Nonlinear dynamics of short traveling capillary-gravity waves
C. H. Borzi,1 R. A. Kraenkel,2 M. A. Manna,3 and A. Pereira3
1Facultad de Ciencias Exactas y Naturales de la Universidad Nacional de Buenos Aires, Pab. II, Nuñez, Buenos Aires, Argentina
2Instituto de Física Teórica-UNESP Rua Pamplona 145, 01405-900 São Paulo, Brazil
3Physique Mathématique et Théorique, CNRS-UMR5825, Université Montpellier II, 34095 Montpellier, France
sReceived 16 July 2004; revised manuscript received 29 November 2004; published 23 February 2005d
We establish a Green-Nagdhi model equation for capillary-gravity waves in s2+1d dimensions. Through the
derivation of an asymptotic equation governing short-wave dynamics, we show that this system possesses
s1+1d traveling-wave solutions for almost all the values of the Bond number u sthe special case u=1/3 is not
studiedd. 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 numberssd: 47.10.1g, 47.20.Ky, 47.35.1i, 04.25.2g
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 sKdVd or the modified KdV smKdVd
equations f1,2g, and intermediate long-wave models like the
various versions of the Boussinesq or modified Boussinesq
equations f3g, the Benjamin-Bona-Mahony-Peregrine equa-
tion f4,5g, the Green-Nagdhi system f6–8g, 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 f9,10g. 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 f10,11g.
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. f12–15g. 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 nonlinear analytical and numerical analysis
of this question for the Benjamin-Bona-Mahony-Peregrine
equation was conducted in Ref. f11g. 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 s2+1d dimensions. In the long-wave limit, the model leads
to the Kadomtsev-Petviaskvili sKPd 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. f16,17g.
The paper is organized as follow. In Sec. II, we derive a
Green-Nagdhi system in s2+1d 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
s1+1d 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 sO ,x ,y ,zd so that sOzd 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=Ssx ,y , td. We assume that
the fluid is ideal, i.e., inviscid, incompressible, and that its
density, s, is uniform and constant. Its surface tension is
denoted by T and the velocity field by v= su ,v ,wd, 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,Ssx ,y , td, we have
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ux + vy + wz = 0, s1ad
ut + uux + vuy + wuz = − px
*/s , s1bd
vt + uvx + vvy + wvz = − py
*/s , s1cd
wt + uwx + vwy + wwz = − pz
*/s − g , s1dd
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, s2ad
St + uSx + vSy − w = 0 at z = Ssx,y,td , s2bd
p * = p0 − T
Sxxs1 + Sy
2
d + Syys1 + Sx
2
d − 2SxySxSy
s1 + Sx
2 + Sy
2
d
3/2
at z = Ssx,y,td . s2cd
Equation s2cd is the Laplace-Young boundary condition
which rules the pressure difference between the two sides of
the interface. We assume in Eq. s2cd that the pressure in the
upper fluid stypically aird remains uniform and equal to p0.
Shallow water equations are usually derived by perform-
ing an asymptotic analysis directly on the Euler equations s1d
and boundary conditions s2d. 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 and v are independent of z, that is,
u = usx,y,td , s3d
v = vsx,y,td . s4d
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
f18g. It was introduced long ago by Green and Nagdhi f6–8g,
not in this form but in the rather different framework of the
Cosserat surface theory ssee references quoted in f7,8gd. 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. s1ad and s2ad, we have
wsx,y,z,td = zqsx,y,td , s5d
where
qsx,y,td = − uxsx,y,td − vysx,y,td . s6d
Equations s1bd and s1cd can then be integrated from z=0 to
z=S and, using Leibnitz’s rule, we obtain
su˙S = − px − T
aSx
b3/2
, s7d
sv˙S = − py − T
aSy
b3/2
, s8d
where the dot stands for the material derivative, i.e.,
u˙ = ut + uux + vuy , s9d
and the functions p , a, and b are defined by
psx,y,td =
E
0
S
p * dz − p0S , s10d
asx,y,td = Sxxs1 + Sy
2
d + Syys1 + Sx
2
d − 2SxySxSy , s11d
bsx,y,td = 1 + Sx
2 + Sy
2
. s12d
Next, we multiply Eq. s1dd by z and integrate it from z=0 to
z=S, which yields
s
S3
3
sq˙ + q2d + sg
S2
2
= p + T
aS
b3/2
. s13d
The pressure p may then be eliminated from Eqs. s7d, s8d,
and s13d, leading to
u˙ = − SSxsq˙ + q2d −
S2
3
sq˙ + q2dx − gSx +
T
s
S
a
b3/2Dx
,
s14ad
v˙ = − SSysq˙ + q2d −
S2
3
sq˙ + q2dy − gSy +
T
s
S
a
b3/2Dy
,
s14bd
S˙ = Sq , s14cd
and, substituting q by its expression fEq. s6dg, we eventually
obtain the equations involving the initial fields,
Ssut + uux + vuyd =
1
3
fS3suxt + uuxx − ux
2 + vyt + vvyy − vy
2
+ vuxy + uvxy − 2uxvydgx − gSSx
+
TS
s
S
a
b3/2Dx
, s15ad
Ssvt + uvx + vvyd =
1
3
fS3suxt + uuxx − ux
2 + vyt + vvyy − vy
2
+ vuxy + uvxy − 2uxvydgy − gSSy
+
TS
s
S
a
b3/2Dy
, s15bd
St + suSdx + svSdy = 0. s15cd
These equations constitute a Green-Nagdhi system, with sur-
face tension, in s2+1d dimensions.
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