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The structure and origin of confined Holmboe waves

Published online by Cambridge University Press:  05 June 2018

Adrien Lefauve Open the ORCID record for Adrien Lefauve [Opens in a new window] ,
J. L. Partridge ,
Qi Zhou Open the ORCID record for Qi Zhou [Opens in a new window] ,
S. B. Dalziel ,
C. P. Caulfield Open the ORCID record for C. P. Caulfield [Opens in a new window]  and
P. F. Linden
Adrien Lefauve*
Affiliation:
Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, UK
J. L. Partridge
Affiliation:
Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, UK
Qi Zhou
Affiliation:
Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, UK
S. B. Dalziel
Affiliation:
Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, UK
C. P. Caulfield
Affiliation:
Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, UK BP Institute, University of Cambridge, Madingley Road, Cambridge CB3 0EZ, UK
P. F. Linden
Affiliation:
Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, UK
*
†Email address for correspondence: [email protected]
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Abstract

Finite-amplitude manifestations of stratified shear flow instabilities and their spatio-temporal coherent structures are believed to play an important role in turbulent geophysical flows. Such shear flows commonly have layers separated by sharp density interfaces, and are therefore susceptible to the so-called Holmboe instability, and its finite-amplitude manifestation, the Holmboe wave. In this paper, we describe and elucidate the origin of an apparently previously unreported long-lived coherent structure in a sustained stratified shear flow generated in the laboratory by exchange flow through an inclined square duct connecting two reservoirs filled with fluids of different densities. Using a novel measurement technique allowing for time-resolved, near-instantaneous measurements of the three-component velocity and density fields simultaneously over a three-dimensional volume, we describe the three-dimensional geometry and spatio-temporal dynamics of this structure. We identify it as a finite-amplitude, nonlinear, asymmetric confined Holmboe wave (CHW), and highlight the importance of its spanwise (lateral) confinement by the duct boundaries. We pay particular attention to the spanwise vorticity, which exhibits a travelling, near-periodic structure of sheared, distorted, prolate spheroids with a wide ‘body’ and a narrower ‘head’. Using temporal linear stability analysis on the two-dimensional streamwise-averaged experimental flow, we solve for three-dimensional perturbations having two-dimensional, cross-sectionally confined eigenfunctions and a streamwise normal mode. We show that the dispersion relation and the three-dimensional spatial structure of the fastest-growing confined Holmboe instability are in good agreement with those of the observed confined Holmboe wave. We also compare those results with a classical linear analysis of two-dimensional perturbations (i.e. with no spanwise dependence) on a one-dimensional base flow. We conclude that the lateral confinement is an important ingredient of the confined Holmboe instability, which gives rise to the CHW, with implications for many inherently confined geophysical flows such as in valleys, estuaries, straits or deep ocean trenches. Our results suggest that the CHW is an example of an experimentally observed, inherently nonlinear, robust, long-lived coherent structure which has developed from a linear instability. We conjecture that the CHW is a promising candidate for a class of exact coherent states underpinning the dynamics of more disordered, yet continually forced stratified shear flows.

JFM classification

Instability: Instability Geophysical and Geological Flows: Stratified flows Turbulent Flows: Stratified turbulence
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 848 , 10 August 2018 , pp. 508 - 544
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Copyright
© 2018 Cambridge University Press 

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Footnotes

‡

Present address: Department of Civil Engineering, University of Calgary, 2500 University Drive NW, Calgary, Alberta T2N 1N4, Canada.

