Nonlinear dynamics of short traveling capillary-gravity waves

Nonlinear dynamics of short traveling capillary-gravity waves

C. H. Borzi,1R. A. Kraenkel,2M. A. Manna,3and A. Pereira3

1

Facultad de Ciencias Exactas y Naturales de la Universidad Nacional de Buenos Aires, Pab. II, Nuñez, Buenos Aires, Argentina 2

Instituto de Física Teórica-UNESP Rua Pamplona 145, 01405-900 São Paulo, Brazil 3

Physique Mathématique et Théorique, CNRS-UMR5825, Université Montpellier II, 34095 Montpellier, France 共Received 16 July 2004; revised manuscript received 29 November 2004; published 23 February 2005兲 We establish a Green-Nagdhi model equation for capillary-gravity waves in共2+1兲 dimensions. Through the derivation of an asymptotic equation governing short-wave dynamics, we show that this system possesses

共1+1兲 traveling-wave solutions for almost all the values of the Bond number␪ 共the special case ␪=1/3 is not

studied兲. These waves become singular when their amplitude is larger than a threshold value, related to the velocity of the wave. The limit angle at the crest is then calculated. The stability of a wave train is also studied via a Benjamin-Feir modulational analysis.

DOI: 10.1103/PhysRevE.71.026307 PACS number共s兲: 47.10.⫹g, 47.20.Ky, 47.35.⫹i, 04.25.⫺g

I. INTRODUCTION

The propagation of surface waves in an ideal incompress-ible fluid is a classical and still open subject of investigation in fluid mechanics. One possible approach to tackle this problem is to elaborate approximate theories from hypoth-eses on the nature of the waves. Thus, the shallow water approximation has produced a lot of interesting and useful nonlinear evolution model equations for small-amplitude long surface waves. These models can be classified in two main categories: extreme long-wave models like the Korteweg–de Vries 共KdV兲 or the modified KdV 共mKdV兲 equations关1,2兴, and intermediate long-wave models like the various versions of the Boussinesq or modified Boussinesq equations 关3兴, the Benjamin-Bona-Mahony-Peregrine equa-tion关4,5兴, the Green-Nagdhi system 关6–8兴, and many others. Roughly speaking, extreme models may be derived from in-termediate ones since the latter allow an asymptotic limit leading to the ubiquitous KdV or mKdV关9,10兴. With regard to long-wave dynamics, intermediate models are therefore more precise concerning dispersion and nonlinearities, and accordingly are more representative of Euler systems than extreme ones. However, this richer description of the waves has a counterpart for it may incorporate also short-wave propagation in the models. It is due to the methods leading from the Euler system to the intermediate models which are not able to filter out completely short waves. This is the case, for example, of some of the Boussinesq-type equations and the Benjamin-Bona-Mahony-Peregrine equation关10,11兴.

The presence of short scales in a long-scale model may constitute a major drawback, especially during its numerical study. Since in that case the final dynamics are a nonlinear superposition of short and long scales, the short waves if unstable can contaminate the whole model. For this reason, nonlinear propagation of short waves in long-wave models has been previously studied in Refs.关12–15兴. Linear theoret-ical estimates on the behaviors of short waves were carried out, based on the linear dispersion relations of the models and thus confirmed by numerical tests performed on the non-linear models. A nonnon-linear analytical and numerical analysis of this question for the Benjamin-Bona-Mahony-Peregrine equation was conducted in Ref. 关11兴. However, we should

not reduce such dynamics to a subsidiary phenomenon, since their study is a way to understand the ultraviolet regime in surface water wave.

The purpose of this paper is to investigate theoretically and numerically nonlinear short-wave behavior in a Green-Nagdhi model with surface tension. Although the model is derived in a limit where large scales dominate, the existence of short waves related to the short scale nature of the capil-lary phenomena cannot be ignored. The study is carried out in共2+1兲 dimensions. In the long-wave limit, the model leads to the Kadomtsev-Petviaskvili共KP兲 equation. Here, we have only considered short waves since, at present, we lack the tools needed to take into account both scales together. This is an important open problem and some progress was made in Refs.关16,17兴.

The paper is organized as follow. In Sec. II, we derive a Green-Nagdhi system in共2+1兲 dimensions, with surface ten-sion. The analysis of the associated linear dispersion relation shows that the model can propagate short waves. To tackle the problem of nonlinear short surface waves, a multiple scale perturbative method is carried out in Sec. III and leads to an asymptotic model equation. The analytical study of its

共1+1兲 traveling-wave solutions is then undertaken in Sec. IV

and, finally, we perform in Sec. V the Benjamin-Feir analysis of the Stokes wave. The last section is devoted to some final remarks.

