Henri-Claude Nataf

A systematic experimental study of rapidly rotating spherical convection in water and liquid gallium

Abstract Results of finite-amplitude convection experiments in a rotating spherical shell are presented. Water (Prandtl number P=7) and liquid gallium (P=0.027) have been used as working fluids. In both liquids, convective velocities could be measured in the equatorial plane using an ultrasonic Doppler velocimetry technique. The parameter space has been systematically explored, for values of the Ekman and Rayleigh numbers E>7×10−7 and Ra

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. (Dormy, E., Cardin, P. and Jault, D., MHD flow in a slightly differentially rotating spherical shell, with conducting inner core, in a dipolar magnetic field, Earth Planet. Sci. Lett., 1998, 160, 15--30). 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.

Rotating spherical Couette flow in a dipolar magnetic field : experimental study of magneto-inertial waves

The magnetostrophic regime, in which Lorentz and Coriolis forces are in balance, has been investigated in a rapidly rotating spherical Couette flow experiment. The spherical shell is filled with liquid sodium and permeated by a strong imposed dipolar magnetic field. Azimuthally travelling hydromagnetic waves have been put in evidence through a detailed analysis of electric potential differences measured on the outer sphere, and their properties have been determined. Several types of waves have been identified depending on the relative rotation rates of the inner and outer spheres: they differ by their dispersion relation and by their selection of azimuthal wavenumbers. In addition, these waves constitute the largest contribution to the observed fluctuations, and all of them travel in the retrograde direction in the frame of reference bound to the fluid. We identify these waves as magneto-inertial waves by virtue of the close proximity of the magnetic and inertial characteristic time scales of relevance in our experiment.

Turbulent geodynamo simulations: a leap towards Earth’s core

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 Earths 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 todays 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 Earths 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 Alfven 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.

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

Abstract In order to explore the magnetostrophic regime expected for planetary cores, in which the Lorentz forces balance the Coriolis forces, 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 just above the rotating outer sphere, and velocity profiles inside the liquid sodium using ultrasonic Doppler velocimetry. This article focuses on the time-averaged axisymmetric part of the flow. The electric potential differences measured at several latitudes can be linked to azimuthal velocities, and are indeed found to be proportional to the azimuthal velocities measured by Doppler velocimetry. 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. A good agreement is found for the angular velocity profiles. In the geostrophic region, a torque-balance model provides very good predictions. Radial velocities change sign with the Rossby number, as expected for an Ekman-pumping dominated flow. For a given Rossby number the amplitude of the measured angular velocity is found to vary by as much as a factor of 3. Comparison with numerical simulations suggests that this is due to variations in the electric coupling between liquid sodium and the inner copper sphere, implying an effect equivalent to a reduction of the inner sphere electric conductivity by as much as a factor 100. We show that the measured electric potential difference can be used as a proxy of the actual fluid velocity. Using this proxy in place of the imposed differential velocity, 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.

A turbulent, high magnetic Reynolds number experimental model of Earth's core

We present new experimental results from the University of Maryland Three Meter Geodynamo experiment. We drive a fully turbulent flow in water and also in sodium at magnetic Reynolds number Rm = ΔΩ(ro−ri)2/η, up to 715 (about half design maximum) in a spherical Couette apparatus geometrically similar to Earths core. We have not yet observed a self-generating dynamo, but we study MHD effects with an externally applied axisymmetric magnetic field. We survey a broad range of Rossby number −68 < Ro = ΔΩ/Ωo< 65 in both purely hydrodynamic water experiments and sodium experiments with weak, nearly passive applied field. We characterize angular momentum transport and substantial generation of internal toroidal magnetic field (the Ω effect) as a function of Ro and find a rich dependence of both angular momentum transport and Ω effect on Ro. Internal azimuthal field generation peaks at Ro = 6 with a gain as high as 9 with weak applied field. At this Rossby number, we also perform experiments with significant Lorentz forces by increasing the applied magnetic field. We observe a reduction of the Ω effect, a large increase in angular momentum transport, and the onset of new dynamical states. The state we reach at maximum applied field shows substantial magnetic field gain in the axial dipole moment, enhancing the applied dipole moment. This intermittent dipole enhancement must come from nonaxisymmetric flow and seems to be a geodynamo-style feedback involving differential rotation and large-scale drifting waves.

Turbulence reduces magnetic diffusivity in a liquid sodium experiment.

The contribution of small scale turbulent fluctuations to the induction of a 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≈100 to obtain radial profiles of the α and β 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.

Magneto–Coriolis waves in a spherical Couette flow experiment

Abstract 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 signatures 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.

Magnetic induction and diffusion mechanisms in a liquid sodium spherical Couette experiment.

We present a reconstruction of the mean axisymmetric azimuthal and meridional flows in the Derviche Tourneur Sodium installation in Grenoble 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-squares 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 nonaxisymmetric 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 superrotation 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.

On the peculiar nature of turbulence in planetary dynamos

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 Lorentz 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. To cite this article: H.-C. Nataf ∇ × � ~ �