Hostname: page-component-7dd5485656-tbj44 Total loading time: 0 Render date: 2025-10-30T13:07:08.333Z Has data issue: false hasContentIssue false
  • English
  • Français

Testing linear marginal stability in stratified shear layers

Published online by Cambridge University Press:  02 February 2018

C. J. Howland Open the ORCID record for C. J. Howland [Opens in a new window] ,
J. R. Taylor Open the ORCID record for J. R. Taylor [Opens in a new window]  and
C. P. Caulfield Open the ORCID record for C. P. Caulfield [Opens in a new window]
C. J. Howland
Affiliation:
Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
J. R. Taylor
Affiliation:
Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
C. P. Caulfield*
Affiliation:
Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK BP Institute, University of Cambridge, Madingley Road, Cambridge CB3 0EZ, UK
*
†Email address for correspondence: c.p.caulfield@bpi.cam.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

We use two-dimensional direct numerical simulations of Boussinesq stratified shear layers to investigate the influence of the minimum gradient Richardson number $Ri_{m}$ on the early time evolution of Kelvin–Helmholtz instability to its saturated ‘billow’ state. Even when the diffusion of the background velocity and density distributions is counterbalanced by artificial body forces to maintain the initial profiles, in the limit as $Ri_{m}\rightarrow 1/4$ , the perturbation growth rate tends to zero and the saturated perturbation energy becomes small. These results imply, at least for such canonical inflectional stratified shear flows, that ‘marginally unstable’ flows with $Ri_{m}$ only slightly less than 1/4 are highly unlikely to become ‘turbulent’, in the specific sense of being associated with significantly enhanced dissipation, irreversible mixing and non-trivial modification of the background distributions without additional externally imposed forcing.

JFM classification

Geophysical and Geological Flows: Geophysical and Geological Flows Instability: Instability Geophysical and Geological Flows: Stratified flows

Information

Type
JFM Rapids
Information
Journal of Fluid Mechanics , Volume 839 , 25 March 2018 , R4
Check for updates
Copyright
© 2018 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.)

