Fluid Dynamics Experiments for Planetary Interiors

Abstract

Understanding fluid flows in planetary cores and subsurface oceans, as well as their signatures in available observational data (gravity, magnetism, rotation, etc.), is a tremendous interdisciplinary challenge. In particular, it requires understanding the fundamental fluid dynamics involving turbulence and rotation at typical scales well beyond our day-to-day experience. To do so, laboratory experiments are fully complementary to numerical simulations, especially in systematically exploring extreme flow regimes for long duration. In this review article, we present some illustrative examples where experimental approaches, complemented by theoretical and numerical studies, have been key for a better understanding of planetary interior flows driven by some type of mechanical forcing. We successively address the dynamics of flows driven by precession, by libration, by differential rotation, and by boundary topography.

Keywords: Instabilities; Planetary cores; Rotational fluid dynamics; Subsurface oceans; Turbulence; Waves.

Fig. 1

Schematic view of a precessing body, and experimental arrangement of Noir et al. (2001): a spheroidal cavity is filled with water and rotates along the spin axis ${\mathit{\Omega }}_{\mathbf{S}}$ tilted versus the precession axis ${\mathit{\Omega }}_{\mathbf{P}}$, set in the experiment by the slowly rotating turntable

Fig. 2

Visualizations in a meridional plane of the experiment by Noir et al. (2001) for increasing (in absolute value) precession forcing, illustrating the transition from the laminar base flow with a uniform vorticity flow along an inclined axis (top left), to the bulk filling turbulence (bottom right). Water is seeded with small reflective flakes called Kalliroscope that align preferentially in the flow due to their anisotropy. Lighted here by a meridional light sheet and observed from an angle of ${90}^{\circ }$, Kalliroscope emphasizes preferential domains of the flow, including zones of strong shear. The two steady, geostrophic bands symmetric with respect to the fluid rotation axis in the top left figure result from the non-linear interaction of the conical shear layers emitted at ${30}^{\circ }$ (oscillating flows within conical shear layers are not directly visible using Kalliroscope)

Fig. 3

Experimental setup at IRPHE, Marseille (Le Reun et al. 2019), derived from the original design of Noir et al. (2009) and Noir et al. (2012). This large ellipsoidal installation allows exploring Ekman numbers down to $Ek=3.7\times {10}^{-6}$ with turbulent flows down to an excitation Rossby number $Ro=3.4\times {10}^{-2}$, defined here as the product of the libration amplitude times the ellipticity in the equatorial plane. Hence, this setup allows characterizing the transition between the 2D geostrophic turbulence at relatively large Ro and the asymptotic, 3D wave turbulence regime at small Ro. See also online movie at https://www.youtube.com/watch?v=Drq2qxX0U90

Fig. 4

Kalliroscope visualizations in a meridional plane of an experimental run in the UCLA ellipsoidal shell setup (Lemasquerier et al. 2017), with a rotation rate of 35 rpm, a libration angle of $7.5^{\circ },$ and a libration frequency of four times the rotation rate: These correspond to an Ekman number $Ek=2.6\times {10}^{-5}$ and an excitation Rossby number $Ro=0.18$ (equal to the product of the libration amplitude times the ellipticity in the equatorial plane). The left picture shows the initial base flow, with the noticeable Stewartson layer jet aligned with the rotation axis and tangent to the solid inner sphere. The right picture shows the turbulent saturation that settles a few tens of seconds later, with noticeable small-scale, 3D wavy patterns. See also online movie at https://www.youtube.com/watch?v=WGe-vLsm9Ho

Fig. 5

The spherical Couette system—two differentially rotating spherical shells with fluid between them. a a schematic of the system, b the Cottbus spherical Couette setup (Source: Michael Hoff)

Fig. 6

Inertial mode observed in the induced magnetic field in the 60-cm spherical Couette device. Here, the magnetic field is represented as an equal area projection at the outer sphere where blue is outward pointing magnetic field and red inward. The left image shows the observed magnetic field in a state with a dominant $l=3$ inertial mode; the right image shows the computed magnetic field that would be induced by a whole-sphere inertial mode of the same l, m. The frequencies of the experimentally observed and analytically computed modes are comparable. Adapted from Kelley et al. (2007)

Fig. 7

Spectrogram from PIV measurements performed on the Cottbus spherical Couette experiment (Hoff et al. 2016). The Ekman number is $Ek=1.52\times {10}^{-5}$. EA means equatorial antisymmetric. Broadband background turbulence appears here when $\Delta \Omega /\Omega \le -1.75$. The figures on the top show flow renderings from simulations (Barik et al. 2018)

Fig. 8

The hydromagnetic spherical Couette experiments: a–c the three Maryland experiments—30 cm, 60 cm, and 3 meter, respectively (Sources: S.A. Triana, Kelley and D. Lathrop). d DTS experiment (picture courtesy: H.-C. Nataf) and e HEDGEHOG experiment (figure from Kasprzyk et al. 2017). See also online movies of the 3-meter setup at https://youtu.be/bm_iqzmR2cE and https://youtu.be/rAYW9n8i-C4

Fig. 9

Experimental study of large-scale topography: flows driven by latitudinal librations in a triaxial ellipsoid (Charles 2018). a Experimental setup. b Amplitude of the uniform vorticity flow as a function of the libration frequency (red circles) showing resonance in agreement with the theoretical prediction (blue curve). The yellow line marks the predicted frequency of the spin-over mode from Vantieghem et al. (2015)

Fig. 10

Investigation of meso-scale topography. Spin-up experiments in a cylinder with bottom topography (Burmann and Noir 2018): a experimental setup; b kinetic energy as a function of time. The decay of the kinetic energy depends on the length scale of the bottom topography ${\lambda }_s$ (ratio between the block size and the radius of the cylindrical container). Topography results in a faster decay of the kinetic energy (all curves where ${\lambda }_s\ne 1$), i.e., enhances the dissipation in the system