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Lagrangian Averages, Averaged Lagrangians, and the Mean Effects of fluctuations in fluid Dynamics

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Lagrangian Averages, Averaged Lagrangians, and the Mean Effects of fluctuations in fluid Dynamics See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/10731045 Lagrangian averages, averaged Lagrangians, and the mean effects of fluctuations in fluid dynamics Article in Chaos (Woodbury, N.Y.) · July 2002 DOI: 10.1063/1.1460941 · Source: PubMed CITATIONS READS 73 517 1 author: Darryl Holm Imperial College London 463 PUBLICATIONS 19,719 CITATIONS SEE PROFILE Some of the authors of this publication are also working on these related projects: Three Dimensional Hydrodynamics Instability View project Stochastic Geometric Mechanics View project All content following this page was uploaded by Darryl Holm on 05 September 2014. The user has requested enhancement of the downloaded file. Lagrangian averages, averaged Lagrangians, and the mean effects of fluctuations in fluid dynamics Darryl D. Holm Citation: Chaos 12, 518 (2002); doi: 10.1063/1.1460941 View online: http://dx.doi.org/10.1063/1.1460941 View Table of Contents: http://chaos.aip.org/resource/1/CHAOEH/v12/i2 Published by the AIP Publishing LLC. Additional information on Chaos Journal Homepage: http://chaos.aip.org/ Journal Information: http://chaos.aip.org/about/about_the_journal Top downloads: http://chaos.aip.org/features/most_downloaded Information for Authors: http://chaos.aip.org/authors Downloaded 30 Sep 2013 to 202.116.1.149. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://chaos.aip.org/about/rights_and_permissions CHAOS VOLUME 12, NUMBER 2 JUNE 2002 Lagrangian averages, averaged Lagrangians, and the mean effects of fluctuations in fluid dynamics Darryl D. Holma) Theoretical Division and Center for Nonlinear Studies, Los Alamos National Laboratory, MS B284, Los Alamos, New Mexico 87545 ͑Received 29 October 2001; accepted 23 January 2002; published 20 May 2002͒ We begin by placing the generalized Lagrangian mean ͑GLM͒ equations for a compressible adiabatic fluid into the Euler–Poincare´ ͑EP͒ variational framework of fluid dynamics, for an averaged Lagrangian. This is the Lagrangian averaged Euler–Poincare´ ͑LAEP͒ theorem. Next, we derive a set of approximate small amplitude GLM equations ͑glm equations͒ at second order in the fluctuating displacement of a Lagrangian trajectory from its mean position. These equations express the linear and nonlinear back-reaction effects on the Eulerian mean fluid quantities by the fluctuating displacements of the Lagrangian trajectories in terms of their Eulerian second moments. The derivation of the glm equations uses the linearized relations between Eulerian and Lagrangian fluctuations, in the tradition of Lagrangian stability analysis for fluids. The glm derivation also uses the method of averaged Lagrangians, in the tradition of wave, mean flow interaction. Next, the new glm EP motion equations for incompressible ideal fluids are compared with the Euler-alpha turbulence closure equations. An alpha model is a GLM ͑or glm͒ fluid theory with a Taylor hypothesis closure. Such closures are based on the linearized fluctuation relations that determine the dynamics of the Lagrangian statistical quantities in the Euler-alpha equations. Thus, by using the LAEP theorem, we bridge between the GLM equations and the Euler-alpha closure equations, through the small-amplitude glm approximation in the EP variational framework. We conclude by highlighting a new application of the GLM, glm, and ␣-model results for Lagrangian averaged ideal magnetohydrodynamics. © 2002 American Institute of Physics. ͓DOI: 10.1063/1.1460941͔ This paper employs the Lagrangian-averaged Euler– in fluid mechanics that arise from Hamilton’s variational Poincare´ theorem to provide a bridge between the gener- principle, these conditions involve the relation between av- alized Lagrangian mean „GLM… equations and the Euler- eraged quantities in the Eulerian and Lagrangian descriptions alpha closure equations. The former „GLM… equations of fluid dynamics. An important advance was the develop- result from an exact theory of nonlinear waves on a La- ment of the generalized Lagrangian mean ͑GLM͒ equations grangian mean flow and are not closed, while the latter by Andrews and McIntyre,1 who introduced a slowϩfast de- „Euler-alpha… equations are closed via linearization that composition of the Lagrangian particle trajectory in general introduces a length scale ␣, which Taylor’s hypothesis form as renders constant if it is initially so. The Euler-alpha clo- ␰͑ ͒ϵ ͑ ͒ϩ␰͑ ͑ ͒ ͒ϭ⌶͑ ͑ ͒ ͒ ͑ ͒ sure equations with an additional Navier–Stokes viscous x x0 ,t x x0 ,t x x0 ,t ,t x x0 ,t ,t , 1.1 dissipation are of interest from the point of view of large where x is the Lagrangian mean trajectory of the particle eddy simulation and the object of the present work is to ␰ labeled by x0 , the fluctuation is and one may replace the show how they relate to a more general theory. To clarify summation in Eq. ͑1.1͒ by the more convenient composition this, we derive them from a small amplitude approxima- of