NPGNonlinear Processes in GeophysicsNPGNonlin. Processes Geophys.1607-7946Copernicus PublicationsGöttingen, Germany10.5194/npg-25-511-2018Exceedance frequency of appearance of the extreme internal waves in the World OceanExceedance frequency of appearance of the extreme internal waves in the
World OceanTalipovaTatyanaPelinovskyEfimhttps://orcid.org/0000-0002-5092-0302KurkinaOxanahttps://orcid.org/0000-0002-4030-2906GiniyatullinAyratKurkinAndreyaakurkin@gmail.comNizhny Novgorod State Technical University n.a. R.E. Alekseev, Nizhny
Novgorod, RussiaInstitute of Applied Physics, Nizhny Novgorod, RussiaAndrey Kurkin (aakurkin@gmail.com)18July20182535115196February201812February20185July20186July2018This work is licensed under the Creative Commons Attribution 4.0 International License. To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/This article is available from https://npg.copernicus.org/articles/25/511/2018/npg-25-511-2018.htmlThe full text article is available as a PDF file from https://npg.copernicus.org/articles/25/511/2018/npg-25-511-2018.pdf
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.
Introduction
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∘ E (in latitudinal band 20–22∘ N). It has been
reported that the most intense packets of internal waves in the South China
Sea are generated in June. Special analysis of wave amplitude distribution
for the tropical part of the western Atlantic, over the northwestern shelf of
Australia and in the eastern Mediterranean Sea was performed in Ivanov et
al. (1993a, b) and Pelinovsky et al. (1995). The authors show that the
Poisson law is a good approximation for amplitude distributions of such
waves. It is known (Gumbel, 1958; Leadbetter et al., 1983) that the Poisson
distribution is widely used in the extreme statistics and very often applied
in the ocean engineering for describing storm waves (Pelinovsky and Kharif,
2016), tsunamis (Kaistrenko, 2014) and rogue waves (Kharif et al., 2009); it
is also used in geophysics for the description of climatic anomalies
(Dischel, 2002). However, the distribution law for temperature deviations in
the tropical part of the eastern Atlantic (Mesopolygon-85) is closer to the
Gaussian distribution (Morozov et al., 1998), which can be explained by lower
energy of internal waves in this region.
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.
Extreme statistics methods for high-amplitude internal waves
Let η(t,|x,y,z) describe the vertical displacement of
any isopycnal surface at fixed point which, in the first approximation, can
be considered as stationary random process. Usually, a few central moments
μr=∫-∞∞(η-η‾)rP(η)dη(r=2,3,4)
are computed for the statistical analysis. Here P(η) is the probability
density function and η‾is the mean value
η‾=∫-∞∞ηP(η)dη
(unperturbed position of isopycnal surface). The second moment μ2=σ2 determines the intensity of internal wave oscillations and
σ is root mean square height of internal waves. The third and fourth
moments determine skewness Sk =μ3/σ3 and kurtosis
Ku =μ4/σ4, which are used to characterize the
deviation of the distribution function from the Gaussian law (note that
Sk = 0 and Ku = 3 for the Gaussian distribution). The sign of the
skewness on our opinion can be explained by the specific shape in the
internal Stokes wave, which is followed from the weakly nonlinear theory of
internal waves. Unlike nonlinear surface waves, which always have narrow and
high crests and flat troughs, internal waves at different depths can have
either narrow crests and flat troughs or vice versa depending on the density
stratification and modal structure, and distance to the bottom or surface. In
the case of energetic internal waves of the first mode, the character of
asymmetry of wave profile with respect to the horizontal axis is determined
by the coefficient of the quadratic nonlinear term in the Korteweg–de Vries
equation, which is strongly variable in the World Ocean (Grimshaw et al.,
2007; Kurkina et al., 2011, 2017a, b). Computations of the skewness and
kurtosis as well as distribution function of nonlinear internal waves are
interesting problems that have been poorly studied to date.
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 η(t) omitting coordinates of the pycnocline in
this function. Both conditions for the outlier beyond the level A in the
interval Δt should be satisfied at once:
η(t)<Aandη(t+Δt)>A.
Due to the smallness of Δt we can assume that η(t+Δt)≈η(t)+w(t)Δt, where
w(t)=dη/ dt > 0 is the vertical
velocity of water particles located within the isopycnal surface, and rewrite
condition (2) as
A-w(t)Δt<η(t)<A.
