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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: [email protected]
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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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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