Total energy and potential enstrophy conserving schemes for the shallow water equations using Hamiltonian methods $$ Part 1: Derivation and properties
The shallow water equations provide a useful analogue of the fully compressible Euler equations since they have similar characteristics: conservation laws, inertiagravity and Rossby waves, and a (quasi) balanced state. In order to obtain realistic simulation results, it is desirable that numerical models have discrete analogues of these properties. Two prototypical examples of such schemes are the 1981 Arakawa and Lamb (AL81) Cgrid total energy and potential enstrophy conserving scheme, and the 2007 Salmon (S07) Zgrid total energy and potential enstrophy conserving scheme. Unfortunately, the AL81 scheme is restricted to logically square, orthogonal grids, and the S07 scheme is restricted to uniform square grids. The current work extends the AL81 scheme to arbitrary nonorthogonal polygonal grids and the S07 scheme to arbitrary orthogonal spherical polygonal grids in a manner that allows for both total energy and potential enstrophy conservation, by combining Hamiltonian methods (work done by Salmon, Gassmann, Dubos, and others) and discrete exterior calculus (Thuburn, Cotter, Dubos, Ringler, Skamarock, Klemp, and others). Lastly, detailed results of the schemes applied to standard test cases are deferred to part 2 of this series of papers.
 Authors:

^{[1]};
^{[2]}
 Univ. of Paris,Villetaneuse (France). Lab. Analysis, Geometry and Applications (LAGA)
 Colorado State Univ., Fort Collins, CO (United States). Dept. of Atmospheric Science
 Publication Date:
 Grant/Contract Number:
 FG0297ER25308
 Type:
 Published Article
 Journal Name:
 Geoscientific Model Development (Online)
 Additional Journal Information:
 Journal Name: Geoscientific Model Development (Online); Journal Volume: 10; Journal Issue: 2; Journal ID: ISSN 19919603
 Publisher:
 European Geosciences Union
 Research Org:
 Krell Inst., Ames, IA (United States)
 Sponsoring Org:
 USDOE; National Science Foundation (NSF)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 58 GEOSCIENCES
 OSTI Identifier:
 1344140
 Alternate Identifier(s):
 OSTI ID: 1366511
Eldred, Christopher, and Randall, David. Total energy and potential enstrophy conserving schemes for the shallow water equations using Hamiltonian methods $$ Part 1: Derivation and properties. United States: N. p.,
Web. doi:10.5194/gmd107912017.
Eldred, Christopher, & Randall, David. Total energy and potential enstrophy conserving schemes for the shallow water equations using Hamiltonian methods $$ Part 1: Derivation and properties. United States. doi:10.5194/gmd107912017.
Eldred, Christopher, and Randall, David. 2017.
"Total energy and potential enstrophy conserving schemes for the shallow water equations using Hamiltonian methods $$ Part 1: Derivation and properties". United States.
doi:10.5194/gmd107912017.
@article{osti_1344140,
title = {Total energy and potential enstrophy conserving schemes for the shallow water equations using Hamiltonian methods $$ Part 1: Derivation and properties},
author = {Eldred, Christopher and Randall, David},
abstractNote = {The shallow water equations provide a useful analogue of the fully compressible Euler equations since they have similar characteristics: conservation laws, inertiagravity and Rossby waves, and a (quasi) balanced state. In order to obtain realistic simulation results, it is desirable that numerical models have discrete analogues of these properties. Two prototypical examples of such schemes are the 1981 Arakawa and Lamb (AL81) Cgrid total energy and potential enstrophy conserving scheme, and the 2007 Salmon (S07) Zgrid total energy and potential enstrophy conserving scheme. Unfortunately, the AL81 scheme is restricted to logically square, orthogonal grids, and the S07 scheme is restricted to uniform square grids. The current work extends the AL81 scheme to arbitrary nonorthogonal polygonal grids and the S07 scheme to arbitrary orthogonal spherical polygonal grids in a manner that allows for both total energy and potential enstrophy conservation, by combining Hamiltonian methods (work done by Salmon, Gassmann, Dubos, and others) and discrete exterior calculus (Thuburn, Cotter, Dubos, Ringler, Skamarock, Klemp, and others). Lastly, detailed results of the schemes applied to standard test cases are deferred to part 2 of this series of papers.},
doi = {10.5194/gmd107912017},
journal = {Geoscientific Model Development (Online)},
number = 2,
volume = 10,
place = {United States},
year = {2017},
month = {2}
}