Current thinking on the Earth's thermal history is marked by notable controversies on the age of the inner core and whether core radiogenic heating is required in order to maintain the magnetic field. A key factor in this balance is the power required to drive the geodynamo, dissipated internally as heat. Present estimates, primarily based on numerical dynamo simulations operating far from the relevant parameter range, vary significantly. The aim of this project is to constrain the Earth's thermal history by finding rigorous lower bounds for the dissipation of the present field. Building on an existing bound based on constraints from observation and models for the Earth's nutation, the proposed addition of the Taylor constraint will impose a significant dynamical restriction long known to be appropriate for the core field. If the resulting bound is sufficiently increased, then contentious scenarios for the lifetime of the inner core without radiogenic heating may well be excluded. Further, this variational formulation exposes some intriguing mathematical issues germane to consideration of the geodynamo as a dynamical system. The optimal fields resulting will be both three-dimensional and time-dependent and thus for the first time give evidence of whether or not the large body of recent observationally-derived findings of large-scale surface features are really representative of the internal field.
Large scale computer simulation has fundamentally altered the way in which science is done. Yet for all the numerous advances, many of the most demanding computations in the area of geophysical fluid dynamics require that key properties, notably viscosity (the fluid "thickness"), be set to artificially high values. The effects of this adjustment on the numerical results are poorly understood and will remain so short of a better theory of turbulence than any presently in prospect. The so-called variational methods outlined in this proposal will both complement and critique numerical simulations of the Earth's dynamo. The results do not rely on artificial parameterizations and their interpretation as providing a bound on heat generated by the dynamo process is thus entirely rigorous. Key to this work will be a collaborative effort to link extensive long term observational data with a mathematically appropriate and numerically tractable formulation of the problem.