Abstract
We study transport of a passive tracer particle in a time dependent turbulent flow in the medium with positive molecular diffusivity. We show that there exists then a probability measure equivalent to the underlying physical probability, corresponding to the Eulerian velocity field, under which the particle Lagrangian velocity observations are stationary. As an application we derive the existence of the Stokes drift and the effective diffusivity—the characteristics of the long time behavior of the particle motion.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
M. B. Isichenko, Percolation, statistical topography and transport in random media, Rev. Mod. Phys. 64:961-1043 (1992).
M. Avellaneda and A. J. Majda, Mathematical models with exact renormalization for turbulent transport, Commun. Math. Phys. 131:381-429 (1990).
A. Fannjiang and T. Komorowski, The fractional Brownian motion limit for motions in turbulence, Ann. of Appl. Prob. 10:1100-1120 (2000).
A. Fannjiang and T. Komorowski, Fractional Brownian motions and enhanced diffusion in a unidirectional wave-like turbulence, J. Statist. Phys. 100:1071-1095 (2000).
A. Fannjiang, Anomalous diffusion in random flows, in Mathematics of Multiscale Materials, Vol. 99, K. M. Golden, G. R. Grimmett, R. D. James, G. W. Milton, and P. N. Sen, eds. (1998), pp. 81-89.
G. M. Zaslavsky, Fractional kinetic equation for Hamiltonian chaos, Physica D 76(1-3):110-122 (1994).
S. Olla, Homogenization of Diffusion Processes in Random Fields (Manuscript of Centre de Mathématiques Appliquées, 1994).
J. L. Lumley, The Mathematical Nature of the Problem of Relating Lagrangian and Eulerian Statistical Functions in Turbulence, Méchanique de la Turbulence. Coll. Int du CNRS a´ Marseille. Ed. du CNRS (Paris, 1962).
S. C. Port and C. Stone, Random measures and their application to motion in an incompressible fluid, J. Appl. Prob. 13:499-506 (1976).
G. C. Papanicolaou and S. R. S. Varadhan, Boundary Value Problems With Rapidly Oscillating Random Coefficients, Vol. 27, J. Fritz and J. L. Lebowitz, eds. (Random Fields Coll. Math. Soc. Janos Bolyai, North-Holland, 1982), pp. 835-873.
S. M. Kozlov, The method of averaging and walks in inhomogeneous environments, Russian Math. Surveys 40:73-145 (1985).
S. A. Molchanov, Lectures on Random Media, Ecole d'été de Probabilités de St. Flour XXII, Lecture Notes in Math., Vol. 1581 (Springer-Verlag, Berlin, 1994), pp. 242-411.
H. Kesten and G. C. Papanicolaou, A limit theorem for turbulent diffusion, Comm. Math. Phys. 65:97-128 (1979).
T. Komorowski, Diffusion approximation for the advection of particles in a strongly turbulent random environment, Ann. of Prob. 24:346-376 (1996).
T. Komorowski and G. Papanicolaou, Motion in a Gaussian, incompressible flow, Annals of Appl. Prob. 17:229-264 (1997).
T. Komorowski and G. Krupa, The Law of Large Numbers for Ballistic, Multi-Dimensional Random Walks on Random Lattices with Correlated Sites. Preprint (2001).
D. G. Aronson and J. Serrin, Local behavior of solutions of quasilinear parabolic equations, Arch. Rat. Mech. Anal. 25:81-122 (1967).
A. Lasota and M. Mackey, Probabilistic Properties of Deterministic Systems (Cambridge University Press, 1985).
A. V. Skorokhod, s-Algebras of events on probability spaces. Similarity and Factorization, Theory Prob. Appl. 36:63-73 (1990).
G. C. Papanicolaou and S. R. S. Varadhan, Boundary Value Problems With Rapidly Oscillating Random Coefficients, Vol. 27, J. Fritz and J. L. Lebowitz, eds. (Random Fields Coll. Math. Soc. Janos Bolyai, North-Holland, 1982), pp. 835-873.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Komorowski, T., Krupa, G. On the Existence of Invariant Measure for Lagrangian Velocity in Compressible Environments. Journal of Statistical Physics 106, 635–651 (2002). https://doi.org/10.1023/A:1013762406825
Issue Date:
DOI: https://doi.org/10.1023/A:1013762406825