II. THE„2+1… GREEN-NAGDHI MODEL EQUATION WITH SURFACE TENSION

Let us consider a fluid layer, initially at rest, in a uniform gravitational field and endow the space with a Cartesian frame 共O,x,y,z兲 so that 共Oz兲 is the upward vertical direc-tion. The fluid domain is contained between a rigid bottom at z = 0 and a upper free surface at z = S共x,y,t兲. We assume that the fluid is ideal, i.e., inviscid, incompressible, and that its density, ␴, is uniform and constant. Its surface tension is denoted by T and the velocity field by v =共u,v,w兲, where each component depends on x , y , z, and t. The motion of the fluid in the bulk is then given by the Euler equations. For 0⬍z⬍S共x,y,t兲, we have

u_x+vy+ wz= 0, 共1a兲 ut+ uux+vuy+ wuz= − px * /␴, 共1b兲 vt+ uvx+vvy+ wvz= − py * /␴, 共1c兲 wt+ uwx+vwy+ wwz= − pz * /␴− g, 共1d兲 where p* is the pressure field in the fluid and g stands for the gravitational acceleration. In the equations above and throughout this paper, the subscripts refer to the partial de-rivatives. These equations in the flow domain must be com-pleted by kinematic and dynamic boundary conditions at the bottom and at the upper free surface. We have

w = 0 at z = 0, 共2a兲 S_t+ uS_x+vSy− w = 0 at z = S共x,y,t兲, 共2b兲 p * = p0− T Sxx共1 + Sy 2_{兲 + S} yy共1 + Sx 2_{兲 − 2S} xySxSy 共1 + Sx 2_{+ S} y 2_兲3/2 at z = S共x,y,t兲. 共2c兲

Equation 共2c兲 is the Laplace-Young boundary condition which rules the pressure difference between the two sides of the interface. We assume in Eq.共2c兲 that the pressure in the upper fluid共typically air兲 remains uniform and equal to p0.

Shallow water equations are usually derived by perform-ing an asymptotic analysis directly on the Euler equations共1兲 and boundary conditions共2兲. The velocity and the pressure fields are then handled perturbatively through the use of asymptotic expansions. Our approach is somewhat different since, instead of studying the entire problem via a perturba-tion theory, we are going to consider first the nonlinear evo-lution of a given initial ansatz for the velocity field. We as-sume indeed that u andv are independent of z, that is,

u = u共x,y,t兲, 共3兲

v = v共x,y,t兲. 共4兲

This ab initio given velocity profile, known as the columnar-flow ansatz, can be justified from linear theoretical arguments or, even better, from direct visualization of the particle tra-jectories of a plane periodic wave in water of fairly depth

关18兴. It was introduced long ago by Green and Nagdhi 关6–8兴,

not in this form but in the rather different framework of the Cosserat surface theory共see references quoted in 关7,8兴兲. Be-sides, these profiles are also obtained through the classical shallow water analysis.

The columnar-flow ansatz enables us to derive equations involving only three fields, namely u ,v, and S, and to

elimi-nate the z dependence. From Eqs.共1a兲 and 共2a兲, we have

w共x,y,z,t兲 = zq共x,y,t兲, 共5兲

where

q共x,y,t兲 = − ux共x,y,t兲 − vy共x,y,t兲. 共6兲

Equations共1b兲 and 共1c兲 can then be integrated from z=0 to z = S and, using Leibnitz’s rule, we obtain

␴u˙S = − p_x− TaSx

b3/2, 共7兲

␴v˙S = − py− T

aSy

b3/2, 共8兲

where the dot stands for the material derivative, i.e., u˙ = ut+ uux+vuy, 共9兲

and the functions p , a, and b are defined by p共x,y,t兲 =

冕

0 S p * dz − p0S, 共10兲 a共x,y,t兲 = Sxx共1 + Sy 2_{兲 + S} yy共1 + Sx 2_{兲 − 2S} xySxSy, 共11兲 b共x,y,t兲 = 1 + S_x2+ S_y2. 共12兲 Next, we multiply Eq.共1d兲 by z and integrate it from z=0 to z = S, which yields ␴S3 3共q˙ + q 2_{兲 +␴g}S 2 2 = p + T aS b3/2. 共13兲 The pressure p may then be eliminated from Eqs.共7兲, 共8兲, and共13兲, leading to u˙ = − SSx共q˙ + q2兲 − S2 3共q˙ + q 2_兲 x− gSx+ T ␴

