Hostname: page-component-848d4c4894-wzw2p Total loading time: 0 Render date: 2024-05-22T01:29:23.655Z Has data issue: false hasContentIssue false
  • English
  • Français

On the normal modes of parallel flow of inviscid stratified fluid

Published online by Cambridge University Press:  29 March 2006

W. H. H. Banks ,
P. G. Drazin  and
M. B. Zaturska
W. H. H. Banks
Affiliation:
School of Mathematics, University of Bristol, England
P. G. Drazin
Affiliation:
School of Mathematics, University of Bristol, England
M. B. Zaturska
Affiliation:
School of Mathematics, University of Bristol, England
Get access Rights & Permissions [Opens in a new window]

Abstract

The overall pattern of normal modes of parallel flow of inviscid stratified fluid is examined. For a given flow and wavenumber the modes are divided into five classes, some of which may be empty: (i) a finite class of non-singular unstable modes; (ii) a conjugate finite class of non-singular damped stable modes; (iii) a finite class of singular stable modes, each of these having a branch point and being the limit of unstable modes; (iv) a discrete class of modified internal gravity waves, these being non-singular stable modes (if the density decreases with height everywhere); (v) a continuous class of singular stable modes. The modified internal gravity waves are described asymptotically for large values of the Richardson number. These asymptotic results are related to and extended by numerical calculations for a sinusoidal basic velocity profile and a Bickley jet. The wave speeds for small values of the Richardson number are found to depend only upon the local behaviour of the mean flow near an overall simple maximum or minimum of the velocity profile. Finally some difficulties in the use of the Howard formula for perturbation at a curve of marginal stability are elucidated.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 75 , Issue 1 , 13 May 1976 , pp. 149 - 171
Copyright
© 1976 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.
Banks, W. H. H. & Drazin, P. G. 1973 Perturbation methods in boundary-layer theory J. Fluid Mech. 58, 763.Google Scholar
Case, K. M. 1960 Stability of an idealized atmosphere. I. Discussion of results Phys. Fluids, 3, 149.Google Scholar
Dikiy, L. A. & Katayev, V. V. 1971 Calculation of the planetary wave spectrum by the Galerkin method Izv. Atmos. Ocean. Phys. 7, 682.Google Scholar
Drazin, P. G. 1958 On the dynamics of a fluid of variable density. Ph.D. thesis, University of Cambridge.
Drazin, P. G. & Howard, L. N. 1966 Hydrodynamic stability of parallel flow of inviscid fluid Adv. Appl. Mech. 9, 1.Google Scholar
Eliassen, A., HÖSILAND, E. & Riis, E. 1953 Two-dimensional perturbation of a flow with constant shear of a stratified fluid. Inst. Weather & Climate Res., Oslo, publ. no. 1.Google Scholar
Friedman, B. 1956 Principles and Techniques of Applied Mathematics. Wiley.
Goldstein, S. 1931 On the stability of superposed streams of fluids of different densities. Proc. Roy. Soc. A 132, 524.Google Scholar
Groen, P. 1948 Contribution to the theory of internal waves. Kon. Ned. Met. Inst., Mededelingen en Verhandelingen, B 125.Google Scholar
Haurwitz, B. 1931 Zur Theorie den Wellenbewegungen in Luft und Wasser Veröff Geophys. Inst., Univ. Leipzig, 6, no. 1.Google Scholar
Hazel, P. 1972 Numerical studies of the stability of inviscid stratified shear flows J. Fluid Mech. 51, 39.Google Scholar
Høsiland, E. 1953 On the dynamic effect of variation in density on two-dimensional perturbations of flow with constant shear Geofys. Publ. 18, no. 10.Google Scholar
Howard, L. N. 1963 Neutral curves and stability boundaries in stratified flow J. Fluid Mech. 16, 333.Google Scholar
Howard, L. N. 1964 The number of unstable modes in hydrodynamic stability problems J. Mécanique, 3, 433.Google Scholar
Howard, L. N. & Maslowe, S. A. 1973 Stability of stratified shear flows Boundary-Layer Met. 4, 511.Google Scholar
Huppert, H. E. 1973 On Howard's technique for perturbing neutral solutions of the Taylor-Goldstein equation J. Fluid Mech. 57, 361.Google Scholar
Miles, J. W. 1967 Internal waves in a continuously stratified atmosphere or ocean J. Fluid Mech. 28, 305.Google Scholar
Taylor, G. I. 1931 Effect of variation of density on the stability of superposed streams of fluid. Proc. Roy. Soc. A 132, 499.Google Scholar
Turner, J. S. 1973 Buoyancy Effects in Fluids. Cambridge University Press.
Yih, C.-S. 1965 Dynamics of Non-homogeneous Fluids. Macmillan.