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
waves on nonlinear surface waves is theoretically analyzed. An analytical and numerical solution of the modulation equations are found under the conditions close to the group velocity resonance. It is shown that smoothing of the down current surface waves is accompanied by a relatively high-frequency modulation while the profile of the opposing current is reproduced by the surface wave’s envelope. Long surface waves can form the wave modulation forerunner ahead of the internal wave, while the relatively short surface waves create the trace of the internal wave. Citation: Lee, K.-J., H.-H. Hwung, R.-Y. Yang, and I. V. Shugan (2007), Stokes waves modulation by internal waves, Geophys. Res. Lett., 34, L23605, doi:10.1029/2007GL031882.
1. Introduction
[2] Internal waves (IW) represent one of the most
inter-esting 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 observa-tions, and is reliably established now. Remote sensing instruments (radars, traditional optical devices, and laser locators – lidars) can detect reliably the IW surface exhibi-tions of the ocean surface. At the same time, the quantifi-cation 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 be-tween 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 s1). (2) The maximum modulation is observed at co-propagation of sea wind waves
and IWs, especially for the sea spectral components whose group velocity cgis close to the IW phase velocity c (cg 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 experi-ments [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
inter-nal 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
decime-ter 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 numer-ical 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 (soli-tons) 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 sufficient-ly 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.
Here
for
Full Article
1Department Naval Architecture and Ocean Engineering, Chosun University, Gwangju, Korea.
2Tainan Hydraulics Laboratory, National Cheng Kung University, Tainan, Taiwan.
3
Also at Prochorov General Physics Institute of Russian Academy of Sciences, Moscow, Russia.
Copyright 2007 by the American Geophysical Union. 0094-8276/07/2007GL031882$05.00
[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
equa-tions 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 var-ious-type of IWs and initial SW parameters are presented in Section 4. Section 5 contains concluding remarks and inferences.
2. Modulational Equations of a One-Dimensional Interaction
[11] Internal waves in the Ocean only produce very small
vertical displacements of the free surface. If N(0) is the Brunt-Vaisala frequency at the sea surface z = 0 (z is the vertical coordinate in the upward direction), then near the surface, the vertical velocity component is w = W(z)exp[i(Kx nt)], W(z) / exp {(N2(0)/n2 1)1/2 Kz}, where n and K – frequency and horizontal wave number of internal wave [Phillips, 1977]. The condition of constant pressure at the free surface leads to
n2Wz gK2W ¼ 0 at z ¼ 0: ð1Þ
and therefore the free surface condition can be fixed as W = 0 at z = 0 (solid cover condition), provided
n2 gK N 2 0 ð Þ=n2 1 1
Since the frequency n of internal waves is always much less than the frequency of a free surface wave with the same wave number, namely (gK)1/2, this condition is usually strongly satisfied.
[12] The typical case here is the existence of a strong
seasonal thermocline which separates two relatively ho-mogeneous water masses. N(z) consequently has a single sharp maximum at the thermocline and is very small elsewhere. By using the continuity equation Ux + Wz = 0
(U and x are the value of horizontal velocity and coordinate, respectively) and the dynamic boundary con-dition (1) we’ll have the following estimation for the vertical velocity near the free surface: W/U n2/gK dr/r0(103 characteristically), where dr and r0 are the
maximal variation of the density inside the thermocline and the mean density, correspondingly.
[13] As about 200 – 500 SW lengths and more can be
settled within one IW length, the interacting waves have very different scales. Therefore, the problem can be con-sidered as the SW propagation in a slowly varying moving medium. The first set of complete equations to describe short waves propagating over much larger scale nonuniform currents were given by Longuet-Higgins and Stewart [1964]. Wave energy is not conserved and the concept of ‘radiation stress’ was introduced to describe the averaged
momentum flux terms which govern the interchange of momentum with the current. In this present model, it is also sufficient to consider the energy exchange in terms of wave action conservation law of the second order and neglect the effect of this momentum transfer on the form of the surface current because it will be an effect of the highest order [Stocker and Peregrine, 1999].
