the Creative Commons Attribution 3.0 License.

the Creative Commons Attribution 3.0 License.

# Note on the directional properties of meter-scale gravity waves

### Alvise Benetazzo

### Fabrice Ardhuin

The directional distribution of the energy of young waves is
bimodal for frequencies above twice the peak frequency; i.e., their
directional distribution exhibits two peaks in different directions and a
minimum between. Here we analyze in detail a typical case measured with a
peak frequency *f*_{p}=0.18 Hz and a wind speed of
10.7 m s^{−1} using a stereo-video system. This technique allows
for the separation of free waves from the spectrum of the sea-surface
elevation. The latter indeed tend to reduce the contrast between the two
peaks and the background. The directional distribution for a given wavenumber
is nearly symmetric, with the angle distance between the two peaks growing
with frequency, reaching 150^{∘} at 35 times the peak wavenumber
*k*_{p} and increasing up to 45 *k*_{p}. When considering only
free waves, the lobe ratio, the ratio of oblique peak energy density over
energy in the wind direction, increases linearly with the non-dimensional
wavenumber *k*∕*k*_{p}, up to a value of 6 at $k/{k}_{\mathrm{p}}\simeq \mathrm{22}$, and possibly more for shorter components. These observations extend to
shorter components' previous measurements, and have important consequences
for wave properties sensitive to the directional distribution, such as
surface slopes, Stokes drift or microseism sources.

Directional properties of waves shorter than the dominant scale play a very important role in many aspects that range from air–sea momentum fluxes (Plant, 1982) to remote sensing, surface drift (Ardhuin et al., 2009) and underwater acoustics (Duennebier et al., 2012). In a landmark paper, Munk (2009) analyzed the linear trends of down-wind and cross-wind mean square slopes of the sea surface, as measured by satellites Bréon and Henriot (2006). These trends cannot be explained by today's understanding of ocean wave spectra, and he proposed that there may be localized sources that could generate oblique propagating waves looking like ship wakes. However, as he put it, the dataset “says nothing about time and space scales” because the reflectance measurements that they present are integrated across all wave scales. Munk further challenged us all: “I look forward to intensive sea-going experiments over the next few years demolishing the proposed interpretations”. We thus went out to sea with the objective of resolving space scales and timescales, and providing further constraints on the wave properties.

Previous time-resolved measurements of ocean waves have clearly established a
prevalence of directional bimodality at frequencies above twice the peak
frequency *f*_{p}, using in situ array
(Long and Resio, 2007; Young et al., 1995) and buoy data
(Ewans, 1998; Wang and Hwang, 2001). These were confirmed by airborne remote
sensing techniques used by Hwang et al. (2000) and
Romero and Melville (2010). All the resolved wavenumber spectra have been
limited to $f/{f}_{\mathrm{p}}<\mathrm{4}$. Numerical modeling by Banner and Young (1994) suggests that the bimodality is caused
by the nonlinear cascade of free wave energy from dominant to high
frequencies. Bimodality is also found after having solved for the nonlinear
evolution equation of the surface elevation field, whether computing it for
Gaussian wave packets according to a nonlinear Schrödinger equation
(Dysthe et al., 2013) or for unimodal wave spectra from the Euler equations
(Toffoli et al., 2010). Alves and Banner (2003) demonstrate the importance
of the parametrizations of wave generation and dissipation in the setting of
bimodality. The model results of Gagnaire-Renou et al. (2010) show that bimodality is followed at smaller
scales by a return to a unimodal directional distribution, somewhere above
$f/{f}_{\mathrm{p}}=\mathrm{10}$, depending on the parametrizations of wave generation
and dissipation.

The distribution of radar backscatter as a function of azimuth clearly shows
that the directional wave spectrum is unimodal above 6 cm
wavelength in the gravity-capillary range (Elfouhaily et al., 1997). Recent backscatter data in L-band presented by
Yueh et al. (2013) show a larger cross-wind than down-wind backscatter,
consistent with a bimodal distribution at scales around 1 m
wavelength, at least for wind speeds around 5 m s^{−1}.

As shown by Leckler et al. (2015), stereo-video imagery is capable of
resolving these waves and providing information on the timescales and space
scales needed to interpret integrated wave parameters such as down-wind and
cross-wind mean square slopes. In that earlier paper, a record with young
wind waves was analyzed (*U*_{23}=13.2 m s^{−1}, *f*_{p}=0.33 Hz). That record revealed the presence of second-order
harmonics and a strong bimodality of the directional distribution. Here we
use the same measurement method and analyze the directional properties of the
free waves in more detail. In particular, we analyze new data that provide a
wider range of frequencies and quantitatively characterize the bimodality
characteristics together with its impacts on several physical variables.

The data and analysis methods are presented in Sect. 2. Directional distributions and bimodality are described in Sect. 3. Discussions and conclusions follow in Sect. 4.

