Toward a PV-Based Algorithm for the Dynamical Core of Hydrostatic Global Models

Ali R. Mohebalhojeh Institute of Geophysics, University of Tehran, Tehran, Iran

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Mohammad Joghataei Institute of Geophysics, University of Tehran, Tehran, and Department of Physics, Yazd University, Yazd, Iran

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David G. Dritschel School of Mathematics and Statistics, University of St. Andrews, St. Andrews, United Kingdom

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Abstract

The diabatic contour-advective semi-Lagrangian (DCASL) algorithms previously constructed for the shallow-water and multilayer Boussinesq primitive equations are extended to multilayer non-Boussinesq equations on the sphere using a hybrid terrain-following–isentropic (σθ) vertical coordinate. It is shown that the DCASL algorithms face challenges beyond more conventional algorithms in that various types of damping, filtering, and regularization are required for computational stability, and the nonlinearity of the hydrostatic equation in the σθ coordinate causes convergence problems with setting up a semi-implicit time-stepping scheme to reduce computational cost. The prognostic variables are an approximation to the Rossby–Ertel potential vorticity Q, a scaled pressure thickness, the horizontal divergence, and the surface potential temperature. Results from the DCASL algorithm in two formulations of the σθ coordinate, differing only in the rate at which the vertical coordinate tends to θ with increasing height, are assessed using the baroclinic instability test case introduced by Jablonowski and Williamson in 2006. The assessment is based on comparisons with available reference solutions as well as results from two other algorithms derived from the DCASL algorithm: one with a semi-Lagrangian solution for Q and another with an Eulerian grid-based solution procedure with relative vorticity replacing Q as the prognostic variable. It is shown that at intermediate resolutions, results comparable to the reference solutions can be obtained.

Denotes Open Access content.

Corresponding author address: Ali R. Mohebalhojeh, Institute of Geophysics, University of Tehran, P.O. Box 14155–6466, Tehran 1435944411, Iran. E-mail: amoheb@ut.ac.ir

Abstract

The diabatic contour-advective semi-Lagrangian (DCASL) algorithms previously constructed for the shallow-water and multilayer Boussinesq primitive equations are extended to multilayer non-Boussinesq equations on the sphere using a hybrid terrain-following–isentropic (σθ) vertical coordinate. It is shown that the DCASL algorithms face challenges beyond more conventional algorithms in that various types of damping, filtering, and regularization are required for computational stability, and the nonlinearity of the hydrostatic equation in the σθ coordinate causes convergence problems with setting up a semi-implicit time-stepping scheme to reduce computational cost. The prognostic variables are an approximation to the Rossby–Ertel potential vorticity Q, a scaled pressure thickness, the horizontal divergence, and the surface potential temperature. Results from the DCASL algorithm in two formulations of the σθ coordinate, differing only in the rate at which the vertical coordinate tends to θ with increasing height, are assessed using the baroclinic instability test case introduced by Jablonowski and Williamson in 2006. The assessment is based on comparisons with available reference solutions as well as results from two other algorithms derived from the DCASL algorithm: one with a semi-Lagrangian solution for Q and another with an Eulerian grid-based solution procedure with relative vorticity replacing Q as the prognostic variable. It is shown that at intermediate resolutions, results comparable to the reference solutions can be obtained.

Denotes Open Access content.

Corresponding author address: Ali R. Mohebalhojeh, Institute of Geophysics, University of Tehran, P.O. Box 14155–6466, Tehran 1435944411, Iran. E-mail: amoheb@ut.ac.ir
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  • Akmaev, R. A., 1991: A direct algorithm for convective adjustment of the vertical temperature profile for an arbitrary critical lapse rate. Mon. Wea. Rev., 119, 24992504, doi:10.1175/1520-0493(1991)119<2499:ADAFCA>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Arakawa, A., C. R. Mechoso, and C. S. Konor, 1992: An isentropic vertical coordinate model: Design and application to atmospheric frontogenesis studies. Meteor. Atmos. Phys., 50, 3145, doi:10.1007/BF01025503.

    • Search Google Scholar
    • Export Citation
  • 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, 19812005, doi:10.1002/qj.49712152810.

    • Search Google Scholar
    • Export Citation
  • Bleck, R., 1984: An isentropic coordinate model suitable for lee cyclogenesis simulation. Riv. Meteor. Aeronaut., 43, 189194.

  • Bleck, R., S. Benjamin, J. Lee, and A. E. MacDonald, 2010: On the use of an adaptive, hybrid-isentropic vertical coordinate in global atmospheric modeling. Mon. Wea. Rev., 138, 21882210, doi:10.1175/2009MWR3103.1.

