## 1. Introduction

[2] Internal waves (IW) represent one of the most interesting phenomena in ocean dynamics. Field investigations carried out by conventional contact oceanography methods are complicated by significant technical problems.

[3] Currently, another line of inquiry is very common: the remote study of IWs by their exhibition at the sea surface along with contact measurements. The ability of IWs to substantially change the wave structure on that surface had long been explored [*Hughes and Grant*, 1978; *Osborne and Burch*, 1980; *Alpers*, 1985], confirmed by many observations, and is reliably established now. Remote sensing instruments (radars, traditional optical devices, and laser locators–lidars) can detect reliably the IW surface exhibitions of the ocean surface. At the same time, the quantification of IW parameters by remote sensing of the sea surface calls for solving a number of theoretical problems. First, construct a model of the sea surface imaging and establish mechanisms of surface wave (SW) perturbation by IWs.

[4] The following are the basic properties of sea surface anomalies established experimentally during combined oceanographic missions as a result of the interaction between IWs and the sea surface [*Basovich et al.*, 1987; *Hughes and Grant*, 1978; *Gasparovic et al.*, 1988]: (1) Wide bands of slicks and rough sea move at a phase velocity of the IW train. These bands are most distinct at moderate and light wind speeds (lower than 5 m · s^{−1}). (2) The maximum modulation is observed at co-propagation of sea wind waves and IWs, especially for the sea spectral components whose group velocity *c*_{g} is close to the IW phase velocity *c* (*c*_{g} ∼ *c*). (3) An individual slick is not uniquely arranged with respect to the IW phase. It can be both above the IW trough and above the IW crest. (4) The distance between the bands usually corresponds to the IW length. (5) A lot of experiments [*Basovich et al.*, 1987; *Bakhanov et al.*, 1994] demonstrated the wind waves anomalies on the long (about 1km) distance from the space filled by the internal waves.

[5] In general, the interaction between surface and internal waves was studied both theoretically and experimentally in the work of the last three decades. Nevertheless, we have no exhaustive theory of this phenomenon up until now, though several mechanisms are proposed for such an interaction in various SW spectral ranges.

[6] A basic modulation mechanism in meter and decimeter SW ranges is the hydrodynamic impact of a subsurface current induced by IWs on the SWs [*Lewis et al.*, 1974; *Phillips*, 1977]. The quasi-steady model of such an impact on a linear surface wavepacket was first proposed by *Gargett and Hughes* [1972] and *Phillips* [1977]. The analysis of modulational equations had shown the packets with a group velocity approaching the IW phase velocity to be most sensitive to the subsurface current. The SW amplitude in the group resonance range grows infinitely, showing inapplicability of the developed linear modulation model to quantitative description of the wave interaction.

[7] Recently the unsteady propagation of short surface waves in the presence of IW current was studied [*Donato et al.*, 1999; *Stocker and Peregrine*, 1999] using both simple ray theory for linear waves and a fully nonlinear numerical potential solver. For ray theory, the occurrence of focusing is examined in some detail and it is shown that in these regions the waves steepened and may break. Comparisons are made between ray theory and the more accurate numerical simulations.

[8] Transformation of gravity capillary surface waves on the current created by a large-amplitude internal wave is observed by *Kropfli et al.* [1999] and *Bakhanov and Ostrovsky* [2002]. In particular, the location of the maxima and minima of the surface wave spectral density with respect to the IW profile is studied. It is shown that for sufficiently large-amplitude internal solitary waves (solitons) propagating in the same direction as the surface wave the minimum of density for all SW lengths is situated over the crest of the soliton. These observations conflict with the expectation that the highest surface roughness would be near the region of the greatest surface gradient [*Gasparovic et al.*, 1988; *Hogan et al.*, 1996]. It can be mentioned, however, that the last conclusions were made for sufficiently small IW currents, ∼1–2 cm/sec, whereas currents may be many times bigger for large amplitude IW. This does clearly emphasize the importance of the wave's nonlinearity in the modeling of an IW impact on the sea surface.

[9] The goal of the present work is to construct a uniformly valid steady model of the IW interaction with nonlinear Stokes surface waves under the condition close to the group velocity resonance and to describe a number of observed experimental effects.

[10] The paper consists of five sections. General equations of the one-dimensional interaction between an SW train and a nonuniform current induced by IWs are derived in section 2. Section 3 is dedicated to an analysis of the traveling solution for the resonant interaction. The data on analytically and numerically calculated interaction for various-type of IWs and initial SW parameters are presented in Section 4. Section 5 contains concluding remarks and inferences.