the Creative Commons Attribution 4.0 License.

the Creative Commons Attribution 4.0 License.

# Effects of current on wind waves in strong winds

### Naohisa Takagaki

### Naoya Suzuki

### Yuliya Troitskaya

### Chiaki Tanaka

### Alexander Kandaurov

### Maxim Vdovin

It is important to investigate the effects of current on wind waves, called
the Doppler shift, at both normal and extremely high wind speeds. Three
different types of wind-wave tanks along with a fan and pump are used to
demonstrate wind waves and currents in laboratories at Kyoto University,
Japan, Kindai University, Japan, and the Institute of Applied Physics,
Russian Academy of Sciences, Russia. Profiles of the wind and current
velocities and the water-level fluctuation are measured. The wave frequency,
wavelength, and phase velocity of the significant waves are calculated, and
the water velocities at the water surface and in the bulk of the water are
also estimated by the current distribution. The study investigated 27 cases
with measurements of winds, waves, and currents at wind speeds ranging
from 7 to 67 m s^{−1}. At normal wind speeds under 30 m s^{−1}, wave
frequency, wavelength, and phase velocity depend on wind speed and fetch.
The effect of the Doppler shift is confirmed at normal wind speeds; i.e.,
the significant waves are accelerated by the surface current. The phase
velocity can be represented as the sum of the surface current and artificial
phase velocity, which is estimated by the dispersion relation of the
deepwater waves. At extremely high wind speeds over 30 m s^{−1}, a similar
Doppler shift is observed as under the conditions of normal wind speeds.
This suggests that the Doppler shift is an adequate model for representing
the acceleration of wind waves by current, not only for wind waves at
normal wind speeds but also for those with intensive breaking at extremely
high wind speeds. A weakly nonlinear model of surface waves at a shear flow
is developed. It is shown that it describes dispersion properties well
not only for small-amplitude waves but also strongly nonlinear and even
breaking waves, which are typical for extreme wind conditions (over 30 m s^{−1}).

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The oceans flow constantly, depending on the rotation of the Earth, tides,
topography, and wind shear. High-speed continuous ocean flows are called
currents. Although the mean surface velocity of the ocean is approximately
0.1 m s^{−1}, the maximum current surface velocity is more than 1 m s^{−1} (e.g., Kawabe, 1988; Kelly et al., 2001). The interaction between
the current and wind waves generated by wind shear has been investigated in
several studies. The acceleration effects of the current on wind waves,
called the Doppler shift, the effects of the current on momentum and
heat transfer across the sea surface, and the modeling of waves and currents
in the Gulf Stream have been the subject of experimental and numerical
investigations (e.g., Dawe and Thompson, 2006; Kara et al., 2007; Fan et
al., 2009; Shi and Bourassa, 2019). Thus, wind waves follow the dispersion
relationship and Doppler shift effect at normal wind speeds. However, these
studies were performed at normal wind speeds only, and few studies have been
conducted at extremely high wind speeds, for which the threshold velocity is
30–35 m s^{−1}, representing the regime shift of air–sea momentum,
heat, and mass transport (Powell et al., 2003; Donelan et al., 2004;
Takagaki et al., 2012, 2016; Troitskaya et al., 2012, 2020; Iwano et al.,
2013; Krall and Jähne, 2014; Komori et al., 2018; Krall et al., 2019).
At such extremely high wind speeds, the water surface is intensively broken
by strong wind shear, along with the foam layer, dispersed droplets, and
entrained bubbles (e.g., Donelan et al., 2004; Troitskaya et al., 2012, 2017,
2018a, b; Takagaki et al., 2012, 2016; Holthuijsen et al., 2012). It is
unclear if the properties of wind waves and the surface foam layer at
extremely high wind speeds are similar to those at normal wind speeds.
Furthermore, in a hurricane, the local ocean flows may be unusually strong,
change rapidly, and strongly affect wind waves. However, the effects of the
current on wind waves have not yet been clarified.

Therefore, the purpose of this study is to investigate the effects of the current on wind waves in strong winds through the application of three different types of wind-wave tanks, along with a pump.

