Nonlinear Hydrophysical Disturbance Infragravity Sea Waves

Abstract

An analysis of the behavior of the infra-gravitational sea wave revealed nonlinear hydrophysical disturbance of the “rogue waves” type. This disturbance is associated with transformation of sea waves when they move along a coastal marine area of decreasing depth, and with interaction with wind waves. These conclusions are confirmed through analysis of in situ data of the laser interference pressure meter of the hydrosphere in this paper and previous works. Full-scale data were obtained on the shelf of the Sea of Japan. During processing of in situ data, we focused on sea waves with periods ranging from 1 to 10 min.

1. Introduction

A wind wave is a surface wave that occurs on the free surface of bodies of water as a result of the wind blowing over the water’s surface. Rogue waves (also known as freak waves or killer waves) are huge and unpredictable surface waves that can be extremely dangerous to ships and isolated structures such as lighthouses. Most of the recorded rogue waves have periods of wind sea waves (1–30 s), with amplitudes several times larger than these waves [1,2]. It is believed that this difference should be at least 2.8 times. Most of the observed rogue waves are described in article Kharif [3], but all of the presented experimental results relate to periods of wind waves. In all these works, nonlinear hydrophysical disturbance is formed within one wave field, namely the wind wave field. They do not consider the interaction of different-scale hydrophysical fields. Various approaches and methods are used to describe the origin of rogue waves. Pelinovsky [4] describes a solution using the dispersion focusing method based on the Korteweg–de Vries equation. Other models describing the origin of rogue waves are based on the Schrodinger and sine-Gordon equations [5,6], which exhibit different properties of modulation instability. Zakharov [7] notes that rogue waves (extreme waves) emerge predictably as a result of evolution of spectrally narrow packets of gravity waves. We can state that rogue waves represent a nonlinear stage of modulation instability. In an article by Zakharov [8], the Euler equation for fluid with free surface in deep water was solved. Periodic and boundary conditions were set as Stokes waves, slightly modulated with a low frequency (10⁻⁵ Hz). Such a wave is inherently unstable, and its modulation should increase over time, consequently generating an extremely high wave (a large-amplitude wave).

Infragravity waves are surface gravity waves with frequencies lower than the wind waves—consisting of both wind sea and swell—thus corresponding with the part of the wave spectrum lower than the frequencies directly generated by forcing through the wind. In the same authors’ works dedicated to rogue waves, there are instances of nonlinear disturbances within the minute range of periods [9,10]. Over the past decade, scientists worldwide have been studying infragravity ocean waves. Of particular interest are observations made along the coast [11,12,13], as it is precisely there that these waves transform, interacting with cross currents and gravity waves. It is on the shelf that they receive additional energy from breaking incident waves [14]. Equally important is the question of mutual exchange of energy between infragravity waves and other wave processes. In an article by Flores [15], the results of studies on the change in the length of short infragravity waves when they move along the surface of longer waves are presented. Unna [16], based on in situ data and numerical models of sea surface gravity waves, demonstrates that in the nearshore zone, during nonlinear wave-wave interaction, energy transfers from low-frequency long waves back to higher-frequency wave processes. Similar results were obtained by Kovalev [17], demonstrating that strong tidal modulation of infragravity waves (periods from 200 to 20 s), observed on the shelf of Southern California, results from nonlinear energy transfer from these low-frequency long waves to waves with higher frequencies. Moreover, this energy transfer process from low-frequency long waves to high frequencies is also present in gravity waves in the atmosphere [18,19]. Recent field research, as presented in articles by Flores and Thomson [15,20], has established a strong correlation of sea infragravity waves with significant water mass movements along the coast. It is this behavior of infragravity sea waves that allows them to be considered as individual free waves.

The complex behavior of sea infragravity waves, their energy intake in the surf zone, interaction with cross currents and gravity waves evokes particular interest in studying these waves. Field research of these waves holds significant value. There are few instruments capable of recording waves in the period range from 30 s to 5 min. One such instrument is the laser interference pressure meter of the hydrosphere, developed basing on modern laser interferometry methods [21]. Research presented by Dolgikh [22] demonstrated the capability of this detector to register not only gravity waves but also sea infragravity waves, as well as its high efficiency in detecting both gravity and infragravity sea waves.