冉

a b3/2

冊x

, 共14a兲 v˙ = − SSy共q˙ + q2兲 − S2 3共q˙ + q 2_兲 y− gSy+ T ␴

冉

a b3/2

冊y

, 共14b兲 S˙ = Sq, 共14c兲

and, substituting q by its expression关Eq. 共6兲兴, we eventually obtain the equations involving the initial fields,

S共u_t+ uu_x+vuy兲 = 1 3关S 3_共u xt+ uuxx− ux 2₊ vyt+vvyy−vy 2 +vuxy+ uvxy− 2uxvy兲兴x− gSSx +TS ␴

冉

a b3/2

冊x

, 共15a兲 S共vt+ uvx+vvy兲 = 1 3关S 3_共u xt+ uuxx− ux 2 +vyt+vvyy−vy 2 +vuxy+ uvxy− 2uxvy兲兴y− gSSy +TS ␴

冉

a b3/2

冊y

, 共15b兲 St+共uS兲x+共vS兲y= 0. 共15c兲

These equations constitute a Green-Nagdhi system, with sur-face tension, in共2+1兲 dimensions.

The linear dispersion relation␻共k,l兲 of the system 共15兲 is derived by expanding u ,v, and S into the following form

共⑀⬍1兲:

u =⑀U + O共⑀2兲, 共16a兲

v =⑀L + O共⑀2兲, 共16b兲 S = h +⑀W + O共⑀2_{兲, 共16c兲} where h is the unperturbed depth. At the order⑀, we obtain

U_t=1 3h 2_共U xtx+ Lytx兲 − gWx+ T ␴共Wxxx+ Wyyx兲, 共17a兲 Lt= 1 3h 2_共U xty+ Lyty兲 − gWy+ T ␴共Wxxy+ Wyyy兲, 共17b兲 Wt= − h共Ux+ Ly兲. 共17c兲

To look for the possible solutions of the preceding linear system, we set U =␣ei␾, L =␤ei␾, and W =␥ei␾, where

␾= kx + ly −␻共k,l兲t, 共18兲 k and l are the wave numbers in the x and y directions, respectively, and␣,␤, and ␥ are arbitrary constants. From Eq. 共17兲, ␣,␤, and ␥ obey a linear homogeneous system which has nontrivial solutions if

␻共k,l兲2_=共l2_{+ k}2_兲 gh +Th ␴ 共l 2_{+ k}2_兲 1 +h 2 3共l 2_{+ k}2_兲 . 共19兲

The Euler equations共1兲 and 共2兲 have the dispersion relation

⍀2_=共k2_{+ l}2_兲1/2

冉

_{g +}T共k 2_{+ l}2_兲

␴

冊

tanh关共k2+ l2兲1/2h兴.

共20兲

We can compare the phase velocity C and the group velocity Cg associated with Eq. 共20兲 with the analogous c and cg

associated with Eq.共19兲. This is done in Fig. 1 and Fig. 2 for

⍀共k兲 and␻共k兲共l=0兲. The pictures show that the linear model 共15a兲, 共15b兲, and 共15c兲 approximates the linear Euler

equa-tions very well for 1⬍kh⬍2. For kh⬎2, the divergence increases.

III. ASYMPTOTIC MODEL FOR SHORT CAPILLARY-GRAVITY WAVES

The system 共15兲 can be seen as a first reduction of the Euler equations via the columnar-flow hypothesis. It incor-porates finite dispersion not only in the long-wave limit but also in the short-wave one since the surface tension has been included in the model. Indeed, the dispersion relation is well behaved whether

冑

k2_{+ l}2_{→0 共long waves兲 or}

冑

_k2_{+ l}2_→⬁

共short waves兲. The long-wave case leads to the KP equation.

To examine more precisely the short-scale behavior, we are

going to look for waves short in x and normal in y, that is, to consider the y dimension as being a lateral weak perturbation with respect to the x one.

