“Although discussion of the Lagrangian aspects of this problem has been brief here, a large body of recent literature emphasizes the direct derivation of Lagrangian mean flow (e.g., Andrews and McIntyre, 1978, Bretherton, 1976). The key difficulty which prevents direct application of this work to turbulent flows is that it assumes particle trajectories to be ‘wave-like’; that is, particles are referenced to their mean trajectory, and their displacements from it are strictly bounded. There is a well-defined averaging time (a wave period). Turbulent flows, by contrast, cause ensembles of particles to disperse widely. The Lagrangian mean flow changes continually with time-since-release of the ensemble.”

- Andrews and McIntyre (1976): An exact theory on nonlinear waves on a Lagrangian-mean flow,
*J. Fluid Mech.*,**89**, 609–646. - Bretherton (1976): Conservation of wave action and angular momentum in a spherical atmosphere, unpublished manuscript.
- Rhines and Holland (1979): A theoretical discussion of eddy-driven mean flows,
*Dyn. Atm. Oceans*,**3**, 289–325.

Eric:

Interesting point that needs more thought. Does ensemble-averaging (that is, considering a large ensemble of realizations) overcome the objection? It is not clear to me yet.

Also, my sense is that even when the mathematical formalism is not strictly correct, it may be pointing in the right direction. This is not good enough, but it is better than nothing.

Eric

Francois:

Re-reading Andrews and McIntyre (1978), the authors do mention that the Eulerian-mean can be an ensemble average. So, we would average over a period P, whatever P, and calculate the Lagrangian-mean flow over a series of period of length P and take the average of it? Or something else?

Eric:

Time-averaging would not be involved at all. It is a matter of doing a thought experiment: imagine a very large number of realizations (e.g., numerical runs) with identical deterministic characteristics (large-scale forcing fields and initializations) and with independent random stochastic aspects (e.g., initialized with different small random noise fields superimposed). All Eulerian averages are ensemble averages, as functions of x and t.

Eric