Statistical Distributions of Crests and Trough of Sea Surface Waves

Толық мәтін

Introduction

The study of sea wave statistical distributions and the identification of rogue waves are among the urgent tasks of modern oceanology [1]. In a linear wave field, which represents a superposition of sinusoidal waves with random phase, provided that the wave spectrum is sufficiently narrow, the distribution of wave heights is described by the Rayleigh distribution [2]. It also describes the distributions of crest heights and trough depths [3]. The Rayleigh distribution is generally regarded as the lower limit giving the lowest probabilities for rogue waves [4]. The linear model also underestimates strongly the probability of high crests [5].

Crest Cr refers to the maximum value of wave record h(t) between the time it crosses the zero level from bottom to top and the time it crosses that level from top to bottom [6]. Similarly, trough Th is minimum value h(t) between two consecutive crossings of the zero level from top to bottom and from bottom to top. Wave height is defined as the sum of consecutive maximum and minimum values between two points where the wave record crosses h(t) the zero level upwards or downwards, i. e., H = Cr + Th [7].

Deviations of sea wave statistical distributions from the linear model are usually described within a second-order nonlinear model which is based on the decomposition of the wave profile into small parameter powers (steepness) [8]. In the above model, the skewness of the sea surface elevation distribution is always higher than zero [9], the crests are higher and the troughs are shallower than predicted by linear theory [10]. Both of these conditions are not always fulfilled in marine environments. Measurements made in different areas of the world ocean have shown that the lower limit of the range where the skewness varies lies in the region of negative values [11, 12]. The second-order nonlinear model describes only average trends of skewness and excess kurtosis changes not allowing to describe the whole variety of situations occurring in the sea [13].

The ratios of crest and trough vary widely. Some situations are observed when the maximum trough is greater than the maximum crest during a measurement session [14, 15]. According to measurements in the Black Sea, the probability of an event in which the trough of the highest wave in the measurement session is greater than its crest reaches 10% [16].

Less attention has been paid to the analysis of the distribution of sea wave troughs than to the statistical description of their crests, although the distribution of troughs is of great importance for a number of engineering applications [9]. The purpose of this paper is to analyse jointly the distributions of crests and troughs of surface waves.

Measurement equipment and conditions

Wave measurement data obtained on the stationary oceanographic platform of Marine Hydrophysical Institute of the Russian Academy of Sciences were used to study the statistical characteristics of surface waves. The stationary oceanographic platform is located in the coastal part of the Black Sea off the Southern Coast of Crimea at a depth of about 30 metres. Two types of wave recorders were used to measure surface waves. The wave recorders of the first type contain a vertically stretched nichrome string as a sensor [17] and the nichrome string is coiled with a constant pitch on a vertically oriented supporting cable-tether in wave recorders of the second type [18].

This paper analyses the measurement data obtained in the summer and autumn of 2006 as well as in the winter of 2018. In 2006, measurements were taken in sessions lasting several hours; in 2018, wave measurements were taken continuously for a month. Continuous recordings of sea surface elevations were divided into 20 min fragments. A total of 2380 twenty-minute fragments were used for the analysis.

For each fragment, crest Cr and trough Th of separate waves were determined and significant wave height н_s, skewness A and excess kurtosis E of surface elevation were calculated. Only waves satisfying the conditions of Cr > 5 cm and Th > 5 cm were considered in the analysis. Hereinafter, parameter Th is equal to the trough modulus.

Wave measurements carried out in different seasons made it possible to cover a wide range of meteorological parameters. The average wind speed during the measurement session varied from conditional zero (the threshold of propeller starting) to 26 m/s. The wind speed reached 35 m/s in gusts. The wave periods calculated from the maximum of the wave spectrum were in the range from 1.1 s to 9 s. Significant wave height varied from 0.1 m to 2.3 m. The values of wave steepness (nonlinearity parameter) ranged mainly from 0.009 to 0.09.

