A unified approach to energy conservation and potential vorticity dynamics for arbitrarily-structured C-grids.

*(English)*Zbl 1307.76054Summary: A numerical scheme applicable to arbitrarily-structured C-grids is presented for the nonlinear shallow-water equations. By discretizing the vector-invariant form of the momentum equation, the relationship between the nonlinear Coriolis force and the potential vorticity flux can be used to guarantee that mass, velocity and potential vorticity evolve in a consistent and compatible manner. Underpinning the consistency and compatibility of the discrete system is the construction of an auxiliary thickness equation that is staggered from the primary thickness equation and collocated with the vorticity field. The numerical scheme also exhibits conservation of total energy to within time-truncation error. Simulations of the standard shallow-water test cases confirm the analysis and show convergence rates between 1st- and 2nd-order accuracy when discretizing the system with quasi-uniform spherical Voronoi diagrams. The numerical method is applicable to a wide class of meshes, including latitude-longitude grids, Voronoi diagrams, Delaunay triangulations and conformally-mapped cubed-sphere meshes.

##### MSC:

76M12 | Finite volume methods applied to problems in fluid mechanics |

76B15 | Water waves, gravity waves; dispersion and scattering, nonlinear interaction |

65M08 | Finite volume methods for initial value and initial-boundary value problems involving PDEs |

65M50 | Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs |

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\textit{T. D. Ringler} et al., J. Comput. Phys. 229, No. 9, 3065--3090 (2010; Zbl 1307.76054)

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##### References:

[1] | Adcroft, A.; Campin, J.; Hill, C.; Marshall, J., Implementation of an atmosphere – ocean general circulation model on the expanded spherical cube, Monthly weather review, (2004) |

[2] | Arakawa, A.; Lamb, V., A potential enstrophy and energy conserving scheme for the shallow water equations, Monthly weather review, (1981) |

[3] | A. Arakawa, Y.J.G. Hsu, Energy conserving and potential-enstrophy dissipating schemes for the shallow-water equations, Monthly Weather Review, 118(10) (1960, 1969) 1981. |

[4] | Arakawa, A.; Lamb, V.R., Computational design of the basic dynamical processes in the ucla general circulation model, Methods in computational physics, 17, 173-265, (1977) |

[5] | Aurenhammer, F., Voronoi diagrams - A survey of a fundamental geometric data structure, ACM computing surveys, 23, 245, 405, (1991) |

[6] | Bonaventura, L.; Ringler, T., Analysis of discrete shallow-water models on geodesic Delaunay grids with c-type staggering, Monthly weather review, (2005) |

[7] | Bretherton, F.P., Critical layer instability in baroclinic flows, Quarterly journal of the royal meteorological society, 92, 393, (1966) |

[8] | Cote, J., A Lagrange multiplier approach for the metric terms of semi-Lagrangian models on the sphere, Quarterly journal of the royal meteorological society, 114, 483, (1988) |

[9] | Du, Q.; Faber, V.; Gunzburger, M., Centroidal Voronoi tessellations: applications and algorithms, SIAM review, 41, 4, 637-676, (1999) · Zbl 0983.65021 |

[10] | Du, Q.; Gunzburger, M.D.; Ju, L., Constrained centroidal Voronoi tessellations for surfaces, SIAM journal on scientific computing, 24, 5, 488-1506, (2003) · Zbl 1036.65101 |

[11] | Du, Q.; Gunzburger, M.; Ju, L., Voronoi-based finite volume methods, optimal Voronoi meshes, and pdes on the sphere, Computer methods in applied mechanics and engineering, 192, 35-36, 3933-3957, (2003) · Zbl 1046.65094 |

[12] | Ham, D.A.; Kramer, S.C.; Stelling, G.S.; Pietrzak, J., The symmetry and stability of unstructured mesh C-grid shallow water models under the influence of Coriolis, Ocean modelling, 16, 1-2, (2007) |

[13] | Heikes, R.; Randall, D., Numerical integration of the shallow-water equations on a twisted icosahedral grid. part i: basic design and results of tests, Monthly weather review, 123, 1862-1880, (1995) |

[14] | Hoskins, B.J.; McIntyre, M.E.; Robertson, A.W., On the use and significance of isentropic potential vorticity maps, Quarterly journal of the royal meteorological society, 111, 466, (1985) |

[15] | Kleptsova, O.; Pietrzak, J.; Stelling, G., On the accurate and stable reconstruction of tangential velocities in C-grid Ocean models, Ocean modelling, 28, (2009) |

