On the relation between Stokes drift and the Gerstner wave

Abrashkin, A. A.; Pelinovskii, E. N.

doi:10.3367/ufnr.2017.03.038089

Cited by 4 publications

(7 citation statements)

References 16 publications

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“…We then presented the explicit connection between the Lagrangian mean flow and wave geometry, kinematics and dynamics for the scenario when the Lagrangian mean flow was weak. This included explicit expressions for the dependence of the Benjamin–Feir instability growth rate (Abrashkin & Pelinovsky 2017, 2018), as well as the range of unstable perturbation wavenumbers and frequencies, on the Lagrangian mean flow. To illustrate these points, we presented results from a recent laboratory demonstration that highlight the striking dependence of finite amplitude wave behaviour on the underlying shear flow.…”

Section: Discussionmentioning

confidence: 99%

“…Next we find the third-order expansions for Stokes waves by expanding the Gerstner waves solutions to include Lagrangian mean flows, (see also the related discussions in Abrashkin & Pelinovsky 2017, 2018). Going through the algebra, we find where and Note, the phase now has dependence on and is unknown at this stage.…”

Section: Weakly Nonlinear Waves On a Weak Shear Flowmentioning

confidence: 99%

“…The form of the phase velocity is enough to provide a heuristic derivation of the nonlinear Schrödinger equation (NLSE) on a weak Lagrangian shear flow (Abrashkin & Pelinovsky 2017), following the classical geometrical optics approach (Benjamin & Feir 1967; Yuen & Lake 1982; Abrashkin & Pelinovsky 2018). Whereas previous, Eulerian derivations of the NLSE in the presence of shear have favoured the special case of constant vorticity which permits potential theory (Baumstein 1998; Thomas, Kharif & Manna 2012); the Lagrangian framework handles arbitrary more readily.…”

Section: Weakly Nonlinear Waves On a Weak Shear Flowmentioning

confidence: 99%

“…Following Abrashkin & Pelinovsky (2018), we take a geometrical optics approach and let and where we have expanded about a carrier wave with frequency and wavenumber . Multiplying the dispersion relationship by , we find that to where and where the group velocity is .…”

Section: Weakly Nonlinear Waves On a Weak Shear Flowmentioning

confidence: 99%

“…The system is unstable when and while the most unstable wavenumber occurs when with growth rate The relations (3.18) and (3.19), although implicit in Abrashkin & Pelinovsky (2018), were not presented there. We see that the growth rate of the instability may be strongly modified (i.e.…”

Section: Weakly Nonlinear Waves On a Weak Shear Flowmentioning

confidence: 99%

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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: Discussionmentioning

confidence: 99%