# Successful online Mini-Symposium on “Energy Transfers in Atmosphere and Ocean”

The mini-symposium, initially scheduled to take place at the SIAM Conference on Mathematics of Planet Earth (MPE20), covering current trends in mathematical geophysical fluid dynamics, was cancelled due to the Corona crisis and therefore moved online. The mini-symposium aimed at reflecting relevant cutting-edge developments in the field and providing a forum for young researchers to communicate their work.

*Online mini-symposium on “Energy Transfers in Atmosphere and Ocean”, June 5+12, 2020*

This minisymposium covered current trends in mathematical geophysical fluid dynamics. Its title, "Energy Transfers in Atmosphere and Ocean", coincides with the name of a collaborative research center in Germany which aims at improving the energetic consistency of ocean and atmosphere models with the hope to further reduce biases and to increase the skill of those models. The research topics within the network at large range from observations, understanding and modeling of physical processes to fundamental questions about coarse-graining, parameterizations, and numerical methods. The minisymposium, in particular, addressed the theoretical end of this circle of questions. It aimed at reflecting relevant cutting-edge developments in the field and also provided a forum for young researchers to communicate their work.

**Abstracts**

**Manita Chouksey** (Universität Hamburg)

*Energy transfers between balanced and unbalanced motions in geophysical flows*

Geophysical flows such as the atmosphere and the ocean are characterized by rotation and stratification, which together give rise to two dominant motions: the slow balanced and the fast unbalanced motions. The interaction between the balanced and unbalanced motions and the energy transfers between them impact the energy and momentum cycle of the flow, and it is therefore crucial to understand the underlying energetics of the atmosphere and the ocean. The exact mechanism of wave generation, however, is not well understood and is hindered to an extent by the challenge of separating the flow field into balanced and unbalanced motions. We achieve this separation using two different balancing procedures in an identical model setup. The first procedure is a non-linear initialisation procedure based on Machenhauer (1977) but extended to higher orders in Rossby number. The second procedure is optimal potential vorticity balance. The results show that the numerics of the model affect the obtained balanced state from the two procedures, and thus the residual signal which we interpret as the unbalanced motions, i.e. internal gravity waves. Further, we assess the energy transfers between balanced and unbalanced motions in experiments. We find that it is crucial to consider the effect of the numerics in models and make a suitable choice of the balancing procedure, as well as diagnose the unbalanced motions at higher orders to precisely detect the internal gravity wave signal.

A recording of the talk can be found here: https://www.youtube.com/watch?v=hRQ6JABnLBc&feature=youtu.be

**Svetlana Dubinkina** (CWI Amsterdam)

*Projected shadowing-based data assimilation*

In numerical shadowing for data assimilation, instead of solving directly for the initial condition as in variational data assimilation, we solve for the whole trajectory at once. We employ (completed) observations as an initial guess and iteratively refine the trajectory to minimize an error. We show that for fully observed linear systems, the solution is unbiased with respect to the truth and the variance is consistent with least-square minimization. For linear systems, well-possedness of the method in algorithmic time limit is guaranteed. Moreover, we show that the shadowing-based DA method can be formulated in a weak constraint form with ensemble approximation.

**Federica Gugole** (CWI Amsterdam)

*Spatial covariance modeling for stochastic sub-grid scales parameterizations*

Numerical models are essential tools to study the underlying drivers of the climate, although many errors are currently present in realistic models due to the lack of knowledge of some physical mechanisms and to the numerical limitations. The atmosphere is a complex dynamical system including phenomena with vastly different temporal and spatial scales, hence the position of the numerical truncation strongly influences the model results. In particular, the energy of the unresolved scales can no longer be backscattered into the resolved scales, and there is a depletion of energy on the resolved modes. To reduce the error due to the numerical truncation, parameterizations representing the sub-grid scales are being introduced. In the framework of the 2-layer Quasi-Geostrophic model, a simplified model for the large scales atmospheric dynamics in the mid-latitudes, I will talk about the definition of the noise covariance for an energy consistent stochastic parameterization. In particular, I analyze the cases when either such a structure is not provided or, if otherwise, when it is defined using Empirical Orthogonal Functions or Dynamic Mode Decomposition. The outcomes revealed that a dynamically consistent structure is fundamental for the model to retain physically meaningful results, and that different techniques lead to different model behaviors. In particular, a dynamically adaptive covariance seems to be more suitable for long time simulations as in case of climate models.

