Scale Analysis and Asymptotic Reduced Models for the Atmosphere

FOR 5528提出了大气-海洋-海冰模式的解理论。这个子项目考虑了地球物理流体动力学中建立的依赖于尺度的模型层次结构，在I. Held (Bull。阿米尔。Meteorol。Soc。， 86,(2005))对我们了解地球系统至关重要。该项目的一个出发点是流体静力原始方程（HPEs）的综合解理论，该方程描述水平长度和时间尺度超过20km/60min的流动。在这方面已澄清了强和弱、局部或全球时间解的存在性，包括适当的先决条件。今天，hpe构成了最先进的全球天气预报和气候模型的基础。因此，hpe将在此作为描述大于20km/60min尺度的流动模型分析的可靠参考，并且工作包WP 1a将旨在严格证明已建立的准地转（QG）和行星地转（PG）模型是hpe的渐近极限动力学。在WP 1b中，WP 1a中采用的分析策略将应用于HPEs本身，尽管包含随机强迫。由于随机建模在天气预报和气候模拟中受到越来越多的关注，并且由于随机噪声可以诱导除假设的耗散确定性过程之外的正则化效应，因此这一点很有趣。WP 1c开发了从20公里/60分钟到行星尺度的多尺度模型。这项工作首先关注的是众所周知的半地转模型，到目前为止，该模型可以嵌入到(R. Klein, Ann。Rev.流体机械师。， 42(2010))只有在特殊的，有效的二维情况下。进一步的目标是在一个统一的渐近多尺度模型中捕获QG-， SG-和pg型过程的相互作用。WPs 2和WPs 3适用于20km/60min以下的尺度。考虑到L. sz<s:1> kelyhidi及其同事关于Euler和Navier-Stokes方程的“野生”非唯一弱解的凸积分理论（CI）的最新结果，人们不能期望在这些较小尺度上的流动有一个经典的解理论。因此，WP 2研究了弱解构造的ci方法是否以及在什么条件下可以应用于hpe以建立非唯一性结果。WP 3侧重于远小于20km/60min的尺度，并研究重力和地球自转的影响是否（如果是，则精确高于哪个尺度）可以稳定模型，并禁止“野”解和非唯一性。由于对小型未解决尺度的基于物理的闭包感兴趣，WP 3还研究了在CI构建中为了方便而使用的相当人工的“Mikado”流是否可以被模拟或实验中实际观察到的小规模流结构所取代。

FOR 5528 advances the solution theory for atmosphere-ocean-sea ice models. This subproject considers the established hierarchy of scale-dependent models in geophysical fluid dynamics which, following I. Held (Bull. Amer. Meteorol. Soc., 86, (2005)) is of outmost importance of our understanding of the Earth system. One point of departure of this project is the comprehensive solution theory for the hydrostatic primitive equations (HPEs) which describe flows on horizontal length and time scales beyond 20km/60min. The existence of strong and weak, local or global in time solutions has been clarified in this context, including appropriate preconditions. Today, the HPEs form the basis of state-of-the-art global weather forecasting and climate models. Thus, the HPEs will serve here as a solid reference for analyses of flow models describing scales larger than 20km/60min, and work package WP 1a will aim to rigorously justify the established quasi-geostrophic (QG) and planetary-geostrophic (PG) models as asymptotic limit dynamics of the HPEs. In WP 1b the analytical strategy adopted in WP 1a shall be applied to the HPEs themselves, albeit with inclusion of stochastic forcing. This is of interest as stochastic modelling is receiving increasing attention in weather forecasting and climate simulation, and because stochastic noise can induce regularizing effects aside from assumed dissipative deterministic processes. WP 1c develops multiscale models for the scale range from 20km/60min to the planetary scales at a formal level. This work first focuses on the well-known semi-geostrophic model, which so far could be embedded in the systematic model hierarchy from (R. Klein, Ann. Rev. Fluid Mech., 42 (2010)) only in special, effectively two-dimensional situations. The further aim is to capture the interaction of QG-, SG- and PG-type processes in a unifying asymptotic multiscale model. WPs 2 and 3 address the scales below 20km/60min. Considering the recent results from convex integration theory (CI) by L. Székelyhidi and colleagues regarding "wild" non-unique weak solutions of the Euler and Navier-Stokes equations, one cannot expect a classical solution theory for flows on these smaller scales. Thus, WP 2 investigates whether, and if so under which conditions, the CI-approach to the construction of weak solutions can be applied to the HPEs to establish non-uniqueness results. WP 3 focuses on scales much smaller than 20km/60min and studies whether, and if so above precisely which scales, the influences of gravity and Earth rotation may stabilize the model and prohibit "wild" solutions and non-uniqueness. Being interested in physically based closures for small unresolved scales, WP 3 also investigates wether the rather artificial "Mikado" flows used for convenience in the CI construction can be replaced with small scale flow structures actually observed in simulations or experiments.