J. Fluid Mech. (2010), vol. 664, pp. 138–149.
c
Cambridge University Press 2010
doi:10.1017/S0022112010004349
The modulational instability in deep water under
the action of wind and dissipation
C. KHARIF
1
†, R. A. KRAENKEL
2
, M. A. MANNA
3
AND R. THOMAS
1
1
Institut de Recherche sur les Ph
´
enom
`
enes Hors
´
Equilibre, 49, rue F. Joliot-Curie, BP 146,
13384 Marseille CEDEX 13, France
2
Instituto de Fisica Teorica, UNESP, R. Pamplona 145, 01405-900 S
˜
ao Paulo, Brazil
3
Laboratoire de Physique Th
´
eorique et Astroparticules, CNRS-UMR 5207, Universit
´
e Montpellier II,
Place Eug
`
ene Bataillon, 34095 Montpellier CEDEX 05, France
(Received 5 February 2010; revised 11 August 2010; accepted 11 August 2010;
first published online 1 November 2010)
The modulational instability of gravity wave trains on the surface of water acted
upon by wind and under influence of viscosity is considered. The wind regime is that
of validity of Miles’ theory and the viscosity is small. By using a perturbed nonlinear
Schr
¨
odinger equation describing the evolution of a narrow-banded wavepacket under
the action of wind and dissipation, the modulational instability of the wave group is
shown to depend on both the frequency (or wavenumber) of the carrier wave and the
strength of the friction velocity (or the wind speed). For fixed values of the water-
surface roughness, the marginal curves separating stable states from unstable states
are given. It is found in the low-frequency regime that stronger wind velocities are
needed to sustain the modulational instability than for high-frequency water waves.
In other words, the critical frequency decreases as the carrier wave age increases.
Furthermore, it is shown for a given carrier frequency that a larger friction velocity
is needed to sustain modulational instability when the roughness length is increased.
Key words: air/sea interactions, surface gravity waves, wind–wave interactions
1. Introduction
Since Stokes (1847), it is well known that the potential water wave problem
admits as solutions uniform wave trains of two-dimensional progressive waves. The
stability of the Stokes’ wave solution started with the work by Lighthill (1965) who
provided a geometric condition for wave instability. Later on, Benjamin & Feir (1967)
showed analytically that Stokes waves of moderate amplitude are unstable to small
long-wave perturbations travelling in the same direction. This instability is called
the Benjamin–Feir instability (or modulational instability). Whitham (1974) derived
the same result independently by using an averaged Lagrangian approach, which
is explained in his book. At the same time, Zakharov (1968), using a Hamiltonian
formulation of the water wave problem obtained the same instability result and
derived the nonlinear Schr
¨
odinger equation (NLS equation). The evolution of a
two-dimensional nonlinear wave train on deep water, in the absence of dissipative
effects, exhibits the Fermi–Pasta–Ulam recurrence phenomenon. This phenomenon
is characterized by a series of modulation–demodulation cycles in which initially
† Email address for correspondence: kharif@irphe.univ-mrs.fr
Modulational instability in deep water under the action of wind and dissipation 139
nearly uniform wave trains become modulated and then demodulated until they are
again nearly uniform. Modulation is caused by the growth of the two dominant
sidebands of the Benjamin–Feir instability at the expense of the carrier. During
the demodulation, the energy returns to the components of the original wave train.
Recently, within the framework of the NLS equation, Segur et al. (2005a) revisited
the Benjamin–Feir instability when dissipation is taken into account. The latter
authors showed that for waves with narrow bandwidth and moderate amplitude,
any amount of dissipation stabilizes the modulational instability. In the wavenumber
space, the region of instability shrinks as time increases. This means that any initially
unstable mode of perturbation does not grow for ever. Damping can stop the growth
of the sidebands before nonlinear interactions become important. Hence, when the
perturbations are small initially, they cannot grow large enough for nonlinear resonant
interaction between the carrier and the sidebands to become important. The amplitude
of the sidebands can grow for a while and then oscillate in time. Segur et al. (2005a)
have confirmed their theoretical predictions by laboratory experiments for waves of
small to moderate amplitude. Later, Wu, Liu & Yue (2006) developed fully nonlinear
numerical simulations which agreed with the theory and experiments of Segur et al.