References

Armi, L. 1986 The hydraulics of two flowing layers with different densities. J. Fluid Mech. 163, 27–58.CrossRefGoogle Scholar
Baines, P. G. & Mitsudera, H. 1994 On the mechanism of shear flow instabilities. J. Fluid Mech. 276, 327–342.CrossRefGoogle Scholar
Bartello, P. & Tobias, S. M. 2013 Sensitivity of stratified turbulence to the buoyancy Reynolds number. J. Fluid Mech. 725, 1–22.CrossRefGoogle Scholar
Browand, F. K. & Winant, C. D. 1973 Laboratory observations of shear-layer instability in a stratified fluid. Boundary-Layer Meteorol. 5 (1–2), 67–77.CrossRefGoogle Scholar
Carpenter, J. R., Lawrence, G. A. & Smyth, W. D. 2007 Evolution and mixing of asymmetric Holmboe instabilities. J. Fluid Mech. 582, 103–132.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.Google Scholar
Carpenter, J. R., Tedford, E. W., Rahmani, M. & Lawrence, G. A. 2010 Holmboe wave fields in simulation and experiment. J. Fluid Mech. 648, 205–223.CrossRefGoogle Scholar
Caulfield, C. P. 1994 Multiple linear instability of layered stratified shear flow. J. Fluid Mech. 258, 255–285.CrossRefGoogle 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.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
Drazin, P. G. 1974 On a model of instability of a slowly-varying flow. Q. J. Mech. Appl. Math. 27 (1), 69–86.CrossRefGoogle Scholar
Farmer, D. & Armi, L. 1999 Stratified flow over topography: the role of small-scale entrainment and mixing. Proc. R. Soc. Lond. A 455, 3221–3258.CrossRefGoogle Scholar
Geyer, W. R., Lavery, A. C., Scully, M. E. & Trowbridge, J. H. 2010 Mixing by shear instability at high Reynolds number. Geophys. Res. Lett. 37, L22607.CrossRefGoogle Scholar
Gibson, C. H. 1980 Fossil temperature, salinity, and vorticity turbulence in the ocean. In Marine Turbulence (ed. Nihoul, J.), pp. 221–257. Elsevier.Google Scholar
Gibson, C. H. 1999 Fossil turbulence revisited. J. Mar. Syst. 21 (1–4), 147–167.CrossRefGoogle Scholar
Gibson, J. F., Halcrow, J. & Cvitanović, P. 2008 Visualizing the geometry of state space in plane Couette flow. J. Fluid Mech. 611, 107–130.CrossRefGoogle Scholar
Haigh, S. P. & Lawrence, G. A. 1999 Symmetric and nonsymmetric Holmboe instabilities in an inviscid flow. Phys. Fluids 11, 1459–1468.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 (23), 8445–8451.CrossRefGoogle Scholar
Hazel, P. 1972 Numerical studies of the stability of inviscid stratified shear flows. J. Fluid Mech. 51, 39–61.CrossRefGoogle Scholar
Helmholtz, H. 1878 Über discontinuierliche Flüssigkeitsbewegungen [On the discontinuous movements of fluids]. Monatsberichte der Königlichen Preussische Akademie der Wissenschaften zu Berlin 23, 215–228.Google Scholar
Hogg, A. McC. & Ivey, G. N. 2003 The Kelvin–Helmholtz to Holmboe instability transition in stratified exchange flows. J. Fluid Mech. 477, 357–375.CrossRefGoogle Scholar
Holmboe, J. 1962 On the behavior of symmetric waves in stratified shear layers. Geophys. Publ. 24, 67–113.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.CrossRefGoogle Scholar
Lawrence, G. A., Browand, F. K. & Redekopp, L. G. 1991 The stability of a sheared density interface. Phys. Fluids A 3, 2360–2370.CrossRefGoogle Scholar
Lefauve, A.2018 Waves and turbulence in sustained stratified shear flows. PhD thesis, University of Cambridge.Google Scholar
Lucas, D. & Caulfield, C. P. 2017 Irreversible mixing by unstable periodic orbits in buoyancy dominated stratified turbulence. J. Fluid Mech. 832, R1.CrossRefGoogle Scholar
Lucas, D., Caulfield, C. P. & Kerswell, R. R. 2017 Layer formation in horizontally forced stratified turbulence: connecting exact coherent structures to linear instabilities. J. Fluid Mech. 832, 409–437.CrossRefGoogle Scholar
Lucas, D. & Kerswell, R. R. 2015 Recurrent flow analysis in spatiotemporally chaotic 2-dimensional Kolmogorov flow. Phys. Fluids 27, 045106.CrossRefGoogle Scholar
Macagno, E. O. & Rouse, H. 1961 Interfacial mixing in stratified flow. J. Engng Mech. Div. Proc. Am. Soc. Civil Engrs 87 (EM5), 55–81.Google Scholar