[14] We construct the model of the IW effect on
propa-gation of the narrow-band weakly nonlinear Stokes packet of gravity SWs based on the following assumptions: (1) Sur-face and internal waves propagate along a common x-direction. (2) The current horizontal velocity in the subsurface layer is set as the traveling wave U = U(Kx) = U(K(x ct)), where x = x ct is the accompanying coordinate. (3) Horizontal subsurface current, U(Kx), is induced by the internal wave we’ll define by the velocity potential:F0 (Kx): dF0(Kx)/dx = U(Kx). Fluid motion is
accepted to be the potential since the horizontal current varies slowly enough (Ux/kU = O(e2), a0is a characteristic
free-surface displacement, 2p/k0is a typical surface
wave-length, and e = a0k0 1 is the conventional small
average wave steepness parameter). The ratio, U/c, is assumed to be small, U/c e, which is usually confirmed by experimental data (U/c 0.1 – 0.3) [Hughes and Grant, 1978; Gasparovic et al., 1988]. This, along with the assumption c cp(cp is the SW phase velocity), satisfies
to the continuity equation in the second order. Finally, the velocity potential function will be represented as a sum of wave’s and current’s potentialf (x, z, t) + F0(Kx).
[15] The set of equations for potential motion of an ideal
incompressible infinite-depth fluid with the free surface is given by the Laplace equation:
fxxþ fzz¼ 0; 1 < z < h x; tð Þ ð2Þ
the boundary conditions at the free surface:
ghþ ftþ 1 2 f 2 xþ f 2 z ¼ 0; z ¼ h x; tð Þ; ð3Þ htþ fxhx¼ fz; z¼ h x; tð Þ; ð4Þ
and at the bottom : f¼ 0; z ¼ 1: ð5Þ
[16] Here,h(x, t) is the free-surface displacement, g is the
gravity acceleration, and t is time.
[17] The variables are normalized as follows:
f¼ a0 ffiffiffiffiffi g k0 r f0¼ e ffiffiffiffiffig k3 0 r f0; h¼ a0h0¼ e k0 h0; t¼ 1ffiffiffiffiffiffiffi gk0 p t0; z¼ z 0 k0 ; x¼x 0 k0 ;U K xð ð ctÞÞ ¼ U0K=k0x0 c=cpt0 cp¼ U0ðe1ðx0 c0t0ÞÞcp; ð6Þ
wheree1= K/k0- another small parameter characterizing the
ratio of the surface and internal wavelengths and the dimensionless quantities are primed. It is noteworthy that
normalization (6) explicitly specifies the principal scales of sought functionsf = O(e) and h = O(e). Then, the set (2) – (5) is reduced to the form
fxxþ fzz¼ 0; 1 < z < eh x; tð Þ ð7Þ h ¼ ftþ U fxþ e 1 2 f 2 xþ f 2 z ; z¼ eh x; tð Þ; ð8Þ htþ U hxþ efxhx¼ fz; z¼ eh x; tð Þ; ð9Þ f¼ 0; z ¼ 1; ð10Þ
where the primes are omitted. The weakly nonlinear surface wave train is described by a solution to equations (7) – (10), expanded into a Stokes series in terms of small parametere. [18] Assuming the wave motion phaseq = q (x, t) in the
presence of a slowly varying current U, we define the local wavenumber k and frequency s as:
k¼ qx; sþ k U ¼ qt ð11Þ
These main wave parameters together with the first-order velocity potential amplitude,f0, will be considered further as slowly varying with the typical scale, O(e1), longer than the primary wavelength and period [Whitham, 1974]:
f0¼ f0ðex; etÞ; k ¼ k ex; etð Þ; s ¼ s ex; etð Þ ð12Þ
On this basis, we attempt to recover the effects of long-scaled current and nonlinear wave dispersion additional (having the same order) to the Stokes’ term with the wave steepness squared.
