User menu

Accès à distance ? S'identifier sur le proxy UCLouvain

Stereographic projection for three-dimensional global discontinuous Galerkin atmospheric modeling

Primary tabs

Blaise, Sébastien [UCL] Lambrechts, Jonathan [UCL] Deleersnijder, Eric [UCL]

A method to solve the three-dimensional compressible Navier-Stokes equations on the sphere is suggested, based on a stereographic projection with a high-order mapping of the elements from the stereographic space to the sphere. The projection is slightly modified, in order to take into account the domain thickness without introducing any approximation about the aspect ratio (deep-atmosphere). In a discontinuous Galerkin framework, the elements alongside the equator are exactly represented using a non-polynomial geometry, in order to avoid the numerical issues associated with the seam connecting the two hemispheres. This is an crucial point, necessary to avoid mass loss and spurious deviations of the velocity. The resulting model is validated on idealized three-dimensional atmospheric test cases on the sphere, demonstrating the good convergence properties of the scheme, its mass conservation, and its satisfactory behavior in terms of accuracy and low numerical dissipation. A simulation is performed on a variable resolution unstructured grid, producing accurate results despite a substantial reduction of the number of elements.

Document type Article de périodique (Journal article) – Article de recherche
Access type Accès restreint
Publication date 2015
Language Anglais
Journal information "Journal of Advances in Modeling Earth Systems" - Vol. 07 (2015), no. 3 (2015)
Peer reviewed yes
Publisher Wiley-Blackwell Publishing, Inc. ((United States) Hoboken)
issn 1942-2466
e-issn 1942-2466
Publication status Publié
Affiliation UCL - SST/IMMC/MEMA - Applied mechanics and mathematics
Links
  1. Baldauf Michael, Reinert Daniel, Zängl Günther, An analytical solution for linear gravity and sound waves on the sphere as a test for compressible, non-hydrostatic numerical models : Analytical Solution for Linear Waves on the Sphere, 10.1002/qj.2277
  2. Bernard P.-E., Remacle J.-F., Comblen R., Legat V., Hillewaert K., High-order discontinuous Galerkin schemes on general 2D manifolds applied to the shallow water equations, 10.1016/j.jcp.2009.05.046
  3. Blaise Sébastien, St-Cyr Amik, A Dynamichp-Adaptive Discontinuous Galerkin Method for Shallow-Water Flows on the Sphere with Application to a Global Tsunami Simulation, 10.1175/mwr-d-11-00038.1
  4. Blaise Sébastien, Comblen Richard, Legat Vincent, Remacle Jean-François, Deleersnijder Eric, Lambrechts Jonathan, A discontinuous finite element baroclinic marine model on unstructured prismatic meshes : Part I: space discretization, 10.1007/s10236-010-0358-3
  5. Blaise Sébastien, St-Cyr Amik, Mavriplis Dimitri, Lockwood Brian, Discontinuous Galerkin unsteady discrete adjoint method for real-time efficient tsunami simulations, 10.1016/j.jcp.2012.08.022
  6. Brdar Slavko, Baldauf Michael, Dedner Andreas, Klöfkorn Robert, Comparison of dynamical cores for NWP models: comparison of COSMO and Dune, 10.1007/s00162-012-0264-z
  7. Chen Yuejuan, Kuo H. L., A global model with overlapping mercator and stereographic grids, 10.1007/bf02678651
  8. Yinchao Chen, Mittra R., Harms P., Finite-difference time-domain algorithm for solving Maxwell's equations in rotationally symmetric geometries, 10.1109/22.506441
  9. Comblen Richard, Legrand Sébastien, Deleersnijder Eric, Legat Vincent, A finite element method for solving the shallow water equations on the sphere, 10.1016/j.ocemod.2008.05.004
