An Energy Consistent Discretization of the Nonhydrostatic Equations in Primitive Variables

Taylor, Mark A.; Guba, Oksana; Steyer, Andrew; Ullrich, Paul A.; Hall, David M.; Eldrid, Christopher

doi:10.1029/2019MS001783

Title: An Energy Consistent Discretization of the Nonhydrostatic Equations in Primitive Variables

Journal Article · 28 December 2019 · Journal of Advances in Modeling Earth Systems

DOI: https://doi.org/10.1029/2019MS001783 · OSTI ID:1631000

Taylor, Mark A. ^[1]; Guba, Oksana ^[1]; Steyer, Andrew ^[1]; Ullrich, Paul A. ^[2]; Hall, David M. ^[3]; Eldrid, Christopher ^[4]

Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
Univ. of California, Davis, CA (United States)
NVIDIA, Santa Clara, CA (United States)
Univ. Grenoble Alpes (France)

We derive a formulation of the nonhydrostatic equations in spherical geometry with a Lorenz staggered vertical discretization. The combination conserves a discrete energy in exact time integration when coupled with a mimetic horizontal discretization. The formulation is a version of Dubos and Tort (2014, https://doi.org/10.1175/MWR-D-14-00069.1) rewritten in terms of primitive variables. It is valid for terrain following mass or height coordinates and for both Eulerian or vertically Lagrangian discretizations. The discretization relies on an extension to Simmons and Burridge (1981, https://doi.org/10.1175/1520-0493(1981)109<0758:AEAAMC>2.0.CO;2) vertical differencing, which we show obeys a discrete derivative product rule. This product rule allows us to simplify the treatment of the vertical transport terms. Energy conservation is obtained via a term-by-term balance in the kinetic, internal, and potential energy budgets, ensuring an energy-consistent discretization up to time truncation error with no spurious sources of energy. We demonstrate convergence with respect to time truncation error in a spectral element code with a horizontal explicit vertically implicit implicit-explicit time stepping algorithm.

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Research Organization:: Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)

Sponsoring Organization:: USDOE Office of Science (SC), Biological and Environmental Research (BER); USDOE National Nuclear Security Administration (NNSA); French National Research Agency

Grant/Contract Number:: AC04-94AL85000; NA0003525; ANR-14-CE23-0010

OSTI ID:: 1631000

Report Number(s):: SAND2020-5242J; 686193

Journal Information:: Journal of Advances in Modeling Earth Systems, Vol. 12, Issue 1; ISSN 1942-2466

Publisher:: American Geophysical Union (AGU)Copyright Statement

Country of Publication:: United States

Language:: English

References (48)

