## Clouds are fractal

Fractal measures provide an elegant mathematical description of cloud structure. Fractals have the same structure at all scales. That may sound exotic, but fractals are actually very common in nature. An instructive example is a photo of a rock, where you cannot see if the rock is 10 cm large or 10 m, without someone or some tool next to it. Other examples of fractals are commodity prices, the branch structure of plants, mountains, coast lines, your lungs and arteries, and of course rain and clouds.The fractal structure of a measurement (time series) of Liquid Water Content (LWC) can be seen by zooming in on the time series. If you zoom in by a factor x, the total variance of this smaller part of the time series will be reduced by a factor y. Each time you again zoom in by factor x, you will find a variance reduction by a factor y, at least on average. This fractal behaviour leads to a power law; the total variance is proportional to the scale (total length of the time series) to the power of a constant; to be precise, this constant is log(x)/log(y). Such power laws can also be seen in the measurements of cloud top height, column integrated cloud liquid water (Liquid Water Path, LWP), the sizes of cumulus clouds, the perimeter of cumulus clouds or showers, and in satellite images of clouds and in other radiative cloud properties.

If you plot such a power law in a graph with logarithmic axis, the power laws looks like a line. Thus, a paper on fractals typically shows a lot of so called log-log-plots and linear fits. To identify scaling you need at least 3 orders of magnitude, thus you need large data sets with little noise.

## Why are clouds fractal?

Meteorologists studying turbulence have used scale invariance long before the word fractal was invented. The only relatively general law in the field of turbulence is Kolmogorovs law that the kinetic energy (E_{k}, proportional to the variance of the wind) of isotropic homogenous turbulence follows a power law, E

_{k}~k-5/3, with k the wave number, a measure of the scale. Furthermore, if one adds a substance to this turbulence that does not influence the flow (passive), nor is influenced by the flow (conservative), the variance of this substance (additive) will follow the same power law. In other words, a passive conservative additive of a turbulent field is also fractal.

Explicitly, or mostly implicitly, most researchers assume that clouds are fractal due to the fractal nature of turbulence. This may be a good approximation for calm stratus and stratocumulus clouds. However, clouds droplets and ice crystals are far from passive or conservative. They influence the flow by releasing heat by condensation, or by cooling the air by evaporation and drops drag the air down as they fall down due to gravity. Not very passive. The flow also influences the amount of cloud liquid water. In an upward flow, the air will become colder and the liquid water content will increase due to condensation. Thus, cloud droplets are also not conservative.

A cloud that is full of water will start raining. Alternatively, if there is a mixture of water and ice present, precipitation will be produced as the ice crystals grow at the expense of the water droplets. In fact, many more clouds are raining than a normal person on the ground would think, as often the rain evaporates before it arrives at the ground. These rain formation processes influence the structure of the clouds. These rain fields have also been shown to have a fractal structure. Certainly, we cannot explain that directly by turbulence.

Atmospheric models are able to generate fractal patterns. However, we do not know which dynamical processes are essential to explain the fractal structure of convective and raining clouds. The models in the framework of self organised criticality (SOC) suggest that the minimal set of dynamical equations may be very small. However, even as the dynamics of SOC-models intuitively looks similar to convection or rain formation processes, I know of no publication with a carefully investigation on the relation between SOC and convection.

## Fractal ideas

Even if you are not using fractal cloud models, there are a few very valuable ideas you can learn from fractal theory. The first is that variations, fluctuations, on all scales are important, and related to each other. Thus one should not denoise a cloud signal, i.e. remove scales with little variance (or apply EOF (empirical orthogonal function) analysis). On the other hand, one should not remove strong events, for example a shower, arguing that this is a special event not related to the rest of the structure. Furthermore, if one uses clouds from dynamical models, the large scale forcing of the models should be realistic, or one should use just small portions of the field.More generally, fractal structures (and the SOC models) suggest, that there is nothing special with strong events. If the dynamics of strong events was different, you would not expect it to scale with all other smaller events. And if the dynamics of the strong events is the same as those of the small ones, you can much better study the small events, which are much more frequent. For example, in stead of studying one large earth quake, better study smaller ones and obtain much better statistics. Or in the atmospheric sciences, studying, modelling, and predicting many small showers will lead to results much faster, as studying one large shower in detail that was headline news.

Concluding, fractal theory provides an elegant way of describing much of the structure of clouds. However, we do not understand fully why clouds have a fractal structure.

