Statistical estimates of internal waves in different regions of the World Ocean are discussed. It is found that the observed exceedance probability of large-amplitude internal waves in most cases can be described by the Poisson law, which is one of the typical laws of extreme statistics. Detailed analysis of the statistical properties of internal waves in several regions of the World Ocean has been performed: tropical part of the Atlantic Ocean, northwestern shelf of Australia, the Mediterranean Sea near the Egyptian coast, and the Yellow Sea.

Internal waves are observed everywhere in the shelf zones of seas. The main source of their generation in the ocean is the semidiurnal tidal wave, which is initially barotropic and generates the baroclinic tidal wave by scattering on the continental shelf. Periodical lunar tide M2 generates internal waves with a period of 12.4 h. This process is well studied and presented in publications (Garret and Kunze, 2007; Morozov, 1995, 2018; Vlasenko et al., 2005). Nevertheless the variability of the magnitude of the lunar tide and variations in the temperature and salinity of the sea water lead to random characteristics of the observed internal wave field; see, for example, the book by Miropolsky (2001) and review paper by Helfrich and Melville (2006). Spectral and correlation methods of random internal wave field are widely applied in science. As a result, climatic spectra of internal waves have been determined. The well-known model of Garret and Munk (1975) became the basis for the background spectra of internal waves in the World Ocean. This model determines the background of oceanic internal wave spectra, over which intense processes of internal wave generation occur leading to the appearance of large-amplitude (up to extreme values of 500 m) internal waves (Alford et al., 2015). Data of the large-amplitude internal waves in various areas of the World Ocean are collected in numerous papers (Apel et al., 1985; Salusti et al., 1989; Holloway et al., 1999; Morozov, 1995, 2018; Ramp et al., 2004; Sabinin and Serebryany, 2007; Shroyer et al., 2011; Xu and Yin, 2012; Kozlov et al., 2014; Xu et al., 2016). For example, extreme waves of high amplitudes in the Strait of Gibraltar and Kara Gates Strait were analyzed in Morozov et al. (2002, 2003, 2008). Large-amplitude internal waves are of interest to researchers due to their dangerous impact on offshore platforms (Fraser, 1999; Song et al., 2011), their influence on safety of submarines and underwater vehicles (Osborn, 2010), and the fact that they also cause phase fluctuations of acoustic signals over large distances (Warn-Varnas et al., 2003; Rutenko, 2010; Si et al., 2012). Special warning systems are developed now in regions of high risk of a pipe and platform damage by intense internal waves (Stöber and Moum, 2011).

Internal waves in the ocean can be considered as a continuous random process,
and their intense large amplitudes can be interpreted as outliers of a random
process and be described by the tails of the distribution functions.
Consequently, the statistics of these processes is usually different from the
Gaussian (normal) distribution. Non-Gaussian character of the observed
internal wave field has been reported in many regions of the World Ocean
(Miropol'sky, 2001; Wang and Gao, 2002). Seasonal and longitudinal
statistical analysis of internal wave field has been reported recently in the
South China Sea (Zheng et al., 2007). It is demonstrated that the largest
number of internal wave packets is observed at a longitude of
116.5

In this paper we apply the methods of extreme statistics to large-amplitude internal waves and present a brief review. First, the theoretical approach is revised in Sect. 2. Then, the results of statistical processing of the internal wave records in various regions of the World Ocean are presented in Sect. 3. Conclusion is given in Sect. 4.

Let

Here we will use the direct method to evaluate the statistics of large-amplitude internal waves. We fix the reference vertical displacement and analyze the statistics of the exceedance of wave oscillations beyond this level (outliers of random process). Let us briefly reproduce the well-known approach for calculating the exceedance frequency for continuous processes (Gumbel, 1958; Stuart, 2001) with application to internal waves.

