Ryzhov Evgeny A, Koshel Konstantin V
Pacific Oceanological Institute of FEB RAS, 43, Baltiyskaya Street, Vladivostok 690041, Russia.
Chaos. 2015 Oct;25(10):103108. doi: 10.1063/1.4930897.
In a two-layer quasi-geostrophic approximation, we study the irregular dynamics of fluid particles arising due to two interacting point vortices embedded in a deformation flow consisting of shear and rotational components. The two vortices are arranged within the bottom layer, but an emphasis is on the upper-layer fluid particle motion. Vortices moving in one layer induce stirring of passive scalars in the other layer. This is of interest since point vortices induce singular velocity fields in the layer they belong to; however, in the other layer, they induce regular velocity fields that generally result in a change in passive particle stirring. If the vortices are located at stagnation points, there are three different types of the fluid flow. We examine how properties of each flow configuration are modified if the vortices are displaced from the stagnation points and thus circulate in the immediate vicinity of these points. To that end, an analysis of the steady-state configurations is presented with an emphasis on the frequencies of fluid particle oscillations about the elliptic stagnation points. Asymptotic relations for the vortex and fluid particle zero-oscillation frequencies are derived in the vicinity of the corresponding elliptic points. By comparing the frequencies of fluid particles with the ones of the vortices, relations between the parameters that lead to enhanced stirring of fluid particles are established. It is also demonstrated that, if the central critical point is elliptic, then the fluid particle trajectories in its immediate vicinity are mostly stable making it harder for the vortex perturbation to induce stirring. Change in the type of the central point to a hyperbolic one enhances drastically the size of the chaotic dynamics region. Conditions on the type of the central critical point also ensue from the derived asymptotic relations.
在两层准地转近似中，我们研究了由嵌入包含剪切和旋转分量的变形流中的两个相互作用的点涡引起的流体粒子的不规则动力学。这两个涡位于底层，但重点是上层流体粒子的运动。在一层中移动的涡会引起另一层中被动标量的搅拌。这是有意义的，因为点涡在它们所属的层中会引起奇异速度场；然而，在另一层中，它们会引起通常会导致被动粒子搅拌变化的规则速度场。如果涡位于驻点，则存在三种不同类型的流体流动。我们研究了如果涡从驻点位移并因此在这些点的紧邻区域循环，每种流动配置的特性将如何改变。为此，给出了对稳态配置的分析，重点是围绕椭圆驻点的流体粒子振荡频率。在相应椭圆点附近推导了涡和流体粒子零振荡频率的渐近关系。通过比较流体粒子的频率与涡的频率，建立了导致流体粒子搅拌增强的参数之间的关系。还证明了，如果中心临界点是椭圆的，那么在其紧邻区域的流体粒子轨迹大多是稳定的，这使得涡扰动更难引起搅拌。中心临界点类型变为双曲型会极大地增加混沌动力学区域的大小。从推导的渐近关系中也得出了关于中心临界点类型的条件。
