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Adaptive volume penalization for ocean modeling

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Abstract

The development of various volume penalization techniques for use in modeling topographical features in the ocean is the focus of this paper. Due to the complicated geometry inherent in ocean boundaries, the stair-step representation used in the majority of current global ocean circulation models causes accuracy and numerical stability problems. Brinkman penalization is the basis for the methods developed here and is a numerical technique used to enforce no-slip boundary conditions through the addition of a term to the governing equations. The second aspect to this proposed approach is that all governing equations are solved on a nonuniform, adaptive grid through the use of the adaptive wavelet collocation method. This method solves the governing equations on temporally and spatially varying meshes, which allows higher effective resolution to be obtained with less computational cost. When penalization methods are coupled with the adaptive wavelet collocation method, the flow near the boundary can be well-resolved. It is especially useful for simulations of boundary currents and tsunamis, where flow near the boundary is important. This paper will give a thorough analysis of these methods applied to the shallow water equations, as well as some preliminary work applying these methods to volume penalization for bathymetry representation for use in either the nonhydrostatic or hydrostatic primitive equations.

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References

  • Adcroft A, Marshall J (1998) How slippery are piecewise-constant coastlines in numerical ocean models? Tellus A 50(1):95–108

    Article  Google Scholar 

  • Adcroft A, Hill C, Marshall J (1997) Representation of topography by shaved cells in a height coordinate ocean model. Mon Weather Rev 125:2293–315

    Article  Google Scholar 

  • Angot P (1999) Analysis of singular perturbations on the Brinkman problem for fictitious domain models of viscous flows. Math Methods Appl Sci 22:1395–412

    Article  Google Scholar 

  • Angot P, Bruneau CH, Fabrie P (1999) A penalization method to take into account obstacles in viscous flows. Numer Math 81(4):497–520

    Article  Google Scholar 

  • Arbic B, Scott R (2008) On quadratic bottom drag, geostrophic turbulence, and oceanic mesoscale eddies. J Phys Oceanogr 38(1):84–103

    Article  Google Scholar 

  • Arquis E, Caltagirone JP (1984) Sur les conditions hydrodynamiques au voisinage d’une interface milieu fluide - milieu poreux : application à la convection naturelle. C R Acad Sci Paris II 299:1–4

    Google Scholar 

  • Blackstock DT (2000) Fundamentals of physical acoustics. Wiley, New York

    Google Scholar 

  • Chen C, Liu H, Beardsley RC (2003) An unstructured grid, finite-volume, three-dimensional, primitive equations ocean model: application to coastal ocean and estuaries. J Atmos Ocean Technol 20(1):159–86

    Article  Google Scholar 

  • Chui C (1997) Wavelets: a mathematical tool for signal analysis. SIAM monographs on mathematical modeling and computation. SIAM, Philadelphia

    Google Scholar 

  • Chung C (1995) Wave propagation and scattering in computational aeroacoustics. Pennsylvania State University, Department of Aerospace Engineering

  • Collins WD, Bitz CM, Blackmon ML, Bonan GB, Bretherton CS, Carton JA, Chang P, Doney SC, Hack JJ, Henderson TB, Kiehl JT, Large WG, McKenna DS, Santer BD, Smith RD (2006) The community climate system model version 3 (CCSM3). J Clim 19:2122–43

    Article  Google Scholar 

  • Danilov S, Kivman G, Schroter J (2004) A finite element ocean model: principles and evaluation. Ocean Model 6(2):125–50

    Article  Google Scholar 

  • Daubechies I (1992) Ten lectures on wavelets. No. 61 in CBMS-NSF series in applied mathematics. SIAM, Philadelphia

    Google Scholar 

  • Donoho DL (1992) Interpolating wavelet transforms. Tech. rep. 408, Department of Statistics, Stanford University

  • Fox-Kemper B (2004) Wind-driven barotropic gyre II: effects of eddies and low interior viscosity. J Mar Res 62:195–232

    Article  Google Scholar 

  • Fox-Kemper B, Pedlosky J (2004) Wind-driven barotropic gyre I: circulation control by eddy vorticity fluxes to an enhanced removal region. J Mar Res 62:169–93

    Article  Google Scholar 

  • Fringer OB, Gerritsen M, Street RL (2006) An unstructured-grid, finite-volume, nonhydrostatic, parallel coastal ocean simulator. Ocean Model 14:139–73

    Article  Google Scholar 

  • Herrnstein A, Wickett M, Rodrigue G (2005) Structured adaptive mesh refinement using leapfrog time integration on a staggered grid for ocean models. Ocean Model 9:283304

    Article  Google Scholar 

  • Iskandarani M, Haidvogel DB, Levin J (2003) A three-dimensional spectral element model for the solution of the hydrostatic primitive equations. J Comput Phys 186(2):397–425

    Article  Google Scholar 

  • Kevlahan KR, Vasilyev OV (2005) An adaptive wavelet collocation method for fluid-structure interaction at high reynolds numbers. SIAM J Sci Comput 26:1894–1915

    Article  Google Scholar 

  • Kevlahan NKR, Vasilyev OV, Cherhabili A (2000) An adaptive wavelet method for turbulence in complex geometries. In: Deville M, Owens R (eds) 16th IMACS world congress 2000 proceedings. Lausanne - August 21–25, 2000, IMACS, vol 411–39

  • Kevlahan NKR, Alam JM, Vasilyev OV (2007) Scaling of space-time modes with Reynolds number in two-dimensional turbulence. J Fluid Mech 570:217–26