## 2.1 Equipment and measurement methods

Wind-wave tanks at Kyoto University, Japan, and the Institute of Applied
Physics, Russian Academy of Sciences (IAP RAS), were used in the experiments
(Fig. 1a, b). For the tank at Kyoto University, the glass test section was
15 m long, 0.8 m wide, and 1.6 m high. The water depth *D* was set at 0.8 m.
For the tank at IAP RAS, the test section in the air side was 15 m long, 0.4 m wide, and 0.4 m high. The water depth *D* was set at 1.5 m. The wind was set
to blow over the filtered tap water in these tanks, generating wind waves.
The wind speeds ranged from 4.7 to 43 m s^{−1} and from 8.5 to 21 m s^{−1} in the tanks at Kyoto and IAP RAS, respectively. Measurements of
the wind speeds, water-level fluctuation, and current were carried out 6.5 m downstream from the edge (*x*=0 m) in both the Kyoto and IAP RAS tanks.
Here, the *x*, *y*, and *z* coordinates are referred to as the streamwise, spanwise,
and vertical directions, respectively, with the origin located at the center
of the edge of the entrance plate. Additionally, the fetch (*x*) is defined as
the distance between the origin and measurement point (*x*=6.5 m).

In Kyoto, a laser Doppler anemometer (Dantec Dynamics LDA) and phase Doppler
anemometer (Dantec Dynamics PDA) were used to measure the wind velocity
fluctuation. A high-power multiline argon-ion (Ar^{+}) laser (Lexel model
95-7; laser wavelengths of 488.0 and 514.5 nm) with a power of 3 W was used.
The Ar^{+} laser beam was shot through the sidewall (glass) of the tank.
Scattered particles with a diameter of approximately 1 µm were produced
by a fog generator (Dantec Dynamics F2010 Plus) and fed into the airflow over the waves (see Takagaki et al., 2012, and Komori et al., 2018, for
details). The wind speed values (*U*_{10}) at a height of 10 m above
the ocean and the friction velocity (*u*^{∗}) were estimated by the eddy
correlation method, by which the mean velocity (*U*) and the Reynolds stress
(−*u**v*) in air were measured. The *u*^{∗} was estimated by an eddy correlation method
as ${u}^{\ast}=(-<uv>{)}^{\mathrm{1}/\mathrm{2}}$ because the shear stress at the interface (*τ*) was defined by $\mathit{\tau}=\mathit{\rho}{u}^{\ast \mathrm{2}}=\mathit{\rho}{C}_{\mathrm{D}}{U}_{\mathrm{10}}^{\mathrm{2}}$. The value of ($-<uv>{)}^{\mathrm{1}/\mathrm{2}}$ was estimated by extrapolating the measured values of
the Reynolds stress to the mean surface of *z*=0 m. The *U*_{10} was
estimated by the log law: ${U}_{\mathrm{10}}-{U}_{\text{min}}={u}^{\ast}/\mathit{\kappa}\mathrm{ln}({z}_{\mathrm{10}}/{z}_{\text{min}}$), where *U*_{min} is the air velocity nearest the water
surface (*z*_{min}) and *z*_{10} is 10 m. Moreover, the drag coefficient
*C*_{D} was estimated by ${C}_{\mathrm{D}}=({u}^{\ast}/{U}_{\mathrm{10}}{)}^{\mathrm{2}}$.

Water-level fluctuations were measured using resistance-type wave gauges
(Kenek CHT4-HR60BNC) in Kyoto. The resistance wire was placed into the
water, and the electrical resistance at the instantaneous water level was
recorded at 500 Hz for 600 s using a digital recorder (Sony EX-UT10). The
energy of the wind waves (*E*) was estimated by integrating the spectrum of the
water-level fluctuations over the frequency (*f*). The values of the wavelength
(*L*_{S}) and phase velocity (*C*_{S}) were estimated using the cross-spectrum
method (e.g., Takagaki et al., 2017) (see details in the Appendix). The
current was measured using the same LDA system.

