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This work aims at acquiring a more complete understanding of how lateral heterogeneities of the CMB heat flux affect the geodynamo while other relevant parameters are pushed towards realistic values. For this purpose, we ran geodynamo simulations with degree 1 and 2 spherical harmonic patterns of heat flux at the CMB. Several geodynamo models are used, ranging from standard numerical dynamos to more extreme parameters, including strong field cases and turbulent cases. We show that heat flux heterogeneities with amplitudes compatible with our knowledge of mantle convection history can favour multipolar dynamos. The multipolar transition is associated with a disruption of westward flows either through eastward thermal winds or through a loss of equatorial symmetry. Strong field dynamo models are found to have larger westward flows and are less sensitive to heat flux heterogeneities. Furthermore, we find that the dipolar fraction of the magnetic field correlates with $M_{Za}^*=\dfrac{\Lambda_{Za}}{Rm_{Za}^2}$ where $\Lambda_{Za}$ is the zonal antisymmetric Elsasser number and $Rm_{Za}$ is the zonal antisymmetric magnetic Reynolds number. Importantly, $M_{Za}^*$ estimated for the Earth's core is consistent with a reversing dipolar magnetic field. Within the range of $M_{Za}^*$ susceptible to reversals, breaking the equatorial symmetry or forcing eastward zonal flows through an equatorial cooling of the core consistently triggers reversals or a transition towards multipolar dynamos in our simulations. Our results support that time variations of heat-flux heterogeneities driven by mantle convection through Earth's history are capable of inducing the significant variations in the reversal frequency observed in the palaeomagnetic record., Comment: 24 pages, 13+ figures, 4 appendices

During the centennial period from 1900 to 2000, the globally averaged unsigned annual rate of the main geomagnetic field experienced a three-episode variation. The maximum annual rates occurred respectively around 1910-1920, 1940-1950, and 1970-1980, showing a 30-year period. In addition, the rising phase in each episode is much shorter than declining phase. The governing factor of this periodic variation is non-dipole field, instead of the dipole field, although the secular variation of the magnetic energy concerned with the dipole is dominative over all other multiples.

The idea of a planetary origin for the solar cycle dates back to the nineteenth century. Despite unsurmounted problems, it is still advocated by some. Stefani, Giesecke, and Weier (2019) thus recently proposed a scenario based on this idea. A key argument they put forward is evidence that the $\sim$11 years-solar cycle is clocked. Their demonstration rests upon the computation of a ratio proposed by Dicke (1978) applied to the solar cycle time series of Schove (1955). I show that their demonstration is invalid, because the assumptions used by Schove to build his time series force a clocked behaviour. I also show that instabilities in a magnetized fluid can produce fluctuation time series that are close to being clocked.

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

We present an attempt to reach realistic turbulent regime in direct numerical simulations of the geodynamo. We rely on a sequence of three convection-driven simulations in a rapidly rotating spherical shell. The most extreme case reaches towards the Earth's core regime by lowering viscosity (magnetic Prandtl number Pm=0.1) while maintaining vigorous convection (magnetic Reynolds number Rm>500) and rapid rotation (Ekman number E=1e-7), at the limit of what is feasible on today's supercomputers. A detailed and comprehensive analysis highlights several key features matching geomagnetic observations or dynamo theory predictions -- all present together in the same simulation -- but it also unveils interesting insights relevant for Earth's core dynamics.In this strong-field, dipole-dominated dynamo simulation, the magnetic energy is one order of magnitude larger than the kinetic energy. The spatial distribution of magnetic intensity is highly heterogeneous, and a stark dynamical contrast exists between the interior and the exterior of the tangent cylinder (the cylinder parallel to the axis of rotation that circumscribes the inner core).In the interior, the magnetic field is strongest, and is associated with a vigorous twisted polar vortex, whose dynamics may occasionally lead to the formation of a reverse polar flux patch at the surface of the shell. Furthermore, the strong magnetic field also allows accumulation of light material within the tangent cylinder, leading to stable stratification there. Torsional Alfv{\'e}n waves are frequently triggered in the vicinity of the tangent cylinder and propagate towards the equator.Outside the tangent cylinder, the magnetic field inhibits the growth of zonal winds and the kinetic energy is mostly non-zonal. Spatio-temporal analysis indicates that the low-frequency, non-zonal flow is quite geostrophic (columnar) and predominantly large-scale: an m=1 eddy spontaneously emerges in our most extreme simulations, without any heterogeneous boundary forcing.Our spatio-temporal analysis further reveals that (i) the low-frequency, large-scale flow is governed by a balance between Coriolis and buoyancy forces -- magnetic field and flow tend to align, minimizing the Lorentz force; (ii) the high-frequency flow obeys a balance between magnetic and Coriolis forces; (iii) the convective plumes mostly live at an intermediate scale, whose dynamics is driven by a 3-term 1 MAC balance -- involving Coriolis, Lorentz and buoyancy forces. However, small-scale (E^{1/3}) quasi-geostrophic convection is still observed in the regions of low magnetic intensity., Comment: 48 pages, 28 figures, accepted for publication in GJI