Two new variables are required to appropriately take into account short waves asymptotically in time: a spatial vari-able, ␨, describing a local pattern and a temporal one, ␶, matching large times. To be compatible with the plane pro-gressive wave solution of angular frequency共19兲, we intro-duce a small parameter⑀and take

k =k0

⑀ and l = l0, 共21兲

where⑀⬍1 and k0and l0are of order 1. The expansion of␻ in terms of⑀reads ␻= k0

冑

3T ␴h

冋

1 ⑀+⑀

冉

g␴ 2Tk₀2+ l₀2 2k₀2− 3 2h2k₀2

冊

+ O共⑀ 3_兲

册

_. 共22兲 It leads to

FIG. 1. Representation of the phase velocity C共k兲 of Eq. 共20兲

共broken line兲 and phase velocity c共k兲 of Eq. 共19兲 共continuous line兲

as a function of kh.

FIG. 2. Representation of the group velocity C_g共k兲 of Eq. 共20兲

共broken line兲 and group velocity cg共k兲 of Eq. 共19兲 共continuous line兲

␨=1

⑀共x − Vt兲, y = y, ␶=⑀t, 共23兲

where V =

冑

3T /␴h; the associated operators are given by ⳵ ⳵x= 1 ⑀ ⳵ ⳵␨, ⳵ ⳵y= ⳵ ⳵y, ⳵ ⳵t= − V ⑀ ⳵ ⳵␨+⑀ ⳵ ⳵␶. 共24兲 Now, using expressions共24兲 in Eqs. 共15兲, together with the expansions

u =⑀2共U0+⑀2U2+ ¯ 兲, 共25a兲

v =⑀共L0+⑀2L2+ ¯ 兲, 共25b兲

S = h +⑀2共W₀+⑀2W₂+ ¯ 兲, 共25c兲 we can isolate nonlinear dynamics of short capillary-gravity waves from the system 共15兲 by performing a perturbative calculation according to⑀.

Equations共15a兲 in order 1/⑀, Eq.共15b兲 in order⑀0, and Eq.共15c兲 in order⑀yield the following system:

V共U0,_␨+ L0,y兲 = 3T ␴h2W0,␨, 共26a兲 VL0= 1 3Vh 2_共U 0,␨y+ L0,yy兲 − T ␴W0,␨y, 共26b兲 VW_0,␨= h共U_0,␨+ L_0,y兲, 共26c兲 whose solution is U₀=V hW0, L0= 0 with V 2₌ 3T ␴h. 共27兲 Next, the orders⑀of Eq.共15a兲 and⑀3_{of Eq.共15c兲 lead to an} evolution equation for U0in␨, y, and␶coordinates. Rewrit-ten with the initial variables, it reads

uxt= 3g 2Vh共1 − 3␪兲u − 1 2uxxu − 1 4ux 2 +3h 2 4Vuxxux 2 −V 2uyy, 共28兲

where u共x,y,t兲 is the fluid velocity at the surface and␪is a dimensionless parameter, the Bond number, given by

␪= T

␴h2g. 共29兲

Equation 共28兲 governs the nonlinear propagation of short waves in the long-wave model共15兲.

IV. ANALYSIS OF THE„1+1… TRAVELING-WAVE SOLUTIONS

As a first investigation in the study of Eq.共28兲, one may begin by looking for its possible traveling-wave solutions. Owing to the difference between the space scales in each direction, we will only consider plane waves propagating in the short-scale one, i.e., we eliminate the y dependence from Eq.共28兲 by removing the u_yyterm,

u_xt= 3g 2Vh共1 − 3␪兲u − 1 2uxxu − 1 4ux 2₊ 3h 2 4Vuxxux 2_{. 共30兲}

We then introduce the following dimensionless variables: u

⬘

= ␭ 3 u V, x

⬘

= ␭ 3 x h, t

⬘

= V 2ht, 共31兲

where␭ is a nonzero dimensionless parameter defined by

␭ =3共1 − 3␪兲

␪ . 共32兲

They lead, if ␪⫽1/3, to the more convenient form of Eq.