(2005a).
Within the framework of random waves, there exists a stochastic counterpart of
the modulational instability discussed above. The stability of nearly Gaussian and
narrow-banded water wave trains was investigated by Alber & Saffman (1978), Alber
(1978) and Crawford, Saffman & Yuen (1980). They found that the modulational
instability on deep water occurs, provided that the relative spectral width is less than
twice the average steepness. Hence, the effect of increasing randomness is to restrict
the instability criterion, to delay the onset of modulational instability, and to reduce
the amplification rate of the modulation.
From the previous studies we could conclude that dissipation and randomness
may prevent the development of the Benjamin–Feir instability (or modulational
instability). These two effects question the occurrence of modulational instability of
water wave trains. Segur, Henderson & Hammack (2005b) speculated about the
effect of dissipation on the early development of rogue waves and asked the question:
can the Benjamin–Feir instability spawn a rogue wave? Since damping affects the
modulational instability of waves in deep water, they assumed that it might affect the
early development of rogue waves. Nevertheless, the latter study did not include wind
effect. What is the role of wind upon modulational instability when dissipative effects
are considered? Waseda & Tulin (1999) showed experimentally that the wind does not
suppress the Benjamin–Feir instability even if the naturally (unseeded experiments)
developed initial sideband energy is reduced. This finding contrasts with that of
Bliven, Huang & Long (1986) who conducted unseeded experiments and found that
sideband growth was reduced in the presence of wind.
The present paper is aimed at reporting on the behaviour of Benjamin–Feir
instability when dissipation and wind input are both taken into account. Following
Miles (1957), but within the framework of modulated wave trains, we assume the
atmospheric pressure at the interface due to wind and the water wave slope to be
in phase. The effect is to produce an exponential growth of the wave amplitude. We
remind the reader that Miles (1957) studied a linear and uniform monochromatic
wave train, whereas we are considering a weakly nonlinear modulated wave train.
In the presence of dissipation it is found that carrier waves of given frequency (or
wavenumber) may suffer modulational instability when the friction velocity is larger
than a threshold value. Conversely, for a given friction velocity it is found that only
140 C. Kharif, R. A. Kraenkel, M. A. Manna and R. Thomas
carrier waves whose frequency (or wavenumber) is less than a threshold value are
unstable. Otherwise dissipation prevents instability developing over time.
In §2, the governing equations and the wave amplification theory of Miles (1957)
are briefly presented. In §3, the NLS equation is introduced when dissipation and
wind forcing are considered and the competition between wave amplification by the
wind and stabilization due to dissipation is considered. The linear stability analysis
of the Stokes-like wave is developed in §4. The final discussion is found in §5.
2. Surface waves under the action of wind and dissipation
We will consider waves on the surface of a fluid whose viscosity is small as given
by Lamb (1993, article no. 349). If g is the acceleration due to gravity, k is the
wavenumber of the surface perturbation and ν is the viscosity, we define a non-
dimensional number L(k)=ν/
g/k
3
, and we say that a fluid is of small viscosity if
L(k) 1. (2.1)
For the free surface problem in water, viscous effects are generally weak producing
a thin rotational layer adjacent to the potential flow. The thickness of this rotational
boundary layer is O(
√
ν/kc), where c is the phase velocity. In this context, it was
shown by Dias, Dyachenko & Zakharov (2008) that the equations governing the
fluid’s motion can be formulated with the help of potential theory. The correction
due to viscosity they derived within the framework of the linearized equations
was heuristically generalized to the nonlinear equations. Using a similar approach,
Lundgren (1989) derived linear versions of the modified boundary equations (2.4)
and (2.5). Note that another variant of the introduction of viscous effects within the
framework of potential theory can be found in the paper by Skandrani, Kharif &
Poitevin (1996).
