Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-19T16:41:03.819Z Has data issue: false hasContentIssue false
  • English
  • Français

Laboratory experiments and a non-harmonic theory for topographic Rossby waves over a linearly sloping bottom on the f-plane

Published online by Cambridge University Press:  09 February 2010

YAIR COHEN ,
NATHAN PALDOR  and
JOËL SOMMERIA
YAIR COHEN
Affiliation:
The Fredy and Nadine Herrmann Institute of Earth Sciences, Edmond J. Safra Campus, Givat Ram, The Hebrew University Jerusalem, Jerusalem 91904, Israel
NATHAN PALDOR*
Affiliation:
The Fredy and Nadine Herrmann Institute of Earth Sciences, Edmond J. Safra Campus, Givat Ram, The Hebrew University Jerusalem, Jerusalem 91904, Israel
JOËL SOMMERIA
Affiliation:
Laboratoire des Ecoulements Géophysiques et Industriels (LEGI, CNRS-INPG-UJF) BP53 38041, Grenoble CEDEX 9, France
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Low-frequency waves that develop in a shallow layer of fluid, contained in a channel with linearly slopping bottom and rotating with uniform angular speed are investigated theoretically and experimentally. Exact numerical solutions of the eigenvalue problem, obtained from the linearized shallow water equations on the f-plane, show that the waves are trapped near the channel's shallow wall and propagate along it with the shallow side on their right in the Northern hemisphere. The phase speed of the waves is slower compared with that of the harmonic theory in which bottom slope is treated inconsistently. A first-order approximation of the cross-channel dependence of the coefficient in the eigenvalue equation yields an approximation of the cross-channel velocity eigenfunction as an Airy function, which, for sufficiently wide channels, yields an explicit expression for the wave's dispersion relation. The analytic solutions of the eigenvalue problem agree with the numerical solutions in both the wave trapping and the reduced phase speed. For narrow channels, our theory yields an estimate of the channel width below which the harmonic theory provides a more accurate approximation. Laboratory experiments were conducted on a 13 m diameter turntable at LEGI-Coriolis (France) into which a linearly sloping bottom of 10 % incline was installed. A wavemaker generated waves of known frequency at one end of the turntable and the wavenumbers of these waves were measured at the opposite end using a particle imaging velocimetry technique. The experimental results regarding the phase speed and the radial structure of the amplitude are in very good agreement with our theoretical non-harmonic predictions, which support the present modification of the harmonic theory in wide channels.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 645 , 25 February 2010 , pp. 479 - 496
Copyright
Copyright © Cambridge University Press 2010

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

Abramowitz, M. & Stegun, I. A. 1972 Handbook of Mathematical Functions. Dover.Google Scholar
Buchwald, V. T. & Adams, J. K. 1968 The propagation of continental shelf waves. Proc. R. Soc. Lond. A 305, 235–250.Google Scholar
Caldwell, D. R., Cutchin, D. L. & Longuet-Higgins, M. S. 1972 Some model experiments on continental shelf waves. J. Marine Res. 30, 38–55.Google Scholar
Cartwright, D. E. 1969 Extraordinary tidal currents near St. Kilda. Nature 223, 928–932.CrossRefGoogle Scholar
Cushman-Roisin, B. 1994 Introduction to Geophysical Fluid Dynamics. Prentice-Hall.Google Scholar
Cutchin, D. L. & Smith, R. 1973 Continental shelf waves: low-frequency variations in sea level and currents over the Oregon continental shelf. J. Phys. Oceanogr. 3, 73–82.2.0.CO;2>CrossRefGoogle Scholar
De-Leon, Y. & Paldor, N. 2009 Linear wave in mid-latitudes on the rotating spherical Earth. J. Phys. Oceanogr. 39, 3204–3215.CrossRefGoogle Scholar
Gill, A. E. 1982 Ocean-Atmosphere Dynamics. Academic Press.Google Scholar
Hamon, B. V. 1962 The spectrum of mean sea level at Sydney, Coff's Barbour and Lord Howe Island. J. Geophys. Res. 67, 5147–5155.CrossRefGoogle Scholar
Hamon, B. V. 1966 Continental shelf waves and the effect of atmospheric pressure and wind stress on the sea level. J. Geophys. Res. 71, 2883–2893.CrossRefGoogle Scholar
Ibbetson, A. & Phillips, N. 1967 Some laboratory experiments on Rossby waves in a rotating annulus. Tellus 19, 81–87.CrossRefGoogle Scholar
Mooers, C. N. K. & Smith, R. L. 1968 Continental shelf waves off Oregon. J. Geophys. Res. 73, 549–557.CrossRefGoogle Scholar
Mysak, L. A. & Hamon, B. V. 1969 Low frequency sea level behavior and continental shelf waves off North Carolina. J. Geophys. Res. 74, 1397–1405.CrossRefGoogle Scholar
Paldor, N., Rubin, S. & Mariano, A. J. 2007 A consistent theory for linear waves of the shallow-water equation on a rotating plane in midlatitudes. J. Phys. Oceanogr. 37, 115–128.CrossRefGoogle Scholar
Paldor, N. & Sigalov, A. 2008 Trapped waves on the mid-latitude β-plane. Tellus A 60 (4), 742–748.CrossRefGoogle Scholar
Pedlosky, J. 1986 Geophysical Fluid Dynamics, 2nd ed. Springer.Google Scholar
Phillips, N. A. 1965 Elementary Rossby waves. Tellus 17, 295–301.CrossRefGoogle Scholar
Pierini, S., Fincham, A. M., Renouard, D., Rosaria, M. & Didelle, H. 2001 Laboratory modeling of topographic Rossby normal modes. Dyn. Atmos. Oceans 35, 205–225.CrossRefGoogle Scholar
Platzman, G. W. 1968 The Rossby wave. Quar. J. Met. Soc. 94, 225–246.CrossRefGoogle Scholar
Rhines, P. 1970 Edge-, bottom-, and Rossby waves in a rotating stratified fluid. Geophys. Fluid. Dyn. 1, 273–302.CrossRefGoogle Scholar
Staniforth, A. N., Williams, R. T. & Neta, B. 1993 Influence of linear depth variation on Poincare, Kelvin, and Rossby waves. J. Atmos. Sci. 50 (7), 929–940.2.0.CO;2>CrossRefGoogle Scholar
Sumathi, S. & Surekha, P. 2007 LabVIEW Based Advanced Instrumentation Systems. Springer.Google Scholar