At IAP RAS, a hot-wire anemometer (E+E Electrinik EE75) was used to
measure the representative mean wind velocity at *x*=0.5 m and *z*=0.2 m.
The three wind velocities (*U*_{10}, *u*^{∗}, *U*_{∞}) at *x*=6.5 m were
taken from Troitskaya et al. (2012) by a Pitot tube. Here, *U*_{∞} is
the free-stream wind speed. The *u*^{∗} was estimated by a profile method
considering the profiles in the constant flux layer and the wake region:

respectively. Here, *δ* is the boundary layer thickness, and *α*
and *β* are the constant values that depend on flow fields and are
calibrated at low wind speeds without the dispersed droplets. At extremely
high wind speeds, measuring the profile in the constant flux layer (Eq. 1)
is difficult because of the large waves; thus, using *β* measured at
low wind speeds, *u*^{∗} is estimated by Eq. (2). The value of *U*_{10} is
estimated by Eq. (1) at *z*_{10}=10 m with measured *α* at normal
wind speeds. The value of *C*_{D} is estimated by ${C}_{\mathrm{D}}={u}^{\ast}/{U}_{\mathrm{10}}{)}^{\mathrm{2}}$. Although the measurement methods for *u*^{∗}, *U*_{10}, and
*C*_{D} at IAP RAS and Kyoto are different, the values approximately
correspond to each other (see Troitskaya et al., 2012, and Takagaki et al., 2012).

The water-level fluctuations were measured using three handmade
capacitive-type wave gauges at IAP RAS. Three wires formed a triangle with
25 mm on a side (*x*-directional distance between wires Δ*x* is 21.7 mm).
The wires were placed in the water, and the output voltages at the
instantaneous water level were recorded at 200 Hz for 5400 s using a digital
recorder through an AD converter (L-Card E14-140). The values (*E*, *f*_{m},
*H*_{S}, *T*_{S}, *C*_{S}, and *L*_{S}) were estimated in the same manner as in
the Kyoto tank. The current was measured through acoustic Doppler velocimetry
(Nortec AS) at *x*=6.5 m and $z=-\mathrm{10}$, −30, −50, −100, −150,
−220, and −380 mm (see Troitskaya et al., 2012, for details).

## 2.2 Artificial current experiments at Kindai University

Additional experiments were performed using a wind-wave tank at Kindai
University with a glass test section 6.5 m long, 0.3 m wide, and 0.8 m high
(Fig. 1c) (e.g., Takagaki et al., 2020). The water depth *D* was set at 0.49 m.
A Pitot tube (Okano Works, LK-0) and differential manometers (Delta Ohm
HD402T) were used to measure the mean wind velocity. The values of *u*^{∗},
*U*_{10}, and *C*_{D} (cases 21–27) were estimated using *U*_{∞} with the
empirical curve by Iwano et al. (2013), which was proposed by the eddy
correlation method used in Kyoto (see Sect. 2.1).

The water-level fluctuations were measured using resistance-type wave gauges
(Kenek CHT4-HR60BNC). To measure *L*_{S} and *C*_{S}, another wave gauge was
fixed downstream at Δ*x*=0.02 m, where Δ*x* is the interval
between the two wave gauges. The values (*E*, *f*_{m}, *H*_{S}, *T*_{S},
*C*_{S}, and *L*_{S}) were estimated in the same manner as in the Kyoto tank. The
current was then measured through electromagnetic velocimetry (Kenek LP3100)
with a probe (Kenek LPT-200-09PS) at *x*=4.0 m. The probe sensing station
was 22 mm long with a diameter of 9 mm. The measurements were performed at
$z=-\mathrm{15}$ to −315 mm at 30 mm intervals. The sampling frequency was 8 Hz,
and the sampling time was 180 s.

