1 Introduction
Ocean surface waves propagating over non-uniform depth are subject to refraction (for example, Massel Reference Massel1996; Goda Reference Goda2000). Their amplitude is governed by the energy flux conservation (for example, Phillips Reference Phillips1977). For the special case of linear and long-crested waves normally incident on straight depth contours this leads to Green’s law (Lamb Reference Lamb1932, Art. 185), stating that their amplitude will in the limit of very shallow water increase slightly and inversely proportional to the fourth root of the depth.
For linear gravity waves with wavenumber
$k$
and angular frequency
$\unicode[STIX]{x1D714}$
on water of depth
$h$
the linear dispersion relation is
$\unicode[STIX]{x1D714}^{2}=gk\tanh kh$
, where
$g$
is the acceleration of gravity, and the group velocity
$c_{g}$
is
$$\begin{eqnarray}\frac{c_{g}\unicode[STIX]{x1D714}}{g}=\frac{1}{2}(\tanh kh+kh\,\text{sech}^{2}kh).\end{eqnarray}$$
The graph of the non-dimensional group velocity is illustrated by the black curve in figure 1. It has a local maximum of approximately 0.6 where
$kh=\coth kh$
or
$kh\approx 1.2$
. For large depths the non-dimensional group velocity approaches 0.5. For small depths the group velocity approaches zero asymptotically as
$c_{g}\unicode[STIX]{x1D714}/g\approx kh$
. Waves approaching a beach for small depths (
$kh<1.2$
) will therefore have their amplitude slightly increased, while waves approaching shallower water from greater depths (
$kh>1.2$
) will first have their amplitude decreased slightly and then increased slightly. At the same time the wavenumber will increase slightly as the waves approach shallower water, therefore the steepness will also increase slightly.
Group velocity indicated with solid curve (——). Experiments indicated by symbols
$\sqcap$
and simulations indicated by symbols
$\sqcup$
with respect to the non-dimensional depth corresponding to the spectral peak frequency. Zeng & Trulsen (Reference Zeng and Trulsen2012) employed waves originating at rather deep water (
$kh=10$
) not shown in this figure. All the mentioned works considered irregular wave statistics, except Beji & Battjes (Reference Beji and Battjes1993), who only considered spectral changes.
The changes of depth, wavelength and amplitude induced by shoaling can modify the statistical distribution of the waves. This was investigated in the pioneering work of Bitner (Reference Bitner1980), reporting a field experiment of waves propagating towards decreasing depth. A similar field experiment was also reported by Cherneva et al. (Reference Cherneva, Petrova, Andreeva and Guedes Soares2005). In both cases they found noticeable deviation from Gaussian statistics.
When waves enter shallower water the spectral shape changes with the amplification of bound harmonics (for example, Grue Reference Grue1992; Beji & Battjes Reference Beji and Battjes1993, Reference Beji and Battjes1994).
Recently it has been found that there can be a significant dynamic response that occurs after the depth change. This dynamic nonlinear behaviour was revealed in experiments as reported by Trulsen, Zeng & Gramstad (Reference Trulsen, Zeng and Gramstad2012), Kashima, Hirayama & Mori (Reference Kashima, Hirayama and Mori2014), Ma, Dong & Ma (Reference Ma, Dong and Ma2014), Bolles, Speer & Moore (Reference Bolles, Speer and Moore2019) and Zhang et al. (Reference Zhang, Benoit, Kimmoun, Chabchoub and Hsu2019), and in numerical simulations as reported by Sergeeva, Pelinovsky & Talipova (Reference Sergeeva, Pelinovsky and Talipova2011), Zeng & Trulsen (Reference Zeng and Trulsen2012), Gramstad et al. (Reference Gramstad, Zeng, Trulsen and Pedersen2013), Viotti & Dias (Reference Viotti and Dias2014), Majda, Moore & Qi (Reference Majda, Moore and Qi2019) and Zhang et al. (Reference Zhang, Benoit, Kimmoun, Chabchoub and Hsu2019). These recent studies suggest the existence of two different regimes: for waves that propagate onto a sufficiently shallow shoal it appears that a localised maximal concentration of extreme waves can occur at or near the shallow incident edge of the sloping bottom. For waves that propagate onto a sufficiently deep shoal it appears that no localised concentration of extreme waves occurs. Among the mentioned experimental and numerical works, Zeng & Trulsen (Reference Zeng and Trulsen2012) definitely belongs to the deeper regime, while the deepest run reported by Trulsen et al. (Reference Trulsen, Zeng and Gramstad2012) may belong to the deeper regime. However, for Zeng & Trulsen (Reference Zeng and Trulsen2012) the slope of the shoal is quite long and mild, which may also have an influence.
