• Bacon, D. P., and Coauthors, 2000: A dynamically adapting weather and dispersion model: The Operational Multiscale Environment Model with Grid Adaptivity (OMEGA). Mon. Wea. Rev., 128 , 20442076.

    • Search Google Scholar
    • Export Citation
  • Behrens, J., 1996: An adaptive semi-Lagrangian advection scheme and its parallelization. Mon. Wea. Rev., 124 , 23862395.

  • Behrens, J., , and J. Zimmermann, 2000: Parallelizing an unstructured grid generator with a space-filling curve approach. Euro-Par 2000, Lecture Notes in Computer Science, Vol. 1900, A. Bode, Ed., Springer-Verlag, 815–823.

    • Search Google Scholar
    • Export Citation
  • Behrens, J., , K. Dethloff, , W. Hiller, , and A. Rinke, 2000: Evolution of small-scale filaments in an adaptive advection model for idealized tracer transport. Mon. Wea. Rev., 128 , 29762982.

    • Search Google Scholar
    • Export Citation
  • Berger, M., , and J. Oliger, 1984: Adaptive mesh refinement for hyperbolic partial differential equations. J. Comput. Phys., 53 , 484512.

    • Search Google Scholar
    • Export Citation
  • Berger, M. J., , and P. Colella, 1989: Local adaptive mesh refinement for shock hydrodynamics. J. Comput. Phys., 82 , 6484.

  • Blayo, E., , and L. Debreu, 1999: Adaptive mesh refinement for finite-difference ocean models: First experiments. J. Phys. Oceanogr., 29 , 12391250.

    • Search Google Scholar
    • Export Citation
  • Boybeyi, Z., , N. N. Ahmad, , D. P. Bacon, , T. J. Dunn, , M. S. Hall, , P. C. S. Lee, , and R. A. Sarma, 2001: Evaluation of the Operational Multiscale Environment Model with Grid Adaptivity against the European tracer experiment. J. Appl. Meteor., 40 , 15411558.

    • Search Google Scholar
    • Export Citation
  • Bryan, G. H., , J. C. Wyngaard, , and J. M. Fritsch, 2003: Resolution requirements for the simulation of deep moist convection. Mon. Wea. Rev., 131 , 23942416.

    • Search Google Scholar
    • Export Citation
  • Carpenter, R. L., , K. K. Droegemeier, , P. R. Woodward, , and C. E. Hane, 1990: Application of the Piecewise Parabolic Method (PPM) to meteorological modeling. Mon. Wea. Rev., 118 , 586612.

    • Search Google Scholar
    • Export Citation
  • Colella, P., , and P. R. Woodward, 1984: The Piecewise Parabolic Method (PPM) for gas-dynamical simulations. J. Comput. Phys., 54 , 174201.

    • Search Google Scholar
    • Export Citation
  • Dennis, J. M., 2003: Partitioning with space-filling curves on the cubed-sphere. Proc. Int. Parallel and Distributed Processing Symposium (IPDPS), Nice, France, IEEE/ACM, CD-ROM, 269a.

  • Doswell, C. A., 1984: A kinematic analysis associated with a nondivergent flow. J. Atmos. Sci., 41 , 12421248.

  • Fox-Rabinovitz, M. S., , G. L. Stenchikov, , M. J. Suarez, , and L. L. Takacs, 1997: A finite-difference GCM dynamical core with a variable-resolution stretched grid. Mon. Wea. Rev., 125 , 29432968.

    • Search Google Scholar
    • Export Citation
  • Fulton, S. R., 1997: A comparison of multilevel adaptive methods for hurricane track prediction. Electron. Trans. Numer. Anal., 6 , 120132.

    • Search Google Scholar
    • Export Citation
  • Fulton, S. R., 2001: An adaptive multigrid barotropic tropical cyclone track model. Mon. Wea. Rev., 129 , 138151.

  • Gombosi, T. I., and Coauthors, 2004: Solution adaptive MHD for space plasmas: Sun-to-Earth simulations. Comput. Sci. Eng., 6 , 1435.

  • Hubbard, M. E., , and N. Nikiforakis, 2003: A three-dimensional, adaptive, Godunov-type model for global atmospheric flows. Mon. Wea. Rev., 131 , 18481864.

