Clouds have distinctive shapes. Or they seem to have distinctive shapes. It turns out that is likely due to the fractal nature of clouds. Fractals, as pretty much anybody these days knows, are geometrical entities that have no intrinsic length scale. They are self-similar – that is, they repeat themselves at any scale you pick. Here’s a very nice picture of something that looks fractal.

The theory of fractals is surprisingly recent. Mandelbrot published his seminal work on fractals only in 1977. Since that, though, people have found and used the theory of fractals in a number of fields. Protein folding, for example, shows fractal behaviour (to be correct, proteins show this behaviour only in a section between the largest and the smallest scales). Turbulence has been shown to show fractal behaviour (although this is still an open problem).

The fractal nature of clouds was first shown in this paper in Science, from 1982. The basic characteristic of a fractal is its fractal dimension. Among the simpler definitions, or ways of calculating, the fractal dimension is the box dimension.

The idea is straightforward: for any ‘regular’ two-dimensional figure (the theory can be extended to more dimensions, but let’s not, for the purposes of this post), the area is related to the square of the characteristic length scale (the perimeter is a good substitute for this) of the figure. Equivalently, the perimeter is proportional to the square-root of the area. In fractals, however, the perimeter is related to the square-root of the area raised to the fractal dimension, D:

P ~ √(A^D)

The way one calculates the box-dimension is to lay out a grid over the figure of interest and count the number of boxes needed to cover the figure as a function of the grid-size. Then you find the perimeter of the boundary of the boxes you’ve included as part of your figure, and plug the area and the perimeter into the relationship above.

The study measures the box-dimension for cloud and rain areas at a wide range of length scales. Satellite pictures of clouds, with a maximum resolution of 4.8 km by 4.8 km are used. To go below this scale, radar ‘shots’ of rain-areas are used (rain-areas because radar scatters off rain particles) – these can give a minimum area of about 1 km by 1 km. The largest cloud the study spots is one with an extent of about 3000 km, with an area of about 1.2 million km.

There is a very nice physical argument to be made here: if clouds were of some fixed length scale, L, one could in principle consider the cloud area to be made up of large-scale structures (length > L) with small-scale noise superimposed (length < L). In which case, when the box-dimension is calculated, the answers for the large and small scales would be different. If, on the other hand, the box-dimension does not depend on the length scale, this would be equivalent to saying that the cloud has no inherent length scale.

It must be noted that it is a property of fractals that their lengths keep on increasing as the resolution is increased (the length scale of measurement is made finer). And because satellite and radar images have different maximum resolutions, the data obtained from the two have to be made commensurate. After this is done, the study finds that in the entire range of length scales corresponding to areas between 1 sq.km and 1.2 *million* sq. km, there is no change in the box dimension, which turns out to be 1.35.

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LOVEJOY, S. (1982). Area-Perimeter Relation for Rain and Cloud Areas Science, 216 (4542), 185-187 DOI: 10.1126/science.216.4542.185

Nice post. Was doing some work on analysis of chaotic data myself. There are a whole set of generalized dimensions and the reason why they remain invariants for a chaotic system is-just as you mentioned-due to their self-similarity. As the boxes get smaller, you discover more points.

One question though. Who names their son/daughter lovejoy? That would’ve ruined any love/joy he/she might have actually received.

@ Nair I believe Lovejoy is the last name