Article purchase

Temporarily unavailable

References

Baker, M. A. & Gibson, C. H. 1987 Sampling turbulence in the stratified ocean: statistical consequences of strong intermittency. J. Phys. Oceanogr. 17 (10), 1817–1836.Google Scholar
Caulfield, C. P. & Peltier, W. R. 2000 The anatomy of the mixing transition in homogeneous and stratified free shear layers. J. Fluid Mech. 413, 1–47.CrossRefGoogle Scholar
Corcos, G. M. & Sherman, F. S. 1976 Vorticity concentration and the dynamics of unstable free shear layers. J. Fluid Mech. 73 (2), 241–264.Google Scholar
Fernando, H. J. S. 1991 Turbulent mixing in stratified fluids. Annu. Rev. Fluid Mech. 23, 455–493.Google Scholar
Goldstein, S. 1931 On the stability of superposed streams of fluids of different densities. Proc. R. Soc. Lond. A 132 (820), 524–548.Google Scholar
van Haren, H. & Gostiaux, L. 2010 A deep-ocean Kelvin–Helmholtz billow train. Geophys. Res. Lett. 37 (3), 1–5.CrossRefGoogle Scholar
Hazel, P. 1972 Numerical studies of the stability of inviscid parallel shear flows. J. Fluid Mech. 51 (1), 39–61.Google Scholar
Howard, L. N. 1961 Note on a paper of John W. Miles. J. Fluid Mech. 10 (4), 509–512.Google Scholar
Ivey, G. N., Winters, K. B. & Koseff, J. R. 2008 Density stratification, turbulence, but how much mixing? Annu. Rev. Fluid Mech. 40, 169–184.Google Scholar
Kaminski, A. K., Caulfield, C. P. & Taylor, J. R. 2014 Transient growth in strongly stratified shear layers. J. Fluid Mech. 758 (R4), 1–12.Google Scholar
Kaminski, A. K., Caulfield, C. P. & Taylor, J. R. 2017 Nonlinear evolution of linear optimal perturbations of strongly stratified shear layers. J. Fluid Mech. 825, 213–244.Google Scholar
Klaassen, G. P. & Peltier, W. R. 1985 Evolution of finite amplitude Kelvin–Helmholtz billows in two spatial dimensions. J. Atmos. Sci. 42 (12), 1321–1339.Google Scholar
Lamb, K. G. 2014 Internal wave breaking and dissipation mechanisms on the continental slope/shelf. Annu. Rev. Fluid Mech. 46, 231–254.Google Scholar
Mashayek, A., Caulfield, C. P. & Peltier, W. R. 2013 Time-dependent, non-monotonic mixing in stratified turbulent shear flows: implications for oceanographic estimates of buoyancy flux. J. Fluid Mech. 736, 570–593.CrossRefGoogle Scholar
Mashayek, A. & Peltier, W. R. 2012a The zoo of secondary instabilities precursory to stratified shear flow transition. Part 1. Shear aligned convection, pairing, and braid instabilities. J. Fluid Mech. 708, 5–44.Google Scholar
Mashayek, A. & Peltier, W. R. 2012b The zoo of secondary instabilities precursory to stratified shear flow transition. Part 2. The influence of stratification. J. Fluid Mech. 708, 45–70.CrossRefGoogle Scholar
Mashayek, A. & Peltier, W. R. 2013 Shear-induced mixing in geophysical flows: does the route to turbulence matter to its efficiency? J. Fluid Mech. 725, 216–261.Google Scholar
Miles, J. W. 1961 On the stability of heterogeneous shear flows. J. Fluid Mech. 10 (4), 496–508.Google Scholar
Moum, J. N., Farmer, D. M., Smyth, W. D., Armi, L. & Vagle, S. 2003 Structure and generation of turbulence at interfaces strained by internal solitary waves propagating shoreward over the continental shelf. J. Phys. Oceanogr. 33 (10), 2093–2112.Google Scholar
Peltier, W. R. & Caulfield, C. P. 2003 Mixing efficiency in stratified shear flows. Annu. Rev. Fluid Mech. 35, 135–167.CrossRefGoogle Scholar
Rohr, J. J., Itsweire, E. C., Helland, K. N. & Atta, C. W. Van 1988 Growth and decay of turbulence in a stably stratified shear flow. J. Fluid Mech. 195, 77–111.Google Scholar
Smyth, W. D. & Moum, J. N. 2012 Ocean mixing by Kelvin–Helmholtz instability. Oceanography 25 (2), 140–149.CrossRefGoogle Scholar
Smyth, W. D. & Moum, J. N. 2013 Marginal instability and deep cycle turbulence in the eastern equatorial Pacific Ocean. Geophys. Res. Lett. 40 (23), 6181–6185.Google Scholar
Smyth, W. D. & Peltier, W. R. 1990 Three-dimensional primary instabilities of a stratified, dissipative, parallel flow. Geophys. Astrophys. Fluid Dyn. 52 (4), 249–261.CrossRefGoogle Scholar
Taylor, G. I. 1931 Effect of variation in density on the stability of superposed streams of fluid. Proc. R. Soc. Lond. A 132 (820), 499–523.Google Scholar
Taylor, J. R.2008 Numerical simulations of the stratified oceanic bottom layer. PhD thesis, University of California, San Diego.Google Scholar
Thorpe, S. A. 1973 Experiments on instability and turbulence in a stratified shear flow. J. Fluid Mech. 61 (4), 731–751.Google Scholar
Thorpe, S. A. & Liu, Z. 2009 Marginal instability? J. Phys. Oceanogr. 39 (9), 2373–2381.Google Scholar
Thorpe, S. A., Smyth, W. D. & Li, L. 2013 The effect of small viscosity and diffusivity on the marginal stability of stably stratified shear flows. J. Fluid Mech. 731, 461–476.Google Scholar
Woods, J. D. 1968 Wave-induced shear instability in the summer thermocline. J. Fluid Mech. 32 (4), 791–800.Google Scholar
Zhou, Q., Taylor, J. R. & Caulfield, C. P. 2017 Self-similar mixing in stratified plane Couette flow for varying Prandtl number. J. Fluid Mech. 820, 86–120.Google Scholar