Presently at ]Department of Mathematics and Statistics, University of Reading,
Reading, UK Presently at ]Department of Engineering, University of Ferrara,
via Saragat 1, I-44122 Ferrara, Italy

# Mean flow modelling in high-order nonlinear Schrödinger equations

Alexis Gomel  [  Group of Applied Physics and Institute for Environmental
Sciences, University of Geneva, 66 Bd Carl-Vogt, CH-1211 Geneva 4, Switzerland
Corentin Montessuit  Group of Applied Physics and Institute for Environmental
Sciences, University of Geneva, 66 Bd Carl-Vogt, CH-1211 Geneva 4, Switzerland
Andrea Armaroli  [  Group of Applied Physics and Institute for Environmental
Sciences, University of Geneva, 66 Bd Carl-Vogt, CH-1211 Geneva 4, Switzerland
Debbie Eeltink  Department of Mechanical Engineering, Massachusetts Institute
of Technology, Cambridge, Massachusetts 02139, USA  Department of Engineering
Science, University of Oxford, Parks Road, Oxford OX1 3PJ, UK  Laboratory of
Theoretical Physics of Nanosystems, EPFL, CH-1015 Lausanne, Switzerland  Amin
Chabchoub  Disaster Prevention Research Institute & Hakubi Center for Advanced
Research, Kyoto University, Kyoto, 606-8501 Kyoto, Japan  School of Civil
Engineering, The University of Sydney, Sydney, NSW 2006, Australia  Jérôme
Kasparian  Maura Brunetti<EMAIL_ADDRESS>Group of Applied Physics
and Institute for Environmental Sciences, University of Geneva, 66 Bd Carl-
Vogt, CH-1211 Geneva 4, Switzerland

###### Abstract

The evaluation and consideration of the mean flow in wave evolution equations
are necessary for the accurate prediction of fluid particle trajectories under
wave groups, with relevant implications in several domains, from the transport
of pollutants in the ocean, to the estimation of energy and momentum exchanges
between the waves at small scales and the ocean circulation at large scale. We
derive an expression of the mean flow at finite water depth which, in contrast
to other approximations in the literature, accurately accords with the deep-
water limit at third order in steepness, and is equivalent to second-order
formulations in intermediate water. We also provide envelope evolution
equations at fourth order in steepness for the propagation of unidirectional
wave groups either in time or space that include the respective mean flow
term. The latter, in particular, is required for accurately modelling
experiments in water wave flumes in arbitrary depths.

##  I Introduction

As waves evolve on the ocean surface, they induce a mean flow to the fluid
particles, particularly when nonlinear effects are taken into account. Near
the surface, fluid particles experience a net horizontal displacement in the
same direction as the water wave propagation, the so-called Stokes drift
Stokes (1847); van den Bremer and Breivik (2018), that decays with depth. A
return flow in the fluid column along vertical and horizontal directions
guarantees the conservation of the water-mass transport, causing localised
variations in the mean water level under wave groups and the associated
propagation of infragravity waves Bjørnestad  _et al._ (2021) at finite water
depth. An accurate description of the mean flow is thus necessary for the
proper reconstruction of the fluid particle trajectories underneath water
waves Calvert  _et al._ (2019); Carter, Curtis, and Kalisch (2020). Since the
wave-induced mean flow is associated to the transport of energy, momentum and
other tracers DiBenedetto (2020), such as pollutants like plastic Cózar  _et
al._ (2014) or offshore oil spill Christensen  _et al._ (2018), it is relevant
for environmental studies. Moreover, it has an impact on the general
circulation of the ocean at large scales and its modelling Babanin (2006);
Onink  _et al._ (2019); van Sebille  _et al._ (2020); Cunningham, Higgins, and
van den Bremer (2022), and on the nearshore circulation Longuet-Higgins and
Stewart (1964); Bjørnestad  _et al._ (2021), in particular in shoaling regions
Janssen, Battjes, and van Dongeren (2003); Battjes  _et al._ (2004), while it
is affected by the presence of background shear currents Monismith  _et al._
(2007); Curtis, Carter, and Kalisch (2018).

Since Stokes drift and return flow are two phenomena occurring at second-order
in steepness, they are taken into account in the finite water depth nonlinear
Schrödinger equation (NLS), which describes the evolution of the envelope of
narrow-banded wave packets and can be obtained using a multi-scale development
of water surface elevation and velocity potential at third-order in steepness
Benney and Roskes (1969); Hasimoto and Ono (1972); Davey and Stewartson
(1974); Mei, Stiassnie, and Yue (2005), or taking the narrow-banded limit of
the Zakharov equation using an Hamiltonian approach Zakharov (1968); Janssen
and Onorato (2007). Indeed, at third-order in steepness, it was shown Davey
and Stewartson (1974) that the mean flow term comes into play as a
modification of the nonlinear coefficient, giving rise to a transition from
focusing to defocusing regimes at the critical value $k_{0}h\approx 1.363$,
where $k_{0}$ is the carrier wavenumber and $h$ the depth. However, this
contribution to the nonlinear coefficient disappears in the deep-water limit.
At fourth-order in steepness, the accurate formulation of the mean flow term
needs to be accounted for, in both, finite and infinite depth waters Dysthe
(1979).

Fourth-order terms in the NLS equation are necessary to explain features like
asymmetrical evolution of spectra Chabchoub  _et al._ (2013), and asymmetries
in the waveform Trulsen and Stansberg (2001); Shemer, Sergeeva, and Slunyaev
(2010); Goullet and Choi (2011); Zhang, Guedes Soares, and Onorato (2014);
Armaroli, Brunetti, and Kasparian (2017), especially when inevitable wave
focusing is at play Slunyaev  _et al._ (2013). Several versions of high-order
wave envelope evolution equations exist in the literature, that can be
obtained when applying the multiple scales development using steepness and
bandwidth with different orders of magnitude Dysthe (1979); Janssen (1983);
Brinch-Nielsen and Jonsson (1986); Trulsen and Dysthe (1996), or the
Hamiltonian approach Stiassnie (1984); Debsarma and Das (2005); Janssen and
Onorato (2007); Gramstad and Trulsen (2011); Gramstad (2014). The main
difference between all these equations is the treatment and approximation of
the mean flow term.

Here, we will focus on narrow-banded unidirectional wave packets where
steepness and bandwidth can be considered as parameters of the same order,
like in the finite-depth developments described in Refs. Sedletsky, 2003
(denoted as Sed03 in the following) and Slunyaev, 2005 (Slu05). The challenge
of such developments is the fact that they do not reduce to the Dysthe
equation (Ref. Dysthe, 1979, Dys79) in the deep-water limit, which has been
shown to well reproduce wave tank experiments Goullet and Choi (2011). We will
show that this convergence depends on how the mean flow is approximated.
Moreover, we will provide finite-depth envelope equations at fourth order in
steepness in both, space-like and time-like formulations, with correct
limiting expressions in deep water. In particular, such unidirectional time-
like equations that allow a continuous scaling from intermediate to the deep
water limit are relevant for reproducing wave tank experiments with high
accuracy in arbitrary depths. In contrast, directional sea states are
typically found in the open ocean, and this restricts the applicability of the
the above unidirectional modelling equations.

The paper is organised as follows. In Sec. II, we will derive the expression
of the mean flow to be inserted in the envelope equation at fourth order in
steepness for the evolution in time. We will compare this expression with the
ones already existing in the literature, and show that it indeed correctly
describes the mean flow in the whole range of water depths, i.e., from
intermediate to deep water regimes. In Sec. III, we will provide an envelope
equation for the water wave evolution in space, relevant for modelling for
instance water tank experiments, and the corresponding mean flow term, at
fourth order in steepness. Again, we will perform the comparison for various
water depth scenarios. Finally, in Sec. IV we will summarise our findings.

##  II Fourth-order equation: propagation in time

The multi-scale approach has been used to derive at fourth order in steepness
the following equation, that describes the evolution of the envelope $U$ of a
uni-directional progressive gravity wave packet propagating in time on the
free surface of a homogeneous liquid with depth $h$ Sedletsky (2003); Slunyaev
(2005); Gandzha, Sedletsky, and Dutykh (2014)

$i\left(\frac{\partial U}{\partial t}+c_{\mathrm{g}}\frac{\partial U}{\partial
X}\right)+\hat{\alpha}\frac{\partial^{2}U}{\partial
X^{2}}\underbrace{-\hat{\beta}|U|^{2}U}_{\text{incl.~{}Mean
Flow}}=i\biggl{(}{\hat{\alpha}_{3}\frac{\partial^{3}U}{\partial
X^{3}}}\underbrace{-\hat{\beta}_{21}|U|^{2}\frac{\partial U}{\partial
X}{-\hat{\beta}_{22}U^{2}\frac{\partial U^{*}}{\partial
X}}}_{\text{incl.~{}Mean Flow}}\biggr{)}.$ (1)

The explicit formulation of the group velocity $c_{\mathrm{g}}$, and of all
dispersive and nonlinear coefficients $\hat{\alpha}$, $\hat{\beta}$,
$\hat{\alpha}_{3}$, $\hat{\beta}_{21}$, and $\hat{\beta}_{22}$ is given in
App. A and B. The surface tension has been neglected and the fluid is
considered as irrotational. The sign convention follows Sed03: the surface
elevation is reconstructed at leading order from the envelope using111Note
that Slu05 uses the opposite convention, giving rise to different signs in
some terms with respect to Sed03.
$\eta(X,t)=\frac{1}{2}[U(X,t)\exp(i(k_{0}X-\omega_{0}t))+{\rm{c.~{}c.}}]$,
where $\omega_{0}$ and $k_{0}$ are the angular frequency and wavenumber of the
carrier wave, respectively.