The Derviche Tourneur Sodium experiment, a spherical Couette magnetohydrodynamics experiment with liquid sodium as the medium and a dipole magnetic field imposed from the inner sphere, recently underwent upgrades to its diagnostics to better characterize the flow and induced magnetic fields with global rotation. In tandem with the upgrades, a set of direct numerical simulations were run with the xshells code to give a more complete view of the fluid and magnetic dynamics at various rotation rates of the inner and outer spheres. These simulations reveal several dynamic regimes, determined by the Rossby number. At positive differential rotation there is a regime of quasigeostrophic flow, with low levels of fluctuations near the outer sphere. Negative differential rotation shows a regime of what appear to be saturated hydrodynamic instabilities at low negative differential rotation, followed by a regime where filamentary structures develop at low latitudes and persist over five to ten differential rotation periods as they drift poleward. We emphasize that all these coherent structures emerge from turbulent flows. At least some of them seem to be related to linear instabilities of the mean flow. The simulated flows can produce the same measurements as those that the physical experiment can take, with signatures akin to those found in the experiment. This paper discusses the relation between the internal velocity structures of the flow and their magnetic signatures at the surface., Comment: 15 pages, 11 figures

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., Comment: European Journal of Mechanics - B/Fluids, Elsevier, 2013

The contribution of small scale turbulent fluctuations to the induction of mean magnetic field is investigated in our liquid sodium spherical Couette experiment with an imposed magnetic field.An inversion technique is applied to a large number of measurements at $Rm \approx 100$ to obtain radial profiles of the $\alpha$ and $\beta$ effects and maps of the mean flow.It appears that the small scale turbulent fluctuations can be modeled as a strong contribution to the magnetic diffusivity that is negative in the interior region and positive close to the outer shell.Direct numerical simulations of our experiment support these results.The lowering of the effective magnetic diffusivity by small scale fluctuations implies that turbulence can actually help to achieve self-generation of large scale magnetic fields., Comment: Rajout d'un erratum

Measurement of the differential rotation of the Sun's interior is one of the great achievements of helioseismology, providing important constraints for stellar physics. The technique relies on observing and analyzing rotationally-induced splittings of p-modes in the star. Here we demonstrate the first use of the technique in a laboratory setting. We apply it in a spherical cavity with a spinning central core (spherical-Couette flow) to determine the mean azimuthal velocity of the air filling the cavity. We excite a number of acoustic resonances (analogous to p-modes in the Sun) using a speaker and record the response with an array of small microphones on the outer sphere. Many observed acoustic modes show rotationally-induced splittings, which allow us to perform an inversion to determine the air's azimuthal velocity as a function of both radius and latitude. We validate the method by comparing the velocity field obtained through inversion against the velocity profile measured with a calibrated hot film anemometer. This 'modal acoustic velocimetry' technique has great potential for laboratory setups involving rotating fluids in axisymmetric cavities. It will be useful especially in liquid metals where direct optical methods are unsuitable and ultrasonic techniques very challenging at best., Comment: Accepted for publication on The New Journal Of Physics