  10. Comblen Richard, Blaise Sébastien, Legat Vincent, Remacle Jean-François, Deleersnijder Eric, Lambrechts Jonathan, A discontinuous finite element baroclinic marine model on unstructured prismatic meshes : Part II: implicit/explicit time discretization, 10.1007/s10236-010-0357-4
  11. Côté, Q. J. R. Meteorol. Soc., 114, 1347 (1988)
  12. Dennis , J. M. M. Levy R. D. Nair H. M. Tufo T. Voran 2005 Proceedings of the 19th IEEE International Parallel and Distributed Processing Symposium (IPDPS'05)-Workshop 13 10.1109/IPDPS.2005.438
  13. Düben Peter D., Korn Peter, Aizinger Vadym, A discontinuous/continuous low order finite element shallow water model on the sphere, 10.1016/j.jcp.2011.11.018
  14. Giraldo F.X., Restelli M., A study of spectral element and discontinuous Galerkin methods for the Navier–Stokes equations in nonhydrostatic mesoscale atmospheric modeling: Equation sets and test cases, 10.1016/j.jcp.2007.12.009
  15. Giraldo F.X., Hesthaven J.S., Warburton T., Nodal High-Order Discontinuous Galerkin Methods for the Spherical Shallow Water Equations, 10.1006/jcph.2002.7139
  16. Giraldo F. X., Restelli M., Läuter M., Semi-Implicit Formulations of the Navier–Stokes Equations: Application to Nonhydrostatic Atmospheric Modeling, 10.1137/090775889
  17. Hesthaven Jan S., Warburton Tim, Nodal Discontinuous Galerkin Methods, ISBN:9780387720654, 10.1007/978-0-387-72067-8
  18. Ivan Lucian, De Sterck Hans, Northrup Scott A., Groth Clinton P.T., Multi-dimensional finite-volume scheme for hyperbolic conservation laws on three-dimensional solution-adaptive cubed-sphere grids, 10.1016/j.jcp.2013.08.008
  19. Jablonowski Christiane, Williamson David L., A baroclinic instability test case for atmospheric model dynamical cores, 10.1256/qj.06.12
  20. Jacobson Mark Z., Fundamentals of Atmospheric Modeling : second Edition, ISBN:9781139165389, 10.1017/cbo9781139165389
  21. Kemmer N., Vector analysis : A physicist's guide to the mathematics of fields in three dimensions, ISBN:9780511569524, 10.1017/cbo9780511569524
  22. Kent James, Ullrich Paul A., Jablonowski Christiane, Dynamical core model intercomparison project: Tracer transport test cases : DCMIP tracer tests, 10.1002/qj.2208
  23. Koren, Advanced Computational Methods in Science and Engineering (2009)
  24. Lamé, Leçons sur les Coordonnées Curvilignes et leurs Diverses Applications (1859)
  25. Lauritzen Peter H., Jablonowski Christiane, Taylor Mark A., Nair Ramachandran D., Rotated Versions of the Jablonowski Steady-State and Baroclinic Wave Test Cases: A Dynamical Core Intercomparison : ROTATED TEST CASES AND DYNAMICAL CORE INTERCOMPARISONS, 10.3894/james.2010.2.15
  26. Majewski Detlev, Liermann Dörte, Prohl Peter, Ritter Bodo, Buchhold Michael, Hanisch Thomas, Paul Gerhard, Wergen Werner, Baumgardner John, The Operational Global Icosahedral–Hexagonal Gridpoint Model GME: Description and High-Resolution Tests, 10.1175/1520-0493(2002)130<0319:togihg>2.0.co;2
  27. Marras, Q. J. R. Meteorol. Soc. (2014)
  28. Martinec , Z. 2003 Continuum Mechanics Charles University in Prague Prague http://geo.mff.cuni.cz/vyuka/Martinec-ContinuumMechanics.pdf
  29. Mayne N. J., Baraffe I., Acreman D. M., Smith C., Wood N., Amundsen D. S., Thuburn J., Jackson D. R., Using the UM dynamical cores to reproduce idealised 3-D flows, 10.5194/gmd-7-3059-2014
  30. Mohseni Kamran, Colonius Tim, Numerical Treatment of Polar Coordinate Singularities, 10.1006/jcph.1999.6382
  31. Murray Ross J., Explicit Generation of Orthogonal Grids for Ocean Models, 10.1006/jcph.1996.0136
