风浪破碎的随机性、非线性和能量耗散特征研究
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摘要
风浪破碎是海气间动量、能量传输和物质交换的重要渠道,在海气相互作用中扮演着主要角色；破碎产生的湍对海洋上混合层各种物理过程有着显著影响；风浪破碎也是维持波浪场能量平衡、限制波高的主要机制。深入研究风浪破碎,对海洋科学研究有重要的意义。
风浪破碎是一种具有高度非线性和强间歇性的复杂物理过程,破碎发生的时间、地点、方式、强度都是随机的。这种非线性、间歇性和随机性成为风浪破碎理论研究的主要障碍。由于缺乏恰当的数学工具,人们对此认识模糊。
随机点过程是描述具有间歇性、阵发性随机现象的有力工具。本文将随机点过程的理论方法引入风浪破碎研究,提出了描述风浪破碎随机性的点过程模型,给出了破碎概率、破碎耗散的严格数学表示。对实验室风浪实验数据的分析表明,点过程理论是研究风浪破碎随机性的有效工具。这为人们深入理解风浪破碎找到了一条新途径。
随机过程的二阶谱能有效地度量过程中的非线性程度,而风浪破碎具有显著的非线性特征。本文研究了破碎风浪的二阶谱、二阶相干谱特征,考查了二阶谱实部、虚部特征量随破碎程度的变化。数据分析结果表明：二阶谱实部最大值、实部积分以及偏度等特征量都随破碎程度的增强而显著增大,能敏感地反映破碎程度的变化,可以用于度量风浪破碎程度；而虚部最大值、积分以及由此定义的水平不对称性等特征量在不同破碎程度下差异并不明显；二阶相干谱中显著部分所占的比例随破碎程度的增强不断增大,提示波波相互作用的范围不断扩大,强度不断增强。
破碎导致的能量耗散是海浪研究的一个热点,也是一个难点。本文借鉴Young和Babanin (2006)的思路,将波面信号划分为破碎段与非破碎段,二者平均谱的差异归因为破碎的影响,由此估计破碎能量耗散。本文也借鉴了Yefimov和Khristoforov (1971a)的思路,利用实测速度谱与波生速度谱的差估计耗散。分析结果表明：前一种方法尽管可能会低估真实的耗散,但估计结果基本合理；后一种方法的结果则明显受深度影响。
可靠的破碎判别是风浪破碎点过程理论实证的一个关键。结合实验数据,本文仔细研究了各种破碎判据的有效性,发现：运动学判据变量、动力学判据变量等局地判据变量能敏感地反映破碎事件的发生,能很好地区分破碎波与非破碎波,存在一个稳定的阈值,依据这些判据的判别结果绝大多数时候是一致的；波陡、峰前波陡、峰后波陡等几何量能够指示破碎事件的发生,但不存在一个稳定的阈值；两种水平不对称因子、垂向不对称因子、波面偏度等不对称几何量在破碎波与非破碎波上的行为没有明显差异,不适合于作为破碎判据变量。
基于47组风浪破碎实验,采用动力学判据,确定破碎波与破碎波群,依次获得每个信号的破碎波间隔、破碎波群间隔、波群内破碎波个数等序列,计算破碎概率与破碎率,风浪破碎点过程理论得到初步验证：
破碎间隔的分布能够完整的表达破碎发生的间歇性,小间隔占得比例大,大间隔出现的概率小,这是破碎间隔分布的共同特征；对间隔分布的Kolmogorov-Smimov检验表明：低风速情形的破碎波间隔服从指数分布。这意味着,低风速情形下,风浪破碎的发生可以视为一齐次Poisson过程；对应的,描述破碎耗散的标值累计过程可以简化为复合Poisson过程。
高风速情形,群发性是风浪破碎的典型特征,波群内破碎波个数的分布能有效地表达群发性。只有在风速超过一定水平时,包含多个破碎波的波群才会渐次出现,风速越大,这种波群占得比例越大,平均的波群内破碎波个数也越大。
破碎发生率与平均破碎间隔呈几近理想的倒数关系,随着风速的增大,破碎间隔减小,破碎发生率增大。事实上,主导波破碎概率、破碎发生率、平均破碎波间隔的倒数以及波群内破碎波平均个数四个指标间存在明显的线性关系,属于一类；波群破碎概率、波群破碎发生率、平均破碎波群间隔则属于另一类。要完整地描述破碎频繁程度,需同时使用两类指标,缺一不可。
Wind wave breaking in deep water is an important component in the process of air-sea interaction, playing a primary role in the exchange of mass, momentum and energy between the atmosphere and the ocean, which have a profound effect on weather and climate. It serves to maintain the energy balance within the continuous wind-wave field and limit the height of surface waves, mix the surface waters, generate ocean current. Its role in the dynamics of the upper ocean is critical.
