Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-19T08:22:28.116Z Has data issue: false hasContentIssue false
  • English
  • Français

Internal waves in a contained rotating stratified fluid

Published online by Cambridge University Press:  20 April 2006

Susan Friedlander  and
William L. Siegmann
Susan Friedlander
Affiliation:
Department of Mathematics, University of Illinois, Chicago Circle, Chicago, Illinois 60680
William L. Siegmann
Affiliation:
Department of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, New York 12181
Get access Rights & Permissions [Opens in a new window]

Abstract

Small-amplitude time-dependent motions of a uniformly rotating, density-stratified, Boussinesq non-dissipative fluid in a rigid container are examined for the case of the rotation axis parallel to gravity. We consider a variety of container shapes, along with arbitrary values for the (constant) Brunt-Väisälä and rotation frequencies. We demonstrate a number of properties of the eigenvalues and eigenfunctions of square-integrable oscillatory motions. Some of these properties hold generally, while others are shown for specific classes of containers (such as with symmetry about the container axis). A full solution is presented for the response of fluid in a cylindrical container to an arbitrary initial disturbance. Features of this solution which are different from the cases of no stratification or no rotation are emphasized. For the situation when Brunt-Väisälä and rotation frequencies are equal, characteristics of the oscillation frequencies and modal structures are found for containers of quite general shape. This situation illustrates, in particular, effects which are possible when rotation and stratification act together and which have been overlooked in previous investigations that assume that the vertical length scale is much smaller than the horizontal scales.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 114 , January 1982 , pp. 123 - 156
Copyright
© 1972 Cambridge University Press

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

Abramowitz, M. & Stegun, I. A. 1965 Handbook of Mathematical Functions. Dover.
Allen, J. S. 1971 Some aspects of the initial value problem for the inviscid motion of a contained rotating, weakly stratified fluid. J. Fluid Mech. 46, 1.Google Scholar
Bryan, G. H. 1889 The waves on a rotating liquid spheroid of finite ellipticity. Phil. Trans. R. Soc. Lond. A 180, 187.Google Scholar
Cairns, J. L. & Williams, G. O. 1976 Internal wave observations from a midwater float. 2. J. Geophys. Res. 81, 1943.Google Scholar
Clarke, A. J. 1977 Wind-forced linear and nonlinear Kelvin waves along an irregular coast line. J. Fluid Mech. 83, 337.Google Scholar
Csanady, G. T. 1976 Topographic waves in Lake Ontario. J. Phys. Oceanog. 6, 93.Google Scholar
Eckart, C. 1960 Hydrodynamics of Oceans and Atmospheres. Pergamon.
Erdelyi, A. 1953 Tables of integral transforms. McGraw-Hill.
Friedlander, S. & Siegmann, W. L. 1982 Internal waves in a rotating stratified fluid in an arbitrary gravitational field. Geophys. Astrophys. Fluid Dyn. (to appear).Google Scholar
Garrett, C. & Munk, W. 1979 Internal waves in the ocean. Ann. Rev. Fluid Mech. 11, 339.Google Scholar
Gradshteyn, I. S. & Ryzhik, I. W. 1965 Table of Integrals, Series, and Products. Academic.
Greenspan, H. P. 1968 The Theory of Rotating Fluids. Cambridge University Press.
Hamblin, R. F. 1978 Internal Kelvin waves in a fjord lake. J. Geophys. Res. 83, 2409.Google Scholar
Howard, L. N. & Siegmann, W. L. 1969 On the initial value problem for rotating stratified flow. Stud. Appl. Math. 48, 153.Google Scholar
Huthnance, J. M. 1978 On coastal trapped waves: analysis and numerical calculation by inverse iteration. J. Phys. Oceanog. 8, 74.Google Scholar
Kamenkovich, V. M. 1977 Fundamentals of Ocean Dynamics. Elsevier.
Krauss, W. 1966 Methoden und Ergebnisse der Theoretischen Oceanographie II: Interne Wellen. Berlin: Gebrueder-Borntraeger.
Kudlick, M. D. 1966 On transient motions in a contained rotating fluid. Ph.D. dissertation, Massachusetts Institute of Technology.
Leblond, P. H. & Mysak, L. A. 1979 Ocean waves: a survey of some recent results. S.I.A.M. Review 21, 289.Google Scholar
London, S. & Shen, M. C. 1979 Free oscillations in a rotating spherical shell. Phys. Fluids 22, 2071.Google Scholar
Malkus, W. V. R. 1967 Hydromagnetic planetary waves. J. Fluid Mech. 28, 793.Google Scholar
Mccreary, J. 1976 Eastern tropical ocean response to changing wind systems: with application to El Niño. J. Phys. Oceanog. 6, 632.Google Scholar
Miles, J. W. 1974 On Laplace's tidal equations. J. Fluid Mech. 66, 241.Google Scholar
Mysak, L. A. 1980a Topographically trapped waves. Ann. Rev. Fluid Mech. 12, 45.Google Scholar
Mysak, L. A. 1980b Recent advances in shelf wave dynamics. Rev. Geophys. Space Phys. 18, 211.Google Scholar
Ou, H. W. & Bennett, J. R. 1979 A theory of the mean flow driven by long internal waves in a rotating basin, with application to Lake Kinneret. J. Phys. Oceanog. 9, 1112.Google Scholar
Philander, S. G. H. 1978 Variability of the tropical oceans. Dynamics Oceans & Atmos. 3, 191.Google Scholar
Phillips, O. M. 1977 The Dynamics of the Upper Ocean. Cambridge University Press.
Reed, M. & Simon, B. 1978 Methods of Modern Mathematical Physics, vol. IV. Academic.
Relton, F. E. 1946 Applied Bessel Functions. Blackie.
Saylor, J., Huang, J. C. K. & Miller, G. 1978 Observations of rotational waves in Southern Lake Michigan. Trans. Am. Geophys. Union 59 (12), 1107.Google Scholar
Smith, R. L. 1978 Poleward propagating perturbations in currents and sea levels along the Peru coast. J. Geophys. Res. 83, 6083.Google Scholar
Spiegel, E. A. & Veronis, G. 1960 On the Boussinesq approximations for a compressible fluid. Astrophys. J. 131, 442.Google Scholar
Stewartson, K. 1971 On trapped oscillations of a rotating fluid. Tellus 23, 506.Google Scholar
Stewartson, K. & Rickard, J. A. 1969 Pathological oscillations of a rotating stratified fluid. J. Fluid Mech. 35, 759.Google Scholar
Stewartson, K. & Walton, I. C. 1976 On waves in a thin shell of stratified rotating fluid. Proc. R. Soc. Land. A 349, 141.Google Scholar
Veronis, G. 1970 The analogy between rotating and stratified fluids. Ann. Rev. Fluid Mech. 2, 37.Google Scholar
Wang, D. P. & Mooers, C. N. K. 1976 Coastal trapped waves in a continuously stratified ocean. J. Phys. Oceanog. 6, 853.Google Scholar
Wunsch, C. 1973 On the mean drift in large lakes. J. Limnol. Oceanog. 18, 793.Google Scholar