This paper investigates in detail a possible mechanism of energy convergence leading to freak waves. We give examples of a freak wave as a (weak) pseudo-maximal wave to illustrate the importance of phase coherence. Given a time signal at a certain position, we identify parts of the time signal with successive high amplitudes, so-called group events, that may lead to a freak wave using wavelet transform analysis. The local coherence of the critical group event is measured by its time spreading of the most energetic waves. Four types of signals have been investigated: dispersive focusing, normal sea condition, thunderstorm condition and an experimental irregular wave. In all cases presented in this paper, it is shown that a high correlation exists between the local coherence and the appearance of a freak wave. This makes it plausible that freak waves can be developed by local interactions of waves in a wave group and that the effect of waves that are not in the immediate vicinity is minimal. This indicates that a local coherence mechanism within a wave group can be one mechanism that leads to the appearance of a freak wave.

Understanding the mechanism of the freak wave
phenomenon is intriguing for scientists, engineers and mariners. The
mechanisms that lead to freak waves are understandably diverse and it is not
surprising that different freak waves exhibit different qualitative features

We consider freak waves in unidirectional wave fields which satisfy the
common definition of a freak wave, namely that the wave height exceeds
approximately 2 times the significant wave height (

In the study of

In unidirectional linear waves, the focusing due to dispersion is one
mechanism that causes a freak wave

According to the study of

In this paper, we will consider the appearance of freak waves in evolving
wave groups in space and time. The waves are generated from a signalling
problem: at the influx position, say

This paper is organized into five sections starting with this introduction.
Section 2 starts with a motivation to investigate the local coherence by
showing the rapid decrease of the maximal amplitude when the coherence is
decreased. Wavelet transformation is then described and shown to be better
capable than Fourier methods to analyze the local phase of a wave. Section 3
starts with the selection of possible freak waves by estimating the critical
group events from the influx signal that can lead to freak waves further
downstream. The propagation of the most energetic group is then simulated to
show the successive local energy convergence. We introduce quantitative
measures of local coherence as one tool to predict the freak wave appearance.
Using numerical simulations of linear and nonlinear waves with the
AB equation described in Appendix

In this section, we will start to motivate and illustrate the role of
coherence by considering maximal, pseudo-maximal (pm) and weak
pseudo-maximal (wpm) signals that can describe freak waves. In

Waves in the ocean at a specific position are described by a time signal. An
irregular signal will have phases that are commonly understood to be
uniformly distributed in

A pseudo-maximal (pm) wave is a partly coherent wave, that is in between a
completely irregular wave and a fully coherent maximal wave. For a given
signal with random phase

The phases of all frequencies in a pm signal are constrained as

A Jonswap spectrum with restricted random phases,

Shown are plots from up to down of a maximal, pseudo-maximal and
weak pseudo-maximal signal corresponding to the same random signal at the
bottom. The random signal corresponds to a Jonswap spectrum with

The maximal temporal amplitude of the linear evolution of the wpm
signal for various values of

The maximal temporal amplitude of the linear evolution of the wpm
signal with

The restriction of wpm signal is typically for frequencies within one (or a
half) standard deviation (SD) around the mean frequency,

In general, the mean frequency is not necessarily equal to the peak frequency because the spectrum of waves that is usually of Jonswap shape is not symmetric around the peak frequency.

Figure

Although the signal coherence can describe and measure the appearance of freak waves, the concepts use the whole interval of the time signal. However, not the whole interval will contribute in generating a freak wave since the waves propagate with their own group and phase velocity. The freak wave will be generated from local waves' interaction. Therefore, we will investigate the local energy propagation using wavelet transformation. This is expected to give a more refined measure of the appearance of the freak wave.

Shown is the distribution of the local energy of the dispersive focusing wave at some positions before the focusing point. The left plots are computed by Fourier transform and the right plots are by wavelet transform. The upper plots are in 3-D view, while the lower plots are in 2-D view.

In Fourier analysis we transform a function that depends on time into a
function that depends on the frequency as a single variable. Given a time
signal

The same as Fig.

The wavelet transform is an extension of Fourier transformation. The basis
function in Fourier transform is a sinusoidal of a specific frequency, and
the

There are many types of mother wavelets: Morlet, Haar, Daubechies, Meyer, etc.
(see

From Definition 2.3, the wavelet transform of a time signal

Similar to Fourier transform, it is possible to rebuild the signal from the
wavelet transform, the so-called inverse wavelet transform. It is given by

The choice of the central frequency

The capability of the wavelet transform to represent a signal in time and frequency domain motivates us to investigate a freak wave locally. For a given signal, we identify group events which are parts of the time signal that may develop into propagating wave groups, i.e., that contain an amount of energy larger than a certain threshold. This threshold is determined such that the group event can build a freak wave if additional conditions are satisfied. We then determine the most energetic waves from each group event to see how the energy is distributed in both time and frequency. The most energetic waves will determine the evolution of the group event and whether its energy will converge or diverge. With these elements, we will be able to define the local coherence which will describe quantitatively the process of freak wave formation from a critical group event.

The normalized spectral shape of the influx signal for the case 202002.

Case 202002.

Case 202002. A filled contour plot of the energy distribution of the
critical group event at position

Case 202002. Zoomed version of the maximal wave; the crest height is 4.65 m and the wave height is 6.56 times the significant wave height.

The same as Fig.

From all the group events determined in this way, we characterize the groups
that may lead to a freak wave. For a given time signal,

Case W100. Initial time signal in the interval

Case W100.

Next, we define the total energy threshold to eliminate group events which
unlikely generate a freak wave. The remaining groups are so-called
critical group events.

Case W100. Zoomed version of the freak wave; the crest height is 1.35 m and the wave height is 2.37 times the significant wave height.

