A case for variational geomagnetic data assimilation: insights from a one-dimensional, nonlinear, and sparsely observed MHD system

TL;DR

This paper demonstrates that variational data assimilation can effectively reconstruct the coupled magnetic and velocity fields in a simplified 1D MHD system, despite only observing magnetic data, offering insights for geomagnetic modeling.

Contribution

It introduces a variational data assimilation approach applied to a 1D nonlinear MHD model, showing potential for inferring core dynamics from magnetic observations.

Findings

Successful reconstruction of both magnetic and velocity fields from partial magnetic data.

Variational data assimilation is effective even with sparse and irregular magnetic observations.

The model captures key features like Alfvén waves, validating the approach's physical relevance.

Abstract

Secular variations of the geomagnetic field have been measured with a continuously improving accuracy during the last few hundred years, culminating nowadays with satellite data. It is however well known that the dynamics of the magnetic field is linked to that of the velocity field in the core and any attempt to model secular variations will involve a coupled dynamical system for magnetic field and core velocity. Unfortunately, there is no direct observation of the velocity. Independently of the exact nature of the above-mentioned coupled system -- some version being currently under construction -- the question is debated in this paper whether good knowledge of the magnetic field can be translated into good knowledge of core dynamics. Furthermore, what will be the impact of the most recent and precise geomagnetic data on our knowledge of the geomagnetic field of the past and future? These questions are cast into the language of variational data assimilation, while the dynamical system considered in this paper consists in a set of two oversimplified one-dimensional equations for magnetic and velocity fields. This toy model retains important features inherited from the induction and Navier-Stokes equations: non-linear magnetic and momentum terms are present and its linear response to small disturbances contains Alfv\'en waves. It is concluded that variational data assimilation is indeed appropriate in principle, even though the velocity field remains hidden at all times; it allows us to recover the entire evolution of both fields from partial and irregularly distributed information on the magnetic field. This work constitutes a first step on the way toward the reassimilation of historical geomagnetic data and geomagnetic forecast.