#  Wind-wave interaction in finite depth: linear and nonlinear approaches,
blow-up and soliton breaking in finite time, integrability perspectives

A. Latifi
Department of Mechanics, Faculty of Physics
Qom University of Technology
Qom, Iran
&M.A. Manna
Université de Montpellier (UM)
and
Conseil National de la Recherche Scientifique (CNRS),
Laboratoire Charles Coulomb UMR 5221,
F-34095, Montpellier, France
&R.A. Kraenkel
Instituto de Física Téorica, UNESP
Universidade Estadual Paulista
01140-070, São Paulo, Brazil

###### Abstract

This work is an analytical investigation of the evolution of surface water
solitary waves in Miles and Jeffreys’ theories of wind wave interaction in
water of finite depth. Although many works have been conducted based on Miles
and Jeffreys’ approach, only a few studies have been carried out on finite
depth. The present review is divided into two major parts. The first
corresponds to the surface water waves in a linear regime and its nonlinear
extensions. In this part, Miles’ theory of wave amplification by wind is
extended to the case of finite depth. The dispersion relation provides a wave
growth rate depending on depth. A dimensionless water depth parameter,
depending on the depth and a characteristic wind speed, induces a family of
curves representing the wave growth as a function of the wave phase velocity
and the wind speed. Our theoretical results are in good agreement with the
data from the Australian Shallow Water Experiment and the data from the Lake
George experiment. In the second part of this study, Jeffreys’ theory of wave
amplification by wind is extended to the case of finite depth, where the fully
nonlinear anti-dissipative Serre-Green-Naghdi (SGN) equation is derived. Anti-
dissipation occurs due to the continuous transfer of wind energy to water
surface waves. We find the solitary wave solution of the system, with an
increasing amplitude under the action of the wind. This continuous increase in
amplitude leads to the soliton breaking and blow-up of the surface wave in
finite time for infinitely large asymptotic space. This dispersive, anti-
dissipative and fully nonlinear phenomenon is equivalent to the linear
instability at infinite time. The theoretical blow-up time is calculated based
on actual experimental data. By applying an appropriate perturbation method,
the SGN equation yields a Korteweg–de Vries–Burger-type equation (KdV-B),
combining weak nonlinearity, dispersion, and anti-dissipation. We show that
the continuous transfer of energy from wind to water results in the growth of
the KdV-B soliton’s amplitude, velocity, acceleration, and energy over time
while its effective wavelength decreases. This phenomenon differs from the
classical results of Jeffreys’ approach due to finite depth. Again, blow-up
and breaking occur in finite time. These times are calculated and expressed
for soliton- and wind-appropriate parameters and values. These values are
measurable in usual experimental facilities. The kinematics of the breaking is
studied, and a detailed analysis of the breaking kinetics and breaking time is
conducted using various criteria. Finally, some integrability perspectives are
presented.

_Keywords_ surface waves $\cdot$ wind-waves $\cdot$ interface waves $\cdot$
Miles’ mechanism $\cdot$ Jeffreys’ mechanism $\cdot$ nonlinear Serre-Green-
Naghdi model $\cdot$ waves in finite depth

##  1 Introduction

The Navier-Stoke equation for air and water is the starting point of wind-
generated water waves. Since the Navier-Stoke equation is analytically
unsolved, various approximations and assumptions are required to obtain
particular solutions.

The wind wave growth is due to the continuous energy and momentum flow from
the air to the water. Winds generate surface waves; in turn, surface water
waves modify the airflow. In this way, the atmosphere depends on the wave
state near the water surface. The physical mechanism behind is anti-
dissipation. The continuous energy and momentum flow from the air to the
surface wave generates an exponential growth of the wave amplitude in time,
more or less quickly, depending on the wind speed and the water depth.

However, waves may lose energy because of dissipation. The action balance
equation, a fundamental equation in fluid dynamics, commonly represents this
temporal and spatial dynamics. In deep water, it reads [1]:

$\frac{\partial}{\partial
t}\mathcal{N}+\overrightarrow{\nabla}.(\overrightarrow{c_{g}}\mathcal{N})=S,$
(1.1)

where $\overrightarrow{c_{g}}$ is the group velocity observed in a frame
moving with the wave, $\mathcal{N}$ is the action density defined as follows:

$\mathcal{N}=\frac{\mathcal{E}}{\omega},$ (1.2)

where $\omega$ is the wave frequency, $\mathcal{E}$ is the energy density

$\mathcal{E}=2\rho_{w}g\eta_{0}^{2},$ (1.3)

where $\eta_{0}$ the small wave amplitude, and $S$ is the source term

$S=S_{in}+S_{nl}+S_{ds}+\ldots,$ (1.4)

where $S_{in}$, $S_{nl}$, $S_{ds}$ are the effects of the wind input, non-
linear interactions and dissipation due to white capping, respectively.

The pioneer theoretical works on surface wind-wave growth were essentially
done by Jeffrys and Miles [2, 3, 4, 5], till some more recent studies[6, 7].
These works essentially aim to compute the term $S_{in}$ in Eq. (1.4). Later,
numerical approaches attempted to campute $S_{in}$, $S_{nl}$, $S_{ds}$ [1].

However, all theories mentioned above are focused on deep water domain and are
not well adapted to correctly describe wind-waves generation near-shore oceans
or shallow lakes. This limitation challenges both the physics and engineering
communities. Indeed, in the finite depth water domain the source term is

$S=S_{in}+S_{nl}+S_{ds}+S_{bf}+S_{tri}+\ldots.$ (1.5)

where $S_{bf}$ is the bottom friction, and $S_{tri}$ is triad nonlinear wave
interactions. Moreover, $S_{in}$ being strongly influenced by the finite depth
$h$, must be recalculated. To our knowledge, although many works have been
conducted based on Miles and Jeffreys’ approach, only a few studies are
carried out on finite depth [9, 10, 8, 11].

This work is an analytical extension of the evolution of surface water
solitary waves in Miles and Jeffreys’ theories of wind wave interaction in
water of finite depth with the Euler equations as outset. The analytic
approach is essential for further numerical investigations due to the scale of
energy dissipation near coasts. Indeed, the scale of energy dissipation is of
the order of a micrometre, which requires $10^{25}$ mesh nodes to produce
correct predictions on scales of 100 km. Hence, no pure numerical modelling of
this problem without recourse to theoretical developments has a chance of
succeeding.

Based on the latest progress [9, 10, 8, 11], this work aims to give l’etat de
l’art  in the field of wind-wave interactions in finite depth, and thereby
supply a theoretical basis allowing to go beyond the empirical laws. A
permanent concern in this work is to compare our theoretical results to
experimental data or check the feasibility of experimentation and field
observations.

This paper is divided into two parts: The first part is dedicated to the
extension of Miles’ approach [5] to finite depth, where the linearized Euler
equations in the water domain are coupled with weak nonlinearity to linearized
Euler equations in the air domain. The problem is solved at the interface, and
the linear dispersion relation of wave amplification at finite depth is
calculated. By introducing dimensionless variables and scaling, the wind
wave’s growth rate is obtained. Our theoretical laws are compared with both
the Young-Verhagen data and plots of empirical relationships from the Lake
George experiment and with Donelan’s data from the AUSWEX program [12, 13,
14].

Introducing a simple modus operandi for the Miles’ mechanism allows us to
derive an anti-diffusive nonlinear Schrödinger equation for the wind-wave in
finite depth and we derive the Akhmediev, Peregrine, and Ma solutions for weak
wind inputs.

The second part of this paper is devoted to the extension of Jeffrey’s
mechanism [2, 3] where in different scales, the anti-dissipative Korteweg-de
Vries-Burger equation and the fully nonlinear Green-Nagdhi are derived. The
anti-dissipative Korteweg-de Vries-Burger equation is obtained by the coupling
of the weakly nonlinear Euler equations in the water domain to the linearized
Euler equations in the air domain. In contrast, the fully nonlinear Green-
Nagdhi is derived by coupling the nonlinear Euler equations in the water
domain to the linearized Euler equations in the air domain.

For both SGN and KdV-B equations, solitary waves solutions exhibiting blow-up
in finite time are found. By considering its kinematics description, the
solition breaking phenomenon is analyzed in detail in the case of Kdv-B
equation.

This paper is organized as follows: Section 2 studies the mutual action of
linearised water and air dynamics through Mile’s mechanism in finite depth.
The wave growth rate is analytically calculated and compared to field
experiment results and empiric laws. A good agreement between theoretical and
experimental results is observed. Section 3 compares our theoretical results
to experimental and empirical laws. In Section 3, we apply the Miles approach
from a quasi-linear perspective. Here, the air dynamics remain linear, while
the water domain is viewed as irrotational and nonlinear. This modification
gives rise to the Nonlinear Schrödinger equation, from which we derive the
Akhmedeiev, Peregrine, and Ma solutions solutions for weak wind input.
Sections 4 and 5 are devoted to the nonlinear Jeffrey’s approach. Namely, in
section 3, the water domain is considered irrotational and nonlinear, and the
air domain is linearized. Jeffrey’s sheltering mechanism allows us to derive
the SGN equation in finite depth, which we solve analytically. The solution is
a solitary wave with the remarkable feature of blowing up in finite time. The
blow-up time is calculated based on experimental measurement. In section 5,
applying an appropriate approximation, we derive the KdV-B equation from the
Serre-Green-Naghdi equation we solve analytically. Again, the solution is a
solitary wave with a blow-up in finite time. In this section, we conduct a
detailed analysis of the wave breaking before the blow-up, which we compare to
existing breaking criteria. In Section 6, we present some integrability
perspectives. Section 7 summarises our results, draws conclusions, and gives
perspectives on further developments. In the Appendix, the direct derivation
of the KdV-B equation Euler equation is shown.

##  2 Mile’s Mechanism

The Miles mechanism in infinite or finite depth is based on a particular
interaction between air flows and surface waves. It’s about a linear mechanism
of resonance between the wind speed and the water phase speed, which will
induce a change in pressure and modify the propagation properties of linear
water surface waves. We therefore model this instability as follows: either a
coordinate system Cartesian dimensions with $x$ the horizontal, $z$ the height
and $y$ the transverse dimension, which we will not consider here. We consider
two fluids, air and water, separated by a free surface $(x,t)$. The constant
densities of air and water are $\rho_{a}$ and $\rho_{w}$. Pressure fields in
water and air are denoted respectively $P_{w}$ and $P_{a}$. The bottom is
located at a depth $z=-h$ and the interface $z=\eta(x,t)$. We prescribe a wind
$U(z)$ having a specific profile, typically logarithmic, as in Figure (1).
Therefore, we will write Euler’s equations for air and water, with an
additional condition of pressure continuity connecting them. We assume the
dynamic to be linear and disregard the air turbulence, building a quasi-
laminar theory.

Figure 1: The Miles mechanism is based on a particular interaction between air
flows and surface waves. It’s about a linear mechanism of resonance between
the wind speed and the water phase speed, which will induce a change in
pressure, which in turn will modify the propagation properties of linear water
surface waves. We consider two fluids, air and water, separated by a free
surface $(x,t)$. The bottom is located at a depth $-h$, and we prescribe a
wind $U(z)$ having a specific profile, typically logarithmic. The interface at
rest. The perturbed air-water interface will be described by $z=\eta(x,t)$.

###  2.1 The linearized water dynamics

In the water domain, we consider the Euler equations for finite depth. The
horizontal and vertical velocities of the fluid are $u(x,z,t)$ and $w(x,z,t)$.
The continuity equation and the linearized equation of motion in the water
domain read [15]:

$u_{x}+w_{z}=0,\quad\rho_{w}u_{t}=-P_{x},\quad\rho_{w}w_{t}=-P_{z}-g\rho_{w},$
(2.1)

where $P(x,z,t)$ is the pressure, $g$ the gravitational acceleration,
$\rho_{w}$ is the water density and subscripts in $u$, $w$, $P_{x}$ and
$P_{z}$ denote partial derivatives of $P$ with respect to $x$ and $z$,
receptively. The boundary conditions at $z=-h$ and at $z=\eta(x,t)$ are

$\displaystyle w(-h)=0,\quad\eta_{t}=w(0),$ (2.2) $\displaystyle
P(x,\eta,t)=P_{a}(x,\eta,t),$ (2.3)

where $P_{a}$ is the air pressure evaluated at $z=\eta$. Thus equation (2.3)
is the continuity of the pressure across the air/water interface. As this is a
vital assumption for the growth mechanism, we give it a more pleasant
expression. So, let us introduce a reduced pressure defined by

$\mathbf{P}(x,z,t)=P(x,z,t)+\rho_{w}gz-P_{0},$ (2.4)

where $P_{0}$ is the atmospheric pressure. In terms of (2.4) equations
(2.1)-(2.3) read

$u_{x}+w_{z}=0,\quad\rho_{w}u_{t}=-\mathbf{P}_{x},\quad\rho_{w}w_{t}=-\mathbf{P}_{z},$
(2.5) $\displaystyle w(-h)=0,\quad\eta_{t}=w(0),$ (2.6)
$\displaystyle\mathbf{P}(x,\eta,t)=P_{a}(x,\eta,t)+\rho_{w}g\eta-P_{0}.$ (2.7)

The linear equations system (2.5)-(2.7) can be solved, assuming normal mode
solutions as

$\displaystyle\mathbf{P}$ $\displaystyle=$
$\displaystyle\mathcal{P}(z)\exp{(i\theta)},\quad
u=\mathcal{U}(z)\exp{(i\theta)},$ $\displaystyle\quad w$ $\displaystyle=$
$\displaystyle\mathcal{W}(z)\exp{(i\theta)},\quad\eta=\eta_{0}\exp{(i\theta)},$
(2.8)

with $\theta=k(x-ct)$ where $k$ is the wavenumber, $c$ the phase speed and
$\eta_{0}$ is a constant. Using equations (2.5), (2.6), (2.7) and (2.1) we
obtain

$\displaystyle w(x,z,t)$ $\displaystyle=$
$\displaystyle\frac{-ikc\sinh{k(z+h)}}{\sinh{kh}}\eta_{0}\exp{(i\theta)},$
(2.9) $\displaystyle u(x,z,t)$ $\displaystyle=$
$\displaystyle\frac{kc\cosh{k(z+h)}}{\sinh{kh}}\eta_{0}\exp{(i\theta)},$
(2.10) $\displaystyle\mathbf{P}(x,z,t)$ $\displaystyle=$
$\displaystyle\frac{k\rho_{w}c^{2}\cosh{k(z+h)}}{\sinh{kh}}\eta_{0}\exp{(i\theta)}.$
(2.11)

The phase speed $c$ is unknown in equations (2.9)-(2.11). To determine $c$ we
have to consider the boundary conditions (2.7), not yet used, and (2.6) which
yields

$\rho_{w}\eta_{0}\exp{(i\theta)}\\{c^{2}k\coth{kh}-g\\}+P_{0}=P_{a}(x,\eta,t).$
(2.12)

In the single-domain problem $P_{a}(x,\eta,t)=P_{0}$ and (2.12) gives the
usual expression for c,

$c^{2}=c_{0}^{2}=\frac{g}{k}\tanh{(kh)}.$ (2.13)

It is not the case in the problem under consideration in the present paper in
which the determination of c needs the use of the air pressure evaluated at
$z=\eta$.

