## Reports Area L

## Decomposition of Vertical Momentum Fluxes in the Tropical Atmosphere

Based on MODES we will develop a tool for the computation of vertical momentum fluxes from high-resolution ERA5 data.

Greetings dear reader! My name is Valentino and I work as a PhD student at the University of Hamburg under the supervision of Dr. Nedjeljka Žagar (Universität Hamburg). I am a part of the TRR subproject L2 named “Quantifying Dynamical Regimes in the Ocean and the Atmosphere”. I am originally from Croatia where I spent all of my education years. I have Bachelor in Physics/Geophysics from the University of Zagreb and a Masters in Meteorology and Physical Oceanography that I acquired with the thesis on “Implementation of the empirical orthogonal functions analysis to determine nonstationarity of time series” from the University of Zagreb.

I started my work as a part of TRR in May of 2021. with the goal of my work being the quantification of vertical momentum fluxes in the tropical atmosphere. To do this we will employ normal mode decomposition ( NMD ) to decompose atmospheric motions to different dynamical regimes. More precisely we will be using the MODES NMD package developed by Žagar et al., for the horizontal velocity and an associated novel spectral approach for the vertical velocity decomposition. Based on MODES we will develop a tool for the computation of vertical momentum fluxes from high-resolution ERA5 data.

Then, we will analyze climate models in the same way and compare the results with those for reanalysis to quantify missing momentum fluxes across scales. The results will be then used to quantify the missing momentum fluxes in climate models that are still running at a much lower resolution. The quantification of vertical momentum fluxes associated with the inertiagravity waves in analysis data can become valuable validation metrics of new parameterizations and upscale transfers in ICON-a and other climate models. The results will provide, among others, a novel scaledependent quantification of the vertical momentum fluxes associated with different atmospheric regimes in analyses and climate models.

## Models, Respecting the Conservation Laws

Recently we have learned how to combine the framework with stochastic turbulent closures, thereby making another step towards realistic turbulence models.

Ideal (i.e. non-dissipative) fluids are characterized by a number of conservation laws, which are the defining features of the motion, such as

- energy conservation;
- mass conservation;
- material conservation of generalized vorticity, e.g. potential vorticity,
- model specific advected quantities, for instance, potential temperature in inviscid primitive equations.

In many applications the dissipation can not be ignored, whereupon all of the above laws must be modified appropriately. However, in order to establish the ideas, it is useful to stick to the non-dissipative case initially. My work in TRR 181 is focused on developing “simplified” models that inherit the conservation laws from their parent system. This branches into two distinct sub-projects.

**1. Lagrangian turbulence models. **Common approach to turbulence modeling is based upon averaging the equations of motion at each spatial location (e.g. Reynolds averaging). While very natural, this approach destroys material conservation laws. This can be remedied by using more elaborate procedures, known as Lagrangian averaging, based on averaging fluid parcels’ trajectories. Combining the recently introduced concept of geometric generalized Lagrangian mean (geometric GLM) with averaging of the variational principles, we developed a turbulence framework, which guarantees the inheritance of the above conservation laws by the model.

Using this framework, we derived a number of idealized turbulence models, namely for the primitive and Euler-Boussinesq equations, Euler’s equations of ideal fluid flow, and the multi-dimensional Burgers’ equations. While working on these models, we found that our framework is highly adaptable to different physical contexts (compressible and incompressible flows, manifolds, various boundary conditions, anisotropy are all treatable within the framework) and leads to models with desirable mathematical properties (well-posedness, filtering of small scales). Another advantage is that the framework is not bound to a particular choice of scalar-averaging (e.g. time-averaging or statistical averaging). Recently we have learned how to combine the framework with stochastic turbulent closures, thereby making another step towards realistic turbulence models.

The bulk of work done up to now provides a proof of concept for our methodology. Several steps still need to be taken, in order to make the concept attractive to applied scientists. One is the inclusion of dissipation. Another avenue is further adapting the framework to a specific physical context (for instance, incorporating the concept of isopycnal averaging for the ocean models) and developing corresponding parametrizations in the closure.

Three papers so far were written for this sub-project: one is published, one is submitted, and one is in preparation.

**2. Variational approximations for rescaled fluid models. **A powerful method of studying equations of mathematical physics is rescaling of equations of motion followed by their asymptotic analysis. However, when applied to Lagrangian system, this approach has an important drawback as it potentially destroys the variational structure of the problem and associated conservation laws.

How to construct asymptotic approximations to the rescaled equations that yield models inheriting conservation laws 1)-4) from the parent system then? The answer is fairly straightforward for an isotropic scaling and is based on first approximating the rescaled variational structure, then computing the equations of motion. However, for anisotropic scaling, the rescaled variational structure becomes more complex and not all its approximations have the desired property.

