Lagrangian averaging theories, most notably the Generalised Lagrangian Mean (GLM) theory of Andrews & McIntyre (1978a), have
been primarily developed in Euclidean space and Cartesian coordinates.
We re-interpret these theories using a geometric, coordinate-free formulation. This gives central roles to the flow map, its decomposition ...
Lagrangian averaging theories, most notably the Generalised Lagrangian Mean (GLM) theory of Andrews & McIntyre (1978a), have
been primarily developed in Euclidean space and Cartesian coordinates.
We re-interpret these theories using a geometric, coordinate-free formulation. This gives central roles to the flow map, its decomposition into
mean and perturbation maps, and the momentum 1-form dual to the
velocity vector. In this interpretation, the Lagrangian mean of any
tensorial quantity is obtained by averaging its pull back to the mean
configuration. Crucially, the mean velocity is not a Lagrangian mean
in this sense. It can be defined in a variety of ways, leading to alternative Lagrangian mean formulations that include GLM and Soward
& Roberts’s (2010) glm. These formulations share key features which
the geometric approach uncovers. We derive governing equations both
for the mean flow and for wave activities constraining the dynamics of
the pertubations. The presentation focusses on the Boussinesq model
for inviscid rotating stratified flows and reviews the necessary tools of
differential geometry.