## 3.1 Waves and current

Figure 2 shows the vertical distributions of the streamwise water velocity.
The water velocities in the three different wind-wave tanks at Kyoto
University, Kindai University, and IAP RAS are separately shown in each
panel. In Fig. 2a, the bulk velocity of water *U*_{BULK} shows negative
values (${U}_{\text{BULK}}=-\mathrm{0.16}$ to −0.01 m s^{−1}) at Kyoto University,
which is generated as the counterflow against the Stokes drift at the wavy
water surface. In Fig. 2b, the bulk velocity of water demonstrates positive
values (*U*_{BULK}=0.019 to 0.044 m s^{−1}) at IAP RAS because the wind-wave
flume is submerged; thus, the Stokes drift on the wavy water surface does
not provide the counterflow for the bulk water, unlike in the closed tank at
Kyoto University. From Fig. 2c, it is clear that the bulk velocities of the
water vary in each case at Kindai University with the use of the pump.
Furthermore, the water bulk velocities change from negative to positive
(${U}_{\text{BULK}}=-\mathrm{0.13}$ to −0.17 m s^{−1}). The bulk velocities of water
were defined as the mean velocity with $z=-\mathrm{0.4}$ to −0.25 m (see dotted
lines in Fig. 2), and the velocities are listed in Table 1. Experiments were
performed under 27 different conditions, with the bulk velocity of water
provided in the three different wind-wave tanks. The surface velocities of
water, *U*_{SURF}, also varied in the three tanks with respect to wind speed
(see Fig. 2). The *U*_{SURF} values were estimated by the linear
extrapolation lines (dashed lines) as the water velocity at the surface (*z*=0 m) shown in Fig. 2, and the velocities are listed in Table 1.

Figure 3 shows the wind velocity dependency of the wave frequency *f*_{m},
wavelength *L*_{S}, phase velocity *C*_{S}, surface velocity of water
*U*_{SURF}, and bulk velocity of water *U*_{BULK}. From Fig. 3a–c, it is
clear that both the Kyoto and IAP RAS data demonstrate that the wind waves
develop with wind shear. Although *f*_{m} values in both cases correspond to each
other, *L*_{S} and *C*_{S} at IAP RAS are different from those in Kyoto. The
disagreement might be caused by the difference in the wind-wave development
or the Doppler effect; this is discussed below. From Fig. 3d and e,
*U*_{SURF} and *U*_{BULK} increase with an increase in *U*_{10} at IAP RAS.
However, in Kyoto, *U*_{SURF} increases, but *U*_{BULK} decreases with an
increase in *U*_{10}. Moreover, *U*_{SURF} at IAP RAS corresponds to
*U*_{SURF} in Kyoto. This is because the Stokes drift generated by the wind
waves, rather than the current, is significant. For the Kindai data,
although *f*_{m}, *U*_{SURF}, and *U*_{BULK} vary, *L*_{S} and *C*_{S} are concentrated at single points at *L*_{S}= 0.1 m and *C*_{S}= 0.4 m s^{−1}, respectively. This shows that the intensity and direction of the
current do not significantly affect *L*_{S} and *C*_{S} but do affect
*f*_{m} and *U*_{SURF}. Thus, this implies that the present artificial current
changes the water flow dramatically but does not affect the development of
wind waves.

Figure 4 shows the dispersion relation and demonstrates that the Kindai data
points depend on the variation in the water velocity of the artificial
current. The plots for the Kyoto University and IAP RAS cases at normal wind
speeds (solid symbols) are concentrated above the solid curve, showing the
dispersion relation of the deepwater waves (*ω*^{2}=*g**k*). Meanwhile, the plots for extremely high wind
speeds (open symbols) are also concentrated above the solid curve. This
implies that the wind waves, along with the intensive breaking at extremely
high wind speeds, are dependent on the Doppler shift. To investigate the
phase velocity trend, Fig. 5 shows the ratio of the measured phase velocity.

*C*_{S} to the phase velocity *C*_{S,0} is estimated by the dispersion relation
of deepwater waves (${C}_{\mathrm{S},\mathrm{0}}=(g{L}_{\mathrm{S}}/\mathrm{2}\mathit{\pi}{)}^{\mathrm{1}/\mathrm{2}}$) against the
wind velocity. From the figure, the ratios at normal wind speeds assume
a constant value (∼1.21 in Kyoto or ∼1.27 at
IAP RAS). Moreover, the ratios at extremely high wind speeds take similar
values of 1.23 and 1.28 for Kyoto and IAP RAS, respectively. This implies
that the phase velocities at extremely high wind speeds are accelerated by the
current just like those at normal wind speeds. However, the Kindai values are
scattered and increase in the following cases and decrease in the opposing
cases. It is clear that the artificial current accelerates (or decelerates)
the phase velocity.

