Selective decay for the rotating shallow-water equations with a structure-preserving discretization
File(s)5.0062573.pdf (4.74 MB)
Published Version
Author(s)
Brecht, R
Bauer, W
Bihlo, A
Gay-Balmaz, F
MacLachlan, S
Type
Journal Article
Abstract
Numerical models of weather and climate critically depend on long-term stability of integrators for systems of hyperbolic conservation laws. While such stability is often obtained from (physical or numerical) dissipation terms, physical fidelity of such simulations also depends on properly preserving conserved quantities, such as energy, of the system. To address this apparent paradox, we develop a variational integrator for the shallow water equations that conserves energy, but dissipates potential enstrophy. Our approach follows the continuous selective decay framework [F. Gay-Balmaz and D. Holm. Selective decay by Casimir dissipation in inviscid fluids. Nonlinearity, 26(2):495, 2013], which enables dissipating an otherwise conserved quantity while conserving the total energy. We use this in combination with the variational discretization method [D. Pavlov, P. Mullen, Y. Tong, E. Kanso, J. Marsden and M. Desbrun. Structure-preserving discretization of incompressible fluids. Physica D: Nonlinear Phenomena, 240(6):443-458, 2011] to obtain a discrete selective decay framework. This is applied to the shallow water equations, both in the plane and on the sphere, to dissipate the potential enstrophy. The resulting scheme significantly improves the quality of the approximate solutions, enabling long-term integrations to be carried out.
Date Issued
2021-11-15
Date Acceptance
2021-10-14
ISSN
1070-6631
Publisher
American Institute of Physics
Journal / Book Title
Physics of Fluids
Volume
33
Copyright Statement
© 2021 Author(s). Published under an exclusive license by AIP Publishing.
Sponsor
Natural Environment Research Council [2006-2012]
Identifier
https://aip.scitation.org/doi/10.1063/5.0062573
Grant Number
NE/R008795/1
Subjects
math.NA
math.NA
cs.NA
math.NA
math.NA
cs.NA
Fluids & Plasmas
01 Mathematical Sciences
02 Physical Sciences
09 Engineering
Publication Status
Published online
Date Publish Online
2021-11-15