Spatial and Temporal Changes of the Geomagnetic Field: Insights from Forward and Inverse Core Field Models Nicolas Gillet

Spatial And Temporal Changes Of The Geomagnetic Field: Insights From Forward And Inverse Core Field Models Nicolas Gillet To cite this version: Nicolas Gillet. Spatial And Temporal Changes Of The Geomagnetic Field: Insights From Forward And Inverse Core Field Models. Geomagnetism, aeronomy and space weather: a journey from the Earth’s core to the sun, In press. hal-02042703 HAL Id: hal-02042703 https://hal.archives-ouvertes.fr/hal-02042703 Submitted on 20 Feb 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Chapter 9 Spatial And Temporal Changes Of The Geomagnetic Field: Insights From Forward And Inverse Core Field Models Nicolas Gillet1 are able to run geodynamo simulations that ressemble to the Earth: some are dipole dominated, and under some Abstract conditions they show polarity reversals [Jones, 2015]. At the same time, as we push the parameters towards Earth- Observational constraints on geomagnetic field changes like values, numerical geodynamos do not show reversals from interannual to millenial periods are reviewed, and anymore [Schaeffer et al., 2017]. We indeed face sev- the current resolution of field models (covering archeo- eral severe obstacles: no simulation is currently able to logical to satellite eras) is discussed. With the perspective mimic all time-scales together, and diffusive processes re- of data assimilation, emphasis is put on uncertainties en- main too large. We also suffer from limited geophysical taching Gauss coefficients, and on the statistical proper- constraints. Direct geomagnetic records [20 yr of contin- ties of ground-based records. These latter potentially call uous satellite data, less than 200 yr of observatory series, for leaving behind the notion of geomagnetic jerks. The four centuries of historical measurements, see Jackson accuracy at which we recover interannual changes also re- and Finlay, 2015] cover a short era in comparison with key quires considering with caution the apparent periodicity time-scales of the core dynamics, while we know from pa- seen in the secular acceleration from satellite data. I then leomagnetic studies that changes occur at all periods up to address the interpretation of recorded magnetic fluctua- some 100 kyr [Constable and Johnson, 2005]. The avail- tions in terms of core dynamics, highlighting the need for able window is also cut towards high frequencies, since models that allow (or pre-suppose) a magnetic energy or- periods shorter than O(1) yr are hidden by signals from ders of magnitudes larger than the kinetic energy at large external (ionospheric and magnetospheric) sources [e.g. length-scales, a target for future numerical simulations of Finlay et al., 2017]. the geodynamo. I finally recall the first attempts at imple- In this chapter I review the main recent findings con- menting geomagnetic data assimilation algorithms. cerning time changes of the geomagnetic field on periods ranging from a couple of years to millenia, covering keywords both observational constraints, dynamical modeling, and the combination of the two through data assimilation stud- geomagnetic secular variation, geodynamo simulations, ies. It is on purpose that I exclude mechanical forcings inverse problem, data assimilation, Earth’s core dynam- (precession, tides...). These are undoubtedly a source of ics core motions, and an altenative scenario to planetary dynamos [Le Bars et al., 2015]. However, up to now they 9.1 Introduction failed at producing Earth-like dynamos. This may just be a question of time, as much more efforts have been put on The past two decades have seen our knwoledge on the ge- the convective side. omagnetic field and the dynamics within the Earth’s core I first recall the main dynamical features captured with strongly modified by the new possibilities offered by mod- geomagnetic data: from interannual to decadal changes ern computers, and the inflow of continuous satellite ob- with modern direct records, to centennial and longer servations, in complement of ground-based records. We fluctuations as seen with historical data, archeological 1Univ. Grenoble Alpes, Univ. Savoie Mont Blanc, CNRS, IRD, IF- artefacts or sediment series (x9.2.3-9.2.5). I highlight STTAR, ISTerre, 38000 Grenoble, France methodological issues associated with the complex sepa- 1 2 CHAPTER 9. ration of magnetic sources, and the difficult estimation of than a few months) is hindered by the slow drift of the model uncertainties (x9.2.1-9.2.2). Then I present some satellite orbit in local time. The associated difficulty in milestones in terms of time-scale separation if one wishes describing slow evolutions of external sources is a ma- to reach Earth-like