Predicting surface dynamic topographies of stagnant lid planetary bodies

Although planetary mantles are viscoelastic media, numerical models of thermal convection in a viscoelastic spherical shell are still very challenging. Here, we examine the validity of simplified mechanical and rheological frameworks classically used to approximate viscoelastic dynamic topography. We compare three simplified approaches to a linear Maxwell viscoelastic shell with a pseudo upper free-surface, considered as the reference model. A viscous model with a free-slip boundary condition at the surface correctly reproduces the final relaxed shape of the viscoelastic body but it cannot reproduce the time evolution of the viscoelastic topography. Nevertheless, characterizing the topography development is important since it can represent a significant fraction of the history for planets having a thick and rigid lithosphere (e.g. Mars). A viscous model with a pseudo free-surface, despite its time-dependency, also systematically fails to describe correctly these transient stages. An elastic filtering of the instantaneous viscous topography is required to capture the essence of the time evolution of the topography. We show that a single effective elastic thickness is needed to correctly reproduce the constant transient viscoelastic topography obtained when the lithosphere corresponds to a step-like viscosity variation, while a time-dependence of the effective elastic thickness must be considered to take account of realistic temperature-dependent viscosity variations in the lithosphere. In this case, the appropriate thickness of the elastic shell can be evaluated, at a given instant, with a simple procedure based on the local Maxwell time. Furthermore, if the elastic filtering is performed using the thin elastic shell formulation, an unrealistic degree-dependence of the thickness of the elastic shell is needed to correctly approximate the viscoelastic topography. We show that a model that fully couples a viscous body to an elastic shell of finite thickness estimated using the local Maxwell time gives the best approximation of the viscoelastic deformation, whatever the degree of the load and the time of loading.