However, it may represent one of the most complex physical phenomena of nature. It is an intermittent random process, with strong non-linearity, very fast by comparison with other processes in the wave system, and the distribution of wave breaking on the water surface is not continuous. The intermittency and randomicity of the wave breaking are the main barriers in the breaking wave research. Due to lack of proper mathematical representation, they are not clear for us by now.
Actually, the intermittency and randomicity can be clearly described by stochastic point processes. The concept and theory of stochastic point process were introduced into wave breaking research. Wind wave breaking observed at a fixed point can be viewed as a one-dimension point process; considering the groupness of breaking waves, cluster point process is the general model for wind wave breaking; marked point process is proper for the expressing energy loss due to breaking. In this theoretical frame, definition and estimation of breaking probability and energy dissipation were discussed strictly. Breaking probability is the mean of a stochastic process and the energy loss due to breaking is a stochastic processes. A series of wind wave breaking experiments and the data analysis results reveal that the theory and method of stochastic point process is effective for wind wave breaking research. This is a new way to understand wind wave breaking.
The bispectrum can measure the non-linearity of a process, and wind wave breaking is a process with strong non-linearity. The bispectra of breaking wind waves under different breaking rate were investigated and come into the following conclusions:the maxima and the integral of the real part of wind wave bispectra, and the skewness of the wave elevation time series increase with dominant wave breaking probability sensitively, and they are suitable for measuring the breaking frequency, however, there is no clear trend in the maxima and the integral of the image part of bispectra at different wind speed.
Energy dissipation due to wave breaking is a focus of the breaking wave research, and there is much difficulty in it. A brief review of breaking dissipation research was presented and two methods for estimating the energy loss of breaking wave were inspected. Young and Babanin (2006) separate the surface waves into two categories: breaking segments and non-breaking segments, the difference between "breaking spectrum" and "non-breaking" spectrum was attributed to the dissipation due to breaking. Yefimov and Khristoforov (1971a) use the difference of the measured velocity spectrum and the wave-induced spectrum as the estimated energy dissipation. Results from our data analysis using first method are reasonable but the other is not.
Reliable breaking criterion is needed in our research, so the classical breaking criteria for surface elevation time series were investigated. Variables for kinematics criterion and for dynamic criterion and instantaneous wave slope are all sensitive to the breaking events, and well consistent in distinguishing the breaking waves. Those geometric variables such as wave steepnees are not so good.
Based on the data of wind wave experiments, using dynamic criterion, the breaking wave and breaking groups were determined, the sequences of time interval between breaking waves and sequences of the number of breaking waves in a group were obtained. The stochastic point process theory for wave breaking was demonstrated as following:
The intermittency of wind wave breaking can be clearly described using the distribution of the intervals between breaking wave. There is a common feature is that the proportion of small intervals is large and those of large intervals are small. The Kolmogorov-Smirnov test for interval sequences show that the probability distribution for intervals between breaking waves in lower wind speed is exponential distribution. This indicates that, in the lower wind speed case, the breaking waves formed a homogenous Poisson process. Correspongingly, the cumulated marked process for breaking dissipation can be simplified into compound Poisson process.
In the higher wind speed case, groupness is the distinct character of wind wave breaking. The groupness of wave breaking can be expressed by the distribution of the number of breaking waves in a group. The groups containing more than one breaking wave occurred only when wind speed come up to certain level. The higher the wind speed is, the more of such groups, and the large is the averaged number of breaking waves in a wave group.
The breaking rate is near with the reciprocal of the averaged breaking interval. In fact, the dominant wave breaking probability, breaking rate, the reciprocal of the averaged interval between breaking waves and the averaged number of breaking waves in a wave group are highly correlated in linear way, they are one kind of index for measuring breaking frequency, the group breaking probability, group breaking rate, and the averaged interval between breaking groups are the other kind of breaking frequency index. It is much better to combine these two kind indexes when describing how frequently the breaking is.
