Theoretical and Computational Fluid Dynamics

, Volume 28, Issue 1, pp 107–128 | Cite as

Simulation of tropical-cyclone-like vortices in shallow-water ICON-hex using goal-oriented r-adaptivity

  • Werner Bauer
  • Martin Baumann
  • Leonhard Scheck
  • Almut Gassmann
  • Vincent Heuveline
  • Sarah C. Jones
Original Article


We demonstrate how efficient r-adapted grids for the prediction of tropical cyclone (TC) tracks can be constructed with the help of goal-oriented error estimates. The binary interaction of TCs in a barotropic model is used as a test case. We perform a linear sensitivity analysis for this problem to evaluate the contribution of each grid cell to an error measure correlated with the cyclone positions. This information allows us to estimate the local grid resolution required to minimize the TC position error. An algorithm involving the solution of a Poisson problem is employed to compute how grid points should be moved such that the desired local resolution is achieved. A hexagonal shallow-water version of the next-generation numerical weather prediction and climate model ICON is used to perform model runs on these adapted grids. The results show that for adequately chosen grid adaptation parameters, the accuracy of the track prediction can be maintained even when a coarser grid is used in regions for which the estimated error contribution is low. Accurate track predictions are obtained only when a grid with high resolution consisting of cells with nearly constant size and regular shape covers the part of the domain where the estimated error contribution is large. The number of grid points required to achieve a certain accuracy in the track prediction can be decreased substantially with our approach.


Binary tropical cyclone interaction Goal-oriented r-adaptivity A posteriori error estimation Geophysical shallow-water equations Hexagonal C-grid model 


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  1. 1.
    Anzt, H., Augustin, W., Baumann, M., Bockelmann, H., Gengenbach, T., Hahn, T., Heuveline, V., Ketelaer, E., Lukarski, D., Otzen, A., Ritterbusch, S., Rocker, B., Ronnås, S., Schick, M., Subramanian, C., Weiss, J.P., Wilhelm, F.: Hiflow3—a flexible and hardware-aware parallel finite element package. In: Proceedings of the 9th Workshop on Parallel/High-Performance Object-Oriented Scientific Computing, POOSC ’10, pp. 4:1–4:6. ACM (2010)Google Scholar
  2. 2.
    Bacon D.P., Ahmad N.N., Boybeyi Z., Dunn T.J., Hall M.S., Lee P.C.S., Sarma R.A., Turner M.D., Waight K.T., Young S.H., Zack J.W.: A dynamically adapting weather and dispersion model: the operational multiscale environment model with grid adaptivity (omega). Mon. Weather Rev. 128(7), 2044–2076 (2000)CrossRefGoogle Scholar
  3. 3.
    Bangerth W., Rannacher R.: Adaptive Finite Element Methods for Differential Equations. Birkhäuser Verlag, Basel (2003)CrossRefzbMATHGoogle Scholar
  4. 4.
    Barros S.R.M., Garcia C.I.: A global semi-implicit semi-Lagrangian shallow-water model on locally refined grids. Mon. Weather Rev. 132(1), 53–65 (2004)CrossRefGoogle Scholar
  5. 5.
    Bauer, W.: Toward Goal-oriented r-adaptive Models in Geophysical Fluid Dynamics using a Generalized Discretization Approach. Ph.D. thesis, Department of Geosciences, University of Hamburg (2013)Google Scholar
  6. 6.
    Baumann, M.: Numerical Simulation of Tropical Cyclones using Goal-Oriented Adaptivity. Ph.D. thesis, Karlsruhe Institute of Technology (KIT), Engineering Mathematics and Computing Lab (EMCL) (2011)Google Scholar
  7. 7.
    Baumann, M., Heuveline, V.: Evaluation of Different Strategies for Goal Oriented Adaptivity in CFD—Part I: The Stationary Case. EMCL Preprint Series (2010)Google Scholar
  8. 8.
