Abstract

Lagrangian averaging plays an important role in the analysis of wave—mean-flow interactions and other multiscale fluid phenomena. Comparing to its Eulerian counterpart Lagrangian averaging has several superiorities. For instance, it removes the doppler shift of wave frequency by strong background flow, which eclipses the separation of time scale between them. Another advantage is that the Lagrangian mean fields usually inherit the material conservation laws (such as the conservation of PV, circulation, and magnetic field) that hold for instantaneous fields. The numerical computation of Lagrangian means, e.g. from simulation data, is however challenging. Typical implementations require tracking a large number of particles to construct Lagrangian time series which are then averaged. This has drawbacks that include large memory demands, particle clustering and complications of parallelisation. We develop a novel approach in which the Lagrangian means of various fields (including particle positions) are computed without tracking particles in time. This approach leads to a set of PDEs that is integrated over successive averaging time intervals. The PDEs can be discretised in a variety of ways, e.g. using the same discretisation as that employed for the governing dynamical equations, and solved on-the-fly to minimise the memory footprint.