Adventures in Modelling Mantle Convection in a Two-Dimensional Spherical Annulus and Discovering the Need for "Anti-Squeeze”

It is common to perform two-dimensional simulations of mantle convection in spherical geometry. These have commonly been performed in axisymmetric geometry, i.e. (r, theta) coordinates, but subsequently we (Hernlund and Tackley, PEPI 2008) proposed using (r, phi) spherical annulus geometry and demonstrated its usefulness for low-viscosity-contrast calculations. 

When performing scaling studies in this geometry, however, strange results that did not match what is expected from Cartesian-geometry calculations were obtained when high-viscosity features (such as slabs) were present. It turns out that this is because the geometrical restriction forces deformation that is not present in 3 dimensions. Specifically, in a 2-D spherical approximation, a downwelling is forced to contract in the plane-perpendicular direction, requiring it to extend in the two in-plane directions. In other words, it is "squeezed" in the plane-perpendicular direction.  If the downwelling has a high viscosity, as a cold slab does, then it resists this forced deformation, sinking much more slowly than in three dimensions, in which it could sink with no deformation. This can cause unrealistic behaviour and scaling relationships for high viscosity contrasts. 

This problem can be solved by subtracting the geometrically-forced deformation ("squeezing") from the strain-rate tensor when calculating the stress tensor. Specifically, components of in-plane and plane-normal strain rate that are required by and proportional to the vertical (radial) velocity are subtracted, a procedure that is here termed "anti-squeeze". It is demonstrated here that this "anti-squeeze" correction results in sinking rates and scaling relationships that are similar to those in 3-D geometry whereas without it, abnormal and physically unrealistic results can be obtained for high viscosity contrasts. This correction has been used for 2-D geometries in the code StagYY (Tackley, PEPI 2008; Hernlund and Tackley, PEPI 2008) since 2010.