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

发展沿海地区海浪自洽统计描述是物理海洋学的一个重要问题。在深水中，主要的非线性效应是由波作用谱的哈塞曼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下也很重要，在浅水中占主导地位。引力波的三波相互作用是非共振的；它们只有在很浅的水面上才会产生共鸣。这一事实使得在有限深度发展一致的、有充分理由的重力波分析统计理论成为一个难题。用对相关函数随时间演化的哈塞曼方程的启发式修正来求解是不可能的。为了得到恰当的描述，必须推导出一对和三重相关函数的时间演化的耦合方程组。本课题将通过以下步骤对这些方程进行推导、论证和研究：(1)推导出对相关函数和三重相关函数的耦合方程组，并确保该方程组能保持能量，并在深水中达到经典的Hasselmann方程；(2)将所得方程推广到底部地形变化和水流存在的情况；(3)建立了求解相关方程的数值代码，包括风的输入和白顶引起的输入耗散；(4)在全三维几何结构中对原始动力学方程进行大量数值模拟，并利用所得数据对统计方程进行论证；(5)在确定性数值实验的基础上，得到了浅水白盖的耗散函数。发展一种用于浅水中波浪统计描述的分析模型将是非线性波浪理论的一个突破，它将对沿海波浪预报等方面产生实际影响。对和三重相关耦合方程的数值解法将是计算地球物理学的一个进步。通过更精确和详细的数值模拟来验证近似解析理论，可以作为将来应用于其他课题的一个模型。更广泛的影响用不同阶相关函数的耦合方程组描述波浪湍流的方法并不局限于浅水重力波。类似的方法可以应用于海洋和大气中的长内重力波理论，以及旋转行星大气中的罗斯比波理论。数值实验也可以提高我们对破碎内波的认识。其中一些结果可能适用于非线性波动动力学的不同分支，如磁流体动力学、等离子体物理学和非线性光学。

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.