[1] The shallow shelf approximation, a balance of membrane stresses for ice flow, is an effective ''sliding law'' for ice sheet modeling. Our use of it as a sliding law becomes a standard model for ice stream flow when the sliding velocity is large (100 m a À1 and faster). Following Schoof (2006a), we describe the basal resistance as plastic till for which the yield stress is given by a Mohr-Coulomb formula. Pore water pressure is related to basal melt rate. The velocity field used in the mass continuity and conservation of energy equations is an average of velocities from the shallow shelf approximation and the nonsliding shallow ice approximation. Using this scheme, our model has realistic, time-dependent ice streams which exhibit the range of surface velocities seen in actual ice streams. We demonstrate the model at high spatial resolution (5 km grid) over multiple millenia using its implementation in the Parallel Ice Sheet Model. Numerical experiments show that the entire scheme is stable with respect to many parameter changes. Some experiments reveal significant ice stream variability in a hypothetical steady climate, with characteristic cycles on the order of 1000 years. We believe this is the first practical whole ice sheet model with a unified treatment of vertical shear stresses and membrane stresses. It is capable of high-resolution, thermomechanically coupled, multimillenia simulations of ice sheets containing ice streams.Citation: Bueler, E., and J. Brown (2009), Shallow shelf approximation as a ''sliding law'' in a thermomechanically coupled ice sheet model,

[1] We used time-lapse imagery, seismic and audio recordings, iceberg and glacier velocities, ocean wave measurements, and simple theoretical considerations to investigate the interactions between Jakobshavn Isbrae and its proglacial ice mélange. The mélange behaves as a weak, granular ice shelf whose rheology varies seasonally. Sea ice growth in winter stiffens the mélange matrix by binding iceberg clasts together, ultimately preventing the calving of full-glacier-thickness icebergs (the dominant style of calving) and enabling a several kilometer terminus advance. Each summer the mélange weakens and the terminus retreats. The mélange remains strong enough, however, to be largely unaffected by ocean currents (except during calving events) and to influence the timing and sequence of calving events. Furthermore, motion of the mélange is highly episodic: between calving events, including the entire winter, it is pushed down fjord by the advancing terminus (at $40 m d À1 ), whereas during calving events it can move in excess of 50 Â 10 3 m d À1 for more than 10 min. By influencing the timing of calving events, the mélange contributes to the glacier's several kilometer seasonal advance and retreat; the associated geometric changes of the terminus area affect glacier flow. Furthermore, a force balance analysis shows that large-scale calving is only possible from a terminus that is near floatation, especially in the presence of a resistive ice mélange. The net annual retreat of the glacier is therefore limited by its proximity to floatation, potentially providing a physical mechanism for a previously described near-floatation criterion for calving.

Comparison of numerically computed solutions to exact (analytical) time-dependent solutions, when possible, is superior to intercomparison as a technique for verification of numerical models. At least two sources of such exact solutions exist for the isothermal shallow ice-sheet equation: similarity solutions and solutions with 'compensatory accumulation'. In this paper, we derive new similarity solutions with non-zero accumulation. We also derive exact solutions with (i) sinusoidalin-time accumulation and (ii) basal sliding. A specific test suite based on these solutions is proposed and used to verify a standard explicit finite-difference method. This numerical scheme is shown to reliably track the position of a moving margin while being characterized by relatively large thickness errors near the margin. The difficulty of approximating the margin essentially explains the rate of global convergence of the numerical method. A transformed version of the ice-sheet equation eliminates the singularity of the margin shape and greatly accelerates the convergence. We also use an exact solution to verify an often-used numerical approximation for basal sliding and we discuss improvements of existing benchmarks.

We describe exact solutions to the thermomechanically coupled shallow-ice approximation in three spatial dimensions. Although artificially constructed, these solutions are very useful for testing numerical methods. In fact, they allow us to verify a finite-difference scheme, that is, to show that the results of our numerical scheme converge to the correct continuum values as the grid is refined in three dimensions. Comparison of numerical results with exact solutions has helped us to precisely quantify and understand some of the numerical errors we are making. Our verified numerical scheme shows the basal temperature spokes which arose in the EISMINT (European Ice Sheet Modelling INiTiative) II intercomparison (Payne and others, 2000). A careful analysis describes these warm spokes as numerical errors which occur when the derivative of the strain-heating term with respect to the temperature is large. On the other hand, the appearance of basal temperature spokes in a verified numerical scheme strongly suggests that they are a feature of the EISMINT II experiment F continuum problem. In fact, they are clear evidence of an unstable equilibrium point of the continuum problem. This paper is a sequel to Bueler and others (2005) which addresses exact solutions and verification in the isothermal case.

International audienceIn this paper we describe a computational methodology that is specifically designed for studying three-dimensional geodynamic processes governed by heterogeneous visco-plastic Stokes flow. The method employs a hybrid spatial discretization consisting of a View the MathML sourceQ2-P1disc mixed finite element formulation for the Stokes problem, coupled to a material-point formulation which is used for representing material state and history-dependent variables. The applicability and practicality of this methodology is realized through the development of an efficient, scalable and robust variable viscosity Stokes preconditioner. In this work, these objectives are achieved through exploiting matrix-free operators and a geometric multigrid preconditioner employing hybrid coarse level operators, Chebyshev smoothers and hybrid Krylov coarse level solvers. The robustness and parallel efficiency of this strategy is demonstrated using an idealized geodynamic model. Lastly, we apply the new methodology to study geodynamic models of continental rifting and break-up in order to understand the diverse range of passive continental margins we observe on Earth today