Hostname: page-component-89b8bd64d-x2lbr Total loading time: 0 Render date: 2026-05-10T00:44:04.143Z Has data issue: false hasContentIssue false
  • English
  • Français

Turbulence and mixing from neighbouring stratified shear layers

Published online by Cambridge University Press:  13 May 2024

Chih-Lun Liu Open the ORCID record for Chih-Lun Liu [Opens in a new window] ,
Alexis K. Kaminski Open the ORCID record for Alexis K. Kaminski [Opens in a new window]  and
William D. Smyth Open the ORCID record for William D. Smyth [Opens in a new window]
Chih-Lun Liu*
Affiliation:
College of Earth, Ocean and Atmospheric Sciences, Oregon State University, Corvallis, OR 97331, USA
Alexis K. Kaminski
Affiliation:
Department of Mechanical Engineering, University of California, Berkeley, Berkeley, CA 94720, USA
William D. Smyth
Affiliation:
College of Earth, Ocean and Atmospheric Sciences, Oregon State University, Corvallis, OR 97331, USA
*
†Email address for correspondence: chihlunl@gmail.com
Get access Rights & Permissions [Opens in a new window]

Abstract

Studies of Kelvin–Helmholtz (KH) instability have typically modelled the initial mean flow as an isolated stratified shear layer. However, geophysical flows frequently exhibit multiple layers. As a step towards understanding these flows, we examine the case of two adjacent stratified shear layers using both linear stability analysis and direct numerical simulations. With sufficiently large layer separation, the characteristics of instability and mixing converge towards the familiar KH turbulence, and similarly when the separation is near zero and the layers add to make a single layer, albeit with a reduced Richardson number. Here, our focus is on intermediate separations, which produce new and complex phenomena. As the separation distance $D$ increases from zero to a critical value $D_c$, approximately half the thickness of the shear layer, the growth rate and wavenumber both decrease monotonically. The minimum Richardson number is relatively low, potentially inducing pairing, and shear-aligned convective instability (SCI) is the primary mechanism for transition. Consequently, mixing is relatively strong and efficient. When $D\sim D_c$, billow length is increased but growth is slowed. Despite the modest growth rate, mixing is strong and efficient, engendered primarily by secondary shear instability manifested on the braids, and by SCI occurring on the eyelids. Shear-aligned vortices are driven in part by buoyancy production; however, shear production and vortex stretching are equally important mechanisms. When $D>D_c$, neighbouring billow interactions suppress the growth of both KH instability and SCI. Strength and efficiency of mixing decrease abruptly as $D_c$ is exceeded. As turbulence decays, layers of marginal instability may arise.

JFM classification

Mixing: Turbulent mixing Turbulent Flows: Stratified turbulence Turbulent Flows: Shear layer turbulence

Information

Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 987 , 25 May 2024 , A8
Check for updates
Copyright
© The Author(s), 2024. Published by 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