**Florian Noethen** (Universit¨at Hamburg)

*Covariant Lyapunov vectors – linear modes for turbulent background flow*

Covariant Lyapunov vectors (CLVs) are intrinsic modes that describe the qualitative behavior of local dynamics near a given background state of a dynamical system. They constitute a basis of the tangent space that characterizes the long-term evolution of every possible perturbation. Contrary to simple eigenmode decompositions, CLVs can be applied to even chaotic states. This makes them a valuable tool for studying instabilities of the turbulent dynamics present in geophysical flows. In this talk I will discuss the mathematical concept behind CLVs, possible applications, and various algorithms to compute them. Special focus will be given to convergence properties.

**Jim Thomas** (University of North Carolina, Chapel Hill)

*Turbulent wave-balance exchanges in the ocean*

Oceanic mesoscales, ranging from 10–100 km horizontal scales, are constrained by the effects of rapid rotation and strong density stratification. Conventional wisdom in the past decades used to be that the turbulence phenomenology at these scales is set primarily by nonlinear interaction of mesoscale eddies, which are approximately in geostrophic and hydrostatic balance. However, satellite altimeter datasets, in situ measurements, and realistically forced global scale oceanic model outputs in recent times point out that oceanic mesoscales are rich with high energy internal gravity waves; consisting of wind generated high baroclinic near-inertial waves and gravitationally generated low baroclinic tides. Not only do these waves have spatial scales comparable to balanced mesoscale eddy field, in multiple parts of the world’s oceans the wave energy levels are seen to significantly exceed balanced energy. In this talk I will describe how interactions between waves and balanced flow shapes geophysical turbulence phenomenology in the ocean. The talk will distinguish and elaborate on how wind generated near-inertial waves, gravitationally generated low baroclinic tides, and a broad-band spectrum of internal gravity waves interact with geostrophically and hydrostatically balanced flow. The focus and goal of this research direction is to develop a comprehensive understanding of oceanic wave-balance energetic interactions and the resulting turbulence phenomenology. An improved understanding of the flow dynamics will assist in developing better parameterization schemes for large scale general circulation models that are still far from capturing intricate fast and small scale oceanic flow dynamics.

A recording of the talk can be found here: https://www.youtube.com/watch?v=-46eWo1oijo&feature=youtu.be

**Golo Wimmer** (Imperial College)

*An energy conserving, upwinded compatible finite element discretisation for the dry compressible Euler equations*

An important aspect of discretisations in numerical weather prediction, particularly for climate simulations, is conservation of quantities such as mass and energy. Conservation of energy requires a careful discretisation of all prognostic equations, ensuring that the energy losses and gains are balanced between the resulting discretised terms. One way to guide this process is to consider the equations in a Hamiltonian framework, where the Hamiltonian is given by the system’s total energy, and the equations are inferred by a Poisson bracket. Conservation of energy then follows easily by the bracket’s antisymmetry, and any space discretisation maintaining this property will then also conserve energy. In this presentation, the Hamiltonian framework is applied to the compatible finite element method, which is the underlying space discretisation for GungHo, the UK Met Office’s next generation dynamical core. We first review the corresponding discretised bracket for the dry compressible Euler equations, and then introduce an extension to include upwind stabilised advection schemes for the different finite element spaces used in this approach, while still retaining energy conservation via the Hamiltonian setup. Finally, we present numerical results, confirming energy conservation as well as an improved field development due to upwinding.