Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-10T06:30:06.621Z Has data issue: false hasContentIssue false
  • English
  • Français

Damping mechanisms of internal waves in continuously stratified rotating basins

Published online by Cambridge University Press:  21 September 2009

K. SHIMIZU  and
J. IMBERGER
K. SHIMIZU*
Affiliation:
Centre for Water Research, University of Western Australia, Crawley, WA 6009, Australia
J. IMBERGER
Affiliation:
Centre for Water Research, University of Western Australia, Crawley, WA 6009, Australia
*
†Email address for correspondence: shimizu@cwr.uwa.edu.au
Get access Rights & Permissions [Opens in a new window]

Abstract

Damping mechanisms, damping rates and the dissipative modal structure of internal waves in stratified rotating circular basins are investigated analytically. The damping is shown to be due to a combination of the internal-wave cancelling, where waves emitted by the oscillatory boundary layers destructively interact with the parent wave and drain energy from it, and spin-down modified by the periodicity, where the energy is drained by the sinks and sources at the bottom corner caused by a discontinuity in the Ekman transport. It is shown that super-inertial Poincaré waves and sub-inertial Kelvin waves are damped predominantly by the internal-wave cancelling and modified spin-down, respectively. These processes also modify the internal-wave structure; for super-inertial waves, the boundary-layer-generated waves intensify the interior flow in the lower part of the water column and delay the phase relative to the isopycnal displacements, but for sub-inertial waves, the Ekman pumping and the corner sinks and sources add a horizontal circular flow that slants the crest and trough backwards near the wall.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 637 , 25 October 2009 , pp. 137 - 172
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