Towards a consistent numerical compressible non-hydrostatic model using generalized Hamiltonian tools Gassmann, Almut; Herzog, Hans-Joachim Quarterly Journal of the Royal Meteorological Society, Vol. 134, Issue 635 https://doi.org/10.1002/qj.297	journal	July 2008
Optimization-based limiters for the spectral element method Guba, Oksana; Taylor, Mark; St-Cyr, Amik Journal of Computational Physics, Vol. 267 https://doi.org/10.1016/j.jcp.2014.02.029	journal	June 2014
A General Method for Conserving Energy and Potential Enstrophy in Shallow-Water Models Salmon, Rick Journal of the Atmospheric Sciences, Vol. 64, Issue 2 https://doi.org/10.1175/JAS3837.1	journal	February 2007
The DOE E3SM Coupled Model Version 1: Overview and Evaluation at Standard Resolution Golaz, Jean‐Christophe; Caldwell, Peter M.; Van Roekel, Luke P. Journal of Advances in Modeling Earth Systems, Vol. 11, Issue 7 https://doi.org/10.1029/2018MS001603	journal	July 2019
Poisson-Bracket Approach to the Construction of Energy- and Potential-Enstrophy-Conserving Algorithms for the Shallow-Water Equations Salmon, Rick Journal of the Atmospheric Sciences, Vol. 61, Issue 16 https://doi.org/10.1175/1520-0469(2004)061<2016:PATTCO>2.0.CO;2	journal	August 2004
Various Vertical Coordinate Systems Used for Numerical Weather Prediction Kasahara, Akira Monthly Weather Review, Vol. 102, Issue 7 https://doi.org/10.1175/1520-0493(1974)102<0509:VVCSUF>2.0.CO;2	journal	July 1974
Energy and Numerical Weather Prediction Lorenz, Edward N. Tellus, Vol. 12, Issue 4 https://doi.org/10.1111/j.2153-3490.1960.tb01323.x	journal	November 1960
Dispersion analysis of the spectral element method Melvin, Thomas; Staniforth, Andrew; Thuburn, John Quarterly Journal of the Royal Meteorological Society, Vol. 138, Issue 668 https://doi.org/10.1002/qj.1906	journal	February 2012
Support-Operator Finite-Difference Algorithms for General Elliptic Problems Shashkov, Mikhail; Steinberg, Stanly Journal of Computational Physics, Vol. 118, Issue 1 https://doi.org/10.1006/jcph.1995.1085	journal	April 1995
Direct Discretization of Planar Div-Curl Problems Nicolaides, R. A. SIAM Journal on Numerical Analysis, Vol. 29, Issue 1 https://doi.org/10.1137/0729003	journal	February 1992
An Overview of the Atmospheric Component of the Energy Exascale Earth System Model Rasch, P. J.; Xie, S.; Ma, P. ‐L. Journal of Advances in Modeling Earth Systems, Vol. 11, Issue 8 https://doi.org/10.1029/2019MS001629	journal	August 2019
A global hexagonal C-grid non-hydrostatic dynamical core (ICON-IAP) designed for energetic consistency Gassmann, Almut Quarterly Journal of the Royal Meteorological Society, Vol. 139, Issue 670 https://doi.org/10.1002/qj.1960	journal	June 2012
The Parabolic Spline Method (PSM) for conservative transport problems Zerroukat, M.; Wood, N.; Staniforth, A. International Journal for Numerical Methods in Fluids, Vol. 51, Issue 11 https://doi.org/10.1002/fld.1154	journal	January 2006
Dispersive behaviour of high order finite element schemes for the one-way wave equation Ainsworth, Mark Journal of Computational Physics, Vol. 259 https://doi.org/10.1016/j.jcp.2013.11.003	journal	February 2014
The “Cubed Sphere”: A New Method for the Solution of Partial Differential Equations in Spherical Geometry Ronchi, C.; Iacono, R.; Paolucci, P. S. Journal of Computational Physics, Vol. 124, Issue 1 https://doi.org/10.1006/jcph.1996.0047	journal	March 1996
A primal–dual mimetic finite element scheme for the rotating shallow water equations on polygonal spherical meshes Thuburn, John; Cotter, Colin J. Journal of Computational Physics, Vol. 290 https://doi.org/10.1016/j.jcp.2015.02.045	journal	June 2015
CAM-SE: A scalable spectral element dynamical core for the Community Atmosphere Model Dennis, John M.; Edwards, Jim; Evans, Katherine J. The International Journal of High Performance Computing Applications, Vol. 26, Issue 1 https://doi.org/10.1177/1094342011428142	journal	November 2011