    Beckers M., Clercx H.J.H., van Heijst G.J.F., Verzicco R.: Dipole formation by two interacting shielded monopoles in a stratified fluid. Phys. Fluids 14(2), 704–720 (2002)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Behrens J., Bader M.: Efficiency considerations in triangular adaptive mesh refinement. Philos. Trans. R. Soc. Ser. A Math. Phys. Eng. Sci. 367(1907), 4577–4589 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Behrens J., Rakowsky N., Hiller W., Handorf D., Lauter M., Papke J., Dethloff K.: amatos: Parallel adaptive mesh generator for atmospheric and oceanic simulation. Ocean Model. 10(1–2), 171–183 (2005)CrossRefGoogle Scholar
  11. 11.
    Birchfield G.E.: Numerical prediction of hurricane movement with the use of a fine grid. J. Meteorol. 17(4), 406–414 (1960)CrossRefGoogle Scholar
  12. 12.
    Bonaventura L., Ringler T.: Analysis of discrete shallow-water models on geodesic Delaunay grids with c-type staggering. Mon. Weather Rev. 133(8), 2351–2373 (2005)CrossRefGoogle Scholar
  13. 13.
    Brand S.: Interaction of binary tropical cyclones of the western north pacific ocean. J. Appl. Meteorol. 9, 433–441 (1970)CrossRefGoogle Scholar
  14. 14.
    Budd C.J., Huang W., Russell R.D.: Adaptivity with moving grids. Acta Numerica 18, 111–241 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Buizza R., Houtekamer P.L., Pellerin G., Toth Z., Zhu Y., Wei M.: A comparison of the ECMWF, MSC, and NCEP global ensemble prediction systems. Mon. Weather Rev. 133(5), 1076–1097 (2005)CrossRefGoogle Scholar
  16. 16.
    Carpio Huertas, J.: Duality Methods for Time-Space Adaptivity to Calculate the Numerical Solution of Partial Differential Equations. Ph.D. thesis, Matemática Aplicada a la Ingeniería Industrial / E.T.S.I. Industriales (UPM) (2008)Google Scholar
  17. 17.
    Cavallo, S.M., Torn, R.D., Snyder, C., Davis, C., Wang, W., Done, J.: Evaluation of the Advanced Hurricane WRF data assimilation system for the 2009 Atlantic hurricane season. Mon. Weather Rev. 141, 523–541 (2012)Google Scholar
  18. 18.
    Chen Q., Gunzburger M., Ringler T.: A scale-invariant formulation of the anticipated potential vorticity method. Mon. Weather Rev. 139, 2614–2629 (2011)CrossRefGoogle Scholar
  19. 19.
    Chevalier, C., Pellegrini, F.: Pt-scotch: A tool for efficient parallel graph ordering. In: 4th International Workshop on Parallel Matrix Algorithms and Applications (PMAA’06), IRISA, Rennes, France (2006)Google Scholar
  20. 20.
    Davis, C., Holland, G.: Realistic simulations of intense hurricanes with the NCEP/NCAR WRF modeling system. In: 10th International Workshop on Wave Hindcasting and Forecasting and Coastal Hazard Symposium, North Shore, Hawaii (2007)Google Scholar
  21. 21.
    Dritschel D.G., Waugh D.W.: Quantification of the inelastic interaction of unequal vortices in two-dimensional vortex dynamics. Phys. Fluids 4, 1737–1744 (1992)CrossRefGoogle Scholar
  22. 22.
    Du Q., Gunzburger M.D., Ju L.: Constrained centroidal Voronoi tessellations for surfaces. SIAM J. Sci. Comput. 24(5), 1488–1506 (2002)CrossRefMathSciNetGoogle Scholar
  23. 23.
    Du Q., Gunzburger M.D., Ju L.: Voronoi-based finite volume methods, optimal Voronoi meshes, and PDEs on the sphere. Comput. Methods Appl. Mech. Eng. 192(35), 3933–3957 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Emanuel K.: Tropical cyclones. Annu. Rev. Earth Planet. Sci. 31(1), 75–104 (2003)CrossRefGoogle Scholar
  25. 25.
