Hostname: page-component-8448b6f56d-t5pn6 Total loading time: 0 Render date: 2024-04-19T08:34:21.249Z Has data issue: false hasContentIssue false
  • English
  • Français

Transient dynamics by continuous-spectrum perturbations in stratified shear flows

Published online by Cambridge University Press:  07 February 2013

R. Camassa  and
C. Viotti
R. Camassa
Affiliation:
Carolina Center for Interdisciplinary Applied Mathematics, Department of Mathematics, University of North Carolina, Chapel Hill, NC 27599, USA
C. Viotti*
Affiliation:
Carolina Center for Interdisciplinary Applied Mathematics, Department of Mathematics, University of North Carolina, Chapel Hill, NC 27599, USA
*
†Email address for correspondence: claudio.viotti@ucd.ie
Get access Rights & Permissions [Opens in a new window]

Abstract

The transient dynamics of the linearized Euler–Boussinesq equations governing parallel stratified shear flows is presented and analysed. Solutions are expressed as integral superpositions of generalized eigenfunctions associated with the continuous-spectrum component of the Taylor–Goldstein linear stability operator, and reveal intrinsic dynamics not captured by its discrete-spectrum counterpart. It is shown how continuous-spectrum perturbations are generally characterized by non-normal energy growth and decay with algebraic asymptotic behaviour in either time or space. This behaviour is captured by explicit long-time/far-field expressions from rigorous asymptotic analysis, and it is illustrated with direct numerical simulations of the whole (non-Boussinesq) stratified Euler system. These results can be helpful in understanding recent numerical observations for parallel and non-parallel perturbed stratified shear flows.

JFM classification

Geophysical and Geological Flows: Geophysical and Geological Flows Instability: Instability Geophysical and Geological Flows: Stratified flows
Type
Rapids
Information
Journal of Fluid Mechanics , Volume 717 , 25 February 2013 , R5
Copyright
©2013 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

Almgren, A. S., Bell, J. B., Colella, P., Howell, L. H. & Welcome, M. L. 1998 A conservative adaptive projection method for the variable density incompressible Navier–Stokes equations. J. Comput. Phys. 142, 1–46.Google Scholar
Banks, W. H. H., Drazin, P. G. & Zaturska, M. B. 1976 On the normal modes of parallel flow of inviscid stratified fluid. J. Fluid Mech. 75, 149–171.Google Scholar
Booker, J. R. & Bretherton, F. P. 1967 The critical layer for internal gravity waves in a shear flow. J. Fluid Mech. 27, 513–539.Google Scholar
Camassa, R. & Viotti, C 2012 On the response of large-amplitude internal waves to upstream disturbances. J. Fluid Mech. 702, 59–88.Google Scholar
Case, K. M. 1960 Stability of an idealized atmosphere. I. Discussion of results. Phys. Fluids 3, 149–154.Google Scholar
Dyson, F. J. 1960 Stability of an idealized atmosphere. II. Zeros of the confluent hypergeometric function. Phys. Fluids 3, 155–157.Google Scholar
Erdelyi, A. 1956 Asymptotic Expansions. Dover.Google Scholar
Faddeev, L. D. 1971 On the theory of the stability of stationary plane-parallel flows of an ideal fluid. Zap. Nauchn. Sem. Leningrad Otdel. Mat. Inst. 21, 164–172.Google Scholar
Farrel, B. F. & Ioannou, P. J. 1993 Transient development of perturbations in stratified shear flows. J. Atmos. Sci. 50, 2201–2214.2.0.CO;2>CrossRefGoogle Scholar
Grosch, C. E. & Salwen, A. 1978 The continuous spectrum of the Orr–Sommerfeld equation. Part 1. The spectrum and the eigenfunctions. J. Fluid Mech. 87, 33–54.Google Scholar
Heaton, C. J. & Peake, N. 2006 Algebraic and exponential instability of inviscid swirling flow. J. Fluid Mech. 565, 279–318.Google Scholar
Miles, J. W. 1961 On the stability of heterogeneous shear flows. Part 1. J. Fluid Mech. 10, 496–508.Google Scholar
Van Duin, C. A. & Kelder, H. 1986 Internal gravity waves in shear flows at large Reynolds number. J. Fluid Mech. 169, 293–306.Google Scholar
Whitham, G. B. 1974 Linear and Nonlinear Waves. Wiley.Google Scholar
Whittaker, E. T. & Watson, G. N. 1927 A Course of Modern Analysis. Cambridge University Press.Google Scholar