Numerical study of a rotating fluid in a spheroidal container.

The motion of an incompressible, viscous rotating fluid contained in a spheroidal conainer is studied by a direct numerical simulation in a oblate speroidal system. An appropriate formalism is first derived which allows us to expand any scalar field in spherical harmonics and to decompose any vector field into its sphero-poloidal and sphero-toroidal scalar parts

Quasi-geostrophic flows responsible for the secular variation of the Earth's magnetic field

International audienceWe present core flows constructed from high resolution secular variation (SV) models for the epochs 2001, 2002.5 and 2004, assuming that these flows are quasi-geostrophic in the core interior and that they do not cross the axial cylindrical surface tangent to the inner core. A large jet encircling the inner core and carrying a significant part of the core angular momentum and axial vortices of ∼700 km diameter mainly clustering around the cylinder tangent to the solid inner core, are inferred from geomagnetic SV. New regularizations are suggested from dynamic considerations. It is found that medium and small-scale velocity fields contribute significantly to the large-scale SV. Accordingly, final models of core flows are calculated after an iterative process, whereby the magnetic field variation produced by small-scale stochastic magnetic fields and medium to small-scale computed velocity fields are incorporated into the inversion itself as modelling errors. This study represents a significant step in an effort to join geomagnetic observations and the fluid core dynamics on short timescales

Forward and adjoint quasi-geostrophic models of the geomagnetic secular variation

International audienceWe introduce a quasi-geostrophic model of core dynamics, which aims at describ- ing core processes on geomagnetic secular variation timescales. It extends the for- malism of Alfv ́en torsional oscillations by incorporating non-zonal motions. Within this framework, the magnetohydrodynamics takes place in the equatorial plane; it involves quadratic magnetic quantities, which are averaged along the direction of ro- tation of the Earth. In addition, the equatorial flow is projected on the core-mantle boundary. It interacts with the magnetic field at the core surface, through the radial component of the magnetic induction equation. That part of the model connects the dynamics and the observed secular variation, with the radial component of the magnetic field acting as a passive tracer. We resort to variational data assimilation to construct formally the relationship between model predictions and observations. Variational data assimilation seeks to minimize an objective function, by computing its sensitivity to its control variables. The sensitivity is efficiently calculated after in- tegration of the adjoint model. We illustrate that framework with twin experiments, performed first in the case of the kinematic core flow inverse problem, and then in the case of Alfv ́en torsional oscillations. In both cases, using the adjoint model allows us to retrieve core state variables which, while taking part in the dynamics, are not directly sampled at the core surface. We study the effect of several factors on the solution (width of the assimilation time window, amount and quality of data), and we discuss the potential of the model to deal with real geomagnetic observations

ensemble inversion of time-dependent core flow models

International audienceQuasi-geostrophic core flow models are built from two secular variation models spanning the periods 1960--2002 and 1997--2008. We rely on an ensemble method to account for the contributions of the unresolved small-scale magnetic field interacting with core surface flows to the observed magnetic field changes. The different core flow members of the ensemble solution agree up to spherical harmonic degree $\ell\simeq 10$, and this resolved component varies only weakly with regularization. Taking into account the finite correlation time of the small-scale concealed magnetic field, we find that the time variations of the magnetic field occurring over short time-scales, such as the geomagnetic jerks, can be accounted for by the resolved -- large scale -- part of the flow to a large extent. Residuals from our flow models are 30 \% smaller for recent epochs, after 1995. This result is attributed to an improvement in the quality of geomagnetic data. The magnetic field models show little frozen-flux violation for the most recent epochs, within our estimate of the apparent magnetic flux changes at the core-mantle boundary arising from spatial resolution errors. We associate the more important flux changes detected at earlier epochs with uncertainties in the field models at large harmonic degrees. Our core flow models show, at all epochs, an eccentric and planetary scale anti-cyclonic gyre circling around the cylindrical surface tangent to the inner core, at approximately 30$^{\circ}$ and 60$^{\circ}$ latitude under the Indian and Pacific oceans, respectively. They account well for the changes in core angular momentum for the most recent epochs

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

Axial invariance of rapidly varying diffusionless motions in the Earth's core interior

Geostrophic jets propagating as Alfv\'en waves are shown to arise ina rapidly rotating spherical shell permeated by a magnetic field among the transient motions set up by an impulsive rotation of the inner core. These axially invariant motions evolve on a time-scale which is short compared to the magnetic diffusion time. The numerical study is taken as illustrative of a more general point: on such a fast time-scale the dimensionless number appropriate to compare the rotation and magnetic forces is independent of the magnetic diffusivity in contrast with the often used Elsasser number. Extension of the analysis to non-axisymmetrical motions is supported by published studies of dynamo models and magnetic instabilities

Towards a rapidly rotating liquid sodium dynamo experiment

The main characteristics of the Earth's dynamo are reviewed. The combined actions of Coriolisand Lorentz forces lead to the so--called ``magnetostrophic'' regime. We derive an estimate of the power needed to sustain the magnetic field in this regime. We show that an experimentwith liquid sodium can be designed to operate in the magnetostrophic regime. Such an experiment would bring most valuable informations on the mechanisms of planetary dynamos. In order toprepare this large--scale experiment and explore the magnetostrophic balance, a smaller scale liquid sodium set--up has been designed and is being built. It consists of a rapidly rotating spherical shell filled with liquid sodium, in which motions are set by spinning at a different rotation rate an inner core permeated by a strong magnetic field. We discuss the processes that can be explored with this new device