Gravity anomalies and convection currents: 1. A sphere and cylinder sinking beneath the surface of a viscous fluid

The effects of a sphere and a cylinder sinking beneath the surface of a viscous fluid are examined, and two results are found: (1) the horizontal divergence of the velocity at the surface of the fluid is related to the measured free-air anomaly if the viscosity of the fluid is uniform, and (2) the vertical load supported at the surface is essentially independent of any vertical variation of the viscosity of the fluid. The vertical load supported at the surface, Fzz, or the magnitude of the depression if the surface is a free surface, is found by an image solution. The solution formally has a pressure term and a velocity term. If the viscosity is constant, we find through an identity of integrals that the contribution to the surface load represented by the pressure term exactly cancels the gravitational attraction of the density inhomogeneities beneath the surface for any arbitrary density distribution. The net attracting mass, which produces the net free-air anomaly, is then related only to the velocity divergence. This theorem is limited in its application because it is sensitive to viscosity variations with depth. It is found that Fzz is independent of the viscosity of the fluid and also of the boundary conditions of the top surface—a free surface or a rigid plate give the same Fzz. A two-layer model in which a fluid of one viscosity lies over a fluid of another viscosity is examined, and it is found that Fzz is relatively independent of the thickness of the top layer or the ratio of the two viscosities, especially if the top layer has the greater viscosity. Whereas the horizontal velocity at the surface is very dependent upon the assumed viscosity pattern, the vertical load Fzz is not, and this enables us to find the approximate mass of a sinking body and its depth beneath the surface independently of the viscosity pattern of the earth.