论文标题
以准轴对称光束形式的内部重力波的反射
Reflection of internal gravity waves in the form of quasi-axisymmetric beams
论文作者
论文摘要
当内部重力波打击倾斜边界时，反射角度的保留会产生聚焦机制，如果入射波传播方向与水平的倾斜方向之间的角度接近斜率倾斜度（近临界反射）。本文提供了对独特的Leray解决方案的引导近似的明确描述，以临界从斜坡上以梁波的形式从斜坡上进行近距离反射。更确切地说，我们的梁波接近允许在特定（非线性）时间尺度内以梁波的形式构造完全一致且Lyapunov稳定的近似解决方案，以$ l^2 $ close构建到Leray解决方案。据我们所知，这是进行内部波的第一个结果，从空间局部的束波进行了数学研究。\\％梁波是快速振荡平面波的线性叠加，其中振动的高频与小型参数的逆向弱点相比，这些参数是弱质量的弱质量板的小参数。 \\％由于快速振荡（和波数的模量的高变化），在物理空间中定位，横梁在物理上比平面波/小数据包的波浪更相关，波浪的波浪几乎是固定的（Microloca \ -li \ -Li \ -ZED）。在数学层面上，这标志着先前的平面波/波浪分析数据包与我们的方法之间的强烈差异。 \\％这项工作的主要新颖性是利用梁波的空间定位以表现出空间局部的，物理相关的解决方案，并从两个角度来改善先前的数学结果：1）我们的梁波近似解决方案是有限的术语的总和，每个术语都不是一个一致的解决方案，并且没有一个一致的解决方案，没有人工/非人工/非人工; 2）由于没有人造校正器（在先前的结果中使用）和非线性术语的特殊结构，我们可以将解决方案的扩展扩展到下一个订单，因此提高准确性并扩大一致性时间尺度。最终，我们的结果在射线上定位在rays上的一组初始条件，以使LERAY解决方案在$ LERAY方案中维持$ L^2 $ l^2 $ lery^2 $ lerive lery lery lery lere lery selutization。
Preservation of the angle of reflection when an internal gravity wave hits a sloping boundary generates a focusing mechanism if the angle between the direction of propagation of the incident wave and the horizontal is close to the slope inclination (near-critical reflection). This paper provides an explicit description of the leading approximation of the unique Leray solution to the near-critical reflection of internal waves from a slope in the form of a beam wave. More precisely, our beam wave approach allows to construct a fully consistent and Lyapunov stable approximate solution, $L^2$ -close to the Leray solution, in the form of a beam wave, within a certain (nonlinear) time-scale. To the best of our knowledge, this is the first result wherea mathematical study of internal waves in terms of spatially localized beam waves is performed.\\%A beam wave is a linear superposition of rapidly oscillating plane waves, where the high frequency of oscillation is proportional to the inverse of a power of the small parameter measuring the weak amplitude of waves. \\%Being localized in the physical space thanks to rapid oscillations (and high variations of the modulus of the wavenumber), beams are physically more relevant than plane waves/packets of waves, whose wavenumber is nearly fixed (microloca\-li\-zed). At the mathematical level, this marks a strong difference between the previous plane waves/packets of waves analysis and our approach. \\%The main novelty of this work is to exploit the spatial localization of beam waves to exhibit a spatially localized, physically relevant solution and to improve the previous mathematical results from a twofold perspective: 1) our beam wave approximate solution is the sum of a finite number of terms, each of them is a consistent solution to the system and there is no artificial/non-physical corrector; 2) thanks to the absence of artificial correctors (used in the previous results) and to the special structure of the nonlinear term, we can push the expansion of our solution to next orders, so improving the accuracy and enlarging the consistency time-scale.Finally, our results provide a set of initial conditions localized on rays, for which the Leray solution maintains approximately in $L^2$ the same localization.