Antenucci, J. P. & Imberger, J. 2001 Energetics of long internal gravity waves in large lakes. Limnol. Oceanogr. 46, 1760–1773.CrossRefGoogle Scholar
Arfken, G. B. & Weber, H. J. 1995 Mathematical Methods for Physicists, 4th edn. Academic.Google Scholar
Barcilon, V. & Pedlosky, J. 1967 a Linear theory of rotating stratified fluid motions. J. Fluid Mech. 29, 1–16.CrossRefGoogle Scholar
Barcilon, V. & Pedlosky, J. 1967 b A unified theory of homogeneous and stratified rotating fluid. J. Fluid Mech. 29, 609–621.CrossRefGoogle Scholar
Benton, E. R. & Clark, A. Jr., 1974 Spin-up. Annu. Rev. Fluid Mech. 6, 257–280.CrossRefGoogle Scholar
Boegman, L., Imberger, J., Ivey, G. N. & Antenucci, J. P. 2003 High-frequency internal waves in large stratified lakes. Limnol. Oceanogr. 45, 895–919.CrossRefGoogle Scholar
Brink, K. H. 1982 The effect of bottom friction on low-frequency coastal trapped waves. J. Phys. Oceanogr. 12, 127–133.2.0.CO;2>CrossRefGoogle Scholar
Brink, K. H. 1988 On the effect of bottom friction on internal waves. Cont. Shelf Res. 8, 397–403.CrossRefGoogle Scholar
Brink, K. H. & Allen, J. S. 1978 On the effect of bottom friction on barotropic motion over the continental shelf. J. Phys. Oceanogr. 8, 919–922.2.0.CO;2>CrossRefGoogle Scholar
Case, K. M. & Parkinson, W. C. 1957 Damping of surface waves in an incompressible liquid. J. Fluid Mech. 2, 172–184.CrossRefGoogle Scholar
Clarke, A. J. & Van Gorder, S. 1986 A method for estimating wind-driven frictional, time-dependent, stratified shelf and slope water flow. J. Phys. Oceanogr. 16, 1013–1028.2.0.CO;2>CrossRefGoogle Scholar
Csanady, G. T. 1967 Large-scale motion in the Great Lakes. J. Geophys. Res. 72, 4151–4162.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, 2182–2191.2.0.CO;2>CrossRefGoogle Scholar
Defant, A. 1961 Physical Oceanography, vol. II. Pergamon.Google Scholar
Dintrans, B., Rieutord, M. & Valdettaro, L. 1999 Gravito-inertial waves in a rotating stratified sphere or spherical shell. J. Fluid Mech. 398, 271–297.CrossRefGoogle Scholar
Dore, B. D. 1968 The viscous damping of internal waves on the rotating earth. Pure Appl. Geophys. 71, 118–131.CrossRefGoogle Scholar
van Dorn, W. G. 1966 Boundary dissipation of oscillatory waves. J. Fluid Mech. 24, 769–779.CrossRefGoogle Scholar
Drijfhout, S. & Maas, L. R. M. 2007 Impact of channel geometry and rotation on the trapping of internal tides. J. Phys. Oceanogr. 37, 2740–2763.CrossRefGoogle Scholar
Duck, P. W. & Foster, M. R. 2001 Spin-up of homogeneous and stratified fluid. Annu. Rev. Fluid Mech. 33, 231–263.CrossRefGoogle Scholar
van Dyke, M. 1964 Perturbation Methods in Fluid Mechanics. Academic.Google Scholar
Gill, A. E. 1982 Atmosphere–Ocean Dynamics. Academic.Google Scholar
Gómez-Giraldo, E. A. 2007 Observation of energy transfer mechanisms associated with internal waves. PhD Thesis. University of Western Australia.Google Scholar
Gómez-Giraldo, A., Imberger, J., Antenucci, J. P. & Yeates, P. S. 2008 Wind-shear-generated high-frequency internal waves as precursors to mixing in a stratified lake. Limnol. Oceanogr. 53, 354–367.CrossRefGoogle Scholar
Greenspan, H. P. 1968 Theory of Rotating Fluid. Cambridge University Press.Google Scholar
Hodges, B. R., Imberger, J., Saggio, A. & Winters, K. B. 2000 Modeling basin-scale internal waves in a stratified lake. Limnol. Oceanogr. 45, 1603–1620.CrossRefGoogle Scholar
Horn, D. A., Imberger, J. & Ivey, G. N. 2001 The degeneration of large-scale interfacial gravity waves in lakes. J. Fluid Mech. 434, 181–207.CrossRefGoogle Scholar
Hunt, J. N. 1952 Viscous damping of waves over an inclined bed in a channel with finite width. Houille Blanche 7, 836–842.CrossRefGoogle Scholar
Hurley, D. G. & Imberger, J. 1969. Surface and internal waves in a liquid of variable depth. Bull. Aust. Math. Soc. 1, 29–46.CrossRefGoogle Scholar
Ivey, G. N. & Nokes, R. I. 1989 Vertical mixing due to the breaking of critical internal waves on sloping boundaries. J. Fluid Mech. 204, 479–500.CrossRefGoogle Scholar
Johns, B. 1968 A boundary layer method for the determination of the viscous damping of small amplitude gravity waves. Quart. J. Mech. Appl. Math. 21, 93–103.CrossRefGoogle Scholar
Kalaba, R., Spingarn, K. & Tesfatsion, L. 1981 Variational equations for the eigenvalues and eigenvectors of nonsymmetric matrices. J. Optim. Theory App. 33, 1–8.CrossRefGoogle Scholar
Lamb, H. 1932 Hydrodynamics, 6th edn. Dover.Google Scholar
