A wave interaction approach to studying non-modal homogeneous and stratified shear instabilities
A wave interaction approach to studying non-modal homogeneous and stratified shear instabilities
复制标题
研究非模态均匀和分层剪切不稳定性的波相互作用方法
DOI：
--
复制
发表时间：
2012
影响因子：
3.7
通讯作者：
G. Lawrence
中科院分区：
文献类型：
--
作者：
A. Guha;G. Lawrence
抽象的Homboe（地球物理。出版物：第24卷，1962年，第121页。67-112）假设两个或多个渐进线性界面波之间的共振相互作用在理想化（折线剖面）、均匀或密度分层的无粘剪切层中产生指数增长的不稳定性。在这里，我们推广了Holmboe的线性剪切不稳定性的机制图（i）最初没有指定波的类型，（ii）提供了非正常增长的选项。我们已经证明了线性剪切不稳定性背后的机制，提出了一个纯粹的运动学模型，由两个线性，多普勒频移，前进界面波在相反的方向移动。此外，我们已经找到了一个必要和充分（N&S）的条件存在的指数增长的不稳定性在理想化的剪切流。这两个界面波，从任意初始条件开始，最终锁相和共振（指数增长），提供了N&S条件得到满足。我们的波相互作用模型的理论基础类似于两个耦合谐振子之间的同步。我们已经将我们的模型重新构建成一个非线性自治动力系统，其稳态配置对应于波相互作用模型的共振配置。当解释的正则简正模理论，稳态/共振配置对应于离散光谱的增长的简正模。在达到稳定状态之前发生的不稳定机制是非模态的，有利于快速瞬态增长。根据波数和初始相移，非模态增益可以超过相应的模态增益许多数量级。不稳定性也被观察到的参数空间，这是被视为稳定的正常模式理论。使用我们的模型，我们推导出离散谱非模态稳定性方程的三个经典的例子剪切不稳定性：瑞利/开尔文-亥姆霍兹，Holmboe和泰勒-考菲尔德。我们已经表明，N&S条件为每种不稳定性类型提供了一个不稳定波数范围，并且这个范围与简正模理论的预测相匹配。
Abstract Homboe (Geophys. Publ., vol. 24, 1962, pp. 67–112) postulated that resonant interaction between two or more progressive, linear interfacial waves produces exponentially growing instabilities in idealized (broken-line profiles), homogeneous or density-stratified, inviscid shear layers. Here we have generalized Holmboe’s mechanistic picture of linear shear instabilities by (i) not initially specifying the wave type, and (ii) providing the option for non-normal growth. We have demonstrated the mechanism behind linear shear instabilities by proposing a purely kinematic model consisting of two linear, Doppler-shifted, progressive interfacial waves moving in opposite directions. Moreover, we have found a necessary and sufficient (N&S) condition for the existence of exponentially growing instabilities in idealized shear flows. The two interfacial waves, starting from arbitrary initial conditions, eventually phase-lock and resonate (grow exponentially), provided the N&S condition is satisfied. The theoretical underpinning of our wave interaction model is analogous to that of synchronization between two coupled harmonic oscillators. We have re-framed our model into a nonlinear autonomous dynamical system, the steady-state configuration of which corresponds to the resonant configuration of the wave interaction model. When interpreted in terms of the canonical normal-mode theory, the steady-state/resonant configuration corresponds to the growing normal mode of the discrete spectrum. The instability mechanism occurring prior to reaching steady state is non-modal, favouring rapid transient growth. Depending on the wavenumber and initial phase-shift, non-modal gain can exceed the corresponding modal gain by many orders of magnitude. Instability is also observed in the parameter space which is deemed stable by the normal-mode theory. Using our model we have derived the discrete spectrum non-modal stability equations for three classical examples of shear instabilities: Rayleigh/Kelvin–Helmholtz, Holmboe and Taylor–Caulfield. We have shown that the N&S condition provides a range of unstable wavenumbers for each instability type, and this range matches the predictions of the normal-mode theory.