共30兲 共dropping the primes兲,

uxt= u − uxxu − 1 2ux 2 +␭ 2uxxux 2 . 共33兲

If␪= 1 / 3, Eq.共30兲 is dispersionless and will not be studied here 共see 关19,20兴 for a detailed analysis of the long-wave dynamics for this special value of␪兲. A traveling wave may be described by a function

u共r兲 = u共x − ct兲, 共34兲

where c denotes the velocity of the wave. In the following, we only consider waves moving to the right, i.e., we assume c⬎0. According to Eq. 共33兲, the wave profile u共r兲 must then obey the ordinary differential equation

冉

␭ 2ur 2 − u + c

冊

urr= − u + 1 2ur 2 . 共35兲

For ␭⫽0, the study of Eq. 共35兲 may be carried out by using the following change of variables:

X =␭ 2ur

2_{− u + c, 共36a兲}

Y = u. 共36b兲

If␭⫽1, it leads to the first-order differential system XXr=共1 − ␭兲

冉

Y +

c

␭ − 1

冊

Yr, 共37a兲

Y_r2=2

␭共X + Y − c兲, 共37b兲

which is equivalent to Eq.共35兲 provided one excludes from Eqs.共37兲 the solutions Yr= 0 with Y⫽0. The phase portrait

of the system共37兲 is then easily obtained since Eq. 共37a兲 can be integrated, giving

X2+共␭ − 1兲

冉

Y + c

␭ − 1

冊

2

=␭␬, 共37a⬘兲

where␬is determined by the initial conditions. For␭=1, Eq.

共37b兲 is still valid, but Eq. 共37a⬘兲 now becomes X2_{+ 2cY =␬.} In the following, we restrict the study of Eq.共35兲 to the solutions u for which there exists r0such that ur共r0兲=0 共the solution u = 0 is left aside兲. We also define u0= u共r0兲 and then have from Eq. 共37a⬘兲 ␬= u₀2+ c2/共␭−1兲 if ␭⫽1 and ␬= u₀2

+ c2 _{if ␭=1. We found two main cases when ␭⬎0. If 兩u} 0兩

⬍c, the solutions are periodic and defined for all r 共Figs. 3

and 4兲. Otherwise, they show a singularity for a finite value of r. In the more specific case 0⬍␭⬍1, the situation is actually a little more involved since the above singularity disappears when兩u0兩⬎c/

冑

1 −␭⬎c: the solution is then de-fined for all r but is not periodic共i.e., not bounded兲. When

␭⬍0, the solutions found are periodic again, similar to the

previous ones, if兩u0兩⬍c/

冑

1 −␭ 共Fig. 5兲. They show a singu-larity if u0⬍−c/

冑

1 −␭ or c/

冑

1 −␭⬍u0⬍c and are not bounded if u0⬎c. The value of the function u at the singu-larity point共see Figs. 3, 4, and 5 for further details兲 is given by u_S±= c 1 −␭± 1 兩␭ − 1兩

冑

␭关共␭ − 1兲u0 2_{+ c}2_{兴 共38兲} if␭⫽1 and uS=共u0 2_{+ c}2_{兲/共2c兲 if ␭=1.}

The preceding analysis shows that Eq.共33兲 has traveling-wave solutions for all values of␭共␭⫽0兲. These waves share the same feature: they only exist if their amplitude 兩u₀兩 is small enough. The threshold value uCdepends on the sign of ␭:uC共␭兲=c if ␭⬎0 and uC共␭兲=c/

冑

1 −␭ if ␭⬍0. Below uC,

the wave is periodic and smooth, whereas as the amplitude reaches the threshold value, a singularity appears in the top

part of the wave. Beyond the threshold, a solution smooth, bounded, and defined for all r no longer exists. However, from the existing solutions of Eq. 共35兲, one may build a periodic piecewise function which may be seen as the evo-lution of the periodic wave after it ceases to exist with a smooth shape, and whose only singularities are located at the crests, periodically distributed on the r axis 共Fig. 6兲. Actu-ally, from a physical point of view such construction is rel-evant only when u₀⬍0. According to Eq. 共38兲, the height of the crest is given by, if␭⫽1,

FIG. 3. Schematic representation of the traveling-wave solutions for␭艌1. u_S± is defined for all values of u₀and is such that 兩u_S±兩

艌兩u0兩.

