表面重力波与空间尺度相当的平均流相互作用。第一部分：约化波动方程
Surface gravity wave-mean flow interaction with comparable spatial scales. Part I: reduced wave equations
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中文总结 AI 辅助
针对空间尺度与波长相当的背景流上的表面重力波，提出等价可解性条件方法，将3D问题约化为2D方程，并推导出窄带波的薛定谔方程形式，消除了快速波周期的时间分辨负担。
中文摘要 AI 辅助
我们考虑在背景流上方传播的深水表面重力波，该背景流的空间尺度与波长相当，重点研究流动速度相对于波的群速度较慢的情形。我们引入一种“等价可解性条件”方法来构造约化方程，要求在进行多时间尺度展开时，约化方程与原系统具有相同的首阶解和第一可解性条件。该方法将完整的3D问题转化为波场的2D约化方程。我们针对深度不变的背景流上的宽带波，以及完全3D背景流上的窄带波，推导了此类约化方程。在后一种情况下，约化方程采用薛定谔方程的形式，仅涉及背景流的近表面涡度，而近表面水平流散度的影响被证明是次主导的。除了空间维度的降低，后一约化方程还描述了波场在背景流缓慢平流时间尺度上的演化，从而消除了对快速波周期进行时间分辨的计算负担。我们通过一个波包在有序流斑块上的弱散射的解析解，以及波包在无序流斑块上的较强散射的数值解，展示了约化方程的能力。
英文摘要
We consider deep-water surface gravity waves propagating above a background flow whose spatial scale is comparable to the wavelength, focusing on the regime where the flow is slow compared to the group velocity of the waves. We introduce an "equivalent solvability condition" method to construct reduced equations, demanding that, upon multiple-timescale expansion, the reduced equations share the same leading-order solution and first solvability condition as the original system. This approach turns the full 3D problem into a 2D reduced equation for the wave field. We derive such reduced equations for broad-band waves above a depth-invariant background flow, and for narrow-band waves above a fully 3D background flow. In the latter case the reduced equation takes the form of a Schrodinger equation involving the near-surface vorticity of the background flow only, with the impact of the near-surface horizontal flow divergence shown to be subdominant. Beyond the reduction in spatial dimensionality, the latter reduced equation describes the evolution of the wave field over the slow advective timescale of the background flow, thereby eliminating the computational burden of time-resolving the fast wave period. We illustrate the capabilities of the reduced equations through an analytical solution for the weak scattering of a wave packet by a patch of organized flow, followed by numerical solutions for stronger scattering of a wave packet by a patch of disorganized flow.
