Induced magnetic moments from a nearly spherical ocean

The five largest planets all have strong intrinsic magnetic fields that interact with their satellites, many of which contain electrically conducting materials on global scales. Conducting bodies exposed to time-varying magnetic fields induce secondary magnetic fields from movement of eddy currents. In the case of spherically symmetric conducting bodies, matching magnetic solutions at the boundary results in relatively simple relations between the excitation field and the induced field. In this work, we determine the more complicated induced magnetic field from a near-spherical conductor, where the outer boundary is expanded in spherical harmonics. Under the approximations that the excitation field is uniform at a single frequency, the product of wavenumber and radius for the body is large, and the average radius of the body is large compared to the perturbation from spherical symmetry, we find that each spherical harmonic in the shape expansion induces discrete magnetic moments that are independent from the other harmonics in the expansion. That is, simple superposition applies to the magnetic moments induced by each perturbation harmonic. We present a table of the magnetic moments induced by each spherical harmonic up to degree 2 in the perturbed shape. We also present a simple formula by which the induced magnetic field may be evaluated for any arbitrary shape described by expanding the radius of the conducting body in spherical harmonics. Unlike the Earth, many moons in the Solar System are tidally locked to their parent bodies, and many also contain saline, subsurface oceans. Conductive material in these moons is therefore expected to be non-spherical. Accounting for the boundary shape of Europa's ocean will be critical for interpretation of Europa Clipper magnetic measurements near the moon, where the effects of quadrupole-and-higher magnetic moments will be most apparent. The results of this work permit magnetic studies considering non-spherical oceans of satellites for the first time.