[19] The solution to the problem, uniformly valid to
O(e3), is found by a two-scale expansion with the differen-tiation: @ @t¼ s þ k Uð Þ @ @qþ e @ @T; @ @x¼ k @ @qþ e @ @X; T ¼ et; X ¼ ex: ð13Þ
Substitution of the velocity potential in its linear form,f = f0ekz sin q, satisfies the Laplace equation (7) in the first
order in e due to (11) and gives the following additional terms in the second order O(e2):
e 2kfð 0X þ kXf0þ 2kkXf0zÞe kz
cosqþ . . . ¼ 0 ð14Þ
To satisfy the Laplace equation in the second order, Yuen and Lake [1982] and Shugan and Voliak [1998] suggested additional phase shifted term with a linear and quadratic z correction in the representation of the potential functionf:
f¼ f0ekzsinq e f0Xzþ kXf0 2 z 2 ekzcosqþ . . . ð15Þ
Exponential decaying of wave’s amplitude with z is accompanied by the second order subsurface jet due to
slow horizontal variations of the wave number and amplitude of the wave packet.
[20] The free-surface displacement h = h (x, t) is also
sought as an asymptotic series,
h¼ h0þ eh1þ e 2
h2þ . . . ; ð16Þ
where h0,h1, and h2are O(1) functions to be determined.
Using expressions (15) – (16) subject to the dynamic boundary condition (8), we consequently find the compo-nents of the free-surface displacement
h0¼ sf0cosq; ð17Þ h1¼ f0Tsinq U f0Xsinqþ 1 2s 2 kf20cos 2qð Þ; ð18Þ h2¼ 3 8sk 3 f30cosq: ð19Þ
Only the self-action term with fundamental wave phaseq is included in the third-order displacement (19).
[21] Substitution of velocity potential (15) and
displace-ment (16) – (19) into the kinematics boundary condition (9) gives the following relationships between the wave modu-lation characteristics: s2¼ k þ e2 k4f20þ e 2 f0TTþ 2U f0XTþ U 2 f0XX =f0; ð20Þ f20s Tþ Uþ 1 2s f20s X ¼ 0: ð21Þ
The first of these formulas represents the wave dispersion relation with the total second-order amplitude-phase dispersion included in the presence of the current, U. Equation (21) yields the known wave action conservation law. Modulation equations (20) – (21) are closed by the equation of wave phase conservation that follows from (11) as a compatibility condition [Phillips, 1977]:
kTþ s þ kUð ÞX¼ 0 ð22Þ
The derived set of equations (20) – (22) in the absence of a current coincide with those of Shugan and Voliak [1998].
3. Traveling Wave Solution
[22] Let us analyze the traveling wave solutions to the
problem (20) – (22) supposing all the unknown functions to depend on the single coordinate x = X cT, where the waves are stationary. Then, after integrating (21) and (22) the problem has the form:
s2¼ k þ e2 k4f20þ e 2 Uð Þ cx ð Þ2f0xx=f0; ð23Þ Uð Þ c þ cx g f20s¼ A; ð24Þ sþ k U xð ð Þ cÞ ¼ W; ð25Þ
where cg = 1/(2s) is the linear group velocity of surface
waves, A and W are the integration free constants with the physical meaning of wave action flux and frequency, respectively, measured in the moving coordinate frame. The v a l u e s o f c o n s t a n t s a r e d e t e r m i n e d f r o m t h e boundary conditions for the uniform Stokes wave at infinity. I n t h e d i m e n s i o n a l f o r m t h e s e v a l u e s a r e A = 1 2 ffiffiffiffiffi g k0 r c a2 0 ffiffiffiffiffiffiffi gk0 p ;W¼ ffiffiffiffiffiffiffigk0 p 1þ ð 1 2ða0k0Þ 2 Þ k0c,
where a0, k0are the constant amplitude and wave number
of the Stokes wave in the absence of current at the infinity. [23] The analysis of the problem in the linear statement
[Gargett and Hughes, 1972; Lewis et al., 1974] where the dispersion relation (23) is independent of the SW amplitude describes the wave modulation under conditions far from the group resonance c cgwell. Amplitude distribution can
be found from the energy equation (24) after solving the ‘‘kinematic’’ closed set of equations (23), (25) for the wavenumber and frequency. In the resonance vicinity.
