Generalized Lagrangian mean
{{Continuum mechanics| cTopic=Fluid mechanics}}
In continuum mechanics, the generalized Lagrangian mean (GLM) is a formalism – developed by {{harvs|txt=yes|last1=Andrews|first1=D.G.|last2=McIntyre|year=1978a|first2=M.E.|author2-link=Michael E. McIntyre|year2=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.{{harvtxt|Craik|1988}}
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, {{harvtxt|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.{{harvtxt|Andrews|McIntyre|1978b}}{{harvtxt|McIntyre|1981}}
The GLM concept can also be incorporated into variational principles of fluid flow.{{harvtxt|Holm|2002}}
Notes
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References
=By Andrews & McIntyre=
{{refbegin}}
- {{Citation
| doi = 10.1017/S0022112078002773
| volume = 89
| issue = 4
| pages = 609–646
| last1 = Andrews
| first1 = D. G.
| last2 = McIntyre
| first2 = M. E.
| author2-link=Michael E. McIntyre
| title = An exact theory of nonlinear waves on a Lagrangian-mean flow
| journal = Journal of Fluid Mechanics
| year = 1978a
| url = http://www.atm.damtp.cam.ac.uk/people/mem/andrews-mcintyre-glm-jfm78.pdf
| postscript = .
|bibcode = 1978JFM....89..609A }}
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| volume = 89
| issue = 4
| pages = 647–664
| last1 = Andrews
| first1 = D. G.
| last2 = McIntyre
| first2 = M. E.
| title = On wave-action and its relatives
| journal = Journal of Fluid Mechanics
| year = 1978b
| url = http://www.atm.damtp.cam.ac.uk/people/mem/andrews-mcintyre-waveac-jfm78.pdf
| postscript = .
|bibcode = 1978JFM....89..647A }}
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| volume = 118
| issue = 1
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| last = McIntyre
| first = M. E.
| title = An introduction to the generalized Lagrangian-mean description of wave, mean-flow interaction
| journal = Pure and Applied Geophysics
| year = 1980
| postscript = .
|bibcode = 1980PApGe.118..152M | s2cid = 122690944
}}
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| pages = 331–347
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| journal = Journal of Fluid Mechanics
| year = 1981
| url = http://www.atm.damtp.cam.ac.uk/people/mem/papers/RECOIL/wave-momentum-myth-scanned.pdf
| postscript = .
|bibcode = 1981JFM...106..331M }}
{{refend}}
=By others=
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| title=Waves and mean flows
| publisher=Cambridge University Press
| edition=2nd
| year=2014
| isbn=978-1-107-66966-6
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- {{Citation
| publisher = Cambridge University Press
| isbn = 9780521368292
| last = Craik
| first = A. D. D.
| title = Wave interactions and fluid flows
| year = 1988
| postscript = .
}} See Chapter 12: "Generalized Lagrangian mean (GLM) formulation", pp. 105–113.
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| doi = 10.1146/annurev.fl.16.010184.000303
| volume = 16
| pages = 11–44
| last = Grimshaw
| first = R.
| title = Wave action and wave–mean flow interaction, with application to stratified shear flows
| journal = Annual Review of Fluid Mechanics
| year = 1984
|bibcode = 1984AnRFM..16...11G }}
- {{Citation
| doi = 10.1063/1.1460941
| volume = 12
| issue = 2
| pages = 518–530
| last = Holm
| first = Darryl D.
| author-link = Darryl Holm
| title = Lagrangian averages, averaged Lagrangians, and the mean effects of fluctuations in fluid dynamics
| journal = Chaos
| year = 2002
| postscript = .
| pmid = 12779582
|bibcode = 2002Chaos..12..518H }}
{{refend}}