###  2.2 The linearized air dynamics

Let us consider the linearized governing equation of a steady airflow, with a
prescribed mean horizontal velocity $U(z)$ depending on the vertical
coordinate $z$. We are going to study perturbations to the mean flow $U(z)$:
$u_{a}(x,z,t)$, $w_{a}(x,z,t)$ and $P_{a}(x,z,t)$ where subscript $a$ stands
for _air_. So with $\mathbf{P}_{a}(x,z,t)=P_{a}(x,z,t)+\rho_{a}gz-P_{0}$,
$\rho_{a}$ the air density, and $U^{\prime}=dU(z)/dz$ we have the following
equations

$\displaystyle u_{a,x}+w_{a,z}=0,$ (2.14)
$\displaystyle\rho_{a}[u_{a,t}+U(z)u_{a,x}+U^{\prime}(z)w_{a}]=-\mathbf{P}_{a,x},$
(2.15) $\displaystyle\rho_{a}[w_{a,t}+U(z)w_{a,x}]=-\mathbf{P}_{a,z},$ (2.16)

which must be completed with the appropriate boundary conditions. The first is
the kinematic boundary condition for air, evaluated at the aerodynamic sea
surface roughness $z_{0}$ located just above the interface. Through this
paper, $z_{0}$ will be regarded as a constant independent from the sea state.
This is a widely used approximation, first proposed by [16]. For the datasets
used later on, the wind speed ranges are such that the roughness may be seen
as a constant [17]. The kinematic boundary condition reads

$\eta_{t}+U(z_{0})\eta_{x}=w_{a}(z_{0}).$ (2.17)

We choose $U(z)$ as the logarithmic wind profile. This is commonly used to
describe the vertical distribution of the horizontal mean wind speed within
the lowest portion of the air side of the marine boundary layer [18]. It can
also be justified with scaling arguments and solution matching between the
near-surface and geostrophic air layer [19].

$U(z)=U_{1}\ln(z/z_{0}),\quad U_{1}=\frac{u_{*}}{\kappa},\quad\kappa\approx
0.41,$ (2.18)

where $u_{*}$ is the friction velocity and $\kappa$ the Von Kármán constant.
So, eq. (2.17) can be reduced to

$\eta_{t}=w_{a}(z_{0}).$ (2.19)

This equation describes the influence of the surface perturbation on the
vertical perturbed wind speed. Next we assume
$\mathbf{P}_{a}=\mathcal{P}_{a}(z)\exp{(i\theta)}$,
$u_{a}=\mathcal{U}_{a}(z)\exp{(i\theta)}$,
$w_{a}=\mathcal{W}_{a}(z)\exp{(i\theta)}$ and we add the following boundary
conditions on $\mathcal{W}_{a}$ and $\mathcal{P}_{a}$,

$\displaystyle\lim\limits_{z\to+\infty}(\mathcal{W}_{a}^{\prime}+k\mathcal{W}_{a})=0,$
(2.20) $\displaystyle\lim\limits_{z\to z_{0}}\mathcal{W}_{a}=W_{0},$ (2.21)
$\displaystyle\lim\limits_{z\to+\infty}\mathcal{P}_{a}=0,$ (2.22)

that is, the disturbance plus its derivative vanish at infinity, and the
vertical component of the wind speed is enforced by the wave movement at the
sea surface. Then, using equations (2.14)-(2.16) and (2.22) we obtain

$\displaystyle w_{a}(x,z,t)$ $\displaystyle=$
$\displaystyle\mathcal{W}_{a}\exp{(i\theta)},$ (2.23) $\displaystyle
u_{a}(x,z,t)$ $\displaystyle=$
$\displaystyle\frac{i}{k}\mathcal{W}_{a,z}\exp{(i\theta)},$ (2.24)
$\displaystyle\mathbf{P}_{a}(x,z,t)$ $\displaystyle=$ $\displaystyle
ik\rho_{a}\exp{(i\theta)}\int^{\infty}_{z}[U-c]\mathcal{W}_{a}dz^{\prime}.$
(2.25)

Removing the pressure from the Euler equations, we find the well-known
Rayleigh equation [20] $\forall z~{}\backslash~{}z_{0}<z<+\infty$ (inviscid
Orr-Sommerfeld equation)

$(U-c)(\mathcal{W}_{a}^{\prime\prime}-k^{2}\mathcal{W}_{a})-U^{\prime\prime}\mathcal{W}_{a}=0$
(2.26)

which is singular at the critical, or matched height
$z_{c}=z_{0}e^{c\kappa/u_{*}}>z_{0}>0$, where $U(z_{c})=c$. We recall that
this model disregards any kind of turbulence, and so the critical height is
set above any turbulent eddies or other non-linear phenomena. In equations
(2.23)-(2.26) neither $\mathcal{W}_{a}(z)$ nor $c$ are known. In order to find
$c$, we have to calculate $P_{a}(x,\eta,t)$. We obtain

$\displaystyle
P_{a}(x,\eta,t)=P_{0}-\rho_{a}g\eta+{ik\rho_{a}\exp{(i\theta)}\int^{\infty}_{z_{0}}[U(z)-c]\mathcal{W}_{a}(z)dz},$
(2.27)

where the lower integration bound is taken at the constant roughness height
$z_{0}$ instead of $z=\eta$ since we are studying the linear problem. Using
equation (2.19) to eliminate the term $ik\rho_{a}\exp(i\theta)$ the equation
(2.27) in (2.12) yields

$g(1-s)+c\frac{sk^{2}}{W_{0}}I_{1}-c^{2}\\{\frac{sk^{2}}{W_{0}}I_{2}+k\coth(kh)\\}=0,$
(2.28)

where $s=\rho_{a}/\rho_{w}$ and the integrals $I_{1}$ and $I_{2}$ are defined
as follow

$I_{1}=\int_{z_{0}}^{\infty}U\mathcal{W}_{a}dz,\quad
I_{2}=\int_{z_{0}}^{\infty}\mathcal{W}_{a}dz.$ (2.29)

Equation (2.28) is the dispersion relation of the problem. The parameter $s$
is small ($\rho_{a}/\rho_{w}\sim 10^{-3}$) and (2.28) may be approximated as

$c=c_{0}+sc_{1}+O(s^{2}).$ (2.30)

The explicit form of $c_{1}$ is calculated in the next section. Therefore, we
can find $\mathcal{W}_{a}(z)$ by solving (2.26) with $c$ replaced by $c_{0}$,
that is to say, of order zero in $s$.

###  2.3 The wave growth rate

The function $\mathcal{W}_{a}(z)$ is complex and consequently $c$ also. Its
imaginary part gives the growth rate of $\eta(x,t)$ defined by

$\gamma=k\Im{(c)},$ (2.31)

where $\Im{(c)}$ is the imaginary part of $c$. The theoretical and numerical
results concerning the growth rate $\gamma$ are studied and computed with two
dimensionless parameters $\delta$ (see [12] and [13]) and $\theta_{dw}$
defined by

$\delta=\frac{gh}{U^{2}_{1}},\quad\theta_{dw}=\frac{1}{U_{1}}\sqrt{\frac{g}{k}}.$
(2.32)

The dimensionless parameter $\delta$, for constant $U_{1}$, measures the
influence of the finite fluid depth on the rate of growth of $\eta(x,t)$. The
parameter $\theta_{dw}$ can be seen as a theoretical analogous of the deep
water wave age. It measures the relative value of the deep water phase speed
about the characteristic wind velocity $U_{1}$. Now a theoretical analogous of
the finite depth wave age $\theta_{fd}$ can be introduced as

$\theta_{fd}=\frac{1}{U_{1}}\sqrt{\frac{g}{k}}\sqrt{\tanh({kh})}=\theta_{dw}T^{1/2},$
(2.33)

where

$T=\tanh(\frac{\delta}{\theta_{dw}^{2}}).$ (2.34)

The form (2.33) is a depth weighted parameter such that for a finite and
constant $\theta_{dw}$ we have $\theta_{fd}\sim\theta_{dw}$ if
$\delta\rightarrow\infty$ and $\theta_{fd}\sim\delta^{1/2}=\sqrt{gh}/U_{1}$ if
$\delta\rightarrow 0$. To obtain the growth rate, we introduce the following
non-dimensional variables and scalings, hats meaning dimensionless quantities

$\displaystyle U$ $\displaystyle=$ $\displaystyle
U_{1}\hat{U},\quad\mathcal{W}_{a}=W_{0}\hat{\mathcal{W}_{a}},\quad
z=\frac{\hat{z}}{k},$ (2.35) $\displaystyle c=U_{1}\hat{c},\quad
t=\frac{U_{1}}{g}\hat{t}.$

Using (2.32) and (2.35) in equation (2.28) and retaining only the terms of
order one in $s$ we obtain $c$,

$\displaystyle\hat{c}=\hat{c}(\delta,\theta_{dw})=\theta_{dw}T^{1/2}-\frac{s}{2}\theta_{dw}T^{1/2}+\frac{s}{2}\\{T\hat{I}_{1}-\theta_{dw}T^{3/2}\hat{I}_{2}\\},$
(2.36)

and with $e^{\gamma
t}=e^{k{\Im}(c)t}=e^{\Im{(\hat{c})}\hat{t}/\theta_{dw}^{2}}$, we have the
dimensionless growth rate $\hat{\gamma}=\frac{U_{1}}{g}\gamma$ as,

$\hat{\gamma}=\frac{s}{2}\\{\frac{T\Im{(I_{1})}}{\theta^{2}_{dw}}-\frac{T^{3/2}\Im{(I_{2})}}{\theta_{dw}}\\},$
(2.37)

So, we can compute it for a given $(\delta,\theta_{dw})$ set. The $\delta$
parameter does not appear explicitly, allowing us to compute $\gamma$ for an
infinite depth, where we have just $T\rightarrow 1$. This gives back to Miles’
theory.

Figure 2: Evolution of the growth rate in semi-logarithmic scale. Every curve
but the rightmost one correspond to finite depth. From left to right, they
match $\delta=1,4,9,25,49,81$. We can observe that for each depth, there is a
$\theta_{fd}$ – limited wave growth. The deep water limit, also computed, is
corresponds to small $\theta_{fd}$ and matches Miles’ results.

The existence of a finite depth $h$ transforms the unique curve of wave growth
rate in deep water in a family of curves indexed by $\delta=gh/U^{2}_{1}$,
i.e., a curve for each value of $\delta$. In Figure 2 shows a family of six
values of $\delta$ against the $\theta_{fd}$ parameter. The limit
$\delta\rightarrow\infty$ is included as well. Small finite $\theta_{fd}$
corresponds to short surface waves. This stage represents the initial growth
of the wave field near the shoreline of a calm lake. As time proceeds, the
surface waves reach moderate $\theta_{fd}$, corresponding to mild or moderate
wavelengths, while long waves are found for large $\theta_{fd}$. Of course, as
the wavelengths increase, the amplitudes keep on growing. Clearly, from a
physical point of view, this means that Figure 2 is a snapshot of the
theoretical dynamical development of the wave, which is growing in amplitude
and wavelength in time.

Figure 2 shows that at small $\theta_{fd}$ the growth rate $\gamma$ is equal
for all values of $\delta$, the limit being the deep water case. The finite-
depth effects appear as $\theta_{fd}$ increases. The growth rate becomes lower
than in the deep water limit for each value of $\delta$. The growth rates are
scaled with $\delta$: for a given $\theta_{fd}$, the bigger the $\delta$ the
larger the $\hat{\gamma}$. Each $\delta$-curve approaches its own (idealized)
_theoretical $\theta_{fd}$-limited growth_ as $\hat{\gamma}$ goes to zero. At
this stage, the wave reaches a final linear progressive wave with zero growth.
In other words, for a given $\delta$ the surface wave does not grow old beyond
a determined $\theta_{fd}$.

In contrast to the usual analysis of wind-wave growth, our results concern the
dimensionless growth rate $\hat{\gamma}$ instead of the $\beta$-Miles
parameter. We have the following transformation rule between this parameter
$\beta$ and dimensionless $\hat{\gamma}$

$\beta=\frac{2\hat{\gamma}}{s}\theta_{dw}^{3}T^{1/2},$ (2.38)

where we took $\beta$ as it is usually defined, with the dimensions,
$\Im(c)=c_{0}\frac{s}{2}\beta(\frac{U_{1}}{c_{0}})^{2}.$ This definition of
Miles’ $\beta$ in finite depth is straightforward. Its evolution is shown
clearly in Figure 3, showing the correct deep water trends and the new finite
depth limits. The effects of depth are critical. As usual, $\beta$ is almost
constant for small $\theta_{fd}$, but it goes dramatically to zero when the
depth limit is close.

Figure 3: Evolution of Miles’ coefficient $\beta$ for several values of the
depth. Each curve is plotted with the same Charnock constant
$\alpha_{c}\approx 0.018$. The finite-depth effect is critical, and high value
of $\delta$ correspond to deep water.

###  2.4 Field experiments on growth of surface wind-waves

For finite depth wave growth the pioneer experiments and numerical studies
were conducted by [21], [22], [14] and particularly the experiments in Lake
George, Australia, described by [12]. They provided one of the first
systematic attempts to understand the physics of wave-wind generation in
finite-depth water.

The results of the field experiments in fetch limited growth have been
presented in references [12] and [13]. These papers described the basin
geometry and bathymetry, experimental design, used instrumentation, and the
adopted scaling parameters. The measurements have confirmed the water depth
dependence of the asymptotic limits to wave growth.

In reference [23] (see also [24]) derived an empirical relation in terms of
appropriate dimensionless parameters able to reproduce the experimental data
of [12]. In particular, the empirical relationship between the fractional
energy increase as a function of the inverse wave age, found by [25] For deep
water, it was extended to the finite depth domain. Experimental results and
plots of the empirical laws have shown that contrary to the deep water case,
the wave age at which the growth rate becomes zero is wind-dependent and
depth-dependent. So, the point of full development is warped from the deep
water case, where established it [26]. As a result, a growth law against the
inverse wave age exists for each value of a parameter, which unites the
dependences on wind intensity and water depth.