Uð Þ c þ cx g¼ 0; ð26Þ
the solution found has singularities related to an infinite amplitude growth at the points of SW ‘‘blockage’’ by the current (see equation (24)), which defines a ‘‘barrier’’ for SW propagation over it. In this case, the linear steady solution becomes incorrect.
[24] After eliminating the wavenumber k and frequencys
the set (23) – (25) transforms into a single equation for the first-order potential amplitudef0:
1 U c ð Þ2 A f20 !2 ¼ ð1þ 4W U cð ÞÞ 4 Uð cÞ2 þ e 2 1 2 A f20 ! 1 U c ð Þ2 þ W U c ð Þ !4 f2 0þ e 2 U c ð Þ2f0xx=f0ð27Þ
[25] The main feature of the model is clearly visible from
equation (27): the solution in the zero order one exists for positive values of expression
1þ 4W U xð ð Þ cÞ; ð28Þ
depending only on the flow velocity and the basic parameters of the problem. For sufficiently large negative values of U(x), expression (27) can become equal to zero or even negative so the highest orders of dispersion relation have to be taken into account. It can be shown that the zero value of expression (28) corresponds to the condition of zero local velocity of the wave action flux and correspond-ingly to unlimited growth of the wave amplitude (see equation (24)).
4. Stokes Surface Wave, Resonantly Modulated
by the Current
[26] First we’ll analyze the derived set (23) – (25) under
the resonant boundary conditions (26). At A = 0 the energy equation immediately yields the following result: the SW energy flux through any sectionx = x0is zero in the frame
of reference related to the IWs. The above mentioned blockage condition (26) defines the direct modulation of the surface wavenumber by the current
k¼ k0 1þ
3Uð Þx c
ð29Þ
Hence, the SW length in the resonant interaction mode is modulated rather weakly by the current induced by IWs. In this case, the SWs shorten and lengthen respectively at down- and countercurrents.
[27] Modulation of an initially homogeneous Stokes
wave will be considered under the following boundary conditions:
x! 1; U xð Þ ! 0; f0! 1; k ! 1: ð30Þ
Free parameters in the set (23) – (25) can be determined by using the boundary condition (30):
c¼1 2 e2 4; W¼ 1 2þ 3e2 4 : ð31Þ
It can be mentioned here, that no dimensional value of velocity c = 1/2 corresponds in the leading order to linear group velocity of surface waves and due to the condition of phase synchronism – to phase velocity of the internal wave. [28] The equation (27) up to O(e2U/c,e12U/c) takes on the
form
G2f0x0x0þ Gð 0uð Þ 1x0 Þf0þ f 3
0¼ 0; ð32Þ
wherex0- is the renormalized variable to the space scale of IW:x0 = (e1/e)x and primes will be omitted further; u(x) = U(x)/jU0j - current velocity function, normalized by its
maximum amplitude, G2= K 2k0 2 1 a0k0 ð Þ2 is the relation-ship between surface and internal typical wavelengths and SW initial steepness, G0 =
2 Uj 0j
a0k0
ð Þ2 O(e
1) is a big
parameter characterizes the value of subsurface current relative to the unperturbed SW steepness.
[29] Equation (32), subject to the boundary conditions
(30), was analyzed analytically and numerically for various values of controlling dimensionless parameters G0and G2.
