Precession, Nutation, Libration and Tides
Precession, Librations and wobbles represent departures from the uniform rotation of a planet due to external gravitational torques created by the sun and the other celestial objects in the solar system. Tides on the other hands represents deformation of the planets induced by gravitational forces exerted by their orbital partners, like the moon on the Earth. All these perturbations of an otherwise uniformly rotating planets of a fixed shape will result in changes of their angular momentum.
How the angular momentum, hence the rotation, changes in time depends on the internal structure of the planet, in particular it is sensitive to the presence of a fluid layer, such as a molten core, an ocean or an atmosphere. At the end of the nineteenth century scientists proposed that one could use the observations of the precession, nutations and librations of planets and moons to detect such fluid envelops. This entreprise requires a detailed understanding of the dynamics of the fluid region. Our group carries out theoretical, numerical and experimental research activities to understand the flows driven by precession, libration and tides and their influence on the rotation of planets and moons.
Precessing spherical shells: flows, dissipation, dynamo and the lunar core
Precession of planets or moons affects internal liquid layers by driving flows, instabilities and possibly dynamos. The energy dissipated by these phenomena can influence orbital parameters such as the planet’s spin rate. However, there is no systematic study of these flows in the spherical shell geometry relevant for planets, and the lack of scaling law prevents convincing extrapolation to celestial bodies. We have run more than 900 simulations of fluid spherical shells affected by precession, to systematically study basic flows, instabilities, turbulence and magnetic field generation. We observe no significant effects of the inner core on the onset of the instabilities. We obtain an analytical estimate of the viscous dissipation, mostly due to boundary layer friction in our simulations. We propose theoretical onsets for hydrodynamic instabilities, and document the intensity of turbulent fluctuations. We extend previous precession dynamo studies towards lower viscosities, at the limits of today’s computers. In the low viscosity regime, precession dynamos rely on the presence of large-scale vortices, and the surface magnetic fields are dominated by small scales. Interestingly, intermittent and self-killing dynamos are observed. Our results suggest that large-scale planetary magnetic fields are unlikely to be produced by a precession-driven dynamo in a spherical core. But this question remains open as planetary cores are not exactly spherical, and thus the coupling between the fluid and the boundary does not vanish in the relevant limit of small viscosity. Moreover, the fully turbulent dissipation regime has not yet been reached in simulations. Our results suggest that the melted lunar core has been in a turbulent state throughout its history. Furthermore, in the view of recent experimental results, we propose updated formulas predicting the fluid mean rotation vector and the associated dissipation in both the laminar and the turbulent regimes.
People involved: David Cebron, R Laguerre, Jerome Noir, N. Schaeffer
Grants: NS and DC were supported by the French Agence Nationale de la Recherche under grant ANR-14-CE33-0012 (MagLune). ISTerre is part of Labex OSUG@2020 (ANR10 LABX56). RL would like to acknowledge support from the European Research Council (ERC Advanced grant 670874 ROTANUT) and the HPC resources of the Royal Observatory of Belgium.
The precession-driven geodynamo
Precession is one aspect of the variability of Earth's rotation. It has been theorised that it could drive the geodynamo, but past investigations have been hampered by the difficulty of solving the governing equations in the aspherical geometries that are (most likely) the problem's key ingredient.
In this work we show how the governing equations within a variably rotating spheroid may be transformed onto a spherical computational domain. Additional coupling between spherical harmonic degrees is the computational cost of such a transformation, but this coupling is shown to be qualitatively similar to the coupling that is already dealt with routinely within implicit implementations of the Coriolis force. On the other hand, the computational gain of this approach is to permit the use of powerful spectral representations of fields on the unit ball, representations that are not readily available in spheroidal geometries. This approach should allow investigation of the precession-driven dynamo at lower Ekman number than previously possible.
People involved: Matthew Maitra, Andrew Jackson, Jerome Noir
Precessing non-axisymmetric ellipsoids: bi-stability and fluid instabilities
Precession driven flows are of great interest for both, industrial and geophysical applications. While cylindrical, spherical and spheroidal geometries have been investigated in great detail, the numerically and theoretically more challenging case of a non-axisymmetric cavity has received less attention.
