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Hydraulic control of continental shelf waves

Published online by Cambridge University Press:  21 April 2021

S. Jamshidi Open the ORCID record for S. Jamshidi [Opens in a new window]  and
E.R. Johnson Open the ORCID record for E.R. Johnson [Opens in a new window]
S. Jamshidi*
Affiliation:
Department of Mathematics, University College London, Gower Street, LondonWC1E 6BT, UK
E.R. Johnson
Affiliation:
Department of Mathematics, University College London, Gower Street, LondonWC1E 6BT, UK
*
†Email address for correspondence: sean.jamshidi.16@ucl.ac.uk
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Abstract

This paper studies the hydraulic control of continental shelf waves using an inviscid barotropic quasi-geostrophic model with piecewise-constant potential vorticity, in which the shelf is represented by a flat step of variable width. A coastal-intensified geostrophic current generates topographic Rossby waves, which can become critical at a local decrease in shelf width when the background current opposes Rossby wave propagation. That is, the shelfbreak perturbation permanently modifies the flow field over arbitrarily large distances and the flow transitions from subcritical to supercritical as it crosses the perturbation. Critically controlled flows also lead to the exchange of significant volumes of water between the shelf and the deep ocean. We derive the boundaries for which critical control occurs in terms of a Froude number and the dimensionless magnitude of the perturbation, and analyse the possible transitions between controlled and far-field flow. When first-order dispersive terms are included in the model, transitions are resolved by dispersive shock waves, which remain attached to the forcing region when the Froude number is close to the boundary for critical flow. Contour dynamic simulations show that the dispersive long-wave model captures the quantitative behaviour of the full quasi-geostrophic system for slowly varying shelves, and replicates the qualitative behaviour even when the long-wave parameter is order one.

JFM classification

Geophysical and Geological Flows: Hydraulic control Geophysical and Geological Flows: Quasi-geostrophic flows Geophysical and Geological Flows: Topographic effects
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 917 , 25 June 2021 , A4
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

