Modeling Mesoscale Circulations by Coupled Nonlinear Systems

低阶模型(LOM)通过关注关键要素和只保留最少的自由度，在理解大气动力学的基本机制方面发挥了重要作用。LOM是由大气动力学基本方程导出的低阶非线性常微分方程组，通常采用Galerkin方法。尽管有许多非常吸引人的特征，但该方法并不能防止LOM违反原始方程的基本守恒性质，这通常会导致非物理行为。在以前的研究中，研究人员开发了一种模块化的方法来构建物理上合理的大气环流LOM，其中经典的机械系统--Volterra陀螺仪(最简单的是著名的二维瑞利-贝纳德对流的洛伦兹模型)--作为基本构件。耦合陀螺仪系统通过总是在无粘性、非受力的极限中守恒能量来避免非物理行为。同时，由于大气动力学中的大多数模型(如原始、浅水和准地转方程)的保守部分是哈密顿的，在LOM中保持哈密顿结构被认为是保持原始系统的基本守恒性质的有效方法。正如首席调查人员的初步研究所表明的那样，陀螺仪方法在这项工作中特别有希望。在这项计划下，我们将尝试从Rayleigh-Benard对流开始，开发适用于重要流体动力系统的哈密顿LOM。相对简单的陀螺仪(特别是哈密顿LOM)具有流体动力学方程的基本守恒性质，将为进一步科学地理解大气动力学中的基本机制及其相互作用提供新的有效工具。从更广泛的影响角度来看，这项研究最终可以为大气可预测性问题提供新的见解，从而限制预测。

Low-order models (LOMs) have played an important role in understanding of basic mechanisms of atmospheric dynamics through a focus on key elements and retaining only minimal number of degrees of freedom. LOMs are low-order systems of nonlinear ordinary differential equations derived from basic equations of atmospheric dynamics, commonly by the Galerkin method. Despite a number of highly attractive features, the method does not prevent a LOM from violations of fundamental conservation properties of the original equations, which often results in unphysical behavior. In previous studies, the investigators have developed a modular approach to constructing physically sound LOMs of atmospheric circulations, in which classical mechanical systems, the Volterra gyrostats (the simplest one being the celebrated Lorenz model of two dimensional Rayleigh-Benard convection), serve as elementary building blocks. Systems of coupled gyrostats avoid unphysical behavior by always conserving energy in the inviscid, unforced limit. At the same time, because the conservative part of most models in atmospheric dynamics (e.g., the primitive, shallow water and quasi-geostrophic equations) is Hamiltonian, maintaining the Hamiltonian structure in a LOM is viewed as an effective way to retain in it the fundamental conservation properties of the original system. As demonstrated in the Principal Investigators' preliminary studies, the gyrostatic approach is particularly promising in this endeavor. Under this project, an attempt will be made to develop useful Hamiltonian LOMs for important fluid dynamical systems, beginning with Rayleigh-Benard convection that provides the principal mechanism for mesoscale shallow convection.Relatively simple gyrostatic (particularly Hamiltonian) LOMs inherently possessing fundamental conservation properties of the fluid dynamical equations will provide a new effective tool to further a scientific understanding of essential mechanisms and their interaction in atmospheric dynamics. From the Broader Impacts perspective, eventually this research could lend new insights into issues of atmospheric predictability and, hence, limits to forecasting.