Mahrt, L. 2014 Stably stratified atmospheric boundary layers. Annu. Rev. Fluid Mech. 46, 23–45.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.CrossRefGoogle 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
Maxworthy, T. & Browand, F. K. 1975 Experiments in rotating and stratified flows: oceanographic application. Annu. Rev. Fluid Mech. 7, 273–305.CrossRefGoogle Scholar
Meyer, C. R. & Linden, P. F. 2014 Stratified shear flow: experiments in an inclined duct. J. Fluid Mech. 753, 242–253.CrossRefGoogle Scholar
Nishida, S. & Yoshida, S. 1990 Influence of the density and velocity profiles on calculated instability characteristics in an inviscid two-layer shear flow. J. Hydrosci. Hydraul. Engng 7.Google Scholar
Olsthoorn, J. & Dalziel, S. B. 2017 Three-dimensional visualization of the interaction of a vortex ring with a stratified interface. J. Fluid Mech. 820, 549–579.CrossRefGoogle Scholar
Partridge, J. L., Lefauve, A. & Dalziel, S. B.2018 A versatile scanning method for volumetric measurements of velocity and density fields. arXiv:1805.01181.CrossRefGoogle Scholar
Portwood, G. D., de Bruyn Kops, S. M., Taylor, J. R., Salehipour, H. & Caulfield, C. P. 2016 Robust identification of dynamically distinct regions in stratified turbulence. J. Fluid Mech. 807, R2.CrossRefGoogle Scholar
Rayleigh, J. W. S. 1879 On the stability, or instability, of certain fluid motions. Proc. Lond. Math. Soc. s1‐11, 57–72.CrossRefGoogle Scholar
Reynolds, O. 1883 An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous, and of the law of resistance in parallel channels. Phil. Trans. R. Soc. Lond. 174, 935–982.Google Scholar
Smyth, W. D., Carpenter, J. R. & Lawrence, G. A. 2007 Mixing in symmetric Holmboe waves. J. Phys. Oceanogr. 37, 1566–1583.CrossRefGoogle Scholar
Salehipour, H., Caulfield, C. P. & Peltier, W. R. 2016 Turbulent mixing due to the Holmboe wave instability at high Reynolds number. J. Fluid Mech. 803, 591–621.CrossRefGoogle Scholar
Smyth, W. D., Klaassen, G. P. & Peltier, W. R. 1988 Finite amplitude Holmboe waves. Geophys. Astrophys. Fluid Dyn. 43 (2), 181–222.CrossRefGoogle 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
Smyth, W. D. & Peltier, W. R. 1991 Instability and transition in finite-amplitude Kelvin–Helmholtz and Holmboe waves. J. Fluid Mech. 228, 387–415.Google 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
Squire, H. B. 1933 On the stability of three-dimensional disturbances of viscous flow between parallel walls. Proc. R. Soc. Lond. A 142, 621–628.Google 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
Tedford, E. W., Pieters, R. & Lawrence, G. A. 2009 Symmetric Holmboe instabilities in a laboratory exchange flow. J. Fluid Mech. 636, 137–153.CrossRefGoogle Scholar
Thomson, W. 1871 Hydrokinetic solutions and observations. Lond. Edinb. Dubl. Phil. Mag. J. Sci. 42 (281), 362–377.CrossRefGoogle Scholar
Thorpe, S. A. 1968 A method of producing a shear flow in a stratified fluid. J. Fluid Mech. 32, 693–704.CrossRefGoogle Scholar
Thorpe, S. A. 1971 Experiments on the instability of stratified shear flows: miscible fluids. J. Fluid Mech. 46, 299–319.CrossRefGoogle Scholar
Woods, J. D. 1968 Wave-induced shear instability in the summer thermocline. J. Fluid Mech. 32, 791–800.CrossRefGoogle Scholar
Yih, C. 1955 Stability of two-dimensional parallel flows for three-dimensional disturbances. Q. Appl. Maths 12 (4), 434–435.CrossRefGoogle Scholar
Zhu, D. Z. & Lawrence, G. A. 2001 Holmboe’s instability in exchange flows. J. Fluid Mech. 429, 391–409.CrossRefGoogle Scholar

Lefauve et al. supplementary movie 1

Spanwise vorticity and density in the mid-plane y=0, showing the full temporal dynamics, to complement the snapshots in figure 2.

Download Lefauve et al. supplementary movie 1(Video)
Video 4.1 MB

Lefauve et al. supplementary movie 2

Isosurfaces of spanwise vorticity, showing the full temporal dynamics, to complement the snapshots in figure 3.

Download Lefauve et al. supplementary movie 2(Video)
Video 8.6 MB

Lefauve et al. supplementary movie 3

Isosurfaces of spanwise vorticity of the CHW and CHI. Panoramic video to complement the views of figure 4c-d and figure 7a-b.

Download Lefauve et al. supplementary movie 3(Video)
Video 11.8 MB