To interpret the relationship among the measured phase velocity *C*_{S},
first phase velocity *C*_{S,0} estimated by the dispersion relation, and
water velocity, two types of phase velocities were evaluated: the sum of
*C*_{S,0} and the surface velocity of water *U*_{SURF} and the sum of *C*_{S,0} and the
bulk velocity of water *U*_{BULK}. Figure 6 shows the relationship of
*C*_{S} to (a) ${C}_{\mathrm{S},\mathrm{0}}+{U}_{\text{SURF}}$ and (b) ${C}_{\mathrm{S},\mathrm{0}}+{U}_{\text{BULK}}$. In
Fig. 6a, we can see that the Doppler shift is confirmed at normal wind
speeds; i.e., significant waves are accelerated by the surface flow, and
the real phase velocity can be represented as the sum of the velocity of the
surface flow and the virtual phase velocity, which is estimated by the
dispersion relation of the deepwater waves. At extremely high wind speeds
over 30 m s^{−1}, a similar Doppler shift is observed as under the
conditions of normal wind speeds, as seen in Fig. 6a. Meanwhile, in Fig. 6b,
although *C*_{S} corresponds to ${C}_{\mathrm{S},\mathrm{0}}+{U}_{\text{BULK}}$ at low phase
velocities, *C*_{S} assumes values larger than ${C}_{\mathrm{S},\mathrm{0}}+{U}_{\text{BULK}}$ at high
phase velocities. This suggests that the Doppler shift is an adequate model
for representing the acceleration of wind waves by the current, not only
for wind waves at normal wind speeds but also for those with intensive
breaking at extremely high wind speeds. Moreover, the Doppler shift of wind
waves occurs due to a very thin surface flow, as the correlation between
*C*_{S} and ${C}_{\mathrm{S},\mathrm{0}}+{U}_{\text{SURF}}$ is higher than the correlation between
*C*_{S} and ${C}_{\mathrm{S},\mathrm{0}}+{U}_{\text{BULK}}$.

## 3.2 The theoretical model of waves at the shear flow

The parameters of the observed Doppler shift can be explained more precisely within the theoretical model of capillary–gravity waves at the surface of the water flows with the velocity profiles prescribed by the experimental data, which are plotted in Fig. 2a–c. Because the dominant wind wave propagates along the wave and water flows, we will consider the 2D wave model in the 2D flow. This flow is described by the system of 2D Euler equations,

and the condition of non-compressibility,

with the kinematical,

and dynamical boundary conditions,

at the water surface. Here, *u* and *w* are the horizontal and vertical velocity
components, *p* is the water pressure, *x* and *z* are the horizontal and upward
vertical coordinates, *g* is the gravity acceleration, and *ρ* is the water
density. The boundary condition at the bottom of the channel is $w{\mathrm{|}}_{z=-D}=\mathrm{0}$. It should be noted that the water depth in almost all the
experimental runs exceeded half of the wavelength of the dominant waves (see
Table 1). In this case, the deepwater approximation is applicable for
describing the surface waves, and the boundary condition of the wave field
vanishing with the distance from the water surface can also be used.

Because the fluid motion under consideration is 2D, the stream function can be introduced as follows:

To derive the linear dispersion relation for the surface waves at the plane
shear flow with the horizontal velocity profile *U*_{w}(*z*), we consider the
solution to Eqs. (3) and (4) in terms of the stream function as the sum of the
undisturbed state with steady shear flow and small-amplitude disturbances.
Then, the stream function *ψ* and pressure *p* are as follows:

where *ε*≪1, and the water elevation value is
also the order of *ε*, namely *ε**η*_{1}(*x*,*t*).

In the linear approximation in *ε*, the system of Eqs. (3) and (4) and
the boundary conditions of Eqs. (5) and (6) take the form

Excluding *p*_{1} with the use of the first equation of the system in Eq. (3) and
eliminating *η*_{1} yields one boundary condition at the water surface
for *ψ*_{1}:

For the harmonic wave disturbance, where

substituting into Eqs. (10) and (11) yields the Rayleigh equation for the complex amplitude of the stream function disturbance,

with the following boundary condition:

Numerically solving the boundary layer problem for Eq. (13) with the
boundary conditions in Eq. (14) enables one to obtain the dispersion
relation *ω*(*k*) for surface waves at inhomogeneous shear flow.
Note that because the phase velocity of the waves significantly exceeded the flow velocity in all experiments (compare Figs. 2 and
3), the Rayleigh equation did not have a singularity, and the calculated
frequency and phase velocity of the wave were real values; i.e., the current
was neutrally stable.

