Equation for the field lines of an axisymmetric magnetic multipole

Abstract

Summary. An exact equation is derived for the magnetic field lines of the general axisymmetric magnetic multipole of arbitrary degree (n). This new result has important applications in studies of the possible nature of solarterrestrial physics during geomagnetic polarity reversals. In the limiting case of a magnetic dipole (n = l), the equation for the magnetic field lines of the general axisymmetric magnetic multipole simplifies correctly to the wellknown dipolar form, which is used extensively in geomagnetism, magnetospheric physics and cosmic-ray physics as a first-order approximation to the actual configuration of the geomagnetic field. It is also shown theoretically that suites of similar magnetic field lines of the general axisymmetric multipole attain their maximum radial distances from the origin on a set of circular conical surfaces, with coincident vertices at the centre of the Earth; this set includes the equatorial plane if the degree (n) of the multipole is odd. The magnetic field is horizontal everywhere on all these surfaces. Palaeomagnetic studies have suggested that during some polarity reversals the magnetic field in the inner magnetosphere can be represented approximately either by a single, non-dipolar, low-degree (2 < n < 4), axisymmetric magnetic multipole or by a linear combination of such multipoles. In this situation, the equation for the field lines of an axisymmetric magnetic multipole of low degree (2 < n < 4) would be as fundamental to a proper understanding of magnetospheric, ionospheric and cosmic-ray physics during polarity reversals as is the equation for dipolar field lines in the case of the contemporary geomagnetic field.