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The length of day variations with periods from five to one hundred years are mainly due to core-mantle interactions. Assuming a differential velocity between the core and the mantle, we investigate the pressure coupling on a core-mantle boundary (CMB) interface with topography. Including rotation, buoyancy, and magnetic effects in local models of the CMB, we provide a taxonomy of the waves radiated by the core flow along the topography. We obtain the local stress with a perturbation approach and a semi-analytical spectral model built upon these waves. We incorporate planetary curvature effects by considering a “non-traditional” beta-plane approximation suited for deep fluid layers and long topography wavelengths. We calculate weakly non-linear flows and characterize the wave drag mechanism. Unlike previous works, our analysis is not restricted to strong stratification or short wavelengths. It reveals the significant impact of the Rossby waves on stress. We also show that these waves are drastically modified when considering two-dimensional topographies instead of simple ridges. For a buoyancy frequency N at least comparable to the rotation frequency, the main factors defining the stress are N and √(U0) for the small velocity amplitudes relevant for the Earth's core. We document the departures from this scaling law as the velocity is increased. The main part of the CMB pressure torque is due to the topography with the largest horizontal length scale. We calculate the minimum stratification for the topographic torque to produce discernible changes in the length-of-day.

Physicists face major challenges in modelling multi-scale phenomena that are observed in geophysical flows (e.g. in the Earth's oceans and atmosphere, or liquid planetary cores). In particular, complexities arise because geophysical fluids are rotating and subject to density variations, but also because the fluid boundaries have complex geometries (e.g. the ocean floor) with wavelengths ranging from metres to thousands of kilometres. Models of planets' fluid layers and their consequences on the dynamics of the whole celestial object are constrained by observations, whose interpretation necessitates a comprehensive understanding of the underlying physics. Geophysical studies often entail combining cutting-edge experiments across a wide range of parameters, together with theory and limited numerical simulations, to derive predictive scaling laws applicable for planetary settings. In this review, we discuss experimental efforts that have contributed to our understanding of geophysical flows with topography. More specifically, we focus on (i) the flow response to mechanical (orbital) forcings in the presence of a large-scale (ellipsoidal) topography, (ii) some effects of small-scale topography onto bulk flows and boundary-layer dynamics, and (iii) the interaction between convection and roughness. For each case, the geophysical context is briefly introduced and some experimental perspectives are drawn.

In planetary fluid cores, the density depends on temperature and chemical composition, which diffuse at very different rates. This leads to various instabilities, bearing the name of double-diffusive convection (DDC). We investigate rotating DDC (RDDC) in fluid spheres. We use the Boussinesq approximation with homogeneous internal thermal and compositional source terms. We focus on the finger regime, in which the thermal gradient is stabilizing whereas the compositional one is destabilizing. First, we perform a global linear stability analysis in spheres. The critical Rayleigh numbers drastically drop for stably stratified fluids, yielding large-scale convective motions where local analyses predict stability. We evidence the inviscid nature of this large-scale double-diffusive instability, enabling the determination of the marginal stability curve at realistic planetary regimes. In particular, we show that in stably stratified spheres, the Rayleigh numbers Ra at the onset evolve like Ra ∼ Ek⁻¹, where Ek is the Ekman number. This differs from rotating convection in unstably stratified spheres, for which Ra ∼ Ek^−4/3. The domain of existence of inviscid convection thus increases as Ek^−1/3. Secondly, we perform non-linear simulations. We find a transition between two regimes of RDDC, controlled by the strength of the stratification. Furthermore, far from the RDDC onset, we find a dominating equatorially antisymmetric, large-scale zonal flow slightly above the associated linear onset. Unexpectedly, a purely linear mechanism can explain this phenomenon, even far from the instability onset, yielding a symmetry breaking of the non-linear flow at saturation. For even stronger stable stratification, the flow becomes mainly equatorially symmetric and intense zonal jets develop. Finally, we apply our results to the early Earth core. Double diffusion can reduce the critical Rayleigh number by four decades for realistic core conditions. We suggest that the early Earth core was prone to turbulent RDDC, with large-scale zonal flows.