Effects of numerical implementations of the impenetrability condition on non-linear Stokes flow: applications to ice dynamics

📝 Abstract

The basal sliding of glaciers and ice sheets can constitute a large part of the total observed ice velocity, in particular in dynamically active areas. It is therefore important to accurately represent this process in numerical models. The condition that the sliding velocity should be tangential to the bed is realized by imposing an impenetrability condition at the base. We study the, in glaciological literature used, numerical implementations of the impenetrability condition for non-linear Stokes flow with Navier’s slip on the boundary. Using the finite element method, we enforce impenetrability by: a local rotation of the coordinate system (strong method), a Lagrange multiplier method enforcing zero average flow across each facet (weak method) and an approximative method that uses the pressure variable as a Lagrange multiplier for both incompressibility and impenetrability. An analysis of the latter shows that it relaxes the incompressibility constraint, but enforces impenetrability approximately if the pressure is close to the normal component of the stress at the bed. Comparing the methods numerically using a method of manufactured solutions unexpectedly leads to similar convergence results. However, we find that, for more realistic cases, in areas of high sliding or varying topography the velocity field simulated by the approximative method differs from that of the other methods by $\sim 1\%$ (two-dimensional flow) and $> 5\%$ when compared to the strong method (three-dimensional flow). In this study the strong method, which is the most commonly used in numerical ice sheet models, emerges as the preferred method due to its stable properties (compared to the weak method in three dimensions) and ability to well enforce the impenetrability condition.

💡 Analysis

The basal sliding of glaciers and ice sheets can constitute a large part of the total observed ice velocity, in particular in dynamically active areas. It is therefore important to accurately represent this process in numerical models. The condition that the sliding velocity should be tangential to the bed is realized by imposing an impenetrability condition at the base. We study the, in glaciological literature used, numerical implementations of the impenetrability condition for non-linear Stokes flow with Navier’s slip on the boundary. Using the finite element method, we enforce impenetrability by: a local rotation of the coordinate system (strong method), a Lagrange multiplier method enforcing zero average flow across each facet (weak method) and an approximative method that uses the pressure variable as a Lagrange multiplier for both incompressibility and impenetrability. An analysis of the latter shows that it relaxes the incompressibility constraint, but enforces impenetrability approximately if the pressure is close to the normal component of the stress at the bed. Comparing the methods numerically using a method of manufactured solutions unexpectedly leads to similar convergence results. However, we find that, for more realistic cases, in areas of high sliding or varying topography the velocity field simulated by the approximative method differs from that of the other methods by $\sim 1\%$ (two-dimensional flow) and $> 5\%$ when compared to the strong method (three-dimensional flow). In this study the strong method, which is the most commonly used in numerical ice sheet models, emerges as the preferred method due to its stable properties (compared to the weak method in three dimensions) and ability to well enforce the impenetrability condition.

📄 Content

Effects of numerical implementations of the im- penetrability condition on non-linear Stokes flow: applications to ice dynamics Christian Helanow1 1Stockholm University, Department of Physical Geography, SE-106 91 Stockholm, Sweden Abstract The basal sliding of glaciers and ice sheets can constitute a large part of the total observed ice velocity, in particular in dynamically active areas. It is therefore important to accu- rately represent this process in numerical models. The condition that the sliding velocity should be tangential to the bed is realized by imposing an impenetrability condition at the base. We study the, in glaciological literature used, numerical implementations of the impenetrability condition for non-linear Stokes flow with Navier’s slip on the boundary. Using the finite element method, we enforce impenetrability by: a local rotation of the coordinate system (strong method), a Lagrange multiplier method enforcing zero average flow across each facet (weak method) and an approximative method that uses the pres- sure variable as a Lagrange multiplier for both incompressibility and impenetrability. An analysis of the latter shows that it relaxes the incompressibility constraint, but enforces impenetrability approximately if the pressure is close to the normal component of the stress at the bed. Comparing the methods numerically using a method of manufactured solutions unexpectedly leads to similar convergence results. However, we find that, for more realistic cases, in areas of high sliding or varying topography the velocity field simu- lated by the approximative method differs from that of the other methods by ∼1% (two dimensional flow) and > 5% when compared to the strong method (three-dimensional flow). In this study the strong method, which is the most commonly used in numerical ice sheet models, emerges as the preferred method due to its stable properties (compared to the weak method in three dimensions) and ability to well enforce the impenetrability condition. 1 Introduction An accurate representation of the dynamics of ice sheets and glaciers is an impor- tant component of increasing our understanding about past and future climate. Glaciers and ice sheets currently contribute to sea-level rise (Church et al., 2013) and can influence large scale weather patterns and ocean circulation (Clark et al., 1999), as well as playing vital roles in triggering abrupt climatic events in the past (Heinrich, 1988). The physical domain that glaciers occupy and how much mass they store or release therefore becomes intricately linked to the climate and possible feedback effects (Zhang et al., 2014). Ice, through a constitutive relation, can be considered to be an incompressible (singular) power-law fluid of very viscous type (Glen, 1955). The dynamics can then, with minimal simplifications, be described as a gravity driven free surface flow governed by the the non-linear Stokes equations. These partial differential equations are considered to be the most accurate representation of the physics of ice deformation, and are often called the Full Stokes (FS) equations in the context of the ice modeling community and are solved for the glacier velocity and pressure fields. The use of numerical models has become an indispensable tool to glaciologists 1 and climate scientists, both to understand paleo-ice sheets and for prognostic simulations of climate. Today, a multitude of models of various complexity exist and multiple of these use the framework of finite element methods (FEM) to solve the FS equations, e.g. Elmer/Ice (Gagliardini et al., 2013), VarGlaS (Brinkerhoff and Johnson, 2013) and ISSM (Larour et al., 2012). The FS equations describe the flow of ice that is due to internal deformation, however when considering the total velocity distribution of ice, aspects of how the glacier slides over the underlying substrate and how this can deform plays an important role (e.g. Cuffey and Paterson, 2010). It is not uncommon that sliding and substrate deformation dominates the total movement of the ice. For instance, Hooke et al. (1997) estimated that the sliding speed at Storglaciären (valley glacier in NW Sweden) accounts for over 85% of the total (surface) veloc- ity, with a similar value given for a land terminating part of the Greenland Ice Sheet (Sole et al., 2013). In general these processes, summed up by the sliding velocity (to be solved for), become a part of the boundary condition necessary to close and solve the partial differential equations. In conjunction with the sliding boundary condition, an impenetrability condition is also specified (i.e. the ice cannot penetrate the bed making velocity tangential). The focus of this study is on different implementations of the impenetrabil- ity condition in FEM and if, or how, the velocity and pressure distribution is affected by three different methods; a strongly imposed, a weakly imposed and an approximative method. We use manufactured solutions from Sargent and Fastook (2010)

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