The evolution of the growth rates is such that at small wave ages, growth
rates are comparable to the deep water limit at large wave ages, the growth
rate is lower the growth rate vanishes beyond a limit wave age in shallow
water than in deep water.

###  2.5 Comparisons with field experiments

In this subsection, we are going to show that our analytical and numerical
results can reproduce qualitatively these experimental facts. At this point,
it is essential to keep in mind that we are studying the linear growth of a
normal Fourier mode $k$ and not the growth of a wave train as the infinite
superposition of wave Fourier modes.

Moreover, results in field or laboratory experiments are commonly given using
the parameter $C_{p}$, the observed phase speed at the peak frequency
$\omega_{p}$. Consequently, qualitative comparison with field observations can
only be made using the phase velocity $c$ or frequency $\omega$ of one mode
instead of $C_{p}$ or $\omega_{p}$.

We are going to show that the theoretical curves for $\hat{\gamma}$ are,
mutatis mutandi, in good qualitative agreement with the empirical curves of
the dimensionless fractional wave energy increase per radian $\hat{\Gamma}$ as
a function of the inverse wave age $U_{10}/Cp$ in [23]. In this reference,
experimental field data for $\hat{\Gamma}$ in the finite depth of the
empirical relationship adequately represents Lake George

$\displaystyle\hat{\Gamma}$ $\displaystyle=$
$\displaystyle\frac{C_{g}}{\omega_{p}}\frac{1}{E}\frac{\partial E}{\partial
x}$ $\displaystyle=$ $\displaystyle
A\left(\frac{U_{10}}{C_{p}}-0,83\right)\tanh^{0,45}{\left(\frac{U_{10}}{C_{p}}-\frac{1,25}{\delta^{0,45}_{Y}}\right)},$

with A constant, $\delta_{Y}=gh/U_{10}^{2}$ the non-dimensional water depth,
$U_{10}$ the wind velocity at $10~{}m$, and $C_{g}$ and $C_{p}$ the group and
phase speeds of the components at the spectral peak frequency $\omega_{p}$.

In order to make a qualitative comparison between $\hat{\Gamma}$ curves in
function of the inverse wave-age $U_{10}/C_{p}$ and theoretical $\hat{\gamma}$
curves in function of $1/\theta_{fd}$ we need to write the empirical
$\hat{\Gamma}$ in terms of theoretical quantities. So, the following changes
are necessary:

measured $\displaystyle
C_{g},C_{p},\omega_{p}\rightarrow\mbox{theoretical}\hskip
5.69054ptc_{g},c,\omega,$ (2.40) and
$\displaystyle\quad\frac{U_{10}C_{10}^{1/2}}{\kappa}=u_{*}/\kappa=U_{1}$
(2.41)

with $C_{10}$ the $10~{}m$ drag coefficient [27]. Thus, from the fact that the
energy growth rate is two times the amplitude growth rate, that is

$\Gamma=2\gamma,$

and using $2c_{g}=c(1+2kh/\sinh(2kh))$, (2.40), (2.41),(2.33) and (2.35) we
obtain

$\hat{\Gamma}=\frac{\theta_{dw}}{T^{1/2}}\hat{\gamma}[1+\frac{2\delta}{\theta_{dw}^{2}\sinh(\frac{2\delta}{\theta_{dw}^{2}})}].$
(2.42)

This expression gives the theoretical equivalent of the empirical
$\hat{\Gamma}$ in function of $\theta_{dw}$, $\delta$ and $\hat{\gamma}$. The
values of $\hat{\gamma}$ for fixed $\delta$’s as a function of $1/\theta_{fd}$
are numerically obtained from Eqs. (2.33) (2.34) and (2.37). Steps (2.40) and
(2.41) transform $\delta_{Y}$ and $C_{p}/U_{10}$ into $\delta$ and
$\theta_{fd}$ according to

$\displaystyle\delta_{Y}$ $\displaystyle=$
$\displaystyle\delta\frac{C_{10}}{\kappa^{2}},$ (2.43)
$\displaystyle\frac{C_{p}}{U_{10}}$ $\displaystyle=$
$\displaystyle\theta_{fd}\frac{C_{10}^{2}}{\kappa}.$ (2.44)

In reference [23] the curves of $\Gamma$ versus $U_{10}/C_{p}$ have been
presented for the $\delta_{Y}$-intervals
$\delta_{Y}\in[0.1-0.2],[0.2-0.3],[0.3-0.4],[0.4-0.5]$, rather than for a
single value of $\delta_{Y}$. The intervals were determined from the
variations in $U_{10}$, the depth $h$ being nearly constant around $2~{}m$.
Consequently we substitute the $\delta_{Y}$-intervals with $\delta$-intervals
using (2.43) and we evaluate the mean value $\overline{\delta}$. For example
$\delta_{Y}\in[0,1-0,2]$ is transformed into $\delta=[13,17-26,35]$ with
$\overline{\delta}=19,76$ in Figure 4(a). Figures 4(a), 4(b), 4(c) and 4(d)
are displaying a fair concordance of the model with the experimental data and
plots of empirical laws for Lake George. The agreement improves as
$\frac{1}{\theta_{fd}}$ increases.

In Figure 5 are plotted, against $\delta$, the critical values of the
parameter $\theta_{fd}^{c}$ for which the growth rate $\gamma$ goes to zero.
They obey the relation

$\theta_{fd}^{c}=\delta^{0,5}.$ (2.45)

The above relation, found numerically, is coherent with the parameter
formulation (2.33). It is indeed a limiting value for $\theta_{fd}$ uniquely
determined by the water depth. In [23] the author has shown from an empirical
relationship (formula (6) in reference above) that $\hat{\Gamma}$ the growth
rate goes to zero as a function of the inverse wave age $U_{10}/C_{p}$ for

$\frac{C_{p}}{U_{10}}=0.8(\frac{gh}{U_{10}^{2}})^{0.45}.$ (2.46)

Using a $C_{10}$ drag coefficient parametrization such as [27]

$C_{10}=(0.065U_{10}+0.8)10^{-3},$ (2.47)

and taking an average $U_{10}=7~{}m/s$ in [23], one finds the $U_{1}$ to
$U_{10}$ relationship

$U_{10}\approx 28,3~{}u_{*}\approx 11,6~{}U_{1},$ (2.48)

So, this limiting law reads

$\frac{C_{p}}{U_{1}}=1,01~{}\delta^{0,45}.$ (2.49)

a result in excellent agreement with the theoretical value (2.45). With
$\theta_{fd}^{c}$ we can calculate the corresponding critical wave length
$\lambda^{c}$. Using (2.45) in (2.33) we obtain

$\frac{\delta}{\theta^{2}_{dw}}=\tanh{(\frac{\delta}{\theta^{2}_{dw}})}.$
(2.50)

Relation (2.50) means the wave has entered the shallow water region. In such a
limit the range of $\delta/\theta^{2}_{dw}$ is:
$0<\delta/\theta^{2}_{dw}<\frac{\pi}{4}$ ([28], [29]). As a result we obtain
$\lambda^{c}=8h$. For values of $\lambda$ such that $\lambda>\lambda^{c}$ the
phase velocity is in the long wave limit i.e., $c=\sqrt{gh}$. Consequently, if
$\lambda>\lambda_{c}$ the wave feels the bottom, the amplitude does not grow
anymore, the resonance wind/phase speed ceases, and the wave reaches its
utmost state as a progressive plane wave.

Finally in Figure 5 are also represented data from [25], from the Australian
Shallow Water Experiment. A fit is also plotted to show the trend. The raw
data consists in the water depth $h$ in meters, the friction velocity $u_{*}$,
the $10$ meters wind velocity $U_{10}$ and the ratio of the former with the
measured phase speed $c_{p}$, $U_{10}/c_{p}$. For example,
$u_{*}=0.44~{}m.s^{-1}$ and $h=0.32~{}m$ gives $\delta=2.7$ and
$\theta_{fd}=1.55$, which gives a small relative error regarding (2.45). All
the points give $(\delta,\theta_{fd})$ coordinates close to the theoretical
limit.

---
Figure 4: Growth rate $\hat{\Gamma}$ as a function of inverse wave age
$1/\theta_{fd}$ for several values of the parameter $\delta$. White squares
correspond to Lake George experiment data, Black squares correspond to the
empirical relationship (eq. (6)) in [23]. Present results correspond to
symbols $+$, $\times$ and $\ast$. a: the dataset covers a range of wind speed
corresponding to $\delta_{Y}=0.1-0.2$, or using (2.43) $\delta=13.17-26.35$,
and an average value $<\delta>=(13.17+26.35)/2$ is used. b: same as “a” with
$\delta_{Y}=0.2-0.3$. c: same as “a” with $\delta_{Y}=0.3-0.4$. d : same as
“a” with $\delta_{Y}=0.4-0.5$. Figure 5: Parameter curves corresponding to
zero growth rate. The theoretical limit is given by (2.45). The AUSWEX data
are experimental results from [25] (the sea state is fairly close to the
finite depth full development).

##  3 A non linear Miles’ approach

In this section we consider the air/water system from quasi-linear point of
view in the sense that the water dynamics is considered nonlinear and
irrotational but the airflow is kept linear . We use the $\beta$-Miles
parameter as given by Eq. 2.38. In Miles’s theory of wave generation [30, 5],
the complex air pressure $P_{a}$ can be separated into two components, one in
phase and one in quadrature with the free surface $\eta$. A phase shift
between those two quantities is necessary to transfer energy from the air flow
to the wave field. The transfer is only due to the part of $P_{a}$ in
quadrature with $\eta$. Hence, we will deal only with the acting pressure
component, that is

$P_{a}(x,t)=\rho_{a}\beta U_{1}^{2}\eta_{x}(x,t),$ (3.1)

###  3.1 The anti-diffusive nonlinear Schrödinger equation in finite depth

Let us consider the air/water system from a quasi-linear point of view i.e.,
the water dynamics is considered nonlinear and irrotational and, as in Miles’
theory, the air flow is kept linear. So with this assumption the complete
irrotational Euler equations and boundary conditions in terms of the velocity
potential $\phi(x,z,t)$ are

$\displaystyle\phi_{xx}+\phi_{zz}=0\quad\mbox{for}\quad-h\leq z\leq\eta,$
(3.2) $\displaystyle\phi_{z}=0\quad\mbox{for}\quad z=-h,$ (3.3)
$\displaystyle\eta_{t}+\phi_{x}\eta_{x}-\phi_{z}=0\quad\mbox{for}\quad
z=\eta,$ (3.4)
$\displaystyle\phi_{t}+\frac{1}{2}\phi_{x}^{2}+\frac{1}{2}\phi_{z}^{2}+g\eta=-\frac{1}{\rho_{w}}P_{a}\quad\mbox{for}\quad
z=\eta.$ (3.5)

Using (3.1) the modified Bernoulli equation reads

$\phi_{t}+\frac{1}{2}\phi_{x}^{2}+\frac{1}{2}\phi_{z}^{2}+g\eta=-s\beta
U_{1}^{2}\eta_{x}\quad\mbox{for}\quad z=\eta.$ (3.6)

From equations (3.2), (3.3), (3.4) and (3.6) we find a wind-forced finite
depth NLS equation for $\eta$ as a function of the standard slow space and
time variables $\xi=\varepsilon(x-c_{g}t)$ and $\nu=\varepsilon^{2}t$,
($\varepsilon<<1$) and $c_{g}$ the group velocity. The perturbed NLS equation
reads

$i\eta_{\nu}+a\eta_{\xi\xi}+b|\eta|^{2}\eta=id\eta$ (3.7)

with $c_{g},a,b$ and $d$ given by

$\displaystyle c_{g}$ $\displaystyle=$
$\displaystyle\frac{c}{2}[1+2kh/\sinh(2kh)],$ $\displaystyle a$
$\displaystyle=$ $\displaystyle-\frac{c_{g}^{2}-gh[1-khT(1-T^{2})]}{2\omega},$
$\displaystyle b$ $\displaystyle=$ $\displaystyle\frac{k^{4}c^{2}}{4\omega
T^{2}}[\frac{9}{T^{2}}-12+13T^{2}-2T^{4}-\frac{2[2c+c_{g}(1-T^{2})]^{2}}{gh-
c_{g}^{2}}],$ $\displaystyle d$ $\displaystyle=$ $\displaystyle
s\frac{\beta}{2}\frac{U_{1}^{2}}{c^{2}}T\omega.$

For more information about the derivation of the coefficients $a$ and $b$ see
[31]. To derive a dimensionless wind-forced NLS equation we use (2.35) and we
obtain in the original laboratory variables $x$ and $t$ (after a Galilean
transformation in order to eliminate the linear term $c_{g}\eta_{x}$ and
dropping the hats)

$i\eta_{t}+A\eta_{xx}+B|\eta|^{2}\eta=iD\eta$ (3.8)

with $c_{g},A,B$, and $D$ now given by

$\displaystyle c_{g}$ $\displaystyle=$
$\displaystyle\frac{1}{2\theta_{fd}}[1+\frac{\delta}{\theta_{dw}^{2}}\frac{1-T^{2}}{T}],$
$\displaystyle A$ $\displaystyle=$
$\displaystyle-\frac{c_{g}^{2}-\delta[1-\delta\theta_{dw}^{-2}(1-T^{2})]}{2\theta_{fd}\theta_{dw}^{2}},$
$\displaystyle B$ $\displaystyle=$
$\displaystyle\frac{1}{4T^{2}\theta_{fd}^{3}\theta_{dw}^{2}}[\frac{9}{T^{2}}-12+13T^{2}-2T^{4}-\frac{2[2\theta_{fd}^{-1}+c_{g}(1-T^{2})]^{2}}{\delta-
c_{g}^{2}}],$ $\displaystyle D$ $\displaystyle=$ $\displaystyle
s\frac{\beta}{2}\frac{T^{1/2}}{\theta_{dw}^{3}}.$

Eq. (3.8) is a wind-forced finite depth NLS equation in dimensionless
variables.

