The construction of exact Taylor states. I: The full sphere

SUMMARY

The dynamics of the Earth's fluid core are described by the so-called magnetostrophic balance between Coriolis, pressure, buoyancy and Lorentz forces. In this regime the geomagnetic field is subject to a continuum of theoretical conditions, which together comprise Taylor's constraint, placing restrictions on its internal structure. Examples of such fields, so-called Taylor states, have proven difficult to realize except in highly restricted cases. In previous theoretical developments, we showed that it was possible to reduce this infinite class of conditions to a finite number of coupled quadratic homogeneous equations when adopting a certain regular truncated spectral expansion for the magnetic field. In this paper, we illustrate the power of these results by explicitly constructing two families of exact Taylor states in a full sphere that match the same low-degree observationally derived model of the radial field at the core–mantle boundary. We do this by prescribing a smooth purely poloidal field that fits this observational model and adding to it an expediently chosen unconstrained set of interior toroidal harmonics of azimuthal wavenumbers 0 and 1. Formulated in terms of the toroidal coefficients, the resulting system is purely linear and can be readily solved to find Taylor states. By calculating the extremal members of the two families that minimize the Ohmic dissipation, we argue on energetic ground that the toroidal field in the Earth's core is likely to be dominated by low order azimuthal modes, similar to the observed poloidal field. Finally, we comment on the extension of finding Taylor states within a general truncated spectral expansion with arbitrary poloidal and toroidal coefficients.