Instabilities across the Scales: Simple models for shear banding, plate subduction and mantle convection in geodynamics

The Earth shows different modes of deformation in response to thermal or gravitational driving forces. The bulk mantle convects like a viscous fluid on the global scale, while the lithosphere is broken into several plates. They show little internal deformation, but change their shapes and relative positions. Oceanic plate material is generated at divergent margins and recycled into the mantle at subduction zones, on a regional scale. The buoyant continental crust resists subduction and develops meter-scale shear bands during deformation. In this article we review Eulerian finite element (FE) schemes and a particle-in-cell (PIC) FE scheme [15]. Focussing initially on models of crustal deformation at a scale of a few tens of km, we choose a Mohr-Coulomb yield criterion based upon the idea that frictional slip occurs on whichever one of many randomly oriented planes happens to be favourably oriented with respect to the stress field. As coupled crust/mantle models become more sophisticated it is important to be able to use whichever failure model is appropriate to a given part of the system. We have therefore developed a way to represent MohrCoulomb failure within a mantle-convection fluid dynamics code. With the modelling of lithosphere deformation we use an orthotropic viscous rheology (a different viscosity for pure shear to that for simple shear) to define a preferred plane for slip to occur given the local stress field. The simple-shear viscosity and the deformation can then be iterated to ensure that the yield criterion is always satisfied. We again assume the Boussinesq approximation - neglecting any effect of dilatancy on the stress field. Subduction is modelled as a Rayleigh-Taylor instability with dense oceanic lithosphere sinking into less dense sublithospheric mantle. We use a linear viscous rheology for the mantle in this case. Parts of the lithosphere are viscous, others brittle. The values of the dynamic viscosity are different for lithosphere and mantle. The brittle behaviour of parts of the lithosphere can be modelled in the continuum limit by using a viscoplastic rheology. Turning to the largest planetary scale, we present an outline of the mechanics of unified models plate-mantle models and then show how computational solutions can be obtained for such models using Escript. The consequent results for different types of convection are presented and the stability of the observed flow patterns with respect to different initial conditions and computational resolutions is discussed.