The fluid layer is limited above by the water surface described by z = η(x,t). We
will consider the case of infinite depth. Under these hypotheses, the Laplace equation,
the bottom boundary condition and the kinematic condition are
φ
xx
+ φ
zz
=0 for −∞6 z 6 η(x,t), (2.2)
∇φ → 0forz →−∞, (2.3)
η
t
+ φ
x
η
x
−φ
z
−2νη
xx
=0 for z = η(x, t). (2.4)
The dynamic boundary condition is modified by wind effect too, and has the form
φ
t
+
1
2
[(φ
x
)
2
+(φ
z
)
2
]+gη = −
1
ρ
P
a
−2νφ
zz
for z = η(x, t), (2.5)
where ρ is the fluid’s density and P
a
is the excess pressure at the free surface. As
proposed by Dias et al. (2008), (2.4) and (2.5) are simply heuristic nonlinear
generalizations of their linear versions when dissipation is introduced.
In this paper we consider water wave trains in the open ocean far from any solid
boundaries. Note that in the case of surface waves generated in wave tanks damping
due to viscous dissipation at the lateral solid boundaries must be introduced (see
Miles 1967).
The dependence of the fluctuating pressure, P
a
,onη is what defines the wind–wave
interactions. Within the framework of linearized equations, Miles (1957) assumed that
the surface elevation and aerodynamic pressure are η(x,t)=ae
ik(x−ct)
and P
a
=(α +
iβ)ρ
a
U
2
1
kη, respectively, where a denotes the amplitude, k is the wavenumber, c is
Modulational instability in deep water under the action of wind and dissipation 141
the phase velocity, α and β are two coefficients depending on both k and c, ρ
a
is
the density of air and U
1
is a characteristic velocity related to the friction velocity,
u
∗
, of wind over the water waves. In the expression of P
a
, there is a component in
phase and a component in quadrature with the water elevation. For an energy flux
to occur from the wind to the water waves, there must be a phase shift between the
fluctuating pressure and the interface. Hence, the transfer of energy is only due to
the component in quadrature with the water surface or in other words in phase with
the slope. To simplify the problem, we consider only the pressure component in phase
with the slope on the interface
P
a
(x,t)=ρ
a
βU
2
1
η
x
(x,t). (2.6)
For a logarithmic velocity profile in the turbulent boundary layer over the wave, we
have U
1
= u
∗
/κ,whereκ is the von K
´
arm
´
an constant. Hence,
P
a
(x,t)=Wη
x
(x,t), (2.7)
where W= ρ
a
(β/κ
2
)u
2
∗
.
The rate of growth of the wave energy is sβω(u
∗
/c)
2
/κ
2
, where s = ρ
a
/ρ is the
air–water density ratio and ω = kc the frequency.
Miles (1957) developed an alternative derivation of the rate of growth of the wave
energy based on a linear stability analysis of the parallel shear flows. The transfer of
energy from a shear flow U (z) to a surface wave of wavenumber, k, and phase velocity,
c, is associated with a singularity at the critical layer z = z
c
at which U(z = z
c
)=c
∂
E
∂t
= −ρ
a
cπ
d
2
U
dz
2
(z
c
)/k
⏐
⏐
⏐
⏐
dU
dz
(z
c
)
⏐
⏐
⏐
⏐
w
2
(z
c
), (2.8)
where
E is the mean surface wave energy and w
2
(z
c
) is the mean-square value of
the wave-induced vertical velocity at z = z
c
. The overbar denotes an average over x.
The vertical velocity, w, is calculated from a Sturm–Liouville differential equation (or
the Rayleigh equation). The Rayleigh equation can be solved numerically once U (z),
k and c are known. Nevertheless, the presence of a singularity at the critical height,
z
c
, complicates the resolution. Conte & Miles (1959) developed a numerical method
to treat this singularity and solve the Rayleigh equation.