###  3.2 The Akhmediev, Peregrine and Ma solutions for weak wind inputs in
finite depth

The wind-forced nonlinear Schrödinger equation allows the study of the wind
influence on the freak waves dynamics [33, 32, 34, 35]. Previous authors have
carried out such studies in deep water. The present work allows, similar
studies in finite depth with the right Miles’ growth rates. In the following
we are going only to consider the so called focusing NLS equation i.e.,
positive $A$ and $B$. Introducing $\eta^{\prime}$ and $x^{\prime}$ as

$\eta^{\prime}=\sqrt{B}\eta,\quad x^{\prime}=\frac{x}{\sqrt{A}},$

Dropping the primes, Eq. (3.8) becomes

$i\eta_{t}+\eta_{xx}+|\eta|^{2}\eta=iD\eta.$ (3.9)

Introducing a function $M(x,t)$ as

$M(x,t)=\eta(x,t)\exp{(-Dt)},$ (3.10)

we obtain from (3.9)

$iM_{t}+M_{xx}+\exp{(2Dt)}|M|^{2}M=0.$ (3.11)

In order to reduce Eq. (3.9) into the standard form of the NLS with constant
coefficients we proceed in the following way. First of all we consider the
wind forcing $(2Dt)$ to be weak, such that the exponential can be approximated
by $\exp{(-2Dt)}\sim 1-2Dt$ so

$iM_{t}+M_{xx}+n|M|^{2}M=0,\quad n=n(t)=1-2Dt.$ (3.12)

Now with a change of coordinates from $(x,t)$ to $(z,\tau)$ defined by

$z(x,t)=xn(t),\quad\tau(x,t)=xn(t),$ (3.13)

and scaling the wave envelope as [35]

$M(z,\tau)=\Psi(z,\tau)\sqrt{n(\tau)}\exp{(\frac{-iDz^{2}}{n(\tau)})},$ (3.14)

we reduce (3.12) to the standard focusing equation

$i\Psi_{\tau}+\Psi_{xx}+|\Psi|^{2}\Psi=0.$ (3.15)

Equation (3.15) admits well known breather solutions that are simple
analytical prototypes for rogue wave events. They are the Akhmediev
($\Psi_{A}$) [36], the Peregrine ($\Psi_{P}$) [37] and the Kuznetsov-Ma
($\Psi_{M}$) [38] breather solutions. [39] investigated whether freak waves in
deep water could be modeled by $\Psi_{A}$, $\Psi_{P}$ or by $\Psi_{M}$. [35]
considered the influence of weak wind forcing and dissipation on these
$\Psi_{A}$, $\Psi_{P}$ or $\Psi_{M}$ solutions in deep water. The present work
allows us to exhibit expressions for $\Psi_{A}$, $\Psi_{P}$ and $\Psi_{M}$
under the influence of weak wind forcing in finite depth $h$ given by the
extended Miles mechanism. These solutions read [39]:

$\eta_{A}=P(\tau)\\{\frac{\cosh(\Omega\tau-2i\omega)-\cos(\omega)\cos(pz)}{\cosh(\Omega\tau)-\cos(\omega)\cos(pz)}\\},$
(3.16)

with $p=2\sin(\omega)$, $\Omega=2\sin(2\omega)$ $\omega$ real and $p$ related
to the spatial period $2\pi/p$

$\eta_{P}=P(\tau)\\{1-\frac{4(1+4i\tau)}{1+4z^{2}+16\tau^{2}}\\},$ (3.17)
$\eta_{M}=P(\tau)\\{\frac{\cos(\Omega\tau-2i\omega)-\cosh(\omega)\cosh(pz)}{\cosh(\Omega\tau)-\cos(\omega)\cos(pz)}\\},$
(3.18)

where $p=2\sinh(\omega)$, $\Omega=2\sinh(2\omega)$ and $\Omega$ real and
related to the time period $2\pi/\Omega$ and

$P(\tau)=n(\tau)\exp{[\frac{-iDz^{2}}{n(\tau)}]}\exp{[2i\tau]}.$

##  4 The Nonlinear Jeffrey’s approach

Jeffreys introduced a second significant mechanism, extensively researched
today, in 1925. The principle was that waves if sufficiently disrupted, could
break the continuity of the airflow and induce a gradient of pressure. This
understanding could have practical implications, allowing the wave to grow
quickly according to the relative distance between the phase speed of the wave
and the wind speed. Jeffreys found the main idea behind this mechanism by
analogizing it with a sphere immersed in a laminar fluid flow. Jeffreys
noticed that, in the viscous case, the fluid particles do not do everything
around the sphere. Those hitting the forehead slide on the sphere and take off
shortly after the halfway point, and on the other side, an area is almost
stagnant and has a relative speed relative to the sphere close to zero. (1925)
When a wave is large enough, it breaks the continuity of the airflow, and a
horizontal axis vortex forms on the side sheltered from the wind by the wave.
Other smaller whirlpools form between the large vortex and the laminar flow.
This phenomenon, called airflow separation, has been observed in waves, see
Figure (6). In this case, the face of the wave opposing the wind will
experience greater pressure, and the sheltered side will experience less
pressure. This gradient induces a reaction at the surface level.

Figure 6: Jeffrys’ mechanism - Airflow separation observed by "Particle Image
Velocimetry" over a breaking wave [40]. The air pressure on the windward face
of the wave is larger than the leeward face, which is the origin of continuous
energy transfer from wind to wave.

###  4.1 The Nonlinear Serre-Green-Naghdi Approximation

We associated water particles, in a system of two-dimensional Cartesian
coordinates $(x,z)$ with origin $0$, where $z$ is the upward vertical
direction. We let $z=0$ at the water–air interface. Hence, the positive values
of $z$, $z\in~{}]0,\infty[$, correspond to the (unperturbed) air domain, while
negative values of $z$, $z\in[-h,0]$ correspond to the (unperturbed) water
domain. Consequently, for the bottom of the water domain of depth $h$, we
obtain $z=-h$. The bottom is considered to be impermeable, and both water and
air, are taken to be inviscid and incompressible. Moreover, the surface
tension effects, at the interface, are not taken into account. The governing
equations are the well known Euler equations, with the mass conservation
equations used in $(x,z,t)$ frame, where $t$ accounts for the time. Namely,

$\displaystyle u_{x}+w_{z}=0,$ (4.1a)
$\displaystyle\rho_{w}(u_{t}+uu_{x}+wu_{z})=-P_{x},$ (4.1b)
$\displaystyle\rho_{w}(w_{t}+uw_{x}+ww_{z})=-P_{z}-g\rho_{w},$ (4.1c)

where $u(x,z,t)$ and $w(x,z,t)$ are the fluid’s horizontal and vertical
velocities, respectively. $P(x,z,t)$ is the Archimedean pressure, $g$ the
gravitational acceleration, $\rho_{w}$ is the water density and subscripts
$x$, $z$ and $t$ denote partial derivatives with respect to $x$, $z$ and $t$,
respectively. The set of Equations (4.1) are completed by the following
boundary conditions:

$\displaystyle w=0,\quad\text{at}\quad z=-h,$ (4.2a)
$\displaystyle\eta_{t}+u\eta_{x}-w=0,\quad\text{at}\quad z=\eta,$ (4.2b)
$\displaystyle P=P_{a},\quad\text{at}\quad z=\eta,$ (4.2c)

where $P_{a}(x,z,t)$ is the air pressure, and Equation (4.2c) expresses the
pressure’s continuity across the air/water interface. Notice that
$z=\eta(x,t)$ is the perturbed water surface. For convenience, we introduced a
reduced pressure $\mathbf{P}$, such that
$\mathbf{P}(x,z,t)=P(x,z,t)+\rho_{w}gz-P_{0}$, where $P_{0}$ denotes the
atmospheric pressure. Using the reduced pressure $\mathbf{P}(x,z,t)$, the set
of Equations (4.1) can be written as follows:

$\displaystyle\rho_{w}(u_{t}+uu_{x}+wu_{z})=-\mathbf{P}_{x},$ (4.3a)
$\displaystyle\rho_{w}(w_{t}+uw_{x}+ww_{z})=-\mathbf{P}_{z},$ (4.3b)
$\displaystyle\mathbf{P}(x,\eta,t)-\rho_{w}g\eta+P_{0}=P_{a}(x,\eta,t).$
(4.3c)

Shallow water model equations, such as the Korteweg-Vries, modified Korteweg-
Vries, and Boussinesq equations and many others, are usually derived by
performing an asymptotic analysis directly from the equations of continuity
(4.1a), the motion Equations (3a,b) and the boundary conditions (2a,b) and
(4.3c) (see [9] and references therein). In this work, our approach was
somewhat different, in the sense that instead of applying a perturbation
theory to the entire problem, we first considered the nonlinear evolution of a
given velocity field profile. Indeed, we assumed the horizontal velocity
$u(x,z,t)$ to be independent of $z$, i.e.,

$u=u(x,t).$ (4.4)

The choice of Equation (4.4), is known as the columnar flow hypothesis, which
was introduced by [41], and [42]. Using Equations (4.1a), (4.2a) and (4.4) we
obtain

$w(x,z,t)=-(z+h)u_{x}(x,t).$ (4.5)

Hence, Equations (4.2b) and (3a,b) read

$\displaystyle\eta_{t}+[u(\eta+h)]_{x}=0,$ (4.6a)
$\displaystyle\rho_{w}(u_{t}+uu_{x})=-\mathbf{P}_{x},$ (4.6b)
$\displaystyle\rho_{w}(z+h)(u_{xt}+uu_{xx}-u^{2}_{x})=\mathbf{P}_{z}.$ (4.6c)

The integration of Equation (4.6c), using Equation (4.3c), yields the pressure
$\mathbf{P}(x,z,t)$ :

$\displaystyle\mathbf{P}(x,z,t)=\frac{1}{2}\rho_{w}[(z+h)^{2}-(\eta+h)^{2}](u_{xt}+uu_{xx}-u^{2}_{x})+P_{a}(x,\eta,t)+\rho_{w}g\eta-
P_{0}.$ (4.7)

The next step consists of substituting Equation (4.7) in Equation (4.6b), and
taking the $z$-average of Equation (4.6b) for $-h\leq z\leq\eta$. Finally,
using Equation (4.3c) we obtained the following system:

$\displaystyle\eta_{t}+[u(\eta+h)]_{x}=0,$ (4.8a) $\displaystyle
u_{t}+uu_{x}+g\eta_{x}-\frac{1}{3(\eta+h)}\\{(\eta+h)^{3}(u_{xt}+uu_{xx}-u^{2}_{x})\\}_{x}=-\frac{1}{\rho_{w}}[P_{a}(x,\eta,t]_{x}.$
(4.8b)

If $P_{a}=P_{0}$, Equations (8a,b) are reduced to the usual Serre–Green–Naghdi
equations. However, in our approach, $P_{a}$ was not taken as equal to
$P_{0}$, and the expression of $P_{a}(x,z,t)$ was found using the sheltering
mechanism.

###  4.2 Jeffreys’ Sheltering Mechanism of Wind Waves Generation Applied to
Serre-Green-Nagdhi Equations

The physical sheltering mechanism assumes that the energy transfer is caused
by pressure drag (also known as “form drag”). The air pressure on the windward
face of the wave is larger than the leeward face, which is the origin of
continuous energy transfer from wind to wave. Using dimensional arguments,
Jeffreys [2, 3] showed that the air pressure perturbation $P_{a}(x,z,t)$
evaluated on the surface can be represented by

$P_{a}(x,z=\eta,t)=\rho_{a}\epsilon(U_{10}-c)^{2}\eta_{x}(x,t),$ (4.9)

where $\epsilon$ is the sheltering coefficient, $\rho_{a}$ is the air density
and $U_{10}$ is the wind velocity at a 10 m height. The sheltering coefficient
is less than $1$ $(\epsilon<1)$. By substituting Equation (4.9) in Equation
(4.8b), we obtained

$\displaystyle\eta_{t}+[u(\eta+h)]_{x}=0,$ (4.10a) $\displaystyle
u_{t}+uu_{x}+g\eta_{x}-\frac{1}{3(\eta+h)}\\{(\eta+h)^{3}(u_{xt}+uu_{xx}-u^{2}_{x})\\}_{x}=-\epsilon
s\Delta^{2}\eta_{xx},$ (4.10b)

where $s=\frac{\rho_{a}}{\rho_{w}}\sim 10^{-3}$ and $\Delta=(U_{10}-c).$ Thus,
Equation (4.10) constitutes the fully nonlinear Serre–Green–Naghdi system
describing surface wave propagation in shallow water under the action of the
wind sheltering mechanism.

For convenience, we introduced new variables $S(x,t)$, $U(x,t)$ and $\alpha$,
defined as follows:

$\displaystyle S(x,t)=\eta(x,t)+h,$ (4.11a) $\displaystyle U(x,t)=u(x,t),$
(4.11b) $\displaystyle\alpha=\epsilon\Delta^{2}.$ (4.11c)

Using the variables (4.11), the system of Equation (10a,b) can be written as
follows:

$\displaystyle S_{t}+U_{x}S+US_{x}=0,$ (4.12a) $\displaystyle
U_{t}+UU_{x}+gS_{x}-\dfrac{1}{3S}\bigg{\\{}S^{3}\big{[}U_{xt}+UU_{xx}-(U_{x})^{2}\big{]}\bigg{\\}}_{x}=-\alpha
sS_{xx}.$ (4.12b)

Considering the following frame $\sigma$, and the slow time $\tau$:

$\displaystyle\sigma=x-vt,$ (4.13a) $\displaystyle\tau=st,$ (4.13b)

and applying Leibniz’s chain rules by considering the change of variables
(4.13), the $x$ and $t$ derivatives can be expressed as follows:

$\partial_{x}=\partial_{\sigma},\quad\partial_{t}=-v\partial_{\sigma}+s\partial_{\tau},\quad\partial^{2}_{xt}=-v\partial^{2}_{\sigma\sigma}+s\partial^{2}_{\tau\sigma},\quad\partial_{xx}=\partial_{\sigma\sigma},\quad\partial_{xxx}=\partial_{\sigma\sigma\sigma}.$
(4.14)

With the parameter $s$ being small ($s\sim 10^{-3}$), Equation (12a,b) can be
expanded in terms of $s$, as follows:

$\displaystyle U=U_{0}+sU_{1}+O(s^{2}),$ (4.15a) $\displaystyle
S=S_{0}+sS_{1}+O(s^{2}).$ (4.15b)

Notice that the expansion should not be continued beyond the first order,
because Equation (4.12b) is an $s^{1}$-order equation. Using the partial
derivate chain rules (4.14) in the SGN Equation (4.12) yields the following:

At order 0

$\displaystyle-vS_{0,\sigma}+U_{0,\sigma}S_{0}+U_{0}S_{0,\sigma}=0,$ (4.16a)
$\displaystyle-
vU_{0,\sigma}+U_{0}U_{0,\sigma}+gS_{0,\sigma}+S_{0}S_{0,\sigma}\big{[}vU_{0,\sigma\sigma}-U_{0}U_{0,\sigma\sigma}+(U_{0,\sigma})^{2}\big{]}$
$\displaystyle\hskip
113.81102pt+\dfrac{1}{3}(S_{0})^{2}\big{[}vU_{0,\sigma\sigma\sigma}-U_{0}U_{0,\sigma\sigma\sigma}+U_{0,\sigma}U_{0,\sigma\sigma}\big{]}=0.$
(4.16b)

