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Time-dependent Kelvin cat-eye structure due to current–topography interaction

Published online by Cambridge University Press:  24 February 2020

Marcelo V. Flamarion Open the ORCID record for Marcelo V. Flamarion [Opens in a new window] ,
André Nachbin Open the ORCID record for André Nachbin [Opens in a new window]  and
Roberto Ribeiro Jr Open the ORCID record for Roberto Ribeiro Jr [Opens in a new window]
Marcelo V. Flamarion
Affiliation:
UFRPE/Rural Federal University of Pernambuco, UACSA/Unidade Acadêmica do Cabo de Santo Agostinho, BR 101 Sul, 5225, Ponte dos Carvalhos, Cabo de Santo Agostinho, PE, 54503-900, Brazil
André Nachbin*
Affiliation:
IMPA/National Institute of Pure and Applied Mathematics, Est. D. Castorina, 110, Rio de Janeiro, RJ, 22460-320, Brazil
Roberto Ribeiro Jr
Affiliation:
UFPR/Federal University of Paraná, Departamento de Matemática, Centro Politécnico, Jardim das Américas, Caixa Postal 19081, Curitiba, PR, 81531-980, Brazil
*
†Email address for correspondence: nachbin@impa.br
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Abstract

Non-stationary, rotational, linear surface waves are considered where the underlying sheared current has constant vorticity. A time-dependent study is presented on the formation and persistence of a Kelvin cat-eye structure in the presence of bottom topography. The flow domain is two-dimensional, which allows for the use of a conformal mapping and working in a computational flat-bottom domain. In some cases an initial disturbance is prescribed, while in others the waves are generated from rest. Submarine particle dynamics numerically captures the horizontal critical layer, defined by closed orbits separating the fluid domain into two disjoint regions. In the wave’s moving frame, these recirculation regions are structured in the form of Kelvin cat-eyes. Owing to the interaction with topography, the usual travelling-wave formulation is abandoned and the critical layer is identified through a non-stationary set of equations. The respective time-dependent Kelvin cat-eye structure dynamically adjusts itself at the onset of wave–topography interaction, without losing its integrity. The formation of a Kelvin cat-eye structure is also studied in the case where the surface is initially undisturbed. Surface waves are generated from either the current–topography interaction or by a pressure distribution suddenly imposed along the free surface. Under the pressure forcing, an isolated cat-eye forms with a single recirculation region beneath the wave.

JFM classification

Waves/Free-surface Flows: Surface gravity waves Waves/Free-surface Flows: Critical layers
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 889 , 25 April 2020 , A11
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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Flamarion et al. supplementary movie 1

Evolution of the Kelvin cat-eye structure due to an initial wave disturbance propagating towards the bottom topography.

Download Flamarion et al. supplementary movie 1(Video)
Video 4 MB

Flamarion et al. supplementary movie 2

Onset and evolution of the Kelvin cat-eye structure due to wave generation by a current-topography interaction.

Download Flamarion et al. supplementary movie 2(Video)
Video 6.2 MB

Flamarion et al. supplementary movie 3

Onset and evolution of the Kelvin cat-eye structure due to wave generation by a surface pressure distribution.

Download Flamarion et al. supplementary movie 3(Video)
Video 1.5 MB