Idealized tropical cyclone simulations of intermediate complexity: A test case for AGCMs Reed, Kevin A.; Jablonowski, Christiane Journal of Advances in Modeling Earth Systems, Vol. 4, Issue 2 https://doi.org/10.1029/2011MS000099	journal	February 2012
The spectral element method on variable resolution grids: evaluating grid sensitivity and resolution-aware numerical viscosity Guba, O.; Taylor, M. A.; Ullrich, P. A. Geoscientific Model Development Discussions, Vol. 7, Issue 3 https://doi.org/10.5194/gmdd-7-4081-2014	journal	January 2014
The Euler Equations of Motion with Hydrostatic Pressure as an Independent Variable Laprise, René Monthly Weather Review, Vol. 120, Issue 1 https://doi.org/10.1175/1520-0493(1992)120<0197:TEEOMW>2.0.CO;2	journal	January 1992
Runge–Kutta IMEX schemes for the Horizontally Explicit/Vertically Implicit (HEVI) solution of wave equations Weller, Hilary; Lock, Sarah-Jane; Wood, Nigel Journal of Computational Physics, Vol. 252 https://doi.org/10.1016/j.jcp.2013.06.025	journal	November 2013
Computational Design of the Basic Dynamical Processes of the UCLA General Circulation Model Arakawa, Akio; Lamb, Vivian R. Methods in Computational Physics: Advances in Research and Applications https://doi.org/10.1016/B978-0-12-460817-7.50009-4	book	January 1977
An Energy and Angular-Momentum Conserving Vertical Finite-Difference Scheme and Hybrid Vertical Coordinates Simmons, A. J.; Burridge, D. M. Monthly Weather Review, Vol. 109, Issue 4 https://doi.org/10.1175/1520-0493(1981)109<0758:AEAAMC>2.0.CO;2	journal	April 1981
A high-order staggered finite-element vertical discretization for non-hydrostatic atmospheric models Guerra, Jorge E.; Ullrich, Paul A. Geoscientific Model Development, Vol. 9, Issue 5 https://doi.org/10.5194/gmd-9-2007-2016	journal	January 2016
High-Resolution Mesh Convergence Properties and Parallel Efficiency of a Spectral Element Atmospheric Dynamical Core Dennis, John; Fournier, Aimé; Spotz, William F. The International Journal of High Performance Computing Applications, Vol. 19, Issue 3 https://doi.org/10.1177/1094342005056108	journal	August 2005
On the use of a coordinate transformation for the solution of the Navier-Stokes equations Gal-Chen, Tzvi; Somerville, Richard C. J. Journal of Computational Physics, Vol. 17, Issue 2 https://doi.org/10.1016/0021-9991(75)90037-6	journal	February 1975
Implicit-Explicit Formulations of a Three-Dimensional Nonhydrostatic Unified Model of the Atmosphere (NUMA) Giraldo, F. X.; Kelly, J. F.; Constantinescu, E. M. SIAM Journal on Scientific Computing, Vol. 35, Issue 5 https://doi.org/10.1137/120876034	journal	January 2013
DCMIP2016: the splitting supercell test case Zarzycki, Colin M.; Jablonowski, Christiane; Kent, James Geoscientific Model Development, Vol. 12, Issue 3 https://doi.org/10.5194/gmd-12-879-2019	journal	January 2019
A unified approach to energy conservation and potential vorticity dynamics for arbitrarily-structured C-grids Ringler, T. D.; Thuburn, J.; Klemp, J. B. Journal of Computational Physics, Vol. 229, Issue 9 https://doi.org/10.1016/j.jcp.2009.12.007	journal	May 2010
Operator-Split Runge–Kutta–Rosenbrock Methods for Nonhydrostatic Atmospheric Models Ullrich, Paul; Jablonowski, Christiane Monthly Weather Review, Vol. 140, Issue 4 https://doi.org/10.1175/MWR-D-10-05073.1	journal	April 2012
Implicit–explicit (IMEX) Runge–Kutta methods for non-hydrostatic atmospheric models Gardner, David J.; Guerra, Jorge E.; Hamon, François P. Geoscientific Model Development, Vol. 11, Issue 4 https://doi.org/10.5194/gmd-11-1497-2018	journal	January 2018
Mixed finite elements for numerical weather prediction Cotter, C. J.; Shipton, J. Journal of Computational Physics, Vol. 231, Issue 21 https://doi.org/10.1016/j.jcp.2012.05.020	journal	August 2012