风浪破碎是一种具有高度非线性和强间歇性的复杂物理过程,破碎发生的时间、地点、方式、强度都是随机的。这种非线性、间歇性和随机性成为风浪破碎理论研究的主要障碍。由于缺乏恰当的数学工具,人们对此认识模糊。
随机点过程是描述具有间歇性、阵发性随机现象的有力工具。本文将随机点过程的理论方法引入风浪破碎研究,提出了描述风浪破碎随机性的点过程模型,给出了破碎概率、破碎耗散的严格数学表示。对实验室风浪实验数据的分析表明,点过程理论是研究风浪破碎随机性的有效工具。这为人们深入理解风浪破碎找到了一条新途径。
随机过程的二阶谱能有效地度量过程中的非线性程度,而风浪破碎具有显著的非线性特征。本文研究了破碎风浪的二阶谱、二阶相干谱特征,考查了二阶谱实部、虚部特征量随破碎程度的变化。数据分析结果表明：二阶谱实部最大值、实部积分以及偏度等特征量都随破碎程度的增强而显著增大,能敏感地反映破碎程度的变化,可以用于度量风浪破碎程度；而虚部最大值、积分以及由此定义的水平不对称性等特征量在不同破碎程度下差异并不明显；二阶相干谱中显著部分所占的比例随破碎程度的增强不断增大,提示波波相互作用的范围不断扩大,强度不断增强。
破碎导致的能量耗散是海浪研究的一个热点,也是一个难点。本文借鉴Young和Babanin (2006)的思路,将波面信号划分为破碎段与非破碎段,二者平均谱的差异归因为破碎的影响,由此估计破碎能量耗散。本文也借鉴了Yefimov和Khristoforov (1971a)的思路,利用实测速度谱与波生速度谱的差估计耗散。分析结果表明：前一种方法尽管可能会低估真实的耗散,但估计结果基本合理；后一种方法的结果则明显受深度影响。
可靠的破碎判别是风浪破碎点过程理论实证的一个关键。结合实验数据,本文仔细研究了各种破碎判据的有效性,发现：运动学判据变量、动力学判据变量等局地判据变量能敏感地反映破碎事件的发生,能很好地区分破碎波与非破碎波,存在一个稳定的阈值,依据这些判据的判别结果绝大多数时候是一致的；波陡、峰前波陡、峰后波陡等几何量能够指示破碎事件的发生,但不存在一个稳定的阈值；两种水平不对称因子、垂向不对称因子、波面偏度等不对称几何量在破碎波与非破碎波上的行为没有明显差异,不适合于作为破碎判据变量。
基于47组风浪破碎实验,采用动力学判据,确定破碎波与破碎波群,依次获得每个信号的破碎波间隔、破碎波群间隔、波群内破碎波个数等序列,计算破碎概率与破碎率,风浪破碎点过程理论得到初步验证：
破碎间隔的分布能够完整的表达破碎发生的间歇性,小间隔占得比例大,大间隔出现的概率小,这是破碎间隔分布的共同特征；对间隔分布的Kolmogorov-Smimov检验表明：低风速情形的破碎波间隔服从指数分布。这意味着,低风速情形下,风浪破碎的发生可以视为一齐次Poisson过程；对应的,描述破碎耗散的标值累计过程可以简化为复合Poisson过程。
高风速情形,群发性是风浪破碎的典型特征,波群内破碎波个数的分布能有效地表达群发性。只有在风速超过一定水平时,包含多个破碎波的波群才会渐次出现,风速越大,这种波群占得比例越大,平均的波群内破碎波个数也越大。
破碎发生率与平均破碎间隔呈几近理想的倒数关系,随着风速的增大,破碎间隔减小,破碎发生率增大。事实上,主导波破碎概率、破碎发生率、平均破碎波间隔的倒数以及波群内破碎波平均个数四个指标间存在明显的线性关系,属于一类；波群破碎概率、波群破碎发生率、平均破碎波群间隔则属于另一类。要完整地描述破碎频繁程度,需同时使用两类指标,缺一不可。
Wind wave breaking in deep water is an important component in the process of air-sea interaction, playing a primary role in the exchange of mass, momentum and energy between the atmosphere and the ocean, which have a profound effect on weather and climate. It serves to maintain the energy balance within the continuous wind-wave field and limit the height of surface waves, mix the surface waters, generate ocean current. Its role in the dynamics of the upper ocean is critical.
