Editorial Type:
Article
Article Type:
Research Article

Assessing the Numerical Accuracy of Complex Spherical Shallow-Water Flows

Ali R. Mohebalhojeh School of Mathematics, University of St Andrews, North Haugh, St Andrews, Scotland, and Institute of Geophysics, University of Tehran, Tehran, Iran

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David G. Dritschel School of Mathematics, University of St Andrews, St Andrews, Scotland

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Print Publication:
01 Nov 2007
DOI:
https://doi.org/10.1175/2007MWR2036.1
Page(s):
3876–3894
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Abstract

The representation of nonlinear shallow-water flows poses severe challenges for numerical modeling. The use of contour advection with contour surgery for potential vorticity (PV) within the contour-advective semi-Lagrangian (CASL) algorithm makes it possible to handle near-discontinuous distributions of PV with an accuracy beyond what is accessible to conventional algorithms used in numerical weather and climate prediction. The emergence of complex distributions of the materially conserved quantity PV, in the absence of forcing and dissipation, results from large-scale shearing and deformation and is a common feature of high Reynolds number flows in the atmosphere and oceans away from boundary layers. The near-discontinuous PV in CASL sets a limit on the actual numerical accuracy of the Eulerian, grid-based part of CASL. For the spherical shallow-water equations, the limit is studied by comparing the accuracy of CASL algorithms with second-order-centered, fourth-order-compact, and sixth-order-supercompact finite differencing in latitude in conjunction with a spectral treatment in longitude. The comparison is carried out on an unstable midlatitude jet at order one Rossby number and low Froude number that evolves into complex vortical structures with sharp gradients of PV. Quantitative measures of global conservation of energy and angular momentum, and of imbalance as diagnosed using PV inversion by means of Bolin–Charney balance, indicate that fourth-order differencing attains the highest numerical accuracy achievable for such nonlinear, advectively dominated flows.

Corresponding author address: A. R. Mohebalhojeh, School of Mathematics, University of St Andrews, North Haugh, St Andrews KY16 9SS, Scotland. Email: arm@mcs.st-and.ac.uk

Abstract

The representation of nonlinear shallow-water flows poses severe challenges for numerical modeling. The use of contour advection with contour surgery for potential vorticity (PV) within the contour-advective semi-Lagrangian (CASL) algorithm makes it possible to handle near-discontinuous distributions of PV with an accuracy beyond what is accessible to conventional algorithms used in numerical weather and climate prediction. The emergence of complex distributions of the materially conserved quantity PV, in the absence of forcing and dissipation, results from large-scale shearing and deformation and is a common feature of high Reynolds number flows in the atmosphere and oceans away from boundary layers. The near-discontinuous PV in CASL sets a limit on the actual numerical accuracy of the Eulerian, grid-based part of CASL. For the spherical shallow-water equations, the limit is studied by comparing the accuracy of CASL algorithms with second-order-centered, fourth-order-compact, and sixth-order-supercompact finite differencing in latitude in conjunction with a spectral treatment in longitude. The comparison is carried out on an unstable midlatitude jet at order one Rossby number and low Froude number that evolves into complex vortical structures with sharp gradients of PV. Quantitative measures of global conservation of energy and angular momentum, and of imbalance as diagnosed using PV inversion by means of Bolin–Charney balance, indicate that fourth-order differencing attains the highest numerical accuracy achievable for such nonlinear, advectively dominated flows.

Corresponding author address: A. R. Mohebalhojeh, School of Mathematics, University of St Andrews, North Haugh, St Andrews KY16 9SS, Scotland. Email: arm@mcs.st-and.ac.uk

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  • Bates, J. R., Y. Li, A. Brandt, S. F. McCormick, and J. Ruge, 1995: A global shallow-water numerical model based on the semi-Lagrangian advection of potential vorticity. Quart. J. Roy. Meteor. Soc., 121 , 1981–2005.

    • Search Google Scholar
    • Export Citation

  • Dritschel, D. G., 1988: Contour surgery: A topological reconnection scheme for extended integrations using contour dynamics. J. Comput. Phys., 77 , 240–266.

    • Search Google Scholar
    • Export Citation

  • Dritschel, D. G., 1989: Contour dynamics and contour surgery: Numerical algorithms for extended, high-resolution modelling of vortex dynamics in two-dimensional, inviscid, incompressible flows. Comput. Phys. Rep., 10 , 77–146.

    • Search Google Scholar
    • Export Citation

  • Dritschel, D. G., 2004: A new spherical shallow-water model: Building in wave-vortex decomposition. Proc. Seminar on Recent Developments in Numerical Methods for Atmospheric and Oceanic Modelling, Reading, United Kingdom, ECMWF, 251–262. [Available online at http://www.ecmwf.int/newsevents/meetings/annual_seminar/2004.].

  • Dritschel, D. G., and M. H. P. Ambaum, 1997: A contour-advective semi-Lagrangian algorithm for the simulation of fine-scale conservative fields. Quart. J. Roy. Meteor. Soc., 123 , 1097–1130.

    • Search Google Scholar
    • Export Citation

  • Dritschel, D. G., and Á Viúdez, 2003: A balanced approach to modelling rotating stably stratified geophysical flows. J. Fluid Mech., 488 , 123–150.

    • Search Google Scholar
    • Export Citation

  • Dritschel, D. G., and M. H. P. Ambaum, 2006: The diabatic contour advective semi-Lagrangian model. Mon. Wea. Rev., 134 , 2503–2514.

