  The Earth acts as a gigantic heat engine driven by decay of radiogenic
isotopes and slow cooling, which gives rise to plate tectonics, volcanoes, and
mountain building. Another key product is the geomagnetic field, generated in
the liquid iron core by a dynamo running on heat released by cooling and
freezing to grow the solid inner core, and on chemical convection due to light
elements expelled from the liquid on freezing. The power supplied to the
geodynamo, measured by the heat-flux across the core-mantle boundary (CMB),
places constraints on Earth's evolution. Estimates of CMB heat-flux depend on
properties of iron mixtures under the extreme pressure and temperature
conditions in the core, most critically on the thermal and electrical
conductivities. These quantities remain poorly known because of inherent
difficulties in experimentation and theory. Here we use density functional
theory to compute these conductivities in liquid iron mixtures at core
conditions from first principles- the first directly computed values that do
not rely on estimates based on extrapolations. The mixtures of Fe, O, S, and Si
are taken from earlier work and fit the seismologically-determined core density
and inner-core boundary density jump. We find both conductivities to be 2-3
times higher than estimates in current use. The changes are so large that core
thermal histories and power requirements must be reassessed. New estimates of
adiabatic heat-flux give 15-16 TW at the CMB, higher than present estimates of
CMB heat-flux based on mantle convection; the top of the core must be thermally
stratified and any convection in the upper core driven by chemical convection
against the adverse thermal buoyancy or lateral variations in CMB heat flow.
Power for the geodynamo is greatly restricted and future models of mantle
evolution must incorporate a high CMB heat-flux and explain recent formation of
the inner core.
