Lagrangian averages, averaged Lagrangians, and the mean effects of fluctuations in fluid dynamics

Chaos

Volumn 12, Issue 2, 2002, Pages 518-530

  Holm, Darryl D ^a  

^a LOS ALAMOS NATIONAL LABORATORY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0036592291     PISSN: 10541500     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.1460941     Document Type: Article

References (31)

1. D.G. Andrews and M.E. McIntyre, "An exact theory of nonlinear waves on a Lagrangian-mean flow," J. Fluid Mech. 89, 609-646 (1978).
2. D.D. Holm, J.E. Marsden, and T.S. Ratiu, "The Euler-Poincaré equations and semidirect products with applications to continuum theories," Adv. Math. 137, 1-81 (1998).
3. D.D. Holm, J.E. Marsden, and T.S. Ratiu, "Euler-Poincaré models of ideal fluids with nonlinear dispersion," Phys. Rev. Lett. 80, 4173-4177 (1998).
4. J.E. Marsden and T.S. Ratiu, Introduction to Mechanics and Symmetry, 2nd ed. (Springer, New York, 1999).
5. R. Camassa and D.D. Holm, "An integrable shallow water equation with peaked solitons," Phys. Rev. Lett. 71, 1661-1664 (1993).
6. V.I. Arnold, "Sur la géometrie differentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits," Ann. Inst. Fourier 16, 319-361 (1966).
7. D. Ebin and J.E. Marsden, "Groups of diffeomorphisms and the motion of an incompressible fluid," Ann. Math. 92, 102-163 (1970).
8. R.S. Rivlin and J.L. Erickson, "Stress-deformation relations for isotropic materials," J. Rat. Mech. Anal. 4, 323-425 (1955).
9. S.Y. Chen, D.D. Holm, L.G. Margolin, and R. Zhang, "Direct numerical simulations of the Navier-Stokes alpha model," Physica D 133, 66-83 (1999).
10. C. Foias, D.D. Holm, and E.S. Titi, "The Navier-Stokes-alpha model of fluid turbulence," Physica D 152, 505-519 (2001).
11. C. Foias, D.D. Holm, and E.S. Titi, "The three-dimensional viscous Camassa-Holm equations, and their relation to the Navier-Stokes equations and turbulence theory," J. Dyn. Diff. Eqns. 14, 1-35 (2001).
12. I. Gjaja and D.D. Holm, "Self-consistent wave-mean flow interaction dynamics and its Hamiltonian formulation for a rotating stratified incompressible fluid," Physica D 98, 343-378 (1996).
13. R. Abraham, J.E. Marsden, and T. Ratiu, Manifolds, Tensor Analysis, and Applications (Addison-Wesley, Reading, 1983).
14. D.D. Holm, "Variational principles, geometry, and topology of Lagrangian-averaged fluid dynamics," An Introduction to the Geometry and Topology of Fluid Flows, edited by R.L. Ricca (Kluwer, Dordrecht, 2001), pp. 271-291.
15. D.D. Holm, "Variational principles for Lagrangian averaged fluid dynamics," J. Phys. A 35, 1-10 (2002).
16. W.R.C. Phillips, "On the pseudomomentum and generalized Stokes drift in a spectrum of rotational waves," J. Fluid Mech. 430, 209-229 (2001).
17. W.R.C. Phillips, "On an instability to Langmuir circulations and the role of Prandtl and Richardson numbers," J. Fluid Mech. 442, 335-358 (2001).
18. J.L. Friedman and B.F. Schtuz, "Lagrangian perturbation theory of non-relativistic fluids," Astrophys. J. 221, 937 (1978).
19. J.L. Friedman and B.F. Schutz, "Secular stability of rotating Newtonian stars," Astrophys. J. 222, 281 (1978).
20. J.E. Marsden and S. Shkoller, "The anisotropic averaged Euler equations," J. Rat. Mech. Anal. (to be published).
21. S.Y. Chen, C. Foias, D.D. Holm, E.J. Olson, E.S. Titi, and S. Wynne, "The Camassa-Holm equations as a closure model for turbulent channel and pipe flows," Phys. Rev. Lett. 81, 5338-5341 (1998).
22. S.Y. Chen, C. Foias, D.D. Holm, E.J. Olson, E.S. Titi, and S. Wynne, "The Camassa-Holm equations and turbulence in pipes and channels," Physica D 133, 49-65 (1999).
23. S.Y. Chen, C. Foias, D.D. Holm, E.J. Olson, E.S. Titi, and S. Wynne, "A connection between the Camassa-Holm equations and turbulence in pipes and channels," Phys. Fluids 11, 2343-2353 (1999).
24. D.D. Holm (submitted).
25. D.D. Holm, "Fluctuation effects on 3D Lagrangian mean and Eulerian mean fluid motion," Physica D 133, 215-269 (1999).
26. J.E. Marsden, T.S. Ratiu, and S. Shkoller, "The geometry and analysis of the averaged Euler equations and a new diffeomorphism group," Geom. Funct. Anal. 10, 582-599 (2000).
27. 1 metric and mean hydrodynamics," J. Funct. Anal. 160, 337-365 (1998).
28. I.B. Bernstein, E.A. Frieman, M.D. Kruskal, and R.M. Kulsrud, "An energy principle for hydromagnetic stability theory," Proc. R. Soc. London, Ser. A 244, 17-40 (1958).
29. I.B. Bernstein, E.A. Frieman, M.D. Kruskal, R.M. Kulsrud, and I.B. Bernstein, "The variational principle for problems of ideal magnetohydrodynamic stability," Handbook of Plasma Physics (North-Holland, Amsterdam, 1983), Vol. 1, pp. 421-449.
30. A. Jeffrey and T. Taniuti, Magnetohydrodynamic Stability and Thermonuclear Containment (Academic, New York, 1966).
31. D.D. Holm, J.E. Marsden, T.S. Ratiu, and A. Weinstein, "Nonlinear stability of fluid and plasma equilibria," Phys. Rep. 123, 1-116 (1985).