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Linear and nonlinear properties of numerical methods for the rotating shallow water equations

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dc.contributor.authorEldred, Chris, author
dc.contributor.authorRandall, David, advisor
dc.contributor.authorBirner, Thomas, committee member
dc.contributor.authorSchubert, Wayne, committee member
dc.contributor.authorEstep, Don, committee member
dc.contributor.authorLauritzen, Peter, committee member
dc.contributor.authorBleck, Rainer, committee member
dc.date.accessioned2015-08-28T14:35:02Z
dc.date.available2015-08-28T14:35:02Z
dc.date.issued2015
dc.description.abstractThe shallow water equations provide a useful analogue of the fully compressible Euler equations since they have similar conservation laws, many of the same types of waves and a similar (quasi-) balanced state. It is desirable that numerical models posses similar properties, and the prototypical example of such a scheme is the 1981 Arakawa and Lamb (AL81) staggered (C-grid) total energy and potential enstrophy conserving scheme, based on the vector invariant form of the continuous equations. However, this scheme is restricted to a subset of logically square, orthogonal grids. The current work extends the AL81 scheme to arbitrary non-orthogonal polygonal grids, by combining Hamiltonian methods (work done by Salmon, Gassmann, Dubos and others) and Discrete Exterior Calculus (Thuburn, Cotter, Dubos, Ringler, Skamarock, Klemp and others). It is also possible to obtain these properties (along with arguably superior wave dispersion properties) through the use of a collocated (Z-grid) scheme based on the vorticity-divergence form of the continuous equations. Unfortunately, existing examples of these schemes in the literature for general, spherical grids either contain computational modes; or do not conserve total energy and potential enstrophy. This dissertation extends an existing scheme for planar grids to spherical grids, through the use of Nambu brackets (as pioneered by Rick Salmon). To compare these two schemes, the linear modes (balanced states, stationary modes and propagating modes; with and without dissipation) are examined on both uniform planar grids (square, hexagonal) and quasi-uniform spherical grids (geodesic, cubed-sphere). In addition to evaluating the linear modes, the results of the two schemes applied to a set of standard shallow water test cases and a recently developed forced-dissipative turbulence test case from John Thuburn (intended to evaluate the ability the suitability of schemes as the basis for a climate model) on both hexagonal-pentagonal icosahedral grids and cubed-sphere grids are presented. Finally, some remarks and thoughts about the suitability of these two schemes as the basis for atmospheric dynamical development are given.
dc.format.mediumborn digital
dc.format.mediumdoctoral dissertations
dc.identifierEldred_colostate_0053A_13043.pdf
dc.identifier.urihttp://hdl.handle.net/10217/167080
dc.languageEnglish
dc.language.isoeng
dc.publisherColorado State University. Libraries
dc.relation.ispartof2000-2019
dc.rightsCopyright and other restrictions may apply. User is responsible for compliance with all applicable laws. For information about copyright law, please see https://libguides.colostate.edu/copyright.
dc.subjectdiscrete exterior calculus
dc.subjectlinear modes
dc.subjectshallow water equations
dc.subjectHamiltonian methods
dc.subjectconservation laws
dc.subjectmimetic finite differences
dc.titleLinear and nonlinear properties of numerical methods for the rotating shallow water equations
dc.typeText
dcterms.rights.dplaThis Item is protected by copyright and/or related rights (https://rightsstatements.org/vocab/InC/1.0/). You are free to use this Item in any way that is permitted by the copyright and related rights legislation that applies to your use. For other uses you need to obtain permission from the rights-holder(s).
thesis.degree.disciplineAtmospheric Science
thesis.degree.grantorColorado State University
thesis.degree.levelDoctoral
thesis.degree.nameDoctor of Philosophy (Ph.D.)

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