texto em 
Inglês (pdf)
Artigo em XML
Referências do artigo
Tradução automática
Enviar este artigo por email
Citado por SciELO 
Similares em
SciELO 
Permalink
Introduction
Freak waves are probably the most dangerous type of waves that appear on the ocean surface. Accidents and damages caused by extreme waves require a better understanding of the origin and physics of these hazardous waves. Knowledge on freak waves is indispensible for the development of a warning system against the attack of extreme waves on maritime structures or their impact on ships.
Freak-type waves, usually defined as extreme waves of height exceeding twice the significant wave height, are probably one of the most complex phenomena occurring during extreme storms. Recently, considerable progress has been made in modeling of freak waves, their origin, and physical properties. Potential mechanisms of the formation of freak waves are a linear superposition of component waves, wave-current interactions, modulational instability of deepwater wave trains, shallow water effects, crossing seas, and wind effects (Didenkulova and Pelinovsky 2011, Sergeeva et al. 2011, Toffoli et al. 2011, Onorato et al. 2013, Xiao et al. 2013). Many theoretical and experimental studies have recently been conducted to extend the understanding of freak waves (e.g., Onorato et al. 2006; Sulisz and Paprota 2006, 2011, 2013; Gramstad and Trulsen 2007; Paplińska-Swerpel et al. 2007; Kharif et al. 2009; Bitner-Gregersen and Toffoli 2012; Chabchoub et al. 2012; Adcock and Taylor 2014; Majewski et al. 2014). Further progress in understanding the origin and physics of freak waves can be achieved by the analysis of data sets containing reliable wind-wave measurements. The problem is that available field measurements that provide insight into the origin and physics of freak waves are very limited. In fact, only few studies based on extreme-wave field data have been performed in the past (e.g., Chien et al. 2002, Mori et al. 2002, Silva et al. 2002, Stansell 2004, Sulisz and Paprota 2005, Didenkulova 2010, Montoya et al. 2013).
A unique set of time series of free-surface elevation records is available at the Institute of Hydroengineering of the Polish Academy of Sciences (IBW PAN), Gdańsk. It consists of measurements obtained by Waverider buoys at several locations along the southern Baltic Sea. It comprises more than 19,025 records including storm conditions. The database can provide useful information on extreme storms, extreme waves, and wave events, and eventually help to better define hazards and protection measures against extreme situations. The problem is serious because during a recent field campaign in Polish territorial waters an individual wave of 12 m in height was measured at a depth of 20.5 m and the waves in the open sea can be far higher (see, e.g., Soomere et al. 2008). Measurements of such extremely large freak-type waves are good arguments to publicize the problem of extreme waves in the Baltic Sea and to conduct more systematic studies on the origin and physics of freak waves.
In this work, time series of free-surface elevation recorded by Waverider buoys in the southern Baltic Sea are analyzed. The main goal of the study is to analyze wind-wave records from the southern Baltic to provide information on parameters and prevailing conditions of freak wave occurrence. First, the available data sets are described and extreme wave records are defined. The wind-wave records are analyzed with emphasis on extreme storm features. Then, the analysis focuses on individual extreme waves and wave events. The analysis includes a wide range of aspects related to extreme storms and individual extreme waves to collect as much information as possible on the occurrence and features of extreme waves, and to eventually contribute to a future warning system. Finally, conclusions arising from the analysis of wave records are specified.
Materials and methods
The wind-wave records analyzed in this study were collected by IBW PAN. The measurements of free-surface elevation were conducted in the southern part of the Baltic Sea at the buoy stations located in the vicinity of Lubiatowo (Fig. 1), where the coastline is straight over a long distance.
The measurements were performed using a Directional Waverider (DWR) buoy. The raw data wave record contains 20-min registration of free-surface elevations measured with a frequency of 1.28 Hz. The data were usually collected every hour. The measurement periods varied from a few to several months. The buoy was deployed 5 km away from the shore at a depth of 20 m. Buoy locations and measurement periods are provided in Table 1. The wind-wave data sets specified in Table 1 constitute a unique and valuable source of information on wind waves in the southern Baltic Sea. This database is the only available collection of wave measurements taken in Polish territorial waters. The total DWR data set for the period 1996-2002 consists of 19,025 wave records. The highest individual wave measured in the data set is 7.6 m and the highest significant wave height is 4.0 m. The water depth of 20 m and wavelengths corresponding to the typical peak periods allow the exclusion of shallow water freak wave generation mechanisms. In a recent field campaign the DWR buoy measured a huge 12-m-high wave at 20.5 m depth, which provides new arguments for conducting studies on extreme waves and wave events in the Baltic Sea.
Table 1 Location of buoy (Directional Waverider) stations in the vicinity of Lubiatowo (Poland) and measurement periods.
