Magnetic induction maps in a magnetized spherical Couette flow experiment

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 Physiqu

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 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

Discrete nonlinear Schrödinger equations for periodic optical systems : pattern formation in \chi(3) coupled waveguide arrays

Discrete nonlinear Schrödinger equations have been used for many years to model the propagation of light in optical architectures whose refractive index profile is modulated periodically in the transverse direction. Typically, one considers a modal decomposition of the electric field where the complex amplitudes satisfy a coupled system that accommodates nearest neighbour linear interactions and a local intensity dependent term whose origin lies in the χ (3) contribution to the medium's dielectric response. In this presentation, two classic continuum configurations are discretized in ways that have received little attention in the literature: the ring cavity and counterpropagating waves. Both of these systems are defined by distinct types of boundary condition. Moreover, they are susceptible to spatial instabilities that are ultimately responsible for generating spontaneous patterns from arbitrarily small background disturbances. Good agreement between analytical predictions and simulations will be demonstrated

An overlapping splitting double sweep method for the Helmholtz equation

We consider the domain decomposition method approach to solve the Helmholtz equation. Double sweep based approaches for overlapping decompositions are presented. In particular, we introduce an overlapping splitting double sweep (OSDS) method valid for any type of interface boundary conditions. Despite the fact that first order interface boundary conditions are used, the OSDS method demonstrates good stability properties with respect to the number of subdomains and the frequency even for heterogeneous media. In this context, convergence is improved when compared to the double sweep methods in Nataf et al. (1997) and Vion et al. (2014, 2016} for all of our test cases: waveguide, open cavity and wedge problems

What is responsible for thermal coupling in layered convection ?

Laboratory experiments have been conducted on convection in a layered system. The system consists in two liquid layers of equal thickness. The liquids are immiscible : the upper one is silicon oil, and the lower one is glycerol. The structure of convection has been analysed, and data obtained both on the temperature field and the velocity field. It is shown that the coupling between the two convecting systems in « thermal », i.e. convection cells are superposed with uprising currents above uprisings. This result is surprising because it contradicts numerical experiments recently obtained for layered convection. These find « mechanical » coupling (cells are superposed but turn in opposite senses) to be the stable mode for the conditions we tried to reproduce in the laboratory. Several tests have been conducted in order to isolate the phenomenon which is responsible for the discrepancy between the two types of analyses. A tentative mechanism is proposed : it involves an equivalent interfacial longitudinal viscosity, whose origin is not yet clearly understood

Turbulence Reduces Magnetic Diffusivity in a Liquid Sodium Experiment

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 erratu

Modes and instabilities in magnetized spherical Couette flow

23 pagesInternational audienceSeveral 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 Bödewadt-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

Measurements of mantle wave velocities and inversion for lateral heterogeneities and anisotropy: 3. Inversion

Lateral heterogeneity in the earth's upper mantle is investigated by inverting dispersion curves of long-period surface waves (100–330 s). Models for seven different tectonic regions are derived by inversion of regionalized great circle phase velocity measurements from our previous studies. We also obtain a representation of upper mantle heterogeneities with no a priori regionalization from the inversion of the degree 6 spherical harmonic expansion of phase and group velocities. The data are from the observation of about 200 paths for Love waves and 250 paths for Rayleigh waves. For both the regionalized and the spherical harmonic inversions, corrections are applied to take into account lateral variations in crustal thickness and other shallow parameters. These corrections are found to be important, especially at low spherical harmonic order the “trench region” and fast velocities down to 250 km under shields. Below 200 km under the oceans, both S velocity and S anisotropy support a model of small-scale convection in which cold blobs detach from the bottom of the lithosphere when its age is large enough. The spherical harmonic models clearly demonstrate (a posteriori) the relation between surface tectonics and S velocity heterogeneities in the first 250 km: all shields are fast; most ridges are slow; below 300 km, a belt of fast mantle follows the Pacific subduction zones. However, at greater depths, large-scale heterogeneities that seem to bear no relationship to surface tectonics are observed. The most prominent feature at 450 km is a fast-velocity region under the South Atlantic Ocean. Smaller-scale heterogeneities that are not related to surface tectonics are also mapped at shallower depths: an anomalously slow region centered in the south central Pacific is possibly linked to intense hot spot activity; a very fast region southeast of South America may be related to subduction of old Pacific plate. Between 200 and 400 km, a belt of SV>SH anisotropy follows part of the ridge and subduction systems, indicating vertical mantle flow in these regions. The spherical harmonic results open new horizons for the understanding of convection in the mantle. Perspectives for the improvement of the models presented are discussed

A robust and adaptive GenEO-type domain decomposition preconditioner for H(curl)\mathbf{H}(\mathbf{curl}) problems in general non-convex three-dimensional geometries

In this paper we develop and analyse domain decomposition methods for linear systems of equations arising from conforming finite element discretisations of positive Maxwell-type equations, namely for $\mathbf{H}(\mathbf{curl})$ problems. It is well known that convergence of domain decomposition methods rely heavily on the efficiency of the coarse space used in the second level. We design adaptive coarse spaces that complement a near-kernel space made from the gradient of scalar functions. The new class of preconditioner is inspired by the idea of subspace decomposition, but based on spectral coarse spaces, and is specially designed for curl-conforming discretisations of Maxwell's equations in heterogeneous media on general domains which may have holes. Our approach has wider applicability and theoretical justification than the well-known Hiptmair-Xu auxiliary space preconditioner, with results extending to the variable coefficient case and non-convex domains at the expense of a larger coarse space