At order 1

$\displaystyle-
vS_{1,\sigma}+U_{0,\sigma}S_{1}+U_{1,\sigma}S_{0}+U_{0}S_{1,\sigma}+U_{1}S_{0,\sigma}=-S_{0,\tau},$
(4.17a) $\displaystyle-
vU_{1,\sigma}+U_{0}U_{1,\sigma}+U_{1}U_{0,\sigma}+gS_{1,\sigma}+(S_{0}S_{1,\sigma}+S_{1}S_{0,\sigma})\left[vU_{0,\sigma\sigma}+(U_{0,\sigma})^{2}-U_{0}U_{0,\sigma\sigma}\right]$
$\displaystyle+S_{0}S_{0,\sigma}\left[vU_{1,\sigma\sigma}-U_{0}U_{1,\sigma\sigma}-U_{1}U_{0,\sigma\sigma}+2U_{0,\sigma}U_{1,\sigma}\right]$
$\displaystyle+\dfrac{1}{3}(S_{0})^{2}\big{[}vU_{1,\sigma\sigma\sigma}-U_{0}U_{1,\sigma\sigma\sigma}-U_{1}U_{0,\sigma\sigma\sigma}+U_{0,\sigma}U_{1,\sigma\sigma}+U_{1,\sigma}U_{0,\sigma\sigma}\big{]}$
$\displaystyle+\dfrac{2}{3}S_{1}S_{0}\left[vU_{0,\sigma\sigma\sigma}+U_{0,\sigma}U_{0,\sigma\sigma}-U_{0}U_{0,\sigma\sigma\sigma}\right]=-\alpha
S_{0,\sigma\sigma}-U_{0,\tau}-\dfrac{1}{3}(S_{0})^{2}U_{0,\tau\sigma\sigma}+S_{0}S_{0,\sigma}U_{0,\tau\sigma}.$
(4.17b)

The set of Equations (4.16) and (4.17) can be reformulated in a matrix.
Indeed, Equation (4.16) can be written equivalently as follows:

$\left({\begin{array}[]{cc}\widehat{A}_{0}&\widehat{B}_{0}\\\
\widehat{C}_{0}&\widehat{D}_{0}\\\
\end{array}}\right)\left({\begin{array}[]{cc}U_{0}\\\ S_{0}\\\
\end{array}}\right)=\left({\begin{array}[]{cc}0\\\ 0\\\ \end{array}}\right),$
(4.18)

where

$\displaystyle\widehat{A}_{0}=S_{0}\partial_{\sigma},$ (4.19a)
$\displaystyle\widehat{B}_{0}=-v\partial_{\sigma}+U_{0}\partial_{\sigma},$
(4.19b)
$\displaystyle\widehat{C}_{0}=-v\partial_{\sigma}+U_{0}\partial_{\sigma}+\dfrac{1}{3}(S_{0})^{2}\left[v\partial_{\sigma\sigma\sigma}-U_{0}\partial_{\sigma\sigma\sigma}+U_{0,\sigma}\partial_{\sigma\sigma}\right],$
(4.19c)
$\displaystyle\widehat{D}_{0}=g\partial_{\sigma}+\big{[}vU_{0,\sigma\sigma}-U_{0}U_{0,\sigma\sigma}+(U_{0,\sigma})^{2}\big{]}S_{0}\partial_{\sigma}.$
(4.19d)

Similarly, Equation (4.17) can be expressed as follows:

$\left({\begin{array}[]{cc}\widehat{A}_{1}&\widehat{B}_{1}\\\
\widehat{C}_{1}&\widehat{D}_{1}\\\
\end{array}}\right)\left({\begin{array}[]{cc}U_{1}\\\ S_{1}\\\
\end{array}}\right)=\left({\begin{array}[]{cc}E_{1}\\\ E_{2}\\\
\end{array}}\right),$ (4.20)

where

$\displaystyle\widehat{A}_{1}=$ $\displaystyle
S_{0}\partial_{\sigma}+S_{0,\sigma},$ (4.21a) $\displaystyle\widehat{B}_{1}=$
$\displaystyle-v\partial_{\sigma}+U_{0}\partial_{\sigma}+U_{0,\sigma},$
(4.21b) $\displaystyle\widehat{C}_{1}=$
$\displaystyle-v\partial_{\sigma}+U_{0}\partial_{\sigma}+U_{0,\sigma}+S_{0}S_{0,\sigma}\left[v\partial_{\sigma\sigma}-U_{0}\partial_{\sigma\sigma}-U_{0,\sigma\sigma}+2U_{0,\sigma}\partial_{\sigma}\right]$
$\displaystyle+\dfrac{1}{3}(S_{0})^{2}\left[v\partial_{\sigma\sigma\sigma}-U_{0}\partial_{\sigma\sigma\sigma}-U_{0,\sigma\sigma\sigma}+U_{0,\sigma}\partial_{\sigma\sigma}+U_{0,\sigma\sigma}\partial_{\sigma}\right],$
(4.21c) $\displaystyle\widehat{D}_{1}=$ $\displaystyle
g\partial_{\sigma}+\big{[}vU_{0,\sigma\sigma}-U_{0}U_{0,\sigma\sigma}+(U_{0,\sigma})^{2}\big{]}(S_{0}\partial_{\sigma}+S_{0,\sigma})$
$\displaystyle+\dfrac{2}{3}S_{0}\left[vU_{0,\sigma\sigma\sigma}-U_{0}U_{0,\sigma\sigma\sigma}+U_{0,\sigma}U_{0,\sigma\sigma}\right],$
(4.21d) $\displaystyle E_{1}=$ $\displaystyle-S_{0,\tau},$ (4.21e)
$\displaystyle E_{2}=$ $\displaystyle-\alpha
S_{0,\sigma\sigma}-U_{0,\tau}+\dfrac{1}{3}(S_{0})^{2}U_{0,\tau\sigma\sigma}+S_{0}S_{0,\sigma}U_{0,\tau\sigma}.$
(4.21f)

###  4.3 Application of Green’s Theorem in One Dimension

Green’s theorem in one dimension has been proved and applied to linear
differential operators ([43, 44, 45]). Here, by extension, we applied this
theorem to matrix differential operators. To do so, we briefly recalled the
following theorem:

$\int_{a}^{b}\left[zP(y)-y\bar{P}(z)\right]dx=\left[P(y,z)\right]_{a}^{b},$
(4.22)

where $P$ is a linear differential operator and $y$ and $z$, any two functions
of $x$ and, $\bar{P}(z)$ and $P(y,z)$, are the adjoint and the bilinear
differential expressions of $P(y)$, respectively [46]. This theorem, in its
usual form, as it is shown in (4.22), was previously used to show the damping
of solitary waves [48, 47].

In our case, we considered operators $\widehat{L}_{0}$ and $\widehat{L}_{1}$,
defined as follows:

$\widehat{L}_{0}=\left({\begin{array}[]{cc}\widehat{A}_{0}&\widehat{B}_{0}\\\
\widehat{C}_{0}&\widehat{D}_{0}\\\
\end{array}}\right),\qquad\widehat{L}_{1}=\left({\begin{array}[]{cc}\widehat{A}_{1}&\widehat{B}_{1}\\\
\widehat{C}_{1}&\widehat{D}_{1}\\\ \end{array}}\right).$

Using $\widehat{L}_{0}$ and $\widehat{L}_{1}$, Equations (4.18) and (4.20)
become

$\widehat{L}_{0}V_{0}=0\quad\text{and}\quad\widehat{L}_{1}V_{1}=E,$ (4.23)

where

$V_{0}=\left({\begin{array}[]{cc}U_{0}\\\ S_{0}\\\ \end{array}}\right),\qquad
V_{1}=\left({\begin{array}[]{cc}U_{1}\\\ S_{1}\\\ \end{array}}\right),\qquad
E=\left({\begin{array}[]{cc}E_{1}\\\ E_{2}\\\ \end{array}}\right).$

Taking into account the symmetric behaviour of $S(x,t)$ and $U(x;t)$ at
$x=\pm\infty$, the extension of Green’s theorem in one dimension to linear
differential matrix operator $\widehat{L}_{1}$ yields

$\int_{-\infty}^{+\infty}\left(V^{\dagger}_{0}\widehat{L}_{1}V_{1}-V^{\dagger}_{1}\widehat{L}_{1}V_{0}\right)d\sigma=0,$
(4.24)

where $V^{\dagger}_{0}$ and $V^{\dagger}_{1}$ are $V_{0}$ and $V_{1}$
transposed, respectively. This extension can easily be proved following the
procedure proposed in the original work of [43].

###  4.4 Blow-Up in Finite Time of the Serre-Green-Naghdi Soliton Solution

Using Equations (4.23) and (4.24), we have:

$\int_{-\infty}^{+\infty}\bigg{[}-U_{0}S_{0,\tau}+S_{0}\left(-\alpha
S_{0,\sigma\sigma}-U_{0,\tau}+\dfrac{1}{3}(S_{0})^{2}U_{0,\tau\sigma\sigma}+S_{0}S_{0,\sigma}U_{0,\tau\sigma}\right)\bigg{]}d\sigma=0,$
(4.25)

which can be written as follows:

$-\int_{-\infty}^{+\infty}\bigg{[}\dfrac{\partial}{\partial\tau}\left(U_{0}S_{0,}\right)\bigg{]}d\sigma+\alpha\int_{-\infty}^{+\infty}(S_{0,\sigma})^{2}d\sigma-\alpha\bigg{[}S_{0}S_{0,\sigma}\bigg{]}_{-\infty}^{+\infty}+\bigg{[}\dfrac{1}{3}(S_{0})^{3}U_{0,\tau\sigma}\bigg{]}_{-\infty}^{+\infty}=0,$
(4.26)

where $U_{0}$ and $S_{0}$ are the unperturbed solutions with a time-dependent
amplitude, namely,

$\displaystyle U_{0}=$ $\displaystyle
c_{0}\left(1+\frac{a(\tau)}{h}\right)^{1/2}\left(1-\frac{h}{S_{0}}\right),$
(4.27) $\displaystyle S_{0}=$
$\displaystyle\frac{a(\tau)}{[\cosh(\beta)]^{2}},\qquad\beta=\sqrt{\frac{3}{4}}\left(\frac{1}{h}\right)\left(\frac{a(\tau)}{a(\tau)+h}\right)^{1/2}\left[x-c_{0}t\left(1+\frac{a(\tau)}{h}\right)^{1/2}\right].$
(4.28)

After inserting (4.28) and (4.27) into Equation (4.26), alongside Equation
(4.13) it can be noticed that the limit of
$\displaystyle{\frac{1}{\cosh(\sigma)}}$ tends to zero, while
$\sigma\to\pm\infty$. The Equation (4.26) yields the following:

$\int_{a(0)}^{a(\tau)}\frac{(a+h)^{1/2}(2a+h)}{a^{3}}da=\frac{4\alpha}{5c_{0}}\left(\frac{1}{h}\right)^{3/2}\tau.$
(4.29)

Using the variable $r$, Equation (4.29) yields:

$\int_{r(0)}^{r(\tau)}\frac{(1+r)^{1/2}(1+2r)}{r^{3}}dr=\frac{4\alpha}{5c_{0}h}\tau.$
(4.30)

The series expansion for a small value of $r$ in the Equation yields:

$\int_{r(0)}^{r(\tau)}\left(\frac{1}{r^{3}}+\frac{5}{2r^{2}}+\frac{7}{8r}-\frac{3}{16}+o(r)\right)dr=\frac{4\alpha}{5c_{0}h}\tau.$
(4.31)

After integration, Equation (4.31) becomes

$\left[-\frac{3r}{16}+\frac{7\ln(r)}{8}-\frac{5}{2r}-\frac{1}{2r^{2}}+o(r^{2})\right]_{r(0)}^{r(\tau)}=\frac{4\alpha}{5c_{0}h}\tau,$
(4.32)

and by keeping the leading term for $r$ small, Equation (4.32) becomes

$\left[-\frac{1}{2r^{2}}\right]_{r(0)}^{r(\tau)}=\frac{4\alpha}{5c_{0}h}\tau.$
(4.33)

Hence, the time dependent amplitude reads as follows:

$a(\tau)=\frac{A_{0}}{\sqrt{1-\displaystyle{\frac{8\alpha
A^{2}_{0}\tau}{5c_{0}h^{3}}}}},$ (4.34)

where $A_{0}=a(0)$. From Equation (4.34), it can be seen that the amplitude
$a(\tau)$ tends to infinity, when the slow time $\tau$ approaches a certain
value $\tau_{b}$, which we call the “slow” blow-up time. Replacing
$\alpha=\epsilon\Delta^{2}$, and $\tau=st$, the blow-up time can be written as
follows:

$t_{b}=\frac{5c_{0}h^{3}}{8\epsilon\Delta^{2}A^{2}_{0}s},$ (4.35)

where $\Delta^{2}=\left(U_{10}-C_{GN}\right)^{2}$, and the dispersion relation
of the SGN solitary wave $C_{GN}$, can be written [49, 10] as follows:

$C_{GN}=\frac{c_{0}}{\sqrt{1+\frac{1}{3}(kh)^{2}}}.$ (4.36)

###  4.5 Blow-up Time’s Effective evaluation

In order to effectively evaluate the blow-up time $t_{b}$ and the growth rate
of wind waves in finite depth, we used detailed measurements of shallow water
parameters in finite depth experiments conducted in the IRPHÉ/Pythéas wind-
wave tank [50]. These measurements were carried out for non-dimensional depth
$kh$ and non-dimensional initial waves’ peak value $kA_{0}$. This led us to
consider the non-dimensional soliton solution $kS_{0}$, instead of $S_{0}$, as
well as the non-dimensional amplitude $ka$ that we denoted as a function of
non dimensional time $\overline{t}$, as follows:

$ka(\overline{t})=\frac{kA_{0}}{\sqrt{1-\displaystyle{\frac{8\epsilon(kA_{0})^{2}s}{5(kh)^{2}}}\hskip
3.0pt\overline{t}}},$ (4.37)

where

$\overline{t}=\frac{\Delta^{2}}{c_{0}h}t.$ (4.38)

Notice that the values of $c_{0}$ as well as $U_{10}$ were also measured
experimentally. For this reason, in what follows, the values of $c_{0}$ differ
slightly from the theoretical values $c_{0}=\sqrt{gh}$.

