Hostname: page-component-7c8c6479df-nwzlb Total loading time: 0 Render date: 2024-03-28T00:24:53.167Z Has data issue: false hasContentIssue false

Horizontally viscous effects in a tidal basin: extending Taylor's problem

Published online by Cambridge University Press:  27 October 2009

P. C. ROOS*
Affiliation:
Water Engineering and Management, Faculty of Engineering Technology, University of Twente, PO Box 217, 7500 AE Enschede, The Netherlands
H. M. SCHUTTELAARS
Affiliation:
Department of Applied Mathematical Analysis, Delft University of Technology, PO Box 5031, 2600 GA Delft, The Netherlands
*
Email address for correspondence: p.c.roos@utwente.nl

Abstract

The classical problem of Taylor (Proc. Lond. Math. Soc., vol. 20, 1921, pp. 148–181) of Kelvin wave reflection in a semi-enclosed rectangular basin of uniform depth is extended to account for horizontally viscous effects. To this end, we add horizontally viscous terms to the hydrodynamic model (linearized depth-averaged shallow-water equations on a rotating plane, including bottom friction) and introduce a no-slip condition at the closed boundaries.

In a straight channel of infinite length, we obtain three types of wave solutions (normal modes). The first two wave types are viscous Kelvin and Poincaré modes. Compared to their inviscid counterparts, they display longitudinal boundary layers and a slight decrease in the characteristic length scales (wavelength or along-channel decay distance). For each viscous Poincaré mode, we additionally find a new mode with a nearly similar lateral structure. This third type, entirely due to viscous effects, represents evanescent waves with an along-channel decay distance bounded by the boundary-layer thickness.

The solution to the viscous Taylor problem is then written as a superposition of these normal modes: an incoming Kelvin wave and a truncated sum of reflected modes. To satisfy no slip at the lateral boundary, we apply a Galerkin method. The solution displays boundary layers, the lateral one at the basin's closed end being created by the (new) modes of the third type. Amphidromic points, in the inviscid and frictionless case located on the centreline of the basin, are now found on a line making a small angle to the longitudinal direction. Using parameter values representative for the Southern Bight of the North Sea, we finally compare the modelled and observed tide propagation in this basin.

Type
Papers
Copyright
Copyright © Cambridge University Press 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.Google Scholar
Brown, P. J. 1973 Kelvin-wave reflection in a semi-infinite canal. J. Mar. Res. 31 (1), 110.Google Scholar
Brown, T. 1987 Kelvin wave reflection at an oscillating boundary with applications to the North Sea. Cont. Shelf Res. 7 (4), 351365.CrossRefGoogle Scholar
Brown, T. 1989 On the general problem of Kelvin wave reflection at an oscillating boundary. Cont. Shelf Res. 9 (10), 931937.CrossRefGoogle Scholar
Cai, S. C., Long, X., Liub, H. & Wanga, S. 2006 Tide model evaluation under different conditions. Cont. Shelf. Res. 26 (1), 104112.CrossRefGoogle Scholar
Carbajal, N. 1997 Two applications of Taylor's problem solution for finite rectangular semi-enclosed basins. Cont. Shelf Res. 17 (7), 803817.CrossRefGoogle Scholar
Davey, M. K., Hsieh, W. W. & Wajsowicz, R. C. 1983 The free Kelvin wave with lateral and vertical viscosity. J. Phys. Oceanogr. 13, 21822191.2.0.CO;2>CrossRefGoogle Scholar
Davies, A. M. & Jones, J. E. 1995 The influence of bottom and internal friction upon tidal currents: Taylor's problem in three dimensions. Cont. Shelf Res. 15 (10), 12511285.CrossRefGoogle Scholar
Davies, A. M. & Jones, J. E. 1996 The influence of wind and wind wave turbulence upon tidal currents: Taylor's problem in three dimensions with wind forcing. Cont. Shelf Res. 16 (1), 2599.CrossRefGoogle Scholar
Dyer, K. R. & Huntley, D. A. 1999 The origin, classification and modelling of sand banks and ridges. Cont. Shelf Res. 19 (10), 12851330.CrossRefGoogle Scholar
Fang, G., Kwok, Y.-K., Yu, K & Zhu, Y. 1999 Numerical simulation of principal tidal constituents in the South China Sea, Gulf of Tonkin and Gulf of Thailand. Cont. Shelf Res. 19 (7), 845869.CrossRefGoogle Scholar
Hendershott, M. C. & Speranza, A. 1971 Co-oscillating tides in long, narrow bays; the Taylor problem revisited. Deep-Sea Res. 18 (10), 959980.Google Scholar
LeBlond, P. H. & Mysak, L. A. 1978 Waves in the Ocean. Elsevier.Google Scholar
Pedlosky, J. 1982 Geophysical Fluid Dynamics. Springer.CrossRefGoogle Scholar
Proudman, J. 1941 The effect of coastal friction on the tides. Mon. Not. R. Astron. Soc. Geophys. Suppl. 1, 2326.CrossRefGoogle Scholar
Rienecker, M. M. & Teubner, M. D. 1980 A note on frictional effects in Taylor's problem. J. Mar. Res. 38, 183191.Google Scholar
Rijksinstituut voor Kust en Zee (RIKZ) 2002 Getijtafels voor Nederland 2003 [in Dutch]. Sdu Uitgevers.Google Scholar
Ripa, P. & Zavala-Garay, J. 1999 Ocean channel modes. J. Geophys. Res. 104 (C7), 15 47915 494.CrossRefGoogle Scholar
Rizal, S. 2002 Taylor's problem – influences on the spatial distribution of real and virtual amphidromes. Cont. Shelf Res. 22 (15), 21472158.CrossRefGoogle Scholar
Sinha, B. & Pingree, R. D. 1997 The principal lunar semidiurnal tide and its harmonics: baseline solutions for M2 and M4 constituents in the North-West European Continental Shelf. Cont. Shelf Res. 17 (11), 13211365.CrossRefGoogle Scholar
Taylor, G. I. 1921 Tidal oscillations in gulfs and rectangular basins. Proc. Lond. Math. Soc. 20, 148181.Google Scholar
Winant, C. D. 2007 Three-dimensional tidal flow in an elongated, rotating basin. J. Phys. Oceanogr. 37, 23452362.CrossRefGoogle Scholar
Xia, Z., Carbajal, N. & Südermann, J. 1995 Tidal current amphidromic system in semi-enclosed basins. Cont. Shelf Res. 15 (2–3), 219240.Google Scholar
Zimmerman, J. T. F. 1982 On the Lorentz linearization of a quadratically damped forced oscillator. Phys. Lett. A 89, 123124.CrossRefGoogle Scholar