Researches on statistical properties of freak waves in uni-directional random waves in deep water
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摘要: 本文基于Longuet-Higgins随机波浪模型和JONSWAP谱，进行了大量深水随机波的模拟，获取了畸形波发生概率稳定的随机波列，并对随机波列中的畸形波进行了分析。结果表明，畸形波发生的概率小于基于Rayleigh分布预测结果，且随谱宽的减小而增大。在固定时间段内，畸形波发生的频次服从泊松分布，时间间隔服从指数分布，且随着谱宽的增大，畸形波的发生频次减小，相邻畸形波的发生时间间隔增加。通过小波变换方法分离随机波中的波群，研究了出现畸形波的波群特征，发现一个波群中最多会出现4个畸形波，但是在发生畸形波的波群中，单个畸形波的概率最大。随着谱宽减小，一个波群中包含多个畸形波的概率增加。另外，出现畸形波的波群时间长度服从广义极值分布，随着谱宽减小，畸形波波群的时间跨度增加。Abstract: Numerous random wave trains are simulated based on the JONSWAP spectrum using the Longuet-Higgins wave model, and then extreme waves are investigated based on the wave trains with stable probabilities of freak waves. The probabilities of freak waves are smaller than those of based on Rayleigh distributions. With the spectra narrower, the probability of freak waves increases. During the fixed times, the frequency of freak waves obeys the Poisson distribution and time intervals satisfy exponential distribution. The most probable occurrence frequency of freak waves decrease and intervals of freak waves are longer with the spectra wider. Wave groups are discriminated based on wavelet spectra and their characteristics are analyzed. There are no more than four freak waves in wave groups. The probability of wave groups containing merely one freak wave is the largest. Numbers of freak waves in wave groups are increasing with the spectral narrower. Furthermore, time lengths of wave groups containing freak waves satisfy Generalized extreme value distribution (GEV distribution), and with spectra narrower, the most probable lengths of the wave groups increase.
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Key words:
- freak waves /
- wave groups /
- wave models
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图 1 γ=1, 3, 5, 7时，时间T内畸形波发生频率的概率密度分布
Fig. 1 Probability distributions of frequency of freak waves for different peak enhancement factors (γ=1, 3, 5, 7)
图 2 泊松分布参数λ与谱宽υ的关系
Fig. 2 Relationships between Poisson distribution parameter λ and spetrum widths υ
图 3 γ=1，3，5，7时，畸形波出现时间间隔分布
Fig. 3 Probability distributions of time intervals of freak waves for different peak enhancement factors (γ=1, 3, 5, 7).
图 4 指数分布参数μ与谱宽υ的关系
Fig. 4 Relationships between exponential distribution parameter μ and spetrum wdths υ
图 5 不同谱峰因子下，不同畸形波特征所占比例
Fig. 5 Proportions of each characteristic of freak waves for different peak enhancement factors
图 6 γ=1，3，5，7时，畸形波群无量纲时间长度概率分布
Fig. 6 Probability distributions of non-dimensional time lengths of freak wave groups for different peak enhancement factors (γ=1, 3, 5, 7)
图 7 GEV系数与谱宽υ的关系
Fig. 7 Relationships between parameters of GEV distribution and spetrum widths υ
表 1 不同谱宽的时间序列长度[27]
Tab. 1 Time series lengths for different spectrum widths
γ 谱宽υ 时间序列
T0/s实际波数 波数相对
误差/%畸形波发生
概率/%1 0.381 11 988 695 1 499 724 0.018 4 0.007 5 2 0.367 12 565 184 1 499 739 0.017 4 0.009 3 3 0.355 12 959 049 1 499 760 0.016 0 0.010 7 4 0.343 13 257 563 1 499 719 0.018 7 0.012 1 5 0.333 13 496 085 1 499 696 0.020 2 0.013 3 6 0.324 13 693 145 1 499 821 0.011 9 0.014 3 7 0.315 13 859 826 1 499 880 0.008 0 0.015 2 
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表 2 不同谱宽下畸形波出现不同频次概率的预测值与数值结果对比
Tab. 2 Comparisons of the numerical and predicted values of possibility for frequencies of freak waves
γ 谱宽υ 出现1次畸形波的概率 相对误差/% 出现2次畸形波的概率 相对误差/% 理论值 数值结果 理论值 数值结果 2 0.367 0.103 4 0.103 3 0.10 0.006 0 0.006 0 0 4 0.343 0.126 9 0.125 5 1.12 0.009 3 0.009 1 2.20 6 0.324 0.144 9 0.146 5 1.09 0.012 5 0.012 7 1.57
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表 3 不同谱宽下相邻畸形波时间间隔预测值与数值结果对比
Tab. 3 Comparisons of the numerical and predicted values of intervals of adjacent freak waves
γ 谱宽υ Td /Tp=0.5×104 相对误差/% Td /Tp=1.5×104 相对误差/% 预测值/10−4 数值结果/10−4 预测值/10−4 数值结果/10−4 2 0.367 0.661 0.637 3.77 0.197 0.203 2.96 4 0.343 0.705 0.691 2.03 0.162 0.170 4.71 6 0.324 0.731 0.761 3.94 0.124 0.119 4.20
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表 4 波群中包含畸形波的类型
Tab. 4 Classifications of freak waves in wave groups
包含畸形波的个数 畸形波的特征和出现位置 命名 1 同时具有最高波峰和最深波谷 Ona 只具有最高波峰或最深波谷 Onb 2 相邻 Twa 间隔 Twn 3 相邻 Tha 间隔 Thn 4 相邻 Foa 间隔 Fon
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表 5 畸形波群无量纲时间长度众数预测值与数值结果对比
Tab. 5 Comparisons of the numerical and predicted modes of the non-dimensional lengths of freak wave groups
γ 谱宽υ 预测值 数值结果 相对误差/% 2 0.367 8.0 8.0 0 4 0.343 9.5 9.2 3.3 6 0.324 10.5 10.5 0
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