A Coupled Atmosphere-Ocean-Sea-Ice Model: Mathematical Analysis, Numerics and Computations

数值天气预报和气候预测是地球物理流体动力学模型的相关应用领域，这些模型是本研究组的主题。天气和气候预报有别于经典地球物理流体动力学的一个特点是使用“耦合”模型，其中质量、物质和能量在耦合模型的组件之间通过可渗透边界进行交换，例如在大气和海洋之间或海洋和海冰之间。这一研究领域是由S.Manabe和K.Hasselmann获得诺贝尔奖的工作开创的，现在继续在气候和天气预报业务中心进行。分隔不同舱室的界面的特征，如大气-海洋界面，没有得到很好的理解，并受到试探性的物理假设和亚网格尺度关闭的影响。天气和气候模拟朝着更高的时空分辨率发展的趋势确实对这些假设中的许多提出了质疑，并为这一项目提供了一般背景。因此，本项目的目标是：1.遵循数学原理建立大气-海洋-海-冰耦合模式；2.研究大气-海洋-海-冰耦合方程的数学基础；3.发展反映这一数学分析的数值方法；4.在大气-海洋-海-冰系统的模拟中对所得程序进行评估。要遵守的一个数学原则是，耦合模型应该可证明地允许在适当的函数空间中存在局部或整体的弱/强解。这些函数空间包含耦合耦合系统的组件的边界条件，因此与通常假定为单独组件的解空间有很大不同。第二个原理是与热力学第一定律和第二定律一致。由于潮湿空气和相变的存在，这对大气和海洋都构成了不小的任务，而对于海洋，由于目前缺乏现实的分析状态方程。由于为各个组件建立了一致的热力学，我们的目标也是证明完全耦合模型的热力学一致性。建立在保持结构的数值方法的基础上，这种建模和分析框架将被转化为保留连续耦合系统的关键性质的有限维近似。连续守恒性质的离散形式的保存在计算物理上合理的解和追求我们的目标方面起着关键作用，即证明这些离散解也满足热力学第一定律和第二定律。最后，我们将对离散耦合方程进行数值实验，并将解与其他耦合模型的解进行比较。

Numerical weather prediction and climate projection are relevant application areas of the geophysical fluid dynamical models that form the subject of this Research Unit. A feature that distinguishes weather and climate forecasting from classical geophysical fluid dynamics is the use of ``coupled'' models in which mass, matter and energy are exchanged between components of a coupled model through a permeable boundary such as, for example, between the atmosphere and the ocean or between the ocean and sea-ice. This research field has been initiated by the Nobel-prize awarded work of S. Manabe and K. Hasselmann and is now continued in operational centres for climate and weather forecasting. The characteristics of the interfaces separating different compartments, such as the atmosphere-ocean interface, are not well understood and subject to heuristic physical assumptions and subgridscale closures. The trend in weather and climate modelling towards higher spatio-temporal resolutions does question many of these assumptions and provides the general background to this project. Accordingly, the goals of this project are to 1. construct a coupled atmosphere--ocean--sea-ice model following mathematical principles, 2. investigate the mathematical underpinnings of coupled atmosphere-ocean-sea-ice equations, 3. develop numerical methods that reflect this mathematical analysis 4. evaluate the resulting codes in simulations of the atmosphere-ocean-sea-ice systems. One mathematical principle to be adhered to is that the coupled model should provably admit local or global existence of weak/strong solutions in suitable function spaces. These function spaces incorporate boundary conditions that couple the components of the coupled system and therefore differ substantially from the solution spaces usually assumed for the individual components. A second principle is consistency with the first and second law of thermodynamics. This constitutes non-trivial tasks already for the atmosphere, due to the presence of moist air and phase changes, and for the ocean, due to the current lack of a realistic analytical equation of state. With consistent thermodynamics established for the individual components, we aim to also demonstrate thermodynamic consistency for the full coupled model. Building upon structure-preserving numerical methods, this modelling and analytical framework will be translated into a finite-dimensional approximation that retains key properties of the continuous coupled system. The preservation of discrete versions of continuous conservation properties plays a pivotal role in computing physically sound solutions and in pursuing our goal to show that these discrete solutions, too, satisfy the first and second law of thermodynamics. Finally, we will carry out numerical experiments of the discrete coupled equations and compare the solutions to solutions of other coupled models.