We perform one of the first studies into the nonlinear evolution of tidally excited inertial waves in a uniformly rotating fluid body, exploring a simplified model of the fluid envelope of a planet (or the convective envelope of a solar-type star) su
bject to the gravitational tidal perturbations of an orbiting companion. Our model contains a perfectly rigid spherical core, which is surrounded by an envelope of incompressible uniform density fluid. The corresponding linear problem was studied in previous papers which this work extends into the nonlinear regime, at moderate Ekman numbers (the ratio of viscous to Coriolis accelerations). By performing high-resolution numerical simulations, using a combination of pseudo-spectral and spectral element methods, we investigate the effects of nonlinearities, which lead to time-dependence of the flow and the corresponding dissipation rate. Angular momentum is deposited non-uniformly, leading to the generation of significant differential rotation in the initially uniformly rotating fluid, i.e. the body does not evolve towards synchronism as a simple solid body rotator. This differential rotation modifies the properties of tidally excited inertial waves, changes the dissipative properties of the flow, and eventually becomes unstable to a secondary shear instability provided that the Ekman number is sufficiently small. Our main result is that the inclusion of nonlinearities eventually modifies the flow and the resulting dissipation from what linear calculations would predict, which has important implications for tidal dissipation in fluid bodies. We finally discuss some limitations of our simplified model, and propose avenues for future research to better understand the tidal evolution of rotating planets and stars.
We consider the classical problem of kinematic dynamo action in simple steady flows. Due to the adjointness of the induction operator, we show that the growth rate of the dynamo will be exactly the same for two types of magnetic boundary conditions:
the magnetic field can be normal (infinite magnetic permeability, also called pseudo-vacuum) or tangent (perfect electrical conductor) to the boundaries of the domain. These boundary conditions correspond to well-defined physical limits often used in numerical models and relevant to laboratory experiments. The only constraint is for the velocity field u to be reversible, meaning there exists a transformation changing u into -u. We illustrate this surprising property using S2T2 type of flows in spherical geometry inspired by Dudley and James (1989). Using both types of boundary conditions, it is shown that the growth rates of the dynamos are identical, although the corresponding magnetic eigenmodes are drastically different.
