Generalized Lagrangian mean
| Continuum mechanics | |||||||
|---|---|---|---|---|---|---|---|
Laws
| |||||||
Solid mechanics
| |||||||
Fluid mechanics
| |||||||
Rheology
| |||||||
Scientists
| |||||||
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]
Contents
1 Background
2 Notes
3 References
3.1 By Andrews & McIntyre
3.2 By others
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
^ Craik (1988)
^ Andrews & McIntyre (1978b)
^ McIntyre (1981)
^ Holm (2002)
References
By Andrews & McIntyre
.mw-parser-output .refbegin{font-size:90%;margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{list-style-type:none;margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li,.mw-parser-output .refbegin-hanging-indents>dl>dd{margin-left:0;padding-left:3.2em;text-indent:-3.2em;list-style:none}.mw-parser-output .refbegin-100{font-size:100%}
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..mw-parser-output cite.citation{font-style:inherit}.mw-parser-output .citation q{quotes:"""""""'""'"}.mw-parser-output .citation .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/12px-Wikisource-logo.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-maint{display:none;color:#33aa33;margin-left:0.3em}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}
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.
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.
