Dynamic soliton–mean flow interaction with non-convex flux
Dynamic soliton–mean flow interaction with non-convex flux
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DOI：
10.1017/jfm.2021.803
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发表时间：
2021-02
影响因子：
3.7
通讯作者：
Kiera van der Sande;G. El;M. Hoefer
中科院分区：
文献类型：
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作者：
Kiera van der Sande;G. El;M. Hoefer
摘要在修正的Korteweg-de Vries （mKdV）方程框架下，研究了局域孤立波与非凸通量下大尺度时变色散平均流的相互作用。mKdV方程是层状流体中内部重力波传播和位涡锋面的典型模型。考虑了大振幅、动态演化的平均流对局域波传播的影响——本质上是平均流的“孤子导向”。最近对这种新型的凸通量动态孤子-平均流相互作用的理论和实验研究揭示了两种情况，即孤子要么通过变化的平均流传输，要么被困在变化的平均流中。在本文中，证明了非凸立方流体动力通量的存在引入了传输和捕获场景的重大修改。利用一组简化的Whitham调制方程，建立了具有非凸通量的孤子-平均流相互作用的一般数学框架。从交叉调制特性的角度阐述了孤立波捕获。非凸性和正色散——在分层流体中是常见的——意味着存在局部的、尖锐的过渡锋（扭结）。扭结扮演着平均流和波的双重角色，分别赋予孤子和色散平均流极性反转。mKdV方程的数值模拟与调制理论预测一致。所开发的数学框架是通用的，不局限于像mKdV这样的完全可积方程，使mKdV设置之外的应用能够应用于其他受非凸通量影响的流体动力学环境，例如在海洋中普遍存在的强烈非线性内波传播。
Abstract The interaction of localised solitary waves with large-scale, time-varying dispersive mean flows subject to non-convex flux is studied in the framework of the modified Korteweg–de Vries (mKdV) equation, a canonical model for internal gravity wave propagation and potential vorticity fronts in stratified fluids. The effect of large amplitude, dynamically evolving mean flows on the propagation of localised waves – essentially ‘soliton steering’ by the mean flow – is considered. A recent theoretical and experimental study of this new type of dynamic soliton–mean flow interaction for convex flux has revealed two scenarios where the soliton either transmits through the varying mean flow or remains trapped inside it. In this paper, it is demonstrated that the presence of a non-convex cubic hydrodynamic flux introduces significant modifications to the scenarios for transmission and trapping. A reduced set of Whitham modulation equations is used to formulate a general mathematical framework for soliton–mean flow interaction with non-convex flux. Solitary wave trapping is stated in terms of crossing modulation characteristics. Non-convexity and positive dispersion – common for stratified fluids – imply the existence of localised, sharp transition fronts (kinks). Kinks play dual roles as a mean flow and a wave, imparting polarity reversal to solitons and dispersive mean flows, respectively. Numerical simulations of the mKdV equation agree with modulation theory predictions. The mathematical framework developed is general, not restricted to completely integrable equations like mKdV, enabling application beyond the mKdV setting to other fluid dynamic contexts subject to non-convex flux such as strongly nonlinear internal wave propagation that is prevalent in the ocean.