Distributions of troughs and crests

For statistical moments h(t), we introduce the following notation

${\mu }_n=⟨{\eta }^n\left(t\right)⟩$,

where $⟨\mathrm{...}⟩$ is averaging. Let us assume that the average value of a random variable is m₁ = 0, then the skewness and excess kurtosis of the surface elevation distribution are equal to А_h = m₃/m₂^3/2 and е_h = m₄/m₂² – 3, respectively.

To compare the statistical distributions of troughs and crests determined in different situations, we will use normalised wave records

$\left(t\right)=\text{η}\left(t\right)/H_S$,   (1)

where н_s is significant wave height connected to the second statistical moment of sea surface elevations by relation $H_S=\text{4}\sqrt{{\text{μ}}_{\text{2}}}$.

The probability density function of the Rayleigh distribution describing distributions of Cr and Th under the linear model is as follows 

$F_R\left(x\right)=\frac{x}{a^{\text{2}}}\text{exp}\left(-\frac{x^{\text{2}}}{\text{2}{a}^{\text{2}}}\right)$,   $x\ge \text{0}$,   (2)

where $a=H_S/\text{4}$. Considering (1), we obtain that in our case mode of distribution (2) is defined as $M{o}_R=\text{0.25}$.

Empirical probability density functions of Cr and Th were calculated based on histograms constructed with equal intervals of 0.05. Fig. 1 shows the empirical probability density functions of crests f_сr (x) and troughs f_Th (x) calculated over the entire measurement data set. It can be seen that modes of empirical distributions $M{o}_{Cr}$ and $M{o}_{Th}$ are shifted relative to the Rayleigh distribution mode towards higher values x, i. е., the following conditions are fulfilled

$M{o}_{Cr}>M{o}_R, M{o}_{Th}>M{o}_R$.

The modes of the сr and Th distributions are located in neighbouring intervals, with their centers $M{o}_{Cr}=\text{0.375}$ and $M{o}_{Th}=\text{0.325}$.

Fig. 1. Probability density functions f (average over an ensemble of situations). The blue curve is f_сr (x), the red curve is f_Th (x), the black curve is f_r (x)

The consideration of nonlinearity leads to the fact that the probability of high crests becomes greater than in the linear model, in which this probability is described by the Rayleigh distribution, and the probability of deep troughs is smaller [9]. It follows from Fig. 1 that inequality f_сr (x) > f_r (x) is true in region $x>M{o}_{Cr}$. The ratio between f_Th (x) and f_r (x) changes at x₀ » 0.8, inequality f_Th (x) > f_r (x) takes place in region $x_{\text{0}}>x>M{o}_{Cr}$, reciprocal ratio f_Th (x) < f_r (x) takes place at x > x₀. Thus, in the area of high crests and deep troughs, deviations f_сr (x) and f_Th (x) from f_r (x) occur in the direction predicted by the second-order nonlinear model [19], i. е., the average distribution of crests and troughs over an ensemble of situations corresponds to this model qualitatively.

Earlier studies of senior statistical moments of sea surface elevations have shown that the second-order nonlinear model makes it possible to describe only average trends of skewness and excess kurtosis, but does not allow describing the whole variety of situations occurring under sea conditions [13]. The skewness and excess kurtosis values vary over a much wider range than the model suggests. In particular, the model estimates of surface elevation distribution skewness А_h and excess kurtosis е_h are always positive [20], while situations in which А_h < 0 and/or е_h < 0 are often observed under sea conditions [12]. Fig. 2 shows at what values of А_h and е_h the wave records analysed in this paper were obtained.

Рис. 2. Изменения асимметрии А_h и эксцесса е_h возвышений морской поверхности

Effect of skewness А_h and excess kurtosis е_h

Usually, the distributions of crests and troughs are analysed within a second-order nonlinear model based on the decomposition of the wave profile into a series of small parameter powers [19–22]. In [8], a simplified second-order nonlinear model, which is the sum of linear h_L (x, t) and nonlinear h_N (x, t) components, is proposed to describe the sea surface statistical characteristics. The model is constructed for waves propagating in deep water in the narrowband spectrum approximation. It is described by the amplitude-modulated Stokes wave equation with average frequency w and random phase e

