Hostname: page-component-848d4c4894-75dct Total loading time: 0 Render date: 2024-06-02T08:11:15.543Z Has data issue: false hasContentIssue false
  • English
  • Français

Geometric generalised Lagrangian-mean theories

Published online by Cambridge University Press:  25 January 2018

Andrew D. Gilbert  and
Jacques Vanneste Open the ORCID record for Jacques Vanneste [Opens in a new window]
Andrew D. Gilbert
Affiliation:
Department of Mathematics, College of Engineering, Mathematics and Physical Sciences, University of Exeter, Exeter EX4 4QF, UK
Jacques Vanneste*
Affiliation:
School of Mathematics and Maxwell Institute for Mathematical Sciences, University of Edinburgh, Edinburgh EH9 3FD, UK
*
†Email address for correspondence: j.vanneste@ed.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

Many fluctuation-driven phenomena in fluids can be analysed effectively using the generalised Lagrangian-mean (GLM) theory of Andrews & McIntyre (J. Fluid Mech., vol. 89, 1978, pp. 609–646) This finite-amplitude theory relies on particle-following averaging to incorporate the constraints imposed by the material conservation of certain quantities in inviscid regimes. Its original formulation, in terms of Cartesian coordinates, relies implicitly on an assumed Euclidean structure; as a result, it does not have a geometrically intrinsic, coordinate-free interpretation on curved manifolds, and suffers from undesirable features. Motivated by this, we develop a geometric generalisation of GLM that we formulate intrinsically using coordinate-free notation. One benefit is that the theory applies to arbitrary Riemannian manifolds; another is that it establishes a clear distinction between results that stem directly from geometric consistency and those that depend on particular choices. Starting from a decomposition of an ensemble of flow maps into mean and perturbation, we define the Lagrangian-mean momentum as the average of the pull-back of the momentum one-form by the perturbation flow maps. We show that it obeys a simple equation which guarantees the conservation of Kelvin’s circulation, irrespective of the specific definition of the mean flow map. The Lagrangian-mean momentum is the integrand in Kelvin’s circulation and distinct from the mean velocity (the time derivative of the mean flow map) which advects the contour of integration. A pseudomomentum consistent with that in GLM can then be defined by subtracting the Lagrangian-mean momentum from the one-form obtained from the mean velocity using the manifold’s metric. The definition of the mean flow map is based on choices made for reasons of convenience or aesthetics. We discuss four possible definitions: a direct extension of standard GLM, a definition based on optimal transportation, a definition based on a geodesic distance in the group of volume-preserving diffeomorphisms, and the ‘glm’ definition proposed by Soward & Roberts (J. Fluid Mech., vol. 661, 2010, pp. 45–72). Assuming small-amplitude perturbations, we carry out order-by-order calculations to obtain explicit expressions for the mean velocity and Lagrangian-mean momentum at leading order. We also show how the wave-action conservation of GLM extends to the geometric setting. To make the paper self-contained, we introduce in some detail the tools of differential geometry and main ideas of geometric fluid dynamics on which we rely. These include variational formulations which we use for alternative derivations of some key results. We mostly focus on the Euler equations for incompressible inviscid fluids but sketch out extensions to the rotating–stratified Boussinesq, compressible Euler, and magnetohydrodynamic equations. We illustrate our results with an application to the interaction of inertia-gravity waves with balanced mean flows in rotating–stratified fluids.