The wave phase velocities ${C}_{\mathrm{S}-\mathrm{theor}-\mathrm{l}}=\mathit{\omega}\left(k\right)/k$ were calculated
for the parameters of those experiments that contained complete information
about the course and characteristics of the waves, namely 1, 3, 5, 7, 9, 11,
13–15, 18, and 21–27 from Table 1. The results are presented in Fig. 7a as
the measured phase velocity *C*s versus calculated phase velocity
${C}_{\mathrm{S}-\mathrm{theor}-\mathrm{l}}$. One can see that the model corresponds to the data
substantially better than the model of linear potential waves at the
homogeneous current *U*_{BULK} (compare Fig. 6b). Considering the structure
of the wave disturbances of the stream function, Ψ_{1}(*z*) was
found as the eigenfunction of the boundary problem in Eqs. (11) and (12). The
profiles of Ψ_{1}(*z*) are presented in Fig. 8. One can see that in all
cases the functions Ψ_{1}(*z*) are close to *e*^{kz} at the background
of the mean velocity profiles. Moreover, for experiment nos. 1, 3, 5, 15,
and 21–27 (see Fig. 8a, b, c, i, and k), the wave field is concentrated
near the surface at a distance less than the scale of the change in the mean
flow, whereby the flow velocity is approximately equal to *U*_{SURF}. This
explains the good correlation in these cases of the observed phase velocity
with the phase velocity of waves at the homogeneous current *U*_{SURF}
presented in Fig. 6a. At the same time, for experiment nos. 7, 9, 5, 11, 13,
14, and 18 (see Fig. 8d–h and j), the scale of the variability of the
flow is significantly smaller than the scale of the wave field. Under these
conditions, a significant difference between the phase velocity of the waves
and that given by the linear dispersion relation can be due to the influence
of nonlinearity.

To estimate the nonlinear addition to the wave phase velocity, we used the
results of the weakly nonlinear theory of surface waves for the current with
a constant shear. Of course, the flow in the experiments of the present work
does not have a constant shift, and this was considered when obtaining the
linear dispersion relation. However, it should be taken into account that
the contributions of the *n*th harmonic to the nonlinear dispersion relation
are determined by wave fields in the *n* power, which have a scale that is *n*
times smaller than the first harmonic. Additionally, the model of constant
shear of the mean current velocity is already approximately applicable for
the second harmonic (see Fig. 8).

We use the nonlinear dispersion relation for waves in the current with a constant shift in the deepwater approximation, which was obtained by Simmen and Saffman (1985):

Here, *ω*_{0} is the solution of the linear dispersion equation. Equation (15) is rewritten in the notation of this work and formulated in a reference
frame in which the surface of the water has the velocity *U*_{w}(0). Note
that the linear part of Eq. (15) coincides with Eq. (14). The results of
solving Eq. (15) are presented in Fig. 7b similarly to Fig. 7a as the
measured phase velocity *C*_{S} versus calculated phase velocity${C}_{\mathrm{S}-\mathrm{theor}-\mathrm{nl}}=\mathit{\omega}\left(k\right)/k$; one can see their good agreement with each other.
Thus, the wave frequency shift can be explained by two factors, including
the Doppler shift at the mean flow and the nonlinear frequency shift, while
the latter can also be interpreted in its physical nature as the wave
frequency shift in the presence of its orbital velocities.

