Kinematic dynamos and the Earth’s magnetic field

Abstract

The Bullard—Gellman formalism is applied to investigate the existence of convergent solutions for steady kinematic dynamos. It is found that the solutions for the Bullard—Gellman dynamo, as well as for Lilley’s modification of it, do not converge. Convergent solutions have been found for a class of spherical convective cells which would be stationary in a perfect fluid in the absence of rotation and of the magnetic field. By calibrating the theoretical magnetic dipole so as to fit the observed value at the Earth’s surface, one can find a dynamo in the above class which also matches the observed equatorial magnetic dipoles. There is a dynamo which has a rate of total ohmic dissipation of only 1.8 x 1016 erg s-1 for an assumed electrical conductivity of 3 x 10~6 e.m.u.'f This is one thousandth the rate of tidal dissipation, and one hundred thousandth the rate of heat outflow from the surface of the Earth. The required velocities are of the order of 10~3 cm s_1, and the average magnetic energy density is 4 erg cm-3. The internal structure of the magnetic field in this model shows a dynamo mechanism situated in the outer part of the liquid core and is thus insensitive to possible rigidity of the material in the * inner core.

The optimal kinematic dynamo driven by steady flows in a sphere

“…In this reduced system, one can study how a flow field amplifies a seed magnetic field. Early numericists found flow fields that are capable of dynamo action in a sphere with electrically insulating boundary conditions (Backus 1958;Pekeris et al 1973;Kumar & Roberts 1975;Gubbins 1973;Dudley & James 1989). However, there is no universal recipe on how to obtain such flow fields; known dynamo solutions do not necessarily share similar spatial structures (Gubbins 2008).…”

Electric Fields Derived from the Earth's Magnetic Field and Their Application to Fluid Motion in the Outer Core

“…As far as the production of the main dipole field is concerned, it may be that there is a toroidal field which is small at the surface of the core and increases in intensity part way into the core (e.g. PEKERIS et al, 1973) but in what follows it is assumed that no toroidal field is present. The ramifications of a small, but finite, electrical conductivity in the mantle (e .g.…”

Three processes of inducing the steady poloidal field in the Earth's core.

“…Near the surface of the core, the dipole field is the most predominant in the harmonics, as at the surface of the Earth. In the deeper part a toroidal field is also regarded as the strongest, with strength of the order of 10 mT (BULLARD and GELLMAN,1954;BRAGINSKY, 1964BRAGINSKY, , 1970HIDE, 1966;KUMAR and ROBERTS, 1975;WATANABE, 1977), although PEKERIS et al (1973), SOWARD (1974), BUSSE (1975BUSSE ( , 1977, KRAUSE (1977) and KRAUSE and RADLER (1979) claim that the toroidal field is of the same order of magnitude as the poloidal field. In this paper, therefore, we take the dipole field and the toroidal field of the second degree as the primary magnetic field.…”