The required probability of finding η(t) in the interval (3) is
P(A-wΔt<η<A)=∫0∞dW∫A-wΔtAf(η,W;t)dη,
where f(η,w; t) is the two-dimensional probability density of
vertical displacement η(t) and vertical velocity w(t) at the same
time moment and the same coordinates. Since Δt is small, we can use
the mean-value theorem to calculate the inner integral in Eq. (4) and write
P(η-wΔt<η<A)=Δt∫0∞wf(A,w;t)dw.
Probability density function (in time) can be easily found from Eq. (5):
p(A;t)=∫0∞wf(A,w;t)dw.
Similarly, the probability of crossing level A from the top down (into the
region of small isopycnal value displacement) is
p′(A;t)=-∫-∞0wf(A,w;t)dw,
since this requires w < 0. This equation can be used to calculate
the probability of large troughs in the vertical displacement, which can be as
dangerous as large crests. It is known that polarity of large internal waves
in the ocean is usually negative; this correlates with the negative sign of
the quadratic nonlinearity parameter in the weakly nonlinear theory based on
the Korteweg–de Vries equation for the deepest parts of the World Ocean
(Grimshaw et al., 2007).
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 Δtj and introduce a random value Nj equal to 1
for an outlier and 0 outside the outlier. Then the total number of outliers
is N(A)=∑Nj and its mean value is the ensemble average, the
probability of this is equal to the probability of crossing level (6). Moving
on to the limit for Δtj→0, we finally have
<N(A)>=∫0T∫0∞wf(A,w;t)dwdt.
In the case of a stationary random process formula (8) is simplified to
<N(A)>=T∫0∞wf(A,w)dw.
Thus, the average number of outliers is proportional to the time interval and
falls with the increase in the outlier level. The same approach can be used
to compute the average number of “negative” outliers (deepest troughs) in
the internal wave field.
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, A→∞),
then their distribution can be regarded as the Poisson law. Then the
probability that at least one outlier appears in the time interval t is
P=1-exp(-νt),
where the mean frequency of outliers
ν= < N > /T is found from Eq. (9) as
ν(A)=∫0∞wf(A,w)dw.
The average frequency of outliers in the first approximation can be used as
an estimate of the internal wave exceedance (cumulative) frequency with the
amplitudes greater than the given value of A.
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:
f(η)=12πδe-(η-η‾)22δ2,
where δ is the standard deviation (mean amplitude of internal waves).
The distribution function of the vertical displacement and the vertical
velocity for the normal process do not correlate; hence their two-dimensional
probability density splits into a product of two Gaussian curves (Eq. 12),
which naturally have different mean-square deviations. Then the average
frequency of outliers can be written as
ν(A)=δw2πδηexp(-A22δη2),
where δw is the mean-square (standard deviation) value of
the vertical velocity in the internal wave and δη is the
mean-square value of the vertical isopycnal displacement. Thus, the internal wave
exceedance frequency depends on the wave amplitude according to the Gaussian
law, which very quickly decreased with an increase in amplitude. We will
demonstrate the Gaussian character of cumulative frequency for tropical zone
of the eastern Atlantic Ocean.
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:
ν=ν0exp-AA0,
where ν0 and A0 are the parameters depending on the specific type
of “tails” of the distribution function in the large amplitude range. This
expression can be used to compute exceedance (cumulative) frequency of large
outliers (positive or negative) in the internal wave field. Obviously, the
predicted amplitude values are also random, and here one can speak only about
its evaluation. Thus to estimate the predicted amplitude A it is necessary
to set the value of the exceedance frequency:
ν=1/T,
where T is the prediction time (or the recording time). Expression (14) is
used to calculate amplitude A of internal wave over prognostic time
interval T:
AT=A0ln(ν0T).
Characteristic values of A0 and ν0 are different in various
regions of the World Ocean, and we shall discuss this in the next section.
Statistics of internal wave fieldProbability density function in the Yellow Sea
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 Nmax=0.067 s-1. Vertical displacements
(double amplitudes!) of high-pass-filtered 25–17.5∘ (16 levels)
isotherms are shown and their histograms are plotted. The maximum wave height
here did not exceed 5 m; nevertheless the process differs from the Gaussian
process. It is found that the standard deviation increases slowly from 0.46
to 0.56 m from the surface to the bottom. Skewness is negative for each
isotherm and its maximum absolute value is 0.5 in absolute value at a depth
of 15 m; it decreases to 0.36 at the surface and to 0.06 close to bottom.