dynamics, and some current limita- jor limit to the recovery of rapid (shorter than a couple tions of geodynamo simulations (x9.3.1). Several reduced of years) internal fiels changes. Another important issue models are next illustrated (x9.3.2-9.3.3), developed on when considering satellite data is the ionosphere being in- purpose to analyse core field changes from interannual to ternal to the spacecraft trajectory: one thus cannot prop- centennial periods, based on Magneto-Coriolis (MC) or erly separate core and ionopsheric field changes by solv- Magneto-Archimedes-Coriolis (MAC) waves. In x9.3.4 I ing Laplace’s equation. Finally, the current-free separa- recall important findings from numerical simulations that tion leading to magnetic potential fields breaks down in integrate primitive equations. I report in x9.4 efforts car- places (such as auroral regions) where the satellite trajec- ried out to merge the information contained into geophys- tory crosses electrical currents [Finlay et al., 2017]. ical observations with that coming from dynamical mod- Separating the signature of the several sources in satel- els, a process known as data assimilation: attempts at us- lite and ground-based stations is carried out by means of ing geodynamo simulations for either static or dynami- field modeling (i.e. space-time interpollation). Complete cal recoveries of the core state (x9.4.1-9.4.2), and applica- descriptions often rely on subtle weighting of the many tions to the construction of field models (x9.4.3). I finally parameters entering this inverse problem [e.g. the com- recall crucial issues in x9.5. prehensive approach by Sabaka et al., 2015], although attempts at objectifying the prior information now appear [Holschneider et al., 2016]. The absence of global deter- 9.2 Geomagnetic Field Modeling ministic physical models able to isolate the external ac- tivity also leads to consider several magnetic indices [e.g. 9.2.1 Separation Of Sources Olsen et al., 2014, Thomson and Lesur, 2007], used to se- lect quiet time data, and onto which rapid external varia- We shall ignore electrical currents in the mantle [a con- tions may be anchored [see Lesur et al., 2010, Finlay et al., dition valid for periods longer than a few months, see 2016]. Jault, 2015], so that the magnetic field B derives from a potential: B = −∇V . From the solenoidal condition r·B = 0, V satisfies to Laplace’s equation, r2V = 0. In 9.2.2 Field Model Uncertainties absence of electrical currents, internal and external solu- Solutions for internal model coefficients are obtained by tions arise, which in spherical coordinates (r; θ; φ) are nat- minimizing a cost function, sum of a measure of the misfit urally expressed by means of spherical harmonics [Olsen m to magnetic data and a measure of the model complexity. et al., 2010] and Gauss coefficients (denoted by gn and m In the case of quadratic norms, the penalty function takes hn , with n and m the degree and order). the form A proper separation of internal and external signals re- quires the knowledge of several components of the field 2 2 J(x) = ky − H(x)kR + kx − xkP ; (9.1) on a closed surface. This is unfortunaltely never com- pletely achieved. Ignoring some components of the field 2 T −1 with the notation kxkM = x M x. Vectors x and y leads to non-uniqueness issues [Sabaka et al., 2010]. The respectively store model parameters and observations. x coverage is obviously un-even for archeomagnetic, obser- is a background model, H is the observation operator, and vatory and historical records, in favor of lands (resp. seas) matrices P and R store the a priori cross-covariances of for ground-based stations [resp. ship logs, see Jonkers respectively the model anomaly to x and the observation et al., 2003]. Such disparities in the spatial sampling error. imply ambiguities between Gauss coefficients [see Gillet Models of the main field (MF) originating from the core et al., 2013, for an application to the observatory era]. The shall be used as input data in re-analyses of the core dy- major North/South discrepancy in the coverage by arche- namics (see x9.4), a reason why it is necessary to estimate ological artifacts and lavas translates into sensitivity ker- uncertainties on Gauss coefficients. With quadratic mea- nels much biased towards the North and the Middle-East sures as in equation (9.1), posterior uncertainties may be [e.g. Korte and Constable, 2011]. described by the inverse of the Hessian matrix (that mea- Magnetic records from space are available continuously sures the curvature of J in the neighborhood of the solu- from low-orbiting satellite data since 1999, with the Oer- tion x∗), sted, SAC-C, CHAMP and Swarm missions [Finlay et al., −1 2017].

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