arXiv subjects

Philippe Cardin

Publications and source records attributed to Philippe Cardin.

17 recordsLinked to original sources

arXiv2020

Acoustic spectra of a gas-filled rotating spheroid

The acoustic spectrum of a gas-filled resonating cavity can be used to indirectly probe its internal velocity field. This unconventional velocimetry method is particularly interesting for opaque fluid or rapidly rotating flows, which cannot be imaged with standard methods. This requires to (i) identify a large enough number of acoustic modes, (ii) accurately measure their frequencies, and (iii) compare with theoretical synthetic spectra. Relying on a dedicated experiment, an air-filled rotating spheroid of moderate ellipticity, our study addresses these three challenges. To do so, we use a comprehensive theoretical framework, together with finite-element calculations, and consider symmetry arguments. We show that the effects of the Coriolis force can be successfully retrieved through our acoustic measurements, providing the first experimental measurements of the rotational splitting (or Ledoux) coefficients for a large collection of modes. Our results pave the way for the modal acoustic velocimetry to be a robust, versatile, and non-intrusive method for mapping large-scale flows. Pages: 1-14

physics.flu-dyn↗

arXiv2018

Convective Lengthscale in Planetary Cores

Convection is a fundamental physical process in the fluid cores of planets because it is the primary transport mechanism for heat and chemical species and the primary energy source for planetary magnetic fields. Key properties of convection, such as the characteristic flow velocity and lengthscale, are poorly quantified in planetary cores due to their strong dependence on planetary rotation, buoyancy driving and magnetic fields, which are all difficult to model under realistic conditions. In the absence of strong magnetic fields, the core convective flows are expected to be in a regime of rapidly-rotating turbulence, which remains largely unexplored to date. Here we use a combination of numerical models designed to explore this low-viscosity regime to show that the convective lengthscale becomes independent of the viscosity and is entirely determined by the flow velocity and planetary rotation. For the Earth's core, we find that the characteristic con-vective lengthscale is approximately 30km and below this scale, motions are very weak. The 30-km cutoff scale rules out small-scale dynamo action and supports large-eddy simulations of core dynamics. Furthermore, it implies that our understanding of magnetic reversals from numerical geodynamo models does not relate to the Earth, because they require too intense flows. Our results also indicate that the liquid core of the Moon might still be in an active convective state despite the absence of a present-day dynamo.

physics.geo-ph↗

arXiv2017

Multiple zonal jets and convective heat transport barriers in a quasi-geostrophic model of planetary cores

We study rapidly-rotating Boussinesq convection driven by internal heating in a full sphere. We use a numerical model based on the quasi-geostrophic approximation for the velocity field, whereas the temperature field is three-dimensional. This approximation allows us to perform simulations for Ekman numbers down to 1e-8, Prandtl numbers relevant for liquid metals (~0.1) and Reynolds numbers up to 3e4. Persistent zonal flows composed of multiple jets form as a result of the mixing of potential vorticity. For the largest Rayleigh numbers computed, the zonal velocity is larger than the convective velocity despite the presence of boundary friction. The convective structures and the zonal jets widen when the thermal forcing increases. Prograde and retrograde zonal jets are dynamically different: in the prograde jets (which correspond to weak potential vorticity gradients) the convection transports heat efficiently and the mean temperature tends to be homogenised; by contrast, in the cores of the retrograde jets (which correspond to steep gradients of potential vorticity) the dynamics is dominated by the propagation of Rossby waves, resulting in the formation of steep mean temperature gradients and the dominance of conduction in the heat transfer process. Consequently, in quasi-geostrophic systems, the width of the retrograde zonal jets controls the efficiency of the heat transfer.

physics.flu-dyn↗

arXiv2016

Subcritical convection of liquid metals in a rotating sphere using a quasi-geostrophic model