    • Search Google Scholar
    • Export Citation
  • Iselin, J. P., , J. M. Prusa, , and W. J. Gutowski, 2002: Dynamic grid adaptation using the MPDATA scheme. Mon. Wea. Rev., 130 , 10261039.

    • Search Google Scholar
    • Export Citation
  • Jablonowski, C., 2004: Adaptive grids in weather and climate modeling. Ph.D. dissertation, University of Michigan, Ann Arbor, 292 pp.

  • Jablonowski, C., , M. Herzog, , J. E. Penner, , R. C. Oehmke, , Q. F. Stout, , and B. van Leer, 2004: Adaptive grids for weather and climate models. Proc. Seminar on Recent Developments in Numerical Methods for Atmosphere and Ocean Modeling, Reading, United Kingdom, ECMWF, 233–250.

  • Jakob-Chien, R., , J. J. Hack, , and D. L. Williamson, 1995: Spectral transform solutions to the shallow water test set. J. Comput. Phys., 119 , 164187.

    • Search Google Scholar
    • Export Citation
  • Kessler, M., 1999: Development and analysis of an adaptive transport scheme. Atmos. Environ., 33 , 23472360.

  • LeVeque, R. J., 2002: Finite Volume Methods for Hyperbolic Problems. Cambridge University Press, 558 pp.

  • Lin, S-J., , and R. B. Rood, 1996: Multidimensional flux-form semi-Lagrangian transport scheme. Mon. Wea. Rev., 124 , 20462070.

  • Lin, S-J., , and R. B. Rood, 1997: An explicit flux-form semi-Lagrangian shallow water model on the sphere. Quart. J. Roy. Meteor. Soc., 123 , 24772498.

    • Search Google Scholar
    • Export Citation
  • Lipscomb, W. H., , and T. D. Ringler, 2005: An incremental remapping transport scheme on a spherical geodesic grid. Mon. Wea. Rev., 133 , 23352350.

    • Search Google Scholar
    • Export Citation
  • MacNeice, P., , K. M. Olson, , C. Mobarry, , R. deFainchtein, , and C. Packer, 2000: PARAMESH: A parallel adaptive mesh refinement community toolkit. Comput. Phys. Comm., 126 , 330354.

    • Search Google Scholar
    • Export Citation
  • Nair, R., , and B. Machenhauer, 2002: The mass-conservative cell-integrated semi-Lagrangian advection scheme on the sphere. Mon. Wea. Rev., 130 , 649667.

    • Search Google Scholar
    • Export Citation
  • Nair, R., , J. Côté, , and A. Stanisforth, 1999: Cascade interpolation for semi-Lagrangian advection over the sphere. Quart. J. Roy. Meteor. Soc., 125 , 14451468.

    • Search Google Scholar
    • Export Citation
  • Nair, R. D., , J. S. Scroggs, , and F. H. M. Semazzi, 2002: Efficient conservative global transport schemes for climate and atmospheric chemistry models. Mon. Wea. Rev., 130 , 20592073.

    • Search Google Scholar
    • Export Citation
  • Nair, R. D., , J. S. Scroggs, , and F. H. M. Semazzi, 2003: A forward-trajectory global semi-Lagrangian transport scheme. J. Comput. Phys., 190 , 275294.

    • Search Google Scholar
    • Export Citation
  • Odman, M. T., , R. Mathur, , K. Alapaty, , R. K. Srivastava, , D. S. McRae, , and R. J. Yamartino, 1997: Nested and adaptive grids for multiscale air quality modeling. Next Generation Environmental Models and Computational Methods, G. Delic and M. F. Wheeler, Eds., Society for Industrial and Applied Mathematics, 59–68.

    • Search Google Scholar
    • Export Citation
  • Oehmke, R. C., 2004: High performance dynamic array structures. Ph.D. dissertation, University of Michigan, Ann Arbor, MI, 93 pp.

  • Oehmke, R. C., , and Q. F. Stout, 2001: Parallel adaptive blocks on a sphere. Proc. 10th Conf. on Parallel Processing for Scientific Computing, Portsmouth, VA, SIAM, CD-ROM.

  • Prusa, J. M., , and P. K. Smolarkiewicz, 2003: An all-scale anelastic model for geophysical flows: Dynamic grid deformation. J. Comput. Phys., 190 , 601622.