Alford, M.H. & Pinkel, R. 2000 Observations of overturning in the thermocline: the context of ocean mixing. J. Phys. Oceanogr. 30 (5), 805–832.2.0.CO;2>CrossRefGoogle Scholar
Bebieva, Y. & Speer, K. 2019 The regulation of sea ice thickness by double-diffusive processes in the Ross Gyre. J. Geophys. Res.: Oceans 124, 7068–7081.CrossRefGoogle Scholar
Brown, J. & Radko, T. 2022 Disruption of Arctic staircases by shear. Geophys. Res. Lett. 49, e2022GL100605.CrossRefGoogle Scholar
Brucker, K. & Sarkar, S. 2007 Evolution of an initially turbulent stratified shear layer. Phys. Fluids 19, 105105.CrossRefGoogle Scholar
Carpenter, J.R., Tedford, E.W., Heifetz, E. & Lawrence, G.A. 2013 Instability in stratified shear flow: review of a physical interpretation based on interacting waves. Appl. Mech. Rev. 64 (6), 060801.CrossRefGoogle Scholar
Caulfield, C.P. 2021 Layering, instabilities, and mixing in turbulent stratified flows. Annu. Rev. Fluid Mech. 53 (1), 113–145.CrossRefGoogle Scholar
Caulfield, C.P. & Peltier, W.R. 2000 Anatomy of the mixing transition in homogeneous and stratified free shear layers. J. Fluid Mech. 413, 1–47.CrossRefGoogle Scholar
Chang, M.-H., et al. 2022 Internal hydraulic transition and turbulent mixing observed in the Kuroshio over the I-Lan ridge off Northeastern Taiwan. J. Phys. Oceanogr. 1, 3179–3198.CrossRefGoogle Scholar
Corcos, G.M. & Sherman, F.S. 1976 Vorticity concentration and the dynamics of unstable free shear layers. J. Fluid Mech. 73, 241–264.CrossRefGoogle Scholar
Davis, P.A. & Peltier, W.R. 1979 Some characteristics of the Kelvin–Helmholtz and resonant overreflection modes of shear flow instability and of their interaction through vortex pairing. J. Atmos. Sci. 36 (12), 2394–2412.2.0.CO;2>CrossRefGoogle Scholar
Desaubies, Y. & Smith, W.K. 1982 Statistics of Richardson number and instability in oceanic internal waves. J. Phys. Oceanogr. 12 (11), 1245–1259.2.0.CO;2>CrossRefGoogle Scholar
Fritts, D.C. et al. 2023 Kelvin Helmholtz instability ‘tube’ & ‘knot’ dynamics, Part I: expanding observational evidence of occurrence and environmental influences. J. Atmos. Sci. 80 (10), 2419–2437.CrossRefGoogle Scholar
Fritts, D.C., Bizon, C., Werne, J.A. & Meyer, C.K. 2003 Layering accompanying turbulence generation due to shear instability and gravity-wave breaking. J. Geophys. Res.: Atmos. 108 (D8).CrossRefGoogle Scholar
Geyer, W.R., Lavery, A., Scully, M.E. & Trowbridge, J.H. 2010 Mixing by shear instability at high Reynolds number. Geophys. Res. Lett. 37, L22607.CrossRefGoogle Scholar
Gregg, M.C., D'Asaro, E.A., Riley, J.J. & Kunze, E. 2018 Mixing efficiency in the ocean. Ann. Rev. Mar. Sci. 10 (1), 443–473.CrossRefGoogle ScholarPubMed
Hazel, P. 1972 Numerical studies of the stability of inviscid parallel shear flows. J. Fluid Mech. 51, 39–62.CrossRefGoogle Scholar
Heifetz, E., Bishop, C.H., Hoskins, B.J. & Methven, J. 2004 The counter-propagating Rossby-wave perspective on baroclinic instability. I: mathematical basis. Q. J. R. Meteorol. Soc. 130, 211–231.Google Scholar
Heifetz, E. & Guha, A. 2019 Normal form of synchronization and resonance between vorticity waves in shear flow instability. Phys. Rev. E 100 (4), 043105.CrossRefGoogle ScholarPubMed
Holleman, R.C., Geyer, W.R. & Ralston, D.K. 2016 Stratified turbulence and mixing efficiency in a salt wedge estuary. J. Phys. Oceanogr. 46, 1769–1783.CrossRefGoogle Scholar
Howard, L.N. 1961 Note on a paper of John W. Miles. J. Fluid Mech. 10, 509–512.CrossRefGoogle Scholar
Kaminski, A.K., D'Asaro, E.A., Shcherbina, A.Y. & Harcourt, R.R. 2021 High-resolution observations of the North Pacific transition layer from a Lagrangian float. J. Phys. Oceanogr. 51 (10), 3163–3181.Google Scholar
