On the magnetic sounding of planetary interiors

Two ways have been suggested for using the magnetic field of a planet to estimate the radius of its electrically conducting core, assumed spherical and concentric with the planet's surface. In the frozen flux method, the core radius is assumed to have the value rFF for which the integrated unsigned radial magnetic flux, from the sphere of that radius, is most nearly constant in time. In the flat spectrum method, the core radius is assumed to have the value rFs for which the power spectrum of the field, defined as the mean square energy density in the lth spherical harmonic component of the field at that radius, is most nearly independent of I. These two methods are here applied to a new geodynamo integration that is a continuation of a recently published simulation (see Glatzmaier and Roberts, 1995a, Phys. Earth Planet. Inter., 91: 63-75; Glatzmaier and Roberts, 1995b, Nature, 377: 203-209) and which, like the Earth, maintains a more strongly dipole dominated magnetic field. The rate of change of the unsigned flux was averaged over two different 300 year intervals at a number of radii, r, from the geocenter. The resulting functions of r were found to have zeros at rv~ of approximately 3550 km and 3477 km, respectively. This demonstrates how sensitive this method is to the time interval selected for the computation. Even if, as is often done when the flat spectrum method is applied to the Earth, the centered dipole (l = 1) is excluded, the spectrum of our model could not be made convincingly flat; but a radius rFS at which it is most flat can be defined. The value of rFs is, however, very sensitive to the number of spherical harmonics retained in the spectrum and to a lesser extent is time dependent. On the basis of these studies, and impressed by the lack of a sound physical justification for the flat spectrum method, we conclude that that method provides a less certain way of estimating the radius of a planetary core than does the frozen flux approximation, and that even the latter should be employed with caution. 1. I n t r o d u c t i o n Two methods have been suggested for estimating the radius of a planetary core from the spatial structure of the magnetic field, B, it creates. We shall call these 'the frozen flux method' and ' the flat spectrum method'. The frozen flux method is based on the idea that the electrical conductivity, cr c, of the planetary core is so large compared with that of the exterior of the core that the latter can be considered to be an electrical insulator and ~r c can be assumed infinite. The field emerging from the core surface (which is supposed spherical, and henceforth termed 'the core-mant le boundary ' or 'CMB' ) is then merely advected by motions on (or more * Corresponding author. 0031-9201/96/$15.00 Copyright © 1996 Elsevier Science B.V. All rights reserved. PH S0031-9201 (96)031 88-3 2 0 8 G.A. Glatzmaier, P.H. Roberts/Physics of the Earth and Planetary Interiors 98 (1996) 207-220 precisely at the inner edge of a thin boundary layer at) the CMB, and matches smoothly to a potential field outside the core; the net unsigned flux, JV(rcMB,t), out of the core is then independent of time t, where ./V( r,t) = ~lBr( r,O,qb,t) IdS (1) Br(r,O,cb,t) being the radial component of the magnetic field and dS = r2sin 0d0d~b, 0 being colatitude and ~b longitude; the integral Eq. (1) is taken over the entire surface (0 _< 0 < 7r, 0 _< ~b < 27r) of the sphere of radius r with origin at the center of the planet. The magnetic field is assumed to be known from observations on and/or above the surface, r = a, of the planet, and it is supposed to be fitted to a potential field