Experimental study of dispersion and modulational instability of surface gravity waves on constant vorticity currents

Steer, James N.; Borthwick, Alistair G. L.; Stagonas, Dimitris; Buldakov, Eugeny; Bremer, Ton S. van den

doi:10.1017/jfm.2019.951

Cited by 11 publications

(6 citation statements)

References 59 publications

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“…The BFI weakening from constant mean shear was observed by (Steer et al. (2020), see also Simmen & Saffman 1985; Baumstein 1998; Thomas et al. 2012); a tailored profile based on the Lagrangian theory enabled this arguably even more striking demonstration.…”

Section: Laboratory Demonstration Of the Impact Of The Lagrangian Mea...mentioning

confidence: 74%

“…This discussion is meant to serve as a demonstration of the ideas presented in this manuscript, particularly in § 3.2. The BFI weakening from constant mean shear was observed by (Steer et al (2020), see also Simmen & Saffman 1985;Baumstein 1998;Thomas et al 2012); a tailored profile based on the Lagrangian theory enabled this arguably even more striking demonstration. A detailed quantitative analysis, beyond the scope of this manuscript, is currently in preparation.…”

Section: Modulation Instabilitymentioning

confidence: 83%

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The role of Lagrangian drift in the geometry, kinematics and dynamics of surface waves

et al. 2023

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The role of the Lagrangian mean flow, or drift, in modulating the geometry, kinematics and dynamics of rotational and irrotational deep-water surface gravity waves is examined. A general theory for permanent progressive waves on an arbitrary vertically sheared steady Lagrangian mean flow is derived in the Lagrangian reference frame and mapped to the Eulerian frame. A Lagrangian viewpoint offers tremendous flexibility due to the particle labelling freedom and allows us to reveal how key physical wave behaviour arises from a kinematic constraint on the vorticity of the fluid, inter alia the nonlinear correction to the phase speed of irrotational finite amplitude waves, the free surface geometry and velocity in the Eulerian frame, and the connection between the Lagrangian drift and the Benjamin–Feir instability. To complement and illustrate our theory, a small laboratory experiment demonstrates how a specially tailored sheared mean flow can almost completely attenuate the Benjamin–Feir instability, in qualitative agreement with the theory. The application of these results to problems in remote sensing and ocean wave modelling is discussed. We provide an answer to a long-standing question: remote sensing techniques based on observing current-induced shifts in the wave dispersion will measure the Lagrangian, not the Eulerian, mean current.

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Section: Laboratory Demonstration Of the Impact Of The Lagrangian Mea...mentioning

confidence: 74%

Section: Modulation Instabilitymentioning

confidence: 83%

The role of Lagrangian drift in the geometry, kinematics and dynamics of surface waves

et al. 2023

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“…Recently, a number of theoretical investigations have been devoted to the water wave problem with non-zero vorticity due to its relevance to oceanography and coastal engineering, where the influence of currents on waves may play an important role (Constantin 2001; Steer et al 2019). Unlike the irrotational case, a full-dimensional computation is in general required to solve for the vorticity field.…”

Section: Introductionmentioning

confidence: 99%

A Hamiltonian Dysthe equation for deep-water gravity waves with constant vorticity

Guyenne

Kairzhan²,

Sulem³

2022

J. Fluid Mech.

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This paper is a study of the water wave problem in a two-dimensional domain of infinite depth in the presence of non-zero constant vorticity. A goal is to describe the effects of uniform shear flow on the modulation of weakly nonlinear quasi-monochromatic surface gravity waves. Starting from the Hamiltonian formulation of this problem and using techniques from Hamiltonian transformation theory, we derive a Hamiltonian Dysthe equation for the time evolution of the wave envelope. Consistent with previous studies, we observe that the uniform shear flow tends to enhance or weaken the modulational instability of Stokes waves depending on its direction and strength. Our method also provides a non-perturbative procedure to reconstruct the surface elevation from the wave envelope, based on the Birkhoff normal form transformation to eliminate all non-resonant triads. This model is tested against direct numerical simulations of the full Euler equations and against a related Dysthe equation derived recently by Curtis, Carter & Kalisch (J. Fluid Mech., vol. 855, 2018, pp. 322–350) in the context of constant vorticity. Very good agreement is found for a range of values of the vorticity.

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“…A positive vorticity, which corresponds to a following current -i.e. U (z) > 0 and U (z) < 0 with U (z) the current oriented along the wave propagation direction, z the vertical coordinate, and a prime denotes the derivative -can remove the modulational instability altogether, demonstrated experimentally by Steer et al [40] and Pizzo et al [41] (the definition of positive/negative shear in ref. [40] is different from ours due to a different choice of the coordinate system).…”

mentioning

confidence: 98%

“…U (z) > 0 and U (z) < 0 with U (z) the current oriented along the wave propagation direction, z the vertical coordinate, and a prime denotes the derivative -can remove the modulational instability altogether, demonstrated experimentally by Steer et al [40] and Pizzo et al [41] (the definition of positive/negative shear in ref. [40] is different from ours due to a different choice of the coordinate system). Francius and Kharif [42] have extended [34] to two-dimensional Stokes waves where new quartet and quintet instabilities have been discovered arising from the presence of a uniform vorticity, while Abrashkin and Pelinovsky [43] derived a nonlinear Schrödinger equation for arbitrary, weak vertical shear in a Lagrangian framework, generalized in ref.…”

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confidence: 98%

Statistics of weakly nonlinear waves on currents with strong vertical shear

Zheng,

Li,

Ellingsen

2023

Preprint

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We investigate how the presence of a vertically sheared current affects wave statistics, including the probability of rogue waves, and apply it to a real-world case using measured spectral and shear current data from the Mouth of the Columbia River. A theory for weakly nonlinear waves valid to second order in wave steepness is derived, and used to analyze statistical properties of surface waves; the theory extends the classic theory by Longuet-Higgins [J. Fluid Mech. 12, 3 (1962)] to allow for an arbitrary depth-dependent background flow, U (z), with U the horizontal velocity along the main direction of wave propagation and z the vertical axis. Numerical statistics are collected from a large number of realisations of random, irregular sea-states following a JONSWAP spectrum, on linear and exponential model currents of varying strengths. A number of statistical quantities are presented and compared to a range of theoretical expressions from the literature; in particular the distribution of wave surface elevation, surface maxima, and crest height; the exceedance probability including the probability of rogue waves; the maximum crest height among N s waves, and the skewness of the surface elevation distribution. We find that compared to no-shear conditions, opposing vertical shear (U (z) > 0) leads to increased wave height and increased skewness of the nonlinear-wave elevation distribution, while a following shear (U (z) < 0) has opposite effects. With the wave spectrum and velocity profile measured in the Columbia River estuary by Zippel & Thomson [J. Geophys. Res: Oceans 122, 3311 (2017)] our second-order theory predicts that the probability of rogue waves is significantly reduced and enhanced during ebb and flood, respectively, adding support to the notion that shear currents need to be accounted for in wave modelling and prediction.

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Experimental study of dispersion and modulational instability of surface gravity waves on constant vorticity currents

Cited by 11 publications

References 59 publications

The role of Lagrangian drift in the geometry, kinematics and dynamics of surface waves

The role of Lagrangian drift in the geometry, kinematics and dynamics of surface waves

A Hamiltonian Dysthe equation for deep-water gravity waves with constant vorticity

Statistics of weakly nonlinear waves on currents with strong vertical shear

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