基于非线性浮力频率分布分层流的三波共振畸形波分析

Rogue waves for resonant triads in stratified fluids with a sharply peaked buoyancy frequency profile

本文对位于海洋上层温跃层中的大幅波动进行了理论分析.该内畸形波可能由具有二次非线性的三波共振而引起.虽然三波共振中的畸形波精确解析解已经在数学物理文献中被推导出,但其在流体动力学中的应用尚未被充分研究.本文计算了在非线性浮力频率分布下三波共振演化方程中的系数.调制不稳定性是指由色散和非线性效应之间的相互作用引起的干扰增长,虽然不稳定性分析是基于线性化研究的,但来自增长率谱的信息可以与畸形波的非线性、数值鲁棒性测试的结果相关联.最后,通过解析延拓将空间变量扩展为复数进行了解析畸形波解的极点分析.对于具有一个或两个峰值的畸形波,物理空间中最大位移的位置与轨迹转折点处的极点的实部相同或近似相等.这些数值和分析研究构成了理解海洋上层大规模瞬态运动的框架.

A formulation for unexpectedly large displacements at the pycnocline of the upper ocean is developed.Such“internal rogue waves”can arise from triad resonance with quadratic nonlinearities.While analytical solutions of rogue waves for triad resonance have been derived in the mathematical physics literature,the applications to fluid dynamics have not been fully examined.Here interaction coefficients of the evolution equations are calculated for a sharply peaked buoyancy frequency profile.Modulation instability refers to the growth of disturbance(s)from the interplay between dispersive and nonlinear effects.Although the instability analysis is studied based on linearization,information from the growth rate spectrum can be correlated with results from the fully nonlinear,numerical robustness tests of rogue waves.Finally,pole analysis of the analytical rogue wave solutions is considered by extending the spatial variable to be complex via analytic continuation.For rogue modes with one or two peaks,locations of maximum displacement in physical space are identical,or approximately equal,to the real parts of poles at the turning points of trajectories.These numerical and analytical studies constitute a framework for understanding large scale,transient motions in the upper ocean.