Using the experimental data of IRPHÉ/Pythéas facilities (Branger et al. [50]),
for

$\displaystyle h=0.14\hskip 3.0pt\text{m},\quad kh=1.54,\quad
kA_{0}=0.114,\quad c_{0}=0.92\hskip 3.0pt\text{m/s},\quad U_{10}=4.82\hskip
3.0pt\text{m/s},$ (4.39)

for the sheltering coefficient, we used $\epsilon=0.5$, and for the small
parameter we used $s=0.001$. Using experimental data (4.39) and Equations
(4.35) and (4.36) assisted in calculating the value of the blow-up time:

$t_{b}\approx 1721\hskip 4.0pt\text{s}.$ (4.40)

The $x$-position of the SGN soliton solution as a function of time is found
using

$x(t)=c_{0}t\left(1+\frac{a(\tau)}{h}\right)^{1/2}.$ (4.41)

The length of the IRPHÉ/Pythéas wind–wave tank facility was 40 m. The growing
solitary wave reaches the tank’s end after 40–45 s. Consequently, the soliton
amplitude is not nearly that of the blow-up. The growth rate of the soliton at
different times and positions, for $h=0.14\hskip 3.0pt\text{m}$, and
$U_{10}=4.48\hskip 3.0pt\text{m/s}$, is given in (Table 1). In these
conditions, when the wave reaches the tunnel’s end, the growth rate is
approximately $0.1$. Hence, it is at the measurability limit of the
IRPHÉ/Pythéas wind–wave tank facility.

Table 1: The growth rate of the SNG solitary wave at different times and positions, for depth $h=0.14\hskip 3.0pt\text{m}$, and wind speed of 10 m $U_{10}=4.82\hskip 3.0pt\text{m/s}.$ _t_ (s) | 0 |  40 |  … |  750 |  1000 |  1250 |  1400
---|---|---|---|---|---|---|---
_x_ (m) | 0 |  35 |  … |  722 |  972 |  1229 |  1329
growth rate |  |  0.09 |  … |  0.15 |  0.55 |  1 |  1.36

And for

$\displaystyle h=0.26\hskip 3.0pt\text{m},\quad kh=2.57,\quad
kA_{0}=0.146,\quad c_{0}=1.0\hskip 3.0pt\text{m/s},\quad U_{10}=4.35\hskip
3.0pt\text{m/s},$ (4.42)

with a sheltering coefficient $\epsilon=0.5$ and small parameter $s=0.001$.
Similarly, experimental data (4.42) and Equations (4.35) and (4.36) led to the
corresponding blow-up time:

$t_{b}\approx 7008\hskip 4.0pt\text{s}.$ (4.43)

The continuous growth of ($kS_{0}$), leads to blow-up at finite time,
$t_{b}\approx$ 7008 s, which of course, is out of reach. Therefore, in this
case, a significant growth in the soliton’s amplitude was not observable in
experimental facilities, but it could be in situ. The growth rate of the
soliton at different times and positions, for $h=0.26$ m, and
$U_{10}=4.35\hskip 3.0pt\text{m/s}$, is given in (Table 2).

Table 2: The growth rate of the SNG solitary wave at different times and positions, for depth $h=0.26\hskip 3.0pt\text{m}$, and wind speed at 10 m $U_{10}=4.35\hskip 3.0pt\text{m/s}.$ t (s) | 0 |  40 |  … |  750 |  1000 |  1250 |  15,000
---|---|---|---|---|---|---|---
x (m) | 0 |  41 |  … |  772 |  1030 |  1288 |  1547
growth rate |  |  0.02 |  … |  0.06 |  0.08 |  0.1 |  0.12

##  5 Korteweg-de Vries-Burger equation

The Jeffreys’ theory allows to compute the linear wave growth of wind-
generated normal Fourier modes of wave-number $k$. The physical mechanism
behind is anti-dissipation. Energy passes continuously from the air to the
surface wave. Consequently the wave amplitude $\eta(x,t,k)$ ($x$ space and $t$
time) grows exponentially in time i.e.; $\eta(x,t,k)\sim\exp{(\gamma_{J}t)}$
more or less quickly according to the coefficient $\gamma_{J}$, which depends
on the wind speed and the water depth $h$. Once the linear and dispersionless
approximation breaks down, non-linear and dispersive processes begin to play a
role. So the issue addressed here is: how to describe the evolution in time of
a normal mode $k$, under the competing actions of (weak) nonlinearity,
dispersion and anti-dissipation? Nonlinearity is likely to balance dispersive
effects, or to stop exponential decay or growth of wave amplitude in time due
to dissipation or anti-dissipation. Equilibrium between nonlinearity and
dispersion can evolve in time to form solitary waves as in the Korteweg-de
Vries equation [51, 52]. Balance between dissipation or anti-dissipation and
nonlinearity creates shock structures as in the Burgers equation [51]. The
standard equation describing competition between weak non-linearity,
dispersion and dissipation is the Korteweg-de Vries-Burgers equation (KdV-B).
It appears in many physical contexts [53, 54, 55, 56, 57, 58]. In this
section, in order to study simultaneous competing effects of weakly
nonlinearity, dispersion and anti-dissipation we derive a KdV-B type equation
with dissipation turned into anti-dissipation.

Green-Naghdi equations under the wind action [59, 49, 10] are given by Eqs
(4.10a) and (4.10b). We introduce dimensionless "primed" variables,
$x^{\prime}$, $t^{\prime}$ and $\eta^{\prime}$ as follows:

$x=\lambda_{0}x^{\prime},\qquad
t=\frac{\lambda_{0}}{c_{0}}t^{\prime},\qquad\eta=a_{0}\eta^{\prime},$ (5.1)

where $c_{0}=(gh)^{1/2}$, $a_{0}$ and $\lambda_{0}$ are initial typical wave
amplitude and wavelength, respectively. In addition, we define two fundamental
parameters, commonly used in the classical water surface studies, namely,
$\nu$ and $\delta$, as follows:

$\nu=\frac{a_{0}}{h},\qquad\delta=\frac{h}{\lambda_{0}}.$ (5.2)

Finally, in order to obtain the dimensionless, scaled Green-Naghdi equations
of motion, the following scaling is required [54]:

$u=\nu u_{0}.$ (5.3)

Introducing Eqs (5.1), (5.2) and (5.3) in Eqs (4.10b) and (4.10a), we obtain
the following dimensionless equations

$\displaystyle u_{0,t}+\nu u_{0}u_{0,x}+\eta_{x}=-\epsilon
s\Delta^{2}\delta\eta_{xx}+\frac{\delta^{2}}{3(1+\nu\eta)}\\{(1+\nu\eta)^{3}(u_{0,xt}+\nu
u_{0}u_{0,xx}-\nu u^{2}_{0,x})\\}_{x},$ (5.4a)
$\displaystyle\eta_{t}+[u_{0}(1+\nu\eta)]_{x}=0,$ (5.4b)

where, for convenience, the "primes" of dimensionless quantities are omitted.
Now, we consider a wave moving from left to right [51]. At the lowest order,
by neglecting the terms of order $\nu$ and $\delta$ and any higher orders, Eqs
(5.4a) and (5.4b) are reduced to

$u_{0,t}+\eta_{x}=0,$ (5.5a) $\eta_{t}+u_{0,x}=0.$ (5.5b)

Eqs (5.5a) and (5.5b) are equivalent to

$\eta_{t}+\eta_{x}=u_{0,t}+u_{0,x}=0,$ (5.6)

and its solution is

$u_{0,x}(x,t)=\eta(x,t).$ (5.7)

Now we look for a perturbed solution with follow form:

$u_{0}=\eta+\nu\mathbb{A}+\delta\mathbb{B}+\delta^{2}\mathbb{C}+O(\delta\nu,\delta^{2}\nu,\nu^{2}),$
(5.8)

where $\mathbb{A}$, $\mathbb{B}$ and $\mathbb{C}$ are functions of $\eta$ and
its derivatives. Inserting Eq. (5.8) in (5.4a) and (5.4b), we obtain

$\displaystyle\eta_{t}+\eta_{x}+\nu(\mathbb{A}_{t}+\eta\eta_{x})+\delta(\mathbb{B}_{t}+\epsilon
s\Delta^{2}\eta_{xx})+\delta^{2}(\mathbb{C}_{t}-\frac{1}{3}\eta_{xxt})+O(\delta\nu,\delta^{2}\nu,\nu^{2})=0,$
(5.9a)
$\displaystyle\eta_{t}+\eta_{x}+\nu(\mathbb{A}_{x}+\eta\eta_{x})+\delta(\mathbb{B}_{x}+\delta^{2}\mathbb{C}_{x}+O(\delta\nu,\delta^{2}\nu,\nu^{2})=0,$
(5.9b)

where

$\eta_{t}=-\eta_{x}+O(\delta\nu,\delta^{2}\nu,\nu^{2}).$ (5.10)

Therefore, in Eq. (5.9a), all the $t$-derivatives may be substituted by
$-\partial_{x}$. Hence, Eqs (5.9a) and (5.9b) are compatible, if

$\mathbb{A}=-\frac{1}{4}\eta^{2},\qquad\mathbb{B}=\frac{1}{2}s\epsilon\Delta^{2}\eta_{x},\qquad\mathbb{C}=\frac{1}{6}\eta_{xx}.$
(5.11)

Substituting Eq. (5.11) in Eqs (5.9a) and (5.9b), yields

$\displaystyle\eta_{t}+\eta_{x}+\frac{3}{2}\nu\eta\eta_{x}+\frac{\delta^{2}}{6}\eta_{xxx}+s\epsilon\frac{\delta}{2}\Delta^{2}\eta_{xx}=0,$
(5.12a) $\displaystyle
u-\eta+\frac{\nu}{4}\eta^{2}-s\epsilon\frac{\delta}{2}\Delta^{2}\eta_{x}-\frac{1}{6}\delta^{2}\eta_{xx}=0.$
(5.12b)

Eq. (5.12a) is the KdV-B equation, while Eq. (5.12b) is a Riemann invariant.

###  5.1 Solution of Korteweg-de Vries-Burger equation

In order to find the solution of Eq.(5.12a), we apply the following change of
variables:

$\sigma=x-t,\qquad t_{1}=\delta^{2}t.$ (5.13)

Hence, Eq.(5.12a) becomes:

$\eta_{t_{1}}+\frac{3}{2}\eta\eta_{\sigma}+\frac{\delta^{2}}{6\nu}\Delta^{2}\eta_{\sigma\sigma\sigma}+\frac{\delta}{2\nu}\Delta^{2}s\eta_{\sigma\sigma}=0.$
(5.14)

It worth noticing that the limit of Eq. (5.14), as $s\to 0$, yields to the
well known KdV equation:

$\eta_{t_{1}}+\frac{3}{2}\eta\eta_{\sigma}+\frac{\delta^{2}}{6\nu}\Delta^{2}\eta_{\sigma\sigma\sigma}=0.$
(5.15)

This result is quite natural, since $s\to 0$, amounts to neglecting the action
of the wind. Therefore, it is possible to assume that the solution KdV-B, Eq.
(5.14), has the same form as the solution of Eq. (5.15), namely

$\eta(\sigma,t_{1})=\frac{a}{\cosh^{2}\left[P(\sigma-c^{\prime}t_{1})\right]}.$
(5.16)

This is the typical soliton solution of KdV equation, with one difference
however, the amplitude $a$ in Eq. (5.16) can be time-dependant, whereas the
amplitude of KdV equation is not.

Inserting (5.16) in (5.14), we obtain

$P=\sqrt{\frac{3\nu}{4\delta^{2}}},\qquad c^{\prime}=\frac{a}{2}.$ (5.17)

Now, the task is to find the time-dependent expression of $a(t)$ in Eq.
(5.16).

Noticing that the anti diffusive term
$\displaystyle{\frac{\delta}{2\nu}\Delta^{2}s\eta_{\sigma\sigma}}$ in Eq.
(5.14), is of order $\delta^{3}$ and small enough at $t=0$, one can find the
solution of Eq. (5.14) by perturbation. For this purpose, we introduce a slow
time $t_{2}$, as follows

$t_{2}=\delta^{2}t_{1}=\delta^{3}t,$ (5.18)

and we expand $\eta$ in terms of $\delta$ as follows

$\eta=\eta_{0}(\sigma,t_{2})+\delta\eta_{1}(\sigma,t_{2})+O(\delta^{2}),$
(5.19)

where $\eta_{0}$ is the solution given by Eq. (5.16).

Introducing $\nu_{0}$ as follows

$\nu=\nu_{0}\delta^{2},$ (5.20)

and inserting Eq. (5.19) in Eq. (5.14), we obtain

at order 0 of $\delta$: $\frac{\partial\eta_{0}}{\partial
t_{1}}+\frac{3}{4}\nu_{0}\eta_{0}\frac{\partial\eta_{0}}{\partial\sigma}+\frac{1}{6}\frac{\partial^{3}\eta_{0}}{\partial\sigma^{3}}=0,$
(5.21a) and, at order 1 of $\delta$: $\frac{\partial\eta_{1}}{\partial
t_{1}}+\frac{3}{2}\nu_{0}\eta_{0}\frac{\partial\eta_{1}}{\partial\sigma}+\frac{3}{2}\nu_{0}\eta_{0,\sigma}\eta_{1}+\frac{1}{4}\nu_{0}\frac{\partial^{3}\eta_{1}}{\partial\sigma^{3}}=-\eta_{0,t_{2}}-\frac{s_{0}}{2}\epsilon\Delta^{2}_{0}\eta_{0,\sigma\sigma}.$
(5.21b)

Introducing operators $\widehat{L}_{0}$ and $\widehat{L}_{1}$ as follows

$\widehat{L}_{0}=\frac{\partial}{\partial
t_{1}}+\frac{3}{4}\nu_{0}\eta_{0}\frac{\partial}{\partial\sigma}+\frac{1}{6}\frac{\partial^{3}}{\partial\sigma^{3}},$
(5.22a) $\widehat{L}_{1}=\frac{\partial}{\partial
t_{1}}+\frac{3}{2}\nu_{0}\left(\eta_{0}\frac{\partial}{\partial\sigma}+\eta_{0,\sigma}\right)+\frac{1}{4}\nu_{0}\frac{\partial^{3}}{\partial\sigma^{3}}.$
(5.22b)

Eqs (5.22a) and (5.22b) read

$\widehat{L}_{0}\eta_{0}=0,$ (5.23a)
$\widehat{L}_{1}\eta_{1}=-\eta_{0,t_{2}}-\frac{s_{0}}{2}\epsilon\Delta^{2}_{0}\eta_{0,\sigma\sigma}.$
(5.23b)

To go further, we apply Green’s theorem in one dimension. One can find the
application of this theorem to linear differential operators in various works
[43, 44, 45]. In particular, the damping of solitary waves [48, 47] has been
shown using this theorem and an extension to matrix differential operators has
been performed by [8].

