Exposure of internal waves on the sea surface

Hwung, Hwung Hweng; Yang, Ray Yeng; Shugan, Igor V.

doi:10.1017/s0022112008004758

Cited by 10 publications

(23 citation statements)

References 27 publications

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“…To satisfy the Laplace equation to second order, Yuen and Lake (1982), Voliak (1998), andHwung et al (2009) suggested an additional phase-shifted term with a linear and quadratic z correction in the representation of the potential function ϕ:…”

Section: Modulation Equations For One-dimensional Interactionmentioning

confidence: 99%

An analytical model of the evolution of a Stokes wave and its two Benjamin–Feir sidebands on nonuniform unidirectional current

Shugan

Hwung

Yang

2015

Nonlin. Processes Geophys.

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Abstract. An analytical weakly nonlinear model of the Benjamin-Feir instability of a Stokes wave on nonuniform unidirectional current is presented. The model describes evolution of a Stokes wave and its two main sidebands propagating on a slowly varying steady current. In contrast to the models based on versions of the cubic Schrödinger equation, the current variations could be strong, which allows us to examine the blockage and consider substantial variations of the wave numbers and frequencies of interacting waves. The spatial scale of the current variation is assumed to have the same order as the spatial scale of the Benjamin-Feir (BF) instability. The model includes wave action conservation law and nonlinear dispersion relation for each of the wave's triad. The effect of nonuniform current, apart from linear transformation, is in the detuning of the resonant interactions, which strongly affects the nonlinear evolution of the system.The modulation instability of Stokes waves in nonuniform moving media has special properties. Interaction with countercurrent accelerates the growth of sideband modes on a short spatial scale. An increase in initial wave steepness intensifies the wave energy exchange accompanied by wave breaking dissipation, resulting in asymmetry of sideband modes and a frequency downshift with an energy transfer jump to the lower sideband mode, and depresses the higher sideband and carrier wave. Nonlinear waves may even overpass the blocking barrier produced by strong adverse current. The frequency downshift of the energy peak is permanent and the system does not revert to its initial state. We find reasonable correspondence between the results of model simulations and available experimental results for wave interaction with blocking opposing current. Large transient or freak waves with amplitude and steepness several times those of normal waves may form during temporal nonlinear focusing of the waves accompanied by energy income from sufficiently strong opposing current. We employ the model for the estimation of the maximum amplification of wave amplitudes as a function of opposing current value and compare the result obtained with recently published experimental results and modeling results obtained with the nonlinear Schrödinger equation.

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Section: Modulation Equations For One-dimensional Interactionmentioning

confidence: 99%

An analytical model of the evolution of a Stokes wave and its two Benjamin–Feir sidebands on nonuniform unidirectional current

Shugan

Hwung

Yang

2015

Nonlin. Processes Geophys.

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“…As discussed in the introduction, several previous works used asymptotic analysis to obtain effective reduced models for addressing the surface signature of an IW (Kawahara et al 1975;Hashizume 1980;Funakoshi & Oikawa 1983;Donato et al 1999;Parau & Dias 2001;Bakhanov & Ostrovsky 2002;Barros & Gavrilyuk 2007;Hwung et al 2009;Craig et al 2012). These reduced models mostly focus on either the dynamics of short-mode waves around the resonant k res at sufficiently long times or the linear modulations and the traveling wave solutions for short-mode surface waves.…”

Section: Conclusion and Discussionmentioning

confidence: 99%

“…(ii) Resonant excitations: The rough region travels along with the internal wave; that is, the SW group velocity is close to the internal-wave phase velocity (Osborne & Burch 1980;Kropfli et al 1999;Hwung et al 2009;Craig et al 2012). To understand the above two surface phenomena, we numerically study the surfacewave and interfacial-wave (IW) manifestation in a two-layer, density-stratified system using a weakly nonlinear Boussinesq-type model.…”

Section: Introductionmentioning

confidence: 99%

Modulation-resonance mechanism for surface waves in a two-layer fluid system

et al. 2019

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We propose a Boussinesq-type model to study the surface/interfacial wave manifestation of an underlying, slowly-varying, long-wavelength, baroclinic flow in a two-layer, density-stratified system. The results of our model show numerically that, under strong nonlinearity, surface waves, with their typical wavenumber being the resonant k res , can be generated locally at the leading edge of the underlying slowly-varying, long-wavelength baroclinic flow. Here, the resonant k res satisfies the class 3 triad resonance condition among two short-mode waves and one long-mode wave in which all waves propagate in the same direction. Moreover, when the slope of the baroclinic flow is sufficiently small, only one spatially-localized large-amplitude surface wave packet can be generated at the leading edge. This localized surface wave packet becomes high in amplitude and large in group velocity after the interaction with its surrounding waves. These results are qualitatively consistent with various experimental observations including resonant surface waves at the leading edge of an internal wave. Subsequently, we propose a mechanism, referred to as the modulation-resonance mechanism, underlying these surface phenomena, based on our numerical simulations. The proposed modulation-resonance mechanism combines the linear modulation (ray-based) theory for the spatiotemporal asymmetric behavior of surface waves and the nonlinear class 3 triad resonance theory for the energy focusing of surface waves around the resonant wavenumber k res in Fourier space.

