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

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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.