Quasi-geostrophic modes in the Earth's fluid core with an outer stably stratified layer

Seismic waves sensitive to the outermost part of the Earth's liquid core seem to be affected by a stably stratified layer at the core-mantle boundary. Such a layer could have an observable signature in both long-term and short-term variations of the magnetic field of the Earth, which are used to probe the flow at the top of the core. Indeed, with the recent SWARM mission, it seems reasonable to be able to identify waves propagating in the core with period of several months, which may play an important role in the large-scale dynamics. In this paper, we characterize the influence of a stratified layer at the top of the core on deep quasi-geostrophic (Rossby) waves. We compute numerically the quasi-geostrophic eigenmodes of a rapidly rotating spherical shell, with a stably stratified layer near the outer boundary. Two simple models of stratification are taken into account, which are scaled with commonly adopted values of the Brunt-V{\"a}is{\"a}l{\"a} frequency in the Earth's core. In the absence of magnetic field, we find that both azimuthal wavelength and frequency of the eigenmodes control their penetration into the stratified layer: the higher the phase speed, the higher the permeability of the stratified layer to the wave motion. We also show that the theory developed by Takehiro and Lister [2001] for thermal convection extends to the whole family of Rossby waves in the core. Adding a magnetic field, the penetrative behaviour of the quasi-geostrophic modes (the so-called fast branch) is insensitive to the imposed magnetic field and only weakly sensitive to the precise shape of the stratification. Based on these results, the large-scale and high frequency modes (1 to 2 month periods) may be detectable in the geomagnetic data measured at the Earth's surface, especially in the equatorial area where the modes can be trapped

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

Quasi-geostrophic kinematic dynamos at low magnetic Prandtl number

International audienceRapidly rotating spherical kinematic dynamos at very low Ekman and Prandtl numbers 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. The QG flow is computed in the equatorial plane of the sphere; it corresponds to Rossby wave instabilities of a geostrophic internal shear layer produced by differential rotation. The induction equation is computed in the whole sphere after the QG flow has been expanded along the rotation axis. Differential rotation and Rossby wave propagation are the key ingredients of this dynamo which can be interpreted in terms of Parker-[Omega] dynamo. Taking into account the quasi-geostrophy of the velocity field enables us to increase time and space resolution to compute the dynamics. For the first time, we report on numerical dynamos with very low Ekman numbers (10- 8). Because the magnetic and velocity fields are computed on different grids, we compute dynamos for very low magnetic Prandtl numbers exhibiting a scale separation between magnetic and velocity field. These dynamos are asymptotically close to rapidly rotating, metallic planetary cores

Can Core Flows inferred from Geomagnetic Field Models explain the Earth's Dynamo?

We test the ability of large scale velocity fields inferred from geomagnetic secular variation data to produce the global magnetic field of the Earth.Our kinematic dynamo calculations use quasi-geostrophic (QG) flows inverted from geomagnetic field models which, as such, incorporate flow structures that are Earth-like and may be important for the geodynamo.Furthermore, the QG hypothesis allows straightforward prolongation of the flow from the core surface to the bulk.As expected from previous studies, we check that a simple quasi-geostrophic flow is not able to sustain the magnetic field against ohmic decay.Additional complexity is then introduced in the flow, inspired by the action of the Lorentz force.Indeed, on centenial time-scales, the Lorentz force can balance the Coriolis force and strict quasi-geostrophy may not be the best ansatz.When the columnar flow is modified to account for the action of the Lorentz force, magnetic field is generated for Elsasser numbers larger than 0.25 and magnetic Reynolds numbers larger than 100.This suggests that our large scale flow captures the relevant features for the generation of the Earth's magnetic field and that the invisible small scale flow may not be directly involved in this process.Near the threshold, the resulting magnetic field is dominated by an axial dipole, with some reversed flux patches.Time-dependence is also considered, derived from principal component analysis applied to the inverted flows.We find that time periods from 120 to 50 years do not affect the mean growth rate of the kinematic dynamos.Finally we notice the footprint of the inner-core in the magnetic field generated deep in the bulk of the shell, although we did not include one in our computations

