Abstract
We study the two-point correlation function of a freely decaying scalar in Kraichnan's model of advection by a Gaussian random velocity field that is stationary and white noise in time, but fractional Brownian in space with roughness exponent 0<ζ<2, appropriate to the inertial-convective range of the scalar. We find all self-similar solutions by transforming the scaling equation to Kummer's equation. It is shown that only those scaling solutions with scalar energy decay exponent a≤(d/γ)+1 are statistically realizable, where d is space dimension and γ=2−ζ. An infinite sequence of invariants J p, p=0, 1, 2,..., is pointed out, where J 0 is Corrsin's integral invariant but the higher invariants appear to be new. We show that at least one of the invariants J 0 or J 1 must be nonzero (possibly infinite) for realizable initial data. Initial datum with a finite, nonzero invariant—the first being J p—converges at long times to a scaling solution Φ p with a=(d/γ)+p, p=0, 1. The latter belongs to an exceptional series of self-similar solutions with stretched-exponential decay in space. However, the domain of attraction includes many initial data with power-law decay. When the initial datum has all invariants zero or infinite and also it exhibits power-law decay, then the solution converges at long times to a nonexceptional scaling solution with the same power-law decay. These results support a picture of a “two-scale” decay with breakdown of self-similarity for a range of exponents (d+γ)/γ<a<(d+2)/γ, analogous to what has recently been found in the decay of Burgers turbulence.
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REFERENCES
T. von Kármán and L. Howarth, On the statistical theory of isotropic turbulence, Proc. Roy. Soc. A 164:192–215 (1938).
A. S. Monin and A. M. Yaglom, Statistical Fluid Mechanics, Vol. II (The MIT Press, Cambridge, MA, 1975).
J. Krug and H. Spohn, Kinetic roughening of growing surfaces, in Solids Far From Equilibrium: Growth, Morphology and Defects, C. Godrèche, ed. (Cambridge University Press, Cambridge, 1991).
A. J. Bray, Theory of phase-ordering kinetics, Adv. Phys. 43:357–459 (1994).
S N. Gurbatov, S. I. Simdyankin, E. Aurell, U. Frisch, and G. Tóth, On the decay of Burgers turbulence, J. Fluid Mech. 344:339–374 (1997).
U. Frisch, Turbulence (Cambridge University Press, Cambridge, 1995).
R. H. Kraichnan, Small-scale structure of a scalar field convected by turbulence, Phys. Fluids 11:945–963 (1968).
R. H. Kraichnan, Anomalous scaling of a randomly advected passive scalar, Phys. Rev. Lett. 72:1016–1019 (1994).
K. Gawedzki and A. Kupiainen, Anomalous scaling of the passive scalar, Phys. Rev. Lett. 75:3834–3837 (1995).
D. Bernard, K. Gawedzki and A. Kupiainen, Anomalous scaling in the N-point functions of a passive scalar, Phys. Rev. E 54:2564–2572 (1996).
B. Shraiman and E. Siggia, Anomalous scaling of a passive scalar in turbulent flow, C. R. Acad. Sci. 321:279–284 (1995).
B. Shraiman and E. Siggia, Symmetry and scaling of turbulent mixing, Phys. Rev. Lett. 77:2463–2466 (1996).
M. Chertkov, G. Falkovich, I. Kolokolov, and V. Lebedev, Normal and anomalous scaling of the fourth-order correlation function of a randomly advected passive scalar, Phys. Rev. E 52:4924–4941 (1995).
M. Chertkov and G. Falkovich, Anomalous scaling exponents of a white-advected passive scalar, Phys. Rev. Lett. 76:2706–2709 (1996).
M. Lesieur, C. Montmory, and J.-P. Chollet, The decay of kinetic energy and temperature variance in three-dimensional isotropic turbulence, Phys. Fluids 30:1278–1286 (1987).
R. H. Kraichnan, Convection of a passive scalar by a quasi-uniform random straining field, J. Fluid Mech. 64:737 (1974).
D. T. Son, Turbulent decay of a passive scalar in the Batchelor limit: Exact results from a quantum-mechanical approach, Phys. Rev. E 59:113811–113814 (1999).
A. J. Majda and P. R. Kramer, Simplified models for turbulent diffusion: Theory, numerical modelling, and physical phenomena, Phys. Rep. 314:237–574 (1999).
L. Onsager, Statistical hydrodynamics, Nuovo Cim. 6 279–287 (1949).
A. M. Polyakov, The theory of turbulence in 2-dimensions, Nucl. Phys. B 396:367–385 (1993).
G. L. Eyink and J. Xin, in preparation.
H. Kunita, Stochastic Flows and Stochastic Differential Equations (Cambridge University Press, Cambridge, 1990).
Y. Le Jan and O. Raimond, Integration of Brownian vector fields, archived as Math. Pr-9909147.
N. Aronszajn and K. T. Smith, Theory of Bessel potentials, I, Ann. Inst. Fourier (Grenoble) 11:385 (1961).
A. Erdélyi (ed.), Higher Transcendental Functions, Vols. I-III, Bateman Manuscript Project (Robert E. Krieger Publishing Co., Malabar, Florida, 1953).