  • Dritschel, D. G., and Á Viúdez, 2006: The persistence of balance in geophysical flows. J. Fluid Mech., 570 , 365–383.

  • Dritschel, D. G., L. M. Polvani, and A. R. Mohebalhojeh, 1999: The contour-advective semi-Lagrangian algorithm for the shallow-water equations. Mon. Wea. Rev., 127 , 1551–1565.

    • Search Google Scholar
    • Export Citation

  • Esfahanian, V., S. Ghader, and A. R. Mohebalhojeh, 2005: On the use of the super compact scheme for spatial differencing in numerical models of the atmosphere. Quart. J. Roy. Meteor. Soc., 131 , 2109–2129.

    • Search Google Scholar
    • Export Citation

  • Ford, R., M. E. McIntyre, and W. A. Norton, 2000: Balance and the slow quasi-manifold: Some explicit results. J. Atmos. Sci., 57 , 1236–1254.

    • Search Google Scholar
    • Export Citation

  • Fornberg, B., 1995: A pseudospectral approach for polar and spherical geometries. Soc. Indust. Appl. Math., J. Sci. Comput., 16 , 1071–1081.

    • Search Google Scholar
    • Export Citation

  • Galewsky, J., R. K. Scott, and L. M. Polvani, 2004: An initial-value problem for testing numerical models of the global shallow-water equations. Tellus, 56A , 429–440.

    • Search Google Scholar
    • Export Citation

  • Hoskins, B. J., M. E. McIntyre, and A. W. Robertson, 1985: On the use and significance of isentropic potential-vorticity maps. Quart. J. Roy. Meteor. Soc., 111 , . 877–946. Corrigendum. 113 , 402–404.

    • Search Google Scholar
    • Export Citation

  • McIntyre, M. E., and W. A. Norton, 2000: Potential vorticity inversion on a hemisphere. J. Atmos. Sci., 57 , 1214–1235. Corrigendum. 58 , 949.

    • Search Google Scholar
    • Export Citation

  • Mohebalhojeh, A. R., and D. G. Dritschel, 2000: On the representation of gravity waves in numerical models of the shallow-water equations. Quart. J. Roy. Meteor. Soc., 126 , 669–688.

    • Search Google Scholar
    • Export Citation

  • Mohebalhojeh, A. R., and D. G. Dritschel, 2001: Hierarchies of balance conditions for the f-plane shallow-water equations. J. Atmos. Sci., 58 , 2411–2426.

    • Search Google Scholar
    • Export Citation

  • Mohebalhojeh, A. R., and D. G. Dritschel, 2004: Contour-advective semi-Lagrangian algorithms for many-layer primitive-equation models. Quart. J. Roy. Meteor. Soc., 130 , 347–364.

    • Search Google Scholar
    • Export Citation

  • Mohebalhojeh, A. R., and M. E. McIntyre, 2007: Local mass conservation and velocity splitting in PV-based balanced models. Part I: The hyperbalance equations. J. Atmos. Sci., 64 , 1782–1793.

    • Search Google Scholar
    • Export Citation

  • Moorthi, S., and R. W. Higgins, 1993: Application of fast Fourier transforms to the direct solution of a class of two-dimensional separable elliptic equations on the sphere. Mon. Wea. Rev., 121 , 290–296.

    • Search Google Scholar
    • Export Citation

  • Nihei, T., and K. Ishii, 2003: A fast solver of the shallow-water equations on a sphere using a combined compact difference scheme. J. Comput. Phys., 187 , 639–659.

    • Search Google Scholar
    • Export Citation

  • Polvani, L. M., D. W. Waugh, and R. A. Plumb, 1995: On the subtropical edge of the stratospheric surf zone. J. Atmos. Sci., 52 , 1288–1309.

    • Search Google Scholar
    • Export Citation

  • Rossmanith, J. A., 2006: A wave propagation method for hyperbolic systems on the sphere. J. Comput. Phys., 213 , 629–658.

  • Smith, R. K., and D. G. Dritschel, 2006: Revising the Rossby–Haurwitz wave test case with contour advection. J. Comput. Phys., 217 , 473–484.

    • Search Google Scholar
    • Export Citation

  • Spotz, W., M. A. Taylor, and P. N. Swarztrauber, 1998: Fast shallow-water equation solvers in latitude–longitude coordinates. J. Comput. Phys., 145 , 432–444.

    • Search Google Scholar
    • Export Citation

  • Thuburn, J., 1997: A PV-based shallow-water model on a hexagonal-icosahedral grid. Mon. Wea. Rev., 125 , 2328–2347.

  • Tolstykh, M. A., 2002: Vorticity-divergence semi-Lagrangian shallow-water model of the sphere based on compact finite differences. J. Comput. Phys., 179 , 180–200.

    • Search Google Scholar
    • Export Citation

  • Whitaker, J. S., 1993: A comparison of primitive and balance equation simulations of baroclinic waves. J. Atmos. Sci., 50 , 1519–1530.

  • Williamson, D. L., J. B. Drake, J. J. Hack, R. Jakob-Chien, and P. N. Swarztrauber, 1992: A standard test set for numerical approximations to the shallow-water equations in spherical equations in spherical geometry. J. Comput. Phys., 102 , 211–224.

    • Search Google Scholar
    • Export Citation
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