Velocity and stress fields in grounded glaciers: a simple algorithm for including deviatoric stress gradients

Abstract

A new and efficient algorithm for computing the three-dimensional stress and velocity fields in grounded glaciers includes the role of deviatoric stress gradients. A consistent approximation of first order in the aspect of ratio of the ice mass gives a set of eight field equations for the five stress and three velocity components and the corresponding boundary conditions. A coordinate transformation mapping the local ice thickness on to unity and approximating the derivatives in the horizontal direction by centered finite-differences yields five ordinary differential and three algebraic equations. This allows use of the method of lines, starting the integration with prescribed stress and velocity components at the base, and a simple iteration procedure converges rapidly.The algorithm can be used for a wide rangе of stress-strain-rate relations, as long as strain only depends on deviatoric and shear stresses and on temperature. Sensitivity tests using synthetic and realistic ice geometries show the relevance of normal deviatoric stresses in the solutions for the velocity components even for ice sheets. Stress and velocity fields may deviate substantially from the widely used shallow-ice approximation.

An investigation of the thermomechanical features of Laohugou Glacier No. 12 on Qilian Shan, western China, using a two-dimensional first-order flow-band ice flow model

“…By assuming the vertical normal stress as hydrostatic and neglecting the bridging effects (Blatter, 1995;Pattyn, 2002), the equation for momentum balance is given as…”

Understanding and Modelling Rapid Dynamic Changes of Tidewater Outlet Glaciers: Issues and Implications

“…However, preliminary experiments of ice tongue removal at JIB using a finiteelement model that solves the full 3-D momentum equations indicate that even for an ice tongue with no lateral drag, terminus retreat resulted in flow acceleration (Luethi et al 2009). Although this needs further investigation, this could imply that in order to include realistic boundary conditions at the marine terminus more sophisticated 3-D models that include the higher-order stresses (Blatter/Pattyn type models, Blatter 1995;Pattyn 2003) or solve the full-Stokes equation are needed. Such models are becoming readily available now (Pattyn et al 2008) but are computationally far more expensive and therefore unlikely to be used for longer-term simulations on an full ice sheet scale in the near future.…”

“…Such models are becoming readily available now (Pattyn et al 2008) but are computationally far more expensive and therefore unlikely to be used for longer-term simulations on an full ice sheet scale in the near future. Thus, it will be crucial to rigorously test and assess the validity of different approximations of flow physics such as the shallow-shelf approximation (MacAyeal (1989); Bueler and Brown (2009); and as used here), a depth-integrated hybrid approximation (Bassis 2010;Schoof and Hindmarsh 2010;Goldberg 2011) or the first-order Blatter/Pattyn type approximation (Blatter 1995;Pattyn 2003) against the full-Stokes case for a marine terminating boundary and in view of computational expense. Importantly, this requires also validation against real world data.…”

Improved convergence and stability properties in a three-dimensional higher-order ice sheet model

“…This gap is filled by models that either superimpose the two shallow approximations (Bueler and Brown, 2009) or by so-called higherorder models (Blatter et al, 1995;Pattyn, 2003;Hindmarsh, 2004;Schoof and Hindmarsh, 2010). In terms of a hierarchy, higher-order models comprise the dynamics of both "shallow" approximations but simplifications to the vertical force balance reduce their complexity compared to FS models.…”

“…However, the term higher-order model is ambiguous and therefore Hindmarsh (2004) introduced a more rigorous classification. Our higher-order model is classified as including Multilayer Longitudinal Stresses (LMLa), generally referred to as the Blatter/Pattyn approximation (Blatter et al, 1995;Pattyn, 2003). In this approximation the crucial simplification is that in the vertical stress balance so-called bridging terms are neglected meaning a glaciostatic assumption.…”