The previous Eq. (1) is referred to as the ‘space-like equation’ since
dispersion is in space. It can also be written in the following equivalent
form Sedletsky (2003) in a reference frame moving with the group velocity
$x=X-c_{g}t$:

$i\frac{\partial U}{\partial
t}+\varepsilon\left(\hat{\alpha}\frac{\partial^{2}U}{\partial
x^{2}}-\hat{\beta}_{D}|U|^{2}U\right)=i\varepsilon^{2}\left({\hat{\alpha}_{3}\frac{\partial^{3}U}{\partial
x^{3}}}{-\omega_{0}k_{0}\tilde{Q}_{41}|U|^{2}\frac{\partial U}{\partial
x}}{-\omega_{0}k_{0}\tilde{Q}_{42}U^{2}\frac{\partial U^{*}}{\partial
x}}\right)+\underbrace{\frac{\mu_{g}k_{0}}{4\sigma}U\frac{\partial\phi_{0}}{\partial
x}}_{\text{Mean Flow}}$ (2)

where $\sigma=\tanh(k_{0}h)$, $\mu_{g}$ is given in Eq. (55), and the high-
order nonlinear coefficients $\tilde{Q}_{41}$ and $\tilde{Q}_{42}$ in Eqs.
(67) and (70), respectively. The nonlinear term $\hat{\beta}_{D}$ is a
positive function (see Fig. 1), given in Eq. (49), and since the dispersion
coefficient $\hat{\alpha}$ is negative, it seems that the characterisation of
the focusing and defocusing regime is somehow lost in the present formulation.

Figure 1: Third-order nonlinear coefficients $\hat{\beta}$ and
$\hat{\beta}_{D}$ in Eqs. (1) and (2), respectively, normalised with respect
to their deep-water values, which are equal. Zero crossing for $\hat{\beta}$
is at $k_{0}h=1.363$, shown by the dotted vertical line.

Notice that we explicitly introduce the scaling parameter $\varepsilon$, a
dummy variable which is set to 1 at the end, that is helpful for grouping
terms of the same order in steepness. The last term, depending on the zero
harmonic of the velocity potential $\phi_{0}$, is the mean flow term that,
from the multi-scale development, takes the following form (see Eqs. (35) and
(57) in Sed03):

$\displaystyle\frac{\partial\phi_{0}}{\partial x}$ $\displaystyle=$
$\displaystyle\varepsilon\frac{\omega_{0}}{2}\frac{\mu_{g}k_{0}}{\sigma\nu}|U|^{2}$
(3) $\displaystyle-$ $\displaystyle
i\varepsilon^{2}\frac{4\omega_{0}\sigma}{\nu}\tilde{q}_{40S}\left(U\frac{\partial
U^{*}}{\partial x}-U^{*}\frac{\partial U}{\partial x}\right)\hskip 28.45274pt$

with $\nu$ given in Eq. (56) and $\tilde{q}_{40S}$ in Eq. (73). Its
contribution can be included into the coefficients $\hat{\beta}_{D}$,
$\tilde{Q}_{41}$, $\tilde{Q}_{42}$ to obtain $\hat{\beta}$,
$\hat{\beta}_{21}$, $\hat{\beta}_{22}$, respectively, in Eq. (1) (see steps
from Eq. (40) to (42) and from Eq. (64) to (66) in Sed03). Thus, the mean flow
term is already taken into account in Eq. (1) at fourth order in steepness
through the nonlinear coefficients.

We remark that from Eq. (3), in the deep-water limit,
$\partial\phi_{0}/\partial x\to 0$ since $\nu\to-\infty$ while all the other
coefficients are finite. This agrees with the developments at third order in
steepness Davey and Stewartson (1974); Peregrine (1983). Note that in the 1D
shallow-water case the mean flow term modifies the nonlinear term, see Eq.
(2.21) in Ref. Davey and Stewartson, 1974, while its contribution vanishes in
the deep-water limit. However, the fact that the contribution of mean flow
remains zero in deep water at higher-order approximation in steepness is not
ideal, as we will discuss in the following subsection.

###  II.1 Mean flow in the deep-water limit

In the Dysthe equation (Dys79), i.e., the evolution equation obtained in the
deep-water limit with the multi-scale method at fourth order in steepness, an
additional term corresponding to the wave-induced mean flow usually appears in
the literature in both, space-like Dysthe (1979); Janssen (1983); Lo and Mei
(1985) and time-like 1D formulations Trulsen and Dysthe (1997); Kit and Shemer
(2002); Onorato  _et al._ (2005); Goullet and Choi (2011); Eeltink  _et al._
(2017). Let us consider for the moment the propagation in time. The Dysthe
equation reads:

$i\frac{\partial U}{\partial
t}-\frac{\omega_{0}}{8k_{0}^{2}}\frac{\partial^{2}U}{\partial
x^{2}}-\frac{\omega_{0}k_{0}^{2}}{2}|U|^{2}U=i\frac{\omega_{0}}{16k_{0}^{3}}\frac{\partial^{3}U}{\partial
x^{3}}-i\frac{\omega_{0}k_{0}}{2}\biggl{(}3|U|^{2}\frac{\partial U}{\partial
x}+\frac{1}{2}U^{2}\frac{\partial U^{*}}{\partial
x}+\underbrace{\frac{2i}{\omega_{0}}U\frac{\partial\phi_{0}}{\partial
x}}_{\text{Mean Flow}}\biggr{)}$ (4)

In this deep-water limit, the mean-flow term is written as Onorato  _et al._
(2005); Goullet and Choi (2011):

$\frac{\partial\phi_{0}}{\partial
x}|_{z=0}=\frac{\omega_{0}}{2}{\mathcal{H}_{x}}\left[\frac{\partial|U|^{2}}{\partial
x}\right],\hskip 28.45274pt$ (5)

where $\mathcal{F}_{x}$ is the Fourier transform in space, and
$\mathcal{H}_{x}$ the Hilbert transform,
$\mathcal{H}_{x}[\eta]=\mathcal{F}^{-1}_{x}\left[+i~{}\mathrm{sgn}(k)\mathcal{F}_{x}\left[\eta\right]\right]$,
$\eta$ being a function of $x$. The role of the mean flow Hilbert term in the
evolution of pulsating wave packets Chabchoub  _et al._ (2013) and narrow-
banded irregular waves Shemer, Sergeeva, and Slunyaev (2010); Goullet and Choi
(2011) has been shown in several experiments in deep water, and its presence
is required for a correct modelling Trulsen and Stansberg (2001).

Janssen Janssen (1983) suggests that the system can be closed by solving the
equations for $\phi_{0}$ as a function of the envelope $U$ in Fourier space.
It is instructive to report here the explicit derivation of the Hilbert term
since we will use analogous developments in the following. The zero harmonic
of the velocity potential $\phi_{0}$ satisfies the Laplace equation in the
entire water column with boundary conditions at the surface and at the bottom,
giving a set of equations that constitutes the following Neumann problem:

$\displaystyle\frac{\partial^{2}\phi_{0}}{\partial
z^{2}}+\frac{\partial^{2}\phi_{0}}{\partial x^{2}}$ $\displaystyle=0$
$\displaystyle-\infty<z\leq 0$ (6a)
$\displaystyle\frac{\partial\phi_{0}}{\partial z}$
$\displaystyle=\frac{\omega_{0}}{2}\frac{\partial|U|^{2}}{\partial x}$
$\displaystyle z=0$ (6b) $\displaystyle\frac{\partial\phi_{0}}{\partial z}$
$\displaystyle=0$ $\displaystyle z\rightarrow-\infty$ (6c)

Substituting the Fourier transform of the mean flow:

$\displaystyle\mathcal{F}_{x}\phi_{0}(x,z,t)=\hat{\phi}_{0}(k,z,t)$ (7)
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\phi_{0}(x,z,t)e^{ikx}dx$

into the Laplace equation gives $\partial^{2}\hat{\phi}_{0}/\partial
z^{2}=k^{2}\hat{\phi}_{0}$ whose solution is

$\hat{\phi}_{0}(k,z,t)=C_{1}e^{|k|z}$ (8)

with $C_{1}$ independent of $z$. This satisfies the bottom boundary condition
for $z\to-\infty$.

Inserting this expression in Eq. (6b) gives:

$|k|C_{1}e^{|k|z}|_{z=0}=|k|C_{1}=\frac{\omega_{0}}{2}\mathcal{F}_{x}\left[\frac{\partial|U|^{2}}{\partial
x}\right],$ (9)

from which $C_{1}$ is obtained, and thus:

$\hat{\phi}_{0}(k,z,t)=\frac{1}{|k|}\frac{\omega_{0}}{2}e^{|k|z}\mathcal{F}_{x}\left[\frac{\partial|U|^{2}}{\partial
x}\right]\,.$ (10)

Now the derivative with respect to $x$ at $z=0$ in Fourier space is given by:

$\displaystyle\mathcal{F}_{x}\left[\frac{\partial\phi_{0}}{\partial
x}\right]|_{z=0}=ik\hat{\phi}_{0}|_{z=0}$ (11) $\displaystyle=$ $\displaystyle
i\frac{\omega_{0}}{2}\mathrm{sgn}(k)\mathcal{F}_{x}\left[\frac{\partial|U|^{2}}{\partial
x}\right]$

and finally, moving back to the direct physical space, the Hilbert term of Eq.
(5) is recovered.

