Large regions of gaseous planets are thought to be stratified with an unstable thermal gradient, but a stabilising gradient of heavy element composition. Fluid in these regions is unstable to semi-convection, with motions driven by differences in the molecular diffusivity of temperature and composition, and could play a role in supporting planetary magnetic fields. Previous studies focus largely on local models in Cartesian boxes; here, we investigate semi-convection in rotating spherical shells. The onset of linear instability shows a transition between the two limits of rotating convection and non-rotating semi-convection. Non-linear simulations evolve into a system of concentric layers of relatively constant density, separated by narrow high-gradient regions. These layers gradually merge, resulting in a statistically steady state dominated by either a single convection region or a narrower convective zone beneath a stably stratified layer (SSL), depending on the strength of the thermal forcing compared to the rotation. When magnetic field generation is considered, our magnetohydrodynamic simulations exhibit self-sustained dynamo action. In cases where the turbulent convective region generates magnetic fields that are smoothed by zonal flows within the overlying SSL, the resulting field is strongly dipolar and axisymmetric, in encouraging agreement with Saturn's observed magnetic field. Within the regimes explored, both the Rossby number and the thickness of the SSL are well predicted by a single combination of control parameters. This enables the identification of a parameter range in which the generated magnetic fields resemble those of planetary dynamos.

Geological data show that, early in its history, the Earth had a large-scale magnetic field with an amplitude comparable to the one of the present geomagnetic field. However, its origin remains enigmatic and various mechanisms have been proposed to explain the Earth's field over geological time scales. Here, we critically evaluate whether tidal forcing could explain the ancient geodynamo, by combining constraints from geophysical models of the Earth-Moon system and predictions from turbulence studies. Our analysis shows that lunar tidal forcing could have been sufficiently strong before-3.25 Gy to trigger turbulence within the Earth's core, and potentially to sustain dynamo action during that interval. Then, we propose new scaling laws for the magnetic field amplitude B. We expect the latter to scale as B proportional to /34/3, where /3 is the equatorial ellipticity of the liquid core, if the turbulence involves weak interactions of three-dimensional inertial waves. Alternatively, in the regime of strong tidal forcing, the expected scaling becomes B proportional to /3. When extrapolated to the Earth's core, it suggests that tidal forcing alone was too weak to possibly explain the ancient geomagnetic field. Therefore, our study indirectly favours another origin for the early Earth's dynamo on long time scales (e.g. exsolution of light elements atop the core, or thermal convection due to secular cooling).

Large earthquakes can trigger translational oscillations of Earth's inner core (Slichter modes), yet their damping remains uncertain. Using simulations, we quantify viscous and Ohmic dissipation in the fluid outer core. Earth's rotation splits the motion into one polar and two equatorial modes. We explore all three and derive scaling laws for the quality factor for each dissipation mechanism. Considering various dissipation regimes (diffusive, skin-layer, and Alfv & eacute;n-wave radiation), we derive scaling laws that capture the damping of translational oscillations across planetary interiors. Viscous effects are negligible, confined to a thin layer at the inner core boundary. Ohmic dissipation dominates, with decay times of 4-16 years. Equatorial modes damp at least twice as fast as the polar mode. Our results suggest that Slichter modes can persist for years. Their continued non-detection is therefore more likely to reflect weak excitation or observational limitations than rapid damping.

