Structure of the terrain and gravitational field of the planets: Kaula’s rule as a consequence of the probability laws by A.N. Kolmogorov and his school

Cover Page


Kaula’s empirical rule has been known for more than 50 years: the coefficients of expansion over spherical harmonics for the fluctuations of the gravitational field and terrain of the planets decrease as the number of the harmonic squared. This was found for Venus, the Moon, Mars, the asteroid Vesta, and very small celestial bodies. The inverse-square line spectra were also found for various types of the Earth’s surface on a scale of up to a hundred kilometers. From this it follows that the spectra of the terrain slope angles are constant, i.e., “white noise”. This, they are delta-correlated horizontally. These are the assumptions under which the random walk laws were derived by A.N. Kolmogorov in 1934. Using them, the equation of the horizontal probability diffusion of the terrain with the linear coefficient diffusion D is derived. Based on the empirical data, D = 1.3 ± 0.3 m for the Earth, while for Venus it is almost an order of magnitude less. The slopes resist the wind; the rock crumbles, and the water flows down the slopes as well. This consideration turns Kaula’s rule into the random walk laws (over terrain) developed by Kolmogorov in 1934.

About the authors

E. B. Gledzer

Obukhov Institute of Atmospheric Physics of the Russian Academy of Sciences


Russian Federation, 3, Pizevsky, Moscow, 119017

G. S. Golitsyn

Obukhov Institute of Atmospheric Physics of the Russian Academy of Sciences

Author for correspondence.

Russian Federation, 3, Pizevsky, Moscow, 119017

Academician of the Russian Academy of Sciences


  1. Kaula W. M. Theory of Satellite Geodesy. Bleinsdell: Waltham, 1966.
  2. Turcotte D. L. Chaos and Fractals in Geology and Geophysics. 2nd ed. Cambridge: Cambridge Univ. Press, 1997. 398 p.
  3. Rexer M., Hirt C. // Surv. Geophys. 2015. V. 36. P. 803.
  4. Konopliv A. S., et al. // Icarus. 2014. V. 240. P. 103-117. 5. McMahon J. W., et al. // Proc. 47th Lunar and Planetary Sci. Conference. 2016.
  5. Kolmogorov A. N. // Ann. Math. 1934. V. 35. P. 116-117.
  6. Монин А. С., Яглом А. М. Статистическая гидромеханика. М.: Физматлит, 1967. Т. 2.
  7. Голицын Г. С. // Физика Земли. 2003. № 7. С. 3-8.
  8. Werner S. C., Harris A.W., Neukum G., Ivanov B.A. // Icarus. 2002. V. 156. P. 287-290.
  9. Голицын Г. С. // Метеорология и гидрология. 2018. № 3. С. 5-15.



Abstract - 206

PDF (Russian) - 149


Copyright (c) 2019 Russian academy of sciences

This website uses cookies

You consent to our cookies if you continue to use our website.

About Cookies