Generalized Lagrangian mean

In continuum mechanics, the generalized Lagrangian mean (GLM) is a formalism – developed by D.G. Andrews and M.E. McIntyre (1978a, 1978b) – to unambiguously split a motion into a mean part and an oscillatory part. The method gives a mixed Eulerian–Lagrangian description for the flow field, but appointed to fixed Eulerian coordinates.^[1]

Background

In general, it is difficult to decompose a combined wave–mean motion into a mean and a wave part, especially for flows bounded by a wavy surface: e.g. in the presence of surface gravity waves or near another undulating bounding surface (like atmospheric flow over mountainous or hilly terrain). However, this splitting of the motion in a wave and mean part is often demanded in mathematical models, when the main interest is in the mean motion – slowly varying at scales much larger than those of the individual undulations. From a series of postulates, Andrews & McIntyre (1978a) arrive at the (GLM) formalism to split the flow: into a generalised Lagrangian mean flow and an oscillatory-flow part.

The GLM method does not suffer from the strong drawback of the Lagrangian specification of the flow field – following individual fluid parcels – that Lagrangian positions which are initially close gradually drift far apart. In the Lagrangian frame of reference, it therefore becomes often difficult to attribute Lagrangian-mean values to some location in space.

The specification of mean properties for the oscillatory part of the flow, like: Stokes drift, wave action, pseudomomentum and pseudoenergy – and the associated conservation laws – arise naturally when using the GLM method.^[2][3]

The GLM concept can also be incorporated into variational principles of fluid flow.^[4]

Notes

1. Craik (1988)
2. Andrews & McIntyre (1978b)
3. McIntyre (1981)
4. Holm (2002)

References

By Andrews & McIntyre

• Andrews, D. G.; McIntyre, M. E. (1978a), "An exact theory of nonlinear waves on a Lagrangian-mean flow" (PDF), Journal of Fluid Mechanics, 89 (4): 609–646, Bibcode:1978JFM....89..609A, doi:10.1017/S0022112078002773.
• Andrews, D. G.; McIntyre, M. E. (1978b), "On wave-action and its relatives" (PDF), Journal of Fluid Mechanics, 89 (4): 647–664, Bibcode:1978JFM....89..647A, doi:10.1017/S0022112078002785.
• McIntyre, M. E. (1980), "An introduction to the generalized Lagrangian-mean description of wave, mean-flow interaction", Pure and Applied Geophysics, 118 (1): 152–176, Bibcode:1980PApGe.118..152M, doi:10.1007/BF01586449, S2CID 122690944.
• McIntyre, M. E. (1981), "On the 'wave momentum' myth" (PDF), Journal of Fluid Mechanics, 106: 331–347, Bibcode:1981JFM...106..331M, doi:10.1017/S0022112081001626.

By others

• Bühler, O. (2014), Waves and mean flows (2nd ed.), Cambridge University Press, ISBN 978-1-107-66966-6
• Craik, A. D. D. (1988), Wave interactions and fluid flows, Cambridge University Press, ISBN 9780521368292. See Chapter 12: "Generalized Lagrangian mean (GLM) formulation", pp. 105–113.
• Grimshaw, R. (1984), "Wave action and wave–mean flow interaction, with application to stratified shear flows", Annual Review of Fluid Mechanics, 16: 11–44, Bibcode:1984AnRFM..16...11G, doi:10.1146/annurev.fl.16.010184.000303
• Holm, Darryl D. (2002), "Lagrangian averages, averaged Lagrangians, and the mean effects of fluctuations in fluid dynamics", Chaos, 12 (2): 518–530, Bibcode:2002Chaos..12..518H, doi:10.1063/1.1460941, PMID 12779582.