Convective instability of a non-uniformly viscous sphere with reference to the earths mantle

Abstract

The principal concern of this paper is with the thermal convective instabilities of a non-uniformly viscous liquid sphere heated internally by some distribution of heat sources. Solutions are first obtained which correspond to an extension of the classic theory for uniformly viscous spheres for (1) radial variations of viscosity and (2) near critical Reynolds numbers. This results in instability growth rates which strongly suggest that the earth's mantle must have been in a highly supercritical state for the present convective pattern to have been established within the earth's lifetime. The paper then proceeds to a different theoretical calculation for incipient growth rates for such supercritical situations. For the earth's mantle this results in typical growth times of the order of 10^5 years. Various outer boundary conditions of the fixed and free type are explored; a true free surface condition is also examined and results in the establishment of a Froude number-like parameter governing distortion of the outer surface. In addition it is demonstrated that the spherical harmonics of degree 3 or 4 which seem to dominate the present convective pattern in the earth's mantle would do so in the theoretical construction if the increase of viscosity with depth within the mantle were such that the deep mantle viscosity was some four times the upper mantle value.