We study nonlinear convection in a rapidly rotating sphere with internal heating for values of the Prandtl number relevant for liquid metals ($Pr\in[10^{-2},10^{-1}]$). We use a numerical model based on the quasi-geostrophic approximation, in which variations of the axial vorticity along the rotation axis are neglected, whereas the temperature field is fully three-dimensional. We identify two separate branches of convection close to onset: (i) a well-known weak branch for Ekman numbers greater than $10^{-6}$, which is continuous at the onset (supercritical bifurcation) and consists of thermal Rossby waves, and (ii) a novel strong branch at lower Ekman numbers, which is discontinuous at the onset. The strong branch becomes subcritical for Ekman numbers of the order of $10^{-8}$. On the strong branch, the Reynolds number of the flow is greater than $10^3$, and a strong zonal flow with multiple jets develops, even close to the nonlinear onset of convection. We find that the subcriticality is amplified by decreasing the Prandtl number. The two branches can co-exist for intermediate Ekman numbers, leading to hysteresis ($Ek=10^{-6}$, $Pr=10^{-2}$). Nonlinear oscillations are observed near the onset of convection for $Ek=10^{-7}$ and $Pr=10^{-1}$.

physics.flu-dyn↗

arXiv2016

Magneto--Coriolis waves in a spherical Couette flow experiment

The dynamics of fluctuations in a fast rotating spherical Couette flow experiment in the presence of a strong dipolar magnetic field is investigated in detail, through a thorough analysis of the experimental data as well as a numerical study. Fluctuations within the conducting fluid (liquid sodium) are characterized by the presence of several oscillation modes, identified as magneto-Coriolis (MC) modes, with definite symmetry and azimuthal number. A numerical simulation provides eigensolutions which exhibit oscillation frequencies and magnetic signature comparable to the observation. The main characteristics of these hydromagnetic modes is that the magnetic contribution has a fundamental influence on the dynamical properties through the Lorentz forces, although its importance remains weak in an energetical point of view. Another specificity is that the Lorentz forces are confined near the inner sphere where the dipolar magnetic field is the strongest, while the Coriolis forces are concentrated in the outer fluid volume close to the outer sphere.

physics.geo-ph↗

arXiv2015

A dynamo driven by zonal jets at the upper surface: Applications to giant planets

We present a dynamo mechanism arising from the presence of barotropically unstable zonal jet currents in a rotating spherical shell. The shear instability of the zonal flow develops in the form of a global Rossby mode, whose azimuthal wavenumber depends on the width of the zonal jets. We obtain self-sustained magnetic fields at magnetic Reynolds numbers greater than 1000. We show that the propagation of the Rossby waves is crucial for dynamo action. The amplitude of the axisymmetric poloidal magnetic field depends on the wavenumber of the Rossby mode, and hence on the width of the zonal jets. We discuss the plausibility of this dynamo mechanism for generating the magnetic field of the giant planets. Our results suggest a possible link between the topology of the magnetic field and the profile of the zonal winds observed at the surface of the giant planets. For narrow Jupiter-like jets, the poloidal magnetic field is dominated by an axial dipole whereas for wide Neptune-like jets, the axisymmetric poloidal field is weak.

physics.geo-ph↗

arXiv2015

Is inner core seismic anisotropy a marker of plastic flow of cubic iron?

This paper investigates whether observations of seismic anisotropy are compatible with a cubic structure of the inner core Fe alloy. We assume that anisotropy is the result of plastic deformation within a large scale flow induced by preferred growth at the inner core equator. Based on elastic moduli from the literature, bcc- or fcc-Fe produce seismic anisotropy well below seismic observations ($\textless{}0.4\%$). A Monte-Carlo approach allows us to generalize this result to any form of elastic anisotropy in a cubic system. Within our model, inner core global anisotropy is not compatible with a cubic structure of Fe alloy. Hence, if the inner core material is indeed cubic, large scale coherent anisotropic structures, incompatible with plastic deformation induced by large scale flow, must be present.

