic, but this is due to the inertial term. Avalos-Zuniga et al (2009) have derived kinematically an α-effect of a different form, driven by Rossby waves arising from the boundary slope. This is a higher-order mechanism than the Ekman boundary suction driving the α-effect in Busse (1975).…”