Physica A 189 (1992) 1-3 North-Holland PHYSICA

Simplified model for fractal dimension of clouds

M a r c u s S c h n e i d e r and T h o m a s W 6 h l k e 1

Gesamtschule Castrop-Rauxel. W-4620 Castrop-Rauxel, Germany

Received 12 July 1992

Projections of three-dimensional random percolation clusters onto a two-dimensional plane give roughly the same fractal dimension 1.3 (for the perimeter versus area relation) as Lovejoy's observation of real clouds and cellular automata simulations of Nagel and Raschke.

Real clouds in our sky are usually observed f rom above or f rom below, and thus their ffactal propert ies are those of their two-dimensional projections or

pictures. Lovejoy [1] measured in real clouds, and Nagel and Raschke [2] found in computer simulations, that the per imeter P of a cloud project ion

varies with some power D / 2 of the area A of that projection:

P ~ a D/2

In a computer simulation on a lattice, the per imeter P is the number of empty sites which have at least one occupied cloud site as a neighbor, and the

area a is the number of occupied sites forming the connected cluster represent- ing the cloud. Both P and a are defined for the two-dimensional projection, not

for the three-dimensional original cluster.

Ref. [2] simulated such clouds by a cellular au tomata approximation which gives also dynamical information. For the static fractal dimension D, Nagel and

Raschke found 1.38_+0.04 in good agreement with the experimental [1] D = 1.33. They suggested already that the static propert ies might have come

out also f rom a simpler percolation model [31. The present note confirms this suggestion.

Thus we simulated on an Amiga computer , p rogrammed in C, with the Lea th algorithm [4] the format ion of a single random percolation cluster on a simple cubic lattice at the percolation threshold 0.3116. This cluster was pro jec ted onto a (100) lattice plane, then the per imeter P and number a of

1 Address for correspondence: T. W6hlke, Stargarderstr. 42, W-4620 Castrop-Rauxel, Germany.

0378-4371/92/$05.00 © 1992- Elsevier Science Publishers B.V. All rights reserved

2 M. Schneider, T. W6hlke / Fractal dimension of clouds

Fig. 1. Example of a two-dimensional projection from a three-dimensional fractal percolation cluster (lattice = 100 x 100 sites).

occupied sites of this two-dimensional projection were determined; our largest area was a = 1836. We then averaged over hundred such clusters. Fig. 1 shows an example of such a projection; obviously due to the projection the interior is quite compact, justifying the above definition of the fractal dimension D

through (x/-d) D. Fig. 2 relates perimeter P to area a; the overall slope of this log-log plot

gives D = 1.28. However , our data show upward curvature, and thus our asymptotic slope D will be slightly larger, close to 1.33 and 1.38 of refs. [1,2]. (Simulations on a bigger and faster computer would be desirable.) This agreement is due to the projections; for purely three-dimensional and purely

perimeter P

100

10

- - I I ~ T ~ T 7 7 - [ I I I I ~ F T ~ 7 - - -

0 1(I 100 1000

number of occupied sites

Fig. 2. Log- log plot of perimeter P versus number a of occupied sites in the projections onto two dimensions.

M. Schneider, T. W6hlke / Fractal dimension of clouds 3

two-dimensional percolation clusters we confirmed the linear relation [4] between the perimeter and the number of occupied sites.

Thus our simple simulation confirmed the suggestion of ref. [2] that projec- tions of three-dimensional percolation clusters should give the desired fractal dimension of clouds. Such projections are useful also for other percolation problems [5]. Of course, clouds have many other properties as well, and not all of them can be expected to be percolation-like.

We thank U. Brauner for guiding us through this work, and D. Stauffer for suggesting it and for help with the manuscript.

References

[1] S. Lovejoy, Science 216 (1982) 185. [2] K. Nagel and E. Raschke, Physica A 182 (1992) 519. [3] D. Stauffer and A. Aharony, Introduction to Percolation Theory (Taylor and Francis, London,

1992). P. Grassberger, Numerical studies of critical percolation in 3 dimensions, preprint.

[4] P.L. Leath, Phys. Rev. B 14 (1976) 5064. [5] A. Margolina and M. Rosso, Illumination: A new method for studying 3D percolation fronts in

a concentration gradient, preprint~