Equations of Atmospheric Motion in Non-Eulerian Vertical Coordinates: Vector-Invariant Form and Quasi-Hamiltonian Formulation Dubos, Thomas; Tort, Marine Monthly Weather Review, Vol. 142, Issue 10 https://doi.org/10.1175/MWR-D-14-00069.1	journal	October 2014
A Terrain-Following Coordinate with Smoothed Coordinate Surfaces Klemp, J. B. Monthly Weather Review, Vol. 139, Issue 7 https://doi.org/10.1175/MWR-D-10-05046.1	journal	July 2011
Numerical analyses of Runge-Kutta implicit-explicit schemes for horizontally explicit, vertically implicit solutions of atmospheric models: Numerical Analyses of RK HEVI Solutions Lock, S. -J.; Wood, N.; Weller, H. Quarterly Journal of the Royal Meteorological Society, Vol. 140, Issue 682 https://doi.org/10.1002/qj.2246	journal	February 2014
A New Terrain-Following Vertical Coordinate Formulation for Atmospheric Prediction Models Schär, Christoph; Leuenberger, Daniel; Fuhrer, Oliver Monthly Weather Review, Vol. 130, Issue 10 https://doi.org/10.1175/1520-0493(2002)130<2459:ANTFVC>2.0.CO;2	journal	October 2002
Conservation of Mass and Energy for the Moist Atmospheric Primitive Equations on Unstructured Grids Taylor, Mark A. Numerical Techniques for Global Atmospheric Models https://doi.org/10.1007/978-3-642-11640-7_12	book	January 2011
DCMIP2016: a review of non-hydrostatic dynamical core design and intercomparison of participating models Ullrich, Paul A.; Jablonowski, Christiane; Kent, James Geoscientific Model Development, Vol. 10, Issue 12 https://doi.org/10.5194/gmd-10-4477-2017	journal	January 2017
A compatible and conservative spectral element method on unstructured grids Taylor, Mark A.; Fournier, Aimé Journal of Computational Physics, Vol. 229, Issue 17 https://doi.org/10.1016/j.jcp.2010.04.008	journal	August 2010
A Generalization of the SLEVE Vertical Coordinate Leuenberger, Daniel; Koller, Marcel; Fuhrer, Oliver Monthly Weather Review, Vol. 138, Issue 9 https://doi.org/10.1175/2010MWR3307.1	journal	September 2010
Impact and importance of hyperdiffusion on the spectral element method: A linear dispersion analysis Ullrich, Paul A.; Reynolds, Daniel R.; Guerra, Jorge E. Journal of Computational Physics, Vol. 375 https://doi.org/10.1016/j.jcp.2018.06.035	journal	December 2018
Conservative Scheme for the Compressible Nonhydrostatic Models with the Horizontally Explicit and Vertically Implicit Time Integration Scheme Satoh, Masaki Monthly Weather Review, Vol. 130, Issue 5 https://doi.org/10.1175/1520-0493(2002)130<1227:CSFTCN>2.0.CO;2	journal	May 2002
DYNAMICO-1.0, an icosahedral hydrostatic dynamical core designed for consistency and versatility Dubos, T.; Dubey, S.; Tort, M. Geoscientific Model Development, Vol. 8, Issue 10 https://doi.org/10.5194/gmd-8-3131-2015	journal	January 2015
A proposed baroclinic wave test case for deep- and shallow-atmosphere dynamical cores: A Baroclinic Wave Test Case for Deep and Shallow-atmosphere Dynamical Cores Ullrich, Paul A.; Melvin, Thomas; Jablonowski, Christiane Quarterly Journal of the Royal Meteorological Society, Vol. 140, Issue 682 https://doi.org/10.1002/qj.2241	journal	December 2013
SUNDIALS: Suite of nonlinear and differential/algebraic equation solvers Hindmarsh, Alan C.; Brown, Peter N.; Grant, Keith E. ACM Transactions on Mathematical Software, Vol. 31, Issue 3 https://doi.org/10.1145/1089014.1089020	journal	September 2005
A “Vertically Lagrangian” Finite-Volume Dynamical Core for Global Models Lin, Shian-Jiann Monthly Weather Review, Vol. 132, Issue 10 https://doi.org/10.1175/1520-0493(2004)132<2293:AVLFDC>2.0.CO;2	journal	October 2004
Numerical representation of geostrophic modes on arbitrarily structured C-grids Thuburn, J.; Ringler, T. D.; Skamarock, W. C. Journal of Computational Physics, Vol. 228, Issue 22 https://doi.org/10.1016/j.jcp.2009.08.006	journal	December 2009
Discretization of generalized Coriolis and friction terms on the deformed hexagonal C-grid Gassmann, Almut Quarterly Journal of the Royal Meteorological Society, Vol. 144, Issue 716 https://doi.org/10.1002/qj.3294	journal	October 2018