# L2: The interior energy pathway: internal wave emission by quasi-balanced flows

Principal investigators: Prof. Jin-Song von Storch (Max Planck Institute for Meteorology), Prof. Gualtiero Badin (Universität Hamburg), Prof. Marcel Oliver (Jacobs University)

## Objectives

Both the large-scale currents and the meso-scale eddies in the ocean are essentially balanced, with geostrophic or gradient-wind balances in the horizontal and hydrostatic balance in the vertical. Moreover, meso-scale eddies tend to transfer energy upward towards larger scales. This situation points to the challenging question of what are the ocean's routes to dissipation that are needed for the interior general circulation, including its related eddy field, to reach an equilibrium state in the presence of the constant atmospheric forcing. This subproject contributes to answering this question by exploring the route to dissipation via spontaneous wave generation: Eddying flows spontaneously emit internal waves. The waves, once generated, are refracted by the eddying flow, and may be captured later. Wave capture is characterised by an exponential increase in wavenumber and wave amplitude and an exponential decrease in the intrinsic group velocity. While capture is well understood theoretically for various types of flows, it is still open whether it occurs in the real ocean. By spontaneous emission and subsequent interaction with the emitted waves, energy is transferred from meso-scale eddies to smaller scales where instabilities and smallscale turbulence complete the downscale cascade to dissipation.

Quantifying this route to dissipation is central for the present CRC. Specically, this subproject aims to address the following questions:

- How important is the route to dissipation via spontaneous wave generation and wave capture?
- What are the key factors that control the internal wave emission by quasi-balanced and

eddying flows and the subsequent interaction between waves and flows? - Is it possible to formally characterise the internal waves emitted by a turbulent geostrophic flow?

We will address these questions using a combination of theory, conceptual models with idealised configurations based on rotating shallow water equations, and existing and specially designed numerical experiments with an ocean general circulation model based on the primitive equations.

## Invited Guests

## Reports

## Balance-imbalance decomposition of the flow field

We are currently working on application of the optimal balance algortihm to the shallow water model and the primitve equations will follow.

I am a PhD student in the subproject L2 at Jacobs University Bremen under supervision of Prof. Marcel Oliver.

Our role in the subproject can be briefly explained as follows: In large-scale ocean models, the ocean circulations are essentially balanced; however, this balance breaks down in small scales due to spontaneous generation of inertia-gravity waves by quasi-balanced circulations, and waves are maybe re-captured in later times. This spontaneous emission and wave capture is considered to contribute to the energy transfer from the essentially balanced large-scale circulation and mesoscale eddy fields down to smaller scales, which is a route to dissipation.

To analyse the role of inertia-gravity waves in interior dissipation, a reasonable approach is to diagnose the inertia-gravity waves by splitting the flow field into balance and imbalance components, which are the ocean circulation and the inertia-gravity waves, respectively. This balance-imbalance decomposition can be achieved by some diagnostic tools such as linear time filters, balance relations, and optimal potential vorticity balance. In this project, we want to provide a new numerical algorithm to separate spontaneously generated imbalanced flows from the vertical flows depending on a prior work called „optimal balance“.

We are currently working on application of the optimal balance algorithm to the shallow water model and the primitive equations will follow. The “optimal balance” algorithm is interesting to us not only for practical aspects but also mathematical features, so that we extensively worked on asymptotics-preserving schemes on a finite dimensional model in the algorithm. There are several other theoretically open questions, which are standing for the algorithm as its existence and uniqueness, to be considered.

## Publications

Conti, G. and

**G. Badin**(2019). Velocity statistics for point vortices in the local α-models of turbulence,*Geophysical and Astrophysical Fluid Dynamics*, doi:10.1080/03091929.2019.1572750Mohamad, H., and

**M. Oliver**(2019). A direct construction of a slow manifold for a semilinear wave equation of Klein–Gordon type.*Journal of Differential Equations*.**Badin, G., M. Oliver and S. Vasylkevych**(2018). Geometric Lagrangian averaged Euler-Boussinesq and Primitive Equations,*Journal of Physics A: Mathematical and Theoretical*, 51, 455501, doi:10.1088/1751-8121/aae1cb**Dräger-Dietel, J.**, Jochumsen, K.,**Griesel, A., and Badin, G.**(2018). Relative dispersion of surface drifters in the Benguela upwelling region.*Journal of Physical Oceanography*.**Franzke, C. L., Oliver, M., Rademacher, J. D.,**and**Badin, G.**(2018). Multi-scale methods for geophysical flows, Springer*(in press)*.**Badin,****G.**and Barry, A., (2018) Collapse of generalized Euler and surface quasi-geostrophic point-vortices, Physical Review E, 98, 023110, doi.org/10.1103/PhysRevE.98.023110Domeisen, D.I.V. ,

**Badin,****G.**and Koszalka, I., (2018) How predictable are the Arctic and North Atlantic Oscillations? Exploring the variability and predictability of the Northern Hemisphere, Journal of Climate, 31, 997-1014, doi.org/10.1175/JCLI-D-17-0226.1Conti, G., &

**Badin, G.**(2017). Hyperbolic Covariant Coherent Structures in Two Dimensional Flows.*Fluids*2017,*2*(4), 50. doi: 10.3390/fluids2040050**Blender, R., & Badin, G.**(2017).Construction of Hamiltonian and Nambu Forms for the Shallow Water Equations.*Fluids 2017*,*2 (2), doi:10.3390/fluids2020024*.**Blender, R.**, &**Badin, G.**(2017). Viscous dissipation in 2D fluid dynamics as a symplectic process and its metriplectic representation.*The European Physical Journal Plus*,*132*(3), 137.