functions ⌶ also introduced in Ref. 1. Andrews and McIn- tion to the GLM equations, the glm equations, by apply- tyre found that the Lagrangian mean of a fluid quantity ing Taylor’s hypothesis of frozen-in fluctuations. This evaluated at the mean particle position x is related to its derivation from first principles is general enough to in- Eulerian mean, evaluated at the displaced fluctuating posi- clude more physics, as we shall illustrate by deriving an tion x␰ by relation ͑2.7͒ below. This relation allows the GLM alpha model for Lagrangian-averaged ideal magnetohy- equations to be expressed directly in the Eulerian represen- drodynamics. tation. The Lagrangian-averaged Euler–Poincare´ ͑LAEP͒ theo- rem discussed here puts the averaged-Lagrangian approach I. INTRODUCTION and the method of GLM-averaged equations onto equal foot- The classical method of averages for dynamical systems ing. This is quite an extension for both approaches to mod- may be extended to the averaged Lagrangian method, but eling fluids. According to this extension, the averaged- only under certain conditions. For the nondissipative systems Lagrangian theory produces dynamics that can be verified directly by averaging the original equations. Likewise, the a͒Electronic mail: [email protected] GLM-averaged equations inherit the conservation laws and 1054-1500/2002/12(2)/518/13/$19.00518 © 2002 American Institute of Physics Downloaded 30 Sep 2013 to 202.116.1.149. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://chaos.aip.org/about/rights_and_permissions Chaos, Vol. 12, No. 2, 2002 Fluctuations in fluid dynamics 519 Kelvin circulation transport properties that arise from the symmetries of the Lagrangian. Of course, this extension and these variational relationships are not possible with the Eu- lerian mean, because the Eulerian mean does not preserve the circulation transport properties of fluid mechanics. The resulting GLM equations are nonlinear and exact. However, they are not closed. The second part of the paper introduces the alpha model closure scheme in this variational framework for GLM. An alpha model is a closure of the GLM equations that is based on applying ͑1͒ Taylor’s fa- mous hypothesis of ‘‘frozen-in’’ turbulence to the Andrews FIG. 1. GLM theory factorizes the Lagrange to Euler map at a given time by and McIntyre slowϩfast decomposition and ͑2͒ Lagrangian first mapping the reference configuration to the mean position, then mapping the mean position to the current position. averaging in Hamilton’s principle. The alpha-model equa- tions then result directly in the Eulerian representation as a 9 closure for GLM from the variational principle given by Chen et al. ͑Physically, alpha is the smallest active length Lagrangian-averaged Euler–Poincare´ theorem. scale participating in the nonlinear interactions—so scales smaller than alpha are swept along by the larger ones.͒ This A. A brief history of the Euler-alpha models is in contrast to the theory of second grade fluids, where The Euler-alpha equations for averaged incompressible alpha is a thermodynamic material parameter. See Foias ideal fluid motion were first derived in Holm, Marsden, and et al.,10,11 and references therein for additional discussions of Ratiu2,3 in the context of the Euler–Poincare´ theory for fluid the properties of alpha models. dynamics. ͑For an introduction to Euler–Poincare´ theory for classical mechanics on Lie groups, see Ref. 4.͒ That deriva- B. Outline of the paper tion proceeded essentially by choosing the kinetic energy to Section II lays out the notation and basic relations in the be the H1 norm of the Eulerian fluid velocity, rather than the GLM theory that we shall need to establish and apply the usual L2 norm. This choice generalized the unidirectional LAEP theorem in the sections that follow. Section III states 5 shallow water equation of Camassa and Holm from one and proves the Lagrangian averaged Euler–Poincare´ theorem dimension to three dimensions. The resulting n-dimensional ͑LAEP͒ theorem. Section IV explains how Lagrangian con- ⌬ϭ ϵ Ϫ␣2 ٌ ͑ Euler-alpha equation is with •u 0, v u u and con- siderations lead to geometric interpretations of the linearized ␣͒ stant length scale relations for spatial trajectory fluctuations in terms of the -displacement fluctuation, ␰. Section V discusses the linear ץ vϩu ٌvϩv ٌu jϩٌpϭ0. ͑1.2͒ ized glm approximation of the exact ͑but not closed͒ GLM t • jץ theory. Section VI then discusses modeling options for pro- This is the EP equation for the Lagrangian ducing a closed glm model, including modeling the pseudo- momentum for the closed theory. Section VII concludes by 1 lϭ ͵ ͉u͉2ϩ␣2ٌ͉u͉2d3x, ͑1.3͒ highlighting the GLM, glm, and ␣-model sequence of results 2 in a new application—a Lagrangian averaged ␣—model clo- for a constant ␣ and divergenceless fluid velocity u. ͓For sure of incompressible ideal magnetohydrodynamics ϭ ␣ ͑MHD͒. Equation ͑7.15͒ provides an expression for the ٌ incompressible flow •u 0 and constant , one may re- pseudomomentum for this new closed theory of Lagrangian 2͉ ٌ͉ place u in this Lagrangian equivalently with 2 tr(e•e), .ϭ 1 ٌ ϩٌ T ͔ averaged ideal MHD and Eq.
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