The analysis of the data set requires a selection of criteria to define extreme waves and wave records. The problem is not trivial because extreme-wave formation and propagation may depend on factors affecting wind waves, including storm severity, its duration and variation, storm direction, etc. In this study, the significant wave height was chosen to define extreme waves, because it is one of the most important parameters used in the description of wind waves. In fact, the significant wave height is widely used to describe and define extreme waves and in the analysis of the occurrence of extreme waves and wave events. Following the studies conducted in the MaxWave project, extreme waves in the Baltic Sea are defined in terms of the maximum wave height (Hmax) and the significant wave height (Hs) as follows (Paprota et al. 2003, Rosenthal and Lehner 2008):
• waves exceeding twice the significant wave height: Hmax/Hs > 2;
• significant wave height is larger than 1 m: Hs > 1 m.
The threshold of 1 m set for significant wave height is exceeded by 4,975 storm wave records. The application of both limits (i.e., Hs > 1 m and Hmax/Hs > 2) results in 261 extreme wave records. The number of extreme wave records may be underestimated taking into account the limits and drawbacks of the applied measuring system. In fact, the wave crests measured during severe conditions are underestimated due to the quasi-Lagrangian motion of the buoy (Magnusson et al. 1999).
Results
Analysis of extreme storms
First, the results from the statistical analysis of the set of the storm and extreme wave records are presented. The relationships between Hmax and Hs for the extreme wave records are presented in Figure 2. The number of extreme wave records corresponding to selected ranges of Hmax and Hs is presented in the form of histograms.
Figure 2 Scatter plot of maximum (Hmax) and significant (Hs) wave heights with histograms for the extreme waves. N = number of cases.
It is worth noting that one can establish a different set of conditions to define extreme waves, including the ratio of the wave crest height to wave height, the ratio of the crest height to trough height, etc. (e.g., Chien et al. 2002). However, many available definitions are more appropriate for defining episodic waves rather than extreme waves or wave events. Extreme waves considered in the present study can be regarded as the wave events or large waves characterized by the extreme values of the tail of the statistical probability distribution. This means that the extreme wave height is defined in such a way that it is the largest value among a particular sample of wave heights arbitrarily chosen from the population of wave heights.
The direction of extreme waves constitutes an important source of information on the occurrence of extreme waves and wave events. Information on the direction of extreme waves, especially if there is any prevailing direction, is important from a practical point of view. The distribution of storm directions and the corresponding distribution of extreme wave directions for the area of Lubiatowo are presented in Figure 3.
Figure 3 Wave rose for storm (blue) and extreme (red) waves from the Directional Waverider buoy measurements: (a) Hs > 1 m; (b) Hs > 2 m. The peak direction is considered.
The plots in Figure 3 show that strong storms and extreme waves come mainly from the W-WNW and N-NNE directions. This outcome is interesting and rather nonintuitive. The analysis shows that wave records with Hs exceeding 1.0 m constitute 26% of conducted measurements and very strong storms with Hs greater than 2 m account for 4.4% of wave records in the analyzed data set. The total set of 261 extreme waves has been observed among 19,025 records, which means that 1.4% of all records can be considered extreme wave records.
Correlations between parameters affecting a sea state are important sources of information on extreme waves. The description of a sea state is usually presented in terms of wave height and wave period (e.g., Tucker 1991, Leyden and Dally 1996). The significant wave height (Hs) and the mean wave period (Tm) are probably the most common parameters used to represent wave height and wave period in a wave record. Statistics of these parameters can be presented in the form of a bivariate histogram, where the number of cases (N) is presented in a two-dimensional grid. Appropriate Hs-Tm histograms for the storm wave records (4,974) and the extreme wave records (261) are shown in Figure 4. The important feature of the histogram is the narrow range of wave periods for high values of Hs, which enables us to estimate Tm for extreme sea states. The Tm value for the southern Baltic Sea is about 5 s. The plots show that strong storms are characterized by a wider range of Tm values.
Figure 4 Bivariate histograms in terms of significant wave height (Hs): (a) storm wave data set and (b) extreme wave data set. Tm = mean wave period; Lm = mean wave length; and N = number of cases.
A different situation is observed in the bivariate Hs-Tm histograms plotted for the storm and extreme wave records (Fig. 4). The plots are characterized by a wide range of mean wave lengths (Lm). The range of Lm values is relatively wide for both severe and extreme sea states. The wave lengths were calculated according to the linear dispersion relation for a given depth and wave period.
Extreme storms are related to higher waves. Wave steepness, together with wave height, is often used to represent higher waves. This dimensionless parameter is very popular and it is of interest to analyze wave steepness in the case of strong and extreme storms. An adequate analysis was performed for the available storm and extreme wave data sets. The relation between wave steepness and Tm is presented in Figure 5.