FIG. 4. Schematic representation of the traveling-wave solutions for 0⬍␭⬍1. u_S±is defined only when兩u0兩艋c/

冑

1 −␭; in that case,

u_S共u₀,␭兲 = c 1 −␭+

1

␭ − 1

冑

␭关共␭ − 1兲u0

2_{+ c}2_{兴, 共39兲}

provided it is defined, and uS共u0, 1兲=共u0

2_{+ c}2_{兲/共2c兲. It may be} worth noting that if ␭⬍0, the lower part of the resulting wave is more extensive than the upper one 关i.e., u_S共u₀,␭兲

⬍−u0兴, whereas it is the opposite if ␭⬎0 关i.e., uS共u0,␭兲⬎ −u₀if␭⬎0兴. Provided it corresponds to the actual evolution of the smooth periodic wave, the piecewise function provides some insight into how the wave becomes singular. We can

distinguish between two scenarios according to whether␭ is positive or not. In the former case, uS共u0,␭兲=−u0 when the threshold is reached 关−u₀= u_C共␭兲兴: the crests become more and more sharp as the amplitude of the wave increases until the top of the wave gives rise to a peak共Fig. 7兲. In the latter case, a breaking occurs in the middle of the wave since we have u_S共u₀,␭兲=−u₀/

冑

1 −␭, and accordingly u_S共u₀,␭兲⬍−u₀, when −u0= uC共␭兲 共Fig. 8兲.

To complete this study, it is interesting to compute the angle of the wave shape at the crest for the critical case u0 = −uC共␭兲. Provided it is defined, the slope at the break point

reads 兩u1S

⬘

兩 =

冑

2 1 −␭

冉

c − 1 ␭

冑

␭关共␭ − 1兲u0 2_{+ c}2_兴

冊

_共40兲 if␭⫽1 and 兩u_1S

⬘

兩=

冑

共u₀2− c2_{兲/2 if ␭=1. When ␭ is negative,} the substitution of u0 by −uC yields 兩u1S

⬘

兩=

冑

2c /共1−␭兲. In contrast, when␭ is positive, we obtain 0, which is not sur-prising since the singularity in that case occurs at the wave crest. The slope of the wave shape at this point is then of little interest and we found it more convenient to evaluate the slope at the inflection point instead. We find共this expression is not valid when␭⬍0 and −u0⬎uC兲

兩u2S

⬘

兩 =

冑

2

1 −␭关c −

冑

共␭ − 1兲u0 2

+ c2兴 共41兲 if␭⫽1 and 兩u_2S

⬘

兩=−u0/

冑

c if␭=1. It leads to, for u0= −uC,

兩u2S

⬘

兩 =

冑

2c

1 +

冑

␭. 共42兲

These expressions show that the limit angle at the crest van-ishes as␭→⬁ 共i.e., T→0兲 while, when ␭=−9 共i.e., T→⬁兲, it remains finite.

V. MODULATIONAL INSTABILITY

In this section, we study the resonant interaction occur-ring in a wave train共Stokes’ wave train兲 with a narrow band of frequencies and wavelengths. Let us consider u共x,t兲 as a

FIG. 5. Schematic representation of the traveling-wave solutions for␭⬍0. u_S± is defined only when兩u0兩艌c/

冑

1 −␭; in that case, u_S±

艋兩u0兩.

FIG. 6. Piecewise continuous function built with solutions of Eq.共35兲 for ␭⬎0. It ranges from u0to u_S.

plane wave. The nonlinear terms in Eq. 共30兲 give rise to harmonics of the fundamental. Assume that a disturbance is present consisting of modes with sideband frequences and wave numbers close to the fundamental. We can have inter-action between harmonics and these sideband modes. This interaction is likely to produce a resonant phenomenon mani-festing itself by the modulation of the plane-wave solution. The exponencial growth in time of the modulation, originat-ing from synchronous resonance between harmonics and sideband modes, leads to the Benjamin-Feir instability关21兴. A formal solution can be given via an asymptotic expansion leading to the nonlinear Schrödinger equation 共NLS兲 关22兴. The particular interest of NLS in the existence of a general and simple criterion enables us to detect the stability or in-stability of the monochromatic wave train. Let us seek a solution of Eq.共30兲 under the form of a Fourier expansion in harmonics of the fundamental exp i共kx−␻st兲 and where the

Fourier components are developed in a Taylor series in

pow-ers of a small parameter ␦ measuring the amplitude of the fundamental, u =

兺

p=1 ⬁

兺

l=−p l=p ␦p_u l p_{共␰,␶兲exp关il共kx −␻} st兲兴. 共43兲

Eq.共43兲, u_−lp = u_l*p共the asterisk denotes complex conjugation兲 and ␰ and ␶ are slow variables introduced through the stretching␰=␦共x−Ct兲 and␶=␦2t and where C will be deter-mined as a solvability condition. The expansions 共43兲 in-clude fast local oscillations through the dependence on the harmonics and slow variation 共modulation兲 in amplitude taken into account by the␰,␶dependence of u_lp. Introducing now this expansion and the slow variables in Eq. 共30兲, we may proceed to collect and solve different order␦ and l. We have with

FIG. 7. Numerical integration of Eq.共35兲 for

␭=2 and c=1. The crest of the wave becomes

sharper as the wave amplitude −u₀ approaches the critical value uC共␭兲=c. The values of −u0for

the three curves are 0.8, 0.9, and 0.99.