Writing ∂E/∂t = γE, we introduce the Miles coefficient β such as
γ =
ρ
a
ρ
kcβ
U
1
c
2
= sωβ
U
1
c
2
, (2.9)
where γ is the rate of growth of the wave energy. Following Miles (1996), the
coefficient β is given by the following expression:
β = −
π
k
(d
2
U/dz
2
)(z
c
)
|(dU/dz)(z
c
)|
w
2
(z
c
)
U
2
1
(∂η/∂x)
2
, (2.10)
where z = η(x) is the equation of the surface wave profile.
For a logarithmic profile of the atmospheric shear flow, the rate of growth γ is
γ =
ρ
a
ρ
kcβ
u
∗
c
2
κ
2
=
s
κ
2
ωβ
u
∗
c
2
. (2.11)
Note that in the classical theory of Miles, the interaction between the wave-induced
motion in the air flow and the turbulence is ignored. The turbulence is introduced
only to sustain a logarithmic wind profile.
142 C. Kharif, R. A. Kraenkel, M. A. Manna and R. Thomas
3. The perturbed NLS equation: amplification versus depletion
In this section, we consider both effects of dissipation and wind amplification
on the Benjamin–Feir instability. In the absence of viscosity and wind action, the
Benjamin–Feir instability may be investigated via an asymptotic expansion leading
to the well-known NLS equation. Our aim is to extend the works of Segur et al.
(2005a) and Leblanc (2007) who investigated this problem by considering damping
and wind effects separately. The derivation of the damped and forced NLS equation
does not present any conceptual difficulty. Hence, the expression of the perturbed
NLS equation can be stated as
i(ψ
t
+ Vψ
x
) −
Ω
0
8k
2
0
ψ
xx
−2Ω
0
k
2
0
|ψ|
2
ψ −i
WΩ
0
k
0
2gρ
ψ = −2iνk
2
0
ψ, (3.1)
where k
0
and Ω
0
are the wavenumber and frequency of the carrier wave, respectively,
satisfying the linear dispersion relation Ω
2
0
= gk
0
and V = Ω
0
/2k
0
is the group velocity
of the carrier wave. Equation (3.1) describes the spatial and temporal evolution of the
envelope, ψ, of the surface elevation, η, of weakly nonlinear and dispersive gravity
waves on deep water when damping and amplification effects are considered. The free
surface elevation is written as follows:
η(x, t)=ψ(x,t)exp[i(k
0
x − Ω
0
t)] + c.c. + O(
2
), (3.2)
where c.c. denotes the complex conjugate. The parameter, , is a small parameter
( 1) used to carry out the multiple scale analysis leading to (3.1). The elevation,
η, and envelope, ψ, are of order O(). We have assumed that the fluid viscosity, ν,
and density ratio, ρ
a
/ρ, are small: ν/
g/k
3
=
2
and ρ
a
/ρ =
2
. These assumptions
are generally used for water and air/sea interface.
We rewrite (3.1) in a standard form by using the following transformations:
ξ =2k
0
(x − Vt),τ= Ω
0
t, Ψ =
√
2k
0
ψ, (3.3)
leading to the following perturbed NLS:
ıΨ
τ
−
1
2
Ψ
ξξ
−|Ψ |
2
Ψ = ıKΨ, (3.4)
with
K =
Wk
0
2gρ
−2
νk
2
0
Ω
0
. (3.5)
The sign of K determines the nature of the perturbation. If K>0wehave
amplification of waves and if K<0 we have depletion. For K<0, this perturbed
NLS equation is similar to the NLS equation considered by Segur et al. (2005a).
It is also similar to the NLS equation used by Bridges & Dias (2007) when their
coefficients a and c vanish. In that case the latter authors demonstrated that there is
no enhancement of the modulational instability, whereas when a>0 the instability is
enhanced.
The condition K>0 implies that
4νκ
2
Ω
0
βsu
2
∗
< 1. (3.6)
For a given friction velocity, this condition states that only carrier wave with frequency
or wavenumber less than a threshold value may suffer modulational instability. Within
the framework of the NLS equation, we consider weakly modulated wave train. Hence,
we can assume that β depends on the frequency (or wavenumber) and phase velocity