## 2.1 Stereo processing

We have chosen one typical stereo record with dominant waves longer than
those described in Leckler et al. (2015). It was acquired on 10 March 2014,
starting at 09:40 UTC, from the Acqua Alta oceanographic research platform,
15 km offshore of Venice, Italy, in the northern Adriatic Sea. The
mean water depth there is approximately *d*=17 m. The experimental
setup has been described in detail in Benetazzo et al. (2015). It is made up
of two digital cameras mounted on a horizontal bar, properly synchronized and
calibrated. The cameras are located *d*=12.5 m above the mean sea
level. The stereo device points in a direction oriented 46^{∘}
clockwise from geographical north, i.e., looking to the northeast. The
cameras' elevation angle is 50^{∘}. This record is 30 min long and
uses a 15 Hz sampling rate.

In the following all variables use the meteorological convention; that is,
the directions are directions from which wave, wind and current come.
Provenance directions, unless otherwise specified, are measured anticlockwise
from the direction along the bar, i.e., 136^{∘} clockwise from
geographical north.

The mean wind speed measured at 10 m above sea level is
10.7 m s^{−1}, with mean direction *θ*_{U}=77^{∘}
(northeasterly). The significant wave height estimated from the stereo system
is *H*_{m0}=1.33 m, with peak frequency
*f*_{p}=0.185 Hz, corresponding to a dominant wavelength
on the order of 45 m. We note that wave
gauges on the platform give independent measurements of
*H*_{m0}=1.36 m and *f*_{p}=0.189 Hz. Dominant
waves and shorter components of the wave spectrum can be considered deep
water waves. An acoustic Doppler current profiler (ADCP) deployed at the sea
floor provides measurements of the horizontal current vector with a vertical
resolution of 1 m.

The raw video images are processed into a three-dimensional surface elevation
matrix $\mathit{\zeta}(x,y,t)$ following the method of Benetazzo et al. (2015). A
local Cartesian reference frame is defined, in which the surface elevation is
reconstructed, with horizontal axes *x* and *y*. By convention, the cameras'
look direction is the *y* axis, increasing away from the cameras, and the *x*
axis is perpendicular, increasing towards the right of the cameras. The sea
surface is discretized with a pixel size $\mathrm{\Delta}x=\mathrm{\Delta}y=\mathrm{20}$ cm. A snapshot of the reconstructed sea-surface elevation map is
presented in Fig. 1. We have selected a 25.6 m by
25.6 m area for Fourier
analysis, delimited by a black square. Its location, close to the cameras, is
chosen to minimize errors in the estimate of the surface elevation. These
errors increase with increasing distance from the cameras, and are dominated
by the quantization error (Benetazzo, 2006). Wavelengths longer than
25 m can be resolved using standard slope array techniques
(Graber et al., 2000) as done by Leckler et al. (2015). These
longer components are not the focus of the present paper.

All of our analysis is based on a three-dimensional power spectral density of
these data (Figs. 2 and 3). This is
obtained by applying a Hann spatiotemporal window with 50 % overlap in
time, and averaging the spectra in time following Welch (1967). The
frequency resolution is Δ*f*=0.015 Hz. The double-sided
Cartesian spectrum $E({k}_{x},{k}_{y},f)$ is normalized so that

is the variance of the surface elevation.

The polar spectrum is more convenient for the study of directional distributions and for working at a given wavenumber. The single-sided polar spectrum is

where $k=({k}_{x}^{\mathrm{2}}+{k}_{y}^{\mathrm{2}}{)}^{\mathrm{1}/\mathrm{2}}$ and $\mathit{\theta}=\mathrm{arctan}({k}_{y},{k}_{x})+\mathit{\pi}$
is the wave provenance direction. For convenience, we use a regular polar
grid whose resolution is set to Δ*k*=0.17 rad m^{−1} and Δ*θ*=1^{∘}.

## 2.2 General properties of the three-dimensional spectrum

The surface elevation spectrum can be interpreted as the distribution of wave energy, which can generally be divided into free and bound waves,

Free waves have a relation between wavenumber and frequency that closely
follows the linear dispersion relation. In the presence of a horizontally
homogeneous and stationary current vertical profile *u*(*z*), and in the limit
of small wave steepness, this dispersion relation is given by
Stewart and Joy (1974):

where

and the effective current *U*(*k*) is approximated by a weighted integral of
the Eulerian current *u*(*z*) over the water column:

Here we have assumed that the current has a constant direction *α* at
all depths. Moreover, Eq. (6) holds only for linear
waves, i.e., waves for which hydrodynamic nonlinearities have been
neglected, although free waves may encompass some weakly nonlinear
contributions – see Leckler et al. (2015) and Janssen (2009). The
depth weighting in the integral of Eq. (6) gives a
stronger influence of surface currents on shorter wave components. In
practice, waves with wavenumber *k* feel the integrated current over a depth
$\sim \mathrm{1}/k$. For convenience, the inverse function providing the wavenumber as
a function of frequency and direction will be denoted *κ* in the
following, namely by definition: if

then

where $\mathit{f}=\left[f\mathrm{cos}\right(\mathit{\theta}),f\mathrm{sin}(\mathit{\theta}\left)\right]$.