###  II.2 Multi-scale development and Hilbert term

As discussed, the multi-scale approach of Sed03 gives Eq. (3), which in the
deep-water limit reduces to

$\frac{\partial}{\partial x}(\phi_{01}+\phi_{02})=0\,.$ (12)

On the other hand, the mean flow can be written as the Hilbert term of Eq. (5)
in the Dysthe equation Dysthe (1979); Janssen (1983). Thus, there is the need
to reconcile these results. This can be done as follows. In the derivation of
the Hilbert term, we use the Laplace equation for $\phi_{0}$ that is the
complete mean flow, and not just its approximation at second order in
steepness as in Eq. (12). Indeed, the Laplace equation at third order for the
mean flow is:

$\frac{\partial^{2}\phi_{03}}{\partial
z^{2}}+\frac{\partial^{2}\phi_{01}}{\partial x^{2}}=0.$ (13)

When integrated in $z$, this gives

$\frac{\partial\phi_{03}}{\partial
z}=-(z+h)\frac{\partial^{2}\phi_{01}}{\partial x^{2}},$ (14)

using the fact that $\partial\phi_{01}/\partial z=0$ and imposing the bottom
boundary condition. From the multi-scale development, the following expression
can be obtained at third-order in steepness (see Eqs. (2.12) and (2.14) in
Ref. Davey and Stewartson, 1974):

$\displaystyle\frac{c_{g}^{2}}{g}\frac{\partial^{2}\phi_{01}}{\partial
x^{2}}+\frac{\partial\phi_{03}}{\partial z}$ $\displaystyle=$
$\displaystyle\frac{\omega_{0}}{2\sigma}(1+C_{FD})\frac{\partial|U|^{2}}{\partial
x}$ (15) $\displaystyle=$ $\displaystyle
D^{\prime}\frac{\partial|U|^{2}}{\partial x},$

with the coefficient $C_{FD}$ defined in Eq. (57) and in Ref. Calvert  _et
al._ , 2019, and where
$D^{\prime}=(1+C_{FD})\,\omega_{0}/(2\sigma)=\mu_{g}\omega_{0}/(8\sigma^{2})$
(see definitions in App. A). Using Eq. (14) at $z=0$, Eq. (15) reduces to

$\frac{\partial^{2}\phi_{01}}{\partial
x^{2}}=-\frac{D}{h}\frac{\partial|U|^{2}}{\partial x},$ (16)

which corresponds to Eqs. (33)-(34) in Sed03 and where $D$ is defined in Eq.
(53). Eq. (16) can also be written, using Eq. (14), as:

$\frac{\partial\phi_{03}}{\partial z}=D\frac{\partial|U|^{2}}{\partial x}.$
(17)

Taking the deep-water limit gives Eq. (6b), i.e., the boundary condition at
the surface in the Neumann problem. Since $\partial\phi_{01}/\partial z=0$ and
$\partial\phi_{02}/\partial z=0$, such expression is valid at third order in
steepness for the mean flow.

Thus, in the derivation of the Hilbert term, the complete mean flow (in the
Laplace equation) is considered together with the mean flow at third order in
steepness in the surface boundary condition. Consequently, a ‘hybrid’ relation
as described by Eq. (5) is obtained, which is different from Eq. (12) where
only the first terms in the development of the mean flow at the surface are
taken into account. In other words, the expression in Eq. (5) is inherently
nonlocal. The same occurs in the nonlinear terms of the super-compact model
Dyachenko, Kachulin, and Zakharov (2017), from which the Dysthe equation can
be derived. The terms in Sed03 and Slu05 are instead local, as they only
involve the surface, and not the entire water column. This is obviously an
approximation, since the mean flow does involve a body of fluid which is not
immediately at the surface, as clearly shown in field measurements Bjørnestad
_et al._ (2021).

###  II.3 Mean flow term in arbitrary depth

We now repeat the procedure used in Sec. II.1 for the case of intermediate
water. We will consider two cases that differ based on the considered surface
boundary condition: the Neumann problem is solved using the condition given by
Eq. (17) in Case 1; using Eq. (15) in Case 2. Replacing the expression of
$\partial\phi_{0}/\partial x$ that is obtained in each case into the last term
of Eq. (2) gives the final high-order NLS equation in arbitrary finite depth
and space-like form.

#### Case 1

Moving as before to the Fourier space, the Laplace equation is
$\partial^{2}\hat{\phi}_{0}/\partial z^{2}=k^{2}\hat{\phi}_{0}$ and its
solution is given by:

$\hat{\phi}_{0}(k,z,t)=C_{1}e^{|k|(z+h)}+C_{2}e^{-|k|(z+h)}.$ (18)

Imposing the boundary condition at the bottom, Eq. (6c), gives:

$\frac{\partial\hat{\phi}_{0}}{\partial z}|_{z=-h}=|k|(C_{1}-C_{2})=0.$ (19)

Thus, we have $C_{1}=C_{2}=C/2$ and:

$\hat{\phi}_{0}(k,z,t)=C\cosh\big{(}|k|(z+h)\big{)}.$ (20)

Inserting this in Eq. (17) gives:

$|k|C\sinh\big{(}|k|(z+h)\big{)}|_{z=0}=D\mathcal{F}_{x}\left[\frac{\partial|U|^{2}}{\partial
x}\right],$ (21)

from which one obtains $C$ and therefore:

$\hat{\phi}_{0}=\frac{1}{|k|}\frac{\cosh\big{(}|k|(z+h)\big{)}}{\sinh\big{(}|k|h\big{)}}D\mathcal{F}_{x}\left[\frac{\partial|U|^{2}}{\partial
x}\right].$ (22)

At $z=0$, this gives:

$\hat{\phi}_{0}|_{z=0}=D\frac{\coth\big{(}|k|h)}{|k|}\mathcal{F}_{x}\left[\frac{\partial|U|^{2}}{\partial
x}\right]\,.$ (23)

Now, the derivative with respect to $x$ at $z=0$ is given by:

$\displaystyle\mathcal{F}_{x}\left[\frac{\partial\phi_{0}}{\partial
x}\right]|_{z=0}=ik\hat{\phi}_{0}|_{z=0}$ (24) $\displaystyle=$ $\displaystyle
iD\frac{\mathrm{sgn}(k)}{\tanh(|k|h)}\mathcal{F}_{x}\left[\frac{\partial|U|^{2}}{\partial
x}\right].$

Moving back to the direct physical space, we finally get:

$\frac{\partial\phi_{0}}{\partial
x}=D\mathcal{F}^{-1}_{x}\Bigg{\\{}\frac{i}{\tanh(kh)}\mathcal{F}_{x}\left[\frac{\partial|U|^{2}}{\partial
x}\right]\Bigg{\\}}.$ (25)

#### Case 2

The surface boundary condition is now Eq. (15). Note that the relation
$\phi_{03z}=-h\phi_{01xx}$ has not been used to simplify the l.h.s. of this
equation and both mean flow terms are thus considered being of the same order,
see Eq. (13) in Ref. Calvert  _et al._ , 2019.

Inserting the generic form of the solution in intermediate water, i.e., Eq.
(20), into the surface boundary condition, Eq. (15), gives:

$C\left.\left[\frac{c_{g}^{2}}{g}(-k^{2})\cosh(|k|(z+h))+|k|\sinh(|k|(z+h))\right]\right\rvert_{z=0}=D^{\prime}\mathcal{F}_{x}\left[\frac{\partial|U|^{2}}{\partial
x}\right],$ (26)

from which we obtain the following expression for $C(k,t)$:

$C(k,t)=\frac{D^{\prime}}{k\tanh(kh)\left[1-c_{g}^{2}k/(g\tanh(kh))\right]}\frac{1}{\cosh(kh)}\mathcal{F}_{x}\left[\frac{\partial|U|^{2}}{\partial
x}\right],$ (27)

where we used $|k|\tanh(|k|h)=k\tanh(kh)$, and $c_{g}$ is the wave group speed
at the carrier wavenumber. Performing analogous steps as in the previous case,
the final expression for the Euler horizontal velocity in $z=0$ is:

$\frac{\partial\phi_{0}}{\partial
x}=D^{\prime}\mathcal{F}^{-1}_{x}\left\\{\frac{i}{\tanh{(kh)}\left[1-c_{g}^{2}k/(g\tanh(kh))\right]}\mathcal{F}_{x}\left[\frac{\partial|U|^{2}}{\partial
x}\right]\right\\}.$ (28)

Note that this expression coincides with Eq. (15) in Ref. Calvert  _et al._ ,
2019 (apart from a sign that is a typo in that latter equation).

By performing the derivative in $x$ on the r.h.s. of Eq. (25) or (28),
considering that $D=D^{\prime}/(1-c_{g}^{2}/(gh))$ and using $\coth
y=1/y+O(y)$, we get in both Case 1 and 2 for small $kh$ numbers:

$\displaystyle\frac{\partial\phi_{0}}{\partial x}$ $\displaystyle\sim$
$\displaystyle
D\mathcal{F}^{-1}_{x}\Bigg{[}i(ik)\coth(kh)\mathcal{F}_{x}[|U|^{2}]\Bigg{]}$
(29) $\displaystyle\sim$ $\displaystyle
D\mathcal{F}^{-1}_{x}\Bigg{[}-\frac{1}{h}\mathcal{F}_{x}[|U|^{2}]\Bigg{]}$
$\displaystyle=$
$\displaystyle-\frac{D}{h}|U|^{2}=\frac{\omega_{0}}{2}\frac{\mu_{g}k_{0}}{\sigma\nu}|U|^{2}\,.$

Thus, we recover the first term on the r.h.s. of Eq. (3). The NLS nonlinear
coefficient is also recovered when the mean flow term is added to the third-
order nonlinear term in Eq. (2), since
$\hat{\beta}=\hat{\beta}_{D}+D^{2}\nu/(2h^{2}\omega_{0})$. Hence, the
defocusing regime is recovered through the inclusion of the mean flow term.