Lamb, K. G. & Yan, L. 1996 The evolution of internal wave undular bores: comparisons with a fully nonlinear numerical model with weakly nonlinear theory. J. Phys. Oceanogr. 26, 2712–2734.2.0.CO;2>CrossRefGoogle Scholar
LeBlond, P. H. 1966 On the damping of internal gravity waves in a continuously stratified ocean. J. Fluid Mech. 25, 121–142.CrossRefGoogle Scholar
Lemckert, C., Antenucci, J., Saggio, A. & Imberger, J. 2004 Physical properties of turbulent benthic boundary layers generated by internal waves. J. Hydraul. Engng 130, 58–69.CrossRefGoogle Scholar
Lighthill, J. 1978. Waves in Fluids. Cambridge University Press.Google Scholar
Maas, L. R. M. & Lam, F.-P. A. 1995 Geometric focusing of internal waves. J. Fluid Mech. 300, 1–41.CrossRefGoogle Scholar
Martinsen, E. A. & Weber, J. E. 1981 Frictional influence on internal Kelvin waves. Tellus 33, 402–410.CrossRefGoogle Scholar
Mei, C. C. & Liu, L. F. 1973 The damping of surface gravity waves in a bounded liquid. J. Fluid Mech. 59, 239–256.CrossRefGoogle Scholar
Mitsudera, H. & Hanawa, K. 1988 Damping of coastal trapped waves due to bottom friction in a baroclinic ocean. Cont. Shelf Res. 8, 113–129.CrossRefGoogle Scholar
Mofjeld, H. O. 1980 Effects of vertical viscosity on Kelvin waves. J. Phys. Oceanogr. 10, 1039–1050.2.0.CO;2>CrossRefGoogle Scholar
Patankar, S. V. 1980 Numerical Heat Transfer and Fluid Flow. Hemisphere.Google Scholar
Pedlosky, J. 1979 Geophysical Fluid Dynamics. Springer.CrossRefGoogle Scholar
Platzman, G. W. 1972 Two-dimensional free oscillations in natural basins. J. Phys. Oceanogr. 2, 117–138.2.0.CO;2>CrossRefGoogle Scholar
Platzman, G. W. 1984 Normal modes of the World Ocean. Part 3. A procedure for tidal synthesis. J. Phys. Oceanogr. 14, 1521–1531.2.0.CO;2>CrossRefGoogle Scholar
Proudman, J. 1929 On a general expansion in the theory of the tides. Proc. Lond. Math. Soc. 29, 527–536.CrossRefGoogle Scholar
Romea, R. D. & Allen, J. S. 1984 The effect of friction and topography on coastal internal Kelvin waves at low latitude. Tellus 36A, 384–400.CrossRefGoogle Scholar
Saggio, A. & Imberger, J. 2001 Mixing and turbulent fluxes in the metalimnion of a stratified lake. Limnol. Oceanogr. 46, 392–409.CrossRefGoogle Scholar
Salwen, H. & Grosch, C. E. 1981 The continuous spectrum of the Orr–Sommerfeld equation. Part 2. Eigenfunction expansions. J. Fluid Mech. 104, 445–465.CrossRefGoogle Scholar
Serruya, S. 1975 Wind, water temperature and motions in Lake Kinneret: general pattern. Verh. Intl Verein. Limnol. 19, 73–87.Google Scholar
Shimizu, K. & Imberger, J. 2008 Energetics and damping of internal waves in a strongly stratified lake. Limnol. Oceanogr. 53, 1574–1588.CrossRefGoogle Scholar
Shimizu, K., Imberger, J. & Kumagai, M. 2007 Horizontal structure and excitation of primary motions in a strongly stratified lake. Limnol. Oceanogr. 52, 2641–2655.CrossRefGoogle Scholar
Spence, G. S. M., Foster, M. R. & Davis, P. A. 1992 The transient response of a contained rotating stratified fluid to impulsively started surface forcing. J. Fluid Mech. 243, 33–50.CrossRefGoogle Scholar
Stewartson, K. 1957 On almost rigid rotations. J. Fluid Mech. 3, 17–26.CrossRefGoogle Scholar
Stocker, R. & Imberger, J. 2003 Energy partitioning and horizontal dispersion in a stratified rotating lake. J. Phys. Oceanogr. 33, 512–529.2.0.CO;2>CrossRefGoogle Scholar
Thorpe, S. A. 1987 Current and temperature variability on the continental slope. Phil. Trans. R. Soc. Lond. A 323, 471–517.Google Scholar
Ursell, F. 1952 Edge waves on a sloping beach. Proc. R. Soc. Lond. A 214, 79–97.Google Scholar
Wake, G. W., Ivey, G. N. & Imberger, J. 2005 The temporal evolution of baroclinic basin-scale waves in a rotating circular basin. J. Fluid Mech. 523, 367–392.CrossRefGoogle Scholar
Walin, G. 1969 Some aspects of time-dependent motion of a stratified rotating fluid. J. Fluid Mech. 36, 289–307.CrossRefGoogle Scholar
Wüest, A., Piepke, G. & Van Senden, D. C. 2000 Turbulent kinetic energy balance as a tool for estimating vertical diffusivity in wind-forced stratified waters. Limnol. Oceanogr. 45, 1388–1400.CrossRefGoogle Scholar
Yeates, Y. & Imberger, J. 2003 Pseudo two-dimensional simulations of internal and buoyancy fluxes in stratified lakes and reservoirs. Intl J. River Basin Manage. 1, 297–319.CrossRefGoogle Scholar
Zou, Q. 2002 An analytical model of wave bottom boundary layers incorporating turbulent relaxation and diffusion effects. J. Phys. Oceanogr. 32, 2441–2456.CrossRefGoogle Scholar