We completely resolved the question for inviscid homogeneous fluids, thereby developing a systematic procedure for constructing conservation laws-preserving approximations to the rescaled systems. It turns out that many known models are trivial to derive using our framework. For instance, the inviscid primitive equations fall within the framework as an approximation to Euler-Boussinesq system.

The problem that initiated this project comes from equatorial dynamics. There is much theoretical interest in the dynamics of planetary-scale Kelvin waves for the purpose of atmospheric and oceanic data assimilation. This calls for a geophysical balance model, which retains equatorial Rossby waves in addition to Kelvin waves. The work on deriving the required balanced model is about to begin. Another direction of our research is to obtain models of front-formation in atmosphere and ocean due to strong meridional temperature gradients. Further applications are likely as the framework should prove useful whenever approximate models are sought in an anisotropic setting.

The theoretical paper for this sub-project will be submitted for publication in the nearest future. Two further papers on equatorial balance models and front-formation models are planned.

## Investigating eddy diffusivitites and eddy-mean flow interactions

Our observational data will serve the model as a reference that includes smaller scales that the model is not able to cover.

In September 2016 I started working as a postdoctoral researcher in the subproject L3 Diagnosing and parameterising the effects of eddies at the Universität Hamburg with Kerstin Jochumsen from Experimental Oceanography and Alexa Griesel from Theoretical Oceanography. Having a background in nonlinear dynamics and statistical physics in application to complex systems, I am strongly attracted by the aim of our research project and by the possibility to work within an inspiring interdisciplinary research network as created by the TRR181 with its great possibilities for exchange with scientists of different fields.

A major link to my former research consists in the analysis of trajectories (derived from in situ experiments or modeled by stochastic processes) and more specifically the analysis of broad (non Gaussian) Langrangian statistics of absolute and relative dispersion. The goal of our research subproject L3 is the quantification of eddy diffusivities and eddy-mean flow interactions by using Langrangian particles statistics in both eddying ocean models and observations. Its aim is to develop and to test energy consistent parameterisations of meso- and sub-mesoscale processes for the global ocean with a focus on 100 km -1 km scales.

At the beginning I developed and tested float deployment strategies by means of the high resolution POP model. In November/December 2016 our cruise with the RV Meteor took place in the atlantic-sea off the african coast. The cold upwelling front off Namibia's coast in the area of Luderitz has a highly irregular structure due to eddies and filaments, finger-like structures of cold upwelling water pushing west into the warm surface waters offshore (see Figure 1). In our field experiment we explored mesoscale and submesoscale structures within a filament by satellite-tracking 37 surface drifters which we released in groups of triplets. As a first result we find that, due to the underlying rich mesoscale system, the dispersion statistics are very different depending on the location of release. While the drifters of the group released at the southern border of the filament separate slower (Figure 2a), the drifters in the group released closer to the upwelling system at the northern border of the filament separate faster from each other and follow distinct paths within the complex surface currents (Figure 2b). Currently our research focuses on the relative dispersion of drifter pairs (and its corresponding probability density function) as its properties depend on the kinetic energy spectrum. The statistical analyzing of single particle dispersion and the comparison of our findings with dispersion statistics of ocean model will be a next step. Here our observational data will serve the model as a reference that includes smaller scales that the model is not able to cover.

More Information about the research cruise on RV Meteor (M132) including reports, posters and videos of the scientific work have a look have a look at our TRR181 homepage.

## An In-Depth Study of Diurnal Warm Layers: Quantification of Air-Sea Interactions

I am fascinated by such theoretical results but also by their various fields of application.

In July 2021 I joined the TRR 181 as a postdoctoral researcher in the project L4, “Energy-Consistent Ocean-Atmosphere Coupling”. Within this project I am studying the phenomenon of diurnal warm layers (DWLs) in the ocean. It describes the warming of the sea surface in certain areas during daytime (by up to 2K, though in particular cases also higher fluctuations have been observed) compared to the surrounding ocean that usually keeps an almost constant surface temperature.

From the point of view of air-sea interactions the appearance of DWLs is of particular interest, since such differential heating can promote a sea breeze like convective movement and, as a result, serve as a cloud building mechanism. Moreover, as such warm spots appear due to solar radiation, one can also expect a feedback behaviour caused by an increase in the cloud cover.

