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.
http://dx.doi.org/10.1201/b15471
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.
http://dx.doi.org/10.1070/PU2004v047n02ABEH001685
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.
http://dx.doi.org/10.3934/krm.2010.3.529
5. Falkovich, G., Gawędzki, K., and Vergassola, M. Particles and fields in fluid turbulence. Rev. Mod. Phys., 2001, 73(4), 913–975.
http://dx.doi.org/10.1103/RevModPhys.73.913
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.
http://dx.doi.org/10.1017/S002211205900009X
8. Aref, H. Stirring by chaotic advection. J. Fluid Mech., 1984, 143, 1–21.
http://dx.doi.org/10.1017/S0022112084001233
9. Ottino, J. M. Mixing, chaotic advection, and turbulence. Annu. Rev. Fluid Mech., 1990, 22(1), 207–254.
http://dx.doi.org/10.1146/annurev.fl.22.010190.001231
10. Sreenivasan, K. R. and Antonia, R. A. The phenomenology of small-scale turbulence. Annu. Rev. Fluid Mech., 1997, 29(1), 435–472.
http://dx.doi.org/10.1146/annurev.fluid.29.1.435
11. Warhaft, Z. Passive scalars in turbulent flows. Annu. Rev. Fluid Mech., 2000, 32(1), 203–240.
http://dx.doi.org/10.1146/annurev.fluid.32.1.203
12. Dimotakis, P. E. Turbulent mixing. Annu. Rev. Fluid Mech., 2005, 37(1), 329–356.
http://dx.doi.org/10.1146/annurev.fluid.36.050802.122015
13. Shraiman, B. I. and Siggia, E. D. Scalar turbulence. Nature, 2000, 405, 639–646.
http://dx.doi.org/10.1038/35015000
14. Toschi, F. and Bodenschatz, E. Lagrangian properties of particles in turbulence. Annu. Rev. Fluid Mech., 2009, 41(1), 375–404.
http://dx.doi.org/10.1146/annurev.fluid.010908.165210
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.
http://dx.doi.org/10.1146/annurev-fluid-011212-140750
16. Kalda, J. Simple model of intermittent passive scalar turbulence. Phys. Rev. Lett., 2000, 84(3), 471–474.
http://dx.doi.org/10.1103/PhysRevLett.84.471
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.
http://dx.doi.org/10.1088/1742-6596/318/5/052045
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.
http://dx.doi.org/10.3176/proc.2015.1.01
http://dx.doi.org/10.1103/PhysRevLett.98.064501