On Ohmic heating in the Earth’s core II: Poloidal magnetic fields obeying Taylor’s constraint

The extremely small Ekman and magnetic Rossby numbers in the Earth’s core make the magnetostrophic limit an attractive approximation to the core’s dynamics. This limit leads to the need for the internal magnetic field to satisfy Taylor’s constraint, associated with the vanishing of the azimuthal component of Lorentz torques averaged over every cylinder coaxial with the rotation axis. A special class of three dimensional poloidal interior magnetic fields is chosen that satisfies Taylor’s constraint identically on every cylinder in a spherical shell exterior to an inner core. This class of fields, which we call small-circle conservative, demonstrates existence of interior fields satisfying Taylor’s constraint, regardless of the morphology of the field on the core surface. These poloidal fields are used to examine the Ohmic dissipation present in the Earth’s core. To address the question of dissipation, we demand that the 3-D core fields agree with recent observations of the core field structure on the core–mantle boundary. We use these poloidal fields to show that the true lower bound on core dissipation must necessarily lie below a value that we calculate. For 2004 we find that this lower bound must lie below 1010 W, and when nutation constraints are also considered the bound must lie below 2 × 1010 W. These numbers are small compared to suggested values of the order of a few TeraWatts. A more restrictive bound may be forthcoming when the time-dependency of the field is considered, using a variational data assimilation technique.