The drag on an oscillating sphere is a classical fluid-mechanics problem, yet no existing theory simultaneously accounts for confinement, rotation, viscosity and magnetic fields. We consider a conducting sphere undergoing translational oscillations inside a rotating spherical cavity, modelling confined magnetohydrodynamic flows relevant to planetary interiors and liquid metal experiments. In planetary settings, these motions correspond to the polar and equatorial Slichter modes of Earth's inner core. Existing theories are restricted to separate asymptotic regimes, including viscous drag in bounded fluids (Stokes 1851), rotational effects in inviscid cavities (Busse 1974), and magnetic coupling through oscillatory boundary layers (Buffett and Goertz 1995). We derive a unified asymptotic framework for oscillatory drag in rotating spherical shells with arbitrary confinement and arbitrary electrical conductivity and magnetic permeability contrasts between the inner sphere, fluid shell and outer solid, applicable to both polar and equatorial oscillations. The theory yields expressions for added mass, viscous and electromagnetic drag, with the associated dissipation. It captures viscous pressure corrections, magnetic pressure and tension, Alfven-wave radiation, and magnetohydrodynamic Stokes-Ekman boundary layers. Classical boundary-layer theories (e.g. Ekman layers) emerge as limiting cases, while confined solutions are also derived for the diffusion-dominated bulk regimes, yielding an explicit closed-form solution for the bounded oscillatory Stokes flow and a confined extension of the inductionless theory. Direct numerical simulations validate the analytical predictions across a broad parameter range. The resulting analytical framework provides quantitative predictions of oscillatory coupling, added mass and dissipation in planetary cores, icy-moon oceans and liquid-metal experiments.

The majority of investigations into planetary core and subsurface ocean dynamics have traditionally assumed a perfectly smooth interface. However, geodynamical models and seismic observations on Earth suggest the presence of topography. This study addresses the role of topography in the simplified but fundamental case of differential rotation between the topography and the fluid within a cylinder. We conducted numerical and experimental analyses, exploring various ranges of Rossby numbers (from 10-1 to 10-4 ) and different wavelengths and heights of topography, always greater than the Ekman boundary layer. Numerical simulations were performed using the spectral elements code Nek5000, while experiments were conducted with water on a rotating table employing particle imagery velocimetry (PIV). Our observations reveal that the topography emits inertial waves into the fluid, and their patterns are correlated with the derivatives of the topography's height, rather than directly with its height. The controlling parameters influence the frequencies and amplitudes of the inertial waves, leading to the derivation of scaling laws in Rossby number, wavelength, and topography height. From these scaling laws, we propose a model for the dynamics of the fluid, including energy transfers.

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. Dynamical models of planetary fluid layers are thus often constrained by observations, whose interpretation necessitates a comprehensive understanding of the underlying physics. To this end, geophysical studies often combine cutting-edge experiments across a wide range of parameters, together with theory and 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. The geophysical context is briefly introduced for each case, and some experimental perspectives are drawn.

Stably stratified fluid layers are common in gaseous planets, stellar interiors, and planetary cores, and have long been considered incapable of sustaining dynamo action. Here, we show that semiconvection - driven by a destabilizing thermal gradient within an overall stably stratified medium - can, in fact, give rise to self-sustained magnetic fields. Motivated by recent models suggesting that large portions of Jupiter and Saturn may be semiconvective, we perform direct numerical simulations in spherical shells, operating in the planetary-relevant regime of low magnetic Prandtl numbers. From a primary semiconvection instability, a layered convection state spontaneously develops, consisting of a convective region beneath a stably stratified layer of comparable thickness. Fluid motions in this convective region are strong enough to produce magnetic fields with key features observed in planetary dynamos, including strong dipolarity, realistic field strengths, and spectral characteristics. These results provide the first direct evidence that semiconvection can drive dynamo action in stably stratified regions of gas giants and stellar interiors, with important implications for understanding astrophysical magnetic field generation.

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” ‐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 at least comparable to the rotation frequency, the main factors defining the stress are and 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.

