Local stochastic subgrid-scale modeling in efficient simulations of geophysical fluid dynamics

由于概念上的原因，有效的大气模型已经很有意义，因为它们可以加深对动态机制的了解，例如在气候变异性方面。然而，它们也可以是气候敏感性研究或古气候调查中的有用工具，其中需要许多或长时间的积分，因此计算效率是一个重要问题。特别是在这样的应用中，必须注意的是，尽可能多的不可避免的子网格尺度参数化的未解决的规模是基于第一性原理尽可能。随机模态约简（SMR）提供了一种相应的策略，其中大部分参数化都是在纸上推导出来的，一旦未解决的模式之间的非线性自相互作用已经拟合到一个简单的随机过程。然而，在迄今为止的应用中，构造的简化模型已经给出了一个谱公式，其中在全球亚网格尺度（SGS）参数化中，所有解析模式都相互作用。这限制了这种方法的适用性非常低维系统。为了规避这个问题，最近的SMR网格点为基础的空间离散化的实施已经开发的结果在当地的随机SGS参数化。到目前为止，这一战略已经在伯格斯方程的框架内得到了检验。在拟议的项目中，将采取重大步骤，将当地SMR战略应用于大气动力学的现实模型。对于正压涡度方程和浅水方程，应在f平面上构造SGS参数化。这两种模式都表现出在将局部SMR应用于大气动力学一般方程时需要考虑的基本特征。新的SGS参数化应满足以下标准：i）它们应该在相对较少的基本假设下以系统的方式从模型方程推导出来ii）它们应该尽可能与模型方程的守恒性质一致以及iii）它们应该要求对所分辨的尺度进行最小的（如果可能的话，根本不需要）回归拟合。目前，在气候模拟中需要物理约束和分辨率无关的随机参数化公式。如本文所建议的，使用SMR的参数化的发展将有助于这种方法。除了气候模拟，大涡模拟中的湍流模拟是另一个可以从这些发展中受益的领域。

Efficient models of the atmosphere are already interesting for conceptual reasons, as they can yield deepened insight into dynamic mechanisms, e.g. with regard to climate variability. They can, however, also be a helpful tool in climate-sensitivity studies, or in investigations of paleoclimate, where many or long integrations are needed, and thus computational efficiency is a matter of importance. Especially in such applications care has to be taken that as much of the inevitable subgrid-scale parameterizations of unresolved scales are based on first principles as possible. Stochastic mode reduction (SMR) offers a corresponding strategy, where most of the parameterization is derived on paper, once the nonlinear self-interactions between the unresolved modes have been fitted to a simple stochastic process. In applications so far, however, the constructed reduced model has been given a spectral formulation, where in the global subgrid-scale (SGS) parameterization all resolved modes interact with each other. This limits the applicability of this approach to very low-dimensional systems. To circumvent this problem, recently an implementation of the SMR to grid-point-based spatial discretizations has been developed which results in a local stochastic SGS parameterization. This strategy has so far been tested within the framework of the Burgers equation. In the proposed project significant steps will be taken towards the application of the local SMR strategy to realistic models of atmospheric dynamics. SGS parameterizations should be constructed for the barotropic vorticity equation and for the shallow water equations on an f-plane. Both models exhibit essential features to be taken into account in the application of the local SMR to the general equations of atmospheric dynamics.The new SGS parameterizations should fulfill the following criteria: i) they should be derived from the model equations in a systematic way under a relatively small number of basic assumptions ii) they should be as consistent as possible with the conservation properties of the model equations and iii) they should require minimal (if possible none at all) regression fitting of the resolved scales. Currently, there is a need in climate modeling for physics constrained and resolution independent formulations of stochastic parameterizations. The development of parameterizations using SMR, as proposed here, will contribute to such methods. Besides climate modeling, turbulence modeling in large eddy simulation is another field, which can benefit from such developments.