Although recent work by Ducrozet & Gouin (Reference Ducrozet and Gouin2017) suggests the enhancement of extreme wave statistics over a shoal is reduced for directional sea states, the early field measurements of Bitner (Reference Bitner1980) and Cherneva et al. (Reference Cherneva, Petrova, Andreeva and Guedes Soares2005) suggest such enhancement can occur under realistic field conditions.
The principal goal of this paper is to clarify the existence of, and transition between, the shallower and deeper regimes for long-crested waves propagating over a shoal in order to determine when an enhancement of extreme wave statistics can be expected. We also extend the consideration of extreme wave statistics to the fluid velocity field and discover that its statistics can be rather different from that of the surface elevation.
2 Experimental set-up
The experiments were carried out in the 24.6 m long and 0.5 m wide tank at the Hydrodynamical Laboratory of the Department of Mathematics at the University of Oslo. At one end there is a wave maker and at the opposite end there is a damping beach. A shoal was installed with height 0.42 m, with a linear up-slope of horizontal length 1.6 m, a horizontal section of length 1.6 m, and a linear down-slope of horizontal length 1.6 m. The depth in the deeper part
$h_{1}$
and the depth on top of the shoal
$h_{2}$
are related by
$h_{1}-h_{2}=0.42~\text{m}$
. We align the
$x$
-axis along the tank with origin at the toe of the shoal (the part of the shoal closest to the wavemaker). The distance from the equilibrium position of the wavemaker to the toe of the shoal is estimated to be 10.78 m. Surface elevation measurements were made with ultrasound probes from General Acoustics. Velocity measurements were made with a ‘Vectrino’ Acoustic Doppler Velocimeter (ADV) from Nortek. The schematic view of the set-up in figure 2 correctly indicates the positions of the probes relative to the shoal, but the wavemaker is farther out to the left than indicated and the damping beach to the right is not shown.
An effort was made to ensure all corners of the shoal were rounded to avoid flow separation and turbulence generation.
Set-up of the two measurement campaigns. Upper part: first campaign (Raustøl Reference Raustøl2014) with an array of 16 probes that was moved to four different locations such that the resolution before the shoal was 0.3 m and the resolution above the shoal was 0.1 m. Lower part: second campaign (Jorde Reference Jorde2018) with ADV placed at depth 48 mm. The ultrasound probe locations are indicated by open circles ○ in the air, while the ADV probe locations are indicated by filled circles ● in the water. The wavemaker is farther out to the left than indicated and the damping beach on the far right side is not shown.
The wavemaker was programmed to generate a JONSWAP spectrum
$$\begin{eqnarray}S(\unicode[STIX]{x1D714})=\frac{\unicode[STIX]{x1D6FC}g^{2}}{\unicode[STIX]{x1D714}^{5}}\text{e}^{-5/4(\unicode[STIX]{x1D714}_{p}/\unicode[STIX]{x1D714})^{4}}\unicode[STIX]{x1D6FE}^{\exp (-1/2(\unicode[STIX]{x1D714}-\unicode[STIX]{x1D714}_{p}/s\unicode[STIX]{x1D714}_{p})^{2})},\end{eqnarray}$$
with peak enhancement factor
$\unicode[STIX]{x1D6FE}=3.3$
, peak frequency
$\unicode[STIX]{x1D714}_{p}=2\unicode[STIX]{x03C0}/T_{p}$
and the parameter
$s=0.07$
for
$\unicode[STIX]{x1D714}<\unicode[STIX]{x1D714}_{p}$
and
$s=0.09$
for
$\unicode[STIX]{x1D714}>\unicode[STIX]{x1D714}_{p}$
. The wave amplitude was tuned by means of the parameter
$\unicode[STIX]{x1D6FC}$
to be as high as possible without visible wave breaking during the experiments.
