Del-Castillo-Negrete Diego
Theoretical Division, Los Alamos National Laboratory, T-15 MS-K717, Los Alamos, New Mexico 87545.
Chaos. 2000 Mar;10(1):75-88. doi: 10.1063/1.166477.
Self-consistent chaotic transport is the transport of a field F by a velocity field v according to an advection-diffusion equation in which there is a dynamical constrain between the two fields, i.e., O(F,v)=0 where O is an integral or differential operator, and the Lagrangian trajectories of fluid particles exhibit sensitive dependence on initial conditions. In this paper we study self-consistent chaotic transport in two-dimensional incompressible shear flows. In this problem F is the vorticity zeta, the corresponding advection-diffusion equation is the vorticity equation, and the self-consistent constrain is the vorticity-velocity coupling z nabla xv=zeta. To study this problem we consider three self-consistent models of intermediate complexity between the simple but limited kinematic chaotic advection models and the approach based on the direct numerical simulation of the Navier-Stokes equation. The first two models, the vorticity defect model and the single wave model, are constructed by successive simplifications of the vorticity-velocity coupling. The third model is an area preserving self-consistent map obtained from a space-time discretization of the single wave model. From the dynamical systems perspective these models are useful because they provide relatively simple self-consistent Hamiltonians (streamfunctions) for the Lagrangian advection problem. Numerical simulations show that the models capture the basic phenomenology of shear flow instability, vortex formation and relaxation typically observed in direct numerical simulations of the Navier-Stokes equation. Self-consistent chaotic transport in electron plasmas in the context of kinetic theory is also discussed. In this case F is the electron distribution function in phase space, the corresponding advection equation is the Vlasov equation and the self-consistent constrain is the Poisson equation. This problem is closely related to the vorticity problem. In particular, the vorticity defect model is analogous to the Vlasov-Poisson model and the single wave model and the self-consistent map apply equally to both plasmas and fluids. Also, the single wave model is analogous to models used in the study of globally coupled oscillator systems. (c) 2000 American Institute of Physics.
自洽混沌输运是指场(F)根据平流扩散方程，由速度场(v)进行的输运，其中这两个场之间存在动力学约束，即(O(F,v)=0)，这里(O)是一个积分或微分算子，并且流体粒子的拉格朗日轨迹对初始条件表现出敏感依赖性。在本文中，我们研究二维不可压缩剪切流中的自洽混沌输运。在这个问题中，(F)是涡度(\zeta)，相应的平流扩散方程是涡度方程，自洽约束是涡度 - 速度耦合(\zeta\nabla\times v = \zeta)。为了研究这个问题，我们考虑了三个介于简单但有限的运动学混沌平流模型和基于纳维 - 斯托克斯方程直接数值模拟的方法之间的中等复杂度的自洽模型。前两个模型，即涡度缺陷模型和单波模型，是通过对涡度 - 速度耦合的连续简化构建的。第三个模型是从单波模型的时空离散化得到的保面积自洽映射。从动力系统的角度来看，这些模型是有用的，因为它们为拉格朗日平流问题提供了相对简单的自洽哈密顿量（流函数）。数值模拟表明，这些模型捕捉到了在纳维 - 斯托克斯方程的直接数值模拟中通常观察到的剪切流不稳定性、涡旋形成和弛豫的基本现象学。本文还讨论了动力学理论背景下电子等离子体中的自洽混沌输运。在这种情况下，(F)是相空间中的电子分布函数，相应的平流方程是弗拉索夫方程，自洽约束是泊松方程。这个问题与涡度问题密切相关。特别是，涡度缺陷模型类似于弗拉索夫 - 泊松模型，单波模型和自洽映射同样适用于等离子体和流体。此外，单波模型类似于用于研究全局耦合振子系统的模型。（c）2000美国物理研究所。
