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

发展近海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由波浪作用谱的哈塞尔曼动力学方程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几乎是共振的。这一事实使得发展一致的、合理的有限深度重力波分析统计理论成为一个困难的问题。这不太可能通过对哈塞尔曼方程的任何启发式修改来解决，哈塞尔曼方程是为对关联函数的时间演化而写的。为了得到正确的描述，我们必须推导出对和三重关联函数的时间演化的耦合方程组。本项目将通过以下步骤来推导、证明和研究这些方程：(1)推导出对和三重相关函数的耦合方程组，并确保该系统守恒能量，并且在深水时进入经典的Hasselmann方程；(2)将所获得的方程推广到海底地形和洋流存在的情况；(3)开发用于求解相关方程的数值代码，包括风的输入和由于白帽的输入的耗散；(4)对原始三维几何动力学方程进行大规模数值模拟，并利用所获得的数据对统计方程进行验证；(5)在确定性数值实验的基础上，找出浅水白帽的耗散函数。对和三重关联耦合方程的数值求解程序将是计算地球物理的一个进步。通过更精确、更详细的数值模拟对近似解析理论进行验证，可以作为未来应用于其他课题的模型。广泛影响用不同阶关联函数的耦合方程组来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，也可以应用于旋转行星大气中的Rossby波理论。数值实验也可以提高我们对破裂内波的理解。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.