Measurements were taken in two campaigns with two different probe arrangements and in 12 runs with different combinations of peak period
$T_{p}$
and water depth. We have sorted these experimental runs by the dimensionless depth
$k_{p}h$
on top of the shoal, where
$k_{p}$
is the peak wavenumber derived from the peak period according to the linear dispersion relation for the relevant depth. The key parameters for the 12 runs are summarised in table 1. The steepness is
$\unicode[STIX]{x1D716}=k_{p}a_{c}$
and the Ursell number is
$Ur=k_{p}a_{c}/(k_{p}h)^{3}$
, where the characteristic amplitude is
$a_{c}=\sqrt{2}\unicode[STIX]{x1D70E}$
and
$\unicode[STIX]{x1D70E}$
is the standard deviation of the surface elevation. The significant wave height is
$H_{s}=4\unicode[STIX]{x1D70E}$
.
Key parameters for all runs. The runs are numbered according to increasing dimensionless depth over the shoal. Run 3 belongs to the second campaign (Jorde Reference Jorde2018), the rest are from the first campaign (Raustøl Reference Raustøl2014). The values of significant wave height
$H_{s}$
, steepness
$\unicode[STIX]{x1D716}$
and Ursell number
$Ur$
are averages over all probes in front of or above the shoal, not including the probes at the edges of the sloping bottom.
In the first measurement campaign (Raustøl Reference Raustøl2014), with probe arrangement illustrated in the upper part of figure 2, eleven runs of only surface elevation measurements were made. An array of 16 ultrasound probes with spacing 0.3 m was moved to four different locations, allowing the surface elevation to be measured at a total of 64 positions with spatial resolution 0.3 m in front of the shoal and resolution 0.1 m above the shoal.
In the second measurement campaign (Jorde Reference Jorde2018) both surface elevation and horizontal velocity were measured at the positions indicated in the lower part of figure 2. Only one run was made in this campaign, with water depth
$h_{1}=0.53~\text{m}$
and peak period
$T_{p}=1.1~\text{s}$
, indicated as run 3. The velocity measurements were made at depth 0.048 m.
In the first campaign we had 16 ultrasound probes available. In the second campaign we had four ultrasound probes and one ADV available. In order to make all the ‘simultaneous’ measurements indicated in figure 2 it was necessary to repeat each run of the first campaign four times, and to repeat the single run of the second campaign 43 times. In order to quantify the repeatability of the experiments we analysed the measurements of the ultrasound probe at
$x=-0.4~\text{m}$
of run 3. For this probe position we have 35 repetitions, and we can compute the corresponding 35 repeated values of skewness and kurtosis. The coefficients of variation (the ratio of standard deviation over mean) of these 35 repeated values were found to be 0.0403 for the skewness and 0.00485 for the kurtosis. We are therefore confident that our experiments are sufficiently repeatable.
3 Results
In the following we present results obtained at each probe with respect to the horizontal position (
$x$
) indicated along the first axis. The edges of the shoal at
$x=0~\text{m}$
, 1.6 m, 3.2 m and 4.8 m are indicated by vertical dashed lines. Time averaging was used to compute the skewness
$\unicode[STIX]{x1D6FE}=\langle (\unicode[STIX]{x1D709}-\langle \unicode[STIX]{x1D709}\rangle )^{3}\rangle /\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D709}}^{3}$
and the kurtosis
$\unicode[STIX]{x1D705}=\langle (\unicode[STIX]{x1D709}-\langle \unicode[STIX]{x1D709}\rangle )^{4}\rangle /\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D709}}^{4}$
, where
$\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D709}}^{2}=\langle (\unicode[STIX]{x1D709}-\langle \unicode[STIX]{x1D709}\rangle )^{2}\rangle$
, for
$\unicode[STIX]{x1D709}$
being either the surface elevation or the horizontal velocity.
In figure 3 we show the intensity of the power spectral density in logarithmic scale. The spectra were computed with the Welch method. The maximum intensity is shown in white colour, weaker intensities are shown in darker colours. The positions along the tank where no measurements were taken are also shown in white colour. It is seen that the presence of the shoal provokes strongly enhanced higher harmonics, in agreement with previous work (Grue Reference Grue1992; Beji & Battjes Reference Beji and Battjes1993, Reference Beji and Battjes1994), provided the shoal is sufficiently shallow. We observe that the appearance of the higher harmonics is part of a dynamic process that has it maximum intensity a short distance inside the shallower part of the shoal and then relaxes.