Current velocity function, u(x), is presented in equation (32) in the normalized form (both space and amplitude scales are presented in the dimensionless parameters G0 and G2).
Therefore in analytical simulations, we’ll assume function u(x) to be a solitary smooth function with a nonzero value inside the interval (1, 1) and maximal amplitude equal to the unit.
[30] Let’s first consider the modulation of SW on a
counter flow when u(x) < 0. The equation (32) contains the big parameter G0 1 and the solution is sought as an
asymptotic series:
f0ð Þ ¼ Gx 1=2
After substitution of expression (33) into (32), the solution in the main order looks like:
f00ð Þ ¼ 1 Gx ð 0uð Þx Þ1=2=G 1=2
0 ð34Þ
The next order of approximation results to the equation
f01xxþ q 2 x ð Þf01þ f00xxG 1=2 0 ¼ 0; ð35Þ
where q2(x) = 2G0f002(x)/G2. This is the linear non-uniform
Schro¨dinger equation, containing the big parameter q2(x) = O(G0) at the potential. The potential function does not
change the sign. Consequently, here we apply a standard method of stationary phase [Nayfeh, 1981] to find its asymptotic solution: f01¼ f00xxG 1=2 0 =q 2þ C 1 Sin Zx 1 qð Þdtt 0 @ 1 A q1=2 þ C2 Cos Zx 1 qð Þdtt 0 @ 1 A q1=2 ; ð36Þ
where C1and C2are the constants that are to be determined
from the boundary conditions for the smooth merging of the solution (36) with the constant velocity potential function: f0= 1,f0x= 0 atx 1. Within o(G01/4), values of constants
are the following: C2= 0, C1= f00x(1)G0 1/2
/q1/2(1).
[31] We consider the data calculated for the modulation
of SW by the typical solitary type IW negative current u(x) = Sech2[x] for the near resonant conditions of wave
inter-action. The results of numerical calculations of the general equation (27) are shown in Figure 1a for two different values of the current intensity parameter G0. The above
asymptotic solution sufficiently describes the basic proper-ties of SW modulation. The main property of the solution is that the envelope of the SW on a counter flow may increase considerably, surpassing its initial value several times. For sufficiently large scale IW (G2 1), the SW envelope
reproduces the shape of the subsurface current. The ampli-tude of modulation grows with the intensity of the IW-induced current and is symmetric relative to the current profile axis.
[32] The data found for shorter IWs (G2 5) and a slight
mismatch from the resonance regime (c < 1/2) are shown in Figure 1b. The following interaction features are notewor-thy: the SW steepness modulation maximum shifts to the IW head slope as the IW horizontal size decreases. Another strong particularity of the solution is the arising of the IW forerunner: ahead of the IW, in the region with no current, an intensive periodic SW train propagates with low-fre-quency modulation exceeding the entering wave amplitude. This means that the relatively long SW (c < 1/2), after crossing the region of internal wave current, keeps the low frequency amplitude modulations. That property is confined by field measurements [Basovich et al., 1987; Bakhanov et al., 1994].
[33] Another possibility of arising the trace of the IW
while crossing the field of short SW (c > 1/2), also in the form of SW low frequency modulations, is shown at Figure 1. Surface wave envelopesf0=sf0(x) at (a) G2= 1, c = 0.5,e = 0.1; G0= 10 curve (I) and G0= 20 curve (II),
respectively; (b) G2= 5, c = 0.45,e = 0.1, G0= 10 curve (I); (c) G2= 5, c = 0.6,e = 0.1, G0= 10 curve (I); (d) G2= 1,
Figure 1c. The intensity of modulations in the trace of the IW is comparable with the main peak modulations.
