The shallow shelf approximation as a 'sliding law' in a thermomechanically coupled ice sheet model

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. ???, XXXX, DOI:10.1029/, The shallow shelf approximation as a “sliding law” in a thermomechanically coupled ice sheet model Ed Bueler [DRAFT AS SUBMITTED; October 25, 2018] Department of Mathematics and Statistics, University of Alaska, Fairbanks, Alaska, USA Jed Brown VAW A-15, ETH Zentrum, 8092 Zurich, Switzerland Abstract. The shallow shelf approximation is a better “sliding law” for ice sheet mod- eling than those sliding laws in which basal velocity is a function of driving stress. The shallow shelf approximation as formulated by Schoof [2006a] is well-suited to this use. Our new thermomechanically coupled sliding scheme is based on a plasticity assump- tion about the strength of the saturated till underlying the ice sheet in which the till yield stress is given by a Mohr-Coulomb formula using a modeled pore water pressure. Using this scheme, our prognostic whole ice sheet model has convincing ice streams. Driv- ing stress is balanced in part by membrane stresses, the model is computable at high spatial resolution in parallel, it is stable with respect to parameter changes, and it pro- duces surface velocities seen in actual ice streams. 1. Introduction A well-known difficulty with numerical ice sheet models is their inability to model the large range of ice flow speeds observed in real ice sheets [Shepard and Wingham, 2007; Truffer and Fahnestock, 2007; Vaughan and Arthern, 2007]. Observed surface speeds for ice flow in the Greenland ice sheet, for example, range from less than 10 meters per year in large areas of the interior [compare Greve, 1997b; Joughin et al., 1997] to more than 10 km per year in three outlet glaciers [Howat et al., 2007; Joughin et al., 2004a]. Exist- ing Greenland ice sheet models have not, however, reported (published) ice surface speeds in excess of 100 m per year [Greve, 1997b, 2000; Ritz et al., 1997; Saito and Abe-Ouchi, 2005; Tarasov and Peltier, 2002]. Fast grounded ice flow, in ice streams and outlet glaciers to differing degrees [Truffer and Echelmeyer, 2003], arises from some combination of sliding, over a rigid or deformable mineral bed, and shear deformation of the lowest wet, dirty layers of ice. Unfortunately and fundamentally, however, re- mote sensing provides no high quality spatially-distributed observations of conditions at or near the ice base with which to constrain models of fast flow. There is a triple need to improve observations, to use existing surface observations more effectively, and to improve models of ice flow including sliding. This paper approaches modeling fast ice stream motion pragmatically, within the high-resolution, comprehensive, thermomechanically-coupled, and time-dependent Parallel Ice Sheet Model [“PISM”; Bueler et al., 2008]. The basal mechanical model we add here is based on a spatially- distributed till friction angle [Paterson, 1994]. We demon- strate that our model responds in a reasonable way to changes in till friction angle and other major parameter choices including grid refinement. We believe that the model is a credible model of shallow ice streams. In our model, ice sheet geometry and thermodynamical fields (ice tempera- ture and effective thickness of basal water) evolve together within a unified shallow framework. Copyright 2018 by the American Geophysical Union. 0148-0227/18/$9.00 Fast-flowing simulated ice is not useful in a model if it arises from unreasonable physics. All ice sheet models in- corporate approximations, and most models, including ours, use the actual shallowness of ice sheets to simplify the equa- tions and reduce computational cost. There are choices in parameterizing the sliding, however. We recall some con- tinuum flow models applied to ice sheets and sliding in the hierarchy in Figure 1. All the illustrated models describe ice as a slow, non-linearly viscous, isotropic fluid, though these qualities are approximations too. The simplest and shallowest models are called the shal- low ice approximation (SIA) [Hutter, 1983; Morland and Johnson, 1980] and the shallow shelf approximation [“SSA”; Morland, 1987; Weis et al., 1999]. Rigorous small-parameter arguments explain how to simplify from Stokes to “higher- order models” [Blatter, 1995; Hindmarsh, 2004], and from the Blatter [1995] model to the SIA and SSA [Schoof and Hindmarsh, submitted]. Thermomechanically coupled, shallow, grounded, and non-sliding (frozen) base ice sheet models based upon the SIA are relatively well understood [Bueler et al., 2007; Payne et al., 2000]. Large portions of actual ice sheets have bases which experience minimal sliding and have modest bed to- pography. For those parts the nonsliding SIA is a second order [Fowler, 1997] theory which predicts a reasonable dis- tribution of flow at rates which compare well to observations (e.g. [Greve, 1997b]). As noted, faster ice flow is a combination of sliding over the mineral base along with deformation of an ice-and-till layer at the base. We necessarily lump the

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