We present a reconstruction of the mean axisymmetric azimuthal and meridional flows in the DTS liquid sodium experiment. The experimental device sets a spherical Couette flow enclosed between two concentric spherical shells where the inner sphere holds a strong dipolar magnet, which acts as a magnetic propeller when rotated. Measurements of the mean velocity, mean induced magnetic field and mean electric potentials have been acquired inside and outside the fluid for an inner sphere rotation rate of 9 Hz (Rm 28). Using the induction equation to relate all measured quantities to the mean flow, we develop a nonlinear least square inversion procedure to reconstruct a fully coherent solution of the mean velocity field. We also include in our inversion the response of the fluid layer to the non-axisymmetric time-dependent magnetic field that results from deviations of the imposed magnetic field from an axial dipole. The mean azimuthal velocity field we obtain shows super-rotation in an inner region close to the inner sphere where the Lorentz force dominates, which contrasts with an outer geostrophic region governed by the Coriolis force, but where the magnetic torque remains the driver. The meridional circulation is strongly hindered by the presence of both the Lorentz and the Coriolis forces. Nevertheless, it contributes to a significant part of the induced magnetic energy. Our approach sets the scene for evaluating the contribution of velocity and magnetic fluctuations to the mean magnetic field, a key question for dynamo mechanisms.

The DTS experiment is a spherical Couette flow experiment with an imposed dipolar magnetic field. Liquid sodium is used as a working fluid. In a series of measurement campaigns, we have obtained data on the mean axisymmetric velocity, the mean induced magnetic field and electric potentials. All these quantities are coupled through the induction equation. In particular, a strong omega-eff ect is produced by di fferential rotation within the fluid shell, inducing a significant azimuthal magnetic field. Taking advantage of the simple spherical geometry of the experiment, I expand the azimuthal and meridional fields into Legendre polynomials and derive the expressions that relate all measurements to the radial functions of the velocity field for each harmonic degree. For small magnetic Reynolds numbers Rm the relations are linear, and the azimuthal and meridional equations decouple. Selecting a set of measurements for a given rotation frequency of the inner sphere (Rm = 9.4), I invert simultaneously the velocity and the magnetic data and thus reconstruct both the azimuthal and the meridional fields within the fluid shell. The results demonstrate the good internal consistency of the measurements, and indicate that turbulent non-axisymmetric fluctuations do not contribute significantly to the axisymmetric magnetic induction., Comment: soumis au Comptes Rendus Physique

Several teams have reported peculiar frequency spectra for flows in a spherical shell. To address their origin, we perform numerical simulations of the spherical Couette flow in a dipolar magnetic field, in the configuration of the DTS experiment. The frequency spectra computed from time-series of the induced magnetic field display similar bumpy spectra, where each bump corresponds to a given azimuthal mode number m. The bumps show up at moderate Reynolds number (2 600) if the time-series are long enough (>300 rotations of the inner sphere). We present a new method that permits to retrieve the dominant frequencies for individual mode numbers m, and to extract the modal structure of the full non-linear flow. The maps of the energy of the fluctuations and the spatio-temporal evolution of the velocity field suggest that fluctuations originate in the outer boundary layer. The threshold of instability if found at Re_c = 1 860. The fluctuations result from two coupled instabilities: high latitude B\"odewadt-type boundary layer instability, and secondary non-axisymmetric instability of a centripetal jet forming at the equator of the outer sphere. We explore the variation of the magnetic and kinetic energies with the input parameters, and show that a modified Elsasser number controls their evolution. We can thus compare with experimental determinations of these energies and find a good agreement. Because of the dipolar nature of the imposed magnetic field, the energy of magnetic fluctuations is much larger near the inner sphere, but their origin lies in velocity fluctuations that initiate in the outer boundary layer., Comment: 23 pages

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., Comment: Phys Rev E, in press

Under the combined constraints of rapid rotation, sphericity, and magnetic field, motions in planetary cores get organized in a peculiar way. Classical hydrodynamic turbulence is not present, but turbulent motions can take place under the action of the buoyancy and Laplace forces. Laboratory experiments, such as the rotating spherical magnetic Couette DTS experiment in Grenoble, help us understand what motions take place in planetary core conditions., Comment: in press in Compte Rendu de l'Academie des Sciences

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., Comment: in press in Physics of the Earth and Planetary Interiors

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.