Abstract
We analyze evolution of passive tracer density field advected in the velocity field governed by the multidimensional Burgers’ equation. A model description is developed and studied at the physical level of rigorousness for 2- and 1-D flows. In the 1-D case, we also consider a more general flow which admits both pressure and dispersion effects for polytropic pressure dependence on density. Fourier-Lagrangian representation and generalized density fields are discussed for multistream regimes and non-smooth Eulerian-to-Lagrangian maps.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
M. Avallaneda, E. Weinan, Statistical properties of shocks in Burgers’ turbulence, Comm. Math. Phys (1994), to appear.
G.T. Csanady, Turbulent Diffusion in the Environment (Reidel, Dordrecht-Boston-London, 1980).
R.E. Davis, On relating Eulerian and Lagrangian velocity statistics: single particles in homogeneous flows, J. Fluid Mech. 114, pp. 1–26 (1982).
R.E. Davis, On relating Eulerian and Lagrangian velocity statistics: single particles in homogeneous flows, J. Marine Res. 41, pp. 163–194 (1983).
R.J. Diperna, Convergence of approximate solutions to conservation laws, Arch. Rat. Math. Mech. 82, pp. 27–70 (1983).
R.J. Diperna, Convergence of the viscosity method for isentropic gas dynamics, Commun. Math. Phys. 91, pp. 1–30 (1983).
J.D. Fournier, U. Frisch, The deterministic and statistical Burgers equation, J. Mec. Theor. Appl. 2, pp. 699–750 (1983).
T. Funaki, D. Surgailis, W.A. Woyczynski, Gibbs-Cox random fields and Burgers’ turbulence, Ann. Appl. Probability 5, (1995) to appear.
E.E. Gossard, U.K. Khuk, Waves in Atmosphere (Nauka, Moscow, 1978).
T. Gotoh, R.H. Kraichnan, Statistics of decaying Burgers’ turbulence, Phys. Fluids A5, p. 2264 (1993).
S. Gurbatov, A. Malakhov, A. Saichev, Nonlinear Random Waves and Turbulence in Nondispersive Media: Waves, Rays and Particles,(Manchester University Press, Manchester, 1991).
S.N. Gurbatov, A.I. Saichev, Inertial nonlinearity and chaotic motion of particle fluxes, Chaos 3, pp. 333–358 (1993).
E. Hopf, The partial differential equation, Commun. Pure Appl. Math. 3, pp. 201–230 (1950).
Y. Hu, W.A. Woyczynski, An extremal rearrangement property of statistical solutions of the Burgers’ equation, Ann. Appl. Probability 4, pp. 838–858 (1994).
R.H. Kraichnan, Models of intermittency in hydrodynamic turbulence, Phys. Rev. Letters 65, pp. 575–578 (1990).
P.D. Lax, C.D. Levermore, The small dispersion limit of the Kortevegde Vries equation, Commun. Pure Appl. Math. 36 I, pp. 253–290; II, pp. 571–593; III, pp. 809–829 (1983).
P.D. Lax, The zero dispersion limit, a deterministic analogue of turbulence, Commun. Pure Appl. Math. 44, pp. 1047–1056 (1991).
J.T. Lipscomb, A.L. Frenkel, D. ter Haar, On the convection of a passive scalar by a turbulent Gaussian velocity field, J. Stat. Phys. 63, pp. 305–313 (1991).
H. Nakazawa, Probabilistic aspects of equations of motion of forced Burgers and Navier-Stokes turbulence, Progr. Th. Phys. 64, pp. 1551–1564 (1980).
O.A. Oleinik, On the Cauchy problem for nonlinear equations in a class of discontinuous functions, Doklady AN USSR 95, pp. 451–454 (1954).
O.A. Oleinik, Discontinuous solutions of nonlinear differential equations, Uspekhi Mat. Nauk 12, pp. 3–73 (1954).
A.I. Saichev, W.A. Woyczynski, Density fields in Burgers’ turbulence and their spectral properties, SIAM J. Appl. Math., to appear.
S.F. Shandarin, B.Z. Zeldovich, Turbulence, intermittency, structures in a self-gravitating medium: the large scale structure of the Universe, Rev. Modern Phys. 61, pp. 185–220 (1989).
YA.G. Sinai, Statistics of shocks in solutions of inviscid Burgers equation, Commun. Math. Phys. 148,pp. 601–622 (1992).
D. Surgailis, W.A. Woyczynski, Scaling limits of solutions of the Burgers’ equation with singular Gaussian initial data, in Chaos Expansions, Multiple Wiener-Ito Integrals, and Their Applications, edited by C. Houdré and V. Pérez-Abreu (Birkhaüser, Boston, 1994), pp. 145–161.
D. Surgailis, W.A. Woyczynski, Long range prediction and scaling limits for statistical solutions of the Burgers’ equation, in Nonlinear Waves and Weak Turbulence, edited by N. Fitzmaurice et al. (Birkhaüser, Boston, 1993), pp. 313–337.
M. Vergassola, B. Dubrulle, U. Frisch, A. Nullez, Burgers equation, Devil’s staircases and the mass distribution for large-scale structures, Astron. Astrophys. (1994), to appear.
D.H. Weinberg, J.E. Gunn, Large scale structure and the adhesion approximation, Monthly Not. Royal Astronom. Soc. 247, pp. 260–286 (1990).
G.B. Witham, Linear and Nonlinear Waves (Wiley, New York, 1974), p. 460.
W.A. Woyczynski, Stochastic Burgers’ flows, in Nonlinear Waves and Weak Turbulence, edited by N. Fitzmaurice et al. (Birkhaüser, Boston, 1993), pp. 279–311.
E. Zambianchi, A. Griffa, Effects of finite scales of turbulence on dispersions estimates, J. Marine Res. 52, pp. 129–148 (1994).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1996 Springer-Verlag New York, Inc.
About this paper
Cite this paper
Saichev, A.I., Woyczynski, W.A. (1996). Model Description of Passive Tracer Density Fields in the Framework of Burgers’ and Other Related Model Equations. In: Funaki, T., Woyczynski, W.A. (eds) Nonlinear Stochastic PDEs. The IMA Volumes in Mathematics and its Applications, vol 77. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-8468-7_11
Download citation
DOI: https://doi.org/10.1007/978-1-4613-8468-7_11
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4613-8470-0
Online ISBN: 978-1-4613-8468-7
eBook Packages: Springer Book Archive