In this subproject we will use the formalism of covariant Lyapunov vectors to investigate the dynamics of simplified multi-scale geophysical fluid systems, in order to gain a fundamentally new understanding of their instabilities and of their statistical mechanics. This will allow for greatly improving our understanding of the link between the micro-mesoscopic and macroscopic properties of turbulent geophysical flows. Asymptotic methods will be used for clarifying the emergence of different regimes of motions and of the corresponding instabilities.
Research visit to “The University of New South Wales” (UNSW) in Sydney, Australia from September to October 2018
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!
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!
Progress on CLVs in PUMA
We would like to understand the multiscale behaviour that is observable in the atmosphere using a spectral primitive equations model.
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.
Noethen, F. (2019). Computing covariant Lyapunov vectors in Hilbert spaces. arXiv: 1907.12458.
Noethen, F. (2019). Well-separating common complements of a sequence of subspaces of the same codimension in a Hilbert space are generic, arXiv:1906.08514
Noethen, F. (2019). A projector-based convergence proof of the Ginelli algorithm for covariant Lyapunov vectors. Physica D: Nonlinear Phenomena, Vol. 396, p. 18-34.
Vissio, G., Lucarini, V., (2018). A proof of concept for scale-adaptive parametrizations: the case of the Lorenz '96 model, Quarterly Journal of the Royal Meteorological Society, 144, 710, 63-75, doi.org/10.1002/qj.3184