###  II.4 Numerical comparisons

We compare the expressions for the horizontal velocity
$\partial\phi_{0}/\partial x$ listed in Table 1 with the sub-harmonic velocity
potential $\phi_{20}$ at second-order in steepness and its horizontal
derivative calculated using the Dalzell analytical method (see the original
paper, Ref. Dalzell, 1999 (Dal99), and the explicit formulae reported in the
Appendix of Ref. Li and Li, 2021).

Table 1:  List of mean flow terms in space-like formulation: $\partial\phi_{0}/\partial x$.  Case 1 | Eq. (25) | 3rd-order, nonlocal
---|---|---
Case 2 | Eq. (28) | 3rd-order, nonlocal
Dys79 Dysthe (1979) | Eq. (5) | 3rd-order, nonlocal
|  | deep water
Sed03 Sedletsky (2003) | Eq. (3) | 3rd-order
Sed03 | Eq. (29) | 2nd-order
Slu05 Slunyaev (2005) | Eq. (86) | 3rd-order

For benchmarking and validation purposes, we use the same parameters as in
Ref. Li and Li, 2021 (case C in their Table II), namely a Gaussian (amplitude)
spectrum $S(k)$ with peak wavenumber $k_{0}=0.0277$ m-1, wavelength
$\lambda_{0}=2\pi/k_{0}$, standard deviation of the spectrum given by
symmetrical values $k_{w}=k_{w1}=k_{w2}=0.27k_{0}$, steepness
$\varepsilon=0.3$ and different values of the normalised depth $k_{0}h$ in the
range $[0.5,50]$, thus, the case of finite and infinite water depth
conditions. The angular frequency is calculated from
$\omega_{0}^{2}=gk_{0}\tanh(k_{0}h)$. In particular, the surface elevation at
first order in steepness given by the superposition of $N=30$ waves is used to
calculate the intensity of wave envelope, i.e. $|U|^{2}$, and then, the
horizontal velocity $\partial\phi_{0}/\partial x$ through the different
relations, as listed in Table 1. We consider a focused wave group, composed of
$N$ individual sinusoidal wave components that are in phase at a single point
in time and space Orszaghova  _et al._ (2014) (the origin in our case), and a
random sea state by using a uniform distribution for the phases. An example of
sea state realization is given in App. D.

Figure 2: Mean flow $\phi_{0x}$ for $N=30$ waves focusing at $x=0$ as
described by the expressions listed in Table 1, for different values of
$k_{0}h$.

Figure 3: Same as Fig. 2 for the evolution of $N=30$ waves with random phases.

Figure 4: Deviation operator with respect to the Dysthe expression (a) and to
the Dalzell one (b) for the derivative in space $\phi_{0x}$, as defined in the
main text, for the different cases listed in Table 1.

The results are summarised in Fig. 2 for waves focusing at $x=0$, and in Fig.
3 for the case of waves superposition with random phases. In both cases, we
see that the horizontal velocity calculated by Eq. (29) (gray line), which
corresponds to the second-order expression in Sed03, reproduces well the
Dalzell waveform (green line) only for $k_{0}h\leq 2$, and cannot be applied
in deep water where it gives a waveform amplitude much smaller than Dysthe’s
result (black line). On the contrary, Dysthe’s expression should not be used
for $k_{0}h<10$ as in our considered cases, especially for large amplitude
waveforms, where it does not attain Dalzell’s accuracy. Although an elementary
consideration of the dispersion relation suggests that $k_{0}h\approx 5$
should be indistinguishable from $k_{0}h\to\infty$, the dynamics of the mean
flow is consistently different from this limit up to $k_{0}h=10$. The
expressions for the horizontal velocity given by Case 1 [Eq. (25)] and Case 2
[Eq. (28)] behave in a similar way at all depths, providing in general an
approximation that corresponds well to the Dalzell solution in intermediate
waters, and to Dysthe in deep waters. It is also important to note that the
expressions provided in Sed03 and Slu05 at third-order in steepness [Eqs. (3)
and (86), respectively], give different results and do not correspond to the
other models at any depth.

More detailed features can be deduced from Fig. 4, where the deviation
operator with respect to the Dysthe expression, defined as
$D_{\rm{Dysthe}}=N^{-1}\int(\phi_{0x}-\phi_{0x}^{\rm{Dysthe}})^{2}\,dx$, and
to the Dalzell one,
$D_{\rm{Dalzell}}=N^{-1}\int(\phi_{0x}-\phi_{0x}^{\rm{Dalzell}})^{2}\,dx$,
with $\phi_{0x}=\partial\phi_{0}/\partial x$, are shown as a function of the
non-dimensional water depth $k_{0}h$. The integral is calculated over
$L=80\lambda_{0}$, and the normalization coefficient is
$N=(\omega_{0DW}\lambda_{0})^{2}L$, with $\omega_{0DW}$ calculated in deep
water (DW). The Stokes series expansion of the velocity potential (and thus
Dalzell’s model) converges in shallow water Rahman (1970) if
$3\varepsilon/(2k_{0}h)^{3}\ll 1$, thus at low Ursell number, which implies in
our case $k_{0}h\gg 0.48$. This region is excluded in Fig. 4. It can be seen
that the expression for the horizontal velocity given by Case 2 [Eq. (28] is
accurate at second-order at all depths, almost superposing to the second-order
Dalzell solution, while the one given by Case 1 [Eq. (25)] is the only
expression that consistently converges to the Dysthe in the deep-water limit
expression, and is equivalent to Case 2 for $k_{0}h<5$. Note that the third-
order corrections provided in Sed03 [Eq. (3)] and in Slu05 [Eq. (86)] are
different, and do not extend the validity to deeper water regimes of the
second-order expression [Eq. (29)], which is accurate for $k_{0}h<2$ (see Fig.
4b). This raises a warning on the quantitative accuracy of these third-order
expressions.

##  III Fourth-order equation: propagation in space

In order to transform Eq. (1) to an expression describing the propagation in
space, that is necessary to describe the nonlinear evolution of waves in a
laboratory flume, a change of variables is needed

$\tau=\varepsilon\left(t-\frac{x}{c_{\mathrm{g}}}\right);\;\xi=\varepsilon^{2}x.$
(30)

Through this transformation, the third-order terms are only subject to a
rescaling,

$\alpha=\frac{\hat{\alpha}}{c_{\mathrm{g}}^{3}};\;\beta=\frac{\hat{\beta}}{c_{\mathrm{g}}}.$
(31)

The higher-order terms are instead dramatically modified. Indeed, a mixed
derivative appears from the second-order dispersion term,
$\partial^{2}{U}/\partial{x}^{2}$, that reads as
$-(\omega^{\prime\prime}/c_{\mathrm{g}})\partial^{2}U/\partial x\partial t$.
In the multiscale spirit, the time-like NLS (i.e., terms at third-order in
steepness) is used to estimate this term, which results in corrections at
fourth-order. The resulting time-like form of the evolution equation does not
appear explicitly in the literature and is given by (setting $\varepsilon=1$,
and changing the notation as $\tau\to t$ and $\xi\to x$):

$i\frac{\partial U}{\partial x}+\alpha\frac{\partial^{2}U}{\partial
t^{2}}\underbrace{-\beta|U|^{2}U}_{\text{incl.~{}Mean
Flow}}=-i\alpha_{3}\frac{\partial^{3}U}{\partial
t^{3}}\underbrace{+i\beta_{21}|U|^{2}\frac{\partial U}{\partial
t}+i\beta_{22}U^{2}\frac{\partial U^{*}}{\partial t}}_{\text{incl.~{}Mean
Flow}},$ (32)

whose dispersion and nonlinear coefficients are given in App. A and B, and
their dependence on $k_{0}h$ is shown in Figs. 5 and 6. High-order dispersion
terms [Inparticular; thefourth-
orderdispersionterm$α_4∂^4U/∂t^4$(tobeaddedonther.h.s.~ofEq.~(\ref{eq:HONLStime}))isgenerallyusedtocanceloutresonancesthatcanmakethemodelnumericallyunstable;
asdiscussedin~]hara_mei_1991 are easily included up to arbitrary order
following Refs. Trulsen  _et al._ , 2000; Eeltink  _et al._ , 2019, reducing
the constraints in bandwidth of the original Dysthe equation.