Figure 5 Bivariate histograms in terms of wave steepness (S): (a) storm wave data set and (b) extreme wave data set. Sm = mean wave steepness; Smax = maximum wave steepness; Tm = mean wave period; Lm = mean wave length; and N = number of cases.
The plots in Figure 5 are characterized by a relatively wide scatter of the mean wave steepnesses (Sm) and Tm values. The results show that the distribution of Sm is similar for the storm and extreme sea states. This is probably due to the particular feature of wave steepness reaching similar values for waves of different height and length.
A similar situation is observed in the bivariate Sm-Lm histograms (Fig. 5). It is worth noting that the presentation of results in terms of Lm instead of Tm has a limited effect on the final plots. Additional information on extreme storm features is provided by the bivariate Smax-Lm histograms shown in Figure 5. The results are plotted for the storm and extreme wave data sets.
The plots in Figure 5 show a relatively wide scatter of the Tm values, as expected from previous analyses. However, the results show that the maximum wave steepness is far higher for the extreme wave data set than for the storm wave data set. The maximum wave steepness for the extreme wave records may reach the level of about 0.1. A higher value of wave steepness may serve as an indication of the presence of extreme waves. It should be emphasized, however, that the above analysis was performed on the basis of a limited data set. Thus, the deduced conclusions cannot be generalized at this stage.
Analysis of individual extreme wave records
The available data set also provides the possibility of analyzing wave groups, correlation between waves, individual wave shapes, etc. The analysis was performed for extreme wave records and focused on records with a high Hmax/Hs ratio. Typical examples of wave records with high Hmax/Hs ratios and distinguished freak-type waves are presented in Figure 6. The plots illustrate strong deviations of freak waves from the average sea state. A significant deviation of freaktype waves from the average sea state makes these waves dangerous even for large vessels.
Figure 6 Examples of extreme wave records with freak-type waves. Hmax = maximum wave height; Hs = significant wave height; and η = free-surface elevation.
A successive example of a freak-type wave is shown in Figure 7. This is a particularly interesting case due to the extremely large Hmax/Hs ratio: 2.41; Hs = 2.57 m, Hmax = 6.2 m. The wave buoy recorded this freak-type wave during a storm on 14 September 2002. These measurements indicate that the Hmax/Hs ratio may occasionally reach 3. In fact, similar results have been reported in other studies on freak waves (Stansell 2004). Because Hs during strong storms in the Baltic Sea may reach 10 m, or even higher values under favorable wave generation conditions (Soomere et al. 2008), freak-type waves during such storms may exceed 20 m.
Figure 7 An extreme wave of large maximum to significant wave height ratio. Hmax = maximum wave height; Hs = significant wave height; and η = free-surface elevation.
Freak-type waves have limited effects on wave energy spectrums that are usually multi-peaked, as indicated in Figure 8. Although more studies are required, the analysis conducted on the basis of our recent multi-point field measurements and laboratory experiments in a wave flume shows that the differences between spectrums with and without freak waves are negligibly small from a practical point of view. This is demonstrated by "subtracting" a freak wave from the wave record and conducting a spectral analysis. This is an important conclusion, because it indicates that the wave energy spectrum cannot be used to predict freak waves. A similar conclusion was derived in the MaxWave project on the basis of wind-wave records from different basins.
Figure 8 Wave energy spectra corresponding to extreme storms. S(f) = spectral density and f = frequency.
Moreover, the analysis indicates that higher order spectral moments cannot be used to predict freak waves, which is in accordance with a similar conclusion derived also for larger basins. This makes the research on extreme waves and warning systems a difficult task because wave energy spectrums are the main outcome of software used to predict wind waves worldwide.
Since the work of Longuet-Higgins (1952), the Rayleigh distribution has been applied to describe the distribution of individual wave heights. Although the use of the Rayleigh distribution is not the most effective for the assessment of statistical properties of freak wave heights, it may serve for comparisons with the outcome of other studies on the probability of occurrence of extreme waves. The probability of occurrence of freak waves can be assessed by applying several distributions including Weibull, Gumbel, etc. (e.g., Goda 2000). In Figure 9, examples are presented of the probability of exceedance of waves for empirical data and corresponding results predicted by the Rayleigh cumulative distribution of wave heights (H):
Figure 9 Probability of exceedance function for the extreme wave record: Rayleigh distribution (straight line); empirical distribution (dashed line). F(H) = cumulative distribution and H = wave height.
