The Viscosity of the Mantle

Abstract

Summary Creep under low stresses is by diffusion and has a linear relation between stress and strain rate; it also obeys the Navier-Stokes equation. Therefore the viscosity of the mantle may be calculated from solid state theory and also from the slow deformations of the Earth. The viscosities derived by these methods are in reasonable agreement, and both show that the viscosity of the lower mantle is - lo5 greater than that of the upper. This high viscosity prevents polar wandering and lower mantle convection. Some suggested modifications of the viscosity depth calculations from post glacial uplift may improve their accuracy considerably. 1. The non-hydrostatic equatorial bulge All calculations concerned with post-glacial uplift (Haskell 1935, McConnell 1965) and with convection within the mantle (Pekeris 1935) have used a linear relation between stress and strain rate. Creep in the material then obeys Stokes’ equation and therefore may be described by a viscosity which is independent of stress. Such creep only takes place at stresses below the yield stress < - p, where p is the shear modulus; at higher stresses the relation between stress and strain rate is non-linear (see Pratt, above). At stresses below the yield point diffusion creep alone can operate, and it has a linear relation between stress and strain rate. The diffusion creep rate also depends exponentially on temperature (Gordon 1965), and therefore is not sufficiently rapid to be measurable under ordinary laboratory conditions. For this reason solids were believed to be unable to deform for stresses below the yield stress, a concept called finite strength. The stresses caused by the melting of ice caps, by the non-hydrostatic bulge, and by regional temperature differences are probably less than the yield stress. Thus the viscosity will be stress independent, and the equation of a Maxwell solid should describe the flow. However, there are regions where earthquakes occur, and where this argument is clearly false. The viscosity of the mantle between the crust and a depth of IOOOkm may be found from the uplift of formerly glaciated areas. McConnell (1965) finds the uplift of Fennoscandia can be explained if the viscosity is least at a depth of about 300 km, increasing both upwards and downwards (Fig. 2). The viscosity of the lower mantle, which is of considerable importance to theories of convection and polar wandering, unfortunately cannot be calculated from the uplift for two reasons. The first is that the depth to which the flow penetrates in a homogeneous half space is about equal to the radius of the applied load, or about 1000 km for Fennoscandia. This difficulty can be overcome when the uplift of the Canadian Shield has been measured. If, however, McConnell is correct and there is a viscosity minimum at a depth of about 300 km a second difficulty arises because the flow in the low viscosity layer will shield the lower mantle from the applied force. It is therefore unlikely that any surface load can be used to estimate the viscosity of the lower mantle.