    Eriksson K., Estep D., Hansbo P., Johnson C.: Introduction to adaptive methods for differential equations. Acta Numerica 4, 105–158 (1995)CrossRefMathSciNetGoogle Scholar
  26. 26.
    Errico R.M., Raeder K.D.: An examination of the accuracy of the linearization of a mesoscale model with moist physics. Q. J. R. Meteorol. Soc. 125(553), 169–195 (1999)CrossRefGoogle Scholar
  27. 27.
    Fujiwhara S.: The natural tendency towards symmetry of motion and its application as a principle of motion. Q. J. R. Meteorol. Soc. 47, 287–293 (1921)CrossRefGoogle Scholar
  28. 28.
    Fujiwhara S.: On the growth and decay of vortical systems. Q. J. R. Meteorol. Soc. 49, 75–104 (1923)CrossRefGoogle Scholar
  29. 29.
    Fujiwhara S.: Short note on the behaviour of two vortices. Proc. Phys. Math. Soc. Japan Ser. 3 13, 106–110 (1931)Google Scholar
  30. 30.
    Gassmann A.: Inspection of hexagonal and triangular c-grid discretizations of the shallow water equations. J. Comput. Phys. 230(7), 2706–2721 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  31. 31.
    Gassmann A.: A global hexagonal c-grid non-hydrostatic dynamical core (ICON-IAP) designed for energetic consistency. Q. J. R. Meteorol. Soc. 139(670), 152–175 (2013)CrossRefGoogle Scholar
  32. 32.
    Giraldo F.X., Warburton T.: A nodal triangle-based spectral element method for the shallow water equations on the sphere. J. Comput. Phys. 207, 129–150 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  33. 33.
    Harrison E.J., Elsberry R.L.: A method for incorporating nested finite grids in the solution of systems of geophysical equations. J. Atmos. Sci. 29(7), 1235–1245 (1972)CrossRefGoogle Scholar
  34. 34.
    Heikes R., Randall D.A.: Numerical integration of the shallow-water equations on a twisted icosahedral grid. part ii. a detailed description of the grid and an analysis of numerical accuracy. Mon. Weather Rev. 123(6), 1881–1887 (1995)CrossRefGoogle Scholar
  35. 35.
    Heinze, T.: An Adaptive Shallow Water Model on the Sphere. PhD thesis, University of Bremen, Germany (2009)Google Scholar
  36. 36.
    Holland G.J.: Tropical cyclone motion: environmental interaction plus a beta effect. J. Atmos. Sci. 40(2), 328–342 (1983)CrossRefGoogle Scholar
  37. 37.
    Holland G.J., Dietachmayer G.S.: On the interaction of tropical-cyclone-scale vortices. iii. Continuous barotropic vortices. Q. J. R. Meteorol. Soc. 119, 1381–1398 (1993)CrossRefGoogle Scholar
  38. 38.
    Jarrell J.D., Brand S., Nicklin D.S.: An analysis of western north pacific tropical cyclone forecast errors. Mon. Weather Rev. 106, 925–937 (1978)CrossRefGoogle Scholar
  39. 39.
    Jones S.C., Harr P.A., Abraham J., Bosart L.F., Bowyer P.J., Evans J.L., Hanley D.E., Hanstrum B.N., Hart R.E., Lalaurette F., Sinclair M.R., Smith R.K., Thorncroft C.: The extratropical transition of tropical cyclones: forecast challenges, current understanding, and future directions. Weather Forecast. 18, 1052–1092 (2003)CrossRefGoogle Scholar
  40. 40.
    Karypis G., Kumar V.: A fast and high quality multilevel scheme for partitioning irregular graphs. SIAM J. Sci. Comput. 20(1), 359–392 (1998)CrossRefMathSciNetGoogle Scholar
  41. 41.
    Lander M., Holland G.J.: On the interaction of tropical-cyclone-scale vortices. i: observations. Q. J. R. Meteorol. Soc. 119, 1347–1361 (1993)CrossRefGoogle Scholar
  42. 42.