Kaminski, A.K. & Smyth, W.D. 2019 Stratified shear instability in a field of pre-existing turbulence. J. Fluid Mech. 863, 639–658.CrossRefGoogle Scholar
Klaassen, G.P. & Peltier, W.R. 1985 The onset of turbulence in finite-amplitude Kelvin–Helmholtz billows. J. Fluid Mech. 155, 1–35.CrossRefGoogle Scholar
Klaassen, G.P. & Peltier, W.R. 1989 The role of transverse secondary instabilities in the evolution of free shear layers. J. Fluid Mech. 202, 367–402.CrossRefGoogle Scholar
Klaassen, G.P. & Peltier, W.R. 1991 The influence of stratification on secondary instability in free shear layers. J. Fluid Mech. 227, 71–106.CrossRefGoogle Scholar
Lasheras, J.C. & Choi, H. 1988 Three-dimensional instability of a plane free shear layer: an experimental study of the formation and evolution of streamwise vortices. J. Fluid Mech. 189, 53–86.CrossRefGoogle Scholar
Lee, V. & Caulfield, C.P. 2001 Nonlinear evolution of a layered stratified shear flow. Dyn. Atmos. Oceans 34, 103–124.CrossRefGoogle Scholar
Lewin, S.F. & Caulfield, C.P. 2021 The influence of far-field stratification on turbulent mixing. J. Fluid Mech. 928, A20.CrossRefGoogle Scholar
Lian, Q., Smyth, W.D. & Liu, Z. 2020 Numerical computation of instabilities and internal waves from in situ measurements via the viscous Taylor–Goldstein problem. J. Atmos. Ocean. Technol. 37 (5), 759–776.CrossRefGoogle Scholar
Liu, C.-L., Kaminski, A.K. & Smyth, W.D. 2022 The butterfly effect and the transition to turbulence in a stratified shear layer. J. Fluid Mech. 953, A43.CrossRefGoogle Scholar
Liu, C.-L., Kaminski, A.K. & Smyth, W.D. 2023 The effects of boundary proximity on Kelvin–Helmholtz instability and turbulence. J. Fluid Mech. 966, A2.CrossRefGoogle 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. 2012 a 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.CrossRefGoogle Scholar
Mashayek, A. & Peltier, W.R. 2012 b 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.CrossRefGoogle Scholar
Miles, J.W. 1961 On the stability of heterogeneous shear flows. J. Fluid Mech. 10, 496–508.CrossRefGoogle Scholar
Moum, J.N., Nash, J.D. & Smyth, W.D. 2011 Narrowband, high-frequency oscillations in the upper equatorial ocean: part 1: intepretation as shear instabilities. J. Phys. Oceanogr. 41, 397–411.CrossRefGoogle Scholar
Newsom, R.K. & Banta, R.M. 2003 Shear flow instability in the stable nocturnal boundary layer as observed by Doppler lidar during CASES-99. J. Atmos. Sci. 60, 16–33.2.0.CO;2>CrossRefGoogle Scholar
Olsthoorn, J., Kaminski, A.K. & Robb, D.M. 2023 Dynamics of asymmetric stratified shear instabilities. Phys. Rev. Fluids 8 (2), 024501.CrossRefGoogle Scholar
Osborn, T.R. 1980 Estimates of the local rate of vertical diffusion from dissipation measurements. J. Phys. Oceanogr. 10, 83–89.2.0.CO;2>CrossRefGoogle Scholar
Peltier, W.R. & Caulfield, C.P. 2003 Mixing efficiency in stratified shear flows. Annu. Rev. Fluid Mech. 35, 136–167.CrossRefGoogle Scholar
Petropoulos, N., Mashayek, A. & Caulfield, C.P. 2023 Turbulent disruption of density staircases in stratified shear flows. J. Fluid Mech. 961, A30.CrossRefGoogle Scholar
Salehipour, H. & Peltier, W.R. 2015 Diapycnal diffusivity, turbulent Prandtl number and mixing efficiency in Boussinesq stratified turbulence. J. Fluid Mech. 775, 464–500.CrossRefGoogle Scholar
Smyth, W.D. 2003 Secondary Kelvin–Helmholtz instability in a weakly stratified shear flow. J. Fluid Mech. 497, 67–98.CrossRefGoogle Scholar
Smyth, W.D. 2006 Secondary circulations in Holmboe waves. Phys. Fluids 18, 06414.CrossRefGoogle Scholar
Smyth, W.D. 2020 Marginal instability and the efficiency of ocean mixing. J. Phys. Oceanogr. 50 (8), 2141–2150.CrossRefGoogle Scholar
