Abstract
A new staggered-grid momentum-based numerical scheme that is equivalent to the non-staggered Z-grid scheme for linearized shallow water equations is proposed. Utilizing the equivalence between the staggered/non-staggered discrete vorticity-divergence fields and the discrete velocity fields, the equivalent vorticity-divergence formulations of some of the existing staggered-grid schemes, namely the C-grid, D-grid, CD-grid, and co-volume schemes are derived. The strengths and weaknesses of these schemes are discussed from the perspectives of their new formulations.
Similar content being viewed by others
References
Adcroft, A.J., Hill, C.N., Marshall, J.C.: A new treatment of the Coriolis terms in C-grid models at both high and low resolutions. Mon. Weather Rev. 127, 1928–1936 (1999)
Arakawa, A., Lamb, V.R.: Computational design of the basic dynamical processes of the UCLA General Circulation Model. Methods Comput. Phys. 17, 173–265 (1977)
Arakawa, A., Lamb, V.R.: A potential enstrophy and energy conserving scheme for the shallow water equations. Mon. Weather Rev. 109(1), 18–36 (1981)
Bonaventura, L., Ringler, T.: Analysis of discrete shallow-water models on geodesic Delaunay grids with C-type staggering. Mon. Weather Rev. 133(8), 2351–2373 (2005)
Bretherton, F.P.: Critical layer instability in baroclinic flows. Q. J. R. Meteorol. Soc. 92(393), 325–334 (1966)
Chen, Q., Ringler, T., Gunzburger, M.: A co-volume scheme for the rotating shallow water equations on conforming non-orthogonal grids. J. Comput. Phys. 240(C), 174–197 (2013)
Du, Q., Faber, V., Gunzburger, M.: Centroidal Voronoi tessellations: applications and algorithms. Siam Rev. 41(4), 637–676 (1999). (electronic)
Du, Q., Gunzburger, M.D., Ju, L.: Constrained centroidal Voronoi tessellations for surfaces. SIAM J. Sci. Comput. 24(5), 1488–1506 (2003). (electronic)
Girault, V., Raviart, P.: Finite Element Methods for Navier–Stokes Equations: Theory and Algorithms. Springer, Berlin (1986)
Heikes, R., Randall, D.A.: Numerical integration of the shallow-water equations on a twisted icosahedral grid. Part I: basic design and results of tests. Mon. Weather Rev. 123(6), 1862–1880 (1995a)
Heikes, R., Randall, D.A.: Numerical integration of the shallow-water equations on a twisted icosahedral grid. Part II. A detailed description of the grid and an analysis of numerical accuracy. Mon. Weather Rev. 123(6), 1881–1887 (1995b)
Hoskins, B., McIntyre, M., Robertson, A.: On the use and significance of isentropic potential vorticity maps. Q. J. R. Meteorol. Soc. 111(470), 877–946 (1985)
Janjić, Z.I., Mesinger, F.: Response to smallscale forcing on two staggered grids used in finitedifference models of the atmosphere. Q. J. R. Meteorol. Soc. 115(489), 1167–1176 (1989)
Marshall, J., Olbers, D., Ross, H.: Potential vorticity constraints on the dynamics and hydrography of the Southern Ocean. J. Phys. 23(3), 465–487 (1993)
Perot, B.: Conservation properties of unstructured staggered mesh schemes. J. Comput. Phys. 159(1), 58–89 (2000)
Randall, D.: Geostrophic adjustment and the finite-difference shallow-water equations. Mon. Weather Rev. 122(6), 1371–1377 (1994)
Ringler, T.D., Randall, D.A.: The ZM grid: an alternative to the Z grid. Mon. Weather Rev. 130(5), 1411–1422 (2002)
Ringler, T.D., Thuburn, J., Klemp, J.B., Skamarock, W.C.: A unified approach to energy conservation and potential vorticity dynamics for arbitrarily-structured C-grids. J. Comput. Phys. 229(9), 3065–3090 (2010)
Sadourny, R.: The dynamics of finite-difference models of the shallow-water equations. J. Atmos. Sci. 32(4), 680–689 (1975)
Skamarock, W.C.: A linear analysis of the NCAR CCSM finite-volume dynamical core. Mon. Weather Rev. 136(6), 2112–2119 (2008)
Staniforth, A., Thuburn, J.: Horizontal grids for global weather and climate prediction models: a review. Q. J. R. Meteorol. Soc. 138(662), 1–26 (2012)
Stuhne, G.R., Peltier, W.R.: A robust unstructured grid discretization for 3-dimensional hydrostatic flows in spherical geometry: a new numerical structure for ocean general circulation modeling. J. Comput. Phys. 213(2), 704–729 (2006)
Thuburn, J.: Some basic dynamics relevant to the design of atmospheric model dynamical cores. Numerical Techniques for Global Atmospheric Models, pp. 3–27. Springer, Berlin (2011)
Thuburn, J., Ringler, T., Skamarock, W., Klemp, J.: Numerical representation of geostrophic modes on arbitrarily structured C-grids. J. Comput. Phys. 228(22), 8321–8335 (2009)
Williamson, D.L., Drake, J.B., Hack, J.J., Jakob, R., Swarztrauber, P.N.: A standard test set for numerical approximations to the shallow water equations in spherical geometry. J. Comput. Phys. 102(1), 211–224 (1992)
Winninghoff. F.J.: On the adjustment toward a geostrophic balance in a simple primitive equation model with application to the problems of initialization and objective analysis. Ph.D. thesis, Thesis (PH.D.), University of California, Los Angeles (1968)
Acknowledgments
The author acknowledges the support of the Simons Foundation through the Collaboration Grants for Mathematicians.