Projet VSQG : Variations Séculaires, approche Quasi-Géostrophique

Variations Séculaires, approche Quasi-Géostrophique

[(

Duration : 2006 - 2010

ISTerre research team involved :
– Geodynamo

ISTerre contact : Nicolas Gillet

)]

 

By combining numerical modelling and magnetic observations, we try to link and infer the variations of the magnetic field and the flow of liquide iron inside the Earth core.

Scientific background and objectives

Geomagnetic data are our main source of information concerning the dynamics and the magnetic induction processes in planetary cores. The short time scale dynamics of the Earth’s core spanning interannual time-scales to centuries, it is important to collect long geomagnetic time series. The first historical data go back to the 16th century (inclination and declination), but we have to wait for Gauss to have the first intensity measurements around 1840 (see e.g. Bloxham et al, 1989). Geomagnetic data are today collected at the Earth’s surface (ground observatories, cf. Intermagnet) or at satellite altitude (cf. missions Oersted, Champ, Sac-C and Swarm). From such measurements it is possible (see Fig. 1) to reconstruct the main field and its secular variation at the core-mantle boundary (review by Jackson and Finlay 2007) : see e.g. the historical model gufm1 (Jackson et al, 2000) and the satellite models of the CHAOS (Olsen et al, 2006) and GRIMM (Lesur et al, 2008) series.

Fig 1.
Sketch of the magnetic Earth (after Fournier et al, in rev) : inward (red) and outward (blue) magnetic flux from the Earth’s surface to the CMB, with in green the equatorial plane, and a schematic fluid column (white).

From such models some attempts have been made to recover the core dynamics. This problem being severely ill-posed, some extra constraint about the flow topology is required to reduce the parameter space (review by Holme 2007). However, this classical approach can lead to inconsistent flow and magnetic field models from the dynamical point of view. In order to tackle this issue, we aim to introduce via the VSQG project some prognostic physics (induction and momentum equations) into the inverse problem, which will advect information through epochs. This is to be done by the use of variational data assimilation, a framework imported from the ocean and atmosphere dynamics communities (Talagrand and Courtier 1987, Ghil 2000). We face today the early days of the geomagnetic data assimilation (see the review by Fournier et al, 2010). Some early results from a 1-D toy model have been obtained : Fournier et al (2007) shown that using a variational formalism, it is possible to backward propagate the high quantity and quality of modern satellite measurements toward earlier epochs (see Fig. 2).

Fig 2.
Time evolution of the error of (a) the magnetic field and (b) the fluid velocity, in a 1-D MHD toy model, for the initial guess (black) and after assimilation : with (green) or without (red) satellite data (after Fournier et al 2007).

The VSQG project aim to interpret the data by the use of quasi-geostrophic motions (cf. ocean/atmosphere dynamics, e.g. the book by Gill 1982), i.e. the flow is columnar and perpendicular to the rotation axis (cf Fig 1). This assumptions, inspired from Hide 1966, is justified in the limit of rapid rotation and diffusionless magnetic field, since the period of Alfen waves due to the presence of magnetic field (Braginski 1970) is much more longer than that of inertial waves due to rotation (Jault 2008). Pais and Jault (2008) used the QG approximation to derive snapshot flow models from satellite geomagnetic models. They discovered a planetary scale, eccentric and anticyclonic gyre circling (see Fig 3). This approach has been extended to the time-dependent problem : Gillet et al (2009) developed an ensemble approach to account for the unresolved small-scales magnetic field (harmonic degrees larger than about 13).

Fig 3.
Top : Stream function in the equatorial plane, illustrating the planetary scale, eccentric and westward anticyclonic gyre (average solution, after Pais and Jault, 2008) ; anticyclonic (cyclonic) circulations in blue (red) ; black radius : projection of the Greenwich meridian. Bottom : several flow solution from an ensemble of possible core states, illustrating the ambiguity in the core flow inverse problem due to unresolved small-scale magnetic field (after Gillet et al, 2009).

However, these kinematic approaches do not include any dynamical system. Canet et al (2009) developed a pronostic QG model suitable for the rapid core dynamics. Using twin experiment, they illustrated the feasability of a dynamical inversion of geomagnetic data. For this strongly nonlinear problem, a good estimate of the background state is crucial (Fig. 4). Gillet et al (in rev) recently applied this framework to geophysical data. They explain with torsional waves the 6-y signal found in the length-of-day series (Abarca del Rio, 2000). This implies intense magnetic field (several mT) within the Earth’s core, and reconciles with estimates from geodynamo simulations (Aubert et al, 2009).

Fig 4.
Assimilation in a synthetic context : the ’true’ state is modified by applying an off-set to a vortex (bottom, view of the equatorial plane). We assimilate to retrieve the fluid flow in the Earth’s core. The misfit function (top) depends on the assimilation windows : extending the time-span generates numerous local minima. It illustrates the need for a good initial guess when assimilating data in a nonlinear system. (After Fournier et al, in rev).