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

虽然在整合简化方程的漩涡解析海洋模型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，快速演变的三维运动不仅对局部动力学而且对大尺度动力学都很重要，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。这些小区域内的流动动力学非常复杂，不允许用简单的参数化进行精确的近似，而需要求解一整套方程。这个项目的目标是建立一个可以同时处理能量活跃运动和大尺度一般循环的建模框架。这个研究教育项目是综合数学和海洋学专业知识的精心策划的合作成果。项目团队的数学、计算和地球物理专业知识的融合是这一努力成功的核心。它对于研究生和本科生的真正跨学科培训也是必不可少的，他们将参与研究项目的所有阶段：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，需要一个真正的多尺度建模框架，该框架将只在剧烈混合的小区域使用计算密集型的Boussinesq方程，而在流体域的其余部分使用计算效率高的原始方程。然而，考虑到海洋流动固有的多尺度性质，使得这种建模框架的发展在数学和计算上都具有挑战性。事实上，人们需要解决突出的开放性问题，例如，两种不同方程组的数学桥接，分辨率差异极大的计算网格的接口，在这个复杂框架中的不确定性的量化和建模，以及在一系列尺度上的海洋流动的建模，其中正向和向后的能量级联共存。这个新框架包括几个重要的数学和计算发展：(i)基于域分解的新的多物理场/多分辨率建模方法，该方法将允许适当处理高度变化的网格分辨率以及非流体静力和流体静力流动状态的接口；（ii）一种新颖的时空滤波方法，通过创建一系列中间模型来填补两组方程在计算效率和物理精度方面的差距，为桥接两组不同的方程提供了一种优雅的数学方法；(3)基于Boussinesq-primitive方程耦合的固有随机性所产生的系统不确定性的新建模策略；（iv）利用近似反褶积方法的数学性质，对Boussinesq和原始流型明显不同的湍流特性进行适当处理的最先进的湍流建模。

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.