AI 中文总结
本文构建表面重力波的耦合随机变分理论框架，将速度势分解为确定性分量与随机噪声项，保留原有哈密顿结构，为海浪简化随机模型及动理学理论提供基础。
AI 中文摘要
本文为表面重力波构建了一套完整的随机变分框架。从无旋不可压缩自由表面流动的Luke变分原理出发，我们将速度势分解为大尺度确定性分量与代表未解析尺度的正则化随机噪声项。路径-wise变分原理给出了拉普拉斯、伯努利及运动边界条件的随机对应形式。为闭合该系统，我们构建了第二个期望形式的变分原理，提供噪声关联函数的演化方程。所得耦合系统保留了Zakharov--Craig--Sulem形式的哈密顿结构。我们进一步通过WKB近似分析显式解，证明噪声关联函数满足带射线追踪动力学的哈密顿-雅可比方程。该框架为海浪的简化随机模型及姊妹篇论文中发展的动理学理论提供了严格基础。
英文摘要
This paper develops a comprehensive stochastic variational framework for surface gravity waves. Starting from Luke's variational principle for irrotational, incompressible free-surface flow, we introduce a decomposition of the velocity potential into a large-scale deterministic component and a regularized stochastic noise term representing unresolved scales. A path-wise variational principle yields the stochastic counterparts of the Laplace, Bernoulli, and kinematic boundary conditions. To close the system, a second variational principle in expectation is formulated, providing evolution equations for the noise correlation functions. The resulting coupled system preserves the Hamiltonian structure of the Zakharov--Craig--Sulem formulation. We further analyze explicit solutions via WKB approximation and show that the noise correlation functions satisfy a Hamilton-Jacobi equation with ray-tracing dynamics. This framework provides a rigorous foundation for reduced stochastic models of ocean waves and for the kinetic theory developed in the companion paper.
