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Stokes Waves Modulation by Internal Waves Kwi-Joo Lee,1 Hwung-Hweng Hwung,2 Ray-Yeng Yang,2 and Igor V

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Stokes Waves Modulation by Internal Waves Kwi-Joo Lee,1 Hwung-Hweng Hwung,2 Ray-Yeng Yang,2 and Igor V GEOPHYSICAL RESEARCH LETTERS, VOL. 34, L23605, doi:10.1029/2007GL031882, 2007 Click Here for Full Article Stokes waves modulation by internal waves Kwi-Joo Lee,1 Hwung-Hweng Hwung,2 Ray-Yeng Yang,2 and Igor V. Shugan2,3 Received 30 August 2007; accepted 29 October 2007; published 4 December 2007. [1] The effect of subsurface currents induced by internal and IWs, especially for the sea spectral components whose waves on nonlinear surface waves is theoretically analyzed. group velocity cg is close to the IW phase velocity c (cg c). An analytical and numerical solution of the modulation (3) An individual slick is not uniquely arranged with respect equations are found under the conditions close to the group to the IW phase. It can be both above the IW trough and velocity resonance. It is shown that smoothing of the down above the IW crest. (4) The distance between the bands current surface waves is accompanied by a relatively high- usually corresponds to the IW length. (5) A lot of experi- frequency modulation while the profile of the opposing ments [Basovich et al., 1987; Bakhanov et al., 1994] current is reproduced by the surface wave’s envelope. Long demonstrated the wind waves anomalies on the long (about surface waves can form the wave modulation forerunner 1km) distance from the space filled by the internal waves. ahead of the internal wave, while the relatively short surface [5] In general, the interaction between surface and inter- waves create the trace of the internal wave. Citation: Lee, nal waves was studied both theoretically and experimentally K.-J., H.-H. Hwung, R.-Y. Yang, and I. V. Shugan (2007), Stokes in the work of the last three decades. Nevertheless, we have waves modulation by internal waves, Geophys. Res. Lett., 34, no exhaustive theory of this phenomenon up until now, L23605, doi:10.1029/2007GL031882. though several mechanisms are proposed for such an interaction in various SW spectral ranges. 1. Introduction [6] A basic modulation mechanism in meter and decime- ter SW ranges is the hydrodynamic impact of a subsurface [2] Internal waves (IW) represent one of the most inter- current induced by IWs on the SWs [Lewis et al., 1974; esting phenomena in ocean dynamics. Field investigations Phillips, 1977]. The quasi-steady model of such an impact carried out by conventional contact oceanography methods on a linear surface wavepacket was first proposed by are complicated by significant technical problems. Gargett and Hughes [1972] and Phillips [1977]. The [3] Currently, another line of inquiry is very common: the analysis of modulational equations had shown the packets remote study of IWs by their exhibition at the sea surface with a group velocity approaching the IW phase velocity to along with contact measurements. The ability of IWs to be most sensitive to the subsurface current. The SW substantially change the wave structure on that surface had amplitude in the group resonance range grows infinitely, long been explored [Hughes and Grant, 1978; Osborne and showing inapplicability of the developed linear modulation Burch, 1980; Alpers, 1985], confirmed by many observa- model to quantitative description of the wave interaction. tions, and is reliably established now. Remote sensing [7] Recently the unsteady propagation of short surface instruments (radars, traditional optical devices, and laser waves in the presence of IW current was studied [Donato et locators–lidars) can detect reliably the IW surface exhibi- al., 1999; Stocker and Peregrine, 1999] using both simple tions of the ocean surface. At the same time, the quantifi- ray theory for linear waves and a fully nonlinear numerical cation of IW parameters by remote sensing of the sea potential solver. For ray theory, the occurrence of focusing surface calls for solving a number of theoretical problems. is examined in some detail and it is shown that in these First, construct a model of the sea surface imaging and regions the waves steepened and may break. Comparisons establish mechanisms of surface wave (SW) perturbation by are made between ray theory and the more accurate numer- IWs. ical simulations. [4] The following are the basic properties of sea surface [8] Transformation of gravity capillary surface waves on anomalies established experimentally during combined the current created by a large-amplitude internal wave is oceanographic missions as a result of the interaction be- observed by Kropfli et al. [1999] and Bakhanov and tween IWs and the sea surface [Basovich et al., 1987; Ostrovsky [2002]. In particular, the location of the maxima Hughes and Grant, 1978; Gasparovic et al., 1988]: (1) Wide and minima of the surface wave