Once the effective current (Eq. 6) is known, the
location of free waves in the (** k**,

*f*) plane can be deduced from Eq. (4), which relates the radian frequency 2

*π*

*f*to the wave vector

**. It is represented in Figs. 2 and 3 by a black solid line. The addition of a current is necessary to fit the observations of energy distribution. The free modes' bimodality is clearly visible, i.e., the fact that two energy patches detach progressively from a main direction as the wave scale decreases.**

*k*Bound waves are dominated by the second-order interaction of free components with wavenumbers
*k*_{1} and *k*_{2}. The sum interaction gives
waves of wavenumber $\mathit{k}={\mathit{k}}_{\mathrm{1}}+{\mathit{k}}_{\mathrm{2}}$, and
frequency $\mathit{\omega}=\mathit{\omega}\left({\mathit{k}}_{\mathrm{1}}\right)+\mathit{\omega}\left({\mathit{k}}_{\mathrm{2}}\right)$, with an energy *E*_{sum}.
The difference interaction gives $\mathit{k}={\mathit{k}}_{\mathrm{1}}-{\mathit{k}}_{\mathrm{2}}$ and
$\mathit{\omega}=\left|\mathit{\omega}\left({\mathit{k}}_{\mathrm{1}}\right)-\mathit{\omega}\left({\mathit{k}}_{\mathrm{2}}\right)\right|$ ,
*E*_{diff}.
These two kinds of interactions have themselves distinct signatures in the surface elevation spectrum, namely

At a given propagation direction, the sum interaction is found at frequencies
higher than the dispersion surface, while the difference interaction
components are found at lower frequencies
(Krogstad and Trulsen, 2010; Leckler et al., 2015). *E*_{bound} can be
deduced from *E*_{free} (Hasselmann, 1962). More specifically,
for a narrow spectrum, the sum interaction component is characterized by a
signature in the (** k**,

*f*) plane (Senet et al., 2001)

also referred to as the first harmonic. The latter corresponds to sum interactions of free waves traveling in the same direction, with the same frequency and propagation direction, for which the interaction cross section is highest (Aubourg and Mordant, 2015). This curve is represented in Figs. 2 and 3 by a white solid line. Its equivalent without a current is also plotted as a white dashed line. Nonlinear components do not exhibit the same directionality as linear waves in general, especially from snapshots at constant frequency. In this case, the harmonic peaks in the dominant wave direction (see Fig. 3b–f), although this is not the only possible behavior.

We also note that waves that are probably reflected by the platform legs are
present, as shown by a white arrow in Fig. 2b, and their
energy decreases with increasing distance. When interpreted as plane waves,
the reflected components appear slightly off the dispersion relation of the
incident waves. Fitting the current for the incident waves gives *U*≃0.22 m s^{−1}, whereas a fit for the reflected components only
would give a current velocity of 0.4 m s^{−1}.

Finally, there are other spectral features that do not correspond to surface
waves, which we shall call noise. We distinguish four kinds of noise.
Firstly, a background noise is present below −50 dB, particularly
visible in Fig. 3d–f and j–l. This noise
practically limits the use of stereo video to *k*<8 rad m^{−1}.
Secondly, some energy propagates at a speed of 0.4 m s^{−1} along
the look direction and at slower speeds for other directions (green dashed
lines in Figs. 2b and 3i–n). For
*k*=2 rad m^{−1} this noise amplitude is comparable in magnitude to
the free wave signature and is distributed around a surface of the type 2*π**f*=0.4sin^{2}(*θ*)*k*, for $\mathit{\theta}\in [\mathrm{0};\mathit{\pi}[$ only. It could be
associated with the difference interaction between incident and reflected
wavenumbers.

Besides these noises, uncertainties in the spectral densities are caused by
the poor spectral resolution close to *k*=0, and quantization error noise,
mostly for *k*>7.5 rad m^{−1} and in the look direction
(Benetazzo, 2006). We thus exclude from our analysis the spectral
components for which any of the following conditions is met

Outside of these components the spectrum is separated into free and bound components. This uses a determination of the effective current that is discussed in the Appendix. Identifying the free wave energy as that close to the linear dispersion relation, the bound components are defined as the rest,

and the same is done for the frequency–direction spectrum.

The spectrum of free waves *E*_{free}(*k*,*θ*) is clearly bimodal
for *k*>4*k*_{p}. Bimodal energy distributions can be characterized
from the knowledge of the position and height of the energy peaks. The
processing starts from the radially integrated directional distributions,
both at a given frequency
${E}_{\mathrm{free}}\left(\mathit{\theta}\right)=\int dk\phantom{\rule{0.33em}{0ex}}{E}_{\mathrm{free}}\left(k,\mathit{\theta}\right)$
and wavenumber
${E}_{\mathrm{free}}\left(\mathit{\theta}\right)=\int df\phantom{\rule{0.33em}{0ex}}{E}_{\mathrm{free}}\left(f,\mathit{\theta}\right)$
(see Fig. 4). The same processing is performed on bound
waves, obtained from Eq. (16). Bound waves are found to stand for
a significant proportion of the overall energy at given slices (62 and
64 % on panels b and d, respectively), but only the free waves are
bimodal. The energy level of bound waves at these small scales is dominated
by the contribution of the more energetic longer waves. The contributions of
the sum and difference interactions are also indicated. Directional
distributions are centered on the spectral mean direction of wave propagation
*θ*_{m}=68^{∘} from Benetazzo et al. (2015).

