Wave modulation: the geometry, kinematics, and dynamics of surface-wave packets
Wave modulation: the geometry, kinematics, and dynamics of surface-wave packets
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DOI：
10.1017/jfm.2016.473
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发表时间：
2015-11
影响因子：
3.7
通讯作者：
N. Pizzo;W. Melville
中科院分区：
文献类型：
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作者：
N. Pizzo;W. Melville
我们研究了由修正非线性Schrödinger方程（Dysthe, Proc. R. Soc.）控制的弱非线性窄带深水波包的几何、运动学和动力学。Lond。A. 1979年第369卷，第105-114页；MNLSE)。通过直接应用Whitham的方法，对空间MNLSE进行了新的推导，阐明了其变分结构。利用这种形式，我们导出了一组守恒量和矩演化方程。接下来，通过考察线性色散消失极限下的MNLSE，可以找到解析解。然后将这些解作为试验函数，将其代入矩演化方程，形成一个封闭的方程集，允许对MNLSE进行定性和定量检查，而无需采用数值求解完整方程。为了检验这一理论，我们首先考虑对称的、啁啾的和非啁啾的波包，选择它们来诱导波聚焦和变陡。通过使用上面讨论的试验函数的ansatz，我们先验地预测了数据包的演化。研究发现，由MNLSE控制的波包的速度取决于它们的振幅，特别是波群在聚焦时加速。接下来，我们描述了波包线的不对称增长，并解释了最初对称波包的前表面变陡。当数据包聚焦时，它的方差会减小，信号的啁啾也会减小。然后将这些理论结果与MNLSE的数值预测进行比较，发现小的fetch值是一致的。最后，我们在现有的理论、数值和实验室研究的背景下讨论了结果。
We examine the geometry, kinematics, and dynamics of weakly nonlinear narrow-banded deep-water wave packets governed by the modified nonlinear Schrödinger equation (Dysthe, Proc. R. Soc. Lond. A., vol. 369, 1979, pp. 105–114; MNLSE). A new derivation of the spatial MNLSE, by a direct application of Whitham’s method, elucidates its variational structure. Using this formalism, we derive a set of conserved quantities and moment evolution equations. Next, by examining the MNLSE in the limit of vanishing linear dispersion, analytic solutions can be found. These solutions then serve as trial functions, which when substituted into the moment evolution equations form a closed set of equations, allowing for a qualitative and quantitative examination of the MNLSE without resorting to numerically solving the full equation. To examine the theory we consider initially symmetric, chirped and unchirped wave packets, chosen to induce wave focusing and steepening. By employing the ansatz for the trial function discussed above, we predict, a priori, the evolution of the packet. It is found that the speed of wave packets governed by the MNLSE depends on their amplitude, and in particular wave groups speed up as they focus. Next, we characterize the asymmetric growth of the wave envelope, and explain the steepening of the forward face of the initially symmetric wave packet. As the packet focuses, its variance decreases, as does the chirp of the signal. These theoretical results are then compared with the numerical predictions of the MNLSE, and agreement for small values of fetch is found. Finally, we discuss the results in the context of existing theoretical, numerical and laboratory studies.