两层剪切流中的布拉格共振和波三波相互作用
On Bragg resonances and wave triad interactions in two-layered shear flows
摘要：
布拉格散射和弱非线性波三波组的标准共振条件传统上是在没有任何背景速度的情况下导出的。在本文中，我们已经研究了这些共振条件得到修改时，均匀的，以及各种分段线性速度剖面，被认为是两层剪切流。背景速度可以通过两种方式影响共振条件：（i）通过引起多普勒频移，以及（ii）通过改变波的固有频率。对于布拉格共振，即使是均匀的速度场也会改变共振条件。速度剪切强烈影响的共振条件，因为除了改变固有频率，它可以导致不平等的多普勒频移之间的表面，密度跃层，和底部。利用多尺度分析和Fredholm替代，我们解析地获得了布拉格共振和波三波组的方程。我们还扩展了高阶谱方法，一个高效的计算工具，通常用于研究三和弦和布拉格共振问题，将分段线性速度分布的影响。一个重要的方面，无论是在理论和数值方面，一直在扩展的势流近似，这是研究这类问题的基础，将分段恒定的背景剪切。
英文摘要：
The standard resonance conditions for Bragg scattering as well as weakly nonlinear wave triads have been traditionally derived in the absence of any background velocity. In this paper, we have studied how these resonance conditions get modified when uniform, as well as various piecewise linear velocity profiles, are considered for two-layered shear flows. Background velocity can influence the resonance conditions in two ways (i) by causing Doppler shifts, and (ii) by changing the intrinsic frequencies of the waves. For Bragg resonance, even a uniform velocity field changes the resonance condition. Velocity shear strongly influences the resonance conditions since, in addition to changing the intrinsic frequencies, it can cause unequal Doppler shifts between the surface, pycnocline, and the bottom. Using multiple scale analysis and Fredholm alternative, we analytically obtain the equations governing both the Bragg resonance and the wave triads. We have also extended the Higher Order Spectral method, a highly efficient computational tool usually used to study triad and Bragg resonance problems, to incorporate the effect of piecewise linear velocity profile. A significant aspect, both in theoretical and numerical fronts, has been extending the potential flow approximation, which is the basis of studying these kinds of problems, to incorporate piecewise constant background shear.
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