Particle dynamics in the KdV approximation

Borluk, Handan; Kalisch, Henrik

doi:10.1016/j.wavemoti.2012.04.007

Cited by 47 publications

(26 citation statements)

References 23 publications

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“…In order then to better understand the interplay of constant vorticity on deep-water nonlinear-surface flows, we study how a constant vorticity shear profile influences the motion and mean properties of particle paths both at and beneath waves in infinitely deep water. Our work complements and expands on the shallow and finite depth results found in Wahlén (2009); Constantin (2011); Borluk & Kalisch (2012) and Ribeiro et al (2017). To do this, we first derive a higher-order nonlinear Schrödinger (NLS) model, which we call the Vor-Dysthe (VD) equation, which describes the long temporal and spatial coupling between the nonlinear carrier wave and the mean fluid depth, in effect extending the now classic results of Dysthe (1979) to include constant vorticity and extending the results of Thomas et al (2012) to the next asymptotic order.…”

Section: Introductionsupporting

confidence: 72%

Particle paths in nonlinear Schrödinger models in the presence of linear shear currents

2018

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We investigate the effect of constant-vorticity background shear on the properties of wavetrains in deep water. Using the methodology of Fokas (A Unified Approach to Boundary Value Problems, 2008, SIAM), we derive a higher-order nonlinear Schrödinger equation in the presence of shear and surface tension. We show that the presence of shear induces a strong coupling between the carrier wave and the mean-surface displacement. The effects of the background shear on the modulational instability of plane waves is also studied, where it is shown that shear can suppress instability, although not for all carrier wavelengths in the presence of surface tension. These results expand upon the findings of Thomas et al. (Phys. Fluids, vol. 24 (12), 2012, 127102). Using a modification of the generalized Lagrangian mean theory in Andrews & McIntyre (J. Fluid Mech., vol. 89, 1978, pp. 609–646) and approximate formulas for the velocity field in the fluid column, explicit, asymptotic approximations for the Lagrangian and Stokes drift velocities are obtained for plane-wave and Jacobi elliptic function solutions of the nonlinear Schrödinger equation. Numerical approximations to particle trajectories for these solutions are found and the Lagrangian and Stokes drift velocities corresponding to these numerical solutions corroborate the theoretical results. We show that background currents have significant effects on the mean transport properties of waves. In particular, certain combinations of background shear and carrier wave frequency lead to the disappearance of mean-surface mass transport. These results provide a possible explanation for the measurements reported in Smith (J. Phys. Oceanogr., vol. 36, 2006, pp. 1381–1402). Our results also provide further evidence of the viability of the modification of the Stokes drift velocity beyond the standard monochromatic approximation, such as recently proposed in Breivik et al. (J. Phys. Oceanogr., vol. 44, 2014, pp. 2433–2445) in order to obtain a closer match to a range of complex ocean wave spectra.

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Section: Introductionsupporting

confidence: 72%

Particle paths in nonlinear Schrödinger models in the presence of linear shear currents

2018

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“…S2). This latter feature points out the main difference between internal progressive waves and ISWs in moving sand along the wave vector: internal progressive waves would cause an oscillating motion, leading to symmetric sand waves, while the passage of an ISW induces a unidirectional momentum to a fluid parcel2122 (Figs 2 and 3, and Supplementary Fig. S2), which agrees with the asymmetric pattern of the sand waves off Capo Rasocolmo.…”

Section: Refracting Internal Solitary Waves For Sand-waves Generationsupporting

confidence: 75%

“…In accordance with the KdV model, particle trajectories induced by the presence of a soliton are not closed during the passage of an ISW, and thus there is no backward motion2122 (Figs 2 and 3d, and Supplementary Fig. S2).…”

Section: Refracting Internal Solitary Waves For Sand-waves Generationsupporting

confidence: 62%

The role of Internal Solitary Waves on deep-water sedimentary processes: the case of up-slope migrating sediment waves off the Messina Strait

Droghei

Falcini

Casalbore

et al. 2016

Sci Rep

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Subaqueous, asymmetric sand waves are typically observed in marine channel/canyon systems, tidal environments, and continental slopes exposed to strong currents, where they are formed by current shear resulting from a dominant unidirectional flow. However, sand-wave fields may be readily observed in marine environments where no such current exists; the physical processes driving their formation are enigmatic or not well understood. We propose that internal solitary waves (ISWs) induced by tides can produce an effective, unidirectional boundary “current” that forms asymmetric sand waves. We test this idea by examining a sand-wave field off the Messina Strait, where we hypothesize that ISWs formed at the interface between intermediate and surface waters are refracted by topography. Hence, we argue that the deflected pattern (i.e., the depth-dependent orientation) of the sand-wave field is due to refraction of such ISWs. Combining field observations and numerical modelling, we show that ISWs can account for three key features: ISWs produce fluid velocities capable of mobilizing bottom sediments; the predicted refraction pattern resulting from the interaction of ISWs with bottom topography matches the observed deflection of the sand waves; and predicted migration rates of sand waves match empirical estimates. This work shows how ISWs may contribute to sculpting the structure of continental margins and it represents a promising link between the geological and oceanographic communities.

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“…Using the Euler equations (8), (9) one can obtain the time evolution of the total drift along L λ t in the form…”

Section: Kinematic Balance Law In Two-dimensional Flowsmentioning

confidence: 99%

“…However, in the present theory, the horizontal velocity is assumed to vary quadratically with the depth. This results in a more complicated set of equations, and may also be used advantageously for kinematic studies of the fluid motion, such as in [6,9,18].…”

Section: D Balance Law In the Green-naghdi Approximationmentioning

confidence: 99%

A kinematic conservation law in free surface flow

2015

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The Green-Naghdi system is used to model highly nonlinear weakly dispersive waves propagating at the surface of a shallow layer of a perfect fluid. The system has three associated conservation laws which describe the conservation of mass, momentum, and energy due to the surface wave motion. In addition, the system features a fourth conservation law which is the main focus of this note. It will be shown how this fourth conservation law can be interpreted in terms of a concrete kinematic quantity connected to the evolution of the tangent velocity at the free surface. The equation for the tangent velocity is first derived for the full Euler equations in both two and three dimensional flows, and in both cases, it gives rise to an approximate balance law in the Green-Naghdi theory which turns out to be identical to the fourth conservation law for this system.

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Particle dynamics in the KdV approximation

Cited by 47 publications

References 23 publications

Particle paths in nonlinear Schrödinger models in the presence of linear shear currents

Particle paths in nonlinear Schrödinger models in the presence of linear shear currents

The role of Internal Solitary Waves on deep-water sedimentary processes: the case of up-slope migrating sediment waves off the Messina Strait

A kinematic conservation law in free surface flow

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