Using the relation Sedletsky (2003)

$\frac{\partial\phi_{01}}{\partial t}=-c_{g}\frac{\partial\phi_{01}}{\partial
x}$ (33)

Eq. (32) can be rewritten in the equivalent form:

$i\frac{\partial U}{\partial x}+\alpha\frac{\partial^{2}U}{\partial
t^{2}}-\beta_{D}|U|^{2}U=-i\alpha_{3}\frac{\partial^{3}U}{\partial
t^{3}}+i\mathcal{B}_{21}|U|^{2}\frac{\partial U}{\partial
t}+i\mathcal{B}_{22}U^{2}\frac{\partial U^{*}}{\partial
t}\underbrace{-\frac{\mu_{g}k_{0}}{4\sigma
c_{g}^{2}}U\frac{\partial\phi_{0}}{\partial t}}_{\text{Mean Flow}}$ (34)

where $\beta_{D}=\hat{\beta}_{D}/c_{g}$,
$\mathcal{B}_{21}=\omega_{0}k_{0}\tilde{Q}_{41}/c_{g}^{2}-4\alpha\beta_{D}c_{g}$,
$\mathcal{B}_{22}=\omega_{0}k_{0}\tilde{Q}_{42}/c_{\mathrm{g}}^{2}-2\alpha\beta_{D}c_{\mathrm{g}}$,
with $\hat{\beta}_{D}$, $\alpha$, $\alpha_{3}$ given in App. A, and
$\tilde{Q}_{41}$, $\tilde{Q}_{42}$ in App. B. The dependence of the nonlinear
coefficients on $k_{0}h$ is illustrated in Fig. 6. Note that
$\beta,\beta_{21},\beta_{22},\mathcal{B}_{21}$ and $\mathcal{B}_{22}$ diverge
quite strongly for $k_{0}h\to 0$, i.e. particularly in the defocusing regime.
That said and as can be also analytically verified, all coefficients correctly
reproduce their deep-water limit at $k_{0}h\to\infty$.

Figure 5: Dispersion coefficients (for $\omega_{0}=1$ Hz). Zero crossings are
marked on the horizontal axis.

Figure 6: Nonlinear coefficients (a) in Eq. (32), and (b) in Eq. (34),
normalized with respect to their deep-water values (in Dysthe equation, Eq.
(37)). Zero crossings are marked in the insets.

###  III.1 Mean flow in time-like equations

At main order in steepness, Eqs. (3) and (33) imply

$\frac{\partial|U|^{2}}{\partial t}=-c_{g}\frac{\partial|U|^{2}}{\partial x}$
(35)

Thus, $\phi_{01}$ and $|U|^{2}$ do not depend on $x$ and $t$ separately, but
only through their combination $(x-c_{g}t)$, so that at leading order the
Fourier transform in space, ${\cal F}_{x}$, can be replaced by the Fourier
transform in time, ${\cal F}_{t}$. From Eq. (5), in the deep-water limit it is
possible to simply exchange the derivative with respect to space with that
with respect to time and vice versa, since the Hilbert transform of the
derivative is the derivative of the Hilbert transform, i.e. these two linear
operators commute. As such:

$\frac{\partial\phi_{0}}{\partial
t}=\frac{\omega_{0}}{2}\mathcal{H}_{t}\left[\frac{\partial|U|^{2}}{\partial
t}\right]$ (36)

The time-like Dysthe equation is given by the deep-water limit of Eq. (34) and
reads, using the above expression for the mean flow term Trulsen and Dysthe
(1997); Kit and Shemer (2002); Onorato  _et al._ (2005); Goullet and Choi
(2011); Eeltink  _et al._ (2017)

$i\frac{\partial U}{\partial
x}-\frac{k_{0}}{\omega_{0}^{2}}\frac{\partial^{2}U}{\partial
t^{2}}-k_{0}^{3}|U|^{2}U=2i\frac{k_{0}^{3}}{\omega_{0}}\biggl{(}4|U|^{2}\frac{\partial
U}{\partial t}+U^{2}\frac{\partial U^{*}}{\partial
t}+\underbrace{iU\mathcal{H}_{t}\left[\frac{\partial|U|^{2}}{\partial
t}\right]}_{\text{Mean Flow}}\biggr{)}$ (37)

where $\alpha_{3}\to 0$, since $k\propto\omega^{2}$, and
$\hat{\beta}_{21}\to\frac{3}{2}\omega_{0}k_{0}$,
$\hat{\beta}_{22}\to\frac{1}{4}\omega_{0}k_{0}$, $\beta_{21}\to
8k_{0}^{3}/\omega_{0}$, and $\beta_{22}\to 2k_{0}^{3}/\omega_{0}$.

For the last term in Eq. (34), the Sed03 expression, as in Eq. (3), can be
written in terms of the time derivative, using Eq. (33):

$\displaystyle\frac{\partial\phi_{0}}{\partial
t}=-c_{g}\frac{\omega_{0}}{2}\frac{k_{0}\mu_{g}}{\sigma\nu}|U|^{2}$ (38)
$\displaystyle-$ $\displaystyle
i\frac{4\omega_{0}\sigma}{\nu}\tilde{q}_{40S}\left(U\frac{\partial
U^{*}}{\partial t}-U^{*}\frac{\partial U}{\partial t}\right).$

However, this expression goes to zero in the deep-water limit, and thus it
does not converge to the Dysthe mean flow given in Eq. (36).

Replacing Eq. (33) in the Laplace equation and the surface boundary
conditions, and repeating the same steps of Sec. II.3, we finally get the
following expressions for the derivative in time of $\phi_{0}$ for Case 1 and
2, respectively, to be inserted in the evolution Eq. (34):

$\displaystyle\frac{\partial\phi_{0}}{\partial t}$ $\displaystyle=$
$\displaystyle D\mathcal{F}^{-1}_{t}\Bigg{\\{}\frac{i}{\tanh(\omega
h/c_{g})}\mathcal{F}_{t}\left[\frac{\partial|U|^{2}}{\partial
t}\right]\Bigg{\\}},\,\rm{for~{}Case~{}1}$ (39)
$\displaystyle\frac{\partial\phi_{0}}{\partial t}$ $\displaystyle=$
$\displaystyle D^{\prime}\mathcal{F}^{-1}_{t}\Bigg{\\{}\frac{i}{\tanh(\omega
h/c_{g})\left[1-c_{g}\omega/(g\tanh(\omega
h/c_{g}))\right]}\mathcal{F}_{t}\left[\frac{\partial|U|^{2}}{\partial
t}\right]\Bigg{\\}},\,\rm{for~{}Case~{}2}$ (40)

Figure 7: Mean flow $\phi_{0t}$ for N = 30 waves focusing at $t=0$ described
by the expressions listed in Table 2, for different values of $k_{0}h$.

###  III.2 Numerical comparisons

Table 2:  List of mean flow terms in time-like formulation: $\partial\phi_{0}/\partial t$.  Case 1 | Eq. (39) | 3rd-order,
---|---|---
|  | non-instantaneous
Case 2 | Eq. (40) | 3rd-order, non-inst.
Dys79 Dysthe (1979) | Eq. (36) | 3rd-order, non-inst.
|  | deep water
Sed03 Sedletsky (2003) | Eq. (38) | 3rd-order
Sed03 | 1st term on | 2nd-order
| r.h.s of Eq. (38) |
Slu05 Slunyaev (2005) | Eq. (87) | 3rd-order

We now compare the expressions for $\partial\phi_{0}/\partial t$ listed in
Table 2 with the sub-harmonic velocity potential $\phi_{20}$ at second-order
in steepness and its time derivative calculated using the Dalzell analytical
method in Refs. Dalzell, 1999; Li and Li, 2021.

We use the same parameters as in Sec. II.4, starting in this case by a
Gaussian (amplitude) spectrum $S(\omega)$ peaked at $\omega_{0}=1$ Hz.

Figure 8: Deviation operator with respect to the Dysthe expression (a) and to
the Dalzell one (b) for the derivative in time $\phi_{0t}$, as defined in the
main text, for the different cases listed in Table 2.

The results for $N=30$ waves focusing at $t=0$ are shown in Fig. 7. We see
that the second-order expression in Sed03 reproduces well the waveform only
for $k_{0}h\leq 1$, showing discrepancies with respect to Dalzell’s solution
at $k_{0}h=2$ and not converging to Dysthe’s solution in deep-water. As for
the space-like form, the expressions for $\phi_{0t}$ given by Case 1 [Eq.
(39)] and Case 2 [Eq. (40)] have a similar behavior at all depths, providing
in general approximations that are as accurate as the Dalzell solution in
intermediate waters, and as the Dysthe expression at deep waters. As before,
the third-order expressions provided by Eqs. (38) and (87) are different and
do not correspond to the other models. Fig. 8 shows the deviation operator
with respect to the Dysthe expression, defined similarly to the spatial
counterpart (see Sec. II.4) as
$D_{\rm{Dysthe}}=N^{-1}\int(\phi_{0t}-\phi_{0t}^{\rm{Dysthe}})^{2}\,dt$, and
to the Dalzell one,
$D_{\rm{Dalzell}}=N^{-1}\int(\phi_{0t}-\phi_{0t}^{\rm{Dalzell}})^{2}\,dt$ as a
function of the non-dimensional water depth $k_{0}h$. The integral is now
calculated over $T=80~{}T_{0}$, and the normalization coefficient is
$N=(\omega_{0}\lambda_{0DW})^{4}T$, with $\lambda_{0DW}$ calculated at deep
water. As for the space-like case, it can be seen that the expression given by
Case 2 [Eq. 40)] is accurate at second-order at all depths as the second-order
Dalzell solution, while the one given by Case 1 [Eq. (39)] is the only
expression that converges to the Dysthe term in the deep-water limit, and is
equivalent to Case 2 for $k_{0}h<3$.