Applying Green’s theorem to our case by replacing Eqs (5.22a) and (5.22b) in
Eq. (4.22), yields

$\int_{-\infty}^{+\infty}\left(\eta_{0}\widehat{L}_{1}\eta_{1}-\eta_{1}\widehat{L}_{0}\eta_{0}\right)d\sigma=0.$
(5.24)

Notice that the right hand side of (5.24) is null due to the symmetric
behaviour of $\eta_{0}$ and $\eta_{1}$ at $\pm\infty$. Replacing $\eta_{0}$ by
Eq. (5.16) with a time dependant amplitude $a(t_{2})$, Eq. (5.24) yields:

$a(t_{2})=\frac{1}{1-\frac{2}{5}\frac{s\epsilon\Delta^{2}}{\delta^{2}}t_{2}}.$
(5.25)

Using the approximation $O(\nu)=O(\delta^{2})$, Eq. (5.18) can equivalently be
written as $t_{2}=\nu\delta t$. Hence, Eq. (5.25) becomes

$a(t)=\frac{1}{1-\frac{t}{t_{b}}},$ (5.26)

where

$t_{b}=\frac{5\delta}{2\epsilon s\Delta^{2}\nu}.$ (5.27)

From Eq. (5.26), it can be seen that the amplitude $a(t)\to\infty$, when $t\to
t_{b}$ which we call the blow-up time. Hence, the solution of (5.12a) reads:

$\eta=a(t)\cosh^{-2}(\theta),$ (5.28)

where

$\theta(x,t)=\alpha
a^{1/2}\left[x-t+\frac{\nu}{2}t_{b}\ln\left(1-\frac{t}{t_{b}}\right)\right],\qquad\alpha=\left(\frac{3}{4}\frac{a_{0}}{h}\right)^{1/2}\frac{\lambda_{0}}{h},$
(5.29)

###  5.2 Blow-up in finite time and the evolution of the soltion’s shape

In this section we shall study the evolution of the soliton’s shape in time ,
Eq. (5.28), before the blow up time $t_{b}$.

Coming back to variables with dimensions, Eq. (5.1), the solution of KdV-B,
i.e. Eq. (5.28) reads:

$\eta(x,t)=\frac{a_{0}}{1-\frac{t}{t_{b}}}\cosh^{-2}\biggl{\\{}\frac{\alpha}{(1-\frac{t}{t_{b}})^{1/2}}\frac{1}{\lambda_{0}}\left[x-c_{0}t+\frac{\nu}{2}c_{0}t_{b}\ln\left(1-\frac{t}{t_{b}}\right)\right]\biggr{\\}}.$
(5.30)

For $t=0$, we have

$\eta(x,0)=a_{0}\cosh^{-2}\left(\frac{\alpha}{\lambda_{0}}x\right),$ (5.31)

where

$\alpha^{2}=\frac{3\nu}{4\delta^{2}}=\frac{3}{4}\frac{a_{0}{\lambda_{0}}^{2}}{h^{3}}.$
(5.32)

Using Eq. (5.32), the blow-up time, $t_{b}$ can also be expressed as follows
for further use:

$t_{b}=\frac{5}{2}\frac{c_{0}h^{2}}{\epsilon sa_{0}\Delta^{2}}.$ (5.33)

The wave number $k$ and the frequency $\omega$, for a monochromatic
progressive wave with a phase $\theta(x,t)=kx-\omega t$, are defined as
follows:

$k=\frac{\partial\theta}{\partial
x},\qquad\omega=-\frac{\partial\theta}{\partial t}.$ (5.34)

These definitions can be generalized for $k$ and $\omega$ depending on $x$ and
$t$:

$k(x,t)=\frac{\partial\theta}{\partial
x}(x,t),\qquad\omega(x,t)=-\frac{\partial\theta}{\partial t}(x,t).$ (5.35)

Using Eqs (5.35), (5.29) and (5.26), the wavelength $\lambda(t)$ and the wave
number $k(t)$ of the soliton solution (5.28) are

$\displaystyle\lambda(t)=\frac{\lambda_{0}}{\alpha}\left(1-\frac{t}{t_{b}}\right)=\left(\frac{4h}{3a_{0}}\right)^{1/2}h\left(1-\frac{t}{t_{b}}\right)^{1/2},$
(5.36) $\displaystyle
k(t)=\left(\frac{3a_{0}}{4h}\right)^{1/2}\frac{1}{h}\left(1-\frac{t}{t_{b}}\right)^{-1/2}=\alpha
a^{1/2}(t).$ (5.37)

We define the effective wave number, noted $\widetilde{k}$, as follows:

$\widetilde{k}=\frac{\alpha}{\lambda_{0}}=\left(\frac{3a_{0}}{4h}\right)^{1/2}\frac{1}{h},$
(5.38)

and we define the associated effective wavelength, noted
$\widetilde{\lambda}$, defined as follows:

$\widetilde{\lambda}=\frac{1}{\widetilde{k}}=\left(\frac{4h}{3a_{0}}\right)^{1/2}h.$
(5.39)

It worth noticing that for $t\to t_{b}$, Eqs (5.36), (5.37) and(5.26) give

$\lim_{t\to t_{b}}\lambda(t)=0,\qquad\lim_{t\to
t_{b}}k(t)=\infty,\qquad\lim_{t\to t_{b}}a(t)=\infty,$ (5.40)

respectively. This is the first indication of the narrowing of the soliton’s
shape while its amplitude $a(t)$ grows.

To go further, we are going to examine the speed of phase planes on either
side of the soliton’s crest. Using Eq. (5.37), it is useful to write Eq.
(5.29) as a function of $k(t)$ as follows

$\theta(x,t)=k(t)(x-t)+\nu t_{b}k(t)\left[\ln(x)-\ln\left(k(t)\right)\right].$
(5.41)

Using Eq. (5.37), we have

$\frac{\partial k(t)}{\partial t}=\frac{k^{3}(t)}{2\alpha^{2}t_{b}}.$ (5.42)

Eqs (5.35) and (5.42) together, yields $\omega$ as a function of $k(t)$:

$\omega(x,t)=-\frac{k^{3}(t)}{2\alpha^{2}t_{b}}(x-t)+k(t)-\nu
t_{b}\frac{k^{3}(t)}{2\alpha^{2}t_{b}}\left[\ln\left(\frac{\alpha}{k(t)}-1\right)\right].$
(5.43)

The dimensionless phase velocity $c(x,t)$ of the soliton solution is

$c(x,t)=\frac{\omega(x,t)}{k(t)}=1+\frac{\nu}{2\alpha^{2}}k^{2}(t)-\frac{k(t)\theta(x,t)}{2\alpha^{2}t_{b}},$
(5.44)

which, by using Eq. (5.37), can equivalently be written as a function of
$a(t)$ as follows

$c(x,t)=1+\frac{\nu}{2}a(t)-\frac{a^{1/2}(t)}{2\alpha t_{b}}\theta(x,t).$
(5.45)

The position of soliton’s crest is found by solving

$\theta(x,t)=0,$ (5.46)

at any time $t$. The dimensionless phase $\theta$ is given by the first of the
equations in Eq. (5.29), while the phase velocity $c$ is given by Eq. (5.45).
Therefore, the position of the soliton’s crest and the velocity of the soliton
at it’s crest are

$\displaystyle x_{crest}(t)=t+\frac{\nu t_{b}}{2}\ln[a(t)],$ (5.47)
$\displaystyle c(x_{crest},t)=1+\frac{\nu}{2}a(t),$ (5.48)

respectively.

Using Eqs (5.45) and (5.29), the phase speed becomes:

$\displaystyle c$
$\displaystyle=c_{crest}-\frac{\alpha}{2}\frac{a(t)}{t_{b}}\left[(x-t)-\frac{\nu}{2}t_{b}\ln\left[a(t)\right]\right]$
$\displaystyle=c_{crest}-\frac{\alpha}{2}\frac{a(t)}{t_{b}}\left[x-x_{crest}\right].$
(5.49)

Therefore, the speed of the phase planes at $(x_{crest}-\Delta x)$ and
$(x_{crest}+\Delta x)$ are

$c_{crest-\Delta x}=c_{crest}+\frac{\alpha}{2}\frac{a(t)}{t_{b}}\Delta x,$
(5.50a) $c_{crest+\Delta x}=c_{crest}-\frac{\alpha}{2}\frac{a(t)}{t_{b}}\Delta
x,$ (5.50b)

respectively. Hence,

$c_{crest-\Delta x}>c_{crest}>c_{crest+\Delta x}.$ (5.51)

This means that the phase planes at the left side of $x_{crest}$ have greater
speed than the phase planes at the right side of $x_{crest}$ resulting to a
narrowing the soliton’s shape while it’s amplitude $a(t)$ grows (Figure 1).

At the first glance, the wave breaking does not result directly from the
soliton solution of KdV-B equation. However, knowing that we have an
accelerating solitary wave at the speed $c_{crest}$, the breaking might result
from phase planes with non equal accelerations. This will be the subject of
fore coming studies.

In the following subsections, various criteria of wave breaking are reviewed.

Figure 7: On the left, the soliton is plotted at $t=0$. The amplitude at crest
is $a_{0}$ and its effective wavelength is $\widetilde{\lambda}$. The origin
of coordinates is placed at its crest position. On the left side, at $t>0$,
the soliton has moved to the right, its shape has sharpened,
$\lambda(t)<\widetilde{\lambda}$, and its amplitude has grown, $a(t)>a_{0}$.
Notice that for convenience, scales are not respected.

###  5.3 Wave breaking criteria

As noticed in the previous section, the wave breaking does not result from the
evolution of the wave as the soliton solution of KdV-B equation. Therefore, we
have to consider other reasons for the wave breaking. The most well-known
criteria of wave breaking for linear monochromatic waves are the McCowan
criterium [60], The Miche criterium [61] and the horizontal velocity criterium
[62]. But, as we shall see, this latest cannot be applied to our case.
Therefore, we propose an alternative criterium that we shall call "alternative
velocity criterium". For further use, we call $t_{d,Mc}$, $t_{d,Mi}$ and
$t_{d,alv}$, the wave breaking times within McCowan criterium, Miche criterium
and alternative velocity criterium, respectively.

In what follows, we shall compute the breaking time $t_{d}$, as well as the
amplitude, the wavelength and the phase velocity of the solitary wave at
$t=t_{d}$ within each of the mentioned criteria.

###  5.4 McCowan criterium

The McCowan criterium is established for linear solitary waves and studies the
highest maximum height that such waves might attain without breaking. McCowan
has shown that the breaking occurs at a given rate between the maximum wave
amplitude $a_{max}$ and the water depth $h$. Namely:

$\frac{a_{max}}{h}\approx 0.78.$ (5.52)

Applying this criterium to the soliton solution Eq. (5.30), where
${a(t)={a_{0}}({1-\frac{t}{t_{b}}}})^{-1}$, Eq. (5.52) yields

$\left(\frac{a_{0}}{h}\right)\frac{1}{1-\frac{t_{d,Mc}}{t_{b}}}\approx 0.78.$
(5.53)

Eq. (5.53) gives the breaking time according to McCowan criterium, which is

$t_{d,Mc}\approx t_{b}(1-1.28\nu).$ (5.54)

Using Eq. (5.30), it is easy to calculate the maximum height reached at
$t=t_{d,Mc}$. This is

$\eta_{max}=a_{max}\approx\frac{a_{0}}{1.28\nu}\approx 0.78h.$ (5.55)

In addition Eq. (5.36), one obtains the soliton’s wavelength at the moment of
wave breaking, namely $\lambda(t_{d,Mc})$:

$\lambda(t_{d,Mc})\approx 1.28\left(\frac{4}{3}a_{0}h\right)^{1/2}.$ (5.56)

###  5.5 Miche Criterium

The Miche criterium is based on empirical observations of linear waves’
dispersion relations in a finite depth $h$ with a wave length $\lambda$. The
Miche criterium fixes the maximum height $a_{max}$ reached by the wave before
the break down as follows

$\left(\frac{a}{\lambda}\right)_{max}=\frac{1}{7}\tanh\left(\frac{2\pi
h}{\lambda}\right).$ (5.57)

In our case, replacing $a$ by ${a(t)={a_{0}}({1-\frac{t}{t_{b}}}})^{-1}$, and
$\lambda$ by Eq. (5.36), in the right side of Eq. (5.57), we have

$\left(\frac{a}{\lambda}\right)_{max}=\frac{\nu}{2}(3\nu)^{1/2}\left(1-\frac{t_{d,Mi}}{t_{b}}\right)^{3/2}.$
(5.58)

Now, replacing $\lambda$ by Eq. (5.36) in the left side of Eq. (5.57), using
$\nu=a_{0}/h$, taking $t=t_{d,Mi}$ and expanding the hyperbolic tangent up of
$\displaystyle{\left(\frac{a}{\lambda}\right)_{max}}$ up to order
$(3\nu)^{3/2}$, we obtain

$\frac{1}{7}\tanh\left(\frac{2\pi
h}{\lambda}\right)=\frac{1}{7}\frac{\pi(3\nu)^{1/2}}{(1-\frac{t_{d,Mi}}{t_{b}})^{1/2}}-\frac{1}{21}\frac{\pi^{3}(3\nu)^{3/2}}{(1-\frac{t_{d,Mi}}{t_{b}})^{3/2}}+O(\nu^{5/2}).$
(5.59)

Using Eqs (5.57), (5.58) and (5.59), and neglecting terms of higher order than
$\nu$, we obtain

$t_{d,Mi}=t_{b}\left(1-\frac{7\nu}{2\pi}-\pi^{2}\nu\right).$ (5.60)

The maximum height reached by the wave at $t=t_{d,Mi}$ is

$\eta_{max}=a_{max}\approx\frac{0.09a_{0}}{\nu}\approx 0.09h.$ (5.61)

Using Eqs (5.58) and (5.60), the soliton’s wavelength can be calculated at
$t=t_{d,Mi}$, which is

$\lambda_{max}\approx 4.7\times 10^{-3}\left(\frac{h}{a_{0}}\right)^{5/2}h.$
(5.62)

###  5.6 Alternative horizontal velocity criterium

The horizontal velocity criterium assumes that the wave breaking occurs when
the group velocity of a water plane wave exceeds the speed of the phase plane
at crest. In our case, this criterium cannot be applied because, for a
solitary wave, at first approach, the group velocity does not mean much.
Therefore, alternatively we replace this criterium by an alternative approach
which assume that the wave breaking occurs when the fluid horizontal velocity
exceeds the phase plane velocity at crest. At this moment, the matter starts
to be ejected from the wave.

