Hostname: page-component-f7d5f74f5-qghsn Total loading time: 0 Render date: 2023-10-05T00:35:53.388Z Has data issue: false Feature Flags: { "corePageComponentGetUserInfoFromSharedSession": true, "coreDisableEcommerce": false, "coreDisableSocialShare": false, "coreDisableEcommerceForArticlePurchase": false, "coreDisableEcommerceForBookPurchase": false, "coreDisableEcommerceForElementPurchase": false, "coreUseNewShare": true, "useRatesEcommerce": true } hasContentIssue false

A limitation on Long's model in stratified fluid flows

Published online by Cambridge University Press:  29 March 2006

Harvey Segur
California Institute of Technology, Pasadena, California 91109


The flow of a continuously stratified fluid into a contraction is examined, under the assumptions that the dynamic pressure and the density gradient are constant upstream (Long's model). It is shown that a solution to the equations exists if and only if the strength of the contraction does not exceed a certain critical value which depends on the internal Froude number. For the flow of a stratified fluid over a finite barrier in a channel, it is further shown that, if the barrier height exceeds this same critical value, lee-wave amplitudes increase without bound as the length of the barrier increases. The breakdown of the model, as implied by these arbitrarily large amplitudes, is discussed. The criterion is compared with available experimental results for both geometries.

Research Article
© 1971 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.)


Ball, F. K. 1959 Long waves, lee waves and gravity waves Quart. J. Roy. Met. Soc. 85, 2430.Google Scholar
Benjamin, T. B. 1970 Upstream influence J. Fluid Mech. 40, 4980.Google Scholar
Davis, R. E. 1969 The two-dimensional flow of a stratified fluid over an obstacle J. Fluid Mech. 36, 127143.Google Scholar
Debler, W. R. 1959 Stratified flow into a line sink J. Eng. Mech. Div., Proc. ASCE, 85, 5165.Google Scholar
Drazin, P. G. & Moore, D. W. 1967 Steady two-dimensional flow of fluid of variable density over an obstacle J. Fluid Mech. 28, 353370.Google Scholar
Dubreil-Jacotin, M. L. 1935 Complément à une note antérieure sur les ondes de type permanent dans les liquides hétérogènes. Atti Accad. Naz. Lincei Rc. (Cl. Sci. Fis. Mat. Nat.) (6), 21, 344346.Google Scholar
Friedman, B. 1956 Principles and Techniques of Applied Mathematics. New York: Wiley.
Grimshaw, R. 1968 A note on the steady two-dimensional flow of a stratified fluid over an obstacle J. Fluid Mech. 33, 293301.Google Scholar
Hardy, G. H., Littlewood, J. E. & Polya, G. 1952 Inequalities. Cambridge University Press.
Kao, T. W. 1965 The phenomenon of blocking in stratified flows J. Geophys. Res. 70, 815822.Google Scholar
Long, R. R. 1953a Some aspects of the flow of stratified fluids, I. Tellus, 5, 4258.Google Scholar
Long, R. R. 1953b Steady motion around a symmetrical obstacle moving along the axis of a rotating liquid J. Meteorology, 10, 197203.Google Scholar
Long, R. R. 1955 Some aspects of the flow of stratified fluids. III. Continuous density gradients Tellus, 7, 341357.Google Scholar
Trustrum, K. 1964 Rotating and stratified fluid flow J. Fluid Mech. 19, 415432.Google Scholar
Yih, C. C. 1965 Dynamics of Non-homogeneous Fluids. New York: Macmillan.