Quantifying long time statistical properties of a few fluid models

主要研究者和他的同事们通过与时间和/或空间近似相关的适当离散动力系统的长时间统计特性，研究了量化一些原型流体系统的长时间统计特性的问题。考虑的物理问题是大普朗特数和/或小allekman数下的雷利-贝纳德对流，以及一些相关的简化模型。特别地，所开发的方法被应用于数值量化一个重要的物理长期统计量，即在一些对流模型中的平均热输运。这里的关键问题是高效和收敛的方案的设计，分析和实现（在某种意义上，离散系统的平稳统计特性收敛于底层系统的统计特性）。近似大型复杂系统的长期行为是一个众所周知的挑战，因为小的误差会累积和放大。与多尺度（由大普朗特数、小埃克曼数、大瑞利数引起）和广义动力系统（如3D Boussinesq系统）相关的其他困难也得到了解决。考虑了流体系统的适当随机扰动，以确保收敛到物理相关的长期行为。长期统计特性的量化在应用中具有重要意义。除了众所周知的经典湍流理论应用外，它也是极其重要的气候研究，因为预测的气候是潜在气候模式的长期统计行为。待研究的模式虽然与实际的气候模式相距甚远，但具有一些对现实气候模式至关重要的机制，如保能非线性平流、旋转、对流、耗散/阻尼和强迫。在这种情况下，对长期统计行为的更清晰的理解有助于我们更好地理解许多地球物理流体现象，并为气候变化的精确数值研究提供指导。该项目还为研究生提供了大量的机会，包括来自代表性不足群体的学生，参与许多物理动机问题的建模，分析和计算。

WangDMS-1008852 The principal investigator and colleagues study the issue ofquantifying the long-time statistical properties of a fewprototype fluid systems via long-time statistical properties ofsuitable discrete dynamical systems related to temporal and/orspatial approximations. The physical problems considered are theRayleigh-Benard convection at large Prandtl number and/or smallEkman number regime, and a few related simplified models. Inparticular, the methodology developed is applied to numericallyquantify an important physical long-time statistical quantity,the averaged heat transport, in a few convection models. The keyissue here is the design, analysis and implementation of schemesthat are efficient and convergent (in the sense that thestationary statistical properties of the discrete system convergeto those of the underlying system). Approximating long-timebehavior of large complex systems is a well-known challengebecause small errors could accumulate and amplify. Additionaldifficulties related to multiple scales (induced by large Prandtlnumber, small Ekman number, large Rayleigh number), andgeneralised dynamical system (such as the 3D Boussinesq system)are also addressed. Suitable random perturbations of the fluidsystems are considered in order to ensure convergence to thephysically relevant long-time behaviour. Quantifying long-time statistical properties is of greatimportance in applications. Besides well-known applications inclassical turbulence theory, it is also extremely important inclimate studies because the predicted climate is the long-timestatistical behaviour of the underlying climate model. Themodels to be investigated, although far from practical climatemodels, share several important mechanisms that are crucial torealistic climate models, such as energy-preserving nonlinearadvection, rotation, convection, dissipation/damping and forcing. A clearer understanding of long-time statistical behavior in thissetting helps us better understand many geophysical fluidphenomena, and provides guidelines for accurate numerical studyof climate changes. The project also provides abundantopportunities for graduate students, including student fromunderrepresented group, to participate in the modeling, analysis,and computation of many physically motivated problems.