physics.geo-ph↗

arXiv2013

Thermal convection in Earth's inner core with phase change at its boundary

Inner core translation, with solidification on one hemisphere and melting on the other, provides a promising basis for understanding the hemispherical dichotomy of the inner core, as well as the anomalous stable layer observed at the base of the outer core - the F-layer - which might be sustained by continuous melting of inner core material. In this paper, we study in details the dynamics of inner core thermal convection when dynamically induced melting and freezing of the inner core boundary (ICB) are taken into account. If the inner core is unstably stratified, linear stability analysis and numerical simulations consistently show that the translation mode dominates only if the viscosity $\eta$ is large enough, with a critical viscosity value, of order $3 10^{18}$ Pas, depending on the ability of outer core convection to supply or remove the latent heat of melting or solidification. If $\eta$ is smaller, the dynamical effect of melting and freezing is small. Convection takes a more classical form, with a one-cell axisymmetric mode at the onset and chaotic plume convection at large Rayleigh number. [...] Thermal convection requires that a superadiabatic temperature profile is maintained in the inner core, which depends on a competition between extraction of the inner core internal heat by conduction and cooling at the ICB. Inner core thermal convection appears very likely with the low thermal conductivity value proposed by Stacey & Davis (2007), but nearly impossible with the much higher thermal conductivity recently put forward. We argue however that the formation of an iron-rich layer above the ICB may have a positive feedback on inner core convection: it implies that the inner core crystallized from an increasingly iron-rich liquid, resulting in an unstable compositional stratification which could drive inner core convection, perhaps even if the inner core is subadiabatic.

physics.geo-ph↗

arXiv2012

On the reflection of Alfvén waves and its implication for Earth's core modeling

Alfvén waves propagate in electrically conducting fluids in the presence of a magnetic field. Their reflection properties depend on the ratio between the kinematic viscosity and the magnetic diffusivity of the fluid, also known as the magnetic Prandtl number Pm. In the special case Pm=1, there is no reflection on an insulating, no-slip boundary, and the wave energy is entirely dissipated in the boundary layer. We investigate the consequences of this remarkable behaviour for the numerical modeling of torsional Alfvén waves (also known as torsional oscillations), which represent a special class of Alfvén waves, in rapidly rotating spherical shells. They consist of geostrophic motions and are thought to exist in the fluid cores of planets with internal magnetic field. In the geophysical limit Pm << 1, these waves are reflected at the core equator, where they are entirely absorbed for Pm=1. Our numerical calculations show that the reflection coefficient at the equator of these waves remains below 0.2 for Pm > 0.3, which is the range of values for which geodynamo numerical models operate. As a result, geodynamo models with no-slip boundary conditions cannot exhibit torsional oscillation normal modes.

physics.geo-ph↗

arXiv2011

Experimental evidence of Alfvén wave propagation in a Gallium alloy

Experiments with a liquid metal alloy, galinstan, are reported and show clear evidence of Alfvén wave propagation as well as resonance of Alfvén modes. Galinstan is liquid at room temperature, and although its electrical conductivity is not as large as that of liquid sodium or NaK, it has still been possible to study Alfvén waves, thanks to the use of intense magnetic fi elds, up to 13 teslas. The maximal values of Lundquist number, around 60, are similar to that of the reference experimental study by Jameson [1]. The generation mechanism for Alfvén waves and their refl ection is studied carefully. Numerical simulations have been performed and have been able to reproduce the experimental results despite the fact that the simulated magnetic Prandtl number was much larger than that of galinstan. An originality of the present study is that a poloidal disturbance (magnetic and velocity fields) is generated, allowing us to track its propagation from outside the conducting domain, hence without interfering.