spectral density with bands of slicks and rough sea move at a phase velocity of respect to the IW profile is studied. It is shown that for the IW train. These bands are most distinct at moderate and sufficiently large-amplitude internal solitary waves (soli- À1 light wind speeds (lower than 5 m Á s ). (2) The maximum tons) propagating in the same direction as the surface wave modulation is observed at co-propagation of sea wind waves the minimum of density for all SW lengths is situated over the crest of the soliton. These observations conflict with the 1 expectation that the highest surface roughness would be Department Naval Architecture and Ocean Engineering, Chosun near the region of the greatest surface gradient [Gasparovic University, Gwangju, Korea. 2Tainan Hydraulics Laboratory, National Cheng Kung University, et al., 1988; Hogan et al., 1996]. It can be mentioned, Tainan, Taiwan. however, that the last conclusions were made for sufficient- 3Also at Prochorov General Physics Institute of Russian Academy of ly small IW currents, 1–2 cm/sec, whereas currents may Sciences, Moscow, Russia. be many times bigger for large amplitude IW. This does clearly emphasize the importance of the wave’s nonlinearity Copyright 2007 by the American Geophysical Union. 0094-8276/07/2007GL031882$05.00 in the modeling of an IW impact on the sea surface. L23605 1of7 L23605 LEE ET AL.: STOKES WAVES MODULATION L23605 [9] The goal of the present work is to construct a momentum flux terms which govern the interchange of uniformly valid steady model of the IW interaction with momentum with the current. In this present model, it is nonlinear Stokes surface waves under the condition close to also sufficient to consider the energy exchange in terms of the group velocity resonance and to describe a number of wave action conservation law of the second order and observed experimental effects. neglect the effect of this momentum transfer on the form [10] The paper consists of five sections. General equa- of the surface current because it will be an effect of the tions of the one-dimensional interaction between an SW highest order [Stocker and Peregrine, 1999]. train and a nonuniform current induced by IWs are derived [14] We construct the model of the IW effect on propa- in section 2. Section 3 is dedicated to an analysis of the gation of the narrow-band weakly nonlinear Stokes packet traveling solution for the resonant interaction. The data on of gravity SWs based on the following assumptions: (1) Sur- analytically and numerically calculated interaction for var- face and internal waves propagate along a common ious-type of IWs and initial SW parameters are presented in x-direction. (2) The current horizontal velocity in the Section 4. Section 5 contains concluding remarks and subsurface layer is set as the traveling wave U = U(Kx)= inferences. U(K(x À ct)), where x = x À ct is the accompanying coordinate. (3) Horizontal subsurface current, U(Kx), is 2. Modulational Equations of a One-Dimensional induced by the internal wave we’ll define by the velocity Interaction potential:F0 (Kx): dF0(Kx)/dx = U(Kx). Fluid motion is accepted to be the potential since the horizontal current 2 [11] Internal waves in the Ocean only produce very small varies slowly enough (Ux/kU = O(e ), a0 is a characteristic vertical displacements of the free surface. If N(0) is the Brunt- free-surface displacement, 2p/k0 is a typical surface wave- Vaisala frequency at the sea surface z =0(z is the vertical length, and e = a0k0 1 is the conventional small coordinate in the upward direction), then near the surface, average wave steepness parameter). The ratio, U/c,is the vertical velocity component is w = W(z)exp[i(Kx À nt)], assumed to be small, U/c e, which is usually confirmed W(z) / exp {(N2(0)/n2 À 1)1/2 Kz}, where n and K – by experimental data (U/c 0.1–0.3) [Hughes and Grant, frequency and horizontal wave number of internal wave 1978; Gasparovic et al., 1988]. This, along with the [Phillips, 1977]. The condition of constant pressure at the assumption c cp (cp is the SW phase velocity), satisfies free surface leads to to the continuity equation in the second order. Finally, the velocity potential function will be represented as a sum of 2 2 n Wz À gK W ¼ 0atz ¼ 0: ð1Þ wave’s and current’s potential f (x, z, t)+F0 (Kx). [15] The set of equations for potential motion of an ideal and therefore the free surface condition can be fixed as W = incompressible infinite-depth fluid with the free surface is 0atz = 0 (solid cover condition), provided given by the Laplace equation: 2 n fxx þ fzz ¼ 0; 1 < z < hðÞx; t ð2Þ N 2ðÞ0 =n2 À 1 1 gK the boundary conditions at the free surface: Since the frequency n of internal waves is always much less 1 ÀÁ than the frequency of a free surface wave with the same gh þ f þ f2 þ f2 ¼ 0; z ¼ hðÞx; t ; ð3Þ wave number, namely (gK)1/2, this condition is usually t 2 x z strongly satisfied.
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