It is known from the vertical structure of internal waves that the largest
amplitudes of the most energetic lowest-mode waves are found in the
pycnocline. For definiteness, we chose the vertical displacement in the
pycnocline and denote it

Here we calculate the average number of “positive” outliers (large crests)
in the wave record. To do so we divide the total time interval into small
subintervals

Only the average number of outliers was discussed above without considering
their probabilistic distribution. A much more difficult problem is to
calculate the latter. It should be noted that if outliers are rather rare
(which is typical for very large-amplitude internal waves,

Detailed calculations of the outlier characteristics in the internal wave
field require the knowledge of two-point (vertical displacement and vertical
velocity) distribution functions of isopycnal variation, which are usually
not measured. If the internal wave random field is assumed normal, the
density of distribution function is described by the Gaussian law:

It should be remembered that usually the statistical distributions of
internal wave field in various regions of the World Ocean are different from
the normal distribution as we have already pointed out in the Introduction (see
the book by Miropolsky, 2001); hence, the result will be different from
Eq. (13) depending on the particular form of the tails of the distribution
function in the large amplitude range. As shown in Leadbetter et al. (1983), the
intermediate asymptotic exceedance frequency for large outliers is described
by the Poisson law:

Data analysis of measurements in the shallow water (Qingdao offshore area) of
the Yellow Sea is reported in Wang and Gao (2002). The authors used a
thermistor chain. The duration of records is 49 h 49 min with a sampling
interval of 6.4 s. The water depth is 33 m. A thin unperturbed pycnocline
is located in the interval from 10 to 16 m with the maximum of the
Brunt–Väisälä frequency

We can explain the sign of the computed skewness applying the weakly nonlinear theory of internal waves based on the Korteweg–de Vries equation (Pelinovsky and Shurgalina, 2017). In this region of the Yellow Sea the sign of quadratic nonlinear term in this equation is negative because the water stratification (see Wang and Gao, 2002) is practically approximated by a two-layer with pycnocline located above the mid-depth (Djorjevich and Redekopp, 1978; Kakutani and Yamasaki, 1978). Nonlinear waves as solutions of the extended Korteweg–de Vries equation with negative quadratic nonlinearity have deepest troughs. For instance, internal wave soliton has negative polarity (Grimshaw et al., 2007). This leads to the negative values of skewness.

The exceedance frequency of internal wave is estimated using the data
obtained during the 39th cruise of the RV

Wave height prediction for the Atlantic tropical zone.

During the time of measurements in the region, the “true” internal wave recording time (considering the ship motion) was about 45 days. According to the prediction for this period, a wave with a height of more than 31 m should be observed once, with a height of more than 23 m – twice, and more than 27 m – three times. In fact, the level of 31 m was exceeded three times, and the level of 27–28 m was exceeded six times, which indicates that a good agreement exists between the measurements and the predictive models.

It is expected that the wave processes in the open ocean are described by the
normal law, which makes it possible to use the theory of normal random
process and estimate the limits of its applicability for internal waves. In
this paper exceedance frequency analysis is undertaken for internal wave
records obtained from a cluster of moorings in the eastern Atlantic Ocean in
1985 (the Mesopolygon-85 experiment; the detailed description of the
experiment is given in Kort (1988). Seventy-six moorings with current and
temperature meters were deployed in the study area called Mesopolygon-85 in
the eastern part of the Atlantic Ocean with the objective of studying
mesoscale variability of hydrophysical processes. The study site was located
between the Canary Basin and the Cabo Verde Basin (19–21

Records of temperature variations (centigrade) at various points of the study
site were used to calculate the average frequency of outliers (temperature
variation exceeding of the set value

Let us discuss internal wave statistics in the seas of low tide where one can
expect the universe statistical characteristics over a short period of time
without correlation to the phases of the moon. The mechanisms of internal
wave generation here can be storms and upwelling as well as the
effect of river discharge. We analyzed internal wave observations in one of
the Mediterranean regions (the Levantine Sea) during the 27th cruise of the RV

The exceedance frequency of internal wave amplitudes in the eastern part of the Mediterranean Sea.

Predicted internal wave heights in the Mediterranean Sea.