physics.flu-dyn↗

arXiv2011

Zonal shear and super-rotation in a magnetized spherical Couette flow experiment

We present measurements performed in a spherical shell filled with liquid sodium, where a 74 mm-radius inner sphere is rotated while a 210 mm-radius outer sphere is at rest. The inner sphere holds a dipolar magnetic field and acts as a magnetic propeller when rotated. In this experimental set-up called DTS, direct measurements of the velocity are performed by ultrasonic Doppler velocimetry. Differences in electric potential and the induced magnetic field are also measured to characterize the magnetohydrodynamic flow. Rotation frequencies of the inner sphere are varied between -30 Hz and +30 Hz, the magnetic Reynolds number based on measured sodium velocities and on the shell radius reaching to about 33. We have investigated the mean axisymmetric part of the flow, which consists of differential rotation. Strong super-rotation of the fluid with respect to the rotating inner sphere is directly measured. It is found that the organization of the mean flow does not change much throughout the entire range of parameters covered by our experiment. The direct measurements of zonal velocity give a nice illustration of Ferraro's law of isorotation in the vicinity of the inner sphere where magnetic forces dominate inertial ones. The transition from a Ferraro regime in the interior to a geostrophic regime, where inertial forces predominate, in the outer regions has been well documented. It takes place where the local Elsasser number is about 1. A quantitative agreement with non-linear numerical simulations is obtained when keeping the same Elsasser number. The experiments also reveal a region that violates Ferraro's law just above the inner sphere.

physics.geo-ph↗

arXiv2010

Numerical Simulations of Dynamos Generated in Spherical Couette Flows

We numerically investigate the efficiency of a spherical Couette flow at generating a self-sustained magnetic field. No dynamo action occurs for axisymmetric flow while we always found a dynamo when non-axisymmetric hydrodynamical instabilities are excited. Without rotation of the outer sphere, typical critical magnetic Reynolds numbers $Rm_c$ are of the order of a few thousands. They increase as the mechanical forcing imposed by the inner core on the flow increases (Reynolds number $Re$). Namely, no dynamo is found if the magnetic Prandtl number $Pm=Rm/Re$ is less than a critical value $Pm_c\sim 1$. Oscillating quadrupolar dynamos are present in the vicinity of the dynamo onset. Saturated magnetic fields obtained in supercritical regimes (either $Re>2 Re_c$ or $Pm>2Pm_c$) correspond to the equipartition between magnetic and kinetic energies. A global rotation of the system (Ekman numbers $E=10^{-3}, 10^{-4}$) yields to a slight decrease (factor 2) of the critical magnetic Prandtl number, but we find a peculiar regime where dynamo action may be obtained for relatively low magnetic Reynolds numbers ($Rm_c\sim 300$). In this dynamical regime (Rossby number $Ro\sim -1$, spheres in opposite direction) at a moderate Ekman number ($E=10^{-3}$), a enhanced shear layer around the inner core might explain the decrease of the dynamo threshold. For lower $E$ ($E=10^{-4}$) this internal shear layer becomes unstable, leading to small scales fluctuations, and the favorable dynamo regime is lost. We also model the effect of ferromagnetic boundary conditions. Their presence have only a small impact on the dynamo onset but clearly enhance the saturated magnetic field in the ferromagnetic parts. Implications for experimental studies are discussed.

physics.geo-ph↗

arXiv2008

Rapidly rotating spherical Couette flow in a dipolar magnetic field: an experimental study of the mean axisymmetric flow

In order to explore the magnetostrophic regime expected for planetary cores, experiments have been conducted in a rotating sphere filled with liquid sodium, with an imposed dipolar magnetic field (the DTS setup). The field is produced by a permanent magnet enclosed in an inner sphere, which can rotate at a separate rate, producing a spherical Couette flow. The flow properties are investigated by measuring electric potentials on the outer sphere, the induced magnetic field in the laboratory frame, and velocity profiles inside the liquid sodium using ultrasonic Doppler velocimetry. The present article focuses on the time-averaged axisymmetric part of the flow. The Doppler profiles show that the angular velocity of the fluid is relatively uniform in most of the fluid shell, but rises near the inner sphere, revealing the presence of a magnetic wind, and gently drops towards the outer sphere. The transition from a magnetostrophic flow near the inner sphere to a geostrophic flow near the outer sphere is controlled by the local Elsasser number. For Rossby numbers up to order 1, the observed velocity profiles all show a similar shape. Numerical simulations in the linear regime are computed, and synthetic velocity profiles are compared with the measured ones. In the geostrophic region, a torque-balance model provides very good predictions. We find that the induced magnetic field varies in a consistent fashion, and displays a peculiar peak in the counter-rotating regime. This happens when the fluid rotation rate is almost equal and opposite to the outer sphere rotation rate. The fluid is then almost at rest in the laboratory frame, and the Proudman-Taylor constraint vanishes, enabling a strong meridional flow. We suggest that dynamo action might be favored in such a situation.

