## Computing Lagrangian Means Without Particle Tracking

**Eulerian**and 2)

**Lagrangian**. The Eulerian mean is the average of flow variables at fixed spatial points, whereas the Lagrangian mean is the average of flow variables along particle trajectories. Lagrangian averaging has several pivotal advantages over its Eulerian counterpart. One of these advantages is demonstrated in the figure above, where a high-amplitude wave is interacting with turbulent shallow water. The strong wave – present in the instantaneous vorticity field (panel (a)) – is filtered out after both types of averaging. However, Lagrangian averaging (panel (b)) has preserved the small-scale features of the flow that are blurred by Eulerian averaging (panel (c)). Lagrangian averaging also removes the Doppler shift that eclipses the separation of time scales between the background flow and waves leading to a more reliable decomposition of the two. Despite its advantages, the widespread adoption of Lagrangian averaging has been hindered by computational complications. To compute the Lagrangian mean in numerical models usually particles are tracked using interpolated velocities at particle positions at every time steps, which requires substantial memory usage and is ill suited for efficient computational parallelisation.

We propose a novel numerical method to compute Lagrangian means that does not require tracking particles and consquently circumvents these difficulties. In this approach, we compute the Lagrangian mean as solutions to a set of Partial Differential Equations (PDEs) integrated over the averaging interval. This paradigm could be a breakthrough in computing Lagrangian means as these PDEs can be discretised in a variety of ways and solved on-the-fly (i.e. simultaneously with the dynamical governing equations). Hence, they do not require storing any time series and substantially reduce the memory footprint compared to particle tracking.

For more details see the following papers:

1. Kafiabad. “Grid-based calculation of the lagrangian mean.” *Journal of Fluid Mechanics*, 940, 2022.

2. Kafiabad and Vanneste. “Computing Lagrangian Mans.” *Under revision for Journal of Fluid Mechanics*, 2022.

Early version of this work is presented in this talk for the University of Hamburg.

## Scattering of Inertia-Gravity Waves by Geostrophic Turbulence

*Two horizontal slices of 3D Boussinesq simulation initiated by geostrophic turbulence superimposed with a monochromatic plane wave. Left: total vertical vorticity, right: wave's vertical velocity. *

For more details see the following papers:

1. Kafiabad, Savva, and Vanneste. “Diffusion of inertia-gravity waves by geostrophic turbulence.” *Journal of Fluid Mechanics*, 869:R7, 2019.

2. Cox, Kafiabad, and Vanneste. “Inertia-gravity-wave diffusion by geostrophic turbulence: the impact of flow time dependence.” *Under revision for the Journal of Fluid Mechanics*, 2022.

3. Savva, Kafiabad, and Vanneste. “Inertia-gravity-wave scattering by three-dimensional geostrophic turbulence.” *Journal of Fluid Mechanics*, 916:A6, 2021.

Considering it a breakthrough, the Journal of Fluid Mechanics selected our work for the Focus on Fluid article, where an invited expert wrote the following review about our findings:

Young. “Inertia-gravity waves and geostrophic turbulence.” *Journal of Fluid Mechanics*, 920, 2021.

## Efficient Computation of Wave Transport and Scattering

## Interaction of Near-Inertial Waves with Vortices

*Horizontal slices of a 3D Boussinesq simulation initialised by a strong inertial wave and an anticyclone. Left: the wave kinetic energy. Right: total vertical vorticity. *

For more details see the following papers:

1. Kafiabad, Vanneste and Young. “Interaction of near-inertial waves with an anticyclonic vortex.” *Journal of Physical Oceanography*, 05 Apr. 2021.

2. Kafiabad, Vanneste and Young. “Wave-averaged balance: a simple example.” *Journal of Fluid Mechanics*, 911:R1, 2021.

## Breakdown of Balance Dynamics from a Turbulence Perspective

*Vertical vorticity at a horizontal and a vertical slice of 3D Boussinesq simulation.Only a part of domain is shown. *

For more details see the following papers:

1. Kafiabad and Bartello. “Spontaneous imbalance in the non-hydrostatic boussinesq equations.” *Journal of Fluid Mechanics*, 847:614-643, 2018.

2. Kafiabad and Bartello. “Rotating stratified turbulence and the slow manifold.” *Computers & Fluids*, 151:23-34, 2017.

3. Kafiabad and Bartello. “Balance dynamics in rotating stratified turbulence. *Journal of Fluid Mechanics*, 795:914-949, 2016.

A part of this project is presented in this talk for the Banff International Research Station.

## Lagrangian Coherent Structure for Detection of Airflow Distrubances over Airports

For more details see:

Kafiabad , Chan, and Haller. “Lagrangian detection of wind shear for landing aircraft.” *Journal of Atmospheric and Oceanic Technology*, 30(12):2808-2819, 2013.