Experiments on Joule heating and the dissipation of energy in the Earth's core

We present measurements of Joule heat production in a fluid gallium vortex permeated by a uniform transverse magnetic field. We find that the Joule heat production increases as the square of the imposed field intensity for weak and moderately strong magnetic fields and magnetic Reynolds numbers up to about one. For stronger magnetic fields, Lorentz forces destroy the 2-D structure of the vortex and the Joule heat production becomes nearly independent of the intensity of the imposed magnetic field. We derive scaling laws relating fluid velocity in the vortex, imposed magnetic field intensity and Joule heat production, for both low and high magnetic Reynolds number regimes. Application of these laws to magnetic induction in the Earth's fluid core indicates that Joule heat production by this mechanism is large enough to limit the intensity of magnetic fields within the core.