The plots in Figure 9 show that for H > 1.5 Hs, the predictions from the Rayleigh model become increasingly poor. The Rayleigh distribution underestimates the probability of exceedance of the largest waves in the extreme wave records, which is in opposition to the common expectation that it over-predicts the probability of occurrence of large wave heights measured in the sea (see, e.g., Stansell 2004). The use of the corrected Rayleigh distribution according to Næss (1985) or the Forristall wave height distribution (Forristall 1978) would result in an even larger disagreement in case of freak wave height statistics.
The Rayleigh distribution does not have an upper bound. The probability density decreases exponentially as the wave height increases, but never becomes zero (Goda 2000). Therefore, the largest wave height is a statistically defined quantity in such a manner that it is the largest value among the population of wave heights. Nevertheless, one has to remember that despite the drawbacks, the Rayleigh distribution and its modifications are widely applied by engineers and scientists and the results of the present study are a source of rare information of significant practical importance.
Application of transform techniques
The occurrence of extreme waves or extreme wave groups is often a highly non-stationary phenomenon. In order to investigate the structure of such waves, a proper method to analyze the temporal and spectral characteristics is required. Standard methods used for spectral analysis are inappropriate for studying non-stationary signals. The wavelet transform method has been proven to be an efficient tool in the analysis of extreme waves and extreme wave events. The method provides the wave energy distribution and enables localizing it simultaneously in the time and frequency domain (see, e.g., Massel 2001, Chien et al. 2002, Mori et al. 2002). The alternative time-frequency analysis techniques that could be applied to evaluate energy in non-stationary wave signals are, for example, the Smoothed Instantaneous Wave Energy History and Hilbert-Huang Transform (see, e.g., Funke and Mansard 1980, Huang and Shen 2005, Dong et al. 2015).
The wavelet transform method was applied to analyze several extreme wave records (Fig. 10). The results of the analysis of a record with extreme waves are presented in Figure 10a and the results of the analysis of an extreme wave record with more emphasis on wave groups are presented in Figure 10b. Both records refer to extreme storms that are usually characterized by multi-peaked wave energy spectrums.
The plots in Figure 10 show that the wavelet transform method detects extreme waves and extreme wave groups fairly well. A similar conclusion was derived in the MaxWave project on the basis of other wind-wave records from the Baltic Sea (Paprota et al. 2003). Obviously, the possibility of detecting extreme waves and wave groups is very important; however, from a practical point of view, we are more interested in a tool that can predict such extreme events.
One should realize that despite some progress that has been made in the studies on freak waves, including present investigations, knowledge on the formation and physics of extreme waves and wave events is still incomplete. One of the problems is the type of available data that are basically limited to waves recorded by Waverider buoys, laser altimeters, and resistance-type wave gauges, and it is difficult to investigate the formation of freak waves, study their evolution and spatial characteristics, learn more about their physics, etc., from single-point wave records. Models and simulations derived on the basis of single-point wave records often provide misleading information on freak waves. There is an obvious need to conduct more studies on the origin and physics of extreme waves and wave events. Investigations should include multi-point wind-wave measurements to provide more information on the physics of freak waves, which may be helpful in developing theoretical models, tools, and finally warning systems.
Discussion
Freak waves are unique and rare phenomena. They pose a threat to boats and even large vessels, as well as offshore and coastal structures. The problem is that available field data that provide insight into the origin and physics of freak waves are very limited. In the present study, a unique set of wave data comprising 19,025 wind-wave records from the southern Baltic Sea was analyzed to provide information on the occurrence, parameters, and prevailing conditions of the formation of freak waves.
The analysis of the wind-wave records revealed a large number of extreme wave and wave events, including waves exceeding 12 m in height. The results show that strong storms with extreme waves come mainly from the W-WNW and N-NNE. This is an interesting and rather non-intuitive outcome. However, a wide range of analyses conducted on wind-wave records do not show any significant effect of freak waves on storm wave characteristics and description. The present analysis shows that a freak-type wave has a weak effect on wave spectrums, higher-order spectral moments, wave instability indexes, etc., and it is hard to detect or predict it from these quantities, which is challenging, because these hazardous waves may exceed 20 m even in the Baltic Sea. The study implies a need for multi-point wind-wave measurements to provide more information about the nature of freak waves.
One should realize that despite considerable progress made in studies on freak waves, including the progress made in recent years, the mechanisms of the formation and physics of extreme waves and wave events still require more investigation. Available data are basically limited to waves recorded by waveriders, laser altimeters, and resistance-type wave gauges, and it is difficult to investigate the formation of freak waves, study their evolution and spatial characteristics, learn more about their physics, etc., from single-point wave records. Models and simulations adjusted to single-point wave measurements are often sources of misleading information on freak waves. More studies on the origin and physics of extreme waves and wave events are needed. These studies should include multi-point wind-wave measurements to provide more information and eventually develop a prediction tool that can be applied to issue a warning against extreme waves or wave events.
