表面重力波与洋流相互作用的一致相平均模型
A consistent phase-averaged model of the interactions between surface gravity waves and currents
AI总结：
本文提出了一种表面重力波与洋流相互作用的相平均模型，通过变分结构和Whitham平均方法，实现了波-洋流系统的动量与能量守恒，并应用于惯性振荡的生成问题。
摘要：
我们提出了一个描述表面重力波与海洋洋流双向相互作用的模型。该模型将波动作传输在四维（水平）位置-波矢相空间中的传输与Craik-Leibovich系统中的洋流传输耦合在一起。耦合是通过波动作传输的色散关系中的多普勒频移以及Craik-Leibovich系统中的波伪动量来实现的。多普勒频移中的速度是洋流拉格朗日平均速度的垂直积分，权重与伪动量的垂直结构一致。这种一致性确保了耦合波-洋流系统中的动量和能量守恒。波-洋流模型的守恒性质源于其底层的变分结构。我们通过引入拉格朗日波-平均分解、进行简化近似以及Whitham平均，从旋转欧拉方程的变分结构中推导出该结构。我们应用波-洋流模型解决由表面波生成惯性振荡的问题，该问题最初由Hasselmann考虑。
英文摘要：
We formulate a model of the two-way interactions between surface gravity waves and ocean currents. The model couples the transport of wave action in the four-dimensional (horizontal) position--wavevector phase space with the Craik--Leibovich system for the currents. Coupling is via the Doppler shift in the dispersion relation governing action transport, and wave pseudomomentum in the Craik--Leibovich system. The velocity in the Doppler shift is a vertical integral of the Lagrangian mean velocity of the currents, with a weight that is consistent with the vertical structure of the pseudomomentum. This consistency ensures conservation of momentum and energy in the coupled wave--current system. The conservation properties of the wave--current model stem from an underlying variational structure. We derive this structure from that of the rotating Euler equations for an incompressible fluid with free surface by introducing a Lagrangian wave--mean decomposition, making simplifying approximations, and Whitham averaging. The wave--mean decomposition is constructed such that the Lagrangian mean velocity is non-divergent and satisfies a rigid-lid boundary condition. We apply the wave--current model to the problem of generation of inertial oscillations by surface waves originally considered by Hasselmann.
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