    Article  Google Scholar 

  • Liandrat J, Tchamitchian P (1990) Resolution of the 1d regularized Burgers equation using a spatial wavelet approximation. Tech. rep., NASA contractor report 187480, ICASE report 90-83. NASA Langley Research Center, Hampton VA 23665-5225

  • Liu Q, Vasilyev OV (2007) A Brinkman penalization method for compressible flow in complex geometries. J Comput Phys 227:946–66

    Article  Google Scholar 

  • Lynch DR, Ip JTC, Naimie CE, Werner FE (1996) Comprehensive coastal circulation model with application to Gulf of Maine. Cont Shelf Res 16(7):875–906

    Article  Google Scholar 

  • Mallat S (1998) A wavelet tour of signal processing. Academic Press, San Diego, CA

    Google Scholar 

  • Marshall J, Adcroft A, Hill C, Perelman L, Heisey C (1997) A finite-volume, incompressible Navier–Stokes model for studies of the ocean on parallel computer. J Geophys Res 102(C3):5753–66

    Article  Google Scholar 

  • Mittal R, Iaccarino G (2005) Immersed boundary methods. Annu Rev Fluid Mech 37:239–61

    Article  Google Scholar 

  • Pain CC, Piggott MD, Goddard AJH, Fang F, Gorman GJ, Marshall DP, Eaton MD, Power PW, de Oliveira CRE (2005) Three-dimensional unstructured mesh ocean modelling. Ocean Model 10(1–2):5–33

    Article  Google Scholar 

  • Pedlosky J (1987) Geophysical fluid dynamics. Springer, New York

    Book  Google Scholar 

  • Pedlosky J (1996) Ocean circulation theory. Springer, New York

    Google Scholar 

  • Peskin CS (2002) The immersed boundary method. Acta Numer 11:479–517

    Article  Google Scholar 

  • Popinet S, Rickard G (2007) A tree-based solver for adaptive ocean modelling. Ocean Model 16(3–4):224–49

    Article  Google Scholar 

  • Reckinger SJ, Livescu D, Vasilyev OV (2010) Adaptive wavelet collocation method simulations of Rayleigh-Taylor instability. Phys Scr T142(014064):1–6

    Google Scholar 

  • Regele JD, Vasilyev OV (2009) An adaptive wavelet-collocation method for shock computations. Int J Comput Fluid Dyn 23(7):503–18

    Article  Google Scholar 

  • Schneider K, Vasilyev OV (2010) Wavelet methods in computational fluid dynamics. Ann Rev Fluid Mech 42:473–503

    Article  Google Scholar 

  • Sweldens W (1998) The lifting scheme: a construction of second generation wavelets. SIAM J Math Anal 29(2):511–46

    Article  Google Scholar 

  • Tseng YH, Fersiger JH (2003) A ghost-cell immersed boundary method for flow in complex geometry. J Comput Phys 192(593–623)

    Google Scholar 

  • Vasilyev OV (2003) Solving multi-dimensional evolution problems with localized structures using second generation wavelets. Int J Comput Fluid Dyn 17(2):151–68

    Article  Google Scholar 

  • Vasilyev OV, Bowman C (2000) Second generation wavelet collocation method for the solution of partial differential equations. J Comput Phys 165:660–93

    Article  Google Scholar 

  • Vasilyev OV, Kevlahan NK (2005) An adaptive multilevel wavelet collocation method for elliptic problems. J Comput Phys 206(2):412–31

    Article  Google Scholar 

  • Vasilyev OV, Kevlahan NKR (2002) Hybrid wavelet collocation–Brinkman penalization method for complex geometry flows. Int J Numer Methods Fluids 40:531–38

    Article  Google Scholar 

  • Vasilyev OV, Yuen DA, Paolucci S (1997) The solution of PDEs using wavelets. Comput Phys 11(5):429–35

    Google Scholar 

  • White L (2007) Accuracy and consistency in finite element ocean modeling. PhD thesis, Universite Catholique De Louvain

Download references

Acknowledgements

This work was supported by DOE-CCPP (DE-FG02-07ER64468). BFK was supported by NSF FRG 0855010. Also, thanks to Scott Reckinger.

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Authors and Affiliations

  1. School of Engineering, Department of Mechanical Engineering, Fairfield University, 1073 North Benson Rd., Fairfield, CT, 06824, USA

    Shanon M. Reckinger

  2. Department of Mechanical Engineering, UCB 427, University of Colorado at Boulder, Boulder, CO, 80309, USA

    Oleg V. Vasilyev

  3. Department of Cooperative Institute for Research in Environmental Sciences (CIRES), and Department of Atmospheric and Oceanic Sciences, UCB 216, University of Colorado at Boulder, Boulder, CO, 80309, USA

    Baylor Fox-Kemper

Authors
  1. Shanon M. Reckinger
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  2. Oleg V. Vasilyev
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  3. Baylor Fox-Kemper
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Correspondence to Shanon M. Reckinger.

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Responsible Editor: Pierre Lermusiaux

This article is part of the Topical Collection on Multi-scale modelling of coastal, shelf and global ocean dynamics

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Reckinger, S.M., Vasilyev, O.V. & Fox-Kemper, B. Adaptive volume penalization for ocean modeling. Ocean Dynamics 62, 1201–1215 (2012). https://doi.org/10.1007/s10236-012-0555-3

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  • DOI: https://doi.org/10.1007/s10236-012-0555-3

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