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“…Internal waves could impact the sea surface by altering the approximately two-dimensional velocity field on the surface plane, for example, see Alpers (1985). The convergent/divergent patterns on the surface driven by underlying IWs may manifest themselves through multiple surface modulation mechanisms (Klemas 2012), such as (1) hydrodynamic modulation through IW strains acting on surface wind-generated capillary-gravity waves (Alpers 1985;Hwung, Yang & Shugan 2009), (2) surface film modulation by forming 'slicks' of oceanic surfactants consisting of organic-rich materials (Ewing 1950a;Ermakov, Salashin & Panchenko 1992) under low-wind conditions, and (3) thermal modulation through thickness changes of the 'cool-skin' temperature sublayer (Osborne 1964(Osborne , 1965Zappa & Jessup 2005). Key to these processes is the horizontal divergence…”

Section: Internal Waves Emitted By Localized Stratified Turbulencementioning

confidence: 99%

Surface manifestation of internal waves emitted by submerged localized stratified turbulence

Zhou

Diamessis

2016

J. Fluid Mech.

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The internal waves (IWs) radiated by the turbulent wake of a sphere of diameter $D$ towed at speed $U$ are investigated using three-dimensional fully nonlinear simulations performed in a linearly stratified Boussinesq fluid with buoyancy frequency $N$. The study focuses on a broad range of wave characteristics in the far field of the turbulent wave source, specifically at the sea surface (as modelled by a free-slip rigid lid) where the IWs reflect. Six simulations are performed at Reynolds number $Re\equiv UD/{\it\nu}\in \{5\times 10^{3},10^{5}\}$ and Froude number $Fr\equiv 2U/(ND)\in \{4,16,64\}$, where ${\it\nu}$ is viscosity. The wave-emitting wake is located at a fixed distance of $9D$ below the surface. As the wake evolves for up to $O(300)$ units of buoyancy time scale $1/N$, IW characteristics, such as horizontal wavelength ${\it\lambda}_{H}$ and wave period $T$, are sampled at the sea surface via wavelet transforms of horizontal divergence signals. The statistics of amplitudes and orientations of IW-induced surface strains are also reported. The mean dimensionless observable wavelength $\overline{{\it\lambda}}_{H}/D$ at the sea surface decays in time as $(Nt)^{-1}$, which is due to the waves’ dispersion. This observation is in agreement with a linear propagation model that is independent of the wake $Re$ and $Fr$. This agreement further suggests that the most energetic waves impacting the surface originate from the early-time wake that is adjusting to buoyancy. The most energetic dimensionless wavelength $\hat{{\it\lambda}}_{H}/D$ is found to scale as $Fr^{1/3}$ and decrease with $Re$, which causes the arrival time (in $Nt$ units) of the strongest waves at the surface to scale as $Fr^{-1/3}$ and increase with $Re$. This wavelength $\hat{{\it\lambda}}_{H}$ is also found to correlate with the vertical Taylor scale of the wake turbulence. IW-driven phenomena at the surface that are of interest to an observer, such as the local enrichment of surfactant and the transport of ocean surface tracers, are also examined. The local enrichment ratio of surface scalar scales linearly with the steepness of IWs that reach the surface, and the ratio often exceeds a possible visibility threshold. The Lagrangian drifts of ocean tracers, which are linked to the nonlinear interaction between incident and reflecting IW packets, create a local divergence in lateral mass transport right above the wake centreline, an effect that intensifies strongly with increasing $Fr$. The findings of this study may serve as a platform to investigate the generation and surface manifestation of IWs radiated by other canonical submerged stratified turbulent flows.

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Exposure of internal waves on the sea surface

Cited by 10 publications

References 27 publications

An analytical model of the evolution of a Stokes wave and its two Benjamin–Feir sidebands on nonuniform unidirectional current

An analytical model of the evolution of a Stokes wave and its two Benjamin–Feir sidebands on nonuniform unidirectional current

Modulation-resonance mechanism for surface waves in a two-layer fluid system

Surface manifestation of internal waves emitted by submerged localized stratified turbulence

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