On Symmetry and Anisotropy of Earth-core Flows

5p.International audienceQuasi-geostrophic (QG) flows are a recently developed and very promising paradigm for modeling decadal secular variation (SV). Here we examine the effects of allowing anisotropy and departures of the flow from quasigeostrophy. We perform dedicated numerical experiments of the flow dynamics and magnetic induction inside the Earth's liquid core at time scales characteristic of secular variation of the geomagnetic field. Obtained results motivate new flow inversion regularization featuring an equatorially anti-symmetric component superimposed to quasi-geostrophic columns, and stronger latitudinal than longitudinal flow gradients. Applying these constraints allows to explain the observed SV for the whole period 1840-2010, and most significantly, provides a clearly improvement in prediction for decadal length-of-day variations for the period 1980-2000. Furthermore, the trace of the inner-core appears clearly without any assumption for the 1997-2010 period covered by satellite geomagnetic data. Our results support QG being the appropriate description of the force balance within the core on decadal time scales and large spatial scales

Exploring the physics of the Earth's core with numerical simulations

In the first chapter of this report, I discuss some of my work of the past 7 years,since I joined the geodynamo team at ISTerre as a CNRS researcher. This workmost often involves numerical simulations with codes that I have written.An important step forward in the efficiency of simulations based on the spherical harmonic transform has come from the matrix-free SHTns library I have designed and written (Schaeffer, 2013).Numerical simulations linked to the magnetized spherical Couette experimentDTS have been performed to understand its peculiar turbulence (Figueroa et al.,2013) and try to characterize the effect of the small turbulent scales on the induction processes (Cabanes, Schaeffer, and Nataf, 2014a; Cabanes, Schaeffer, andNataf, 2014b).Related to the Earth’s core dynamics, my simulations helped to characterizethe effects of the magnetic field on short timescale flows, leading to strong arguments for quasi-geostrophic (columnar) flows at large spatial scales (& 10 km)and short timescales (. 10 years) in the core (Gillet, Schaeffer, and Jault, 2011).Smaller length scales evade the rotational constraint because inertial waves aregetting too slow. At longer length scales more research is needed, but we havealready explored the implications of a deformation of the columns by magneticfields (Schaeffer, Lora Silva, and Pais, 2016). We have also shown that near theequator, where the columnar flows were expected to wither, the quasi-geostrophyseems strong in the Earth’s core (Schaeffer and Pais, 2011).Prompted by observations, we studied the propagation and reflection of torsional Alfvén waves in the core, and showed the importance of the value of themagnetic Prandtl number (Schaeffer, Jault, et al., 2012). Moreover, an extensionof this work to include a conducting solid layer at the top of the core suggests theexistence of such a layer with a rather strong conductance in the Earth.More recently, and as a logical follow-up, I have produced turbulent geodynamo simulations, with interesting implications that will be reported in a futurepublication, but several facts are already presented in appendix D and will alsobe discussed here. These simulations have also fed the reflection that resulted inour chapter on core turbulence in the second edition of the Treatise on Geophysics(Nataf and Schaeffer, 2015).In the second chapter, I present synthetically my ongoing work and projectsthat develop even further the topics above, but also new projects on the dynamoof the early Moon and further numerical developments

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

Efficient Spherical Harmonic Transforms aimed at pseudo-spectral numerical simulations

In this paper, we report on very efficient algorithms for the spherical harmonic transform (SHT). Explicitly vectorized variations of the algorithm based on the Gauss-Legendre quadrature are discussed and implemented in the SHTns library which includes scalar and vector transforms. The main breakthrough is to achieve very efficient on-the-fly computations of the Legendre associated functions, even for very high resolutions, by taking advantage of the specific properties of the SHT and the advanced capabilities of current and future computers. This allows us to simultaneously and significantly reduce memory usage and computation time of the SHT. We measure the performance and accuracy of our algorithms. Even though the complexity of the algorithms implemented in SHTns are in $O(N^3)$ (where N is the maximum harmonic degree of the transform), they perform much better than any third party implementation, including lower complexity algorithms, even for truncations as high as N=1023. SHTns is available at https://bitbucket.org/nschaeff/shtns as open source software.Comment: 8 page

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

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 incoming wave energy is entirely dissipated in the boundary layer. We investigate the consequences of this remarkable behaviour for the numerical modelling 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, but 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 mode

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