Abstract
The generalized Lagrangian-mean (GLM) description formulated by Andrews and McIntyre is extended to the general coordinate system. Four-dimensional Lagrangian coordinates are introduced to obtain a general relationship between the Lagrangian coordinate mean (LCM) and the GLM. It is shown that the choice of an initial hypersurface in space-time is essential in the determination of the relationship. The Eulerian mean and the GLM tensors are defined referring to a given coordinate system, so that mean quantities are dependent on the choice of the coordinate system. Symmetries in the Lagrangian density for fluids and related conservation laws are discussed for energy-momentum, pseudoenergy-pseudomomentum and wave-action. The extended GLM description provides a wider applicability in practice, owing to the less stringent assumptions for the initial conditions.
