Michel G, Chini G P
Laboratoire de Physique Statistique, École Normale Supérieure, CNRS, Université P. et M. Curie, Université Paris Diderot, Paris 75005, France.
Department of Mechanical Engineering and Program in Integrated Applied Mathematics, University of New Hampshire, Durham, NH 03824, USA.
Proc Math Phys Eng Sci. 2019 Mar;475(2223):20180630. doi: 10.1098/rspa.2018.0630. Epub 2019 Mar 13.
This article illustrates the application of multiple scales analysis to two archetypal quasi-linear systems; i.e. to systems involving fast dynamical modes, called fluctuations, that are not directly influenced by fluctuation-fluctuation nonlinearities but nevertheless are strongly coupled to a slow variable whose evolution may be fully nonlinear. In the first case, fast waves drive a slow, spatially inhomogeneous evolution of their celerity field. Multiple scales analysis confirms that, although the energy , the angular frequency and the modal structure of the waves evolve, the wave action / is conserved in the absence of forcing and dissipation. In the second system, the fast modes undergo an instability that is saturated through a feedback on the slow variable. A new multi-scale analysis is developed to treat this case. The key technical point, confirmed by the analysis, is that the fluctuation energy and mode structure evolve slowly to ensure that the slow field remains in a state of near marginal stability. These two model systems appear to be generic, being representative of many if not all quasi-linear systems. In each case, numerical simulations of both the full and reduced dynamical systems are performed to highlight the accuracy and efficiency of the multiple scales approach. Python codes are provided as electronic supplementary material.
本文阐述了多尺度分析在两个典型拟线性系统中的应用；即涉及快速动力学模式（称为涨落）的系统，这些涨落不受涨落 - 涨落非线性的直接影响，但仍与一个慢变量强烈耦合，而该慢变量的演化可能是完全非线性的。在第一种情况下，快波驱动其波速场的缓慢、空间非均匀演化。多尺度分析证实，尽管波的能量、角频率和模态结构会演化，但在没有强迫和耗散的情况下，波作用/是守恒的。在第二个系统中，快模式经历一种不稳定性，这种不稳定性通过对慢变量的反馈而饱和。为此开发了一种新的多尺度分析方法来处理这种情况。分析证实的关键技术要点是，涨落能量和模态结构缓慢演化，以确保慢场保持在接近边际稳定的状态。这两个模型系统似乎具有普遍性，代表了许多（如果不是全部的话）拟线性系统。在每种情况下，都对完整动力学系统和简化动力学系统进行了数值模拟，以突出多尺度方法的准确性和效率。Python代码作为电子补充材料提供。