    Läuter M., Handorf D., Rakowsky N., Behrens J., Frickenhaus S., Best M., Dethloff K., Hiller W.: A parallel adaptive barotropic model of the atmosphere. J. Comput. Phys. 223(2), 609–628 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  43. 43.
    Liao G., Anderson D.: A new approach to grid generation. Appl. Anal. Int. J. 44(3), 285–298 (1992)CrossRefzbMATHMathSciNetGoogle Scholar
  44. 44.
    Liao, G., de la Pena, G., Liao, G.: A deformation method for moving grid generation. In: Proceedings, 8th International Meshing Roundtable, pp. 155–162. South Lake Tahoe, CA, USA (1999)Google Scholar
  45. 45.
    MacDonald, A.E., Middlecoff, J., Henderson, T., Lee, J.L.: A general method for modeling on irregular grids. Int. J. High Perform. Comput. Appl. 25(4), 392–403 (2011)Google Scholar
  46. 46.
    Melander M.V., McWilliams J.C., Zabusky N.J.: Symmetric vortex merger in two dimensions: causes and conditions. J. Fluid Mech. 195, 303–340 (1988)CrossRefzbMATHMathSciNetGoogle Scholar
  47. 47.
    Molinari J., Vollaro D.: Rapid intensification of a sheared tropical storm. Mon. Weather Rev. 138(10), 3869–3885 (2010)CrossRefGoogle Scholar
  48. 48.
    Moser J.: On the volume elements on a manifold. Trans. Am. Math. Soc. 120(2), 286–294 (1965)CrossRefzbMATHGoogle Scholar
  49. 49.
    Persson P.O.: Mesh size functions for implicit geometries and PDE-based gradient limiting. Eng. Comput. 22, 95–109 (2006)CrossRefGoogle Scholar
  50. 50.
    Prieto R., McNoldy B.D., Fulton S.R., Schubert W.H.: A classification of binary tropical cyclone—like vortex interactions. Mon. Weather Rev. 131, 2656–2666 (2003)CrossRefGoogle Scholar
  51. 51.
    Rauser F., Korn P., Marotzke J.: Predicting goal error evolution from near-initial-information: a learning algorithm. J. Comput. Phys. 230(19), 7284–7299 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  52. 52.
    Reynolds C.A., Peng M.S., Chen J.H.: Recurving tropical cyclones: singular vector sensitivity and downstream impacts. Mon. Weather Rev. 137, 1320–1337 (2009)CrossRefGoogle Scholar
  53. 53.
    Ringler T., Ju L., Gunzburger M.: A multiresolution method for climate system modeling: application of spherical centroidal Voronoi tessellations. Ocean Dyn. 58(5-6), 475–498 (2008)CrossRefGoogle Scholar
  54. 54.
    Ringler T.D., Jacobsen D., Gunzburger M., Ju L., Duda M., Skamarock W.: Exploring a multi-resolution modeling approach within the shallow-water equations. Mon. Weather Rev. 139, 3348–3368 (2011)CrossRefGoogle Scholar
  55. 55.
    Ringler T.D., Thuburn J., Klemp J.B., Skamarock W.C.: A unified approach to energy conservation and potential vorticity dynamics for arbitrarily-structured c-grids. J. Comput. Phys. 229, 3065–3090 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  56. 56.
    Rípodas P., Gassmann A., Förstner J., Majewski D., Giorgetta M., Korn P., Kornblueh L., Wan H., Zängl G., Bonaventura L., Heinze T.: Icosahedral shallow water model (ICOSWM): results of shallow water test cases and sensitivity to model parameters. Geosci. Model Dev. 2(2), 231–251 (2009)CrossRefGoogle Scholar
  57. 57.
    Ritchie E.A., Holland G.J.: On the interaction of two tropical cyclone scale vortices. ii: discrete vortex patches. Q. J. R Meteorol. Soc. 119, 1363–1379 (1993)CrossRefGoogle Scholar
  58. 58.
    Schieweck F.: A-stable discontinuous Galerkin-Petrov time discretization of higher order. J. Numer. Math. 18(1), 25–57 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  59. 59.