Kurtosis changes from 3.24 close to bottom to 5.05 at the 14 m of depth and
4.12 near the surface. It is shown that the distribution of large internal
wave amplitudes does not coincide with the Gaussian distribution.
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.
Exceedance frequency of internal waves in the tropical zone of the
western Atlantic
The exceedance frequency of internal wave is estimated using the data
obtained during the 39th cruise of the RV Akademik Vernadsky in the
northwestern Atlantic tropical zone near the mouth of the Amazon River
(2–15∘ N, 38–52∘ W; Ivanov et al., 1993b). Internal waves
of moderate (2.5–10 m) amplitudes are observed in this region. The ship echo
sounder was used to obtain long-term internal wave records. This device
allowed the authors to study the fluctuations of the sound-scattering layer
at depths up to 100 m. It is known that fluctuations of this layer can be
caused by various processes, but in the range of periods up to 3 h they are
mainly related to internal waves. The wave period on the sonar records varies
in a wide range from 3 to 30 min on the sonar recordings. Since the
measurements were made from the ship moving at a velocity of about V=15 kn and as the maximum of internal wave speed is c=3 kn, in the
first approximation, the internal wave pattern can be considered frozen in
the first approximation. In this case, the “true” wave period increases in
comparison to the observed one with a ratio of V/c=5. The amplitude of
sound-scattering layer fluctuations was associated with the
internal wave amplitude in the pycnocline everywhere. The total record duration was
about 218 h. Wave height (defined as the fluctuation swing between adjacent
extremes) and fluctuation duration of the recording were considered the main
characteristics. Primary echogram processing results by day are presented in
Ivanov et al. (1993b). These data are used to estimate the exceedance
(cumulative) frequency. They well agree with the regression line
ν=9.2exp(-0.3H),
where H is the wave height measured in meters, and the dimension of ν
is in h-1, except for the heights greater than 25 m, where the total
number of wave observations does not exceed six. Expression (17) is used to
estimate the predicted wave height versus predicted time function
H=18+3.3lnT.
Predicted values of internal wave heights versus time are summarized in
Table 1.
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.
Temperature fluctuations caused by internal waves at
Mesopolygon-85 in the Atlantic Ocean
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∘ N and
36–38∘ E). The moorings operated approximately two months from
April to May. The instruments were set at four levels, but the most
representative measurements were gathered at the height level of 200 m. The
total size of the study site was approximately 148.2 by 148.2 km. The
sampling interval was 15 min. In the Mesopolygon area, the bottom is covered
with hills from 500 to 1000 m high over the floor. Such hills are located
every 10 or 20 miles. They form a corrugated bottom topography over which the
horizontal streamlines of barotropic currents are deformed. Thus, the
internal tide is generated immediately in this area over the deep-sea bottom
topography. It should be mentioned that there 49 buoy records available in
the region, and it is a unique possibility to estimate the horizontal
variability of the internal wave amplitude distribution function.
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 ΔT). All processed records are
very well described by the Gaussian distribution:
ν=(0.79±0.17)exp-(10.61±4.5)(ΔT)2,
where the parameters vary from station to station. The dimension of ν is
h-1 and ΔT is in centigrade. It should be noted that the
deviation in ν0 consists of 21 and 42 % in the exponent, over a
study site of 48 000 km2 in the tropical part of eastern Atlantic.
More details of the experiment data are given in Morozov et al. (1998).
Internal wave heights in the Mediterranean Sea
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
Professor Kolesnikov (July–August 1991). These old data briefly
presented in Ivanov et al. (1993a) have been revised. During the period from 27
to 29 July 1991 a special experiment to record internal waves was performed
in the study site near the Egyptian shelf. A distributed temperature sensor
25 m long (MHI 4106) was towed in the thermocline along the tacks
located as a star. The temperature data were later recalculated into the
vertical isopycnal displacement. The basin depth in the study site
varies from 200 to 1100 m. The vertical profile of the Brunt–Väisälä
frequency contained in the pycnocline presence at a depth of about 25 m is
characterized by a frequency of 17 cycle h-1. Below the pycnocline the
mean value of the Brunt–Väisälä frequency is 4 cycle h-1. The wave height
distribution function has been calculated from these data. The vessel speed
was approximately V=5 kn, which significantly exceeds the internal wave
propagation speed in this region (c= 2 kn). Therefore, in the first
approximation, the internal wave field can be considered frozen. This means
that the “true” time recording can be increased by V/c=2.5 times. The
applicability of the Gaussian (red) and Poisson (blue) laws for the
exceedance (cumulative) frequency, as one can see from Fig. 1, is well
applied for the observed data: they are approximated by the formulas
ν=4exp(-1.9A),ν=2exp(-A2),
where A is a wave amplitude measured in meters, and ν
is in h-1. Distribution (20) can be used for the prediction of
relatively large-amplitude waves. The predicted values of internal wave
amplitudes calculated using formula (20a) that can occur in the Mediterranean
Sea near the Egyptian shelf over different return periods are summarized in
Table 2. Return period is the time period when the wave of predicted
amplitude appears at least once.