In a series of publications we have studied precession-driven flows in a non-axisymmetric ellipsoids. Using a combination of numerical and experimental approaches, we focus on two aspects of the flow: the base flow of uniform vorticity and the development of fluid instabilities. In contrast to a preceding paper (J. Fluid. Mech., vol. 932, 2022, A24), where the ellipsoid rotated around its shortest axis, we do not observe bi-stability or hysteresis of the base flow, but a continuous transition from small to large differential rotation and tilt of the fluid rotation axis. We then use the model developed by Noir & Cébron (J. Fluid. Mech., vol. 737, 2013, pp. 412–439) to numerically determine regions in the parameter space of axial and equatorial deformations for which bi-stability may exist. Concerning fluid instabilities, we use three independent observations to track the onset of both boundary layer and parametric instabilities. Our results clearly show the presence of a parametric instability, yet the exact nature of the underlying mechanism (conical shear layer instability, shear instability and elliptical instability) is not unambiguously identified. A coexisting boundary layer instability, although unlikely, cannot be ruled out based on our experimental data. To make further progress on this topic, a new generation of experiments at significantly lower Ekman numbers (ratio of rotation and viscous time scales) is clearly needed.
People involved: Fabian Burmann, Lennart Kira, Jerome Noir
Grants: Founded through Swiss National Science Foundation grant 200021_140708, Swiss National Science Foundation grant no. 200021_165641, Swiss National Science Foundation grant no. 200021_185088 and ETH Research Grant ETH-26 15-1 and ETH-04 22-1. This research is supported by funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement no. 833848, UEMHP project)
Experimental study of the flows in a non-axisymmetric ellipsoid under precession
Precession driven flows are of great interest for both, industrial and geophysical applications. While cylindrical, spherical and spheroidal geometries have been investigated in great detail, the numerically and theoretically more challenging case of a non-axisymmetric cavity has received less attention. We report experimental results on the flows in a precessing triaxial ellipsoid, with a focus on the base flow of uniform vorticity, which we show to be in good agreement with existing theoretical models. As predicted, the uniform vorticity component exhibits two branches of solutions leading to a hysteresis cycle as a function of the Poincaré number. The first branch is observed at low forcing and characterized by large amplitude of the total fluid rotation and a moderate tilt angle of the fluid rotation axis. In contrast, the second branch displays only a moderate fluid rotation and a large tilt angle of the fluid rotation axis, which tends to align with the precession axis. In addition, we observe the occurrence of parametric instabilities early in the first branch, which saturate in the second branch, where we observe the same order of the kinetic energy in the base flow and instabilities.
People involved: Fabian Burmann, Jerome Noir
Grants: This research was supported by Swiss National Science Foundation grant 200021_140708, Swiss National Science Foundation grant no. 200021_165641, Swiss National Science Foundation grant no. 200021_185088 and ETH Research Grant ETH-26 15-1
Resonant and non-resonant flows in longitudinally and latitudinally librating spheres
We investigate the linear response to longitudinal and latitudinal libration of a rapidly rotating fluid-filled sphere. Asymptotic methods are used to explore the structure of resonant modes in both cases, provided that the nondimensional libration frequency is in the range \(\omega \in \left[0,2 \right]\). High-resolution numerics are then used to map out this entire frequency range, picking out both the resonant peaks as well as the non-resonant troughs in between. The kinetic energy is independent of the Ekman number \(E\) at the peaks and scales as \(E^{1/2}\) at the troughs. As the Ekman number is reduced, down to $E=10^{-10}$ for longitudinal libration and \(E=10^{-9}\) for latitudinal libration, the frequency response also exhibits an increasingly fractal structure, with more and more peaks and troughs emerging. The spacing between peaks is seen to follow an \(E=10^{1/2}\) self-similarity factor. However, detailed examinations of some of the more prominent troughs shows that their widths follow an \(E=10^{\sim0.23}\) self-similarity factor.
People involved: Yufeng Lin, Rainer Hollerbach, Jerome Noir Stijn Vantieghem
Grants: Funded through from SNSF Grant Nos. 200021_140708 and 200021_185088.
