Geomagnetic secular variation generated by a tangentially geostrophic flow under the frozen-flux assumption - I. Necessary conditions

Existing geomagnetic main field models do not take adequate account of constraints implied by the magnetohydrodynamics of the core surface. Yet the flow is likely to be tangentially geostrophic in a region that is located around the magnetic equator. This is called the geostrophic region. This region could be quite large if the Lorentz force does not exceed its presently accepted upper bounds. We first show that if the frozen-flux assumption also applies within this region, the secular variation must satisfy a continuous set of curvilinear integral constraints. We then prove that three sets of surface integral constraints, some of them having been derived by others, can be formally derived from these curvilinear constraints. We discuss the connections between these new constraints and the ones previously derived by other authors. The question whether the curvilinear integral constraints are in fact sufficient (i.e. whether any secular variation satisfying these conditions within a given region can be thought of as being generated by a tangentially geostrophic flow under the frozen-flux assumption within this region) is to be addressed in the second paper of the series.