Intensity of power spectral density of the surface elevation. White colour indicates maximum power or lack of measurement. For enhanced visibility the thickness of each spectrogram is doubled for the probes in front of the shoal (
$x<0$
).
In figure 4 we show the skewness of the surface elevation. The skewness can have a local maximum some distance inside the shallower part of the shoal and can have a local minimum some distance onto the down-slope on the lee side. There is a tendency that the location of the maximum skewness approaches the incident edge of the shallower part of the shoal with increasing depth of the shoal.
Skewness of surface elevation.
In figure 5 we show the kurtosis of the surface elevation. The kurtosis can also have a local maximum some distance inside the shallower part of the shoal, but does not have a local minimum some distance into the lee side. The locations of the maxima of the kurtosis and the skewness appear to coincide, including the tendency that the location of the maximum approaches the incident edge of the shallower part of the shoal with increasing depth of the shoal.
Kurtosis of surface elevation.
In figure 6 we show a comparison between the statistics of the surface elevation and of the horizontal velocity for run 3. The skewness of the surface elevation and the skewness of the horizontal velocity follow each other. However, for the kurtosis the behaviour of the surface elevation and of the horizontal velocity is rather different. The horizontal velocity does not have a maximum of kurtosis on top of the shoal, but has a local maximum on the lee side.
Skewness (a) and kurtosis (b) of surface elevation (*) and horizontal velocity (
$+$
).
4 Discussion
Our experiments seem to confirm the existence of two regimes for wave statistics when long-crested waves propagate over a shoal. When the shoal is shallower than some threshold depth the skewness and the kurtosis can be significantly enhanced over the shallower part of the shoal while the skewness can have a local minimum on the lee side of the shoal. The maximum of the skewness and the kurtosis is achieved some distance into the shallower part of the shoal, moving closer to the incident edge of the shallower part of the shoal as the depth of the shoal increases.
The power spectrum shows strongly enhanced higher harmonic behaviour at the same locations where the skewness and the kurtosis of the surface elevation are amplified. The enhanced higher harmonic behaviour also depends on the shoal being sufficiently shallow, our results seem to suggest the threshold depth for enhanced higher harmonics may be smaller than the threshold depth for enhanced extreme wave statistics.
Our results suggest that the threshold depth for enhanced extreme wave statistics may be close to
$kh=1.3$
for the shallower part of the shoal; however, it should be anticipated that this may depend on the steepness and the bandwidth of the waves and on the slope of the shoal.
The behaviour of the kurtosis of the horizontal velocity is surprising, with the maximum achieved on the lee side rather than above the shallower part of the shoal. Recently, we have carried out numerical simulations showing the same behaviour as that seen in figure 6; these simulations will be published in the future.
As far as giving an explanation for the anomalous behaviour of the skewness and the kurtosis is concerned, we suspect it is not due to flow separation around the sharp edges of the shoal, since an effort was made to round the corners to avoid separation. We also suspect it is not due to wave breaking provoked by the shoal, since an effort was made to reduce the amplitude such that breaking did not occur. Our best guess is that the phenomenon is related to a relaxation process: when a wave field enters a new environment the process of reaching a new equilibrium state can give extreme excursions in the values of certain statistical parameters.
5 Conclusion
We have carried out laboratory experiments of long-crested irregular water surface waves propagating over a shoal. For a sufficiently shallow shoal we find that the surface elevation can have a local maximum of skewness and kurtosis above the shallower part of the shoal close to the edge on the incoming side, and a local minimum of skewness over the downward slope on the lee side of the shoal. We also find that the horizontal fluid velocity can have a local maximum and minimum of skewness at the same locations as those for the surface elevation. The kurtosis of the horizontal fluid velocity can have a local maximum over the lee side of the shoal different from the location of the maximum of kurtosis of the surface elevation.
Acknowledgements
Dedicated to K. B. Dysthe on the occasion of his 80th birthday. We thank C. Lawrence for sharing his numerical simulations, O. Gundersen for building and installing the experimental set-up, A. Jensen for assistance with the experiments, and H. E. Krogstad for valuable feedback. This work has been supported by the Research Council of Norway (RCN) through grants 214556 and 256466.
Supplementary information
The laboratory data for this paper are now available from the Norwegian Research Data Archive. In order to access the published data, please go tohttps://archive.sigma2.no/ and search for ‘Extreme wave statistics of longcrested irregular waves over a shoal’, which is the title of this paper.