As the directional distributions are noisy, they need to be fitted by an
appropriate shape function. Inspired by Ewans (1998), the double
pseudo-Voigt function with a positive floor has been chosen. The fit is
performed using the Python lmfit package (Newville et al., 2014). The double
pseudo-Voigt function allows for more various curve shapes than only two
Gaussian beams, with its 9 degrees of freedom, when the Lorentzian fraction
*x* is nonzero,

where

with ${\mathit{\sigma}}_{g}=\mathit{\sigma}/\sqrt{\mathrm{2}\mathrm{ln}\mathrm{2}}$. The use of a double Voigt profile
does not strictly ensure a smooth periodic distribution (around $\mathit{\theta}={\mathit{\theta}}_{\mathrm{m}}\pm \mathit{\pi}$). However, in practice, due to the relative
directional narrowness of the bimodal profiles, the constant energy floor is
quickly reached away from the mean wave propagation direction. Bimodality can
then be characterized using a set of three remarkable points in the double
Voigt profile (see Fig. 4b, d) (Wang and Hwang, 2001),
i.e., the central minimum (*θ*_{0},*E*_{0}) and the two peaks
(*θ*_{1},*E*_{1}) and (*θ*_{2},*E*_{2}), with *θ*_{1}<*θ*_{2}.

The present-case bimodality is characterized by plotting the positions of the
two peaks and the so-called lobe ratios as a function of normalized
wavenumber *k*∕*k*_{p} (see Fig. 5).
Full markers (triangles, disks and stars) correspond to estimates from
constant wavenumber snapshots, while empty markers (circles, diamonds and
upside down triangles) correspond to estimates from constant frequency
snapshots. For the latter, the *x*-axis is *κ*(*f*)∕*k*_{p} – see
Eq. (7). Bimodal profiles are first detected at
*f*=0.43 Hz and *k*=0.7 rad m^{−1}, corresponding
approximately to $k/{k}_{\mathrm{p}}=\mathrm{5}$. The previously mentioned direction
*θ*_{m} is the best compromise for centering the bimodality. An
empirical parametrization is found for the constant wavenumber estimates of
the directional distributions, that is,

where the value *a*=0.039 was found after a least squares fit of constant
wavenumber data points in the range $\mathrm{5}<k/{k}_{\mathrm{p}}<\mathrm{45}$. This
parametrization fits most of the measurements, except the position of the
peak furthest from the current direction (*θ*_{1}), particularly for the
estimates from constant frequency snapshots above $k/{k}_{\mathrm{p}}=\mathrm{22}$ (see
the black arrow in Fig. 5). At this
location, the peak is progressively moved towards the center of the
directional distribution.

The lobe ratios *r*_{i} are conventionally defined as the ratios of the
energy of each peak of the bimodal directional distribution to the one of the
central minimum (Wang and Hwang, 2001), namely

We can note that they are particularly sensitive to the background energy
level. This level is given by the constant term *C*^{st} of the
fitting function (17), without knowing whether this level is an
actual surface wave signal or noise. The lobe ratios of the current record
are plotted in Fig. 5b, from estimates at
constant wavenumber only, with and without this background term. The overall
tendency consists in their linear and symmetric increase at intermediate wave
scales, until $k/{k}_{\mathrm{p}}\simeq \mathrm{22}$. As for peak positions in
Fig. 5a, the lobe ratios from constant
frequency estimates exhibit a more pronounced asymmetry. A fit is performed
over constant wavenumber lobe ratios (full markers) for which
$\mathrm{4}<k/{k}_{\mathrm{p}}<\mathrm{22}$, providing the parametrization

Above $k/{k}_{\mathrm{p}}\simeq \mathrm{22}$, the lobe ratios progressively decrease,
except if the background term *C*^{st}>0 is removed (transparent
markers in Fig. 5). The lobe ratio
decrease is natural since the lobe ratios without background are

as long as *r*_{i}>1 and the proportion of background noise increases towards
shorter scales. We cannot however formally associate this noise with an
actual surface wave signal.

