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International Journal for Multiscale Computational Engineering
IF: 1.016 5-Year IF: 1.194 SJR: 0.452 SNIP: 0.68 CiteScore™: 1.18

ISSN Print: 1543-1649
ISSN Online: 1940-4352

International Journal for Multiscale Computational Engineering

DOI: 10.1615/IntJMultCompEng.2013005024
pages 463-495


Omer San
Department of Engineering Science and Mechanics, Virginia Tech, Blacksburg, Virginia 24061, USA
Anne E. Staples
Department of Engineering Science and Mechanics, Virginia Tech, Blacksburg, Virginia 24061, USA


This paper puts forth a coarse grid projection (CGP) multiscale method to accelerate computations of quasigeostrophic (QG) models for large-scale ocean circulation. These models require solving an elliptic subproblem at each time step, which takes the bulk of the computational time. The method we propose here is a modular approach that facilitates data transfer with simple interpolations and uses black-box solvers for solving the elliptic subproblem and potential vorticity equations in the QG flow solvers. After solving the elliptic subproblem on a coarsened grid, an interpolation scheme is used to obtain the fine data for subsequent time stepping on the full grid. The potential vorticity field is then updated on the fine grid with savings in computational time due to the reduced number of grid points for the elliptic solver. The method is applied to both single-layer barotropic and two-layer stratified QG ocean models for mid-latitude oceanic basins in the beta plane, which are standard prototypes of more realistic ocean dynamics. The method is found to accelerate these computations while retaining the same level of accuracy in the fine-resolution field. A linear acceleration rate is obtained for all the cases we consider due to the efficient linear-cost, fast Fourier transform-based elliptic solver used. We expect the speed-up of the CGP method to increase dramatically for versions of the method that use other suboptimal, elliptic solvers, which are generally quadratic cost. It is also demonstrated that numerical oscillations due to lower grid resolutions, in which the Munk scales are not resolved adequately, are effectively eliminated with CGP method.


  1. Alam, J., Towards a multiscale approach for computational atmospheric modelling. DOI: 10.1175/2011MWR3533.1

  2. Allen, J., Models of wind-driven currents on the continental shelf. DOI: 10.1146/annurev.fl.12.010180.002133

  3. Arakawa, A., Computational design for long-term numerical integration of the equations of fluid motion: Two-dimensional incompressible flow. Part I. DOI: 10.1016/0021-9991(66)90015-5

  4. Barth, T., Chan, T., and Haimes, R., Multiscale and Multiresolution Methods: Theory and Applications.

  5. Berloff, P. and McWilliams, J., Large-scale, low-frequency variability in wind-driven ocean gyres. DOI: 10.1175/1520-0485(1999)029<1925:LSLFVI>2.0.CO;2

  6. Berloff, P., Kamenkovich, I., and Pedlosky, J., A mechanism of formation of multiple zonal jets in the oceans. DOI: 10.1017/S0022112009006375

  7. Berselli, L., Iliescu, T., and Layton, W., Mathematics of Large Eddy Simulation of Turbulent Flows.

  8. Brandt, A., Multi-level adaptive solutions to boundary value problems. DOI: 10.1090/S0025-5718-1977-0431719-X

  9. Brandt, A., Multiscale scientific computation: Review 2001. DOI: 10.1007/978-3-642-56205-1_1

  10. Brandt, A., Multiscale solvers and systematic upscaling in computational physics. DOI: 10.1016/j.cpc.2005.03.097

  11. Brandt, A., Principles of systematic upscaling. DOI: 10.1093/acprof:oso/9780199233854.003.0007

  12. Brezina, M., Falgout, R., MacLachlan, S., Manteuffel, T., McCormick, S., and Ruge, J., Adaptive smoothed aggregation (α SA) multigrid. DOI: 10.1137/050626272

  13. Briggs, W., Henson, V., and McCormick, S., A Multigrid Tutorial.

  14. Bryan, K., A numerical investigation of a nonlinear model of a wind-driven ocean. DOI: 10.1175/1520-0469(1963)020<0594:ANIOAN>2.0.CO;2

  15. Campin, J., Hill, C., Jones, H., and Marshall, J., Super-parameterization in ocean modeling: Application to deep convection. DOI: 10.1016/j.ocemod.2010.10.003

  16. Chang, K., Ghil, M., Ide, K., and Lai, C., Transition to aperiodic variability in a wind-driven double-gyre circulation model. DOI: 10.1175/1520-0485(2001)031<1260:TTAVIA>2.0.CO;2

  17. Cummins, P., Inertial gyres in decaying and forced geostrophic turbulence. DOI: 10.1357/002224092784797548

  18. DiBattista, M. and Majda, A., Equilibrium statistical predictions for baroclinic vortices: The role of angular momentum. DOI: 10.1007/s001620050142

  19. Dijkstra, H., Nonlinear Physical Oceanography. DOI: 10.1007/1-4020-2263-8

  20. Dijkstra, H. and Ghil, M., Low-frequency variability of the large-scale ocean circulation: A dynamical systems approach. DOI: 10.1029/2002RG000122

  21. E, W., Principles of Multiscale Modeling.

  22. E, W., Engquist, B., Li, X., Ren, W., and Vanden-Eijnden, E., The heterogeneous multiscale method: A review.

  23. Fish, J., Multiscale Methods: Bridging the Scales in Science and Engineering.

  24. Fish, J. and Chen, W., Discrete-to-continuum bridging based on multigrid principles. DOI: 10.1016/j.cma.2003.12.022

  25. Fox-Kemper, B., Reevaluating the roles of eddies in multiple barotropic wind-driven gyres. DOI: 10.1175/JPO2743.1

  26. Ghil, M., Chekroun, M., and Simonnet, E., Climate dynamics and fluid mechanics: Natural variability and related uncertainties. DOI: 10.1016/j.physd.2008.03.036

  27. Gottlieb, S. and Shu, C., Total variation diminishing Runge-Kutta schemes. DOI: 10.1090/S0025-5718-98-00913-2

  28. Greatbatch, R. and Nadiga, B., Four-gyre circulation in a barotropic model with double-gyre wind forcing. DOI: 10.1175/1520-0485(2000)030<1461:FGCIAB>2.0.CO;2

  29. Griffa, A. and Salmon, R., Wind-driven ocean circulation and equilibrium statistical mechanics. DOI: 10.1357/002224089785076235

  30. Gupta, M., Kouatchou, J., and Zhang, J., Comparison of second-and fourth-order discretizations for multigrid Poisson solvers. DOI: 10.1006/jcph.1996.5466

  31. Hackbusch, W., Multi-Grid Methods and Applications.

  32. Holland, W., The role of mesoscale eddies in the general circulation of the ocean-numerical experiments using a wind-driven quasi-geostrophic model. DOI: 10.1175/1520-0485(1978)008<0363:TROMEI>2.0.CO;2