However, it may represent one of the most complex physical phenomena of nature. It is an intermittent random process, with strong non-linearity, very fast by comparison with other processes in the wave system, and the distribution of wave breaking on the water surface is not continuous. The intermittency and randomicity of the wave breaking are the main barriers in the breaking wave research. Due to lack of proper mathematical representation, they are not clear for us by now.
Actually, the intermittency and randomicity can be clearly described by stochastic point processes. The concept and theory of stochastic point process were introduced into wave breaking research. Wind wave breaking observed at a fixed point can be viewed as a one-dimension point process; considering the groupness of breaking waves, cluster point process is the general model for wind wave breaking; marked point process is proper for the expressing energy loss due to breaking. In this theoretical frame, definition and estimation of breaking probability and energy dissipation were discussed strictly. Breaking probability is the mean of a stochastic process and the energy loss due to breaking is a stochastic processes. A series of wind wave breaking experiments and the data analysis results reveal that the theory and method of stochastic point process is effective for wind wave breaking research. This is a new way to understand wind wave breaking.
The bispectrum can measure the non-linearity of a process, and wind wave breaking is a process with strong non-linearity. The bispectra of breaking wind waves under different breaking rate were investigated and come into the following conclusions:the maxima and the integral of the real part of wind wave bispectra, and the skewness of the wave elevation time series increase with dominant wave breaking probability sensitively, and they are suitable for measuring the breaking frequency, however, there is no clear trend in the maxima and the integral of the image part of bispectra at different wind speed.
Energy dissipation due to wave breaking is a focus of the breaking wave research, and there is much difficulty in it. A brief review of breaking dissipation research was presented and two methods for estimating the energy loss of breaking wave were inspected. Young and Babanin (2006) separate the surface waves into two categories: breaking segments and non-breaking segments, the difference between "breaking spectrum" and "non-breaking" spectrum was attributed to the dissipation due to breaking. Yefimov and Khristoforov (1971a) use the difference of the measured velocity spectrum and the wave-induced spectrum as the estimated energy dissipation. Results from our data analysis using first method are reasonable but the other is not.
Reliable breaking criterion is needed in our research, so the classical breaking criteria for surface elevation time series were investigated. Variables for kinematics criterion and for dynamic criterion and instantaneous wave slope are all sensitive to the breaking events, and well consistent in distinguishing the breaking waves. Those geometric variables such as wave steepnees are not so good.
Based on the data of wind wave experiments, using dynamic criterion, the breaking wave and breaking groups were determined, the sequences of time interval between breaking waves and sequences of the number of breaking waves in a group were obtained. The stochastic point process theory for wave breaking was demonstrated as following:
The intermittency of wind wave breaking can be clearly described using the distribution of the intervals between breaking wave. There is a common feature is that the proportion of small intervals is large and those of large intervals are small. The Kolmogorov-Smirnov test for interval sequences show that the probability distribution for intervals between breaking waves in lower wind speed is exponential distribution. This indicates that, in the lower wind speed case, the breaking waves formed a homogenous Poisson process. Correspongingly, the cumulated marked process for breaking dissipation can be simplified into compound Poisson process.
In the higher wind speed case, groupness is the distinct character of wind wave breaking. The groupness of wave breaking can be expressed by the distribution of the number of breaking waves in a group. The groups containing more than one breaking wave occurred only when wind speed come up to certain level. The higher the wind speed is, the more of such groups, and the large is the averaged number of breaking waves in a wave group.
The breaking rate is near with the reciprocal of the averaged breaking interval. In fact, the dominant wave breaking probability, breaking rate, the reciprocal of the averaged interval between breaking waves and the averaged number of breaking waves in a wave group are highly correlated in linear way, they are one kind of index for measuring breaking frequency, the group breaking probability, group breaking rate, and the averaged interval between breaking groups are the other kind of breaking frequency index. It is much better to combine these two kind indexes when describing how frequently the breaking is.
引文
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