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The long-wave potential-vorticity dynamics of coastal fronts

Published online by Cambridge University Press:  07 February 2020

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, London WC1E 6BT, UK
E. R. Johnson
Affiliation:
Department of Mathematics, University College London, London WC1E 6BT, UK
*
†Email address for correspondence: sean.jamshidi.16@ucl.ac.uk
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Abstract

This paper studies the propagation of free, long waves on a potential vorticity front in the presence of a vertical coast, using a $1{\textstyle \frac{1}{2}}$-layer, quasi-geostrophic model with piecewise-constant potential vorticity. The coastal boundary induces flow through image vorticity and a Kelvin wave, either of which can reinforce or oppose the Rossby wave dynamics at the front. The behaviour of the front depends strongly on the relative strengths of these three mechanisms, which are explicit in our model. The richest behaviour, which includes kink solitons (under-compressive shocks) and compound-wave structures, occurs in the regime where vortical effects are dominant. The evolution of the front is described by a fully nonlinear finite-amplitude equation including first-order dispersive effects, which is related to the modified Korteweg–de Vries equation. The different behaviours are classified using the canonical example of the Riemann problem, which we analyse using El’s technique of ‘dispersive shock-fitting’. Contour-dynamic simulations show that the dispersive long-wave theory captures the behaviour of the full quasi-geostrophic system to a high degree of accuracy.

JFM classification

Geophysical and Geological Flows: Quasi-geostrophic flows Geophysical and Geological Flows: Waves in rotating fluids Waves/Free-surface Flows: Solitary waves
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 888 , 10 April 2020 , A19
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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References

Congy, T., El, G. A., Hoefer, M. A. & Shearer, M. 2019 Nonlinear Schrödinger equations and the universal description of dispersive shock wave structure. Stud. Appl. Maths 142 (3), 241–268.CrossRefGoogle Scholar
Cushman-Roisin, B., Pratt, L. & Ralph, E. 1993 A general theory for equivalent barotropic thin jets. J. Phys. Oceanogr. 23 (1), 91–103.2.0.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. & Kamchatnov, A. M. 2007 Evolution of solitary waves and undular bores in shallow-water flows over a gradual slope with bottom friction. J. Fluid Mech. 585, 213–244.CrossRefGoogle Scholar
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, 187–214.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
Esler, J. G. & Pearce, J. D. 2011 Dispersive dam-break and lock-exchange flows in a two-layer fluid. J. Fluid Mech. 667, 555–585.CrossRefGoogle Scholar
Grimshaw, R. H. J. & Yi, Z. 1990 Finite-amplitude long waves on coastal currents. J. Phys. Oceanogr. 20 (1), 3–18.2.0.CO;2>CrossRefGoogle Scholar
Gurevich, A. V. & Pitaevskii, L. P. 1973 Nonstationary structure of a collisionless shock wave. Zh. Eksp. Teor. Fiz. 65, 590–604.Google Scholar
Hermann, A. J., Rhines, P. B. & Johnson, E. R. 1989 Nonlinear Rossby adjustment in a channel: beyond Kelvin waves. J. Fluid Mech. 205, 469–502.CrossRefGoogle Scholar
Hoefer, M. A. & Ablowitz, M. 2009 Dispersive shock waves. Scholarpedia 4 (11), 5562.CrossRefGoogle Scholar
Horner-Devine, A. R., Hetland, R. D. & MacDonald, D. G. 2015 Mixing and transport in coastal river plumes. Annu. Rev. Fluid Mech. 47, 569–594.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
Johnson, E. R. & Clarke, S. R. 1999 Dispersive effects in Rossby-wave hydraulics. J. Fluid Mech. 401, 27–54.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.Google ScholarPubMed
Lee, T. & Cornillon, P. 1996 Propagation of Gulf Stream meanders between 74 and 70 W. J. Phys. Oceanogr. 26 (2), 205–224.2.0.CO;2>CrossRefGoogle Scholar
Maiden, M., Franco, N., Webb, E., El, G. & Hoefer, M. 2020 Solitary wave fission of a large disturbance in a viscous fluid conduit. J. Fluid Mech. 883, A10.CrossRefGoogle Scholar
Nycander, J., Dritschel, D. G. & Sutyrin, G. G. 1993 The dynamics of long frontal waves in the shallow-water equations. Phys. Fluids A 5 (5), 1089–1091.CrossRefGoogle Scholar
Pickart, R. S., Weingartner, T. J., Pratt, L. J., Zimmermann, S. & Torres, D. J. 2005 Flow of winter-transformed Pacific water into the Western Arctic. Deep-Sea Res. Part II 52 (24–26), 3175–3198.CrossRefGoogle Scholar
Pimenta, F. M., Kirwan, A. D. Jr. & Huq, P. 2011 On the transport of buoyant coastal plumes. J. Phys. Oceanogr. 41 (3), 620–640.CrossRefGoogle Scholar
Pratt, L. J. 1988 Meandering and eddy detachment according to a simple (looking) path equation. J. Phys. Oceanogr. 18 (11), 1627–1640.2.0.CO;2>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
Spall, M. A., Pickart, R. S., Fratantoni, P. S. & Plueddemann, A. J. 2008 Western Arctic shelfbreak eddies: formation and transport. J. Phys. Oceanogr. 38 (8), 1644–1668.CrossRefGoogle Scholar
Stern, M. E. & Helfrich, K. R. 2002 Propagation of a finite-amplitude potential vorticity front along the wall of a stratified fluid. J. Fluid Mech. 468, 179–204.CrossRefGoogle Scholar
Stern, M. E. & Pratt, L. á J. 1985 Dynamics of vorticity fronts. J. Fluid Mech. 161, 513–532.CrossRefGoogle Scholar
Tracey, K. L., Watts, D. R., Donohue, K. A. & Ichikawa, H. 2012 Propagation of Kuroshio Extension meanders between 143° and 149° E. J. Phys. Oceanogr. 42 (4), 581–601.CrossRefGoogle Scholar
Tsujino, H., Usui, N. & Nakano, H. 2006 Dynamics of Kuroshio path variations in a high-resolution general circulation model. J. Geophys. Res. 111, C11.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