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We perform one of the first studies into the nonlinear evolution of tidallyexcited inertial waves in a uniformly rotating fluid body, exploring asimplified model of the fluid envelope of a planet (or the convective envelopeof a solar-type star) subject to the gravitational tidal perturbations of anorbiting companion. Our model contains a perfectly rigid spherical core, whichis surrounded by an envelope of incompressible uniform density fluid. Thecorresponding linear problem was studied in previous papers which this workextends into the nonlinear regime, at moderate Ekman numbers (the ratio ofviscous to Coriolis accelerations). By performing high-resolution numericalsimulations, using a combination of pseudo-spectral and spectral elementmethods, we investigate the effects of nonlinearities, which lead totime-dependence of the flow and the corresponding dissipation rate. Angularmomentum is deposited non-uniformly, leading to the generation of significantdifferential rotation in the initially uniformly rotating fluid, i.e. the bodydoes not evolve towards synchronism as a simple solid body rotator. Thisdifferential rotation modifies the properties of tidally excited inertialwaves, changes the dissipative properties of the flow, and eventually becomesunstable to a secondary shear instability provided that the Ekman number issufficiently small. Our main result is that the inclusion of nonlinearitieseventually modifies the flow and the resulting dissipation from what linearcalculations would predict, which has important implications for tidaldissipation in fluid bodies. We finally discuss some limitations of oursimplified model, and propose avenues for future research to better understandthe tidal evolution of rotating planets and stars.

Nonlinear evolution of tidally forced inertial waves in rotating fluid bodies