    • Search Google Scholar
    • Export Citation
  • Charney, J. G., 1962: Integration of the primitive and balance equations. Proc. Int. Symp. on Numerical Weather Prediction, Tokyo, Japan, Meteorological Society of Japan, 131152.

  • Cook, A. W., and W. H. Cabot, 2005: Hyperviscosity for shock–turbulence interactions. J. Comput. Phys., 203, 379385, doi:10.1016/j.jcp.2004.09.011.

    • 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, 240266, doi:10.1016/0021-9991(88)90165-9.

    • 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, 77146, doi:10.1016/0167-7977(89)90004-X.

    • Search Google Scholar
    • Export Citation
  • Dritschel, D. G., and M. H. P. Ambaum, 1997: The diabatic contour advective semi-Lagrangian model. Quart. J. Roy. Meteor. Soc., 123, 10971130, doi:10.1002/qj.49712354015.

    • Search Google Scholar
    • Export Citation
  • Durran, D. R., 2010: Numerical Methods for Fluid Dynamics. 2nd ed. Springer, 516 pp.

  • 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, 21092129, doi:10.1256/qj.04.73.

    • Search Google Scholar
    • Export Citation
  • Galewsky, J. R., R. K. Scott, and L. M. Polvani, 2004: An initial-value problem for testing numerical models of the global shallow-water equations. Tellus, 56A, 429440, doi:10.1111/j.1600-0870.2004.00071.x.

    • Search Google Scholar
    • Export Citation
  • Hoskins, B. J., 1991: Towards a PV-θ view of the general circulation. Tellus, 43AB, 2735, doi:10.3402/tellusb.v43i4.15396.

  • 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, 877946, doi:10.1002/qj.49711147002.

    • Search Google Scholar
    • Export Citation
  • Hsu, Y.-J. G., and A. Arakawa, 1990: Numerical modeling of the atmosphere with an isentropic vertical coordinate. Mon. Wea. Rev., 118, 19331959, doi:10.1175/1520-0493(1990)118<1933:NMOTAW>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Jablonowski, C., and D. L. Williamson, 2006a: A baroclinic instability test case for atmospheric model dynamical cores. Quart. J. Roy. Meteor. Soc., 132, 29432975, doi:10.1256/qj.06.12.

    • Search Google Scholar
    • Export Citation
  • Jablonowski, C., and D. L. Williamson, 2006b: A baroclinic wave test case for dynamical cores of general circulation models: Model intercomparisons. NCAR Tech. Note NCAR/TN-469+STR, 75 pp., doi:10.5065/D6765C86.

  • Jablonowski, C., and D. L. Williamson, 2011: The pros and cons of diffusion, filters and fixers in atmospheric general circulation models. Numerical Techniques for Global Atmospheric Models, P. H. Lauritzen et al., Eds., Springer-Verlag, 381–493, doi:10.1007/978-3-642-11640-7_13.

  • Kasahara, A., 1974: Various vertical coordinate systems used for numerical weather prediction. Mon. Wea. Rev., 102, 509522, doi:10.1175/1520-0493(1974)102<0509:VVCSUF>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Konor, C. S., and A. Arakawa, 1997: Design of an atmospheric model based on a generalized vertical coordinate. Mon. Wea. Rev., 125, 16491673, doi:10.1175/1520-0493(1997)125<1649:DOAAMB>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Lauritzen, P. H., C. Jablonowski, M. A. Taylor, and R. D. Nair, 2010: Rotated versions of the Jablonowski steady-state and baroclinic wave test cases: A dynamical core intercomparison. J. Adv. Model. Earth Syst., 2, doi:10.3894/JAMES.2010.2.15.

    • Search Google Scholar
    • Export Citation
  • Lele, S. K., 1992: Compact finite difference schemes with spectral-like resolution. J. Comput. Phys., 103, 1642, doi:10.1016/0021-9991(92)90324-R.

    • Search Google Scholar
    • Export Citation
  • Li, Y., J. Ruge, J. R. Bates, and A. Brandt, 2000: A proposed adiabatic formulation of 3-dimensional global atmospheric models based on potential vorticity. Tellus, 52, 129139, doi:10.1034/j.1600-0870.2000.00004.x.

    • Search Google Scholar
    • Export Citation
  • McIntyre, M. E., and W. A. Norton, 2000: Potential vorticity inversion on a hemisphere. J. Atmos. Sci., 57, 12141235, doi:10.1175/1520-0469(2000)057<1214:PVIOAH>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • MirRokni, S. M., A. R. Mohebalhojeh, and D. G. Dritschel, 2011: Revisiting vacillations in shallow-water models of the stratosphere using potential-vorticity-based numerical algorithms. J. Atmos. Sci., 68, 10071022, doi:10.1175/2011JAS3622.1.