The Stokes drift current for linear waves in deep water is (Kenyon, 1969)

where

is plotted in Fig. 6a, and where the impact of the wave field directionality is included in the factor

plotted in Fig. 6b with

and

the Longuet-Higgins coefficients with respect to the mean wave propagation direction, where

and

The resulting Stokes drift vertical profile has been plotted in
Fig. 6e, together with two profiles compatible with the
effective current measured from stereo video (see Appendix). Waves slightly
shorter than peak waves are the main contributors to the Stokes drift
(Fig. 6a). Half of the Stokes drift is carried by waves with
frequencies greater than 0.4 Hz approximately (wavelength
10 m). In order to correct for the stereo device field of view
limitation (long waves are indeed not spatially resolved), the wavenumber
spectrum for *k*<Δ*k* has been evaluated from their frequency spectrum
using the Jacobian transform. In particular, the short wave bimodality
substantially reduces the contribution of those waves to the Stokes drift. At
a given wave scale, contributions symmetric with respect to the mean wave
propagation direction cancel out laterally, resulting in a decrease in the
Stokes drift at those scales (Fig. 6b). In particular, the
Stokes drift at *z*=0 is reduced by 44 % (from 0.11 to
0.06 m s^{−1}), which is greater than the approximately 20 %
reduction reported in Ardhuin et al. (2009) and Breivik et al. (2014). Mean
square slopes in the up-wind and cross-wind direction are defined by

and

and are of particular interest for ocean remote sensing (Munk, 2009). Due to the wave field bimodality, the mean square slope is rather carried by cross-wind propagating waves than up-wind ones (Fig. 6c) at short scales, as was qualitatively described in Elfouhaily's delta ratio (Elfouhaily et al., 1997). Bound waves cause a slight increase in the mean square slopes in the up-wind direction. Finally, short wave directional distributions are critical in understanding the source of seismo-acoustic noise (Farrell and Munk, 2010), caused by quasi-stationary pressure oscillations at the sea surface (Longuet-Higgins, 1950). The spectrum of stationary pressure waves can be written as

where the free wave contribution is proportional to the overlap integral *I*
(Wilson et al., 2003),

given by

The correction arising from bound harmonics *F*_{p,bound} has never
been rigorously considered in past studies, but should remain weak. The
overlap integral (Eq. 34) has been plotted in
Fig. 6d. For the same energy level at a given wave scale, the
overlap integral is increased from a unimodal to a bimodal directional
distribution. In particular, at short enough scales, more energy should be
radiated by a bimodal surface wave field than by an equivalent isotropic wave
field (for which the value 1∕(2*π*) is reached). The parametrization of
Duennebier et al. (2012) is also superimposed.

The characteristics of a bimodal short surface wave energy distribution are extracted from the spectrum of a single stereo-video reconstruction of the sea surface at the Acqua Alta platform. Peak positions and lobe ratios are computed which can quantitatively summarize the observations, with associated parametrizations.

The domain of surface waves which can be measured with this system depends on
the configuration of the device. Stereo video has a wide scale coverage and
an upper bound that is not limited by the Nyquist frequency and wavelength
(here ${f}_{\mathrm{s}}/\mathrm{2}=\mathrm{7.5}$ Hz and $\mathrm{1}/\left(\mathrm{2}\mathrm{\Delta}x\right)=\mathrm{15.7}$ rad m^{−1}), but rather by the accuracy of reconstruction
of short waves of small amplitudes. The effective directional resolution can
be computed using

In our case, for 1 Hz waves, Δ*θ*∼5^{∘}, and
for 0.5 Hz waves, it reaches 15^{∘}.

Bimodality has been characterized by extracting the positions of the two
bimodal peaks and the central minimum from directional distributions of the
free waves, either at constant frequency or constant wavenumber (see
Fig. 4). Free waves only are affected by bimodality at
both a given wavenumber and frequency. Moreover, bound waves' distribution
can be deduced from one of the free waves (Leckler et al., 2015). The short
waves' field bimodality starts growing between $k/{k}_{\mathrm{p}}=\mathrm{3.6}$ and
$k/{k}_{\mathrm{p}}=\mathrm{4.3}$ from constant wavenumber snapshots, or between
$k/{k}_{\mathrm{p}}=\mathrm{4.8}$ ($f/{f}_{\mathrm{p}}=\mathrm{2.16}$) and $k/{k}_{\mathrm{p}}=\mathrm{5.2}$
($f/{f}_{\mathrm{p}}=\mathrm{2.23}$) from constant frequency snapshots. Bimodality may be
initiated at even larger scales and not detected, due to a directional
resolution at those scales which is smaller than the peak distance –
Eq. (35). The two peaks then detach from the
main direction *θ*_{m}. Apart from the asymmetry introduced by
the current, the three points characterizing bimodality sensibly fluctuate
around their positions, reaching a distance of ∼160^{∘} towards
$k/{k}_{\mathrm{p}}=\mathrm{45}$, the latter being the accepted limit for stereo-video
measurement validity. The real bimodal directional distribution differs from
its parametrizations mainly at wave scales smaller than $k/{k}_{\mathrm{p}}=\mathrm{22}$.
This is particularly the case for the peak furthest from the current
direction at a given frequency (see the arrow in
Fig. 5a) which gets away from the
parametrization by slowly moving closer to the
center of the directional distribution, while the constant wavenumber
estimates remain close to the parametrization, with an almost perfectly
symmetric distribution. This difference might come from the effect of the
current. Indeed, the two peaks at a given wavenumber do not appear at the
same frequency because of the presence of the current. For example, in
Fig. 4d, the waves at *k*=4.0 rad m^{−1} exhibit
a bimodal behavior which is symmetric with respect to the main wave
propagation direction *θ*_{m}. In the absence of a current, the
two peaks would appear at the same frequency: *f*=1.0 Hz. In this
snapshot, the peak furthest from the current, i.e., *θ*_{1}, is
located at a frequency *f*=0.95 Hz, while the other peak is located
at a frequency *f*=1.022 Hz. The shift is larger for the former,
*θ*_{1}, than for *θ*_{2}; hence, the current is a cause of
asymmetry in bimodality characteristics. As a consequence, the wavenumber
parametrization is more robust against currents than is the frequency one, as
was already observed by Wyatt (2012). This is the one chosen throughout
this paper. This asymmetry is also visible in
Fig. 5b.

