Scale Analysis and Asymptotic Reduced Models for the Atmosphere

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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.