    Semper B., Liao G.: A moving grid finite-element method using grid deformation. Numer. Methods Partial Differ. Equ. 11, 603–615 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  60. 60.
    Shin S.E., Han J.Y., Baik J.J.: On the critical separation distance of binary vortices in a nondivergent barotropic atmosphere. J. Meteorol. Soc. Japan 84(5), 853–869 (2006)CrossRefGoogle Scholar
  61. 61.
    Smith R.K., Ulrich W., Dietachmayer G.: A numerical study of tropical cyclone motion using a barotropic model. I: the role of vortex asymmetries. Q. J. R. Meteorol. Soc. 116(492), 337–362 (1990)CrossRefGoogle Scholar
  62. 62.
    St-Cyr A., Jablonowski C., Dennis J.M., Tufo H.M., Thomas S.J.: A comparison of two shallow-water models with nonconforming adaptive grids. Mon. Weather Rev. 136(6), 1898–1922 (2008)CrossRefGoogle Scholar
  63. 63.
    Thuburn J., Ringler T.D., Skamarock W.C., Klemp J.B.: Numerical representation of geostrophic modes on arbitrarily structured c-grids. J. Comput. Phys. 228, 8321–8335 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  64. 64.
    Tomita H., Satoh M., Goto K.: An optimization of the icosahedral grid modified by spring dynamics. J. Comput. Phys. 183(1), 307–331 (2002)CrossRefzbMATHGoogle Scholar
  65. 65.
    Valcke S., Verron J.: Interactions of baroclinic isolated vortices: The dominant effect of shielding. J. Phys. Oceanogr. 27(4), 524–541 (1997)CrossRefGoogle Scholar
  66. 66.
    Walko R.L., Avissar R.: A direct method for constructing refined regions in unstructured conforming triangular-hexagonal computational grids: application to OLAM. Mon. Weather Rev. 139, 3923–3937 (2011)CrossRefGoogle Scholar
  67. 67.
    Wan, H.: Developing and Testing a Hydrostatic Atmospheric Dynamical Core on Triangular Grids. Ph.D. thesis, Reports on Earth System Science at International Max Planck Research School, Hamburg (2009)Google Scholar
  68. 68.
    Weller H., Weller H.G., Fournier A.: Voronoi, Delaunay, and block-structured mesh refinement for solution of the shallow-water equations on the sphere. Mon. Weather Rev. 137(12), 4208–4224 (2009)CrossRefGoogle Scholar
  69. 69.
    White, B.S., McKee, S.A., de Supinski, B.R., Miller, B., Quinlan, D., Schulz, M.: Improving the computational intensity of unstructured mesh applications. In: Proceedings of the 19th Annual International Conference on Supercomputing, ICS ’05, pp. 341–350. ACM, New York, NY, USA (2005)Google Scholar
  70. 70.
    Williamson D.L., Drake J.B., Hack J.J., Jakob R., Swarztrauber P.N.: A standard test set for numerical approximations to the shallow water equations in spherical geometry. J. Comput. Phys. 102(1), 211–224 (1992)CrossRefzbMATHMathSciNetGoogle Scholar
  71. 71.
    Wloka, J.: Partial Differential Equations, engl. ed., reprinted edn. Cambridge University Press Cambridge (1992)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Werner Bauer
    • 1
    • 5
  • Martin Baumann
    • 2
  • Leonhard Scheck
    • 3
  • Almut Gassmann
    • 4
  • Vincent Heuveline
    • 2
  • Sarah C. Jones
    • 3
    • 6
  1. 1.Max Planck Institute for MeteorologyHamburgGermany
  2. 2.Engineering Mathematics and Computing LabKarlsruhe Institute of TechnologyKarlsruheGermany
  3. 3.Institute for Meteorology and Climate ResearchKarlsruhe Institute of TechnologyKarlsruheGermany
  4. 4.Leibniz-Institute of Atmospheric PhysicsUniversity of RostockKühlungsbornGermany
  5. 5.KlimaCampusUniversity of HamburgHamburgGermany
  6. 6.Deutscher WetterdienstOffenbachGermany

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