The phase plane velocity is given by Eq. (5.45). The phase velocity at crest,
i.e. for $\theta(x,t)=0$, is given be Eq. (5.48). The horizontal water
velocity is given by Eq. (5.12b). Using Eqs (5.19) and (5.3), as well as
(5.18) or equivalently $t_{2}=\nu\delta t$, the water velocity can be expanded
as follows

$u_{0}=\eta-\frac{\nu}{4}\eta^{2}+O(\delta,\delta^{2}).$ (5.63)

It can be seen from Eq. (5.28) that at crest $\eta(\theta=0)=a(t)$. Hence the
alternative velocity criterium reads

$1+\frac{\nu}{2}a\geq a-\frac{\nu}{4}a^{2}$ (5.64)

the above inequality yields the following quadratic equation

$a^{2}+\left(2-\frac{4}{\nu}\right)a+\frac{4}{\nu}\geq 0$ (5.65)

Taking into account that $\nu$ is a small parameter, the solution of Eq.
(5.65) are $a\lessapprox\frac{1}{2}$ and
$a\gtrapprox\frac{4}{\nu}-\frac{5}{2}$. the first part of the solution is not
acceptable since it corresponds to negative times, while the latest, using Eq.
(5.26) and neglecting terms of order $\nu^{2}$ , gives the breaking time
$t_{d,alv}$, as follows

$t_{d,alv}\approx t_{b}\left(1-\frac{\nu}{4}\right).$ (5.66)

Using Eq. (5.30), the maximum height reached at $t=t_{d,alv}$ is given by

$a_{max}(t_{d,alv})\approx 4h$ (5.67)

The wave length at $t=t_{d,a}$ is obtained from Eq. (5.36), and reads

$\lambda(t=t_{d,alv})=\left(\frac{1}{3}\right)^{1/2}h.$ (5.68)

##  6 Integrability perspectives

Recently, the integrability of a general form of the Kdv and NLS equations
with forcing involving product of eigenfunctions has been proved [65], [66].

More specifically, in [65] it is proved that the following forced version of
NLS is integrable:

$\displaystyle\mathrm{i}q_{t}+\frac{\alpha}{2}q_{xx}-\alpha\lambda|q|^{2}q=\frac{2\mathrm{i}}{\pi}\int_{\mathbb{R}}\frac{g(t,k)}{a_{0}(k)}\Phi_{1}^{+}(x,t,k)$
$\displaystyle\Psi^{+}_{1}(x,t,k)\mathrm{e}^{\mathrm{i}\lambda\left(HG(t,k)|b_{0}|^{2}(k)\right)}dk,$
$\displaystyle x\in\mathbb{R},\qquad t>0,$ (6.1)

where $\alpha$ is a constant real parameter, $\lambda=\pm 1$, $\Phi^{+}_{1}$,
$\Psi^{+}_{1}$ are appropriate eigenfunctions,

$\displaystyle G(t,k)=\int_{0}^{t}g(\tau,k)d\tau,\quad$ $\displaystyle
b_{0}(k)=\lim_{x\rightarrow-\infty}\mathrm{e}^{2\mathrm{i}kx}\Psi^{+}_{10}(x,k),$
$\displaystyle a_{0}(k)=\lim_{x\rightarrow-\infty}\Psi^{+}_{10}(x,k),$ (6.2)

and $(\Psi_{10}^{+}(x,k),\Psi_{20}^{+}(x,k))$ are defined in terms of
$q_{0}(x)=q(x,0)$ by

$\displaystyle\Psi_{10}^{+}(x,k)=-\int_{x}^{\infty}d\xi\mathrm{e}^{2\mathrm{i}k(\xi-x)}q_{0}(\xi)\Psi_{20}^{+}(\xi,k),$
$\displaystyle\Psi_{20}^{+}(x,k)=1-\int_{x}^{\infty}d\xi\lambda\overline{q}_{0}(\xi)\Psi_{10}^{+}(\xi,k),\qquad-\infty<x<\infty\quad\text{Im}k\geq
0.$ (6.3)

Eq. (6) possesses the following Lax pair:

$X_{x}+\mathrm{i}k[\sigma_{3},X]-QX=0,\qquad
Q=\begin{pmatrix}0&q\cr\lambda\overline{q}&0\end{pmatrix},$ (6.4)
$\displaystyle X_{t}+$ $\displaystyle\mathrm{i}\alpha
k^{2}[\sigma_{3},X]-\alpha\left(kQ-\frac{\mathrm{i}}{2}(Q_{x}+Q^{2})\sigma_{3}\right)X$
$\displaystyle=\frac{1}{2\mathrm{i}}(Hg)X\sigma_{3}-\frac{1}{2\mathrm{i}}(HgX\sigma_{3}X^{-1})X,\qquad
x\in\mathbbm{R},\quad t>0,\quad k\in\mathbbm{R},$ (6.5)

where $Hf$ denotes the Hilbert transform

$(Hf)(k)=\frac{1}{\pi}p\int_{\mathbb{R}}\frac{f(l)}{l-k}dl,\qquad
k\in\mathbb{R}.$ (6.6)

And in [66], it is shown that the forced integrable extension of the KdV,
namely the following equation is integrable.

$u_{t}+\alpha(u_{xxx}+6uu_{x})=d_{x}(x,t)+2h_{x}(x,t),$ (6.7)

where $\alpha$ is introduced in order to consider the $\alpha=0$ limit and $d$
and $h$ are defined by

$\displaystyle
d(x,t)=\frac{1}{\pi}\int_{\mathbb{R}}g(t,l)(v_{11}v_{22}+v_{12}v_{21})(x,t,l)dl,$
(6.8a) $\displaystyle
h(x,t)=\frac{1}{\pi}\int_{\mathbb{R}}g(t,l)v_{21}v_{22}(x,t,l)dl,$ (6.8b)

is integrable. The functions $v_{ij}$, $i,j=1,2$ are the $ij$ components of
the matrix $v$ given by

$v(x,t,k)=\frac{1}{2}\begin{pmatrix}\phi(x,t,-k)-\displaystyle{\frac{1}{\mathrm{i}k}\phi_{x}(x,t,-k)}&\phi(x,t,k)-\displaystyle{\frac{1}{\mathrm{i}k}\phi_{x}(x,t,k)}\cr\cr\phi(x,t,-k)+\displaystyle{\frac{1}{\mathrm{i}k}\phi_{x}(x,t,-k)}&\phi(x,t,k)+\displaystyle{\frac{1}{\mathrm{i}k}\phi_{x}(x,t,k)}\end{pmatrix},$
(6.9)

where $\phi$ is an appropriate solution of the associated Lax pair. Namely,
$\phi$ satisfies

$\phi_{xx}+(u+k^{2})\phi=0,$ (6.10)

and

$\displaystyle\phi_{t}+\alpha(4\mathrm{i}k^{3}-u_{x})\phi+\alpha(2u-4k^{2})\phi_{x}+\frac{1}{2\mathrm{i}}(Hg)\phi-\frac{1}{4\mathrm{i}}[Hg(\phi\widehat{\phi}+\frac{1}{k^{2}}\phi_{x}\widehat{\phi}_{x})]\frac{\phi_{x}}{\mathrm{i}k}$
$\displaystyle+\frac{1}{4\mathrm{i}}[Hg(\phi\widehat{\phi}-\frac{1}{k^{2}}\phi_{x}\widehat{\phi}_{x})]\phi-\frac{1}{4\mathrm{i}}[Hg(\frac{1}{\mathrm{i}k}{(\phi\widehat{\phi})}_{x})]\frac{\phi_{x}}{\mathrm{i}k}=0,$
(6.11)

where the hat denotes evaluation at $-k$.

We have strong indications to think that both the KdV-Burger equation (5.12a)
and the NLS equation (3.15), with appropriate air and water dynamics and
wind/wave interaction can be integrable. This will be the subject of fore
coming studies.

##  7 Conclusions

In the first part of this review paper, our aim was exclusively to derive a
linear Miles’ theory for waves propagating at finite depth $h$. Hence, we
extended the well-known Miles’ theory to the finite depth context under breeze
to moderate wind conditions. We have linearized the equations of motion
governing the dynamics of the air/water interface problem in finite depth and
have studied the linear instability in time of a normal Fourier mode $k$. The
prediction of exponential growth of wave amplitude (or energy) is well
confirmed by field and laboratory experiments. As an extension, we have
derived an anti-diffusive nonlinear Schrödinger equation in finite depth and
found the Akhmediev, Peregrine and Ma solutions for weak wind inputs in finite
depth.

In the second part, We derived a SGN fully nonlinear, dispersive and anti-
dissipative system of equations in the context of the nonlinear dynamics of
surface water waves under wind forcing, in finite depth. We found that its
soliton solution with amplitude, velocity, and effective wavelength increased
with time. Antidissipation due to wind action through the sheltering Jeffreys’
mechanism increases the amplitude of the solitary wave and leads to blow-up
which occurs in finite time for infinitely large asymptotic space. The blow-up
time is calculated. As an extension, the anti-diffusive Korteweg-de Vries-
Burger equation is derived. we have studied in detail the kinetics of the
breaking of the wind/wave solitary wave and reviewed various break-up
criteria. Experimentally, it is clear that the breaking will occur before the
blow up. The experimental confirmation of the present theory can be tested in
existing facilities. We will conduct further investigations in this direction
in forecoming studies.

Finally, the perspectives of integrability for the NLS equation and the KdV-B
equation have been exposed which also be investigated in forecoming studies.

## Appendix: Direct derivation of the anti-diffusive Korteweg-de Vries-Burger
equation from Euler Equations

Let us consider a quasi-linear air/water system with the air dynamics
linearized and the water dynamics considered nonlinear and irrotational. The
system is $(2+1)$ dimensional ($x,z,t$) with $x$ and $z$ the vertical and the
horizontal space coordinates. The aerodynamic air pressure $P_{a}(x,z,t)$
evaluated at the free surface $z=\eta(x,t)$ has 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 energy transfer 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, following
references [3], [30] and [63], only the pressure component in phase with the
slope on the interface i.e.,

$P_{a}=\epsilon\rho_{a}\Delta^{2}\eta_{x}\quad\mbox{with}\quad\Delta=[\frac{\kappa
U_{1}}{\sqrt{C_{10}}}-c]$ (A.1)

where $\epsilon<1$ is the sheltering coefficient, $c=\sqrt{g/k}\tanh(kh)$,
$U_{1}=u_{*}/\kappa$ with $u_{*}$ the friction velocity, $\kappa\sim 0,41$ the
Von Kàrmàn constant, $C_{10}$ the wind-stress coefficient and $g$ the
gravitational acceleration. This is nothing more that Jeffrey’s sheltering
mechanisms. In order to adimensionalize the equations of motion we introduce
dimensionless primed variables:
$x=lx^{\prime},z=hz^{\prime},t=lt^{\prime}/c_{0},\eta=a\eta^{\prime},\phi=gla\phi^{\prime}/c_{0},U_{1}=c_{0}U_{1}^{\prime}$
with $\phi$ the velocity potential and $a$ and $l$ typical wave amplitude and
wavelength and $c_{0}=\sqrt{gh}$. We define two dimensionless parameters
$\nu=a/h<1$ and $\delta=h/l<1.$ So with this assumption the complete
irrotational Euler equations and boundary conditions are (dropping the primes)

$\displaystyle\delta^{2}\phi_{xx}+\phi_{zz}=0,\quad-1\leq z\leq\nu\eta,$ (A.2)
$\displaystyle\phi_{z}=0,\quad z=-1,$ (A.3)
$\displaystyle\eta_{t}+\nu\phi_{x}\eta_{x}-\frac{1}{\delta^{2}}\phi_{z}=0,\quad
z=\nu\eta,$ (A.4)
$\displaystyle\phi_{t}+\frac{\nu}{2}\phi_{x}^{2}+\frac{\nu}{2\delta^{2}}\phi_{z}^{2}+\eta+\delta\epsilon
s\Delta^{2}\eta_{x}=0,\quad z=\nu\eta,$ (A.5)

where $s=\rho{{}_{a}}/\rho_{w}\sim 10^{-3}$ with $\rho_{a}$ ($\rho_{w}$) the
air (water) density. We solve the Laplace equation and its boundary conditions
with an expansion in powers of $(z+1)$, namely

$\phi=\sum_{m=0}^{m=\infty}(z+1)^{m}\delta^{m}q_{m}(x,t).$ (A.6)

Substituting (A.6) in Eq.(A.2) and using Eq.(A.3) we obtain

$\phi=\sum_{m=0}^{m=\infty}(-1)^{m}\frac{(z+1)^{2m}}{(2m)!}\delta^{2m}q_{0,2mx}.$
(A.7)

Using the kinematic and dynamics boundary conditions Eq.(A.4) and Eq.(A.5) and
disregarding terms in $\mathcal{O}(\nu\delta^{2})$ and
$\mathcal{O}(\delta^{4})$ we find, with $r=q_{0,x}$, the system

$\displaystyle\eta_{t}+\\{(1+\nu\eta)r\\}_{x}-\frac{1}{6}\delta^{2}r_{xxx}=0,$
(A.8) $\displaystyle\eta_{x}+r_{t}+\nu
rr_{x}-\frac{1}{2}\delta^{2}r_{xxt}+\delta\epsilon s\Delta^{2}\eta_{xx}=0.$
(A.9)

The linear wave solution of (A.8) and (A.9) moving to the right is
$r(\xi)=\eta(\xi)$, $\xi=x-t$, with $\eta$ (or $r$) an arbitrary function of
$\xi$. Now we look for a solution with nonlinear corrections to the orders
$\mathcal{O}(\nu)$, $\mathcal{O}(s\delta)$, and $\mathcal{O}(\delta^{2})$.
Following procedure in reference [51] we obtain

$\displaystyle
r=\eta-\frac{1}{4}\eta^{2}\nu+\frac{\epsilon}{2}\Delta^{2}\eta_{x}s\delta+\frac{1}{3}\eta_{xx}\delta^{2}+\mathcal{O}(\nu\delta^{2},s^{2}\delta^{2},\delta^{4}),$
(A.10)

Substituting (A.10) in (A.8) and (A.9) we obtain an anti-dissipative KdV-B
equation

$\eta_{t}+\eta_{x}+\frac{3}{2}\nu\eta\eta_{x}+\frac{1}{6}\delta^{2}\eta_{xxx}+\frac{s}{2}\delta\epsilon\Delta^{2}\eta_{xx}=0.$
(A.11)

For traveling wave solutions, the action of dissipation or anti-dissipation in
KdV-B is not of great matter except for the sign of the slope [64]. But
soliton solutions under anti-dissipation exhibits a blow-up and breaking in
finite time.

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