The compressional and compositional stratifications in Maxwell earth models : the gravitational overturning and the long-period tangential flux

We analyse self-gravitating compressible Maxwell earth models with compressional and compositional stratifications by taking into account the continuous variations of the initial density and of the bulk modulus. We derive the analytical solution in the Laplace domain of an earth model composed of a homogeneous inviscid core and a viscoelastic mantle characterized by a specific Darwin-law density profile. This makes us discover a new class of relaxation modes, the compositional C-modes, of which the previously discussed RT-modes are a subclass. Compositional stratifications present a denumerable set of C-modes situated on the positive and negative real axis depending on the sign of the square of the Brunt–Vaisala frequency ω and converging to the origin of the Laplace domain, while compressional stratifications do not present such modes. If ω^2 < 0 the C-modes are instable and describe the gravitational overturning. If ω^2 > 0 the C-modes are stable, but they affect the tangential perturbation in such a way that the tangential fluid limit diverge. We interpret this perturbation as a long-period tangential flux of material. By analysing the tangential velocities, we show that this flux fades in the fluid limit controlled by the viscosity. Our findings shed new light on the isostatic criterion, which we reformulate in a new way. Successively, by means of a numerical algorithm, we consider earth models including also the continuous variations of the bulk modulus. In this case, the differential system describing the conservation of the momentum and the self-gravitation is not uniformly Lipschitzian in the region of the Laplace domain associated with the inverse compressional relaxations times. This implies a continuous spectrum in the transient region that the normal mode approach cannot deal with. Nevertheless, the buoyancy mode regions is not characterized by a continuous spectrum and the C-modes are still present for the compositional stratifications.