[34] Resonant solution for the SW modulation on a down
flow u(x) > 0 can be found in the following form:
f0ð Þ ¼ 1 þ fx 01ð Þ þ . . .x ð37Þ
wheref01(x) = o(1). Substitution of the expression (37) into
(32) gives for f01(x):
f01xxþ q 2
x
ð Þf01þ G0=G2uð Þ ¼ 0;x ð38Þ
where q2(x) = (2 + G0u(x))/G2. This is the linear
non-uniform Schro¨dinger equation, containing the big parameter q2(x) = O(G0) O(e1) at the potential. The potential
function does not change in sign and consequently, we can apply a standard method of stationary phase [Nayfeh, 1981] to find an asymptotic solution on the interval (1, 1):
f01¼ C1 Sin Zx 1 qð Þdtt 0 @ 1 A q1=2 þ C2 Cos Zx 1 qð Þdtt 0 @ 1 A q1=2 G0u 2þ G0u : ð39Þ
where C1 and C2 are the constants, determined from the
boundary conditions for the smooth merging of the solution (39) with the constant velocity potential function: f0= 1,
f0x= 0 at x 1: C2= 0, C1= G0ux(1)q1/2(1)/(2 +
G0u(1)).
[35] Numerical calculations of the SW variation at the
positive current u(x) = Sech2[x] are shown in Figure 1d. The analytical solution to the problem conforms well to numer-ical calculations. Here, the basic effect is a strong smoothing of the sea surface over the developed current. The wave amplitude decreases and accompanies by formation of the SW envelope local maxima, the surface smoothing strength-ens, and some maxima grow with the intensity for the IW-induced current. As the interaction length grows (G2
decreases), the smoothing strengthens increasingly. More-over, high-frequency modulation of the SW envelope grows as well. Numerical simulations show that a significant forerunner or trace of the IW will not appear for a positive subsurface currents.
[36] Numerical simulations of the SW modulation by
IW-induced currents where also performed for various ranges of SW frequencies on the basis of the general modulation equation (27). Results clearly show that the maximal modulations take place for the SW in the vicinity of resonance conditions cg c and sharply decreased far from
the group resonance frequencies.
5. Conclusions
[37] The account of SW nonlinearity allows constructing
of the uniformly valid stationary solutions to the problem of the SW – IW interaction in the vicinity of their group velocity resonance and to describe a number of observed experimental effects. The basic properties of the solution correspond to the results of observations of SW modulations by large amplitude internal waves [Kropfli et al., 1999;
Bakhanov and Ostrovsky, 2002] and by the relatively short internal waves [Basovich et al., 1987; Bakhanov et al., 1994].
[38] We note the basic features of the near resonance SW
modulation:
[39] (1) The surface wave number vector in the current
region varies insignificantly and the SWs shorten and lengthen on following and counter currents, respectively.
[40] (2) The interaction with large-scale solitary IWs lead
to sufficiently strong modulations of the SW envelope. The surface countercurrent causes the SW steepness growth and the envelope shape follows the current profile. The steep-ness of modulation grows with the intensity of IW-induced current and is symmetric relative to the current profile axis. A moderate high-frequency oscillation is imposed on an algebraic part of the solution.
[41] (3) Modulation by a large-scale positive solitary
IW-induced current leads to the surface smoothing accompanied by relatively high-frequency modulation.
[42] (4) An IW forerunner can arise ahead of the IW train
for a long SW, manifesting itself as a modulated SW train with the period comparable to the IW spatial scale. Leaving the internal wave field, the surface waves save the obtained modulation.
[43] (5) A trace of IW in the case of negative subsurface
current can also arise for the relatively short SW in the form of a modulated SW train with a comparable intensity.
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H.-H. Hwung, I. V. Shugan, and R.-Y. Yang, Tainan Hydraulics Laboratory, National Cheng Kung University, 5th Floor, 500, Section 3, Anming Road, Tainan 709, Taiwan. (ishugan@yahoo.com)
K.-J. Lee, Department of Naval Architecture and Ocean Engineering, Chosun University, 375 Seosuk-Dong, Dong-Gu, Gwangju 501-759, Korea.