$\text{η}\left(x,t\right)={\text{η}}_L\left(x,t\right)+{\text{η}}_N\left(x,t\right)=a_r\left(x,t\right)\text{cosθ}+\frac{\text{1}}{\text{2}}{k}_p{a}_r^{\text{2}}\left(x,t\right)\text{cos}\left(\text{2θ}\right)$,   (3)

where а_r (x, t) is envelope; q = k_p x – wt + e; k_p is wave number corresponding to the wave spectrum peak. The local maxima of nonlinear term h_N (x, t) coincide with the crest and trough of the linear wave h_L (x, t), hence, the ridge and trough within model (3) are equal [9]

$C{r}_N=a_r+\frac{\text{1}}{\text{2}}{k}_p{a}_r^{\text{2}}, T{h}_N=a_r-\frac{\text{1}}{\text{2}}{k}_p{a}_r^{\text{2}}$.

In order to assess how applicable this model is to the description of statistical distributions of crests and troughs, it is necessary to analyse how functions f_сr (x) and f_Th (x) change in different situations, in particular, when the skewness or excess kurtosis is negative. Fig. 3 shows the results of this analysis.

Fig. 3. Probability density functions f calculated for three ranges of skewness А_h. The blue curve is f_сr (x), the red curve is f_Th (x), the black curve is f_r (x)

It follows from Fig. 3 that when changing the sign of skewness А_h, the type of function f_Th (x) changes significantly. If condition А_h < 0 takes place, equality f_Th (x) » f_сr (x) is observed. Note that equality f_Th (x) = f_сr (x) takes place within the linear model when the Cr and Th distributions are described by the Rayleigh distribution. In this case, the difference from the linear model at А_h < 0 is that inequalities f_сr (x) > f_r (x) and f_Th (x) > f_r (x) are fulfilled in region x > 0.45.

As follows from Fig. 4, changes in the excess kurtosis have a lesser effect on the type of functions f_сr (x) and f_Th (x).

Fig. 4. Probability density functions f calculated for four ranges of excess kurtosis е_h. The blue curve is f_сr (x), the red curve is f_Th (x), the black curve is f_r (x)

Conclusion

Based on direct wave measurements carried out under sea conditions, the distributions of troughs Th and crests Cr of sea surface waves were analysed. On average over an ensemble of situations, the greater values of crests calculated from the measurement data have higher probability than the Rayleigh distribution suggests, and the probability of deep troughs is smaller. Such distributions of crests and troughs correspond to a second-order nonlinear model qualitatively.

At the same time, the second-order nonlinear model fails to describe f_Th (x) and f_сr (x) when sea surface elevation distribution skewness А_h is negative. It is shown that functions f_Th (x) and f_сr (x) are approximately equal at А_h < 0.

Changes in the excess kurtosis of the distribution of sea surface elevations have a lesser effect on the probability density functions of Th and Cr than changes in the skewness of the distribution.

Авторлар туралы

A. Zapevalov

Хат алмасуға жауапты Автор.
Email: sevzepter@mail.ru
ORCID iD: 0000-0001-9942-2796
SPIN-код: 6784-7782
Scopus Author ID: 7004433476
ResearcherId: V-7880-2017

главный научный сотрудник, доктор физико-математических наук

Әдебиет тізімі

Қосымша файлдар

Қосымша файлдар

Әрекет

1. JATS XML

Жүктеу

2. Fig. 1. Probability density functions f (average over an ensemble of situations). The blue curve is fсr (x), the red curve is fTh (x), the black curve is fr (x)

Жүктеу (42KB)

Метадеректер

3. Fig. 2. Changes in the skewness Аh and excess kurtosis еh of sea surface elevations

Жүктеу (51KB)

Метадеректер

4. Fig. 3. Probability density functions f calculated for three ranges of skewness Аh. The blue curve is fсr (x), the red curve is fTh (x), the black curve is fr (x)

Жүктеу (85KB)

Метадеректер

5. Fig. 4. Probability density functions f calculated for four ranges of excess kurtosis еh. The blue curve is fсr (x), the red curve is fTh (x), the black curve is fr (x)

Жүктеу (119KB)

Метадеректер