In previous research [23], we analyzed the records of the laser interference pressure meter of the hydrosphere (supersensitive detector of hydrosphere pressure variations), which was installed in the coastal marine area in various bays of the Primorsky Territory. We analyzed data bulk, accumulated between 2007 and 2019. As a result, it was established that interaction of infragravity and gravity sea waves in the coastal marine area leads to formation of nonlinear disturbances related to “rogue waves”. In this work, we focused on wind sea waves.

In this study, using in situ data, we will explore the mechanisms of sea infragravity wave destruction on the coastal marine area of the Sea of Japan. In the paper, we investigate the processes of infragravity sea wave destruction on the shelf of the Sea of Japan, recorded by the supersensitive detector of hydrosphere pressure variations. The records from this laser-interference instrument are provided prior to, in the process of, and after the occurrence of nonlinear hydrophysical disturbances. The experimental research establishes behavior patterns of infragravity sea waves as they propagate along the shelf.

The study of infragravity is important because they have an impact on coastal and coastal infrastructure, such as ships. And the appearance of nonlinear hydrophysical disturbances near these objects can cause significant damage. After all, the amplitudes of these waves, compared with wind waves, are much higher. For a wind wave with a height of 0.1 m, this disturbance may be 0.5 m and will not cause significant damage. Then, in the range of infragravity waves, their amplitude may be several times smaller.

2. Method and Place of Experimental Research

The experimental studies on registering hydrosphere pressure variations were held in the south of the Primorsky Territory of Russia. That is where the marine experimental station of the V.I. Il’ichev Pacific Oceanological Institute of the Far Eastern Branch of the Russian Academy of Sciences, known as “Shultz Cape”, is located. The laser interference pressure meter of the hydrosphere was installed on the coastal marine area of the Sea of Japan, off the southern coast of Shultz Cape, at a distance of 200 m from the shore and at a depth of 25 m. A layout map of the instrument installation is shown in Figure 1. The instrument was installed on 1 August 2022 and operated for over 3 months. The laser interferometer was deployed on the seabed and was powered through a cable line, enabling real-time data transmission. The detector’s sensitive element consists of a circular membrane firmly secured around its perimeter. One side of the membrane comes into contact with water and, under the influence of variations in hydrosphere pressure, bends in one direction or another. The reverse side of the membrane acts as a reflective element within the interferometer’s measuring arm. Pressure changes alter the length of the measuring arm; the change is captured by the digital recording system housed within the instrument. These data are then transmitted via cable line to the onshore observation post. The procedure of deploying the device onto the seabed includes several steps. Initially, it is placed within a protective cage to prevent interference from seabed sediments. During the descent, air from the external reservoir is pumped into the compensation chamber located inside the instrument. This step ensures the membrane’s placement in the neutral position at the initial time of measurements. Once the laser interference pressure meter of the hydrosphere reaches the operational depth, the air supply stops, and the device is ready for operation.

In [21], a detailed description of the working principles of this laser-interference instrument is provided. The laser interference pressure meter of the hydrosphere allows recording pressure within frequency range from 0 (conventionally) to 1000 Hz with an accuracy of 0.24 mPa, and it can submerge to a maximum depth of 50 m. Data from the laser interferometer, after preliminary processing, were compiled into one-hour files and stored in the experimental database for subsequent analysis and interpretation. In this study, we consider ocean waves in the period range from 1 to 10 min. For better analysis, the initial data underwent filtering using a low-pass filter with a Hamming window of 3000 length and were downsampled with a frequency averaging 1 Hz. The choice of this filter is due to the fact that even with a window length of 3000, it has no smooth transition at the filtering coefficient boundaries; it cuts off all unnecessary frequencies [24]. Figure 2 illustrates the graph of the frequency dependence of the coefficient for the low-pass filter with a Hamming window length of 3000. It is evident from the plot that the filter’s boundary is nearly vertical and yields practically no smoothed frequencies.

