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The magnetic field inside the Earth's fluid and electrically conducting outer core cannot be directly probed. The root-mean-squared (r.m.s.) intensity for the resolved part of the radial magnetic field at the core-mantle boundary is 0.3 mT, but further assumptions are needed to infer the strength of the field inside the core. Recent diagnostics obtained from numerical geodynamo models indicate that the magnitude of the dipole field at the surface of a fluid dynamo is about ten times weaker than the r.m.s. field strength in its interior, which would yield an intensity of the order of several millitesla within the Earth's core. However, a 60-year signal found in the variation in the length of day has long been associated with magneto-hydrodynamic torsional waves carried by a much weaker internal field. According to these studies, the r.m.s. strength of the field in the cylindrical radial direction (calculated for all length scales) is only 0.2 mT, a figure even smaller than the r.m.s. strength of the large-scale (spherical harmonic degree n

The core of a terrestrial-type planet consists of a spherical shell of rapidly rotating, electrically conducting, fluid. Such a body supports two distinct classes of quasi-geostrophic (QG) eigenmodes: fast, primarily hydrodynamic, inertial modes with period related to the rotation time scale and slow, primarily magnetic, magnetostrophic modes with much longer periods. Here, we investigate the properties of these hydromagnetic quasi-geostrophic modes as a function of non-dimensional parameters controlling the strength of the background magnetic field, the planetary rotation rate, and the amount of magnetic dissipation.We first present analytic solutions that illustrate the essential parameter dependences of the modes and provide a convenient benchmark for our numerical scheme. A comparison between known three-dimensional inertial modes in a sphere and our axially invariant QG modes shows encouraging agreement at low azimuthal wavenumbers, particularly for the slowest modes. The container geometry and background magnetic field structure are found to influence the radial structure of the modes, but not the scaling of their frequency with the control parameters. When the background magnetic field decreases toward the outer boundary in a spherical shell, QG modes tend to be compressed towards the outer boundary. Including magnetic dissipation, we find a continuous transition from diffusionless slow magnetic modes into quasi-free decay magnetic modes. During that transition (which is controlled by the magnitude of the Elsasser number), we find that slow magnetic modes weakly modified by diffusion exhibit a distinctive spiralling planform. When magnetic diffusion is significant (Elsasser number much smaller than unity), we find quasi-free decay slow magnetic modes whose decay time scale is comparable to, or shorter than, their oscillation time scale.Based on our analysis, we expect Mercury to be in a regime where the slow magnetic modes are of quasi-free decay type. Earth and possibly Ganymede, with their larger Elsasser numbers, may possess slow modes that are in the transition regime of weak diffusion, depending on the details of their poorly known internal magnetic fields. Fast QG modes, that are almost unaffected by the background magnetic field, are expected in the cores of all three bodies.

International audienceData assimilation in geomagnetism designates the set of inverse methods for geomagnetic data analysis which rely on an underlying prognostic numerical model of core dynamics. Within that framework, the time-dependency of the magnetohydrodynamic state of the core need no longer be parameterized: The model trajectory (and the secular variation it generates at the surface of the Earth) is controlled by the initial condition, and possibly some other static control parameters. The primary goal of geomagnetic data assimilation is then to combine in an optimal fashion the information contained in the database of geomagnetic observations and in the dynamical model, by adjusting the model trajectory in order to provide an adequate fit to the data. The recent developments in that emerging field of research are motivated mostly by the increase in data quality and quantity during the last decade, owing to the ongoing era of magnetic observation of the Earth from space, and by the concurrent progress in the numerical description of core dynamics. In this article we review briefly the current status of our knowledge of core dynamics, and elaborate on the reasons which motivate geomagnetic data assimilation studies, most notably (a) the prospect to propagate the current quality of data backward in time to construct dynamically consistent historical core field and flow models, (b) the possibility to improve the forecast of the secular variation, and (c) on a more fundamental level, the will to identify unambiguously the physical mechanisms governing the secular variation. We then present the fundamentals of data assimilation (in its sequential and variational forms) and summarize the observations at hand for data assimilation practice. We present next two approaches to geomagnetic data assimilation: The first relies on a three-dimensional model of the geodynamo, and the second on a quasi-geostrophic approximation. We also provide an estimate of the limit of the predictability of the geomagnetic secular variation based upon a suite of three-dimensional dynamo models. We finish by discussing possible directions for future research, in particular the assimilation of laboratory observations of liquid metal analogs of Earth's core