CMG Collaborative Research: Ocean Modeling by Bridging Primitive and Boussinesq Equations

虽然在结合了简化方程的涡旋分辨海洋模式的框架内可以准确地处理海洋的广大区域，但在相对较小的区域中，快速演变的三维运动不仅对局部而且对 大尺度动力学，从而在沿海和全球海洋流动的多尺度动力学中发挥重要作用。在这些小区域中的流动动力学足够复杂，不允许通过简单的参数化进行精确的近似，而是要求解全套方程。本计画的目标是建立一个能同时处理大尺度大气环流与能量主动运动的模式架构。这个研究教育项目是一个精心策划的努力，综合数学和海洋学方面的专业知识。项目团队的数学、计算和地球物理专业知识的融合是这一奋进成功的关键。这也是至关重要的研究生和本科生，谁将参与研究项目的所有阶段的真正跨学科的培训：建模、数学分析、离散化、验证、计算和数据分析。为了模拟这些具有挑战性的海洋流动，一个真正的多-规模的建模框架，将采用计算密集型Boussinesq方程只在小区域的激烈混合和需要在流体域的其余部分中的计算有效的原始方程。然而，所考虑的海洋流动的固有的多尺度性质，使得这样一个建模框架的发展具有挑战性，无论是在数学上还是在计算上。 事实上，一个需要解决悬而未决的问题，如数学桥接两个不同的方程组，界面的计算网格的分辨率差异很大，量化和建模的不确定性，在这个复杂的框架和海洋流动的建模范围内的尺度向前和向后的能量级联共存。 这个新的框架包括几个重要的数学和计算发展：（i）一个新的多物理场/多分辨率模拟方法的基础上区域分解，这将允许一个适当的处理高度变化的网格分辨率和接口的非流体静力学和流体静力学流动制度;（ii）一个新的空间─一种时间滤波方法，提供了一种优雅的数学方法，用于通过创建一系列中间模型填充两组方程之间在计算效率和物理精度方面的差距;（iii）针对由Boussinesq原始方程耦合的固有随机性质产生的系统中的不确定性的新建模策略;及（iv）有关的状况─利用下列方法，适当处理Boussinesq和原始流态明显不同的湍流特性的先进湍流模型：近似解卷积方法的数学性质。

While vast regions of the ocean can be treated accurately within the framework of eddy-resolving ocean models integrating simplified equations, there are comparatively small regions in which rapidly-evolving three dimensional motions are important not only for local but also for large-scale dynamics, thereby playing an important role in the multi-scale dynamics of both coastal and global oceanic flows. The flow dynamics in these small regions are complex enough not to permit an accurate approximation by simple parameterizations, but rather demand solution of the full set of equations. The objective of this project is to build a modeling framework which can handle both energetically active motions and large scale general circulations simultaneously. This research-education project is an orchestrated effort of a collaboration synthesizing expertise in both mathematics and oceanography. The blend of mathematical, computational, and geophysical expertise of the project team is central to the success of this endeavor. It is also essential to the truly interdisciplinary training of graduate and undergraduate students, who will be involved in all the stages of a research project: Modeling, mathematical analysis, discretization, validation, computation, and data analysis.To model these challenging oceanic flows, a truly multi-scale modeling framework that will employ the computationally intensive Boussinesq equations only in the small regions of intense mixing and the computationally efficient primitive equations in the rest of the fluid domain is needed. The inherent multi-scale nature of the oceanic flows considered, however, makes the development of such a modeling framework challenging, both mathematically and computationally. Indeed, one needs to address outstanding open questions, such as, the mathematical bridging of two different systems of equations, the interfacing of computational meshes of vastly varying resolutions, the quantification and modeling of uncertainty in this complex framework and the modeling of oceanic flows over a range of scales where forward and backward energy cascades coexist. This new framework comprises several significant mathematical and computational developments: (i) a new multiphysics/multiresolution modeling approach based on domain decomposition that will allow an appropriate treatment of highly varying mesh resolutions and the interfacing of the non-hydrostatic and hydrostatic flow regimes; (ii) a novel spatio-temporal filtering methodology that provides an elegant mathematical approach for bridging two different sets of equations by creating a spectrum of intermediate models filling the gap between the two sets of equations in terms of computational efficiency and physical accuracy; (iii) new modeling strategies for the uncertainty in the system generated by the inherently stochastic nature of the Boussinesq-primitive equations coupling; and (iv) state-of-the-art turbulence modeling for an appropriate treatment of the markedly different turbulence character of the Boussinesq and primitive flow regimes by taking advantage of the mathematical nature of approximate deconvolution approaches.