Smyth, W.D. & Carpenter, J.R. 2019 Instability in Geophysical Flows. Cambridge University Press.CrossRefGoogle Scholar
Smyth, W.D., Mayor, S.D. & Lian, Q. 2023 The role of ambient turbulence in canopy wave generation by Kelvin–Helmholtz instability. Boundary-Layer Meteorol. 187, 501–526.CrossRefGoogle Scholar
Smyth, W.D. & Moum, J.N. 2000 Length scales of turbulence in stably stratified mixing layers. Phys. Fluids 12, 1327–1342.CrossRefGoogle Scholar
Smyth, W.D. & Moum, J.N. 2001 3D turbulence. In Encyclopedia of Ocean Sciences (ed. S. Thorpe, J. Steele & K. Turekian). Academic.Google Scholar
Smyth, W.D., Moum, J.N. & Caldwell, D.R. 2001 The efficiency of mixing in turbulent patches: inferences from direct simulations and microstructure observations. J. Phys. Oceanogr. 31, 1969–1992.2.0.CO;2>CrossRefGoogle Scholar
Smyth, W.D. & Peltier, W.R. 1989 The transition between Kelvin–Helmholtz and Holmboe instability: an investigation of the overreflection hypothesis. J. Atmos. Sci. 46 (24), 3698–3720.2.0.CO;2>CrossRefGoogle Scholar
Smyth, W.D. & Winters, K.B. 2003 Turbulence and mixing in Holmboe waves. J. Phys. Oceanogr. 33, 694–711.2.0.CO;2>CrossRefGoogle Scholar
Staquet, C. 1995 Two-dimensional secondary instabilities in a strongly stratified shear layer. J. Fluid Mech. 296, 73–126.CrossRefGoogle Scholar
Taylor, J.R. 2008 Numerical simulations of the stratified oceanic bottom boundary layer. Thesis, University of California, San Diego, CA.Google Scholar
Taylor, J.R. & Zhou, Q. 2017 A multi-parameter criterion of layer formation in a stratified shear flow using sorted buoyancy coordinates. J. Fluid Mech. 823, R5.CrossRefGoogle Scholar
Timmermans, M.-L., Toole, J., Krishfield, R. & Winsor, P. 2008 Ice-tethered profiler observations of the double-diffusive staircase in the Canada Basin thermocline. J. Geophys. Res. 113, C00A02.CrossRefGoogle Scholar
Tseng, Y. & Ferziger, J.H. 2001 Mixing and available potential energy in stratified flows. Phys. Fluids 13 (5), 1281–1293.CrossRefGoogle Scholar
Tu, J., Fan, D., Liu, Z. & Smyth, W. 2022 Scaling the mixing efficiency of sediment-stratified turbulence. Geophys. Res. Lett. 49 (13), e2022GL099025.CrossRefGoogle 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.CrossRefGoogle Scholar
Winters, K., Lombard, P.N., Riley, J.J. & D'Asaro, E.A. 1995 Available potential energy and mixing in density-stratified fluids. J. Fluid Mech. 289, 115–128.CrossRefGoogle Scholar
Woods, J.D. 1968 Wave-induced shear instability in the summer thermocline. J. Fluid Mech. 32, 791–800.CrossRefGoogle Scholar
Supplementary material: File

Liu et al. supplementary movie 1

A cross-sectional view of the buoyancy field, showing the separation distance D=0.5. Initially, the stationary primary Kelvin-Helmholtz (KH) billows are formed, followed by turbulent pairing. Eventually, the KH billows are broken down by turbulence.
Download Liu et al. supplementary movie 1(File)
File 7.7 MB
Supplementary material: File

Liu et al. supplementary movie 2

A cross-sectional view of the buoyancy field, showing the separation distance D=2. Two trains of Kelvin-Helmholtz (KH) billows are formed, creating oscillation. Eventually, the billows are broken down by turbulence.
Download Liu et al. supplementary movie 2(File)
File 21.8 MB
Supplementary material: File

Liu et al. supplementary movie 3

A cross-sectional view of the buoyancy field, showing the separation distance D=1. Large stationary Kelvin-Helmholtz (KH) billows are formed, followed by secondary shear instability (SSI). Eventually, billows become turbulent.
Download Liu et al. supplementary movie 3(File)
File 4.8 MB