FIG. 8. Numerical integration of Eq.共35兲 for

␭=−3 and c=1. As the wave amplitude −u0

reaches the critical value u_C共␭兲=c/

冑

1 −␭, the wave breaks at uS(−uC共␭兲,␭)=c/共1−␭兲. The

val-ues of −u0for the three curves are 0.3, 0.4, and

u₁1=␺共␰,␶兲, A =3g共1 − 3␪兲 2Vh , B =

3h2

4V, 共44兲 the following conditions of solvability:

u₀1= 0, ␻_s= kV +A k, C = 2V − ␻s k , 共45兲 u₀2= − k 2 2A兩␺兩 2_{, u} 2 2 = k 2 4A␺ 2_{, u} 1 2 = i k␺␰. 共46兲 At order␦= 3 , l = 1 we obtain as a solvability condition the NLS for␺共␰,␶兲, − i␺_␶− A k3␺␰␰+

冉

1 8A− B

冊

k 3_␺兩␺兩2_{= 0. 共47兲} The nature of solutions of NLS depends drastically on the sign of the product between the coefficient of␺_␰␰and that of ␺兩␺兩2_{. In this case, this product is positive for}

␪⬍ 3

10, 共48兲

and according to a well known stability criterion 共see, for example, 关1兴兲, Stokes’ wave train is unstable, that is, any slight deformation of the plane wave experiences an expo-nential growth. In the case of water at room temperature共T = 0.074 N m−1_{,␴= 10}3_{kg m}−3_{兲, we obtain that a short-wave} train is unstable for a depth h⬎0.49 cm.

Last but not least, the value ␪= 3 / 10 corresponds to ␭ = 1 in Eq.共33兲. Precisely,

␪⬍ 0.3⇒␭ ⬎ 1. 共49兲

VI. FINAL REMARKS

The purpose of this paper was to investigate the behavior of short waves in a long-wave model. The presence of these waves may have different origins. First, in modeling real physical systems, we are often led to use these models be-yond the precise range of validity under which they were

derived. In that situation, we will very probably be in a short-wave region. Second, whenever we realize numerical dis-cretizations of them, short waves are introduced as secondary effects coming from truncations and finite-difference meth-ods, when the wavelength is of the order of the grid spacing. Third, initial wave-packet solutions, in terms of Fourier in-tegrals, contain in general short-wave components. Instabili-ties of these waves can cause instability in the entire system. In this paper, we have investigated the short-wave dynamics in a Green-Naghdi model with surface tension. We have found that the traveling plane-wave solutions exist for all the values of the surface tension parameter共except␪= 1 / 3兲 and have determined an amplitude threshold beyond which these waves become singular. The limit angle value of the resulting wave is supplied. We have also studied the Benjamin-Feir instability and have specified the regions in which a wave train is modulationally stable or unstable.

Some points remain to be developed. Equation共30兲 was already studied in Ref.关23兴 in relation to its integrability and the solutions going to zero for x→⬁. It has been shown that in that case the system is completely integrable and con-nected with the sine-Gordon or the sinh-Gordon equations depending on the value of␭. It would be interesting to know if this property is still valid for Eq.共28兲.

Finally, let us note that the study of wave dynamics at short scales in real fluids constitutes an open problem signi-fiant for theoretical or practical reasons. Dissipative phenom-ena take place at small scales. They ultimately appear from the turbulent motion of the fluid. The turbulent fluctuations of the fluid tend to drain energy and momentum from large scales of motion to short scales of motion where viscosity can act directly. Equation共28兲 was derived under the hypoth-esis that the system is dissipation-free. However, viscosity cannot always be neglected in real fluids, and this will cer-tainly affect the asymptotic dynamics of short waves in long-wave models and their stability. The solution of this problem must enlighten our knowledge of turbulence phenomena.

ACKNOWLEDGMENTS

C.H.B. and M.A.M. wish to thank the IFT-UNESP for their hospitality and FAPESP-Brasil for financial support.

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