[16] | Lin, S.J.; Rood, R.B., An explicit flux-form semi-Lagrangian shallow-water model on the sphere, Quarterly journal of the royal meteorological society, 123, 544, 2477-2498, (1997) |

[17] | Lipscomb, W.; Ringler, T., An incremental remapping transport scheme on a spherical geodesic grid, Monthly weather review, (2005) |

[18] | Marshall, J.; Olbers, D.; Ross, H.; Wolf-Gladrow, D., Potential vorticity constraints on the dynamics and hydrography of the southern Ocean, Journal of physical oceanography, 23, 3, 465-487, (1993) |

[19] | McWilliams, J., A note on a consistent quasigeostrophic model in a multiply connected domain, Dynamics of atmospheres and oceans, 1, 5, 427-441, (1977) |

[20] | Ničković, S.; Gavrilov, M.B.; Tošić, I.A., Geostrophic adjustment on hexagonal grids, Monthly weather review, 130, 3, 668-683, (2002) |

[21] | Nicolaides, R.A., Direct discretization of planar div – curl problems, SIAM journal on numerical analysis, 32-56, (1992) · Zbl 0745.65063 |

[22] | Panton, R.L., Incompressible flow, (1996), John Wiley & Sons, Inc. New York, pp. 1-840 |

[23] | Perot, B., Conservation properties of unstructured staggered mesh schemes, Journal of computational physics, 159, 1, 58-89, (2000) · Zbl 0972.76068 |

[24] | Ringler, T.D.; Heikes, R.; Randall, D., Modeling the atmospheric general circulation using a spherical geodesic grid: a new class of dynamical cores, Monthly weather review, 128, 7, 2471-2490, (2000) |

[25] | Ringler, T.D.; Randall, D.A., A potential enstrophy and energy conserving numerical scheme for solution of the shallow-water equations on a geodesic grid, Monthly weather review, 130, 5, 1397-1410, (2002) |

[26] | Sadourny, R., The dynamics of finite-difference models of the shallow-water equations, Journal of the atmospheric sciences, 32, 4, 680-689, (1975) |

[27] | Sadourny, R.; Basdevant, C., Parameterization of subgrid scale barotropic and baroclinic eddies in quasi-geostrophic models: anticipated potential vorticity method, Journal of the atmospheric sciences, 42, 13, 1353-1363, (1985) |

[28] | W. Skamarock, A linear analysis of the ncar ccsm finite-volume dynamical core, Monthly Weather Review, January 2008. |

[29] | R.D. Smith, S. Kortas, Curvilinear coordinates for global ocean models, Los Alamos National Laboratory Technical Report, LA-UR-95-1146, 1995. |

[30] | 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, Journal of computational physics, 213, 2, (2006) · Zbl 1136.86303 |

[31] | Swarztrauber, P., Spectral transform methods for solving the shallow water equations on the sphere, Monthly weather review, 124, 730-744, (1996) |

[32] | Thuburn, J., Multidimensional flux-limited advection schemes, Journal of computational physics, 123, 1, 74-83, (1996) · Zbl 0840.76063 |

[33] | Thuburn, J., A pv-based shallow-water model on a hexagonal – icosahedral grid, Monthly weather review, 125, 9, 2328-2347, (1997) |

[34] | Thuburn, J., Some conservation issues for the dynamical cores of nwp and climate models, Journal of computational physics, 227, 7, 3715-3730, (2008) · Zbl 1132.86314 |

[35] | Thuburn, J.; Ringler, T.; Klemp, J.; Skamarock, W., Numerical representation of geostrophic modes on arbitrarily structured C-grids, Journal of computational physics, 228, 8321-8335, (2009) · Zbl 1173.86304 |

[36] | Tomita, H.; Tsugawa, M.; Satoh, M.; Goto, K., Shallow water model on a modified icosahedral geodesic grid by using spring dynamics, Journal of computational physics, (2001) · Zbl 1056.76058 |

[37] | Williamson; Drake, J.; Hack, J.; Jakob, R.; Swarztrauber, P., A standard test set for numerical approximations to the shallow water equations in spherical geometry, Journal of computational physics, 102, 211-224, (1992) · Zbl 0756.76060 |

[38] | Zalesak, S.T., Fully multidimensional flux-corrected transport algorithms for fluids, Journal of computational physics, 31, 3, 335-362, (1979) · Zbl 0416.76002 |

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