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Shear instability in mode-2 internal Kelvin waves

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Abstract

Shear instabilities of stratified fluids are a classical topic with a broad literature. The classic instability takes the form of Kelvin-Helmholtz billows that initially develop in two dimensions, one of which is the vertical. Spanwise instability develops later as part of the transition to a three-dimensionalized state. We simulate mode-2 internal waves on the laboratory scale in a rotating frame of reference that, in the absence of rotation, form spanwise aligned billows on the wave flanks. Rotation breaks the symmetry of the classical shear instability because the wave amplitude decays away from the focussing wall (i.e. the waves generated are internal Kelvin waves). We document the development of the wave and the shear instabilities as the Rossby number is varied, finding that (i) even weak rotation (high Rossby number) leads to a significant modification of the billow three-dimensionalization, (ii) strong rotation (low Rossby number) leads to a strong near wall focussing of turbulence transition that is clearly evident in the second invariant of the velocity gradient, Q, of turbulence theory. For low rotation rates, or intermediate to high Rossby numbers, we identify novel instabilities with billow cores aligned in the along-tank direction, rather than the typical spanwise direction.

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Data availability

The model used is public domain. Case files and data are both available from the corresponding author upon reasonable request.

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Acknowledgements

This work was supported by NSERC Discovery Grant RGPIN-311844-37157.

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Authors and Affiliations

  1. Department of Applied Mathematics, University of Waterloo, 200 University Ave. West, Waterloo, Ontario, N2L 3G1, Canada

    Marek Stastna & Andrew Grace

  2. Research Computing Services, IT, University of Calgary, 2500 University Dr. NW, Calgary, Alberta, T2N 1N4, Canada

    David Deepwell

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  1. Marek Stastna
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Contributions

MS and DD designed the structure of the study and the simulations. MS ran the simulations and created the figures. MS and AG wrote and edited the various versions of the text. DD provided editorial comments throughout.

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Correspondence to Marek Stastna.

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Stastna, M., Deepwell, D. & Grace, A. Shear instability in mode-2 internal Kelvin waves. Environ Fluid Mech 23, 407–428 (2023). https://doi.org/10.1007/s10652-022-09895-w

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