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

Abstract

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.

Authors:
 [1];  [1];  [1];  [2];  [3];  [4]
  1. Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
  2. Univ. of California, Davis, CA (United States)
  3. NVIDIA, Santa Clara, CA (United States)
  4. Univ. Grenoble Alpes (France)
Publication Date:
Research Org.:
Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
Sponsoring Org.:
USDOE Office of Science (SC), Biological and Environmental Research (BER)
OSTI Identifier:
1631000
Report Number(s):
SAND2020-5242J
Journal ID: ISSN 1942-2466; 686193
Grant/Contract Number:  
AC04-94AL85000
Resource Type:
Accepted Manuscript
Journal Name:
Journal of Advances in Modeling Earth Systems
Additional Journal Information:
Journal Volume: 12; Journal Issue: 1; Journal ID: ISSN 1942-2466
Publisher:
American Geophysical Union (AGU)
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING

Citation Formats

Taylor, Mark A., Guba, Oksana, Steyer, Andrew, Ullrich, Paul A., Hall, David M., and Eldrid, Christopher. An Energy Consistent Discretization of the Nonhydrostatic Equations in Primitive Variables. United States: N. p., 2019. Web. https://doi.org/10.1029/2019MS001783.
Taylor, Mark A., Guba, Oksana, Steyer, Andrew, Ullrich, Paul A., Hall, David M., & Eldrid, Christopher. An Energy Consistent Discretization of the Nonhydrostatic Equations in Primitive Variables. United States. https://doi.org/10.1029/2019MS001783
Taylor, Mark A., Guba, Oksana, Steyer, Andrew, Ullrich, Paul A., Hall, David M., and Eldrid, Christopher. Sat . "An Energy Consistent Discretization of the Nonhydrostatic Equations in Primitive Variables". United States. https://doi.org/10.1029/2019MS001783. https://www.osti.gov/servlets/purl/1631000.
@article{osti_1631000,
title = {An Energy Consistent Discretization of the Nonhydrostatic Equations in Primitive Variables},
author = {Taylor, Mark A. and Guba, Oksana and Steyer, Andrew and Ullrich, Paul A. and Hall, David M. and Eldrid, Christopher},
abstractNote = {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.},
doi = {10.1029/2019MS001783},
journal = {Journal of Advances in Modeling Earth Systems},
number = 1,
volume = 12,
place = {United States},
year = {2019},
month = {12}
}

Journal Article:
Free Publicly Available Full Text
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Figures / Tables:

Figure 1 Figure 1: The terrain following s coordinate levels and level indexing. There are n midpoint levels indexed by i = 1, 2, ... , n and n + 1 interface levels indexed by i + $^{1}_{2}$, i = 0, 1, ... , n.

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