physics.geo-ph↗

arXiv2005

Experimental study of super-rotation in a magnetostrophic spherical Couette flow

We report measurements of electric potentials at the surface of a spherical container of liquid sodium in which a magnetized inner core is differentially rotating. The azimuthal angular velocities inferred from these potentials reveal a strong super-rotation of the liquid sodium in the equatorial region, for small differential rotation. Super-rotation was observed in numerical simulations by Dormy et al. [1]. We find that the latitudinal variation of the electric potentials in our experiments differs markedly from the predictions of a similar numerical model, suggesting that some of the assumptions used in the model - steadiness, equatorial symmetry, and linear treatment for the evolution of both the magnetic and velocity fields - are violated in the experiments. In addition, radial velocity measurements, using ultrasonic Doppler velocimetry, provide evidence of oscillatory motion near the outer sphere at low latitude: it is viewed as the signature of an instability of the super-rotating region.

physics.geo-ph↗

arXiv2005

Visco-magnetic torque at the core mantle boundary

A magneto-hydrodynamic model of boundary layers at the Core-Mantle Boundary (CMB) is derived and used to compute the viscous and electromagnetic torques generated by the Earth's nutation forcing. The predicted electromagnetic torque alone cannot account for the dissipation estimated from the observations of the free core nutation. The presence of a viscous boundary layer in the electromagnetic skin layer at the CMB, with its additional dissipative torques, may explain the geodetic data. An apparent Ekman number at the top of the core between 3 and $5 10^{-11}$ is inferred depending on the electrical conductivity of the mantle.

physics.geo-ph↗

arXiv2004

Quasi-geostrophic model of the instabilities of the Stewartson layer

We study the destabilization of a shear layer, produced by differential rotation of a rotating axisymmetric container. For small forcing, this produces a shear layer, which has been studied by Stewartson and is almost invariant along the rotation axis. When the forcing increases, instabilities develop. To study the asymptotic regime (very low Ekman number $E$), we develop a quasi-geostrophic two-dimensional model, whose main original feature is to handle the mass conservation correctly, resulting in a divergent two-dimensional flow, and valid for any container provided that the top and bottom have finite slopes. We use it to derive scalings and asymptotic laws by a simple linear theory, extending the previous analyses to large slopes (as in a sphere), for which we find different scaling laws. For a flat container, the critical Rossby number for the onset of instability evolves as $E^{3/4}$ and may be understood as a Kelvin-Helmoltz shear instability. For a sloping container, the instability is a Rossby wave with a critical Rossby number proportional to $βE^{1/2}$, where $β$ is related to the slope. We also investigate the asymmetry between positive and negative differential rotation and propose corrections for finite Ekman and Rossby numbers. Implemented in a numerical code, our model allows us to study the onset over a broad range of parameters, determining the threshold but also other features such as the spatial structure. We also present a few experimental results, validating our model and showing its limits.

physics.flu-dyn↗

arXiv2004

Quasi-geostrophic kinematic dynamos at low magnetic Prandtl number

Rapidly rotating spherical kinematic dynamos are computed using the combination of a quasi geostrophic (QG) model for the velocity field and a classical spectral 3D code for the magnetic field. On one hand, the QG flow is computed in the equatorial plane of a sphere and corresponds to Rossby wave instabilities of a geostrophic internal shear layer produced by differential rotation. On the other hand, the induction equation is computed in the full sphere after a continuation of the QG flow along the rotation axis. Differential rotation and Rossby-wave propagation are the key ingredients of the dynamo process which can be interpreted in terms of $αΩ$ dynamo. Taking into account the quasi geostrophy of the velocity field to increase its time and space resolution enables us to exhibit numerical dynamos with very low Ekman (rapidly rotating) and Prandtl numbers (liquid metals) which are asymptotically relevant to model planetary core dynamos.

physics.class-ph↗