JFM classification

Mathematical Foundations: Mathematical Foundations Waves/Free-surface Flows: Waves/Free-surface Flows
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 839 , 25 March 2018 , pp. 95 - 134
Copyright
© 2018 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Andrews, D. G. & McIntyre, M. E. 1978a An exact theory of nonlinear waves on a Lagrangian-mean flow. J. Fluid Mech. 89, 609–646.CrossRefGoogle Scholar
Andrews, D. G. & McIntyre, M. E. 1978b On wave-action and its relatives. J. Fluid Mech. 89, 647–664.CrossRefGoogle Scholar
Arnold, V. I. 1966 Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l’hydrodynamique des fluides parfaits. Ann. Inst. Fourier 16, 316–361.CrossRefGoogle Scholar
Arnold, V. I. & Khesin, B. A. 1998 Topological methods in hydrodynamics. In Applied Mathematical Sciences, vol. 125. Springer.Google Scholar
Arnold, V. I., Kozlov, V. V. & Neishtadt, A. I. 1988 Mathematical aspects of classical and celestial mechanics. In Encyclopedia of Mathematical Sciences: Dynamical Systems III. Springer.Google Scholar
Bhat, H. S., Fetecau, R. C., Marsden, J. E., Mohseni, K. & West, M. 2005 Lagrangian averaging for compressible fluids. Multiscale Model. Simul. 3, 818–837.CrossRefGoogle Scholar
Berger, M. 2003 A Panoramic View of Riemannian Geometry. Springer.CrossRefGoogle Scholar
Besse, N. & Frisch, U. 2017 Geometric formulation of the Cauchy invariants for incompressible Euler flow in flat and curved spaces. J. Fluid Mech. 825, 412–478.CrossRefGoogle Scholar
Boyland, P. 2001 Fluid mechanics and mathematical structures. In An Introduction to the Geometry and Topology of Fluid Flows (ed. Ricca, R. L.), NATO Science Series, pp. 105–134. Kluwer Academic Publishers.CrossRefGoogle Scholar
Brenier, Y. 1989 The least action principle and the related concept of generalized flows for incompressible perfect fluids. J. Amer. Math. Soc. 2, 225–255.CrossRefGoogle Scholar
Brenier, Y. 2003 Topics in hydrodynamics and volume-preserving maps. In Handbook of Mathematical Fluid Dynamics (ed. Friedlander, S. J. & Serre, D.), vol. 2, pp. 55–86. Elsevier Sience BV.CrossRefGoogle Scholar
Bretherton, F. P. 1971 The general linearized theory of wave propagation. In Mathematical Problems in the Geophysical Sciences, Lect. Appl. Math., vol. 13, pp. 61–102. Pub. Am. Math. Soc.Google Scholar
Bretherton, C. S. & Schär, C. 1993 Flux of potential vorticity substance: a simple derivation and uniqueness property. J. Atmos. Sci. 50, 1834–1836.2.0.CO;2>CrossRefGoogle Scholar
Bühler, O. 2014 Waves and Mean Flows, 2nd edn. Cambridge University Press.CrossRefGoogle Scholar
Bühler, O. & McIntyre, M. E. 1998 On non-dissipative wave–mean interactions in the atmosphere or oceans. J. Fluid Mech. 354, 301–343.CrossRefGoogle Scholar
Bühler, O. & McIntyre, M. E. 2005 Wave capture and wave-vortex duality. J. Fluid Mech. 354, 67–95.CrossRefGoogle Scholar
Dewar, R. L. 1970 Interaction between hydromagnetic waves and a time-dependent inhomogeneous medium. Phys. Fluids 13, 2710–2720.CrossRefGoogle Scholar
Ebin, D. G. & Marsden, J. 1970 Groups of diffeomorphisms and the motion of an incompressible fluid. Ann. Maths 92, 102–163.CrossRefGoogle Scholar
Frankel, T. 1997 The Geometry of Physics. Cambridge University Press.Google Scholar
Gjaja, I. & Holm, D. D. 1996 Self-consistent wave-mean flow interaction dynamics and its Hamiltonian formulation for rotating stratified incompressible fluids. Physica D 98, 343–378.Google Scholar
Grimshaw, R. 1985 Wave action and wave–mean flow interaction, with application to stratified shear flows. Annu. Rev. Fluid Mech. 16, 11–44.CrossRefGoogle Scholar
Grimshaw, R. H. J. 1975 Nonlinear internal gravity waves in a rotating fluid. J. Fluid Mech. 71, 497–512.CrossRefGoogle Scholar
Holm, D. D. 2002a Variational principles for Lagrangian-averaged fluid dynamics. J. Phys. A: Math. Gen. 35, 679–688.CrossRefGoogle Scholar
Holm, D. D. 2002b Lagrangian averages, averaged Lagrangians, and the mean effects of fluctuations in fluid dynamics. Chaos 12, 518–530.CrossRefGoogle ScholarPubMed