It should be noted that the observed internal waves in this region have much smaller amplitudes than over the ridges, for example the Mascarene Ridge (Morozov et al., 1996) and in the Luzon Strait, where 100 m waves are recorded (Ramp et al., 2004). This fact is well known in the seas with low tides and is reflected in the large value of the return period for internal wave of 5 m amplitude in this part of the Mediterranean Sea. Hence the observed height distribution is in the “middle” between the Gaussian statistics (for weak-amplitude waves) and Poisson statistics (for large-amplitude waves).

Relatively long internal wave records were obtained from moorings on the
northwestern shelf of Australia (Pelinovsky et al., 1995). The water depth is
approximately 123 m. We shall analyze the velocities in the internal wave
range, recorded at a level of 3 m above the ocean bottom. The time sampling was 2 min, and the duration of
measurements was 10 days. Only the velocity component, which contains the
strongest wave fluctuations, was analyzed in the transverse to the isobath
direction (45

The regression formulae presented above can be used to calculate the exceedance probability of large-amplitude internal waves as a function of the amplitude of velocity caused by internal waves and time duration. The results of calculation for the northwestern shelf of Australia are shown in Fig. 2.

Probability of occurrence of internal waves at the northwestern shelf of Australia.

Areas where we consider statistical characteristics of internal waves.

We have considered statistical characteristics of the internal wave field in several zones of the World Ocean: the tropical part of the western Atlantic Ocean near the mouth of the Amazon, the part of the eastern Atlantic, the western part of the Mediterranean Sea, the northwestern shelf of Australia and the Yellow Sea shelf (Fig. 3).

It is difficult to compare directly the results of exceedance frequency calculations for various regions of the World Ocean. The observations were not performed using similar methods. One of the difficulties is that different characteristics were measured. In particular, in the tropical zone of the Atlantic, the vertical displacement of the sound-scattering layers was measured; in the Mediterranean Sea it was the amplitude of displacement of the thermocline, while on the Australian shelf the records of flow velocity fluctuations were analyzed. At the Mesopolygon-85 it was the temperature fluctuations. To recalculate these values into the amplitude of internal wave displacement we should know additional information such as the temperature gradient. The next difficulty is that all measurements were produced at different levels. The internal mode structures and hydrology were never analyzed in these measurements, and we cannot say what value of the internal wave amplitude we can expect at the comparison level. So, now we can predict the internal wave amplitude only at the level of measurements.

We find that the Poisson law is valid for internal wave amplitude distribution at the three study sites, but in the Mediterranean, where the internal wave amplitudes did not exceed 2 m, we find that the Gaussian distribution is also appropriate here as the Poisson distribution. The Gaussian law is valid for small amplitudes, and we also obtain this law in Mesopolygon-85.

Meanwhile, the value of

Currently, the numerical methods to predict internal wave field characteristics in different regions of the World Ocean are widely applied (Kurkina and Talipova, 2011; Talipova et al., 2014). They demonstrate that such characteristics are very sensitive to the density stratification of the ocean. The influence of variation of water stratification on the internal wave dynamics can be illustrated by the seasonal maps of kinematic parameters of internal waves (Kurkina et al., 2011, 2017a, b). Statistical estimates of internal waves existing in various regions under different background conditions using numerical models can be calculated. The authors have started to do this work, which will be analyzed further.

The data used by this study are extracted from the GDEM database.

All authors made the same contribution to this work.

The authors declare that they have no conflict of interest.

This study was initiated in the framework of the state task programme in the sphere of scientific activity of the Ministry of Education and Science of the Russian Federation (project nos. 5.4568.2017/6.7 and 5.1246.2017/4.6) and financially supported by this programme, grants of the President of the Russian Federation (NSh-2685.2018.5 and MK-1124.2018.5) and Russian Foundation for Basic Research (grant no. 16-05-00049).

Authors thank Eugene Morozov and Yury Stepanyants for useful critical comments. Edited by: Kateryna Terletska Reviewed by: Eugene Morozov and Yury Stepanyants