We have reported on new stereo-video recordings of ocean waves that offer a
wider range of resolved scales than previous datasets, up to $k/{k}_{\mathrm{p}}=\mathrm{45}$. Looking at free waves, the bimodal nature of their directional
distribution is more pronounced at the shorter scales, with a separation of
the two peaks that exceeds 160^{∘}. This distribution was found to
reduce the Stokes drift by over 40 % compared to a unidirectional wave
field, with a significant source of acoustic noise due to waves in opposing
directions, typically larger than an isotropic spectrum for $k/{k}_{\mathrm{p}}>\mathrm{20}$. These effects are partly compensated for by the importance of bound
harmonics which have directions closer to the mean wave direction. The
analysis of the contribution of these nonlinear components to the Stokes
drift and acoustic noise is beyond the scope of the present paper.

This stereo-video record is part of a larger dataset that is intended to be made available in the near future.

The extraction of free wave components from the surface elevation spectrum is
detailed here. Looking at snapshots of Figs. 2 and
3, there is no ambiguity in the distinction between
free (along the dispersion line, black) and bound (white line) waves, except
if the spatial resolution is limiting. From
Eqs. (4) to (6), their
location in the (** k**,

*f*) space is determined by the value of the effective current

**, Eq. (6), at each wave scale. It depends on the true near-surface current vertical profile**

*U**u*(

*z*).

## A1 Effective current measurement

Starting from a snapshot of the surface elevation spectrum at a given frequency or wavenumber (see Fig. 4 for example), the estimate of the effective current which minimizes the cost function

is retained, where (*U*_{x},*U*_{y}) are the coordinates of the effective
current vector in the local frame and *w*_{j} are empirical weights,
normalized so that ${\sum}_{j=\mathrm{1}}^{N}{w}_{j}=N$, and where *χ* is the
expected standard deviation of model and data, adapted from
Senet et al. (2001):

The flexibility of this method relies upon a careful choice of data points and weights.

This choice is exposed here for the case of a constant frequency snapshot.
First, a rough estimate of the current is required in order to approximately
locate the dispersion relation. For this experiment, the current vector does
not vary much with wave scales. This estimate is obtained by manually
selecting data points on the dispersion relation of ∼1 Hz waves
(10 are enough) and by minimizing the cost function with equal weights. The
index 0 is put on the current value obtained
(*U*_{0}=0.22 m s^{−1} and *α*_{0}=102^{∘}). A second
cost function is computed by keeping only data points for which

Then, among the rest, the ones with the lowest signal-to-noise ratio are removed:

The weights are

and then normalized, where the index *j* runs over remaining data points, and
*d**S*_{j} stands for the elementary surface around data point *j*. The
minimization algorithm is initiated with values *U*_{0} and *α*_{0}, and
run until convergence at each frequency, providing a more accurate result
than the rough estimate. Finally, free waves are isolated using this more
accurate estimate. Only the points with coordinates (*k*_{j},*θ*_{j}) are
kept if they fall in the interval

The same procedure can be applied to constant wavenumber snapshots. It is the
same as the one of the previous paragraph, after having exchanged *k* with
*f* and *κ* with *ω*∕(2*π*).

## A2 Current profile

The effective current values at all wave scales from the extraction of free
waves are plotted in Fig. A1. Either as a function of
frequency or wavenumber, both estimates show a gradual increase in the
effective current magnitude towards *U*_{0}=0.22 m s^{−1} and
*α*_{0}=102^{∘}. These values are in agreement with ADCP
measurements indicating a current of 0.19 m s^{−1} flowing from
the direction 110^{∘} at 2 m below the surface, which is
already too deep to significantly influence the effective current. Effective
currents for typical wind drift profiles $u\left(z\right)={u}_{a}+{u}_{b}{e}^{z/\mathit{\delta}}$ are
plotted in Fig. A1 for various values of *δ*. We assume
that *u*_{b}=0.1 m s^{−1}, i.e., 1 % of the surface wind
speed, and ${u}_{a}={U}_{\mathrm{0}}-{u}_{b}=\mathrm{0.12}$ m s^{−1}, yielding a surface
vertical shear ${u}_{b}/\mathit{\delta}=\mathrm{0.36}\phantom{\rule{0.33em}{0ex}}{\mathrm{s}}^{-\mathrm{1}}$. Two plausible profiles
have been plotted in Fig. 6e, for which

denoted Stereo 1, and

Stereo 2.

