Low‐frequency, motionally induced electromagnetic fields in the ocean: 1. Theory

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Abstract
The theory of electromagnetic induction by motional sources in the ocean is examined from a first principles point of view. The electromagnetic field is expanded mathematically in poloidal and toroidal magnetic modes based on the Helmholtz decomposition. After deriving a set of Green functions for the modes in an unbounded ocean of constant depth and conductivity underlain by an arbitrary one‐dimensional conducting earth, a set of exact integral equations are obtained which describe the induction process in an ocean of vertically varying conductivity. Approximate solutions are constructed for the low‐frequency (subinertial) limit where the horizontal length scale of the flow is large compared to the water depth, the effect of self induction is weak, and the vertical velocity is negligible, explicitly yielding complex relationships between the vertically‐integrated, conductivity‐weighted horizontal water velocity and the horizontal electric and three component magnetic fields and accounting for interactions with the conductive earth. After introducing geophysically reasonable models for the conductivity structures of the ocean and earth, these reduce to a spatially smoothed proportionality between the electromagnetic field components and the vertically‐integrated, conductivity‐weighted horizontal water velocity. An upper bound of a few times the water depth for the lateral averaging scale of the horizontal electric field is derived, and its constant of proportionality is shown to be nearly 1 for most of the deep ocean based on geophysical arguments. The magnetic field is shown to have a similar form but is a relatively weak, larger‐scale average of the velocity field. Because vertical variations in the conductivity of seawater largely reflect its thermal structure and are weak beneath the thermocline, the horizontal electric field is a spatially filtered version of the true water velocity which strongly attenuates the influence of baroclinicity and accentuates the barotropic component. This is quantified using conductivity profiles and velocity information from a variety of locations.