## Clouds are not fractal

Fractal structure is not all there is to cloud structure. In cloud fields one can find waves (in various directions and often only in part of the atmosphere), and daily cycles (especially clear in convective cumulus clouds, also showers typically occur in the late afternoon and evening), and also yearly cycles. Thus in these cases there is a clear dominating frequency (scale), contrary to the scale free fractal structures. Furthermore, changes in surface properties (e.g. sea to land) can be seen back in the cloud field and very well known are orographic clouds. At the scales where such spatial patterns influence cloud dynamics, the structure will not be fractal. At smaller scales turbulence may make the structure fractal again. Care has to be taken in analysing such cases. A step function, representing a sea to land change, has a power spectrum that follows a power law with exponent -2, still one cannot call a step function self-similar or fractal.To see the fractal structure of a cloud field, one needs to average. If you have a measured time series of cloud liquid water, the small scales typically show good scaling behaviour. However, at large scales, the scaling normally becomes worse, as the amount of averaging decreases. In the extreme case, at the scale of the length of the measurement, you only have one estimate of the variance at that scale. On the contrary, all the methods I know to make theoretical cloud time series or fields, make fields that scale perfectly up to the largest scale. Such perfect fractal clouds are too perfect to be real.

Especially Altocumulus, but also stratocumulus and other cloud types, can show highly regular Bernard cells of one specific size. One afternoon, I saw a cloud with cells that gradually became smaller towards the edges. Such regular features are not fractal. Furthermore, two-point statistics are probably not sufficient to describe such cells. My paper on iterative surrogate cloud fields shows that the Bernard cells of a stratocumulus fields are not present in a field with the same power spectrum, which describes the two-point structure with linear spatial correlations. Fortunately, this removal of the Bernard cells was not important for the clouds radiative properties. Thus for radiative transfer problems, two-point statistics may be sufficient.

The temporal structure of cloud fields can be asymmetric. The build up of a cloud can have a different dynamics as its dying out. The build up of a cumulus cloud is, for example, determined by the temperature profile of the atmosphere and the condensation heat. The dying out is caused by, e.g., rain falling out and turbulent mixing. To describe such asymmetric time series one needs three-point statistics. Fractal mathematics is normally based on two-point statistics; at least I do not know of any extensions to three-point statistics.

Concluding, the fractal structure describes much of the structure of cloud fields. Many of the questions raised here can be solved without going too far away from the fractal framework by allowing deviations from perfect scaling. For example, a gravity wave would introduce a special scale, the wavelength of this wave, but below and above this scale, one would expect good scaling behaviour. The problems mentioned in the two paragraphs on Bernard cells and asymmetry would probably need new mathematics.

## Measurements of cloud structure

The pioneers in fractal cloud studies have done groundbreaking work showing that clouds are good examples of fractals, that they, thus, should not be modelled by spheres and cubes floating in the air, nor by cloud models with artificially regular boundary conditions. I would say: we are convinced by now. Even if still the occasional paper with unrealistic cloud structures get through the review.To proof that cloud structure is close to fractal, large amounts of high quality, noise-free data had to be processed and averaged. Data that was preferably taken above oceans, which are ideal for remote sensing cloud measurements and provide stable, stationary conditions.

I feel it is time to go further, and analyse data from more difficult regions, with data analysis method aimed at finding deviations from the fractal pattern. It may be hard to find nonfractal structures. We will still need to average, but will have to do this in way that does not remove waves, which may only be present in part of the field and in case of airplane measurements, may only be obvious in certain flight directions. How would we know that other structures are present? Would we, in our standard fractal analysis, notice deviations due to Bernard cells or jumps at transitions?

The final words of such a text should of course spell out how all these thoughts show that the writers work is especially important. I work on two themes: 1) scanning measurements of cloud liquid water and 2) the generation of surrogate cloud fields. Such synthetic cloud fields have (statistical) properties of measured cloud fields.

The iterative surrogate cloud fields I am generating have a measured power spectrum (not an idealised fractal one) and a measured cloud liquid water distribution (not some beautiful well-known mathematical distribution). Together with the scanning measurements, this combination can be used to make cloud fields with are almost fractal, but have a wave in them, or cloud fields with a jump in cloud properties at a land-sea interface (if you happen to have measured on the beach). With constrained surrogate fields one can probably use almost arbitrary structure measures, also asymmetric or multifractal ones. Thus, this surrogate clouds allow for more realist cloud fields, which go beyond the fractal framework.

## Related post

**Radiative transfer and cloud structure**- A primer on the relation between radiation and cloud structure or, more generally, between nonlinear processes and variability.

This post was first published on my homepage on 22 April 2004.

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