  32. Murray Ross J., Reason C.J.C., Fourier filtering and coefficient tapering at the North Pole in OGCMs, 10.1016/s1463-5003(01)00009-9
  33. Nair Ramachandran D., Jablonowski Christiane, Moving Vortices on the Sphere: A Test Case for Horizontal Advection Problems, 10.1175/2007mwr2105.1
  34. Nair Ramachandran D., Thomas Stephen J., Loft Richard D., A Discontinuous Galerkin Global Shallow Water Model, 10.1175/mwr2903.1
  35. Park Sang-Hun, Skamarock William C., Klemp Joseph B., Fowler Laura D., Duda Michael G., Evaluation of Global Atmospheric Solvers Using Extensions of the Jablonowski and Williamson Baroclinic Wave Test Case, 10.1175/mwr-d-12-00096.1
  36. Phillips N. A., A Map Projection System Suitable for Large-scale Numerical Weather Prediction, 10.2151/jmsj1923.35a.0_262
  37. Putman William M., Lin Shian-Jiann, Finite-volume transport on various cubed-sphere grids, 10.1016/j.jcp.2007.07.022
  38. Rahman, High Order Finite Elements for Microsystems Simulation (2008)
  39. Rančić M., Purser R. J., Mesinger F., A global shallow-water model using an expanded spherical cube: Gnomonic versus conformal coordinates, 10.1002/qj.49712253209
  40. Ronchi C., Iacono R., Paolucci P.S., The “Cubed Sphere”: A New Method for the Solution of Partial Differential Equations in Spherical Geometry, 10.1006/jcph.1996.0047
  41. SADOURNY ROBERT, Conservative Finite-Difference Approximations of the Primitive Equations on Quasi-Uniform Spherical Grids, 10.1175/1520-0493(1972)100<0136:cfaotp>2.3.co;2
  42. Snyder, Map Projections: A Working Manual (1987)
  43. St-Cyr, Comput. Sci., 5545, 243 (2009)
  44. St-Cyr Amik, Jablonowski Christiane, Dennis John M., Tufo Henry M., Thomas Stephen J., A Comparison of Two Shallow-Water Models with Nonconforming Adaptive Grids, 10.1175/2007mwr2108.1
  45. Straka J. M., Wilhelmson Robert B., Wicker Louis J., Anderson John R., Droegemeier Kelvin K., Numerical solutions of a non-linear density current: A benchmark solution and comparisons, 10.1002/fld.1650170103
  46. Stuhne G.R., Peltier W.R., A robust unstructured grid discretization for 3-dimensional hydrostatic flows in spherical geometry: A new numerical structure for ocean general circulation modeling, 10.1016/j.jcp.2005.08.031
  47. Stuhne G.R., Peltier W.R., An unstructured C-grid based method for 3-D global ocean dynamics: Free-surface formulations and tidal test cases, 10.1016/j.ocemod.2008.11.005
  48. Ullrich Paul A., Melvin Thomas, Jablonowski Christiane, Staniforth Andrew, A proposed baroclinic wave test case for deep- and shallow-atmosphere dynamical cores : A Baroclinic Wave Test Case for Deep and Shallow-atmosphere Dynamical Cores, 10.1002/qj.2241
  49. Wood Nigel, Staniforth Andrew, White Andy, Allen Thomas, Diamantakis Michail, Gross Markus, Melvin Thomas, Smith Chris, Vosper Simon, Zerroukat Mohamed, Thuburn John, An inherently mass-conserving semi-implicit semi-Lagrangian discretization of the deep-atmosphere global non-hydrostatic equations, 10.1002/qj.2235
  50. Zängl Günther, Reinert Daniel, Rípodas Pilar, Baldauf Michael, The ICON (ICOsahedral Non-hydrostatic) modelling framework of DWD and MPI-M: Description of the non-hydrostatic dynamical core, 10.1002/qj.2378
Bibliographic reference Blaise, Sébastien ; Lambrechts, Jonathan ; Deleersnijder, Eric. Stereographic projection for three-dimensional global discontinuous Galerkin atmospheric modeling. In: Journal of Advances in Modeling Earth Systems, Vol. 07 (2015), no. 3 (2015)
Permanent URL http://hdl.handle.net/2078.1/162970