Abstract
We consider the compressible Kraichnan model of turbulent advection with small molecular diffusivity and velocity field regularized at short scales to mimic the effects of viscosity. As noted in ref 5, removing those two regularizations in two opposite orders for intermediate values of compressibility gives Lagrangian flows with quite different properties. Removing the viscous regularization before diffusivity leads to the explosive separation of trajectories of fluid particles whereas turning the regularizations off in the opposite order results in coalescence of Lagrangian trajectories. In the present paper we re-examine the situation first addressed in ref 6 in which the Prandtl number is varied when the regularizations are removed. We show that an appropriate fine-tuning leads to a sticky behavior of trajectories which hit each other on and off spending a positive amount of time together. We examine the effect of such a trajectory behavior on the passive transport showing that it induces anomalous scaling of the stationary 2-point structure function of an advected tracer and influences the rate of condensation of tracer energy in the zero wavenumber mode.
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REFERENCES
D. Bernard, K. Gaw?dzki,and A. Kupiainen,Slow modes in passive advection,J.Stat. Phys. 90:519–569 (1998).
A. Borodin and P. Salminen,Handbook of Brownian Motion:Facts and Formulae, (Birkh¨auser, Boston,1996).
L. Breiman,Probability, (Addison-Wesley, Reading MA,1968).
M.Chaves,P.Horvai,K.Gawe ¸dzki,A.Kupiainen,and M. Vergassola, Lagrangian Dispersion in Gaussian Self-similar Ensembles arXiv:nlin.CD/0303031, J.Stat.Phys. (to appear).
E.W.,and Vanden-Eijnden, E.,Generalized flows,intrinsic stochasticity,and turbulent transport,Proc.Natl.Acad.Sci.USA 97:8200–8205 (2000).
E.W.,and Vanden-Eijnden, E.,Turbulent Prandtl number effect on passive scalar advec-tion,Physica D 152–153:636–645 (2001).
G. Falkovich, K. Gaw?dzki,and M. Vergassola, Particles and fields in uid turbulence. Rev.Mod.Phys. 73:913–975 (2001).
W. Feller,The parabolic differential equations and the associated semi-groups of trans-formations,Ann.Math. 55:468–519 (1952).
M.I. Freidlin and A.D. Wentzell, Necessary and Suficient Conditions for Weak Con-vergence of One-Dimensional Markov Processes.in The Dynkin Festschrift,Markov pro-cesses and their Applications, M.I. Freidlin,ed.(Birkh ¨auser, Boston, 1994)pp.95–109.
K. Gaw?dzki,Unpublished.
K. Gaw?dzki and M. Vergassola,Phase Transition in the Passive Scalar Advection, Phy-sica D 138:63–90 (2000).
I.S. Gradsteyn and I.M. Ryzhik,Table of Integrals:Series and Products, Academic Press, New York,1980.
K. Itô, H.P. McKean,Diffusion Processes and Their Sample Paths, (Springer, Berlin, 1965).
A.P. Kazantzev,Enhancement of a magnetic eld by a conducting uid, Sov.Phys. JETP 26:1031–1034 (1968).
R.H. Kraichnan,Small-scale structure of a scalar eld convected by turbulence,Phys. Fluids 11:945–963 (1968).
Y. Le Jan and O. Raimond,Integration of Brownian vector elds,Ann.Probab. 30:826–873 (2002).
Y. Le Jan and O. Raimond,Flows,Coalescence and Noise, arXiv:math.PR/0203221,Ann. Probab.(to appear).
Y. Le Jan and O. Raimond,Sticky Flows on the Circle, arXiv:math.PR/0211387.
L. Onsager,Statistical hydrodynamics,Nuovo Cim.Suppl. 6:279–287 (1949).
D. Revuz and M. Yor,Continuous Martingales and Brownian Motion, (Springer, Berlin, 1991).
L.F. Richardson,Atmospheric diffusion shown on a distance-neighbour graph,Proc.R. Soc.Lond. A 110:709–737 (1926).
L.C.G. Rogers and D. Williams,Diffusions,Markov Processes and Martingales, (Cambridge University Press, Cambridge, 2000).
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Gawędzki, K., Horvai, P. Sticky Behavior of Fluid Particles in the Compressible Kraichnan Model. Journal of Statistical Physics 116, 1247–1300 (2004). https://doi.org/10.1023/B:JOSS.0000041740.90705.d5
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DOI: https://doi.org/10.1023/B:JOSS.0000041740.90705.d5