This subproject is concerned with a mathematical analysis of quantitative and qualitative aspects of dynamics in turbulent geophysical flows. Bounds for energy dissipation and heat transfer will be considered and complicated dynamics will be identified in geophysical model systems. Transferring novel mathematical methods, which have been used so far for simple highly idealized models only, to more complex geophysically relevant unstable flow dynamics has proven to be very useful in this CRC, e.g. in subproject M1. We focus in the new subproject M7 on the existence of chaotic attractors in models of lateral and vertical convection, and optimal upper bounds for e.g. lateral or vertical turbulent heat flux which will also provide scaling laws that can be compared to the observed ones.
In particular, we aim at working in turbulent regimes, where the aforementioned quantities are supposed to be enhanced. Two complementary approaches will be used: one, based on dynamical system techniques, will describe chaotic motion in selected parameter regimes and the other one will be employed to estimate ensemble averages directly from the equation of motion. Thus, on the one hand, we pursue a bottom-up approach that provides lower bounds on the complexity by identifying specific dynamical features, in particular by giving sufficient conditions for the occurrence of certain types of chaotic motion. On the other hand, this is complemented by a top-down approach that finds optimal upper bounds for complexity by studying a priori – “worst-case” – bounds from the structure of the partial differential equations. These bounds provide scaling laws and can thus be used for diagnostics and to validate parameterizations used throughout the CRC. The main challenge in the geophysical context is in the treatment of the Coriolis terms and the resulting anisotropic convection.
Beyond scaling laws, such bounds should capture important features of the flows (for example, the structure of the boundary layers). For the quantities we will be interested in (heat transfer, energy), any “good” upper bound will be universal, but possibly overestimate the intrinsic dynamics, while lower bounds will depend sensitively on the initial data and the dynamics of individual solutions or ensembles in regions of phase space. Although lower bounds are very hard to derive, when some restrictions on the parameters are imposed, it is possible to identify specific solutions or invariant set, or even attractors. Our focus in this bottom-up approach lies in chaotic attractors, which possess features of turbulence. We propose an analysis of channel flows via rigorous reductions (centre manifolds) to finitely many modes. The main challenge is to prove specific forms of chaotic dynamics in the reduced model, which requires further extensions of the mathematical theory of chaotic dynamical systems.
Finally, we aim at identifying models where the bottom-up and top-down approaches can be synthesized. In particular, we propose to study the saturation of upper bounds by the dynamics of solutions.