3. Processing and Analysis of In Situ Data

Let us analyze the in situ data after preliminary filtering and downsampling to 1 Hz. We examine the hydrosphere pressure variations obtained by the supersensitive detector of hydrosphere pressure variations, caused by sea waves’ activity in the minute range periods. Our primary focus is on observing manifestations of nonlinear hydrophysical disturbances. For instance, Figure 3 shows a fragment of the laser-interference instrument recording of 2 September 2022, where such a disturbance is clearly present against the background of wind waves (marked with an arrow). For a more detailed analysis of the minute range sea waves’ activity, we filter out wind waves and waves with longer periods. We apply a bandpass filter with a 3000-point Hamming window in the period range from 1 to 10 min (Figure 3b). In the filtered record from the laser interference pressure meter of the hydrosphere, it is evident that approximately 30 min before a nonlinear hydrophysical disturbance, the amplitude of the sea waves’ oscillations increases fivefold. Subsequently, hydrophysical disturbance emerges, with a height roughly 20 times greater than the initial sea waves’ amplitude. The duration of this disturbance is about 5.5 min. Following this event, the amplitude of the waves in the minute range periods abruptly decreases to its initial value.

The dynamic spectrogram in Figure 4 shows a more detailed increase in the sea wave periods. In the spectrogram, the nonlinear hydrophysical disturbance is located almost in the center. To the left of this disturbance, there is an increase in the amplitude of infragravity waves, and the amplitude values are much smaller thereafter.

The next nonlinear hydrophysical disturbance with similar characteristics was detected on 3 September 2022. Figure 5 shows a fragment of the record of the supersensitive detector of hydrosphere pressure variations, where the manifestation of infragravity sea waves and nonlinear disturbance is already visible against the background of wind waves (marked with an arrow). Let us filter the initial record of the instrument using the same bandpass filter. As a result, the filtered record shown in Figure 5b clearly shows sea waves in the minute range period, the amplitude of which has increased by about fivefold compared to the regular infragravity waves. Further, a wave with a height almost 15 times greater than the amplitude of the regular waves was registered, and the duration of this disturbance was about 5 min. As in the previous case, after registration of the nonlinear hydrophysical disturbance, a decrease in the sea waves’ amplitude in the minute range period was observed.

The dynamic spectrogram in Figure 6 shows the change in sea waves’ amplitude in more detail. In the spectrogram, this disturbance is located to the right of the center, and we can see that the sea waves’ amplitude in the minute range increased prior to this disturbance. After this, the amplitude of the infragravity waves decreased abruptly. Also, in the upper part of the dynamic spectrogram, there are wind waves with periods of about 12 s.

The records of the laser interference pressure meter of the hydrosphere revealed nonlinear hydrophysical disturbances not only into the positive but also into the negative area. Thus, on the fragments of the instrument records shown in Figure 7, the nonlinear hydrophysical disturbance is directed downward (marked with an arrow). In the fragment of the field record of the instrument of 4 September 2022, the infragravity waves with periods of about 5 min are well distinguished against the background of the wind disturbances with periods of 6–7 s.

As a result of analysis of the fragment of the laser interference pressure meter of the hydrosphere record of 4 September, it was established that approximately 30 min before registration of the nonlinear hydrophysical disturbance, the amplitude of the infragravity sea waves increased threefold, and the height of the hydrophysical disturbance was ten times greater than the amplitude of the initial sea waves in the minute range. The duration of this disturbance was slightly more than 3 min. The dynamic spectrogram in Figure 8 confirms that prior to the nonlinear hydrophysical disturbance, the amplitude of the infragravity sea waves is several times higher than the amplitude after the disturbance.

Let us analyze the fragment of the supersensitive detector of hydrosphere pressure variations record of 2 September 2022, shown in Figure 9, and the dynamic spectrogram (Figure 10). However, in the fragment of the record of 2 September 2022 (Figure 9), the infragravity sea waves are not so vivid. There is no clear increase in the amplitude of the infragravity sea waves before the record of the nonlinear hydrophysical disturbance. However, about 10 min before the disturbance, a solitary wave with a duration equal to the duration of this disturbance but with a height 2 times lower was registered. The height of the nonlinear hydrophysical disturbance was approximately 13 times greater than the amplitude of the infragravity sea waves. The same phenomenon is confirmed by the dynamic spectrogram shown in Figure 10. Prior to the nonlinear hydrophysical disturbance (marked with an arrow), the amplitude of the infragravity sea waves is larger than after it, especially at the moment of registration of the solitary wave and disturbance. Further, after the “surge” in the infragravity waves’ energy in the form of the registered disturbance, the amplitude of the sea waves in the periods of minute range abruptly decreases.

