F. Siso-Nadal and P.A. Davidson
Department of Engineering, University of Cambridge, UK
Within the outer core, the low magnetic Reynolds number ($R_m$) approximation seems appropriate for the study of small length-scale motion (of the order of a few tens of kilometers). Despite the smallness of these structures, the Rossby number is less than unity and the characteristic velocity of the flow is, most likely, smaller than the phase speed of inertial waves. Therefore, the evolution of small scale motion might be dominated by inertial wave propagation, particularly for times scales comparable to the rotational period. We have studied the low $R_m$ evolution of vortices and other disturbances in a rotating fluid. Particular attention was given to the interaction between the inertial waves which radiate from the disturbances and the field. Inertial waves do not necessarily propagate solely along the axis of rotation. For disturbances of geophysical interest as plumes or eddies, energy radiates forming `fans' as opposed to columns. The direction of the field lines are devoid of oscillations due to the dissipative action of the field; motion is simply extruded along this direction at a rate of $(t/\tau)^{1/2}$, where $\tau$ is the Joule damping time. Motion appears highly anisotropic as well as fragmented, exponential decay of angular momentum and wave motion precluding the latter effect. The fragmentation occurs even in the absence of buoyancy, which should be contrasted with the larger time-scale buoyant flows discussed by Braginsky \& Meytlis [GAFD 1990]. Of particular interest is the fact that $\tau$ determines the rate at which energy is dissipated and it is of the same order of magnitude as the rotational period, that is, small in comparison with the rest of the meaningful geological time scales. Motion at the small scales might occur in the form of transient clouds surrounding the longer-lived steady buoyant flows, radiating from them, fragmenting and being quickly damped out. The relative orientation of the field with respect to the rotational axis affects how efficiently the field dissipates energy; suggesting that, throughout the core, Joule heating might not be homogeneously distributed.