The exceedance frequency of internal wave amplitudes in the eastern
part of the Mediterranean Sea.
Predicted internal wave heights in the Mediterranean Sea.
Time period1 day1 week1 month3 monthsA (m)2.63.64.55.1
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).
Exceedance frequency in the current velocity from the mooring data
(northwestern shelf of Australia)
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∘ northeast). The time series were processed by a
high-frequency filter to remove the tidal component. Each record was divided
into equal intervals of 4000 min. The analysis of time series processing
results is reported by Pelinovsky et al. (1995). The calculated values of
exceedance frequency for different amplitudes are approximated by the
expression
ν=1.33exp(-0.071U),
where the dimension of ν is h-1 and the dimension of the amplitude
of horizontal velocity variation U is in cm s-1.
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.
Discussion and conclusion
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 ν0 has universal character and should not
depend on the measured characteristics of internal waves. In the Australian
shelf ν0=1.3 h-1, in the tropical zone of the Atlantic
ν0=9.2 h-1, in Mesopolygon-85 ν0 lies between 0.62 and
0.96 h-1 and in the Mediterranean (Levantine Sea) ν0=4 h-1. In fact the T0= 2π/ν0 can be interpreted
as the mean time of the internal wave appearance in the study site. Each
value is characteristic of the specific ocean region. The scatter of these
values is sufficiently high. So, in Mesopolygon-85 the value of T0 lies
between 6.5 and 10.1 h. In the Australian shelf the semidiurnal tide is the
main factor of the internal wave generation, and T0 is about 5 h here.
The time periods of 6.5 and 5 h are close to the time period of maximal
tidal–ebb flow in zone of internal wave generation. The Levantine Sea into
the Mediterranean is very intensive in the wave generation, T0=1.6 h. The tide is not the main
reason for internal wave generation here and in the tropical zone of the
Atlantic T0=40 min, which means that the internal waves appear here
very often, mainly due to their propagation from zones of generation.
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.
Acknowledgements
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
References
Alford, M. H., Peacock, T., MacKinnon, J. A., Nash, J. D., Buijsman, M. C.,
Centurioni, L. R., Chao, S. Y., Chang, M. H., Farmer, D. M., Fringer, O. B., Fu,
K. H., Gallacher, P. C., Graber, H. C., Helfrich, K. R., Jachec, S. M., Jackson,
C. R., Klymak, J. M., Ko, D. S., Jan, S., Johnston, T. M. S., Legg, S., Lee,
I. H., Lien, R. C., Mercier, M. J., Moum, J. N., Musgrave, R., Park, J. H.,
Pickering, A. I., Pinkel, R., Rainville, L., Ramp, S. R., Rudnick, D. L.,
Sarkar, S., Scotti, A., Simmons, H. L., St Laurent, L. C., Venayagamoorthy,
S. K., Wang, Y. H., Wang, J., Yang, Y. J., Paluszkiewicz, T., and Tang, T. Y.:
The formation and fate of internal waves in the South China Sea, Nature,
521, 65–69, 2015.
Apel, J. R., Holbrock, J. R., Kiu, A. K., and Tsai, J. J.: The Sulu sea internal
soliton experiment, J. Phys. Oceanogr., 15, 1625–1651, 1985.
Dischel, R. S.: Climate Risk and the Weather Market, London, Risk Waters
Group Ltd, 300 pp., 2002.
Fraser, N.: Surfing an oil rig, Energy Rev., 4, 20, 1999.
Grimshaw, R., Pelinovsky, E., and Talipova, T.: Modeling internal solitary
waves in the coastal ocean, Surv. Geophys., 28, 273–298, 2007.
