The Cryosphere Discussions www.the-cryosphere-discuss.net 1994-0432 1994-0440 4 2 2010 10.5194/tcd-4-495-2010 http://www.the-cryosphere-discuss.net/4/495/2010/ http://www.the-cryosphere-discuss.net/4/495/2010/tcd-4-495-2010.html http://www.the-cryosphere-discuss.net/4/495/2010/tcd-4-495-2010.pdf 495 560 2010-04-08 Manufactured analytical solutions for isothermal full-Stokes ice sheet models A. Sargent asarge21maine.edu J. L. Fastook Climate Change Institute, University of Maine, Orono, USA We present the detailed construction of an exact solution to time-dependent and steady-state isothermal full-Stokes ice sheet problems. The solutions are constructed for two-dimensional flowline and three-dimensional full-Stokes ice sheet models with variable viscosity. The construction is done by choosing for the specified ice surface and bed a velocity distribution that satisfies both mass conservation and the kinematic boundary conditions. Then a compensatory stress term in the conservation of momentum equations and their boundary conditions is calculated to make the chosen velocity distributions as well as the chosen pressure field into exact solutions. By substituting different ice surface and bed geometry formulas into the derived solution formulas, analytical solutions for different geometries can be constructed. <br><br> The boundary conditions can be specified as essential Dirichlet conditions or as periodic boundary conditions. By changing a parameter value, the analytical solutions allow investigation of algorithms for a different range of aspect ratios as well as for different, frozen or sliding, basal conditions. The analytical solutions can also be used to estimate the numerical error of the method in the case when the effects of the boundary conditions are eliminated, that is, when the exact solution values are specified as inflow and outflow boundary conditions. Blatter, H.: Velocity and stress fields in grounded glaciers: a simple algorithm for including deviatoric stress gradients, J. Glaciol, 41, 333–344, 1995. Bueler, E. and Brown, J.: On exact solutions and numerics for cold, shallow, and thermocoupled ice sheets, Tech. rep., physics.geo-ph, arXiv:physics/0610106, 2006. Bueler, E., Lingle, C S., KallenBrown, J A., Covey, D N., and Bowman, L N.: Exact solutions and verification of numerical models for isothermal ice sheets, J. Glaciol., 51, 291–306, 2005. Bueler, E., Brown, J., and Lingle, C.: Exact solutions to the thermocoupled shallow ice approximation: effective tools for verification, J. Glaciol., 53, 499–516, 2007. der Veen, C V.: Fundamentals of glacier dynamics, A.A. Balkema, Rotterdam, 1999. Gudmundsson, G. H.: Transmission of basal variability to a glacier surface, J. Geophys. Res., 108(B5), 2253, doi:10.1029/2002JB002107, 2003. Halfar, P.: On the dynamics of the ice-sheets, J. Geophys. Res., 86(C11), 11065–11072, 1981. Halfar, P.: On the dynamics of the ice-sheets 2, J. Geophys. Res., 88(C10), 6043–6051, 1983. Hutter, K.: Time-dependent surface elevations of an ice slope, J. Glaciol., 25, 247–266, 1980. Hutter, K.: Theoretical glaciology; material science of ice and the mechanics of glaciers and ice sheets, D. Reidel, 1983. Huybrechts, P., Payne, T., Abe-Ouchi, A., Calov, R., Fastook, J., Greve, R., Hindmarsh, R., Hoydal, O., Johannesson, T., MacAyeal, D., Marsiat, I., Ritz, C., Verbitsky, M., Waddington, E., and Warner, R.: The EISMINT benchmarks for testing ice-sheet models, Chamonix, France. IGS/EISMINT International Conference on Ice Sheet Modeling, Ann. Glaciol., 23, 1–12, 1995. Johannesson, T.: Landscape of temperate ice caps, Ph.D. thesis, University of Washington, Seattle, 1992. Nye, J F.: A flow model for the polar caps of Mars, J. Glaciol., 46, 438–444, 2000. Pattyn, F.: A new three-dimensional higher-order thermomechanical ice sheet model: Basic sensitivity, ice stream development, and ice flow across subglacial lakes, J. Geophys. Res., 108(B8), 2382, doi:10.1029/2002JB002329, 2003. Pattyn, F., Perichon, L., Aschwanden, A., Breuer, B., de Smedt, B., Gagliardini, O., Gudmundsson, G. H., Hindmarsh, R. C. A., Hubbard, A., Johnson, J. V., Kleiner, T., Konovalov, Y., Martin, C., Payne, A. J., Pollard, D., Price, S., Rückamp, M., Saito, F., Soucek, O., Sugiyama, S., and Zwinger, T.: Benchmark experiments for higher-order and full-Stokes ice sheet models (ISMIP–HOM), The Cryosphere, 2, 95–108, 2008. Raymond, M. and Gudmundsson, G H.: On the relationship between surface and basal properties on glaciers, ice sheets, and ice streams, J. Geophys. Res., 110, B08411, \doi10.1029\/2005JB003681, 2005. Reeh, N.: Steady-state three-dimensional ice flow over an undulating base: first-order theory with linear ice rheology, J. Glaciol., 33, 177–185, 1987.