Viscous dissipation by tidally forced inertial modes in a 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. By numerically solving the linear equations of this problem, including viscosity, we first confirm some analytical results obtained on a two-dimensional model by Ogilvie (J. Fluid Mech., vol. 543, 2005, p. 19); some additional properties of this model are uncovered like the existence of narrow resonances associated with periodic orbits of characteristics. 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. The three-dimensional solutions show, like their two-dimensional 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 emit- ted at the critical latitude on the inner sphere. The asymptotic regime, where the dissipation is independent of the viscosity, is reached when an attractor has been excited by this shear layer. This property is not shared by the two-dimensional model where shear layers around attractors are independent of those emitted at the critical latitude. 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 weakly contracting webs 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.