Numerical solution of a two-dimensional rock-glacier flow model via the pressure method

Literature confirms the crucial influence on glacier and rock glacier flow of non-viscous deformations together with temperature impact. This observation suggests numerical glaciologists ought to reconsider the established mathematical modeling based on the representation of ice as a power-law viscous fluid and the Glen's law. Along this line, we propose the numerical solution of a two-dimensional rock-glacier flow model, based on a constitutive law of second grade of complexity two, as just published for a one-dimensional set-up by two of the authors. With the representation of the composition of the rocky ice as a mixture of ice and rock and sand grains, and the inclusion of the local impact of pressure and of thermal effects, this model has allowed the reproduction of borehole measurement data from alpine glacier internal sliding motion via a similarity solution of the flow governing equations. Here, the adopted numerical procedure uses a second order finite difference scheme and imposes the incompressibility constrain up to computer accuracy via the pressure method, that we have extended from Newtonian computational fluid dynamics. This method solves the governing equations for the flow in primitive variables with the advantage that no pre-/post-processing is required; in addition, it avoids splitted solution of the Poisson equation for pressure which might be source of undesired numerical mass unbalancing. The results of a numerical test on the Murtel-Corvatsch alpine glacier flow, reporting satisfactory matching with published on-field observations, are presented.