    • Search Google Scholar
    • Export Citation
  • Rančić, M., 1992: Semi-Lagrangian piecewise biparabolic scheme for two-dimensional horizontal advection of a passive scalar. Mon. Wea. Rev., 120 , 13941406.

    • Search Google Scholar
    • Export Citation
  • Rasch, P., 1994: Conservative shape-preserving two-dimensional transport on a spherical reduced grid. Mon. Wea. Rev., 122 , 13371350.

  • Sarma, A., , N. Ahmad, , D. P. Bacon, , Z. Boybeyi, , T. J. Dunn, , M. S. Hall, , and P. C. S. Lee, 1999: Application of adaptive grid refinement to plume modeling. Air Pollution VII, C. A. Brebbia, M. Jacobson, and H. Powell, Eds., Computational Mechanics Publications, 59–68. [Also available from WIT Press as Advances in Air Pollution, Vol. 7.].

    • Search Google Scholar
    • Export Citation
  • Skamarock, W. C., 1989: Truncation error estimates for refinement criteria in nested and adaptive models. Mon. Wea. Rev., 117 , 872886.

    • Search Google Scholar
    • Export Citation
  • Skamarock, W. C., , and J. B. Klemp, 1993: Adaptive grid refinements for two-dimensional and three-dimensional nonhydrostatic atmospheric flow. Mon. Wea. Rev., 121 , 788804.

    • Search Google Scholar
    • Export Citation
  • Skamarock, W. C., , J. Oliger, , and R. L. Street, 1989: Adaptive grid refinements for numerical weather prediction. J. Comput. Phys., 80 , 2760.

    • Search Google Scholar
    • Export Citation
  • Srivastava, R. K., , D. S. McRae, , and M. T. Odman, 2000: An adaptive grid algorithm for air-quality modeling. J. Comput. Phys., 165 , 437472.

    • Search Google Scholar
    • Export Citation
  • Stevens, D. E., , and S. Bretherton, 1996: A forward-in-time advection scheme and adaptive multilevel flow solver for nearly incompressible atmospheric flows. J. Comput. Phys., 129 , 284295.

    • Search Google Scholar
    • Export Citation
  • Stout, Q. F., , D. L. DeZeeuw, , T. I. Gombosi, , C. P. T. Groth, , H. G. Marshall, , and K. G. Powell, 1997: Adaptive blocks: A high-performance data structure. Proc. SC1997, San Jose, CA, ACM/IEEE, CD-ROM.

  • Taylor, M., , J. Tribbia, , and M. Iskandarani, 1997: The spectral element method for the shallow water equations on the sphere. J. Comput. Phys., 130 , 92108.

    • Search Google Scholar
    • Export Citation
  • Temperton, C., 2004: Horizontal representation by double Fourier series on the sphere. Proc. Seminar on Recent Development in Numerical Methods for Atmosphere and Ocean Modeling, Reading, United Kingdom, ECMWF, 135–137.

  • Tomlin, A., , M. Berzins, , J. Ware, , J. Smith, , and M. J. Pilling, 1997: On the use of adaptive gridding methods for modelling chemical transport from multi-scale sources. Atmos. Environ., 31 , 29452959.

    • Search Google Scholar
    • Export Citation
  • van Leer, B., 1974: Towards the ultimate conservative difference scheme. II. Monotonicity and conservation combined in a second-order scheme. J. Comput. Phys., 14 , 361370.

    • Search Google Scholar
    • Export Citation
  • van Leer, B., 1977: Towards the ultimate conservative difference scheme. IV. A new approach to numerical convection. J. Comput. Phys., 23 , 276299.

    • Search Google Scholar
    • Export Citation
  • van Leer, B., 1985: Upwind-difference methods for aerodynamic problems governed by the Euler equations. Lectures Appl. Math., 22 , 327336.

    • Search Google Scholar
    • Export Citation
  • Veldman, A. E. P., , and R. W. C. P. Verstappen, 1998: Symmetry-conserving discretization with application to the simulation of turbulent flow. Numerical Methods for Fluid Dynamics VI, M. J. Baines, Ed., Will Print, 539–545.

    • Search Google Scholar
    • Export Citation
  • Williamson, D. L., , J. B. Drake, , J. J. Hack, , R. Jakob, , and P. N. Swarztrauber, 1992: A standard test set for numerical approximations to the shallow water equations in spherical geometry. J. Comput. Phys., 102 , 211224.

