Stochastic modelling of regional archaeomagnetic series

SUMMARY We report a new method to infer continuous time series of the declination, inclination and intensity of the magnetic field from archeomagnetic data. Adopting a Bayesian perspective, we need to specify a priori knowledge about the time evolution of the magnetic field. It consists in a time correlation function that we choose to be compatible with present knowledge about the geomagnetic time spectra. The results are presented as distributions of possible values for the declination, inclination or intensity. We find that the methodology can be adapted to account for the age uncertainties of archeological artefacts and we use Markov Chain Monte Carlo to explore the possible dates of observations. We apply the method to intensity datasets from Mari, Syria and to intensity and directional datasets from Paris, France. Our reconstructions display more rapid variations than previous studies and we find that the possible values of geomagnetic field elements are not necessarily normally distributed. Another output of the model is better age estimates of archeological artefacts

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

Planetary gyre, time-dependent eddies, torsional waves, and equatorial jets at the Earth's core surface

We report a calculation of time-dependent quasi-geostrophic core flows for 1940-2010. Inverting recursively for an ensemble of solutions, we evaluate the main source of uncertainties, namely the model errors arising from interactions between unresolved core surface motions and magnetic fields. Temporal correlations of these uncertainties are accounted for. The covariance matrix for the flow coefficients is also obtained recursively from the dispersion of an ensemble of solutions. Maps of the flow at the core surface show, upon a planetary-scale gyre, time-dependent large-scale eddies at mid-latitudes and vigorous azimuthal jets in the equatorial belt. The stationary part of the flow predominates on all the spatial scales that we can resolve. We retrieve torsional waves that explain the length-of-day changes at 4 to 9.5 years periods. These waves may be triggered by the nonlinear interaction between the magnetic field and sub-decadal non-zonal motions within the fluid outer core. Both the zonal and the more energetic non-zonal interannual motions were particularly intense close to the equator (below 10 degrees latitude) between 1995 and 2010. We revise down the amplitude of the decade fluctuations of the planetary scale circulation and find that electromagnetic core-mantle coupling is not the main mechanism for angular momentum exchanges on decadal time scales if mantle conductance is 3 10 8 S or lower

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

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

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

Zonal shear and super-rotation in a magnetized spherical Couette flow experiment

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 pres

Can core-surface flow models be used to improve the forecast of the Earth's main magnetic field?

[1] Geomagnetic main field models used for navigation are updated every 5 years and contain a forecast of the geomagnetic secular variation for the upcoming epoch. Forecasting secular variation is a difficult task. The change of the main magnetic field is thought to be principally due to advection of the field by flow at the surface of the outer core on short timescales and when large length scales are considered. With accurate secular variation (SV) and secular acceleration (SA) models now available from new satellite missions, inverting for the flow and advecting it forward could lead to a more accurate prediction of the main field. However, this scheme faces two significant challenges. The first arises from the truncation of the observable main field at spherical harmonic degree 13. This can however be handled if the true core flow is large scale and has a rapidly decaying energy spectrum. The second is that even at a given single epoch the instantaneous SV and SA cannot simultaneously be explained by a steady flow. Nevertheless, we find that it may be feasible to use flow models for an improved temporal extrapolation of the main field. A medium-term (≈10 years) hindcast of the field using a steady flow model outperforms the usual extrapolation using the presently observed SV and SA. On the other hand, our accelerated, toroidal flow model, which explains a larger portion of the observed average SA over the 2000–2005 period, fails to improve both the short-term and medium-term hindcasts of the field. This somewhat paradoxical result is related to the occurrence of so-called geomagnetic jerks, the still poorly known dynamical nature of which remains the main obstacle to improved geomagnetic field forecasts