# Reports

## Nonlinear waves, dissipation and (quasi-)geostrophic balances

I can show that for certain type of backscatter there are exact exponentially growing solutions, which shows that energy can get concentrated at some scales, rather than be transferred across scales.

Hi, my name is Artur. I am a PhD student at the University of Bremen and I am a member of the subproject M2 “Systematic multi-scale modelling and analysis for geophysical flow” since April 2017. My supervisor is Professor Jens Rademacher and I work in his research group “Applied Analysis”.

My research is about investigating the effects of various damping and driving realisations on the dynamics of different geophysical fluid models. Waves in linearisations of these models often characterise large scale phenomena in the ocean and atmosphere. I am interested in finding and analysing solutions of the full nonlinear equations.

Exact steady solutions for instance can bifurcate from trivial solutions by changing some parameters. In various cases we can prove analytically when these bifurcating waves occur and we can also determine some of their properties, such as their stability. With numerical tools we can corroborate these results and obtain additional insights into their structure and further properties.

Somewhat surprisingly, there are also linear waves that solve the full nonlinear problem. I was able to extend the class of known solutions of this type and for the first time took into account backscatter. For instance, I can show that for certain type of backscatter there are exact exponentially growing solutions, which shows that energy can get concentrated at some scales, rather than be transferred across scales.

For the investigations I use simple models like the single-layer and two-layer shallow water model as well as more complex ones like the Boussinesq approximation and the Navier–Stokes equations. In order to analyse the dynamics numerically I use the Matlab package “pde2path”.

With our investigations of idealised cases we hope to gain a better understanding of the different models used in the ocean and climate research. It is not only important from the mathematical perspective, but also could help to evaluate and improve numerical prediction models for weather and climate.

## Developing stochastic parameterisations for different flow regimes

My work is focused on developing a stochastic parameterisation for the interaction between different flow regimes that will still preserve the total energy of the system and require not too much computational time.

Hi, I am Federica and I am a PhD student in project M2. I studied mathematics and now I am working at the Meteorologic Institute of the University of Hamburg. Mathematics and physics are very intertwined one with the other: improvements in understanding the mathematics leads to physics advances to new realizations which in turn help developing more accurate mathematical models. The invention of the computer and the improvement of this technology, allowed more and more mathematical theories to find useful applications and one field for those applications is also climate (and ocean) modeling. Even though a lot has already been accomplished in this field, there is still room for development. There are material limitations (as computer processors, memory storage, computer precision, etc…) as well as theory limitations (not all the phenomena taking place are fully understood at the moment and the techniques used to model them are not always accurate enough) that have to be overcome.

This is where my work takes place. To be more specific most climate models include only the slow and most energetic modes since including also the fast modes would require too much computational time. However, in real atmosphere and ocean there is energy, enstrophy and momentum transfer between the resolved and the unresolved scales. Most current deterministic parameterisation schemes do not re-inject energy into the resolved scales; instead they are effectively an energy drain. Similarly, current stochastic parameterisations are operated mainly ad hoc without consideration of energy and momentum consistency. Recent studies showed that neglecting the fast unresolved modes induce also error growth, uncertainty and biases in the model therefore they should be included somehow. My work is focused on developing a stochastic parameterisation for the interaction between different flow regimes that will still preserve the total energy of the system and require not too much computational time.

Stochastic processes have some nice features that make them suitable in climate and ocean modeling. However, when dealing with stochastic processes, extra care should be employed. Their main feature is to have different realizations with same initial conditions. Therefore, if you find something worked fine once, you should be sure it was not just luck!

## Working on fundamental mathematical questions

It is important to understand fundamental mathematical questions for evaluating and improving numerical weather and climate prediction models.

Hi, I am Gözde. I am a PhD student in the subproject M2 Systematic multi-scale modeling and analysis for geophysical flow at the Jacobs University with Marcel Oliver. This subproject is splitted into three parts. Our part is variational model reduction. The purpose is to look at balance, multi-scale phenomena and variational approaches.

We mainly focus on foundational aspects of the problem. Then, we expect to encounter a variety of models where fundamental mathematical questions, such as well-posedness, regularity of solutions, validity of limits, and the analysis of associated numerical schemes is open and possibly nontrivial to resolve. It is important to understand them for evaluating and improving numerical weather and climate prediction models.

We started to examine the well-posedness for balance models for stratified flow. We will compare some models such as L1 balance and classical semigeostrophic model from shocks. Both models are formally derived in the same distinguished scaling limit and to the same order of expansion. The shocks in semigeostrophic theory are thought to be representative of the frontal dynamics regime, but this has not been tested in direct comparison.

I am really happy to be part of this project. The best advantage of it is to contribute with the other groups.