##  IV Conclusion

During the evolution of surface gravity waves, fluid particles experience
Stokes drift along the propagation direction of the waves, and a return flow
in vertical and horizontal directions, which closes the water-mass transport.
Both processes give rise to a mean flow at the surface that needs to be
accounted for in order to accurately predict the transport of pollutants
DiBenedetto (2020); Cózar  _et al._ (2014); Christensen  _et al._ (2018) (such
as oil, microplastics, etc) and, more in general, the impact of waves
evolution at small scale on the ocean circulation at large scale Babanin
(2006); Onink  _et al._ (2019); van Sebille  _et al._ (2020); Cunningham,
Higgins, and van den Bremer (2022). We have derived nonlocal (viz. non-
instantaneous) expressions of the mean flow [Eqs. (25) and (39)] that
correctly converge to the deep-water limit at third-order in steepness, while
being equivalent to second-order formulations in intermediate waters. We have
included these expressions in an envelope evolution equation at fourth-order
in steepness in both, space-like [Eq. (2)] and time-like [Eq. (34)]
formulations. We emphasize that the time-like form is relevant to study the
evolution of unidirectional wave groups in space, thus, for modelling
experiments in water wave flumes at high accuracy in arbitrary depths. Future
work will be focusing on the experimental validation of our results in
different water depth regimes.

###### Acknowledgements.

A. G., J. K. and M. B. acknowledge financial support from the Swiss National
Science Foundation (Project No. 200020-175697). D.E. acknowledges financial
support from the Swiss National Science Foundation (Fellowship P2GEP2-191480).
A. C. acknowledges support from Kyoto University’s Hakubi Center for Advanced
Research.

## Author declarations

### Conflict of Interest Statement

The authors have no conflicts to disclose.

### Author Contributions

Alexis Gomel: conceptualization (equal); formal analysis (equal);
investigation (equal); writing/original draft (supporting); writing/review and
editing (equal). Corentin Montessuit: formal analysis (equal); investigation
(equal). Andrea Armaroli: conceptualization (equal); formal analysis (equal);
investigation (equal); writing/review and editing (equal). Debbie Eeltink:
formal analysis (equal); writing/review and editing (equal). Amin Chabchoub:
supervision (equal); writing/review and editing (equal). Jérôme Kasparian:
supervision (equal); funding acquisition (lead); writing/review and editing
(equal). Maura Brunetti: conceptualization (equal); formal analysis (equal);
investigation (equal); methodology (lead); supervision (equal);
writing/original draft (lead); writing/review and editing (equal).

## Data Availability Statement

The data that support the findings of this study and the matlab scripts are
available from the corresponding author upon reasonable request.

##  Appendix A Dispersion and nonlinear coefficients

The dispersion coefficients are:

$\displaystyle\hat{\alpha}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\omega^{\prime\prime}(k_{0})=\alpha\,c_{\mathrm{g}}^{3}$
(41) $\displaystyle\alpha$ $\displaystyle=$
$\displaystyle-\frac{1}{2}k^{\prime\prime}(\omega_{0})=-\frac{1}{2\omega_{0}c_{g}}\left[1-\frac{gh}{c^{2}_{g}}(1-\kappa\sigma)(1-\sigma^{2})\right]$
(42) $\displaystyle\hat{\alpha}_{3}$ $\displaystyle=$
$\displaystyle\frac{1}{6}\omega^{\prime\prime\prime}(k_{0})$ $\displaystyle=$
$\displaystyle\frac{\omega_{0}}{48k_{0}^{3}\sigma}\left[3\sigma+\kappa(1-\sigma^{2})\left(-3+\kappa\left(-\frac{3}{\sigma}+\frac{3\kappa}{\sigma^{2}}+13\kappa-15\kappa(1-\sigma^{2})-9\sigma\right)\right)\right]$
$\displaystyle\alpha_{3}$ $\displaystyle=$
$\displaystyle-\frac{1}{6}k^{\prime\prime\prime}(\omega_{0})=\frac{\hat{\alpha}_{3}}{c_{\mathrm{g}}^{4}}-2\alpha^{2}c_{\mathrm{g}}$
(44) $\displaystyle k^{\prime\prime\prime}(\omega_{0})$ $\displaystyle=$
$\displaystyle-\frac{1}{c_{\mathrm{g}}^{4}}\frac{\partial^{2}c_{\mathrm{g}}}{\partial
k^{2}}+\frac{3}{c_{\mathrm{g}}^{5}}\left(\frac{\partial
c_{\mathrm{g}}}{\partial k}\right)^{2}$ (45) $\displaystyle\hat{\alpha}_{4}$
$\displaystyle\equiv$
$\displaystyle\frac{1}{24}\frac{\partial^{4}\omega}{\partial
k^{4}}=\frac{\omega_{0}}{384k_{0}^{4}}[-15+12\kappa^{2}+46\kappa^{4}$
$\displaystyle+4\kappa(3-5\kappa^{2})\coth\kappa+2\kappa^{2}(3+10\kappa^{2})\coth^{2}\kappa+12\kappa^{3}\coth^{3}\kappa-15\kappa^{4}\coth^{4}\kappa$
$\displaystyle+\kappa\sigma(-12+68\kappa^{2}+3\kappa\sigma(-6-52\kappa^{2}+5\kappa\sigma(-4+7\kappa\sigma)))]$
$\displaystyle\alpha_{4}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{24}\frac{\partial^{4}k}{\partial\omega^{4}}=-\frac{1}{24c_{\mathrm{g}}^{5}}\frac{\partial^{3}c_{\mathrm{g}}}{\partial
k^{3}}+\frac{5}{12c_{\mathrm{g}}^{6}}\frac{\partial^{2}c_{\mathrm{g}}}{\partial
k^{2}}\frac{\partial c_{\mathrm{g}}}{\partial
k}-\frac{5}{8c_{\mathrm{g}}^{7}}\left(\frac{\partial c_{\mathrm{g}}}{\partial
k}\right)^{3}$ (47) $\displaystyle=$
$\displaystyle-\frac{\hat{\alpha}_{4}}{c_{\mathrm{g}}^{5}}+\frac{5}{c_{g}^{6}}\hat{\alpha}_{3}\hat{\alpha}-\frac{5}{c_{\mathrm{g}}^{7}}\hat{\alpha}^{3}$

where $\kappa=k_{0}h$, $\sigma=\tanh\kappa$ and $c_{\mathrm{g}}$ is the group
velocity:

$c_{g}\equiv\frac{\partial\omega}{\partial
k}=\frac{g}{2\omega_{0}}[\sigma+\kappa(1-\sigma^{2})]$ (48)

The third-order nonlinear coefficients are:

$\displaystyle\hat{\beta}_{D}$ $\displaystyle=$
$\displaystyle-\frac{\omega_{0}k_{0}^{2}}{16\sigma^{4}}(2\sigma^{6}-13\sigma^{4}+12\sigma^{2}-9)$
(49) $\displaystyle\hat{\beta}$ $\displaystyle=$
$\displaystyle\beta\,{c_{\mathrm{g}}}=\hat{\beta}_{D}+\omega_{0}k_{0}^{2}\frac{\mu_{g}^{2}}{8\sigma^{2}\nu}=\hat{\beta}_{D}-\frac{k_{0}\mu_{g}}{4\sigma
h}D$ (50) $\displaystyle\beta_{D}$ $\displaystyle=$
$\displaystyle\frac{\hat{\beta}_{D}}{c_{g}}$ (51) $\displaystyle\beta$
$\displaystyle=$
$\displaystyle\frac{\omega_{0}k_{0}^{2}}{16\sigma^{4}c_{g}}\left\\{9-10\sigma^{2}+9\sigma^{4}-\frac{2\sigma^{2}c^{2}_{g}}{gh-
c_{g}^{2}}\left[4\frac{c^{2}_{p}}{c^{2}_{g}}+4\frac{c_{p}}{c_{g}}(1-\sigma^{2})+\frac{gh}{c^{2}_{g}}(1-\sigma^{2})^{2}\right]\right\\}$
(52)

where, interestingly, the coefficients in the curly brackets can be expressed
in terms of $\kappa$: $c_{g}/c_{p}=(\sigma+\kappa(1-\sigma^{2}))/(2\sigma)$,
$gh/c_{g}^{2}=(c_{p}^{2}/c_{g}^{2})\kappa/\sigma$, $2\sigma^{2}c^{2}_{g}/(gh-
c_{g}^{2})=2\sigma(c_{g}^{2}/c_{p}^{2})/(\kappa/\sigma-c_{g}^{2}/c_{p}^{2})$
and

$\displaystyle D$ $\displaystyle=$
$\displaystyle-h\frac{\omega_{0}}{2}\frac{k_{0}\mu_{g}}{\sigma\nu}=\frac{\omega_{0}}{2}\frac{\kappa}{2\sigma}\frac{2+(1-\sigma^{2})c_{g}/c_{p}}{\kappa-\sigma
c_{g}^{2}/c_{p}^{2}}=\frac{D^{\prime}}{1-c_{g}^{2}/(gh)}$ (53) $\displaystyle
D^{\prime}$ $\displaystyle=$
$\displaystyle\frac{\omega_{0}}{2\sigma}(1+C_{FD})$ (54)
$\displaystyle\mu_{g}$ $\displaystyle=$
$\displaystyle\frac{2\sigma}{\omega_{0}}(2\omega-
kc_{g}(\sigma^{2}-1))=(\sigma^{2}-1)^{2}\kappa-\sigma(\sigma^{2}-5)=4\sigma(1+C_{FD})$
(55) $\displaystyle\nu$ $\displaystyle=$
$\displaystyle\frac{4k_{0}\sigma}{g}(c_{g}^{2}-gh)=[(\sigma+1)^{2}\kappa-\sigma][(\sigma-1)^{2}\kappa-\sigma]$
(56) $\displaystyle C_{FD}$ $\displaystyle=$
$\displaystyle\frac{\omega_{0}c_{g}}{g\sinh(2\kappa)}$ (57)