Gumbel, E. J.: Statistics of extremes, Columbia Univ. Press, New York,
375 pp., 1958.
Helfrich, K. R. and Melville, W. K.: Long nonlinear internal waves, Annu.
Rev. Fluid Mech., 38, 395–425, 2006.
Holloway, P., Pelinovsky, E., and Talipova, T.: A generalised Korteweg-de
Vries model of internal tide transformation in the coastal zone, J. Geophys.
Res., 104, 18333–18350, 1999.
Ivanov, V. A., Pelinovsky, E. N., and Talipova, T. G.: Recurrence frequency of
internal wave amplitudes in the Mediterranean, Oceanology, 33, 180–184,
1993a.
Ivanov, V. A., Pelinovsky, E. N., and Talipova, T. G.: The long-time prediction
of intense internal wave heights in the tropical region of Atlantic, J. Phys.
Oceanography, 23, 2136–2142, 1993b.
Kaistrenko, V.: Tsunami, Recurrence function: structure, methods of
creation, and application for tsunami hazard estimates, Pure Appl.
Geophys., 171, 3527–3538, 2014.
Kakutani, T. and Yamasaki, N.: Solitary waves on a two-layer fluid, J.
Phys. Soc. Jpn., 45, 674–679, 1978.
Kharif, Ch., Pelinovsky, E., and Slunyaev, A.: Rogue waves in the ocean,
Springer, 216 pp., 2009.
Kort, V. G. (Ed.): Hydrophysical studies on the program “Mesopolygon”,
Nauka, 256 pp., 1988.
Kozlov, I., Romanenkov, D., Zimin, A., and Chapron, B.: SAR observing
large-scale nonlinear internal waves in the White Sea, Remote Sens.
Environ., 147, 99–107, 2014.Kurkina, O. E. and Talipova, T. G.: Huge internal waves in the vicinity of
the Spitsbergen Island (Barents Sea), Nat. Hazards Earth Syst. Sci., 11,
981–986, 10.5194/nhess-11-981-2011, 2011.
Kurkina, O., Talipova, T., Pelinovsky, E., and Soomere, T.: Mapping the
internal wave field in the Baltic Sea in the context of sediment transport in
shallow water, J. Coast. Res., 64, 2042–2047, 2011.
Kurkina, O., Rouvinskaya, E., Talipova, T., and Soomere, T.: Propagation
regimes and populations of internal waves in the Mediterranean Sea basin,
Estuar. Coast. Shelf Sci., 185, 44–54, 2017a.
Kurkina, O., Talipova, T. , Soomere, T., Kurkin, A., and Rybin, A.: The
impact of seasonal changes in stratification on the dynamics of internal
waves in the Sea of Okhotsk, Est. J. Earth Sci., 66, 238–255, 2017b.
Leadbetter, M. R., Lindgren, G., and Rootzen, H.: Extremes and related
properties of random sequences and processes, Springer, 336 pp., 1983.
Miropolsky, Yu. Z.: Dynamics of internal gravity waves in the ocean, edited
by: Shishkina O., Springer, 321 pp., 2001.
Morozov, E. G.: Semidiurnal internal wave global field, Deep Sea Res., 42,
135–148, 1995.
Morozov, E. G. and Vlasenko, V. I.: Extreme tidal internal waves near the
Mascarene Ridge, J. Mar. Syst., 9, 203–210, 1996.
Morozov, E., Pelinovsky, E., and Talipova, T.: Exceedance frequency for
internal waves during the Mesopolygon-85 experiment in the Atlantic,
Oceanology, 38, 470–475, 1998.
Morozov, E. G., Trulsen, K., Velarde, M. G., and Vlasenko, V. I., Internal
tides in the Strait of Gibraltar, J. Phys. Oceanogr., 32, 3193–3206, 2002.
Morozov, E. G., Parrilla-Barrera, G., Velarde, M. G., and Scherbinin, A. D.:
The Straits of Gibraltar and Kara Gates: A Comparison of Internal Tides,
Oceanol. Ac., 26, 231–241, 2003.Morozov, E. G., Paka, V. T., Bakhanov, V. V.: Strong internal tides in the
Kara Gates Strait, Geophys. Res. Lett., 35, L16603,
10.1029/2008GL033804, 2008.
Morozov, E. G.: Oceanic internal tides, observations, analysis, and modeling.
A global view, Springer, 291 pp., 2018.