REFERENCES

Brink, K.H. 1991 Coastal-trapped waves and wind-driven currents over the continental shelf. Annu. Rev. Fluid Mech. 23 (1), 389–412.CrossRefGoogle Scholar
Dale, A.C. & Barth, J.A. 2001 The hydraulics of an evolving upwelling jet flowing around a cape. J. Phys. Oceanogr. 31 (1), 226–243.2.0.CO;2>CrossRefGoogle Scholar
Dritschel, D.G. 1988 Contour surgery: a topological reconnection scheme for extended integrations using contour dynamics. J. Comput. Phys. 77 (1), 240–266.CrossRefGoogle Scholar
El, G.A. 2005 Resolution of a shock in hyperbolic systems modified by weak dispersion. Chaos 15 (3), 037103.CrossRefGoogle ScholarPubMed
El, G.A., Grimshaw, R.H.J. & Smyth, N.F. 2006 Unsteady undular bores in fully nonlinear shallow-water theory. Phys. Fluids 18 (2), 027104.CrossRefGoogle Scholar
El, G.A., Grimshaw, R.H.J. & Smyth, N.F. 2009 Transcritical shallow-water flow past topography: finite-amplitude theory. J. Fluid Mech. 640, 187.CrossRefGoogle Scholar
El, G.A., Hoefer, M.A. & Shearer, M. 2017 Dispersive and diffusive-dispersive shock waves for nonconvex conservation laws. SIAM Rev. 59 (1), 3–61.CrossRefGoogle Scholar
Gill, A.E. 1977 The hydraulics of rotating-channel flow. J. Fluid Mech. 80 (04), 641–671.CrossRefGoogle Scholar
Gill, A.E. & Schumann, E.H. 1979 Topographically induced changes in the structure of an inertial coastal jet: application to the Agulhas Current. J. Phys. Oceanogr. 9 (5), 975–991.2.0.CO;2>CrossRefGoogle Scholar
Gordon, R.L. & Huthnance, J.M. 1987 Storm-driven continental shelf waves over the Scottish continental shelf. Cont. Shelf Res. 7 (9), 1015–1048.CrossRefGoogle Scholar
Grimshaw, R.H.J. 1987 Resonant forcing of barotropic coastally trapped waves. J. Phys. Oceanogr. 17 (1), 53–65.2.0.CO;2>CrossRefGoogle Scholar
Haynes, P.H., Johnson, E.R. & Hurst, R.G. 1993 A simple model of Rossby-wave hydraulic behaviour. J. Fluid Mech. 253, 359–384.CrossRefGoogle Scholar
Hughes, R.L. 1985 Multiple criticalities in coastal flows. Dyn. Atmos. Oceans 9 (4), 321–340.CrossRefGoogle Scholar
Jamshidi, S. & Johnson, E.R. 2019 Coastal outflow currents into a buoyant layer of arbitrary depth. J. Fluid Mech. 858, 656–688.CrossRefGoogle Scholar
Jamshidi, S. & Johnson, E.R. 2020 The long-wave potential-vorticity dynamics of coastal fronts. J. Fluid Mech. 888, A19.CrossRefGoogle Scholar
Johnson, E.R. & Clarke, S.R. 1999 Dispersive effects in Rossby-wave hydraulics. J. Fluid Mech. 401, 27–54.CrossRefGoogle Scholar
Johnson, E.R. & Clarke, S.R. 2001 Rossby wave hydraulics. Annu. Rev. Fluid Mech. 33 (1), 207–230.CrossRefGoogle Scholar
Johnson, E.R. & McDonald, N.R. 2006 Vortical source-sink flow against a wall: the initial value problem and exact steady states. Phys. Fluids 18 (7), 076601.CrossRefGoogle Scholar
Johnson, E.R., Southwick, O.R. & McDonald, N.R. 2017 The long-wave vorticity dynamics of rotating buoyant outflows. J. Fluid Mech. 822, 418–443.CrossRefGoogle Scholar
Kamchatnov, A.M. 2019 Dispersive shock wave theory for nonintegrable equations. Phys. Rev. E 99 (1), 012203.CrossRefGoogle ScholarPubMed
Kubokawa, A. 1991 On the behaviour of outflows with low potential vorticity from a sea strait. Tellus A 43 (2), 168–176.CrossRefGoogle Scholar
Martell, C.M. & Allen, J.S. 1979 The generation of continental shelf waves by alongshore variations in bottom topography. J. Phys. Oceanogr. 9 (4), 696–711.2.0.CO;2>CrossRefGoogle Scholar
Miller, A.J., Lermusiaux, P.F.J. & Poulain, P.-M. 1996 A topographic–Rossby mode resonance over the Iceland–Faeroe Ridge. J. Phys. Oceanogr. 26 (12), 2735–2747.2.0.CO;2>CrossRefGoogle Scholar
Mitsudera, H. & Grimshaw, R.H.J. 1990 Resonant forcing of coastally trapped waves in a continuously stratified ocean. Pure Appl. Geophys. 133 (4), 635–664.CrossRefGoogle Scholar
Pratt, L.J. & Stern, M.E. 1986 Dynamics of potential vorticity fronts and eddy detachment. J. Phys. Oceanogr. 16 (6), 1101–1120.2.0.CO;2>CrossRefGoogle Scholar
Pratt, L.J. & Whitehead, J.A. 2008 Rotating Hydraulics, vol. 1. Springer.Google Scholar
Saldías, G.S. & Allen, S.E. 2020 The influence of a submarine canyon on the circulation and cross-shore exchanges around an upwelling front. J. Phys. Oceanogr. 50 (6), 1677–1698.CrossRefGoogle Scholar
Stern, M.E. 1986 On the amplification of convergences in coastal currents and the formation of squirts. J. Mar. Res. 44 (3), 403–421.CrossRefGoogle Scholar
Strub, P.T., Kosro, P.M. & Huyer, A. 1991 The nature of the cold filaments in the California Current System. J. Geophys. Res.: Oceans 96 (C8), 14743–14768.CrossRefGoogle Scholar
Tansley, C.E. & Marshall, D.P. 2000 On the influence of bottom topography and the Deep Western Boundary Current on Gulf Stream separation. J. Mar. Res. 58 (2), 297–325.CrossRefGoogle Scholar
Troupin, C., Mason, E., Beckers, J.-M. & Sangrà, P. 2012 Generation of the Cape Ghir upwelling filament: a numerical study. Ocean Modell. 41, 1–15.CrossRefGoogle Scholar
Wåhlin, A.K., Kalen, O., Assmann, K.M., Darelius, E., Ha, H.K., Kim, T.-W. & Lee, S.H. 2016 Subinertial oscillations on the Amundsen Sea shelf, Antarctica. J. Phys. Oceanogr. 46 (9), 2573–2582.CrossRefGoogle Scholar
Zhang, W.G. & Lentz, S.J. 2017 Wind-driven circulation in a shelf valley. Part I: mechanism of the asymmetrical response to along-shelf winds in opposite directions. J. Phys. Oceanogr. 47 (12), 2927–2947.CrossRefGoogle Scholar
Zhang, W.G. & Lentz, S.J. 2018 Wind-driven circulation in a shelf valley. Part II: dynamics of the along-valley velocity and transport. J. Phys. Oceanogr. 48 (4), 883–904.CrossRefGoogle Scholar