eesti teaduste
akadeemia kirjastus
SINCE 1952
Proceeding cover
of the estonian academy of sciences
ISSN 1736-7530 (Electronic)
ISSN 1736-6046 (Print)
Impact Factor (2021): 1.024
Implications of the theory of turbulent mixing for wave propagation in media with fluctuating coefficient of refraction; pp. 285–290
PDF | doi: 10.3176/proc.2015.3.12

Jaan Kalda, Mihkel Kree

Based on ray tracing approach, light propagation in inhomogeneous media with fluctuating coefficient of refraction n = n (r) can be interpreted as a chaotic mixing of the wavefront in the 6-dimensional phase space where the spatial coordinates are complemented by the respective wave vector components. According to ray tracing, the evolution of wave vectors follows Hamiltonian dynamics and hence, according to the Liouville’s theorem, the mixing of the wave front takes place in an incompressible flow field. We use this approach to show that the brightest light speckles in inhomogeneous media follow a power law intensity distribution, and to derive the relevant scaling exponents.


  1. Wheelon, A. D. Electromagnetic Scintillation: Volume 1, Geometrical Optics. Cambridge University Press, Cambridge, UK, 2001.

  2. Korotkova, O. Random Light Beams: Theory and Applications. Taylor and Francis, Hoboken, NJ, 2013.

  3. Klyatskin, V. I. Electromagnetic wave propagation in a randomly inhomogeneous medium as a problem in mathematical statistical physics. Physics–Uspekhi, 2004, 47(2), 169–186.

  4. Bal, G., Komorowski, T., and Ryzhik, L. Kinetic limits for waves in a random medium. Kinetic and Related Models, 2010, 3(4), 529–644.

  5. Falkovich, G., Gawędzki, K., and Vergassola, M. Particles and fields in fluid turbulence. Rev. Mod. Phys., 2001, 73(4), 913–975.

  6. Richardson, L. F. Atmospheric diffusion shown on a distance-neighbour graph. P. Roy. Soc. Lond. A Mat., 1926, 110, 709–737.

  7. Batchelor, G. K. Small-scale variation of convected quantities like temperature in turbulent fluid. Part 1. General discussion and the case of small conductivity. J. Fluid Mech., 1959, 5, 113–133.

  8. Aref, H. Stirring by chaotic advection. J. Fluid Mech., 1984, 143, 1–21.

  9. Ottino, J. M. Mixing, chaotic advection, and turbulence. Annu. Rev. Fluid Mech., 1990, 22(1), 207–254.

10. Sreenivasan, K. R. and Antonia, R. A. The phenomenology of small-scale turbulence. Annu. Rev. Fluid Mech., 1997, 29(1), 435–472.

11. Warhaft, Z. Passive scalars in turbulent flows. Annu. Rev. Fluid Mech., 2000, 32(1), 203–240.

12. Dimotakis, P. E. Turbulent mixing. Annu. Rev. Fluid Mech., 2005, 37(1), 329–356.

13. Shraiman, B. I. and Siggia, E. D. Scalar turbulence. Nature, 2000, 405, 639–646.

14. Toschi, F. and Bodenschatz, E. Lagrangian properties of particles in turbulence. Annu. Rev. Fluid Mech., 2009, 41(1), 375–404.

15. Grabowski, W. W. and Wang, L.-P. Growth of cloud droplets in a turbulent environment. Annu. Rev. Fluid Mech., 2013, 45(1), 293–324.

16. Kalda, J. Simple model of intermittent passive scalar turbulence. Phys. Rev. Lett., 2000, 84(3), 471–474.

17. Kalda, J. On the multifractal properties of passively con\-vected scalar fields. In Paradigms of Complexity. Fractals (Novak, M., ed.). World Scientific, Singapore, 2000, 193–201.

18. Kalda, J. k-spectrum of decaying, aging and growing passive scalars in Lagrangian chaotic fluid flows. J. Phys. Conf. Ser., 2011, 318(5), 052045.

19. Ainsaar, S. and Kalda, J. On the effect of finite-time correlations on the turbulent mixing in smooth chaotic compressible velocity fields. Proc. Estonian Acad. Sci., 2015, 64, 1–7.

20. Kalda, J. Sticky particles in compressible flows: aggregation and Richardson’s law. Phys. Rev. Lett., 2007, 98(6), 064501.

Back to Issue