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Instability of sheared density interfaces

Published online by Cambridge University Press:  03 December 2018

T. S. Eaves Open the ORCID record for T. S. Eaves [Opens in a new window]  and
N. J. Balmforth Open the ORCID record for N. J. Balmforth [Opens in a new window]
T. S. Eaves*
Affiliation:
Department of Mathematics, University of British Columbia, 1984 Mathematics Road, Vancouver BC, V6T 1Z2, Canada
N. J. Balmforth
Affiliation:
Department of Mathematics, University of British Columbia, 1984 Mathematics Road, Vancouver BC, V6T 1Z2, Canada
*
†Email address for correspondence: tse23@math.ubc.ca
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Abstract

Of the canonical flow instabilities (Kelvin–Helmholtz, Holmboe-wave and Taylor–Caulfield) of stratified shear flow, the Taylor–Caulfield instability (TCI) has received relatively little attention, and forms the focus of the current study. First, a diagnostic of the linear instability dynamics is developed that exploits the net pseudomomentum to distinguish TCI from the other two instabilities for any given flow profile. Second, the nonlinear dynamics of TCI is studied across its range of unstable horizontal wavenumbers and bulk Richardson numbers using numerical simulation. At small bulk Richardson numbers, a cascade of billow structures of sequentially smaller size may form. For large bulk Richardson numbers, the primary nonlinear travelling waves formed by the linear instability break down via a small-scale, Kelvin–Helmholtz-like roll-up mechanism with an associated large amount of mixing. In all cases, secondary parasitic nonlinear Holmboe waves appear at late times for high Prandtl number. Third, a nonlinear diagnostic is proposed to distinguish between the saturated states of the three canonical instabilities based on their distinctive density–streamfunction and generalised vorticity–streamfunction relations.

JFM classification

Geophysical and Geological Flows: Geophysical and Geological Flows Instability: Instability Geophysical and Geological Flows: Stratified flows
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 860 , 10 February 2019 , pp. 145 - 171
Copyright
© 2018 Cambridge University Press 

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References

Abarbanel, H. D. I., Holm, D. D., Marsden, J. E. & Ratiu, T. S. 1986 Nonlinear stability analysis of stratified fluid equilibria. Phil. Trans. R. Soc. Lond. A 318, 349–409.Google Scholar
Balmforth, N. J., Roy, A. & Caulfield, C. P. 2012 Dynamics of vorticity defects in stratified shear flow. J. Fluid Mech. 694, 292–331.Google Scholar
Bühler, O. 2014 Waves and Mean Flows. Cambridge University Press.Google Scholar
Carpenter, J. R., Balmforth, N. J. & Lawrence, G. A. 2010 Identifying unstable modes in stratified shear layers. Phys. Fluids 22, 054104.Google Scholar
Carpenter, J. R., Tedford, E. W., Heifetz, E. & Lawrence, G. A. 2011 Instability in stratified shear flow: review of a physical interpretation based on interacting waves. Appl. Mech. Rev. 64, 060801.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. 412, 1–47.Google Scholar
Caulfield, C. P., Peltier, W. R., Yoshida, S. & Ohtani, M. 1995 An experimental investigation of the instability of a shear flow with multilayered density stratification. Phys. Fluids 7, 3028–3041.Google Scholar
Eaves, T. S. & Caulfield, C. P. 2017 Multiple instability of layered stratified plane Couette flow. J. Fluid Mech. 813, 250–278.Google Scholar
van Haren, H., Gostiaux, L., Morozov, E. & Tarakanov, R. 2014 Extremely long Kelvin–Helmholtz billow trains in the Romanche fracture zone. Geophys. Res. Lett. 41, 8445–8451.Google Scholar
Helmholtz, H. 1868 On discontinuous movements of fluids. Phil. Mag. 36, 337–346.Google Scholar
Holmboe, J. 1962 On the behavior of symmetric waves in stratified shear layers. Geophys. Publ. 24, 67–113.Google Scholar
Howard, L. N. 1961 Note on a paper of John W. Miles. J. Fluid Mech. 10, 509–512.Google Scholar
Kelvin, Lord 1871 Hydrokinetic solutions and observations. Phil. Mag. 42, 362–377.Google Scholar
Lee, V. & Caulfield, C. P. 2001 Nonlinear evolution of a layered stratified shear flow. Dyn. Atmos. Oceans 34, 103–124.Google Scholar
Lindzen, R. S. & Barker, R. S. 1985 Instability and wave over-reflection in stably stratified shear flow. J. Fluid Mech. 151, 189–217.Google Scholar
Long, R. R. 1953 Some aspects of the flow of stratified fluids. Part I. A theoretical investigation. Tellus 5, 42–57.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.Google 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.Google 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, 496–508.Google Scholar
Peltier, W. R. & Caulfield, C. P. 2003 Mixing efficiency in stratified shear flows. Annu. Rev. Fluid Mech. 35 (1), 135–167.Google Scholar
Salehipour, H., Caulfield, C. P. & Peltier, C. P. 2016 Turbulent mixing due to the Holmboe wave instability at high Reynolds number. J. Fluid. Mech. 803, 591–621.Google Scholar
Scinocca, J. F. 1995 The mixing of mass and momentum by Kelvin–Helmholtz billows. J. Atmos. Sci. 52, 2509–2530.Google Scholar
Smyth, W. D., Carpenter, J. R. & Lawrence, G. A. 2007 Mixing in symmetric Holmboe waves. J. Phys. Oceanogr. 37, 1566–1583.Google Scholar
Smyth, W. D. & Winters, K. B. 2003 Turbulence and mixing in Holmboe waves. J. Phys. Oceanogr. 33, 694–711.Google Scholar
Stuart, J. T. 1967 On finite amplitude oscillations in laminar mixing layers. J. Fluid Mech. 29, 417–440.Google Scholar
Synge, J. L. 1933 The stability of heterogeneous liquid. Trans. R. Soc. Can. 27, 1–18.Google Scholar
Taylor, G. I. 1931 Effect of variation in density on the stability of superposed streams of fluid. Proc. R. Soc. Lond. 132, 449–523.Google Scholar
Taylor, J. R.2008 Numerical simulations of the stratified oceanic bottom boundary layer. PhD thesis, Mech. Eng., UCSD.Google Scholar
Tedford, E. W., Pieters, R. & Lawrence, G. A. 2009 Symmetric Holmboe instabilities in a laboratory exchange flow. J. Fluid Mech. 636, 137–153.Google Scholar