    • Search Google Scholar
    • Export Citation
  • Mirzaei, M., A. R. Mohebalhojeh, and F. Ahmadi-Givi, 2012: On imbalance generated by vortical flows in a two-layer spherical Boussinesq primitive equation model. J. Atmos. Sci., 69, 28192834, doi:10.1175/JAS-D-11-0318.1.

    • 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, 669688, doi:10.1002/qj.49712656314.

    • 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, 24112426, doi:10.1175/1520-0469(2001)058<2411:HOBCFT>2.0.CO;2.

    • 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, 347364, doi:10.1256/qj.03.49.

    • Search Google Scholar
    • Export Citation
  • Mohebalhojeh, A. R., and D. G. Dritschel, 2007: Assessing the numerical accuracy of complex spherical shallow water flows. Mon. Wea. Rev., 135, 38763894, doi:10.1175/2007MWR2036.1.

    • Search Google Scholar
    • Export Citation
  • Mohebalhojeh, A. R., and D. G. Dritschel, 2009: The diabatic contour-advective semi-Lagrangian algorithms for the spherical shallow water equations. Mon. Wea. Rev., 137, 29792994, doi:10.1175/2009MWR2717.1.

    • Search Google Scholar
    • Export Citation
  • Mohebalhojeh, A. R., and J. Theiss, 2011: The assessment of the equatorial counterpart of the quasi-geostrophic model. Quart. J. Roy. Meteor. Soc., 137, 13271339, doi:10.1002/qj.835.

    • Search Google Scholar
    • Export Citation
  • Neale, R. B., and Coauthors, 2012: Description of the NCAR Community Atmosphere Model (CAM 5.0). NCAR Tech. Note NCAR/TN-486+STR, 274 pp. [Available online at www.cesm.ucar.edu/models/cesm1.0/cam/docs/description/cam5_desc.pdf.]

  • Simmons, A. J., and D. M. Burridge, 1981: An energy and angular-momentum conserving vertical finite-difference scheme and hybrid vertical coordinates. Mon. Wea. Rev., 109, 758766, doi:10.1175/1520-0493(1981)109<0758:AEAAMC>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Skamarock, W. C., 2011: Kinetic energy spectra and model filters. Numerical Techniques for Global Atmospheric Models, P. H. Lauritzen et al., Eds., Springer-Verlag, 495–512, doi:10.1007/978-3-642-11640-7_14.

  • Skamarock, W. C., J. B. Klemp, M. G. Duda, L. D. Fowler, and S. H. Park, 2012: A multiscale nonhydrostatic atmospheric model using centroidal Voronoi tesselations and C-grid staggering. Mon. Wea. Rev., 140, 30903105, doi:10.1175/MWR-D-11-00215.1.

    • Search Google Scholar
    • Export Citation
  • Smith, R. K., and D. G. Dritschel, 2006: Revisiting the Rossby–Haurwitz wave test case with contour advection. J. Comput. Phys., 217, 473484, doi:10.1016/j.jcp.2006.01.011.

    • Search Google Scholar
    • Export Citation
  • Taylor, M. A., 2011: Conservation of mass and energy for the moist atmospheric primitive equations on unstructured grids. Numerical Techniques for Global Atmospheric Models, P. H. Lauritzen et al., Eds., Springer-Verlag, 357–380, doi:10.1007/978-3-642-11640-7_12.

  • Temperton, C., 1984: Orthogonal vertical modes for a multilevel model. Mon. Wea. Rev., 112, 503509, doi:10.1175/1520-0493(1984)112<0503:OVNMFA>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Thuburn, J., 2011: Some basic dynamics relevant to the design of atmospheric model dynamical cores. Numerical Techniques for Global Atmospheric Models, P. H. Lauritzen et al., Eds., Springer-Verlag, 3–27, doi:10.1007/978-3-642-11640-7_1.

  • Toy, M. D., and D. A. Randall, 2009: Design of a nonhydrostatic atmospheric model based on a generalized vertical coordinate. Mon. Wea. Rev., 137, 23052330, doi:10.1175/2009MWR2834.1.

    • Search Google Scholar
    • Export Citation
  • Williams, P. D., 2009: A proposed modification to the Robert–Asselin time filter. Mon. Wea. Rev., 137, 25382546, doi:10.1175/2009MWR2724.1.

    • Search Google Scholar
    • Export Citation
  • Williams, P. D., 2011: The RAW filter: An improvement to the Robert–Asselin filter in semi-implicit integrations. Mon. Wea. Rev., 139, 19962007, doi:10.1175/2010MWR3601.1.

    • Search Google Scholar
    • Export Citation
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