薄界面内孤立波的最佳瞬态增长
Optimal transient growth in thin-interface internal solitary waves
摘要：
本文用线性最优瞬态增长法分析了薄界面双层流动中大振幅内孤立波的扰动动力学。最佳扰动计算通过直接伴随迭代的纳维-斯托克斯方程线性化周围的无粘，稳定的ISW从Dubreil-Reintin-Long（DJL）方程。最佳扰动被发现作为一个功能的ISW相速度$c$（或者振幅）的一个代表性的分层。这些干扰被发现是本地化的波一样的包，起源于上游的ISW自感应区（足够大的$c$）的潜在不稳定的理查森数，$Ri < 0.25$。它们作为相干包通过基波传播，相干包的总能量增益随着$c$而迅速增加。最佳扰动也被证明是相关的DJL孤立波已被修改的粘度代表的实验室实验。最佳的干扰相比，当地WKB近似空间增长的开尔文-亥姆霍兹（KH）波通过$Ri < 0.25$区。WKB方法能够捕获属性（例如，载波频率，波数和能量增益）的最佳干扰，除了初始阶段的非正常增长，由于奥尔机制。非正常增长可以是总增益的相当大的一部分，特别是对于对K-H波弱不稳定的ISW。高斯包的线性自由波的线性演化具有相同的载波频率的最佳扰动的结果在更少的能量增益比发现的最佳扰动或WKB近似由于非正常效应，导致吸收的干扰能量到波的前导面。
英文摘要：
The dynamics of perturbations to large-amplitude Internal Solitary Waves (ISW) in two-layered flows with thin interfaces is analyzed by means of linear optimal transient growth methods. Optimal perturbations are computed through direct-adjoint iterations of the Navier-Stokes equations linearized around inviscid, steady ISWs obtained from the Dubreil-Jacotin-Long (DJL) equation. Optimal perturbations are found as a function of the ISW phase velocity $c$ (alternatively amplitude) for one representative stratification. These disturbances are found to be localized wave-like packets that originate just upstream of the ISW self-induced zone (for large enough $c$) of potentially unstable Richardson number, $Ri < 0.25$. They propagate through the base wave as coherent packets whose total energy gain increases rapidly with $c$. The optimal disturbances are also shown to be relevant to DJL solitary waves that have been modified by viscosity representative of laboratory experiments. The optimal disturbances are compared to the local WKB approximation for spatially growing Kelvin-Helmholtz (K-H) waves through the $Ri < 0.25$ zone. The WKB approach is able to capture properties (e.g., carrier frequency, wavenumber and energy gain) of the optimal disturbances except for an initial phase of non-normal growth due to the Orr mechanism. The non-normal growth can be a substantial portion of the total gain, especially for ISWs that are weakly unstable to K-H waves. The linear evolution of Gaussian packets of linear free waves with the same carrier frequency as the optimal disturbances is shown to result in less energy gain than found for either the optimal perturbations or the WKB approximation due to non-normal effects that cause absorption of disturbance energy into the leading face of the wave.
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