Reports

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.

Being halfway done with my PhD, I felt the need to talk to other mathematicians working on the same topic as me: the analysis of algorithms for covariant Lyapunov vectors. However, with such an exotic topic, it is hard to find experts to discuss with. Thus, I had to search outside of local conferences and workshops. Looking into literature, I found an author whose work is closely related to mine. After speaking to him and my supervisors, we were convinced that a research stay would be perfect. The stay should not only serve as a means to communicate my recent findings, but the primary goal was to obtain new research questions and contacts to help advance the second half of my PhD.

When organizing the trip, I applied for visa well in advance and even got a next-day response. Everything else was planned on more short notice. Hence, I did not find a place to stay near university and ended up booking a hotel a bit farther away but with a good connection via public transport. My hotel was located near Green Square Station, which has a train running to the airport, the central station, and the inner city. The university can be reached by bus in about 15 minutes. Unfortunately, there is no train connection as of yet. However, constructions on a new line stopping at the university are expected to finish in 2019. My relatively short visit of three weeks did not require any prior arrangements with the university itself, although for longer stays I recommend filling out the visit request form online to gain access to special rooms, such as printer rooms. The visitor's room provided basis necessities like desks and computers. After preparing the assigned workspace on my first day, I had lunch with my host at a café on campus.

Following lunch, I presented my recent work as a basis for discussions, which ensued the next weeks. Sadly, one of my host's students, whom I wished to meet, was no longer at the institute. Nevertheless, the discussions were very fruitful. Besides the helpful comments on my presentation, I got to ask questions that always bugged me and talked about various research ideas. Some turned out to be worth pursuing, while others seemed no more than an interesting thought. This kind of feedback was exactly what I was hoping for. Even more so, we came up with new ideas during the stay. It left me with the impression that there is still much to discover about my topic. Moreover, my host told me of applications that were previously unknown to me. In particular, the computation of long-time coherent sets in ocean dynamics is an application that I find fascinating. One topic we initially planned to collaborate in turned out to be already well-answered. Nevertheless, we agreed upon keeping in touch for further exchange.

Next to the exchange with my host, I was lucky to meet a lot of friendly and interesting people inside and outside of university. For one, a professor staying with me in the visitor's room gave me useful tips on leisure activities. Although in spring the ocean is still a bit cold, a trip to Coogee Beach is a must, as it is only a 15-minute walk from university. A bit farther away, but still reachable with Sydney's Opal card for public transport, is the Blue Mountains National Park. From the heritage center near Blackheath Station there are several hiking trails leading through the beautiful nature. Another nice place is the Royal National Park south of Sydney. It boasts a long walk along the coast that occasionally passes by sandy beaches. However, watch out for blue bottle jellyfish and bring enough sun screen to protect yourself from the strong sun of Australia.

All in all, I had a wonderful time in Sydney that has been enriching on both a professional and personal level. Sydney is a modern city, where people are welcoming and always glad to help. Countless possible activities make it hard deciding on where to spend your free time. The three weeks were over so fast that I still wonder how I managed to explore Sydney and reach my goals. Arriving back in Hamburg, I am filled with new motivation and ideas on how to continue my PhD. Without a doubt, I will always remember the impressions of my research stay in Sydney. Thank you MINGS for making this experience possible!

I am Sebastian Schubert and I am a postdoc in sub project M1 “Instabilities across scales and statistical mechanics of multi-scale GFD systems”.

We would like to understand the multi-scale behaviour that is observable in the atmosphere using a spectral primitive equations models. For this, we use PUMA, a spectral primitive equation model, that is the dynamical core of PLASIM (Planet Simulator). For this purpose, we are studying instability of linear Progress on CLVs in PUMA years. Our results show that there is convergence towards a rate function which describes the behavior of large fluctuations. Nevertheless, we did not find a growth dependent variation of the rate function. This means in order to find discriminating perturbations in a generalized framework which develop on chaotic backgrounds.

For this, we make use of the splitting of tangent linear space into a covariant Lyapunov basis as described by Osedelecs theorem. Recently, we have studied the existence of a large fluctuation theorem for the Lyapunov exponents. The investigation is difficult because the computational effort only allows “short” time series of about 25 years. Our results show that there is convergence towards a rate function which describes the behavior of large fluctuations. Nevertheless, we did not find a growth dependent variation of the rate function. This means in order to find discriminating properties that are growth dependent we really have to study the scale dependency of the CLVs. As a first step, we are investigating the fastest growing instabilities in comparison to their presence in the actual non-linear background state. We see a clear detachment of the scales present in the first CLVs after going to a resolution of T85 (128x256, 1.39° at the equator). Our objective is now to expand this analysis to leading linear instabilities (the CLVs) and see if there are trends of the dominating waves towards larger scales.

Since September 2016, I work as a Post-Doctoral Researcher for the TRR. Previously, I was working as part of the DFG funded project MERCI after finishing my PhD at the International Max-Planch Research School at the Max-Planck-Institute for Meteorology in Hamburg.

My research is mainly focused on the various applications of dynamical system theory to geophysical models of simple to intermediate complexity. In particular, I have applied the theory of Covariant Lyapunov Vectors to a quasi-geostrophic two layer model and studied the connection between the unstable and stable directions to their baroclinic and barotropic energy conversions (Schubert & Lucarini, 2015). This type of analysis also allowed it to illuminate some features of simple blocking like patterns (Schubert & Lucarini, 2016).

For the project M1, I will study the properties of multi-scale instabilities using CLVs. The presence of multi-scale features usually impedes efforts to make good predictions. I will investigate the connection between multi-scale instabilities and energy transfers between atmosphere and ocean using firstly a rather simple quasi geostrophic model of the atmosphere and ocean (MAOOAM). Secondly, my interest lies in exploring the multi-scale properties of linear instabilities in a primitive equation model (PUMA). With these investigations using new tools from dynamical system theory, we expect to foster the understanding of multi-scale processes that are slow evolving and are usually “hidden” behind the faster dynamics.