Geoscientific Model Development Discussions www.geosci-model-dev-discuss.net 1991-9611 1991-962X 2 1 2009 10.5194/gmdd-2-81-2009 http://www.geosci-model-dev-discuss.net/2/81/2009/ http://www.geosci-model-dev-discuss.net/2/81/2009/gmdd-2-81-2009.html http://www.geosci-model-dev-discuss.net/2/81/2009/gmdd-2-81-2009.pdf 81 158 2009-02-06 Derivation of a numerical solution of the 3D coupled velocity field for an ice sheet – ice shelf system, incorporating both full and approximate stress solutions T. J. Reerink t.reerink@uu.nl R. S. W. van de Wal P.-P. Borsboom Institute for Marine and Atmospheric Research Utrecht, Utrecht University, The Netherlands To overcome the mechanical coupling of an ice sheet with an ice shelf, one single set of velocity equations is presented covering both the sheet and the shelf. This set is obtained by applying shared sheet-shelf approximations. The hydrostatic approximation and a constant density are the only approximations that are applied to the full-Stokes momentum equations. The remaining stress terms from the momentum equations and the stress terms from the usual ice-flow law are multiplied by coefficients which can be put to zero or one, facilitating several stress approximations per domain within one model. In addition we derived a matrix format for the discretized set of the fully coupled velocity equations on a three-dimensional vertically scaled grid, in which all linear derivative terms are treated implicitly. The compact vector format of this sparse matrix equation is developed, including the boundary conditions. Acheson, D.: Elementary Fluid Dynamics, Cambridge University Press, Oxford, 1st Edn., 2003. Amestoy, P R., Duff, I S., and L'Excellent, J.-Y.: Multifrontal parallel distributed symmetric and unsymmetric solvers, Comput. Methods in Appl. Mech., 184, 501–520, 2000. Balay, S., Buschelman, K., Eijkhout, V., Gropp, W D., Kaushik, D., Knepley, M G., McInnes, L C., Smith, B F., and Zhang, H.: PETSc Users Manual, Tech. Rep. ANL-95/11 – Revision 2.1.5, Argonne National Laboratory, 2004. Deponti, A., Pennati, V., De~Biase, L., Maggi, V., and Berta, F.: A new fully three-dimensional numerical model for ice dynamics, J. Glaciol., 52, 365–376, 2006. Herterich, K.: On the flow within the transition zone between ice sheet and ice shelf, in: Dynamics of the West Antarctic ice sheet, edited by: Van der Veen, C. J. and Oerlemans, J., D. Reidel, Dordrecht, 185–202, 1987. Hutter, K.: Theoretical Glaciology, material science of ice and the mechanics of glaciers and ice sheets, D. Reidel Publishing, Dordrecht, Terra Scientific Publishing, 1983. Huybrechts, P.: The Antartic ice sheet and enviremental change: a three-dimensional modelling study, Ph.D. thesis, Alfred-Wegener-Institute for Polar and Marine Research, 1991. Mercer, J H.: West Antarctic ice sheet and CO₂ greenhouse effect: a threat of disaster, Nature, 271, 321–325, 1978. Paterson, W.: The physics of glaciers, Pergamon, Copenhagen, 3rd Edn., 1994. Pattyn, F.: Ice-sheet modelling at different spatial resolutions: focus on the grounding zone, Ann. Glaciol., 31, 211–216, 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, \doi10.1029/2002JB002329, 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., Souček, 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. Schoof, C.: Ice sheet grounding line dynamics: Steady states, stability, and hysteresis, J. Geophys. Res. (Earth Surface), 112, F03S28, \doi10.1029/2006JF000664, 2007. van Tuyll, C I., van de Wal, R S W., and Oerlemans, J.: The response of a simple Antarctic ice-flow model to temperature and sea-level fluctuations over the Cenozoic era, Ann. Glaciol., 46, 69–77, 2007. Vaughan, D G.: West Antarctic Ice Sheet collapse � the fall and rise of a paradigm, Climatic Change, 65–79, \doi10.1007/s10584-008-9448-3, 2007. Vieli, A. and Payne, A.: Assessing the ability of numerical ice sheet models to simulate grounding line migration, J. Geophys. Res., 110, F01003, \doi10.1029/2004JF000202, 2005. Zachos, J., Pagani, M., Sloan, S., Thomas, E., and Billups, K.: Trends, Rhythms, and Aberrations in Global Climate 65 Ma to Present, Science, 292, 686–693, 2001.