Instability of layered systems in the field of gravity. II

Abstract Equations have been developed for the dynamics of layered systems in the field of gravity, the assumptions being (1) Newtonian viscosity, (2) small amplitude/wavelength ratio, (3) negligible inertial forces, (4) two-dimensional flow, and finally (5) orthogonal gravity-field symmetry. The theory applies to systems with any number of layers provided that each layer is homogeneous and shows uniform thickness. The various layers in a model may or may not differ in all or some of the properties: viscosity, density and thickness. By means of a computer program employed numerous models belonging to six main types have been analysed numerically. The six main types are distinguished by the boundary conditions at the top and bottom. Type I has a free flexible upper surface and infinite half space at bottom. Type II has also a free flexible upper surface, but is bounded by a straight rigid bottom. Type III is bounded both upward and downward by infinite half spaces. Type IV has an infinite half space on the top and a straight rigid bottom. Type V is characterized by boundary conditions opposite to those of type IV, and finally type VI contains models whose upper and lower boundaries both are straight and rigid. Each type can contain any number of layers with flexible interfaces between the bounding layers or half spaces on the top and bottom. So far we have studied systems with up to eleven layers and ten flexible boundaries. In simple systems consisting of a single layer overlain by a more dense half space, or overlaying a less dense half space, the wavelength/thickness ratio for the dominant perturbation increases with increasing ratio between the viscosities of the half space and of the layer with limited thickness. In multilayers with only one layer that is less dense than the immediately overlaying medium the wavelength/thickness ratio for the dominant perturbation increases with increasing viscosity of the overburden and with decreasing viscosity of the substratum all other things being equal. The dominant wavelength in such multilayers is also controlled by the density-contrast ratios, the viscosity ratios and the thickness ratios of several layers both below and above the buoyant one. If two or more layers in a system are less dense than the media immediately above the possibility exists that two sets of perturbations develop with unlike wavelengths, thus giving rise to structures of type anticlinorium and synclinorium. The theory predicts that the low-velocity layer, even if only a few percent less dense than the overlying sima, would rise rapidly measured in geologic time, the exact rate depending upon the viscosities, densities and geometric dimensions assumed. Instable multilayers with a large number of layers (6 and 10) have been studied numerically as dynamically equivalent simulators for thermal convection cells in the mantle. Based on reasonable thermal- and density data such models give horizontal velocity at the earth's surface of the order 1 cm/year. Applied to isostatic adjustment in a layered globe the theory predicts how internal discontinuity boundaries are deflected in response to movements of the free surface, or how the free surface is deflected due to movement of internal boundaries.