Osborne, A.: Nonlinear ocean waves and the inverse scattering transform,
Academic Press, 944, 2010.
Pelinovsky, E., Holloway, T., and Talipova, T.: A statistical analysis of
extreme events in current variations due to internal waves from the
Australian North West Shelf, J. Geophys. Res., 100, 24831–24839, 1995.Pelinovsky, E. and Kharif, C. (Eds.): Extreme ocean waves, 2nd Edition,
Springer, 236 pp., 10.1007/978-3-319-21575-4, 2016.
Pelinovsky, E. and Shurgalina, E.: KDV soliton gas: interactions and
turbulence. Book: Challenges in complexity: dynamics, patterns, cognition,
edited by: Aronson, I., Rulkov, N., Pikovsky, A., and Tsimring, L., Series:
Nonlinear Systems and Complexity, Springer, 20, 295–306, 2017.
Ramp, S. R., Tang, T. Y., Duda, T. F., Lynch, J. F., Liu, A. K., Chiu, C. S.,
Bahr, F. L., Kim, H. R., and Yang, Y. J.: Internal solitons in the
northeastern South China Sea – Part I: Sources and deep water propagation,
IEEE J. Ocean. Eng., 29, 1157–1181, 2004.
Rutenko, A. N.: The influence of internal waves on losses during sound
propagation on a shelf, Acoust. Phys., 56, 703–713, 2010.
Sabinin, K. D. and Serebryany, A. N.: “Hot spots” in the field of internal
waves in the ocean, Acoust. Phys., 53, 357–380, 2007.
Salusti, F., Lascaratos, A., and Nittis, K.: Changes of polarity in marine
internal waves: Field evidence in eastern Mediterranean Sea, Ocean Model.,
82, 10–11, 1989.Shroyer, E. L., Moum, J. N., and Nash, J. D.: Nonlinear internal waves over
New Jersey's continental shelf, J. Geoph. Res., 116, C03022,
10.1029/2010JC006332, 2011.
Si, Z., Zhang, Y., and Fan, Z.: A numerical simulation of shear forces and
torques exerted by large-amplitude internal solitary waves on a rigid pile in
South China Sea, Appl. Ocean Res., 37, 127–132, 2012.
Song, Z. J., Teng, B., Gou, Y., Lu, L., Shi, Z. M., Xiao, Y., and Qu, Y.:
Comparisons of internal solitary wave and surface wave actions on marine
structures and their responses, Appl. Ocean Res., 33, 120–129, 2011.
Stöber, U. and Moum, J. N.: On the potential for automated realtime
detection of nonlinear internal waves from seafloor pressure measurements,
Appl. Ocean Res., 33, 275–285, 2011.
Stuart, C.: An introduction to statistical modeling of extreme values,
Springer, 242 pp., 2001.
Talipova, T. G., Kurkina, O. E., Terletska, E. V., Kurkin, A. A., and
Rouvinskaya, E. A.: Modeling of internal wave field in the coastal zone of
the Barents Sea, Ecol. Syst. Dev., 3, 26–38, 2014.
Xu, J., Chen, Zh., Xie, J., and Cai, Sh.: On generation and evolution of
seaward propagating internal solitary waves in the north western South China
Sea, Commun. Nonlinear Sci. Numer. Simulat., 32, 122–136, 2016.
Xu, Zh. and Yin, B.: Variability of internal solitary waves in the Northwest
South China Sea, Oceanography, edited by: Marcelli, M., InTech., 131–146,
2012.
Vlasenko, V., Stashchuk, N., and Hutter, K.: Baroclinic tides: theoretical
modeling and observational evidence, Cambridge University Press, 351 pp.,
2005.
Wang, T. and Gao, T.: Statistical properties of high-frequency internal waves
in Qingdao offshore area of the Yellow Sea, Chinese J. Oceanol. Limnol., 20,
16–21, 2002.Warn-Varnas, A. C., Chin-Bing, S. A., King, D. B., Hallock, Z., and Hawkins,
J. A.: Ocean-acoustic solitary wave studies and predictions, Surv. Geophys.,
24, 39–79, 2003.
Zheng, Q., Susanto, R. D., Ho, Ch.-R., Song, Y. T., and Xu, Q.: Statistical
and dynamical analyses of generation mechanisms of solitary internal waves in
the northern South China Sea, J. Geophys. Res., 112, C03021,
10.1029/2006JC003551, 2007.