Holm, D. D., Marsden, J. E. & Ratiu, T. 1998 The Euler–Poincaré equations and semi-direct products with applications to continuum theories. Adv. Math. 137, 1–81.CrossRefGoogle Scholar
Holm, D. D., Schmah, T. & Stoica, C. 2009 Geometric Mechanics and Symmetry: From Finite to Infinite Dimensions. Oxford University Press.CrossRefGoogle Scholar
Holmes–Cerfon, M., Bühler, O. & Ferrari, R. 2011 Particle dispersion by random waves in the rotating Boussinesq system. J. Fluid Mech. 670, 150–175.CrossRefGoogle Scholar
Hawking, S. W. & Ellis, G. F. R. 1973 The Large Scale Structure of Space-time. Cambridge University Press.CrossRefGoogle Scholar
Haynes, P. H. & McIntyre, M. E. 1987 On the evolution of vorticity and potential vorticity in the presence of diabatic heating and frictional or other forces. J. Atmos. Sci. 44, 828–841.2.0.CO;2>CrossRefGoogle Scholar
Haynes, P. H. & McIntyre, M. E. 1990 On the conservation and impermeability theorems for potential vorticity. J. Atmos. Sci. 47, 2021–2031.2.0.CO;2>CrossRefGoogle Scholar
Marsden, J. E. & Hughes, T. J. R. 1983 Mathematical Foundations of Elasticity. Dover.Google Scholar
Marsden, J. E. & Shkoller, S. 2003 The anisotropic Lagrangian averaged Euler and Navier–Stokes equations. Arch. Rat. Mech. Anal. 166, 27–46.CrossRefGoogle Scholar
McCann, R. J. 2001 Polar factorization of maps on Riemannian manifolds. Geom. Funct. Anal. 11, 589–608.CrossRefGoogle Scholar
McIntyre, M. E. 1980 Towards a Lagrangian-mean description of stratospheric circulations and chemical transport. Phil. Trans. R. Soc. Lond. A 296, 129–148.Google Scholar
McIntyre, M. E. 1988 A note on the divergence effect and the Lagrangian-mean surface elevation in periodic water waves. J. Fluid Mech. 189, 235–242.CrossRefGoogle Scholar
Nayfeh, A. 1963 Perturbation Methods. Wiley.Google Scholar
Pennec, A. 2006 Intrinsic statistics on Riemannian manifolds: basic tools for geometric measurements. J. Math. Imag. Vis. 25, 127–154.CrossRefGoogle Scholar
Roberts, P. H. & Soward, A. M. 2006a Eulerian-Lagrangian means in rotating magnetohydrodynamic flows I. General results. Geophys. Astrophys. Fluid Dyn. 100, 457–483.CrossRefGoogle Scholar
Roberts, P. H. & Soward, A. M. 2006b Covariant description of non-relativistic magnetohydrodynamics. Geophys. Astrophys. Fluid Dyn. 100, 485–502.CrossRefGoogle Scholar
Salmon, R. 2013 An alternative view of generalized Lagrangian mean theory. J. Fluid Mech. 719, 165–182.CrossRefGoogle Scholar
Salmon, R. 2016 Variational treatment of inertia-gravity waves interacting with a quasi-geostrophic mean flow. J. Fluid Mech. 809, 502–529.CrossRefGoogle Scholar
Sanders, J. A. & Verhulst, F. 1985 Averaging Methods in Nonlinear Dynamical Systems. Springer.CrossRefGoogle Scholar
Shnirelman, A. 1994 Generalized fluid flows, their approximation and applications. Geom. Funct. Anal. 4, 586–620.CrossRefGoogle Scholar
Schutz, B. 1980 Geometrical Methods of Mathematical Physics. Cambridge University Press.CrossRefGoogle Scholar
Soward, A. M. 1972 A kinematic theory of large magnetic Reynolds number dynamos. Phil. Trans. R. Soc. Lond. A 272, 431–462.Google Scholar
Soward, A. M. & Roberts, P. H. 2010 The hybrid Euler–Lagrange procedure using an extension of Moffatt’s method. J. Fluid Mech. 661, 45–72.CrossRefGoogle Scholar
Soward, A. M. & Roberts, P. H. 2014 Eulerian and Lagrangian means in rotating, magnetohydrodynamic flows II. Braginsky’s nearly axisymmetric dynamo. Geophys. Astrophys. Fluid Dyn. 108, 269–322.CrossRefGoogle Scholar
Villani, C. 2003 Topics in Optimal Transportation, Graduate Studies in Mathematics, vol. 58. American Mathematical Society.Google Scholar
Wagner, G. L. & Young, W. R. 2015 Available potential vorticity and wave-averaged quasi-geostrophic flow. J. Fluid Mech. 785, 401–424.CrossRefGoogle Scholar
Xie, J.-H. & Vanneste, J. 2015 A generalised-Lagrangian-mean model of the interactions between near-inertial waves and mean flow. J. Fluid Mech. 774, 143–169.CrossRefGoogle Scholar