  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.

Geophysics