4. Discussion of the Obtained Results

As a result of the studies, in all cases of registration of the nonlinear hydrophysical process, an increase in the sea waves’ amplitude in the minute-range periods is observed. Immediately after this disturbance, the sea waves’ amplitude abruptly decreases. This phenomenon is connected to the “surge” in sea waves’ energy in the form of nonlinear hydrophysical disturbances. These disturbances were observed from 2 to 4 September 2022. At that time, there was a local vortex over the site of the laser interference pressure meter of the hydrosphere installation. Figure 11 shows the wind direction over the sea surface for this time interval. The first image (Figure 11a) was taken on 2 September 2022 at 8:00 (UTC). Subsequent images are taken every 8 h. The trajectory of the local vortex can be traced from these images.

In addition to the energy “surge” mechanism of wave processes, which is directly involved in formation of nonlinear hydrophysical disturbances, there may also be a “modulation” mechanism emerging from the interaction between long waves and shorter waves. Thus, ref. [25] describes a method for identifying modulation processes in the interaction of wind and swell waves with tidal and seiche oscillations. This method is based on regression analysis and a general period change function [25], which describes the dispersion of waves as they propagate from the source to the point of registration. Using this method, some anomalies related to the modulation of ripples on tidal oscillations were identified. For comparison, Figure 12 shows examples of normal and anomalous modulations obtained by the above method.

In Figure 12a, there is a clearly modulation of both wind waves’ period and wind waves’ amplitude on tidal oscillations. As a result of this modulation, waves with a larger period and amplitude are concentrated in the points of tide with maximum values, which is consistent with general understanding of this process and with the two-scale sea waves model used in radar and satellite methods for monitoring the ruffled surface. In Figure 12b, we can observe the opposite pattern, where the waves’ periods and amplitude are in antiphase, which is not the norm for natural wave modulation processes. Below, Figure 13 shows the fragment of the supersensitive detector of hydrosphere pressure variations record with this anomalous process.

Figure 13a shows an example of a complex oscillatory process, which includes wind waves with periods of 8–10 s, seiche (18–20 min) and tidal oscillations with a period of 12 h. The wind waves are modulated not only by the tide but also by seiche. As a result, the cycles of period and amplitude modulation are in antiphase (Figure 12b). Below, Figure 13b shows an enlarged fragment of anomalously large oscillations, after the occurrence of which the amplitudes and periods of wind waves and seiche decreased, and the modulation process returned to the normal state, i.e., waves with a large period and amplitude are concentrated in the areas with maximum amplitude of low-frequency processes that modulate them. The described phenomenon is not frequent, but we have repeatedly registered it. In view of the above, we believe that there is a mechanism of formation of nonlinear hydrophysical phenomena (extremely large solitary waves) as a result of modulation interactions of different-scale wave processes. This mechanism is as follows. When complex oscillating systems occur, there can be a phase shift between modulations of period and amplitude. As a result, the oscillating system of wave processes makes a single “surge” in wave energy in the form of a single disturbance, which allows the stabilization of modulation of the period and amplitude in a phase and a return to the state of hydrodynamic equilibrium.

5. Conclusions

As a result of processing data from the laser interference pressure meter of the hydrosphere installed on the shelf of the Sea of Japan, nonlinear hydrophysical disturbances were identified. Analysis of in situ data of sea waves’ activity in the period range from 1 to 10 min revealed an increase in amplitude during registration of a disturbance. Immediately after the disturbance, the amplitude of the sea waves abruptly dropped back to its initial value. This process can be characterized as a “surge” in energy of the infragravity sea waves as they move along the coastal marine area of decreasing depth and interact with the wind waves. The greatest manifestation of nonlinear hydrophysical disturbances was during the passage of a local vortex over the instrument installation site. It was this vortex that contributed to the increase in the wind waves’ amplitude and their subsequent interaction with the waves of large periods, and the subsequent “surge” in energy in the form of a nonlinear hydrophysical disturbance.

Also, one of the physical mechanisms of the appearance of extremely large solitary waves may be the emergence of complex oscillating systems consisting of different-scale wave processes, which, as a result of nonlinear interaction, lead to breaks in the modulation of periods and amplitudes of one of the processes. This, in turn, leads to a single “surge” in wave energy and restoration of the state of hydrodynamic equilibrium.

Further work in this field will be devoted to the study of the mechanisms driving nonlinear hydrophysical disturbances, the dependence of the appearance of local vorticity on the frequency of occurrence of disturbances, and the behavior of these disturbances in various situations.