Improved Hard Lower Bounds on the Geodynamo Power Requirements

The Earth's thermal history is currently undergoing renewed scrutiny, the age of the inner core taking center stage of the debate. A key element of the controversy is estimates of the power required to drive the geodynamo or, equivalently, the power lost to Ohmic dissipation in the core. A recent prominent estimate of 1-2 TW (Roberts et al. 2003) presents substantial difficulty in forming a consistent Earth model, as the associated high heat flux leaking out of the core- mantle boundary would cool the core too fast, leading to an inner core far younger than expected. A current high- profile theory that sidesteps this difficulty suggests the existence of radiogenic heating in the core, reducing the solidification rate for a prescribed heat flux. Other lower estimates of 0.1-0.5 TW (Christensen & Tilgner, 2004) on the Ohmic dissipation do not require such a source of energy to explain an old inner core, of age comparable to that of the geomagnetic field. The current estimates of Ohmic dissipation are based on numerical geodynamo simulations, computations currently performed with parameter values far from those that are geophysically realistic. Despite the resulting magnetic and velocity fields being arguably at least Earth-like, the extrapolation of such models to the real Earth is nonetheless no more than suggestive. We present results concerning rigorous lower bounds on the Ohmic dissipation associated with the present day field. A lower bound exceeding any existing estimate would immediately rule that estimate as invalid; a sufficiently high value would lead to the inescapable inference of radiogenic heating in order to explain an old inner core. We build on previous work (Jackson and Livermore, 2007), where we considered the minimum Ohmic dissipation subject to various constraints: the field structure on the core-mantle boundary derived from observations of the present day geomagnetic field, and Earth-nutation models (Buffett et al. 2002) that predict rms field strengths on both the inner and outer core boundaries. With additional assumptions on the spectrum of the field, the lower bound was found to be approximately 0.1 TW, very close to the lower-end estimates derived from numerical simulations. However, such constraints effectively supply only boundary conditions on the field structure. We now add in a dynamical condition that the geomagnetic field must satisfy in the entire fluid core: Taylor's constraint. This supplies an infinite number of nonlocal constraints on the permissible structure of the field and must necessarily raise the previous lower bound. Although the improved lower bound will itself be of considerable interest, the associated optimizing field will provide the first example of a 3D magnetic field that is associated with the same dynamical regime as the core, namely, one that satisfies Taylor's constraint.