Case W100. A filled contour plot of the energy distribution of the
critical group event at various positions. At each position, the red solid
lines show the time of maximal energy at each wave frequency. The ++
lines show the wave frequency as function of time. Both are estimated by the
most energetic waves in time and frequency, respectively. Both lines show a
decreasing frequency before the freak wave and an increasing frequency after
the freak wave, while the freak wave occurs at

The same as Fig.

Case TS10000. Initial time signal in the interval

Case TS10000.

Case TS10000. Zoomed version of the freak wave; the crest height is 1.22 m and the wave height is 2.23 times the significant wave height.

We start from the complex value of the wavelet transform of

can distinguish the two cases:

Moreover, we can also look at the most energetic waves as a function of wave frequency. This leads to a local time of each wave contribution. In the case of a dispersive focusing wave, focusing of the energy occurs when all wave contributions are in phase at one local time.

Motivated by this, for each critical group event in a local time interval

Case TS10000. A filled contour plot of the energy distribution of the critical group event at various positions. At each position, the red solid lines show the time of maximal energy at each wave frequency. The ++ lines show the wave frequency as function of time. Both are estimated by the most energetic waves in time and frequency, respectively.

The same as Fig.

The observations of the most energetic waves in either time or frequency can
be used to see whether a freak wave may appear in forward or backward time,
but the generation of a freak wave is still not assured, since the amplitude
is not determined yet. The local information of the energy and phase gives a
method to investigate locally the relation between the local coherence and
freak wave occurrence. In this subsection, we measure the local coherence of
the group event along its evolution and we will show that the highest
amplitude occurs when the local coherence is maximum in the restricted
frequency interval. As the wavelet transformation gives a function of
frequency and time, we define a time spreading of the most energetic waves

This section presents the investigations of four study cases: an experimental
dispersive focusing wave, a synthetic normal wave condition (W100), a
synthetic thunderstorm condition (TS10000) and an experimental irregular
wave (IW12). For each case, we start to characterize the critical group
events, then we investigate the local features of these groups, namely the
most energetic wave and its time spreading. We investigate the evolution of
the local energy and the time spreading of each case, particulary around the
critical group events, and measure the local coherence. Furthermore, we
compute the correlation between the local coherence and the maximum amplitude
of the group event that generates a freak wave. It will give an impression of
the relevance of the parameters

The case is a focusing wave that will lead to a maximal wave. We consider a
dispersive focusing wave with significant wave height 0.013

Measure of the local coherence of the dispersive focusing wave.

Referring to Fig.

During the evolution, the changes of the distribution of the local energy in
the time–frequency frame are described well by the filled contour plot of the
local energy. The local energy distribution from one group event is squeezed
into a maximal wave. This is also shown by the decreasing width of the time
intervals towards the focusing point in Fig.

In order to show that the occurrence of the freak wave is related to a local
coherence, and to illustrate the three different measures of coherence
introduced above, we show the evolution of these coherence measures for the
linear and nonlinear evolution in Table

The second and third case are synthetic signals of irregular waves that are
generated from a Jonswap spectrum with normal and thunderstorm sea
conditions at a water depth of 480

The initial time signal is generated from a Jonswap spectrum with time period
11.3

Figure

In this case, the occurrence of the freak wave can also be observed from the
most energetic wave in either time or frequency (see Fig.

Measure of the local coherence of the normal sea condition wave.

Furthermore, we investigate the change of the local coherence of the critical
group event during its 3 km linear wave evolution. The measure of coherence
at various positions is shown in Table

The other synthetic signal is generated from a Jonswap spectrum with time
period 13.6

Figure

Measure of the local coherence of the thunderstorm condition wave.

We measure the local coherences of the critical group event along its linear
evolution and the results are presented in Table

The fourth case is an irregular wave, for which measurements at several
positions are available from MARIN experiment with a water depth of
0.6

Case IW12. The upper plot shows the influx signal. Four critical group events are shown in the shaded areas. The lower plot shows the local energy signal of group events compared to the local energy threshold (dashed line). The local energy signal of the critical group events are above the threshold.

Case IW12.

The evolution of the time signal around the critical group event and its
energy distribution at several positions are shown in Fig.

Case IW12. Zoomed version of the freak wave; the crest height is 1.31 m and the wave height is 2.15 times the significant wave height.

Measure of the local coherence of IW12.

Case IW12. A filled contour plot of the energy distribution of the group event at various positions. At each position, the red solid lines show the time of maximal energy at each wave frequency. The ++ lines show the wave frequency as function of time. Both are estimated by the most energetic waves in time and frequency, respectively.

In this paper, we showed the relevance of phase coherence by
illustrations of signals with increasingly less restrictions on the phase
function. Then, the wavelet transform was used to determined the time–frequency
spectrum of a time signal. We used the wavelet transform to identify critical
group events of the influx signal and it is shown that the group event with
the largest local energy signal is the most probable group to generate a
freak wave. We remarked that the identification of a group event is dependent
on the choice of the threshold value (

The data used by this study are experimental and synthetic data. The data are freely available but not otherwise published in any publicly accessible database. The experimental data can nonetheless be provided on request by MARIN hydrodynamic laboratory, Wageningen, the Netherlands. The synthetic data can nonetheless be provided on request via email to the corresponding author Arnida L. Latifah (a.l.latifah@utwente.nl).

The AB equation proposed by

We describe the dynamics by the surface elevation,

This work was funded by the Netherlands Organisation for Scientific Research, Technology Foundation STW, number 7216. We acknowledge the MARIN hydrodynamic laboratory for their measurement data 202002 and 103001 used in this paper. Edited by: R. Grimshaw Reviewed by: E. Pelinovsky and one anonymous referee