Recent studies have indicated a regime shift in the momentum, heat, and mass
transfer across an intensive broken wave surface along with the amount of
dispersed droplets and entrained bubbles at extremely high wind speeds over 30 m s^{−1} (e.g., Powell et al., 2003; Donelan et al., 2004; Takagaki et
al., 2012, 2016; Troitskaya et al., 2012; Iwano et al., 2013; Krall and
Jähne, 2014; Komori et al., 2018; Krall et al., 2019). Thus, there is
the possibility of a similar regime shift in the Doppler shift of wind waves
by the current at extremely high wind speeds. However, the present study
reveals that such a Doppler shift is observed under the conditions of
normal wind speeds. In this case, the weakly nonlinear approximation turns
out to be applicable for describing the dispersion properties of not only
small-amplitude waves but also nonlinear and even breaking waves. This
implies that intensive wave breaking at extremely high wind speeds occurs
with the saturation (or dumping) of the wave height rather than the
wavelength. This evidence might be helpful in investigating and modeling
wind-wave development at extremely high wind speeds.

The effects of the current on wind waves were investigated through
laboratory experiments in three different wind-wave tanks with a pump at
Kyoto University, Japan, Kindai University, Japan, and IAP RAS. The study
investigated 27 cases with measurements of winds, waves, and currents
at wind speeds ranging 7–67 m s^{−1}. We observed that the wind
waves do not follow the dispersion relation at either normal or
extremely high wind speeds in the three tanks (Fig. 4) – excluding case 25,
in which the artificial current experiment used the Kindai tank. In case 25,
*U*_{SURF} is approximately zero (Fig. 3); thus, the Doppler shift does not
occur. Then, using 18 datasets (Kyoto and IAP RAS tanks) (Fig. 5), we found
that the ratio of ${C}_{\mathrm{S}}/{C}_{\mathrm{S},\mathrm{0}}$ is constant at both normal and extremely
high wind speeds. Moreover, in the artificial current experiment in Kindai,
we observed that the ratio varies (Fig. 5). The evidence from the three tank
experiments implies that the same wave–current interaction occurs at normal
and extremely high wind speeds.

To develop an adequate model for wave–current interaction at normal and
extremely high wind speeds, we validated four models (Figs. 6 and 7). At
normal wind speeds under 30 m s^{−1}, the wave frequency, wavelength,
and phase velocity of waves, as well as the surface velocity of the water depended on the
wind speed (Fig. 3). However, the bulk velocity of the water showed a
dependence on the tank type, i.e., a large tank with a submerged wind-wave
flume (IAP RAS) or wind flume above a tank (general type of wind-wave tank)
(Kyoto University) (Fig. 3). The effect of the Doppler shift was confirmed
at normal wind speeds; i.e., significant waves were accelerated by the
surface flow, and the phase velocity was represented as the sum of the
surface velocity of water and the phase velocity, which is estimated by the
dispersion relation of deepwater waves (Fig. 6). At extremely high wind
speeds over 30 m s^{−1}, a Doppler shift was observed similar to that
under the conditions of normal wind speeds (Figs. 4 and 5). This suggests
that the Doppler shift is an adequate model for representing the
acceleration of wind waves by the current, not only for wind waves at
normal wind speeds but also for those with intensive breaking at extremely
high wind speeds. The data obtained by the artificial current experiments
conducted at Kindai University were used to explain how the artificial
current accelerates (or decelerates) significant waves. A weakly
nonlinear model of surface waves at a shear flow was developed (Fig. 7). It
was shown that it describes dispersion properties well not only for
small-amplitude waves but also strongly nonlinear and even breaking waves, which are
typical for extreme wind conditions, with speeds, *U*_{10}, exceeding 30 m s^{−1}.

It is important to estimate the phase velocity and wavelength of
significant wind waves using the water-level fluctuation data. Here, we
explain the method, called the cross-spectrum method. The water-level
fluctuation *η* (*x*,*t*) at an arbitral location *x* and time *t* is shown as the
equation

where *ω* is the angular frequency, *A*(*ω*) is the complex
amplitude, *k*(*ω*) is the wavenumber of waves having *ω*,
and Ω is the maximum angular frequency of the surface waves. *F*_{η}(*ω*) is the Fourier transformation of *η* (*x*, *t*) when the
measurement time (*t*_{m}) and Ω are sufficiently large. Using the
inverse Fourier transformation of *F*_{η}(*ω*), *η*(*x*, *t*) is
shown as

