Generalised Lagrangian mean: formulation and computation

Many fluid dynamical phenomena are the result of interactions between small-scale or high-frequency fluctuations – associated with waves or turbulence – and mean flows. Modelling these requires some form of averaging to obtain coarse-grained models in which the effect of fluctuations is parameterised. In this context, Lagrangian averaging, whereby averages are computed along fluid trajectories, has advantages over the more straightforward Eulerian averaging, mainly because it preserves the advective structure of the equations of motion. The theory of generalised Lagrangian mean (GLM) formulated by Andrews & McIntyre provides the basis for the derivation of convenient Lagrangian-averaged models. In this talk, I will introduce this theory and show how a geometric perspective helps its interpretation and generalisation. I will then discuss recently introduced numerical methods for the computation of (temporal) Lagrangian averages from numerical simulation data. These methods avoid the explicit tracking of particles, replacing it by the solution of partial different equations that can be carried out on-the-fly, in tandem with the simulation.