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M1: Dynamical Systems Methods and Reduced Models in Geophysical Fluid Dynamics

Principal investigators: Prof. Ingenuin Gasser (Universität Hamburg), Prof. Jens Rademacher (Universität Hamburg)

We study the multi-scale nature of geophysical flows from a mathematical perspective


Energy is distributed across a wide range of scales which are not all properly resolved and whose interactions are only partially understood.

  • We use asymptotic methods for clarifying the emergence of different regimes of motion and their corresponding instabilities to greatly improve our understanding of the link between the micro-mesoscopic and macroscopic properties of turbulent geophysical flows.
  • Central in our analyses are so-called covariant Lyapunov vectors (CLVs) which are objects that describe instabilities and statistical mechanics based on rigorous mathematical theory.
  • We foster the mathematical foundation of CLV algorithms to validate numerical results, to improve existing algorithms, and to develop new algorithms that help to describe scale-interactions.

Climate models cannot resolve the smallest scales due to a lack of computational power. To incorporate the missing scales some of the TRR181’s target operational models for large scales entail small-scale ‘subgrid’ models and parametrizations.

  • We address the mathematical foundation and properties of parameterized PDEs (PPDEs) to provide a more fundamental understanding of the TRR181’s target operational models and to gain new insight into their crucial dynamical features.
  • Based on a unified framework of PPDE models, we pave the way towards implementing CLV methods into more comprehensive climate models used in the TRR181.

Research Stay in Texas by Paul Holst (April 23)

In April this year, I had the unique opportunity to enjoy a 2-week stay in College Station, USA, at Texas A&M University to visit Edriss Titi, who is a professor of nonlinear mathematical science. Professor Edriss Titi is a worldwide renowned applied mathematician who specializes in the mathematical study of problems from fluid dynamics, nonlinear partial differential equations, and in a dynamical systems approach to turbulence. His contributions to these areas are of the highest calibre and practical impact. 

After hearing about Edriss Titi's expertise a few months before my research stay and after a simple email to him in which I briefly introduced myself and my research work, a good exchange of information with him developed, I was already very excited to do research together with him and learn from him during the research stay planned shortly thereafter in April. When I arrived in Texas shortly after Easter in April, we started working together on my research topic almost immediately. We talked daily at selected times from then on and made good progress. I greatly benefited from the conversations with him and learned new methods and aspects of my research topic and, even more generally, of the topics in my research area. The conversations I had with Xin Liu, one of the postdocs in Edriss Titi's research group at Texas A&M University, were also very helpful. At the end of my time in Texas, our conversations even resulted in a meaningful outcome regarding my research topic.

In between the meetings I had with Edriss Titi, I worked a lot on the exercises he gave me as well as a lot on my research topic. Away from my research work, I mostly explored the town of College Station on a bicycle in my scarce free time. During these explorations, I especially remembered a couple of beautiful green parks that College Station has to offer, as well as the impressive mansion district. There were also long distances that I had to travel to get from one place to another in College Station. This then gave me a sense, in those moments when I was out of breath, of why even many local people thought you needed a car to get from A to B in Texas.

After my return, I was very happy with the experience, knowledge, and results I was able to gain in Texas. The whole experience I had there has advanced me both scientifically and personally. I am very grateful for this and can therefore only recommend that everyone take advantage of the opportunity of a research stay.

Screening The Coupled Atmosphere-Ocean System Based On Covariant Lyapunov Vectors

I use the tangent linear version of the coupled atmosphere-ocean quasi-geostrophic model MAOOAM, and calculate the CLVs based on the so-called Ginelli method.

Melinda Galfi, Postdoc in M1

Covariant Lyapunov vectors (CLVs) reveal the local geometrical structure of the systems‘s attractor, thus providing valuable information about the basic dynamics. They are physically meaningful since they point into the directions of linear perturbations applied to the trajectory of the system. CLVs are linked to Lyapunov exponents, which describe the growth or decay rate of linear perturbations.

My name is Melinda Galfi, and I am a postdoc in the M1 subproject. I am continuing the work on CLV analysis started by Sebastian Schubert. I use the tangent linear version of the coupled atmosphere-ocean quasi-geostrophic model MAOOAM, and calculate the CLVs based on the so-called Ginelli method. I compute the CLVs in the phase space of the model, spanned by the spectral model variables, which can be grouped into four different categories: atmospheric dynamic and thermodynamic variables, as well as oceanic dynamic and thermodynamic variables.

The spectrum of Lyapunov exponents of our systems reveals the existence of a central or slow manifold. This is a basic property of coupled ocean-atmosphere models, and has to do with the multiscale character of this type of chaotic system. Based on the CLVs, we hope to understand more deeply the dynamical properties of the system itself, and especially of the slow manifold. To achieve this, one can use several CLV based indicators. One of these indicators is the variance of CLVs, showing the contribution of each model variable to the growth or decay of perturbations. By computing the variance of the CLVs in MAOOAM, we see that the atmospheric variables have the strongest contribution to the evolution of perturbations in our system. However, we detect an exception in case of instabilities growing or decaying on long time scales, where the contribution of the oceanic thermodynamic variables is approximately as strong as the one of the atmosphere. This shows that the d y n a m i c s of the slow manifold is governed by interactions between atmosphere and ocean, with the main coupling taking place through the ocean thermodynamics. The contribution of the ocean is the strongest in case of perturbations decaying over long time scales. Another useful indicator is the angle between CLVs, revealing the local structure of the attractor. Our results show that the angle between the CLVs corresponding to the slow manifold is dominantly very near zero, hinting to multiscale instabilities and geometrical degeneracies.

As a next step, we would like to repeat the CLV analysis for a substantially higher model resolution. The currently used resolution consists of 5x5 atmospheric and 5x5 oceanic modes. Our final goal is to study the energy transfers between atmosphere and ocean based on CLVs.