Eaves and Balmforth supplementary movie 1

Movie of the flow evolution of the simulations of Group 1 (k,J) = (0.8,0.14) showing the density field next to the vorticity field for the (Re,Pr) = (300 000,0.6), first row, (Re,Pr) = (180 000,1), second row, (Re,Pr) = (60 000,3), third row, and (Re,Pr) = (20 000,10), forth row. Timer shows advective time units.

Download Eaves and Balmforth supplementary movie 1(Video)
Video 1.5 MB

Eaves and Balmforth supplementary movie 2

Movie of the flow evolution of the simulations of Group 2 (k,J) = (4/3,0.14) showing the density field next to the vorticity field for the (Re,Pr) = (300 000,0.6), first row, (Re,Pr) = (180 000,1), second row, (Re,Pr) = (60 000,3), third row, and (Re,Pr) = (20 000,10), forth row. Timer shows advective time units.

Download Eaves and Balmforth supplementary movie 2(Video)
Video 1.5 MB

Eaves and Balmforth supplementary movie 3

Movie of the flow evolution of the simulations of Group 3 (k,J) = (4/3,0.23) showing the density field next to the vorticity field for the (Re,Pr) = (300 000,0.6), first row, (Re,Pr) = (180 000,1), second row, (Re,Pr) = (60 000,3), third row, and (Re,Pr) = (20 000,10), forth row. Timer shows advective time units.

Download Eaves and Balmforth supplementary movie 3(Video)
Video 1.9 MB

Eaves and Balmforth supplementary movie 4

Movie of the flow evolution of the simulations of Group 4 (k,J) = (4/3,0.3) showing the density field next to the vorticity field for the (Re,Pr) = (300 000,0.6), first row, (Re,Pr) = (180 000,1), second row, (Re,Pr) = (60 000,3), third row, and (Re,Pr) = (20 000,10), forth row. Timer shows advective time units.

Download Eaves and Balmforth supplementary movie 4(Video)
Video 2 MB

Eaves and Balmforth supplementary movie 5

Movie of the flow evolution of the simulations of Group 5 (k,J) = (2,0.3) showing the density field next to the vorticity field for the (Re,Pr) = (300 000,0.6), top left, (Re,Pr) = (180 000,1), top right, (Re,Pr) = (60 000,3), bottom left, and (Re,Pr) = (20 000,10), bottom right. Timer shows advective time units.

Download Eaves and Balmforth supplementary movie 5(Video)
Video 1.3 MB

Eaves and Balmforth supplementary movie 6

Movie of the flow evolution of the simulations of Group 6 (k,J) = (4,0.5) showing the density field next to the vorticity field for the (Re,Pr) = (300 000,0.6), top left, (Re,Pr) = (180 000,1), top right, (Re,Pr) = (60 000,3), bottom left, and (Re,Pr) = (20 000,10), bottom right. Timer shows advective time units.

Download Eaves and Balmforth supplementary movie 6(Video)
Video 837.6 KB