Comparing Eqs. (A1) and (A2), *F*_{η}(*ω*) is ${F}_{\mathit{\eta}}\left(\mathit{\omega}\right)=\mathrm{2}\mathit{\pi}A\left(\mathit{\omega}\right){e}^{-ik\left(\mathit{\omega}\right)x}$. Assuming that
the wind waves change the shape little between two wave probes set upstream
and downstream, we can set the upstream and downstream water-level
fluctuations ${\mathit{\eta}}_{\mathrm{1}}\left(t\right)=\mathit{\eta}(\mathrm{0},t)$ and ${\mathit{\eta}}_{\mathrm{2}}\left(t\right)=\mathit{\eta}(\mathrm{\Delta}x,t)$, respectively, with Δ*x* downstream from the first
probe. The Fourier transformations *F*_{η1}(*ω*) and *F*_{η2}(*ω*) for *η*_{1}(*t*) and *η*_{2}(*t*), respectively, are
shown as

Then, the power spectra *S*_{η1η1}(*ω*) and *S*_{η2η2}(*ω*) for *η*_{1}(*t*) and *η*_{2}(*t*), respectively, are
shown as

Here, the superscript ^{∗} indicates the complex conjugate number. The
cross-spectrum Cr(*ω*) for *η*_{1}(*t*) and *η*_{2}(*t*) is shown
as

Using Euler's theorem, Eq. (A7) transforms to

The co-spectrum Co(*ω*) and quad spectrum *Q*(*ω*) are defined as the
real and imaginary parts of Cr(*ω*), respectively, shown as
Cr(*ω*) = Co(*ω*) + iQ(*ω*). Moreover, the phase *θ*(*ω*) is defined as *θ*(*ω*)= tan${}^{-\mathrm{1}}\left(Q\right(\mathit{\omega})/\text{Co}(\mathit{\omega}\left)\right)$. Thus, *θ*(*ω*) can be calculated as

Generally, the velocity of the wind waves *C* is defined as

where *L* is the wavelength and *T* is the wave period. From Eqs. (A9) and (A10),
*C*(*ω*) and *L*(*ω*) can be transformed to

When we estimate the phase *θ*_{m}(*ω*_{m}) at the angular
frequency of significant wind waves *ω*_{m} (=2*π**f*_{m}), the
phase velocity of significant wind waves *C*_{S} (=*C*(*ω*_{m}))
and significant wavelength *L*_{S} (=*L*(*ω*_{m})) are calculated by

In the study, *C*_{S} and *L*_{S} are estimated by Eqs. (A13) and (A14) using the
cross-spectrum method.

All analytical data used in this study are compiled in Table 1.

NT and NS planned the experiments, evaluated the data, and contributed equally to writing the paper excluding Sect. 3.2. YT planned the Russia experiment, provided the linear and nonlinear models, prepared figures in Sect. 3.2, and contributed to writing Sect. 3.2. CT prepared all figures excluding Sect. 3.2. NT performed the wind, current, and wave measurements in the Kyoto experiment. NT, NS, and CT performed the wind, current, and wave measurements in the Kindai experiment. AK and MV performed the wind, current, and wave measurements in the Russia experiment.

The authors declare that they have no conflict of interest.

This work was supported by the Ministry of Education, Culture, Sports, Science and Technology (Grant-in-Aid nos. 18H01284, 18K03953, and 19KK0087). This project was supported by the Japan Society for the Promotion of Science and the Russian Foundation for Basic Research (grant nos. 18-55-50005, 19-05-00249, and 20-05-00322) under the Japan–Russia Research Cooperative Program. The experiments of IAP RAS were partially supported by the RSF (project 19-17-00209). We thank Takumi Tsuji and Satoru Komori for their help in conducting the experiments and for useful discussions. The experiments of IAP RAS were performed at the Unique Scientific Facility “Complex of Large-Scale Geophysical Facilities” (http://www.ckp-rf.ru/usu/77738/, last access: 2 September 2020).

This research has been supported by the Ministry of Education, Culture, Sports, Science and Technology (grant nos. 18H01284, 18K03953, and 19KK0087), the Japan Society for the Promotion of Science, and the Russian Foundation for Basic Research (grant nos. 18-55-50005, 19-05-00249, and 20-05-00322).

This paper was edited by Judith Wolf and reviewed by two anonymous referees.

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