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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: lefauve@damtp.cam.ac.uk
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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.

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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