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Viscous dissipation by tidally forced inertial modes in a rotating spherical shell

We investigate the properties of forced inertial modes of a rotating fluid inside a spherical shell. Our forcing is tidal like, but its main property is that it is on the large scales. Our solutions first confirm some analytical results obtained on a two-dimensional model by Ogilvie (2005). We also note that as the frequency of the forcing varies, the dissipation varies drastically if the Ekman number E is low (as is usually the case). We then investigate the three-dimensional case and compare the results to the foregoing model. These solutions show, like their 2D counterpart, a spiky dissipation curve when the frequency of the forcing is varied; they also display small frequency intervals where the viscous dissipation is independent of viscosity. However, we show that the response of the fluid in these frequency intervals is crucially dominated by the shear layer that is emitted at the critical latitude on the inner sphere. The asymptotic regime is reached when an attractor has been excited by this shear layer. This property is not shared by the two-dimensional model. Finally, resonances of the three-dimensional model correspond to some selected least-damped eigenmodes. Unlike their two-dimensional counter parts these modes are not associated with simple attractors; instead, they show up in frequency intervals with a weakly contracting web of characteristics. Besides, we show that the inner core is negligible when its relative radius is less than the critical value 0.4E^{1/5}. For these spherical shells, the full sphere solutions give a good approximation of the flows (abridged abstract).