The higher-order nonlinear coefficients are:

$\displaystyle\beta_{21}$ $\displaystyle=$
$\displaystyle\frac{\hat{\beta}_{21}}{c_{\mathrm{g}}^{2}}-4\alpha\beta
c_{\mathrm{g}}$ (58) $\displaystyle\beta_{22}$ $\displaystyle=$
$\displaystyle\frac{\hat{\beta}_{22}}{c_{\mathrm{g}}^{2}}-2\alpha\beta
c_{\mathrm{g}}$ (59) $\displaystyle\hat{\beta}_{21}$ $\displaystyle=$
$\displaystyle\omega_{0}k_{0}Q_{41S}$ (60) $\displaystyle\hat{\beta}_{22}$
$\displaystyle=$ $\displaystyle\omega_{0}k_{0}Q_{42S}$ (61)
$\displaystyle\mathcal{B}_{21}$ $\displaystyle=$
$\displaystyle\omega_{0}k_{0}\tilde{Q}_{41}/c_{g}^{2}-4\alpha\beta_{D}c_{g}$
(62) $\displaystyle\mathcal{B}_{22}$ $\displaystyle=$
$\displaystyle\omega_{0}k_{0}\tilde{Q}_{42}/c_{\mathrm{g}}^{2}-2\alpha\beta_{D}c_{\mathrm{g}}$
(63)

where $Q_{41S}$, $Q_{42S}$, $\tilde{Q}_{41}$, $\tilde{Q}_{42}$ are given in
App. B.

##  Appendix B Notation used in Sedletsky (2003) (Sed03)

The main coefficients in Sedletsky’s notation are Sedletsky (2003):

$\displaystyle Q_{41}$ $\displaystyle=$
$\displaystyle\tilde{Q}_{41}-\frac{\mu_{g}}{\nu}\tilde{q}_{40}$ (64)
$\displaystyle Q_{42}$ $\displaystyle=$
$\displaystyle\tilde{Q}_{42}+\frac{\mu_{g}}{\nu}\tilde{q}_{40}$ (65)
$\displaystyle\tilde{q}_{40}$ $\displaystyle=$
$\displaystyle\frac{1}{32\sigma^{3}\nu}[(\sigma^{2}-1)^{5}\kappa^{4}-4\sigma(2\sigma^{4}+9\sigma^{2}+5)(\sigma^{2}-1)^{2}\kappa^{3}$
(66)
$\displaystyle+2\sigma^{2}(9\sigma^{4}+16\sigma^{2}-9)(\sigma^{2}-1)\kappa^{2}$
$\displaystyle-4\sigma^{3}(4\sigma^{4}-9\sigma^{2}-7)\kappa+5\sigma^{4}(\sigma^{2}-5)]$
$\displaystyle\tilde{Q}_{41}$ $\displaystyle=$ $\displaystyle\tilde{q}_{41}$
(67) $\displaystyle=$
$\displaystyle\frac{1}{16\sigma^{5}\nu}[(2\sigma^{6}-11\sigma^{4}-10\sigma^{2}+27)(\sigma^{2}-1)^{3}\kappa^{3}-\sigma(6\sigma^{8}-21\sigma^{6}+9\sigma^{4}-43\sigma^{2}+81)(\sigma^{2}-1)\kappa^{2}$
$\displaystyle+\sigma^{2}(6\sigma^{8}-15\sigma^{6}-77\sigma^{4}+71\sigma^{2}-81)\kappa-\sigma^{3}(\sigma^{2}+1)(2\sigma^{4}-7\sigma^{2}-27)]$
$\displaystyle\tilde{q}_{42}$ $\displaystyle=$
$\displaystyle\frac{1}{32\sigma^{5}\nu}[(4\sigma^{6}-13\sigma^{4}+10\sigma^{2}-9)(\sigma^{2}-1)^{3}\kappa^{3}-\sigma(12\sigma^{8}-51\sigma^{6}+17\sigma^{4}-\sigma^{2}-9)(\sigma^{2}-1)\kappa^{2}$
(68)
$\displaystyle+\sigma^{2}(12\sigma^{8}-67\sigma^{6}+33\sigma^{4}-\sigma^{2}-9)\kappa-\sigma^{3}(4\sigma^{6}-29\sigma^{4}+42\sigma^{2}-9)]$
$\displaystyle q_{3}$ $\displaystyle=$
$\displaystyle-\frac{\hat{\beta}}{\omega_{0}k_{0}^{2}}$ (69)
$\displaystyle\tilde{Q}_{42}$ $\displaystyle=$
$\displaystyle\tilde{q}_{42}-2\frac{c_{g}}{c_{p}}q_{3}$ (70)

The final expressions for $Q_{41}$ and $Q_{42}$ are given in Eqs. (67) and
(68) in Ref. Sedletsky, 2003 and reported here for completeness:

$\displaystyle\mathcal{Q}_{41}$ $\displaystyle=$
$\displaystyle\frac{1}{32\sigma^{5}\nu^{2}}\\{(3\sigma^{6}-20\sigma^{4}-21\sigma^{2}+54)(\sigma^{2}-1)^{5}\kappa^{5}$
(71)
$\displaystyle-\sigma(11\sigma^{8}-99\sigma^{6}-61\sigma^{4}+7\sigma^{2}+270)(\sigma^{2}-1)^{3}\kappa^{4}$
$\displaystyle+2\sigma^{2}(\sigma^{2}-1)(7\sigma^{10}-58\sigma^{8}+38\sigma^{6}+52\sigma^{4}-181\sigma^{2}+270)\kappa^{3}$
$\displaystyle-2\sigma^{3}(3\sigma^{10}+18\sigma^{8}-146\sigma^{6}-172\sigma^{4}+183\sigma^{2}-270)\kappa^{2}$
$\displaystyle-\sigma^{4}(\sigma^{8}-109\sigma^{6}+517\sigma^{4}+217\sigma^{2}+270)\kappa$
$\displaystyle+\sigma^{5}(\sigma^{6}-40\sigma^{4}+193\sigma^{2}+54)\\}$
$\displaystyle\mathcal{Q}_{42}$ $\displaystyle=$
$\displaystyle\frac{1}{32\sigma^{5}\nu^{2}}\\{-(3\sigma^{6}+7\sigma^{4}-11\sigma^{2}+9)(\sigma^{2}-1)^{5}\kappa^{5}$
(72)
$\displaystyle+\sigma(11\sigma^{8}-48\sigma^{6}+66\sigma^{4}+8\sigma^{2}+27)(\sigma^{2}-1)^{3}\kappa^{4}$
$\displaystyle-2\sigma^{2}(\sigma^{2}-1)(7\sigma^{10}-79\sigma^{8}+282\sigma^{6}-154\sigma^{4}-\sigma^{2}+9)\kappa^{3}$
$\displaystyle+2\sigma^{3}(3\sigma^{10}-63\sigma^{8}+314\sigma^{6}-218\sigma^{4}+19\sigma^{2}+9)\kappa^{2}$
$\displaystyle+\sigma^{4}(\sigma^{8}+20\sigma^{6}-158\sigma^{4}-28\sigma^{2}-27)\kappa-\sigma^{5}(\sigma^{6}-7\sigma^{4}+7\sigma^{2}-9)\\}$

We have verified that they are equivalent to the expressions obtained using
Eqs. (64)-(65).

From Eqs. (71)-(72), in the deep-water limit $\kappa\to\infty$, $\sigma\to 1$,
$\nu\to 1-4\kappa$, $\mu_{g}\to 4$, $\mathcal{Q}_{41}\to 768/(32\cdot 16)=3/2$
and $\mathcal{Q}_{42}\to 128/(32\cdot 16)=1/4$ recovering the Dysthe result
for such terms.

As suggested in Ref. Gandzha, Sedletsky, and Dutykh, 2014, the above
expressions can be modified to agree with the results presented in Ref.
Slunyaev, 2005. However, we have verified that the (small) modification
suggested in Ref. Gandzha, Sedletsky, and Dutykh, 2014 missed a factor 2, the
right one being the following:

$\displaystyle\tilde{q}_{40S}$ $\displaystyle=$
$\displaystyle\tilde{q}_{40}+\frac{\Delta}{2}\frac{\nu}{\mu_{g}}$ (73)
$\displaystyle Q_{41S}$ $\displaystyle=$ $\displaystyle
Q_{41}-\frac{\Delta}{2}$ (74) $\displaystyle Q_{42S}$ $\displaystyle=$
$\displaystyle Q_{42}+\frac{\Delta}{2}$ (75)

where the term $\Delta$ is defined in Ref. Gandzha, Sedletsky, and Dutykh,
2014:

$\displaystyle\Delta$ $\displaystyle=$
$\displaystyle-\frac{\sigma^{2}-1}{16\sigma^{3}\nu}[(\sigma^{2}-1)^{3}(3\sigma^{2}+1)\kappa^{3}-\sigma(\sigma^{2}-1)(5\sigma^{4}-18\sigma^{2}-3)\kappa^{2}$
(76) $\displaystyle+$
$\displaystyle\sigma^{2}(\sigma^{2}-1)(\sigma^{2}-9)\kappa+\sigma^{3}(\sigma^{2}-5)]$

In the deep-water limit $\kappa\to\infty$, $Q_{41S}\to 3/2$ and $Q_{42S}\to
1/4$, since $\Delta\to 0$, thus recovering the Dysthe result for such terms.

