Computing Lagrangian means in multi-timescale fluid flows

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Time averaging is one of the most essential tools in analysing fluid flows with multiple time scales, which are ubiquitous in nature and industries. A prominent use of time averaging is the flow decomposition into fast and slow parts to understand different phenomena associated with each time scale. For instance, in geophysical flows the wave dynamics is associated with the fast part, and the slow dynamics can be reduced to a balance between a few forces (geostrophic and hydrostatic balance). Another application of time averaging is to filter out the fast variations that are not fully captured in numerical simulations or observations to make meaningful inferences from the remaining slow dynamics. Similarly, the noise and error inherent in measurements or simulations are removed by averaging. For fluids, time-averaging can be performed in two different ways. The most straightforward approach is to average time series of flow variables at fixed spatial points, to obtain the so-called Eulerian mean. Another approach is to average flow variables along particle trajectories instead of fixed positions, which gives the Lagrangian mean. Lagrangian averaging has several pivotal advantages over its Eulerian counterpart as illustrated by a growing number of studies. For instance, it removes the Doppler shift that eclipses the separation of time scales between the background flow and waves. However, 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. This is a computational challenge that requires extra memory space (considering time series of variables are stored at all grid points) and is ill-suited for efficient computational parallelisation. In this project, we develop a numerical approach that circumvents these difficulties. This new approach is based on the evolution of partial means instead of particle tracking. Partial means can be viewed as means over shorter intervals than the total averaging period. In this approach, we compute the Lagrangian means as solutions to a set of Partial Differential Equations (PDEs) that describe the evolution of these partial means. 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. The basic idea of this numerical technique is put forward by the PI. The goals of this proposal are to 1) expand the theoretical underpinnings of this novel method, 2) develop and implement a set of numerical schemes to solve the associated PDEs, and 3) apply them to 3D fluid models and laboratory data.