A. J. Majda, Eplicit inertial range renormalization theory in a model of turbulent diffusion, J. Stat. Phys. 73:515–542 (1993).
B. Shraiman and E. D. Siggia, Lagrangian path integrals and fluctuations in random flow, Phys. Rev. E 49:2912–2927 (1994).
K. Gawedzki and A. Kupiainen, Universality in turbulence: An exactly soluble model, Low-Dimensional Models in Statistical Physics and Quantum Field Theory, Proceedings of the 34 Internationale Universitätwochen fur Kern-and Teilchenphysik, Schladming, Austria, March 4-11, 1995, H. Grosse and L. Pittner, eds., Lecture Notes in Physics, Vol. 469 (Springer, Berlin, 1996). Archived as chao-dyn-9504002.
I. Proudman and W. H. Reid, On the decay of a normally distributed and homogeneous turbulent velocity field, Phil. Trans. Roy. Soc. A 247:163–189 (1954).
W. H. Reid, On the stretching of material lines and surfaces in isotropic turbulence with zero fourth cumulants, Proc. Camb. Phil. Soc. 51:350–362 (1955).
R. Wong, Asymptotic Approximations of Integrals (Academic Press, San Diego, CA, 1989).
S. Corrsin, The decay of isotropic temperature fluctuations in an isotropic turbulence, J. Aeronaut. Sci. 18:417–423 (1951).
S. Corrsin, On the spectrum of isotropic temperature fluctuations in an isotropic turbulence, J. Appl. Phys. 22:469–473 (1951).
N. Goldenfeld, Lectures on Phase Transitions and the Renormalization Group (Addison-Wesley, Reading, MA, 1992).
G. K. Batchelor, Diffusion in a field of homogeneous turbulence, II. The relative motion of particles, Proc. Camb. Philos. Soc. 48:345–362 (1952).
L. F. Richardson, Atmospheric diffusion shown on a distance-neighbor graph, Proc. Roy. Soc. A 110(756):709–737 (1926).
E. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces (Princeton University Press, Princeton, NJ, 1971).
P. Lévy, Théorie de l'Addition des Variables Aléatoires, 2nd Ed. (Gauthier-Villars, Paris, 1937).
M. E. Feldheim, Étude de la stabilitédes lois de probabilité, Thèse de la Faculté des Sciences de Paris (1937).
V. M. Zolotarev, One-Dimensional Stable Distributions (American Mathematical, Society, Providence, RI, 1986).
S. Bochner, Stable laws of probability and completely monotone functions, Duke Math. J. 3:726 (1937).
R. Olshen and L. J. Savage, A generalized unimodality, J. Appl. Prob. 7:21–34 (1970).
A. Dharmadhikari and K. Joag-Dev, Unimodality, Convexity, and Applications (Academic Press, San Diego, CA, 1988).
A. Wintner, On a class of Fourier transforms, Amer. J. Math. 58:45–90 (1936).
M. Yamazato, Unimodality of infinitely divisible distribution functions of class L, Ann. Probab. 6:523–531 (1978).
S. J. Wolfe, On the unimodality of spherically symmetric stable distribution functions, J. Multivar. Anal. 5:236–242 (1975).
D. Thomson, private communication.
D. Thomson, Backwards dispersion of particle pairs and decay of scalar fields in isotropic turbulence, in preparation.
G. Szegö, Orthogonal Polynomials, 4th Ed. (American Mathematical Society, Providence, RI, 1975).
D. Bernard, K. Gawedzki, and A. Kupiainen, Slow modes in passive advection, J. Stat. Phys. 90:519–569 (1998).
I. I. Gihman and A. V. Skorohod, Stochastic Differential Equations (Springer-Verlag, NY, 1972).
P. G. Saffman, The large scale structure of homogeneous turbulence, J. Fluid Mech. 27:581–593 (1967).
L. G. Loitsyanskii, Some basic laws of isotropic turbulent flow, Trudy Tsentr. Aero. Giedrodin. Inst. 440:3–23 (1939).
G. Eyink and D. Thomson, Free decay of turbulence and breakdown of self-similarity, submitted to Phys. Fluids. Archived as chao-dyn-9908006.
M. Conti, B. Meerson, and P. V. Sasorov, Breakdown of scale invariance in the phase ordering of fractal clusters, Phys. Rev. Lett. 80:4683–4696 (1998).
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Eyink, G.L., Xin, J. Self-Similar Decay in the Kraichnan Model of a Passive Scalar. Journal of Statistical Physics 100, 679–741 (2000). https://doi.org/10.1023/A:1018675525647
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DOI: https://doi.org/10.1023/A:1018675525647