The authors declare that they have no conflict of interest.

This work is supported by LabexMer via grant ANR-10-LABX-19-01, and the
Copernicus Marine Environment Monitoring Service (CMEMS) as part of the
Service Evolution program. Installation of the stereo system was supported by
the funding from the RITMARE flagship project. The Italian Research for the
Sea was coordinated by the Italian National Research Council and funded by
the Italian Ministry of Education, University and Research within the
National Research Program 2011–2015.

Edited by:
Andreas Sterl

Reviewed by: Frederic Dias and one anonymous
referee

Alves, J. H. G. and Banner, M. L.: Performance of a saturation-based dissipation rate source term in modeling the fetch-limited evolution of wind waves, J. Phys. Oceanogr., 33, 1274–1298, https://doi.org/10.1175/1520-0485(2003)033<1274:poasds>2.0.co;2, 2003. a

Ardhuin, F., Marié, L., Rascle, N., Forget, P., and Roland, A.: Observation and estimation of Lagrangian, Stokes and Eulerian currents induced by wind and waves at the sea surface, J. Phys. Oceanogr., 39, 2820–2838, https://doi.org/10.1175/2009jpo4169.1, 2009. a, b

Aubourg, Q. and Mordant, N.: Nonlocal resonances in weak turbulence of gravity-capillary waves, Phys. Rev. Lett., 114, https://doi.org/10.1103/PhysRevLett.114.144501, 2015. a

Banner, M. L. and Young, I. R.: Modeling spectral dissipation in the evolution of wind waves. Part I: assessment of existing model performance, J. Phys. Oceanogr., 24, 1550–1570, https://doi.org/10.1175/1520-0485(1994)024<1550:msdite>2.0.co;2, 1994. a

Benetazzo, A.: Measurements of short water waves using stereo matched image sequences, Coastal. Eng., 53, 1013–1032, 2006. a, b

Benetazzo, A., Barbariol, F., Bergamasco, F., Torsello, A., Carniel, S., and Sclavo, M.: Observation of extreme sea waves in a space-time ensemble, J. Phys. Oceanogr., 45, 2261–2275, https://doi.org/10.1175/JPO-D-15-0017.1, 2015. a, b, c

Breivik, Ø., Janssen, P. A. E. M., and Bidlot, J.-R.: Approximate Stokes Drift Profiles in Deep Water, J. Phys. Oceanogr., 44, 2433–2445, https://doi.org/10.1175/jpo-d-14-0020.1, 2014. a, b

Bréon, F. M. and Henriot, N.: Spaceborne observations of ocean glint reflectance and modeling of wave slope distributions, J. Geophys. Res., 111, C06005, https://doi.org/10.1029/2005jc003343, 2006. a

Duennebier, F. K., Lukas, R., Nosal, E.-M., Aucan, J., and Weller, R. A.: Wind, Waves, and Acoustic Background Levels at Station ALOHA, J. Geophys. Res., 117, C03017, https://doi.org/10.1029/2011JC007267, 2012. a, b

Dysthe, K. B., Trulsen, K., Krogstad, H. E., and Socquet-Juglard, H.: Evolution of a narrow-band spectrum of random surface gravity waves, J. Fluid Mech., 478, 1–10, https://doi.org/10.1017/s0022112002002616, 2003. a

Elfouhaily, T., Chapron, B., Katsaros, K., and Vandemark, D.: A unified directional spectrum for long and short wind-driven waves, J. Geophys. Res., 102, 15781–15796, 1997. a, b

Ewans, K. C.: Observations of the Directional Spectrum of Fetch-Limited Waves, J. Phys. Oceanogr., 28, 495–512, https://doi.org/10.1175/1520-0485(1998)028<0495:ootdso>2.0.co;2 1998. a, b

Farrell, W. E. and Munk, W.: Booms and busts in the deep, J. Phys. Oceanogr., 40, 2159–2169, 2010. a

Gagnaire-Renou, E., Benoit, M., and Forget, P.: Ocean wave spectrum properties as derived from quasi-exact computations of nonlinear wave-wave interactions, J. Geophys. Res., 115, C12058, https://doi.org/10.1029/2009JC005665, 2010. a

Graber, H. C., Terray, E. A., Donelan, M. A., Drennan, W. M., Leer, J. C. V. and Peters, D. B. : ASIS–A New Air-Sea Interaction Spar Buoy: Design and Performance at Sea, J. Atmos. Ocean. Tech., 17, 708–720, https://doi.org/10.1175/1520-0426(2000)017<0708:AANASI>2.0.CO;2, 2000. a

Hasselmann, K.: On the non-linear energy transfer in a gravity wave spectrum, part 1: general theory, J. Fluid Mech., 12, 481–501, https://doi.org/10.1017/s0022112062000373, 1962. a

Hasselmann, K.: Feynman diagrams and interaction rules of wave-wave scattering processes, Rev. Geophys., 4, 1–32, https://doi.org/10.1029/rg004i001p00001, 1966.

