Biological and Environmental Research - Earth and Environmental System Sciences
Earth and Environmental System Modeling

An Energy Consistent Discretization of the Nonhydrostatic Equations in Primitive Variables

TitleAn Energy Consistent Discretization of the Nonhydrostatic Equations in Primitive Variables
Publication TypeJournal Article
Year of Publication2020
JournalJournal of Advances in Modeling Earth Systems
Volume12
Number1
Abstract / Summary

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)1092.0.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.

URLhttp://dx.doi.org/10.1029/2019ms001783
DOI10.1029/2019ms001783
Journal: Journal of Advances in Modeling Earth Systems
Year of Publication: 2020
Volume: 12
Number: 1
Publication Date: 01/2020

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)1092.0.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
Citation:
Taylor, M, O Guba, A Steyer, P Ullrich, D Hall, and C Eldrid.  2020.  "An Energy Consistent Discretization of the Nonhydrostatic Equations in Primitive Variables."  Journal of Advances in Modeling Earth Systems 12(1).  https://doi.org/10.1029/2019ms001783.