The presence of DWLs as well as their influence on the cloud amount is well documented in the literature, at least in the qualitative sense. Moreover, this phenomenon has been confirmed in idealised simulation studies. However, most modern global coupled simulations do not capture this mechanism, since it requires a high vertical resolution of the sea levels in order to correctly represent the heat transport (involving only about 20m in the vertical), but also a high horizontal resolution in the atmosphere that would permit to directly resolve convection. My work in the project consists in implementing a simulation that would incorporate both these features. This has been made possible thanks to recent model development advances for the ICON models at the Max Planck Institute for Meteorology. A subsequent analysis of the output will improve the understanding of the phenomenon itself, in particular permitting to quantify the feedback mechanisms, but it will also clarify how significant of an influence the correct representation of DWLs has on the global cloud amount, and, as a consequence, on the climate described by the simulation. Such results would, moreover, enable a parametrisation of this phenomenon such that it can be included in lower resolution models in order to improve their performance.

Within the project L4 I work under supervision of Cathy Hohenegger at the MPI for Meteorology and collaborate mainly with Nils Brüggemann, Lars Umlauf and Mira Schmitt who already implemented a set of thin layer ocean simulations and contributed significantly to my understanding of the mechanisms involved.

Prior to joining the TRR 181 I spent several years doing research in Probability Theory. After obtaining my Master’s degree at the HU Berlin I went on to complete my PhD at the TU Dortmund with a research stay at the University of Lille. During my doctorate I studied stochastic (partial) differential equations driven by random processes or fields with long memory, i.e. such that the increment correlation decays only slowly over time. An example is the fractional Brownian motion. Using techniques from the Malliavin-Stein toolkit (providing a definition for multiple stochastic integrals with respect to Gaussian processes and many limiting results for those) I proved in several collaborations limit theorems for certain functionals of the solutions of such equations. From the practical point of view, this enabled me to derive results in mathematical statistics and provide estimators for different quantities in such equations as well as show their asymptotic properties.

After defending my dissertation I stayed at the TU Dortmund as a postdoctoral researcher. During this time I studied (in another collaboration) random fields on a sphere. Such objects are used in cosmology to describe cosmic microwave background, but they can also be applied to analyse other random spherical observations such as, for instance, temperature defects.

I am fascinated by such theoretical results but also by their various fields of application. I hope to be able to use some of the models that I studied in order to assess the impact of DWLs and/or to describe other phenomena in the atmosphere and ocean that would help advance the understanding and modelling of physical processes on different scales.

## The Impact of Submesoscales on the Air-Sea Exchange

I will investigate on the potential impact of submesoscale dynamics on the sea surface temperature or the influence of wind on instabilities at ocean fronts.

Hello everyone, my name is Moritz Epke and I am pleased to give you a small impression of my work at TRR. I am part of the subproject L4 „Energy consistent ocean atmosphere coupling”, which investigates small scale and balanced processes and their impact on feedback mechanism between atmosphere and ocean. Before I go into more detail, maybe a few words about my background. I moved to Hamburg to study theoretical mechanical engineering at the Hamburg University of Technology. My interest in the physics of fluids grew and grew through my studies and drove me to focus on this topic and related numerical solution approaches. In my thesis I developed and implemented a lattice Boltzmann scheme to efficiently simulate non-isothermal flows, which I benchmarked on standard testcases like Rayleigh-Bénard convection in a cavity and which I used to simulate the internal cooling of a turbine blade by a turbulent flow.

While most engineering applications have setups with scales from less than a centimeter as in a pipe flow, or up to a few hundred meters as in a large ship, the ocean and the atmosphere have scales that are orders of magnitude higher. Even if we make use of clever approximation techniques to simplify the governing equations in order to reduce the computational effort, we can only carry out coupled climate simulations with roughly tenkilometer (ocean) grid spacing on a modern supercomputer. In such a simulation an 80km ocean eddy would only be coarsely resolved. The computational surplus to resolve more scales in long-term simulations is simply too high. What cannot be resolved is usually parameterized or neglected. If parameterized, a model is developed which is based at best on a physical relationship between the relevant parameters. These parameterizations are then tested and optimized in idealized or regional setups. If now such parameterizations or insufficient parameterizations are used, the model is most likely subject to biases. These types of biases might have a strong impact on the energy consistency.

In the first phase of my PhD I am using an ICON submesoscale telescope simulation, which is based on an unstructured grid and allows us (for a short time period) to use an extremely fine spatial resolutions of up to 600m in the focus region. If we look again at an 80km ocean eddy, which is now well resolved, we can see small scale coherent structures that we associate with the submesoscale (see figure) and define to be smaller than the first baroclinic Rossby radius of deformation. It is an objective to understand and quantify the impact of submesoscale dynamics like baroclinic and symmetric instabilities on the downward heat and energy transfer and their role for ocean-atmosphere interactions. Here, I will investigate on the potential impact of submesoscale dynamics on the sea surface temperature or the influence of wind on instabilities at ocean fronts. Therewith, I aim to obtain a better understanding of submesoscale dynamics and their role in the coupled ocean-atmosphere system. This improved understanding might ultimately lead to improved parameterizations and therewith less biases in the coupled climate models.