Libration-driven inertial waves and mean zonal flows in spherical shells
Several planetary bodies in our solar system undergo a forced libration owing to gravitational interactions with their orbital companions, leading to complex fluid motions in their metallic liquid cores or subsurface oceans. In this study, we numerically investigate flows in longitudinally librating spherical shells.
We focus on the Ekman number dependencies of several shear layers when the libration frequency is less than twice of the rotation frequency and the libration amplitude is small. Time-dependent flows mainly consist of inertial waves excited at the critical latitudes due to the Ekman pumping singularities, forming conical shear layers.
In particular, previous theoretical studies have proposed different scalings for the conical shear layers spawned from the critical latitudes at the inner boundary.
Our numerical results favour the velocity amplitude scaling \(\mathcal{O}(\epsilon E^{1/2})\) predicted by Le Dizès & Le Bars (J. Fluid Mech. 2017, 826, 653) over the scaling \(\mathcal{O}(\epsilon E^{1/6})\) initially proposed by Kerswell (J. Fluid Mech. 1995, 298, 311), though the Ekman numbers in our calculations are not sufficiently small to pin down this scaling. Non-linear interactions in the boundary layers drive a mean zonal flow with several geostrophic shears. Our numerical results show that geostrophic shears associated with the critical latitudes at the inner and outer boundaries exhibit the same scalings, i.e. an amplitude of \(\mathcal{O}(\epsilon^2 E^{-1/10})\) over a width of \(\mathcal{O}(E^{1/5})\). Apart from the geostrophic shear associated with the critical latitude, our numerical results show that the reflection of inertial waves can induce a geostrophic shear with an amplitude of \(\mathcal{O}(\epsilon^2 E^{-1/6})\) over a width of \(\mathcal{O}(E^{1/3})\). As the amplitude of the geostrophic shears increases as reducing the Ekman number, the geostrophic shears in the mean flows may be significant in planetary cores and subsurface oceans given small Ekman numbers of these systems.
People involved: Jerome Noir
Grants: Funded through Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung (SNF) grants #20021-140708 and #20021-165641. This study has received funding from the European Research Council (ERC) through Grant No. 247303 (MFECE) under the European Union's Horizon 2020 research and innovation programme at ETH Zurich.
Flows driven by librations in latitude in triaxial ellipsoids
Many planets and moons in our solar system possess subsurface oceans and/or liquid metallic cores. Orbital perturbations such as precession, nutation and libration (a planet’s oscillation with respect to its rotation axis) can affect these liquid layers, which might in turn impact angular momentum exchange, energy dissipation and magnetic field generation. While precession and libration in longitude have been at the center of many studies, librations in latitude (latitudinal oscillation of a planet’s figure axis with respect to its spin axis) have only received minor attention. With this in mind, we investigate experimentally the influence of librations in latitude within a fluid-filled ellipsoidal vessel on the development and stability of its base flow.
We begin by studying the flow that develops for small librations of the container. Our measurements confirm the analytical results of previous studies, bringing the first experimental characterization of a direct resonance between latitudinal librations and an inertial mode, namely the spin-over flow. They also show quantitative changes in the frequency and magnitude of the resonant base flow, at low forcing, compared to the predictions of linear theories. Although these non-linear effects are not expected to be significant at the very low amplitude of planetary librations, they may be pertinent for numerical and experimental work since most studies to date typically lie within the concerned parameter range.
We then investigate instabilities developing on top of the non-resonant and resonant base flow. In the former case, we observe a transition between a weakly and strongly energetic instability, both characterized by inertial-waves dynamics. In the latter case, we report on a complex bifurcation pattern. In particular we observe a transition from a strong branch to a weak branch at a forcing amplitude several times supercritical. We also observe transient dynamics in the form of periodic growth and collapse of parametric instabilities, and a hysteretic behavior around the resonance frequency where a strong instability may develop instead of the weak branch given a strong enough initial perturbation. Finally, we emphasize that we only witnessed flow regimes in which instabilities were present. This underlines the lack of stability of this system.
People involved: Jerome Noir