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The propagation and decay of a coastal vortex on a shelf

Published online by Cambridge University Press:  29 September 2021

Matthew N. Crowe Open the ORCID record for Matthew N. Crowe [Opens in a new window]  and
Edward R. Johnson Open the ORCID record for Edward R. Johnson [Opens in a new window]
Matthew N. Crowe*
Affiliation:
Department of Mathematics, University College London, London WC1E 6BT, UK
Edward R. Johnson
Affiliation:
Department of Mathematics, University College London, London WC1E 6BT, UK
*
†Email address for correspondence: m.crowe@ucl.ac.uk
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Abstract

A coastal eddy is modelled as a barotropic vortex propagating along a coastal shelf. If the vortex speed matches the phase speed of any coastal trapped shelf wave modes, a shelf wave wake is generated leading to a flux of energy from the vortex into the wave field. Using a simple shelf geometry, we determine analytic expressions for the wave wake and the leading-order flux of wave energy. By considering the balance of energy between the vortex and wave field, this energy flux is then used to make analytic predictions for the evolution of the vortex speed and radius under the assumption that the vortex structure remains self-similar. These predictions are examined in the asymptotic limit of small rotation rate and shelf slope and tested against numerical simulations. If the vortex speed does not match the phase speed of any shelf wave, steady vortex solutions are expected to exist. We present a numerical approach for finding these nonlinear solutions and examine the parameter dependence of their structure.

JFM classification

Geophysical and Geological Flows: Topographic effects Geophysical and Geological Flows: Waves in rotating fluids Vortex Flows: Vortex dynamics
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 927 , 25 November 2021 , A38
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

REFERENCES

Bretherton, F.P. 1967 The time-dependent motion due to a cylinder moving in an unbounded rotating or stratified fluid. J. Fluid Mech. 28 (3), 545–570.CrossRefGoogle Scholar
Burns, K.J., Vasil, G.M., Oishi, J.S., Lecoanet, D. & Brown, B.P. 2020 Dedalus: a flexible framework for numerical simulations with spectral methods. Phys. Rev. Res. 2, 023068.CrossRefGoogle Scholar
Crowe, M.N. & Johnson, E.R. 2020 The effects of vertical mixing on nonlinear Kelvin waves. J. Fluid Mech. 903, A22.CrossRefGoogle Scholar
Crowe, M.N., Kemp, C.J.D. & Johnson, E.R. 2021 The decay of Hill's vortex in a rotating flow. J. Fluid Mech. 919, A6.CrossRefGoogle Scholar
Deremble, B., Johnson, E.R. & Dewar, W.K. 2017 A coupled model of interior balanced and boundary flow. Ocean Model. 119, 1–12.CrossRefGoogle Scholar
Dewar, W.K., Berloff, P. & Hogg, A.M. 2011 Submesoscale generation by boundaries. J. Mar. Res. 69 (4–5), 501–522.CrossRefGoogle Scholar
Dewar, W.K. & Hogg, A.C. 2010 Topographic inviscid dissipation of balanced flow. Ocean Model. 32 (1), 1–13.CrossRefGoogle Scholar
Flierl, G.R. & Haines, K. 1994 The decay of modons due to Rossby wave radiation. Phys. Fluids 6 (10), 3487–3497.CrossRefGoogle Scholar
Ford, R., McIntyre, M.E. & Norton, W.A. 2000 Balance and the slow quasimanifold: some explicit results. J. Atmos. Sci. 57 (9), 1236–1254.2.0.CO;2>CrossRefGoogle Scholar
Fraenkel, L.E. 1956 On the flow of rotating fluid past bodies in a pipe. Proc. R. Soc. A 233 (1195), 506–526.Google Scholar
Gill, A.E. & Schumann, E.H. 1974 The generation of long shelf waves by the wind. J. Phys. Oceanogr. 4, 83–90.2.0.CO;2>CrossRefGoogle Scholar
Hill, M.J.M. 1894 On a spherical vortex. Phil. Trans. R. Soc. A 185, 213–245.Google Scholar
Hogg, A.M., Dewar, W.K., Berloff, P. & Ward, M.L. 2011 Kelvin wave hydraulic control induced by interactions between vortices and topography. J. Fluid Mech. 687, 194–208.CrossRefGoogle Scholar
Huthnance, J.M. 1974 On trapped waves over a continental shelf. J. Fluid Mech. 69, 689–704.CrossRefGoogle Scholar
Isern-Fontanet, J., García-Ladona, E. & Font, J. 2006 Vortices of the Mediterranean Sea: an altimetric perspective. J. Phys. Oceanogr. 36, 87–103.CrossRefGoogle Scholar
Johnson, E.R. 1979 Finite depth stratified flow over topography on a beta-plane. Geophys. Astrophys. Fluid Dyn. 12, 35–43.CrossRefGoogle Scholar
Johnson, E.R. 1989 Topographic waves in open domains. Part 1. Boundary conditions and frequency estimates. J. Fluid Mech. 200, 69–76.CrossRefGoogle Scholar
Johnson, E.R. & Crowe, M.N. 2021 The decay of a dipolar vortex in a weakly dispersive environment. J. Fluid Mech. 917, A35.CrossRefGoogle Scholar
Johnson, E.R. & Rodney, J.T. 2011 Spectral methods for coastal-trapped waves. Cont. Shelf Res. 31 (14), 1481–1489.CrossRefGoogle Scholar
LeBlond, P.H. & Mysak, L.A. 1978 Waves in the Ocean. Elsevier.Google Scholar
Lighthill, M.J. 1967 On waves generated in dispersive systems by travelling forcing effects, with applications to the dynamics of rotating fluids. J. Fluid Mech. 27 (4), 725–752.CrossRefGoogle Scholar
Long, R.R. 1953 Steady motion around a symmetrical obstacle moving along the axis of a rotating liquid. J. Met. 10, 197–203.2.0.CO;2>CrossRefGoogle Scholar
Machicoane, N., Labarre, V., Voisin, B., Moisy, F. & Cortet, P.-P. 2018 Wake of inertial waves of a horizontal cylinder in horizontal translation. Phys. Rev. Fluids 3, 034801.CrossRefGoogle Scholar
de Marez, C., Carton, X., Morvan, M. & Reinaud, J. 2017 The interaction of two surface vortices near a topographic slope in a stratified ocean. Fluids 2 (4), 57.CrossRefGoogle Scholar
de Marez, C., Morvan, M., L'Hégaret, P., Meunier, T. & Carton, X. 2020 Vortex-wall interaction on the beta-plane and the generation of deep submesoscale cyclones by internal Kelvin waves-current interactions. Geophys. Astrophys. Fluid Dyn. 114 (4–5), 588–606.CrossRefGoogle Scholar
Meleshko, V.V. & van Heijst, G.J.F. 1994 On Chaplygin's investigations of two-dimensional vortex structures in an inviscid fluid. J. Fluid Mech. 272, 157–182.CrossRefGoogle Scholar
Moffatt, H.K. 1969 The degree of knottedness of tangled vortex lines. J. Fluid Mech. 35 (1), 117–129.CrossRefGoogle Scholar
Penduff, T., Juza, M., Barnier, B., Zika, J., Dewar, W.K., Treguier, A.-M., Molines, J.-M. & Audiffren, N. 2011 Sea level expression of intrinsic and forced ocean variabilities at interannual time scales. J. Clim. 24 (21), 5652–5670.CrossRefGoogle Scholar

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