    • Search Google Scholar
    • Export Citation
  • Zalesak, S. T., 1979: Fully multidimensional flux-corrected transport algorithms for fluids. J. Comput. Phys., 31 , 335362.

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Block-Structured Adaptive Grids on the Sphere: Advection Experiments

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  • 1 Department of Atmospheric, Oceanic, and Space Sciences, University of Michigan, Ann Arbor, Michigan
  • | 2 Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, Michigan
  • | 3 Department of Aerospace Engineering, University of Michigan, Ann Arbor, Michigan
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Abstract

A spherical 2D adaptive mesh refinement (AMR) technique is applied to the so-called Lin–Rood advection algorithm, which is built upon a conservative and oscillation-free finite-volume discretization in flux form. The AMR design is based on two modules: a block-structured data layout and a spherical AMR grid library for parallel computer architectures. The latter defines and manages the adaptive blocks in spherical geometry, provides user interfaces for interpolation routines, and supports the communication and load-balancing aspects for parallel applications. The adaptive grid simulations are guided by user-defined adaptation criteria. Both statically and dynamically adaptive setups that start from a regular block-structured latitude–longitude grid are supported. All blocks are logically rectangular, self-similar, and independent data units that are split into four in the event of refinement requests, thereby doubling the horizontal resolution. Grid coarsenings reverse this refinement principle. Refinement and coarsening levels are constrained so that there is a uniform 2:1 mesh ratio at all fine–coarse-grid interfaces. The adaptive advection model is tested using three standard advection tests with increasing complexity. These include the transport of a cosine bell around the sphere, the advection of a slotted cylinder, and a smooth deformational flow that describes the roll-up of two vortices. The latter two examples exhibit very sharp edges and gradients that challenge not only the numerical scheme but also the AMR approach. The adaptive simulations show that all features of interest are reliably detected and tracked with high-resolution grids. These are steered by either a threshold- or gradient-based adaptation criterion that depends on the characteristics of the advected tracer field. The additional resolution clearly helps preserve the shape and amplitude of the transported tracer while saving computing resources in comparison to uniform-grid model runs.

* Current affiliation: Geophysical Fluid Dynamics Laboratory, Princeton, New Jersey

Corresponding author address: Dr. Christiane Jablonowski, University of Michigan, Department of Atmospheric, Oceanic, and Space Sciences, 2455 Hayward St., Ann Arbor, MI 48109. Email: cjablono@umich.edu

Abstract

A spherical 2D adaptive mesh refinement (AMR) technique is applied to the so-called Lin–Rood advection algorithm, which is built upon a conservative and oscillation-free finite-volume discretization in flux form. The AMR design is based on two modules: a block-structured data layout and a spherical AMR grid library for parallel computer architectures. The latter defines and manages the adaptive blocks in spherical geometry, provides user interfaces for interpolation routines, and supports the communication and load-balancing aspects for parallel applications. The adaptive grid simulations are guided by user-defined adaptation criteria. Both statically and dynamically adaptive setups that start from a regular block-structured latitude–longitude grid are supported. All blocks are logically rectangular, self-similar, and independent data units that are split into four in the event of refinement requests, thereby doubling the horizontal resolution. Grid coarsenings reverse this refinement principle. Refinement and coarsening levels are constrained so that there is a uniform 2:1 mesh ratio at all fine–coarse-grid interfaces. The adaptive advection model is tested using three standard advection tests with increasing complexity. These include the transport of a cosine bell around the sphere, the advection of a slotted cylinder, and a smooth deformational flow that describes the roll-up of two vortices. The latter two examples exhibit very sharp edges and gradients that challenge not only the numerical scheme but also the AMR approach. The adaptive simulations show that all features of interest are reliably detected and tracked with high-resolution grids. These are steered by either a threshold- or gradient-based adaptation criterion that depends on the characteristics of the advected tracer field. The additional resolution clearly helps preserve the shape and amplitude of the transported tracer while saving computing resources in comparison to uniform-grid model runs.

* Current affiliation: Geophysical Fluid Dynamics Laboratory, Princeton, New Jersey

Corresponding author address: Dr. Christiane Jablonowski, University of Michigan, Department of Atmospheric, Oceanic, and Space Sciences, 2455 Hayward St., Ann Arbor, MI 48109. Email: cjablono@umich.edu

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