Abstract
In this paper the stages in formulating a Galerkin-Spectral model are illustrated by developing a spectral model of the vertical profile of oscillatory flow. Such a simple model is chosen so that the steps in the method can be clearly illustrated. References to the literature are given for general background information; the reader is directed to the second section of this chapter [Davies, 1986b] for the extension to three dimensions. Some results from a three-dimensional tidal model are presented to illustrate oceanographic applications. The objective is to introduce (by references to the literature and a simple example) the Galerkin method to someone new to the topic and to illustrate its applications.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bowden, K. F., L. A. Fairbairn, and P. Hughes, The distribution of shearing stresses in a tidal current, Geophys. J. R. Astr. Soc.,2, 288–305, 1959.
Cheng, R. T., T. M. Powell, and T. M. Dillon, Numerical models of wind-driven circulation in lakes, Applied Mathematical Modelling, 1, 141–158, 1976.
Connor, J. J., and C. A. Brebbia, Finite element techniques for fluid flow, 307 pp., London, Newnes - Butterworths, 1976.
Davies, A. M., A numerical model of the North Sea and its use in choosing locations for the deployment of offshore tide gauges in the JONSDAP ‘76 oceanographic experiment, Dtsch. Hydrogr. Z., 29, 11–24, 1976.
Davies, A. M., Application of the Galerkin method to the formulation of a three-dimensional non-linear hydrodynamic numerical sea model, Applied Mathematical Modelling, 4, 245–256, 1980.
Davies, A. M., Formulation of a linear three-dimensional hydrodynamic sea model using a Galerkin-Eigenfunction method, Int. J. Num. Meth. in Fluids, 3, 33–60, 1983.
Davies, A. M., On determining current profiles in oscillatory flows, in press, Applied Mathematical Modelling 1985.
Davies, A. M., Mathematical formulation of a spectral circulation model, this volume, 1986b.
Davies, A. M., and G. K. Furnes, Observed and computed M2 tidal current in the North Sea, J. Phys. Oceanogra., 10, 237–257, 1980.
Davies, A. M., and I. D. James, Three-dimensional Galerkin-Spectral sea models of the North Sea and German Bight, in North Sea Dynamics, edited by J. Sundermann and W. Lenz, pp. 85–95, Springer-Verlag, 1983.
Davies, A. M., and A. Owen, Three-dimensional numerical sea model using the Galerkin method with a polynomial basis set, Appl. Math. Modelling, 3, 421–428, 1979.
Fang, G., and T. Ichiye, On the vertical structure of tidal currents in a homogeneous sea, Geophys. J. R. Astr. Soc., 73, 65–82, 1983.
Flather, R. A., A tidal model of the northwest European continental shelf, Mem. Soc. Roy. Sci. Liege, 10, 141–164, 1976.
Finlayson, B. A., The method of weighted residuals and variational principles, Academic Press, New York, 1972.
Gottlieb, D., and S. Orszag, A numerical analysis of spectral methods, NSF-CBMS Monograph No. 26, Soc. Ind. and Appl. Math., Philadelphia, 1977.
Gray, W. G., Some inadequacies of finite element models as simulators of two-dimensional circulation, Adv. Water Resources, 5, 171–177, 1982.
Grotkop, G., Finite element analysis of long-period water waves, Comput. Meth. Appl. Mech. Eng., 2, 133–146, 1973.
Heaps, N. S., and J. E. Jones, Three-dimensional model for tides and surges with vertical eddy viscosity prescribed in two layers. H. Irish Sea with bed friction layer, Geophys. J. R. Astr. Soc., 64, 303–320, 1981.
Mathisen, J P., and 0. Johansen, A numerical tidal and storm surge model of the North Sea, Marine Geodesy, 6, 267–291, 1983.
Owen, A., A three-dimentional model of the Bristol Channel, J. Phys. Oceanography, 10, 1290–1302, 1980.
Prandle, D., The Vertical structure of tidal currents and other oscillatory flows, Continental Shelf Research, 1, 191–207, 1982.
Proctor, R., Tides and residual circulation in the Irish Sea: a numerical modelling approach. Ph.D. thesis, Liverpool University, 1981.
Schwiderski, E. W., Atlas of ocean tidal Charts and maps, Part 1: the semidiurnal principal lunar tide M2, Marine Geodesy, 6, 219–265, 1983.
Smith, J. D., Modelling of sediment transport on continental shelves, in The Sea, 6, edited by E. D. Goldberg et al., pp. 539–578, Wiley-Interscience, 1977.
Soulsby, R. L., The bottom boundary layer of shelf seas, in Physical Oceanography of Coastal and Shelf Seas, edited by B. Johns, pp. 189–266, Elsevier Oceanography Series, No. 35, 1983.
Strang, G., and G. J. Fix, An analysis of the finite element method, Prentice Hall Inc., Englewood Cliffs, 1973.
Wolf, J., Estimation of shearing stresses in a tidal current with application to the Irish Sea, in Marine Turbulence, Proceedings of the 11th’Liege Colloquium on Ocean Hydrodynamics edited by J.C.J. Nihoul, pp. 319–344, Elsevier Oceanography, Series No. 28, 1980.
Wolf, J., The variability of currents in shallow seas, Ph.D. thesis, Liverpool University, 1984.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1986 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Davies, A.M. (1986). Mathematical Formulation of a Spectral Tidal Model. In: O’Brien, J.J. (eds) Advanced Physical Oceanographic Numerical Modelling. NATO ASI Series, vol 186. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-0627-8_21
Download citation
DOI: https://doi.org/10.1007/978-94-017-0627-8_21
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-8428-6
Online ISBN: 978-94-017-0627-8
eBook Packages: Springer Book Archive