Flows in rapidly spinning bodies, such as the iconic libration-induced flow, are key ingredients of the dynamics of stars and planetary interiors. Laboratory experiments of such flows experience a strong centrifugal acceleration, which hinders the use of classical velocimetry methods relying on particle tracking. Modal acoustic velocimetry was introduced by Triana et al. (New J Phys 16(11):113005, 2014) as a new particle-free method, inspired from helioseismology, to alleviate this problem. In this method, acoustic modes are excited in the fluid and recorded in the spinning container. Rotation and fluid flow modify the characteristics of these modes, lifting the degeneracy of non-axisymmetric modes. To date, this method has only been applied to stationary or statistically stationary flows, by measuring frequency splittings in the spectral domain. Here, we analyze time-varying libration-induced flows. We propose and test two data acquisition strategies. The first strategy operates in the frequency domain and relies on the periodicity of the flow, while the second strategy involves a high-resolution algorithm applied in the time domain. The retrieved mode frequency splittings are compared to those computed for a classical linear libration-induced flow model as reported (Greenspan The theory of rotating fluids, Cambridge University Press, Cambridge, 1968). A very good agreement is obtained, but we observe an unexpected time delay, which we attribute to the buildup time of acoustic modes. We retrieve more than 50 splitting measurements at 10 successive libration phases. Inverting these data with the SOLA method, often used in helioseismology, we derive profiles (1D inversion) and maps (2D inversion) of the azimuthally averaged fluid rotation rate. The inversions recover the main characteristics of this time-dependent flow. The 2D inversion confirms the invariance of the flow along the rotation axis. Resolution kernels show that flow can be mapped on patches that spread over approximately 5 % of a meridian quarter-plane. Our study paves the way to the investigation of more exotic regimes of precession- or libration-induced flows.

Paleomagnetic data show that the Earth has owned a dynamo magnetic field for at least 3.5 Ga [1]. The geodynamo thus appeared well before the inner core nucleation [2], but its origin remains puzzling. Indeed, given the contradictory estimates of the thermal conductivity of liquid iron at core conditions [3], secular cooling may have not been strong enough to sustain it. Finding mechanisms capable of sustaining the geodynamo is thus crucial to understand the early Earth evolution. In particular, it has been proposed that the geodynamo could have been sustained by exsolution (or precipitation) of light elements near the core-mantle boundary (CMB) [4,5]. This mechanism could be powerful enough to sustain dynamo action (from an energetic viewpoint [6]), but its relevance has to be assessed using fluid dynamics models. Using global simulations, we explore the flow dynamics driven by exsolution of light elements in the early Earth. When the thermal conductivity is assumed to be large, we report and characterize two different flow regimes [7], which depend on the strength of thermal stratification in the core. Next, we assess the dynamo capability of these two regimes for the first time. We show that they are associated with a dipolar-multipolar transition for dynamo action. Our simulations thus constrain the thermal stratification of the core to be likely weak in the early Earth (to be compatible with the large-scale magnetic field evidenced by paleomagnetic data). [1] Tarduno et al., 2020, PNAS, 117(5), 2309-2318[2] Zhou et al., 2022, Nat. Comm., 13(1), 4161[3] Pozzo et al., 2022. EPSL, 584, 117466.[4] Badro et al., 2016, Nature, 536(7616), 326-328[5] Hirose et al., 2017, Nature, 543(7643), 99-102.[6] Landeau et al., 2022, Nat. Rev. Earth Environ., 3(4), 255-269[7] Monville, Vidal et al., 2019, GJI, 219(S1), S195-S218

A new diamond anvil cell experimental approach has been implemented at the European x-ray Free Electron Laser, combining pulsed laser heating with MHz x-ray diffraction. Here, we use this setup to determine liquidus temperatures under extreme conditions, based on the determination of time-resolved crystallization. The focus is on a Fe-Si-O ternary system, relevant for planetary cores. This time-resolved diagnostic is complemented by a finite-element model, reproducing temporal temperature profiles measured experimentally using streaked optical pyrometry. This model calculates the temperature and strain fields by including (i) pressure and temperature dependencies of material properties, and (ii) the heat-induced thermal stress, including feedback effect on material parameter variations. Making our model more realistic, these improvements are critical as they give 7000 K temperature differences compared to previous models. Laser intensities are determined by seeking minimal deviation between measured and modeled temperatures. Combining models and streak optical pyrometry data extends temperature determination below detection limit. The presented approach can be used to infer the liquidus temperature by the appearance of SiO2 diffraction spots. In addition, temperatures obtained by the model agree with crystallization temperatures reported for Fe–Si alloys. Our model reproduces the planetary relevant experimental conditions, providing temperature, pressure, and volume conditions. Those predictions are then used to determine liquidus temperatures at experimental timescales where chemical migration is limited. This synergy of novel time-resolved experiments and finite-element modeling pushes further the interpretation capabilities in diamond anvil cell experiments.

