Material versus local incompressibility and its influence on glacial-isostatic adjustment

We present an analytical form of the layer propagator matrix for the respon se of a locally incompressible, layered, linear-viscoelastic sphere to an e xternal load assuming that the initial density stratification rho (o)(r) wi thin each layer is parametrized by Darwin's law. From this, we show that th e relaxation of a sphere consisting of locally incompressible layers is gov erned by a discrete set of viscous modes. The explicit dependence of the la yer propagator matrix on the Laplace transform variable allows us to determ ine the amplitudes of the viscous modes analytically. Employing Darwin's pa rametrization, we construct three simplified earth models with different in itial density gradients that are used to compare the effects of the local i ncompressibility constraint, div (rho (o)u)=0, and the material incompressi bility constraint, div u=0, on viscoelastic relaxation. We show that a loca lly incompressible earth model relaxes faster than a materially incompressi ble model. This is a consequence of the fact that the perturbations of the initial density are zero during viscoelastic relaxation of a locally incomp ressible medium, so that there are no internal buoyancy forces associated w ith the continuous radial density gradients, only the buoyancy forces gener ated by internal density discontinuities. On the other hand, slowly decayin g internal buoyancy forces in a materially incompressible earth model cause it to reach the hydrostatic equilibrium after a considerably longer time t han a locally incompressible model. It is important to note that the approx imation of local incompressibility provides a solution for a compressible e arth model that is superior to the conventional solutions for a compressibl e earth with homogeneous layers because it is based on an initial state tha t is consistent with the assumption of compressibility.