Note, however, that the final equations that include the mean flow (namely,
Eqs. (2) and (34)) are not affected by such ambiguity, since their high-order
nonlinear coefficients only depend on $\tilde{Q}_{41}$ and $\tilde{Q}_{42}$.

##  Appendix C Notation used in Slunyaev (2005) (Slu05)

One can use the notation in Ref. Slunyaev, 2005, where
$\hat{\beta}_{21}=\omega_{0}k_{0}Q_{41S}$ and
$\hat{\beta}_{22}=\omega_{0}k_{0}Q_{42S}$, and
$Q_{41S}=(h^{3}\omega_{0})\tilde{\alpha}_{21}/(\kappa^{3}\sigma^{2})$,
$Q_{42S}=(h^{3}\omega_{0})\tilde{\alpha}_{22}/(\kappa^{3}\sigma^{2})$, with
$\tilde{\alpha}_{21}=\tilde{\rho}_{21}-\tilde{\rho}_{12}\gamma_{2}$ and
$\tilde{\alpha}_{22}=\tilde{\rho}_{22}+\tilde{\rho}_{12}\gamma_{2}$,
$\tilde{\rho}_{21}=P_{21}+s\beta_{1}\gamma_{1}$,
$\tilde{\rho}_{22}=P_{22}-s\beta_{1}\gamma_{1}$. Here
$\beta_{1}=-\hat{\alpha}$ and the other coefficients are given by222We have
expressed Slunyaev’s coefficients in terms of Sedletsky’s expressions defined
in App. A and B whenever possible.

$\displaystyle h^{3}\omega_{0}P_{21}$ $\displaystyle=$
$\displaystyle\left(\kappa^{2}\frac{(\sigma^{2}-1)(-4\sigma^{4}+3\sigma^{2}+1)}{8\sigma^{2}}+\kappa\frac{4\sigma^{4}-9\sigma^{2}+3}{4\sigma}+\frac{-4\sigma^{2}+19}{8}\right)h^{3}\omega_{0}\gamma_{1}$
(77)
$\displaystyle+\kappa^{2}\frac{-\sigma^{4}+3}{2(\sigma^{2}+1)}h\omega_{0}\chi_{2}+\left(\kappa^{2}\frac{-3\sigma^{6}+7\sigma^{4}-9\sigma^{2}-3}{4\sigma(\sigma^{2}+1)}+3\kappa\frac{\sigma^{4}-5}{4(\sigma^{2}+1)}\right)h^{2}\omega_{0}\chi_{1}$
$\displaystyle+\kappa^{4}\frac{(\sigma^{2}-1)(11\sigma^{4}-12\sigma^{2}-3)}{16\sigma}+\kappa^{3}\frac{-11\sigma^{4}+40\sigma^{2}-9}{16}$
$\displaystyle h^{3}\omega_{0}P_{22}$ $\displaystyle=$
$\displaystyle\left(-\kappa^{2}\frac{(\sigma^{2}-1)^{2}}{8}+\kappa\frac{\sigma^{4}-5\sigma^{2}+2}{4\sigma}+\frac{-\sigma^{2}+8}{8}\right)h^{3}\omega_{0}\gamma_{1}$
(78)
$\displaystyle+\left(\kappa^{2}\frac{(\sigma^{2}-1)(\sigma^{4}+3)}{4\sigma(\sigma^{2}+1)}-\kappa\frac{\sigma^{4}+3}{4(\sigma^{2}+1)}\right)h^{2}\omega_{0}\chi_{1}$
$\displaystyle+\kappa^{4}\frac{(\sigma^{2}-1)(-3\sigma^{4}-8\sigma^{2}+3)}{32\sigma}+3\kappa^{3}\frac{\sigma^{4}-1}{32}$
$\displaystyle h^{2}\omega_{0}s$ $\displaystyle=$
$\displaystyle\kappa^{2}\frac{\sigma^{2}-1}{2}$ (79) $\displaystyle V_{d}^{2}$
$\displaystyle=$ $\displaystyle gh-c_{g}^{2}=-\frac{g\nu}{4k_{0}\sigma}$ (80)
$\displaystyle h^{3}\omega_{0}\gamma_{1}$ $\displaystyle=$ $\displaystyle
h^{3}\omega_{0}\frac{k_{0}^{2}c_{g}(\sigma^{2}-1)-2\omega_{0}k_{0}}{4V_{d}^{2}}=\frac{\kappa^{3}\sigma\mu_{g}}{2\nu}$
(81) $\displaystyle\gamma_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{V_{d}^{2}}\left[2c_{g}\gamma_{1}\beta_{1}+k_{0}^{2}\beta_{1}\frac{(\sigma^{2}-1)}{4}+\frac{\omega^{2}-k_{0}^{2}c_{g}^{2}(\sigma^{2}-1)}{4\omega_{0}}\right]$
(82) $\displaystyle\tilde{\rho}_{12}$ $\displaystyle=$
$\displaystyle\frac{2\omega_{0}k_{0}-k_{0}^{2}c_{g}(\sigma^{2}-1)}{2\omega_{0}}=\frac{k\mu_{g}}{4\sigma}$
(83) $\displaystyle h^{2}\omega_{0}\chi_{1}$ $\displaystyle=$ $\displaystyle
3\kappa^{2}\frac{\sigma^{4}-1}{8\sigma^{2}}$ (84) $\displaystyle
h\omega_{0}\chi_{2}$ $\displaystyle=$
$\displaystyle\left(\kappa\frac{-\sigma^{3}+3}{\sigma}+1\right)\frac{h^{2}\omega_{0}\chi_{1}}{\kappa}+\frac{3\kappa^{2}(\sigma^{2}-1)(3\sigma^{2}+1)}{16\sigma}+9\kappa\frac{1-\sigma^{4}}{16\sigma^{2}}$
(85)

Using the same notation as in Sed03 for the reconstruction at leading order of
the surface elevation,
$\eta(\chi,t)=\frac{1}{2}[U(x,t)\exp(i(k_{0}\chi-\omega_{0}t))+{\rm{c.~{}c.}}]$,
the mean flow term is written as (see Eqs. (32), (43), (45) in Ref. Slunyaev,
2005):

$\frac{\partial\phi_{0}}{\partial
x}=\frac{\omega_{0}}{2}\frac{k_{0}\mu_{g}}{\sigma\nu}|U|^{2}-i\frac{c_{p}^{2}}{\sigma^{2}}\left(\gamma_{2}+\frac{\beta_{1}\gamma_{1}}{c_{g}}\right)\left(U\frac{\partial
U^{*}}{\partial x}-U^{*}\frac{\partial U}{\partial x}\right)$ (86)

Note that, while the first term is equal to the corresponding term in Eq. (3)
from Sedletsky’s derivation, the second one has a different coefficient, since
${4\omega_{0}\sigma}\tilde{q}_{40}/\nu\not=c_{p}^{2}{\sigma^{2}}(\gamma_{2}+\beta_{1}\gamma_{1}/c_{g})/{\sigma^{2}}$
(even if we take into account the correction $\tilde{q}_{40S}$ in Eq. (73)).

Using Eq. (33), the derivative in time of $\phi_{0}$ is

$\frac{\partial\phi_{0}}{\partial
t}=-c_{g}\frac{\omega_{0}}{2}\frac{k_{0}\mu_{g}}{\sigma\nu}|U|^{2}-i\frac{c_{p}^{2}}{\sigma^{2}}\left(\gamma_{2}+\frac{\beta_{1}\gamma_{1}}{c_{g}}\right)\left(U\frac{\partial
U^{*}}{\partial t}-U^{*}\frac{\partial U}{\partial t}\right)$ (87)

##  Appendix D Example of sea state realization

Following the definitions in Ref. Li and Li, 2021, we consider a Gaussian
amplitude spectrum, denoted by $S(k)$, given by

$S(k)=\exp\left(-\frac{(k-k_{0})^{2}}{2k_{w}^{2}}\right)$ (88)

for $k>0$, where $k_{0}=0.0277$ m-1 and $k_{w}=0.27~{}k_{0}$ is the
dimensional bandwidth. Other spectra, like JONSWAP or Pierson-Moskowitz, can
also be used. This yields the following surface elevation at first order in
steepness:

$\eta_{1}(x,t)=\frac{A_{p}}{2}\frac{\int_{0}^{\infty}S(k)e^{i[k(x-x_{f})-\omega(t-t_{f})])}dk}{\int_{0}^{\infty}S(k)dk}+c.c.$
(89)

where $A_{p}$, $x_{f}$, $t_{f}$ are the amplitude, position and time for the
group at linear focus, with steepness given by $\varepsilon=A_{p}k_{0}=0.3$.
An example of sea state realization at second order in steepness, obtained
using the Dalzell development Dalzell (1999), is shown in Fig. 9 for the case
of a focused wave group at $x_{f}=0$, $t_{f}=0$ for $k_{0}h=1.5$. The power
spectrum can be obtained as Young (2020) $P(k)=|\hat{\eta}|^{2}/(2dk)$, where
$\hat{\eta}$ is the Fourier transform in space of the surface elevation, from
which the significant wave height $H_{s}$ can be obtained as
$H_{s}=4\sqrt{m_{0}}=5.2$ m, $m_{0}$ being equal to the area under the power
spectrum curve.

Figure 9: Sea surface elevation $\eta$ and its components at first
($\eta_{1}$) and second order in steepness ($\eta_{22}$ and the set-down
$\eta_{20}$) in the case of a focused wave group at the origin for
$k_{0}h=1.5$: (a) full range of considered spatial values; (b) same as (a),
zoomed on $30\lambda_{0}$.

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