Motivated by modelling rotating turbulence in planetary fluid layers, we investigate precession-driven flows in ellipsoids subject to stress-free boundary conditions (SF-BC). The SF-BC could indeed unlock numerical constraints associated with the no-slip boundary conditions (NS-BC), but are also relevant for some astrophysical applications. Although SF-BC have been employed in the pioneering work of Lorenzani & Tilgner ( J. Fluid Mech. , vol. 492, 2003, pp. 363–379), they have scarcely been used due to the discovery of some specific mathematical issues associated with angular momentum conservation. We revisit the problem using asymptotic analysis in the low-viscosity regime, which is validated with numerical simulations. First, we extend the reduced model of uniform-vorticity flows in ellipsoids to account for SF-BC. We show that the long-term evolution of angular momentum is affected by viscosity in triaxial geometries, but also in axisymmetric ellipsoids when the mean rotation axis of the fluid is not the symmetry axis. In a regime relevant to planets, we analytically obtain the primary forced flow in triaxial geometries, which exhibits a second inviscid resonance. Then, we investigate the bulk instabilities existing in precessing ellipsoids. We show that using SF-BC would be useful to explore the non-viscous instabilities (e.g. Kerswell, Geophys. Astrophys. Fluid Dyn. , vol. 72, 1993, pp. 107–144), which are presumably relevant for planetary applications but are often hampered in experiments or simulations with NS-BC.

Earth’s magnetic field is generated by fluid motions in the outer core. This geodynamo has operated for over 3.4 billion years. However, the mechanism that has sustained the geodynamo for over 75% of Earth’s history remains debated. In this Review, we assess the mechanisms proposed to drive the geodynamo (precession, tides and convection) and their ability to match geomagnetic and palaeomagnetic observations. Flows driven by precession are too weak to drive the geodynamo. Flows driven by tides could have been strong enough in the early Earth, before 1.5 billion years ago, when tidal deformation and Earth’s spin rate were larger than they are today. Evidence that the thermal conductivity of Earth’s core could be as high as 250 W m −1 K −1 calls the ability of convection to maintain the dynamo for over 3.4 billion years into question. Yet, convection could supply enough power to sustain a long-lived geodynamo if the thermal conductivity is lower than 100 W m −1 K −1 . Exsolution of light elements from the core increases this upper conductivity limit by 15% to 200%, based on the exsolution rates reported so far. Convection, possibly aided by the exsolution of light elements, remains the mechanism most likely to have sustained the geodynamo. The light-element exsolution rate, which remains poorly constrained, should be further investigated.

The acoustic modes of a rotating fluid-filled cavity can be used to determine the effective rotation rate of a fluid (since the resonant frequencies are modified by the flows). To be accurate, this method requires a prior knowledge of the acoustic modes in rotating fluids. Contrary to the Coriolis force, centrifugal gravity has received much less attention in the experimental context. Motivated by on-going experiments in rotating ellipsoids, we study how global rotation and buoyancy modify the acoustic modes of fluid-filled ellipsoids in isothermal (or isentropic) hydrostatic equilibrium. We go beyond the standard acoustic equation, which neglects solid-body rotation and gravity, by deriving an exact wave equation for the acoustic velocity. We then solve the wave problem using a polynomial spectral method in ellipsoids, which is compared with finite-element solutions of the primitive fluid-dynamic equations. We show that the centrifugal acceleration has measurable effects on the acoustic frequencies when MΩ≳0.3, where MΩ is the rotational Mach number defined as the ratio of the sonic and rotational time scales. Such a regime can be reached with experiments rotating at a few tens of Hz by replacing air with a highly compressible gas (e.g., SF6 or C4F8).