Hwang, P. H., Wang, D. W., Walsh, E. J., Krabill, W. B., and Swift, R. N.: Airborne measurement of the wavenumber spectra of ocean surface waves. Part II: directional distribution, J. Phys. Oceanogr., 30, 2768–2787, https://doi.org/10.1175/1520-0485(2001)031<2768:AMOTWS>2.0.CO;2, 2000. a

Janssen, P. A. E. M.: On some consequences of the canonical transformation in the Hamiltonian theory of water waves, J. Fluid Mech., 637, 1–44, https://doi.org/10.1017/s0022112009008131, 2009. a

Kenyon, K. E.: Stokes drift for random gravity waves, J. Geophys. Res., 74, 6991–6994, https://doi.org/10.1029/jc074i028p06991, 1969. a

Krogstad, H. E. and Trulsen, K.: Interpretations and observations of ocean wave spectra, Ocean Dynam., 62, 973–991, https://doi.org/10.1007/s10236-010-0293-3, 2010. a

Leckler, F., Ardhuin, F., Peureux, C., Benetazzo, A., Bergamasco, F., and Dulov, V.: Analysis and interpretation of frequency-wavenumber spectra of young wind waves, J. Phys. Oceanogr., 45, 2484–2496, https://doi.org/10.1175/JPO-D-14-0237.1, 2015. a, b, c, d, e

Long, C. E. and Resio, D. T.: Wind wave spectral observations in Currituck Sound, North Carolina, J. Geophys. Res., 112, C05001, https://doi.org/10.1029/2006JC003835, 2007. a

Longuet-Higgins, M. S.: A theory of the origin of microseisms, Phil. Trans. Roy. Soc. London A, 243, 1–35, https://doi.org/10.1098/rsta.1950.0012, 1950. a

Munk, W.: An Inconvenient Sea Truth: Spread, Steepness, and Skewness of Surface Slopes, Annu. Rev. Mar. Sci., 1, 377–415, https://doi.org/10.1146/annurev.marine.010908.163940, 2009. a, b

Newville, M., Stensitzki, T., Allen, D. B., and Ingargiola, A.: LMFIT: Non-Linear Least-Square Minimization and Curve-Fitting for Python, https://doi.org/10.5281/zenodo.11813, 2014. a

Plant, W. J.: A relationship between wind stress and wave slope, J. Geophys. Res., 87, 1961–1967,https://doi.org/10.1029/jc087ic03p01961, 1982. a

Romero, L. and Melville, K. W.: Airborne Observations of Fetch-Limited Waves in the Gulf of Tehuantepec, J. Phys. Oceanogr., 40, 441–465, https://doi.org/10.1175/2009jpo4127.1, 2010. a

Senet, C. M., Seemann, J., and Zeimer, F.: The near-surface current velocity determined from image sequences of the sea surface, IEEE T. Geosci. Remote, 39, 492–505, https://doi.org/10.1109/36.911108, 2001. a, b

Stewart, R. H. and Joy, J. W.: HF radio measurements of surface currents, Deep-Sea Res., 21, 1039–1049, https://doi.org/10.1016/0011-7471(74)90066-7, 1974. a

Toffoli, A., Onorato, M., Bitner-Gregersen, E. M., and Monbaliu, J.: Development of a bimodal structure in ocean wave spectra, J. Geophys. Res., 115, C03006, https://doi.org/10.1029/2009JC005495, 2010. a

Wang, D. W. and Hwang, P. A.: Evolution of the Bimodal Directional Distribution of Ocean Waves, J. Phys. Oceanogr., 31, 1200–1221, https://doi.org/10.1175/1520-0485(2001)031<1200:eotbdd>2.0.co;2, 2001. a, b, c

Welch, P. D.: The use of fast Fourier transform for the estimation of power spectra: a method based on time averaging over short, modified periodograms, IEEE T. Acoust. Speech, 15, 70–73, https://doi.org/10.1109/TAU.1967.1161901, 1967. a

Wilson, D. K., Frisk, G. V., Lindstrom, T. E., and Sellers, C. J.: Measurement and prediction of ultralow frequency ocean ambient noise off the eastern U.S. coast, J. Acoust. Soc. Am., 113, 3117–3133, https://doi.org/10.1121/1.1568941, 2003. a

Wyatt, L. R.: Shortwave Direction and Spreading Measured with HF Radar, J. Atmos. Ocean. Tech., 29, 286–299, https://doi.org/10.1175/jtech-d-11-00096.1, 2012. a

Young, I. R., Verhagen, L. A., and Banner, M. L.: A note on the bimodal directional spreading of fetch-limited wind waves, J. Geophys. Res., 100, 773–778, https://doi.org/10.1029/94jc02218, 1995. a

Yueh, S. H., Tang, W., Fore, A. G., Neumann, G., Hayashi, A., Freedman, A., Chaubell, J., and Lagerloef, G. S. E.: L-Band Passive and Active Microwave Geophysical Model Functions of Ocean Surface Winds and Applications to Aquarius Retrieval, IEEE T. Geosci. Remote, 51, 4619–4632, https://doi.org/10.1109/tgrs.2013.2266915, 2013. a