In this CRC colloquium, organized by Maximilan Engel, we're happy to welcome:
Cedrick Ansorge (FU)
Title: Exploiting Scale Similarity in the Atmospheric Boundary Layer
Despite the complexity of the atmospheric system and its size as measured by the Reynolds number, meteorology has found avenues to decipher the most relevant processes in the system. A key in this respect is scale similarity – allowing us to transfer insight across scales (given certain conditions). In this talk, I demonstrate how scale similarity can be combined with DNS, that is, turbulence-resolving simulation at finest scales. Indeed, this allows one to approach problems from a new perspective, namely based on first principles. Doing so, I will delve into three difficulties throughout my talk: First, I will introduce a new similarity hypothesis yielding a closed-form solution for wind veering in turbulent Ekman flow. Second, I will discuss new approaches to the study of the lower constant-flux layer and interaction of turbulence with a rough surface, and conclude on some generic insight to intermittently turbulent flow.
Alex Blumenthal (FU MATH+)
Title: Tractability of chaotic dynamics of noisy systems
One of the major advances in 20th century dynamical systems theory was the development of an abstract framework for the description of chaotic, disordered dynamics-- the multiplicative ergodic theorem, Lyapunov exponents, and other related tools of smooth ergodic theory. However, it is a notoriously challenging problem to mathematically verify the presence of chaos for given systems of practical interest, even for low-dimensional toy models such as the periodically-kicked rotor, also known as the Chirikov standard map. Remarkably however many of these problems become tractable in the presence of even a small amount of nondegenerate noise. My talk will discuss this somewhat surprising principle, highlighting applications to complicated dynamical systems in fluid mechanics out of reach of existing techniques for purely-deterministic dynamics. Joint work with a variety of coauthors, including Jacob Bedrossian (UCLA) and Sam Punshon-Smith (Tulane).
Tommaso Rosati (University of Warwick, UK)
Title: Lower bounds to Lyapunov exponents for stochastic PDEs
We introduce an approach to control the projective dynamic of dissipative linear stochastic PDEs with adapted initial data. Our results apply to vector valued stochastic heat equations and hyperviscous equations. Our proof relies on the introduction of a novel Lyapunov functional for the projective process associated to the equation, based on the study of dynamics of the energy median and on a notion of non-degeneracy of the noise that leads to high-frequency stochastic instability. This technique is applied to obtain – for the first time in a setting without order preservation – lower bounds to Lyapunov exponents of the equation with adapted initial data, and – under more stringent conditions on the model – their uniqueness via Furstenberg–Khasminskii formulas. Joint work with Martin Hairer.