Planetary magnetic fields are generated by motions of electrically conducting fluids in their interiors. The dynamo problem has thus received much attention in spherical geometries, even though planetary bodies are non-spherical. To go beyond the spherical assumption, we develop an algorithm that exploits a fully spectral description of the magnetic field in triaxial ellipsoids to solve the induction equation with local boundary conditions (i.e. pseudo-vacuum or perfectly conducting boundaries). We use the method to compute the free-decay magnetic modes and to solve the kinematic dynamo problem for prescribed flows. The new method is thoroughly compared with analytical solutions and standard finite-element computations, which are also used to model an insulating exterior. We obtain dynamo magnetic fields at low magnetic Reynolds numbers in ellipsoids, which could be used as simple benchmarks for future dynamo studies in such geometries. We finally discuss how the magnetic boundary conditions can modify the dynamo onset, showing that a perfectly conducting boundary can strongly weaken dynamo action, whereas pseudo-vacuum and insulating boundaries often give similar results.

We revisit the generation of mean zonal flows in fluid planetary interiors subjected to precession. The main effect of precession on a (nearly) spherical fluid envelope is to make the fluid rotate along an axis tilted with respect to the rotation axis of the solid mantle. This is the so-called "spin-over" response of the fluid. also shows that a steady shear flow develops on top of the spin-over mode due to non-linear effects in the boundary layer equation. This mean zonal shear flow has been studied theoretically and numerically by . With faster computers and more efficient codes, we compute this flow down to very low viscosity and compare with the inviscid theory of Busse (1968). In addition we investigate the width and the intensity of the detached shear layer, which is controlled by viscosity and therefore not present in the theory. We also use this problem as a benchmark to assess the benefits of using a semi-lagrangian numerical scheme, where solid-body rotation is treated exactly.

Changes in the Earth's rotation are deeply connected to fluid dynamical processes in the outer core. This connection can be explored by studying the associated Earth eigenmodes with periods ranging from nearly diurnal to multi-decadal. It is essential to understand how the rotational and fluid core eigenmodes mutually interact, as well as their dependence on a host of diverse factors, such as magnetic effects, density stratification, fluid instabilities or turbulence. It is feasible to build detailed models including many of these features, and doing so will in turn allow us to extract more (indirect) information about the Earth's interior. In this article, we present a review of some of the current models, the numerical techniques, their advantages and limitations and the challenges on the road ahead.

The generation of mean flows is a long-standing issue in rotating fluids. Motivated by planetary objects, we consider here a rapidly rotating fluid-filled spheroid, which is subject to weak perturbations of either the boundary (e.g. tides) or the rotation vector (e.g. in direction by precession, or in magnitude by longitudinal librations). Using boundary-layer theory, we determine the mean zonal flows generated by nonlinear interactions within the viscous Ekman layer. These flows are of interest because they survive in the relevant planetary regime of both vanishing forcings and viscous effects. We extend the theory to take into account (i) the combination of spatial and temporal perturbations, providing new mechanically driven zonal flows (e.g. driven by latitudinal librations), and (ii) the spheroidal geometry relevant for planetary bodies. Wherever possible, our analytical predictions are validated with direct numerical simulations. The theoretical solutions are in good quantitative agreement with the simulations, with expected discrepancies (zonal jets) in the presence of inertial waves generated at the critical latitudes (as for precession). Moreover, we find that the mean zonal flows can be strongly affected in spheroids. Guided by planetary applications, we also revisit the scaling laws for the geostrophic shear layers at the critical latitudes, and the influence of a solid inner core.