Slow modes in passive advection

Bernard, Denis; Gawedzki, Krzysztof; Kupiainen, Antti

doi:10.1023/A:1023212600779

Slow modes in passive advection

Jun 4, 1997

41 pages

Published in:

J.Statist.Phys. 90 (1998) 519

e-Print:

cond-mat/9706035 [cond-mat]

DOI:

10.1023/A:1023212600779 (publication)

Report number:

IHES/P/97/47 and SPhT-97-053

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Abstract: (arXiv)

The anomalous scaling in the Kraichnan model of advection of the passive scalar by a random velocity field with non-smooth spatial behavior is traced down to the presence of slow resonance-type collective modes of the stochastic evolution of fluid trajectories. We show that the slow modes are organized into infinite multiplets of descendants of the primary conserved modes. Their presence is linked to the non-deterministic behavior of the Lagrangian trajectories at high Reynolds numbers caused by the sensitive dependence on initial conditions within the viscous range where the velocity fields are more regular. Revisiting the Kraichnan model with smooth velocities we describe the explicit solution for the stationary state of the scalar. The properties of the probability distribution function of the smeared scalar in this state are related to a quantum mechanical problem involving the Calogero-Sutherland Hamiltonian with a potential.

Note:

41 pages, latex, no figures

References(25)

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[1]

: The Local Structure of Turbulence in Incompressible Viscous Fluid for Very Large Reynolds’ Numbers

A.N. Kolmogorov

- Compt.Rend.Acad.Sci.URSS 30 (1941) 301305

[2]

: Small-Scale Structure of a Scalar Field Convected by Turbulence

R.H. Kraichnan

- Phys.Fluids 11 (1968) 945-963

[3]

: Anomalous Scaling of a Passive Scalar in Turbulent Flow. C.R

B. Shraiman
,
E. Siggia

- Science 321 (1995) 279-284

[4]

Normal and anomalous scaling of the fourth-order correlation function of a randomly advected passive scalar

M. Chertkov
(
- Weizmann Inst. and
- Milan U. and
- Novosibirsk, IYF and
- Landau Inst.
)
,
G. Falkovich
(
- Weizmann Inst. and
- Milan U. and
- Novosibirsk, IYF and
- Landau Inst.
)
,
I. Kolokolov
(
- Weizmann Inst. and
- Milan U. and
- Novosibirsk, IYF and
- Landau Inst.
)
,
V. Lebedev
(
- Weizmann Inst. and
- Milan U. and
- Novosibirsk, IYF and
- Landau Inst.
)

- Phys.Rev.E 52 (1995) 4924-4941
•
e-Print:
- chao-dyn/9503001
•
DOI:
- 10.1103/PhysRevE.52.4924

[5]

Anomalous Scaling of the Passive Scalar

- Phys.Rev.Lett. 75 (1995) 3834-3837
•
e-Print:
- chao-dyn/9506010
•
DOI:
- 10.1103/PhysRevLett.75.3834

[6]

K. Gaw¸ edzki and A. Kupiainen: Anomalous Scaling in the N-Point Functions

D. Bernard

- Phys.Rev.E 54 (1996) 2564-2572

[7]

: Lagrangian Path Integrals and Fluctuations in Random Flow

B. Shraiman
,
E. Siggia

- Phys.Rev.E 49 (1994) 2912-2927

[8]

: Symmetry and Scaling of Turbulent Mixing

B. Shraiman
,
E. Siggia

- Phys.Rev.Lett. 77 (1996) 2463-2466

[9]

: Statistics of a Passive Scalar Advected by a Large-Scale Two-Dimensional Velocity Field: Analytic Solution

M. Chertkov
,
G. Falkovich
,
I. Kolokolov
,
V. Lebedev

- Phys.Rev.E 51 (1995) 5609-5627

[10]

: Exact Field-Theoretical Description of Passive Scalar Convection in an N-Dimensional Long-Range

M. Chertkov
,
A. Gamba
,
I. Kolokolov

- Phys.Lett.A 192 (1994) 435-443

[11]

: The Lyapunov Spectrum of Continuous Product of Random Matrices

A. Gamba
,
I. Kolokolov

- J.Statist.Phys. 85 (1996) 489-499

[12]

: The Fourth-Order Correlation Function of a Randomly Advected Passive Scalar. Pis’ma v

E. Balkovsky
,
M. Chertkov
,
I. Kolokolov
,
V. Lebedev

- Zh.Eksp.Teor.Fiz. 61 (1995) 1012-1017

[13]

: Instantons and Intermittency

G. Falkovich
,
I. Kolokolov
,
V. Lebedev
,
A. Migdal

- Phys.Rev.E 54 (1996) 4896-4907

[14]

Instanton for random advection

Michael Chertkov
(
- Weizmann Inst.
)

- Phys.Rev.A 55 (1997) 2722
•
e-Print:
- chao-dyn/9606011
•
DOI:
- 10.1103/PhysRevE.55.2722

[15]

Particle Dispertion in a Multidimensional Random Flow with Arbitrary Temporal Correlations. To appear in J. Phys. A

G. Falkovich
,
V. Kazakov
,
V. Lebedev

[16]

V. S. L’vov: A Scale Invariant

V.I. Belinicher

- Sov.Phys.JETP 66 (1987) 303-313

[17]

Anomalous scaling exponents of a white advected passive scalar

M. Chertkov
(
- Weizmann Inst.
)
,
G. Falkovich
(
- Weizmann Inst.
)

- Phys.Rev.Lett. 76 (1996) 2706
•
e-Print:
- chao-dyn/9509007
•
DOI:
- 10.1103/PhysRevLett.76.2706

[18]

edzki: Turbulence under a Magnifying Glass

K. Gaw

e-Print:
- chao-dyn/9610003

[19]

: Products of Random Matrices

H. Furstenberg
,
H. Kesten

- Annals Math.Statist. 31 (1960) 457-469

[20]

Gol’dshein, S. A

I. Ya

- Funct.Anal.Appl. 11 (1977) 1-8

[21]

edzki and A. Kupiainen: Universality in Turbulence: an Exactly Soluble Model. In: Low-dimensional models in statistical physics and quantum field theory. H. Grosse and L. Pittner (eds.)

K. Gaw

[22]

: Diffusion Processes with Continuous Coefficients, I

D.W. Stroock
,
S.R.S. Varadhan

- Commun.Pure Appl.Math. 22 (1969) 345-400

[23]

Statistical Dynamics of Classical Systems

P.C. Martin
(
- Harvard U. and
- Orsay and
- SPhN, DAPNIA, Saclay
)
,
E.D. Siggia
(
- Harvard U. and
- Orsay and
- SPhN, DAPNIA, Saclay
)
,
H.A. Rose
(
- Harvard U.
)

- Phys.Rev.A 8 (1973) 423-437
•
DOI:
- 10.1103/PhysRevA.8.423

[24]

Exact results for a quantum many body problem in one-dimension. 2.

Bill Sutherland
(
- Utah U.
)

- Phys.Rev.A 5 (1972) 1372-1376
•
DOI:
- 10.1103/PhysRevA.5.1372

[25]

Quantum Integrable Systems Related to Lie Algebras

M.A. Olshanetsky
(
- Moscow, ITEP
)
,
A.M. Perelomov
(
- Moscow, ITEP
)

- Phys.Rept. 94 (1983) 313-404
•
DOI:
- 10.1016/0370-1573(83)90018-2