Finn John M.
Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545.
Chaos. 2002 Jun;12(2):508-517. doi: 10.1063/1.1468246.
The effect of an externally imposed perturbation on an unstable or weakly stable shear flow is investigated, with a focus on the role of Lagrangian chaos in the bifurcations that occur. The external perturbation is at rest in the laboratory frame and can form a chain of resonances or cat's eyes where the initial velocity v(x0)(y) vanishes. If in addition the shear profile is unstable or weakly stable to a Kelvin-Helmholtz instability, for a certain amplitude of the external perturbation there can be an unlocking bifurcation to a nonlinear wave resonant around a different value of y, with nonzero phase velocity. The interaction of the propagating nonlinear wave with the external perturbation leads to Lagrangian chaos. We discuss results based on numerical simulations for different amplitudes of the external perturbation. The response to the external perturbation is strong, apparently because of non-normality of the linear operator, and the unlocking bifurcation is hysteretic. The results indicate that the observed Lagrangian chaos is responsible for a second bifurcation occurring at larger external perturbation, locking the wave to the wall. This bifurcation is nonhysteretic. The mechanism by which the chaos leads to locking in this second bifurcation is by means of chaotic advective transport of momentum from one chain of resonances to the other (Reynolds stress) and momentum transport to the vicinity of the wall via chaotic scattering. These results suggest that locking of waves in rotating tank experiments in the presence of two unstable modes is due to a similar process. (c) 2002 American Institute of Physics.
研究了外部施加的扰动对不稳定或弱稳定剪切流的影响，重点关注拉格朗日混沌在发生的分岔中所起的作用。外部扰动在实验室坐标系中静止，并且可以形成共振链或猫眼，其中初始速度v(x0)(y)消失。此外，如果剪切剖面对于开尔文 - 亥姆霍兹不稳定性是不稳定的或弱稳定的，对于一定幅度的外部扰动，可能会发生解锁分岔，形成围绕不同y值共振的非线性波，其相速度不为零。传播的非线性波与外部扰动的相互作用导致拉格朗日混沌。我们基于不同幅度外部扰动的数值模拟讨论结果。对外部扰动的响应很强，显然是由于线性算子的非正规性，并且解锁分岔是滞后的。结果表明，观察到的拉格朗日混沌导致在较大外部扰动时发生第二次分岔，使波锁定在壁上。这次分岔是非滞后的。在第二次分岔中混沌导致锁定的机制是通过动量从一个共振链到另一个共振链的混沌平流输运（雷诺应力）以及通过混沌散射将动量输运到壁附近。这些结果表明，在存在两种不稳定模式的旋转水箱实验中波的锁定是由于类似的过程。（c）2002美国物理研究所。
