Collaborative Research: Deterministic and Statistics Theory of Wind Driven Sea of Finite Depth.

合作研究：风驱动有限深度海洋的确定性和统计理论。

• 批准号: 1130450
• 负责人: Vladimir Zakharov
• 金额: $ 18.51万
• 依托单位: University of Arizona
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-08-15 至 2015-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1130450&HistoricalAwards=false
• 关键词: Collaborative; Research; Deterministic; Statistics; Theory

Development of self-consistent statistical description of ocean waves in the coastal area is an important problem in physical oceanography. On deep water the main nonlinear effect is the four-wave resonant interaction described by Hasselmann kinetic equation for spectrum of wave action. Three-wave interaction becomes also important at finite depth, and comes to dominate in shallow water. Three-wave interactions of gravity waves are non-resonant; they become almost resonant on very shallow water only. This fact makes the development of consistent, well-justified analytical statistical theory of gravity waves at finite depth a difficult problem. It is unlikely to be solved by any heuristic modification of the Hasselmann equation that is written for time evolution of the pair correlation function. For the proper description, one has to derive a coupled system of equations for time evolution of pair and triple correlation functions. This project will derive, justify, and study these equations through the following steps: (1) derive the coupled system of equations for pair and triple correlation functions and make sure that this system preserves energy and on deep water goes to the classical Hasselmann equation; (2) generalize the obtained equation for the case of varying bottom topography and presence of current; (3) develop the numerical code for solution of equation for correlations, including into equations the input from wind and the dissipation of this input due to white-capping; (4) perform a massive numerical simulation of primordial dynamic equations in full 3-dimension geometry and use the obtained data for justification of statistical equations; and (5) on the base of deterministic numerical experiments find the function of dissipation due to white-capping on shallow water.Intellectual MeritThe development of an analytical model for statistical description of waves in shallow water will be a breakthrough in the theory of nonlinear waves, which would have practical consequences such as for coastal wave forecasting. The numerical codes for solution of the coupled equation for pair and triple correlations will be an advancement in computational geophysics. The verification of approximate analytical theory by a more exact and detailed numerical simulation could be a model applied to other topics in the future.Broader ImpactsThe method of using the coupled system of equations for correlation functions of different orders for description of wave turbulence is not limited to gravity waves on shallow water. Similar methods can be applied to the theory of long internal gravity waves in ocean and atmosphere, to the theory of Rossby waves in atmosphere of rotating planets. Numerical experiments can improve our understanding of breaking internal waves as well. Some of the results will likely be applicable to different branches of nonlinear wave dynamics such as magnetohydrodynamics, plasma physics, and nonlinear optics.

发展海岸地区海浪的自洽统计描述是物理海洋学中的一个重要问题。在深水中，主要的非线性效应是波作用谱的Hasselmann动力学方程所描述的四波共振相互作用。三波相互作用在有限深度也变得重要，并在浅水中占主导地位。重力波的三波相互作用是非共振的;它们只在非常浅的水域才变得几乎共振。这一事实使得发展一致的，合理的分析统计理论的重力波在有限深度的一个困难的问题。这是不可能解决的任何启发式修改的哈塞尔曼方程是书面的时间演变的对相关函数。为了正确的描述，人们必须推导出一个耦合系统的方程的时间演变的对和三重相关函数。本计画将透过下列步骤推导、验证及研究这些方程式：（1）推导成对及三重相关函数的耦合方程式系统，并确保此系统能保存能量，且在深水上可达经典的Hasselmann方程式;（2）将所得方程式推广至不同海底地形及有海流存在的情形;（3）开发了求解相关方程的数值程序，将风的输入和白顶对输入的耗散纳入方程中：（4）对原始动力方程进行了大规模的全三维数值模拟，并利用所获得的数据对统计方程进行了验证;（5）在确定性数值试验的基础上，得到了浅水区白顶的耗散函数。智力价值浅水区波浪统计描述的解析模式的发展将是非线性波浪理论的一个突破，这将对海岸波浪预报等具有实际意义。对关联和三重关联耦合方程的数值解是计算物理学的一个新进展。近似解析理论的验证通过一个更精确和详细的数值模拟可以是一个模型，适用于其他主题在future.Broader ImpactsThe方法使用耦合系统的方程描述的波动湍流的不同阶的相关函数不限于重力波浅水。类似的方法也可用于海洋和大气中的重力长波理论，以及旋转行星大气中的Rossby波理论。数值实验也可以提高我们对破碎内波的理解。一些结果将可能适用于非线性波动力学的不同分支，如磁流体力学，等离子体物理和非线性光学。