Introduction to fractal drainage systems

Tree-shaped structures are ubiquitous

A number of models have been use to describe the formation of tree-shaped structures.

These include:

Diffusion-limited aggregation is responsible for the patterns observed in snowflakes, frost, bacterial colonies, and mineral dendrites;
Branching instabilities can be seen as being behind the formation of tree branches, and perhaps branching in some metal crystals;
Fractal drainage systems govern patterns formed by streams and rivers, and by ionised discharges in plasma;
Iterated function systems is a bit of an umbrella term, which covers a number of artificial systems, including the mandelbrot set;
Injection-percolation models describe liquids forced into porus materials;
Dielectric breakdown is a model that is used when modelling electrical discharges, and cracks;

This page is devoted to fractal drainage systems - and the patterns they produce.

Fractal drainage systems

As an example of a fractal drainage system consider erosion caused by rain on a landscape. Falling rain turns into little rivulets, which move down local gradients, forming streams and rivers - and corroding the surface across which they flow as they go. The resulting patterns are known as fractal drainage patterns.

Fractal drainage patterns can easily be observed at home:

Take a glass of milk. Empty its contents down the sink and then stand the empty glass upright on a flat surface. In a short period of time, a detailed tracery of fine lines will be formed as the milk flows down the sides of the glass under gravity.

Fractal drainage systems are common

Fractal drainage systems are rather widespread. They arise in both organic and inorganic systems.

Another system that shows fractal drainage patterns is the flow of ionised particles in plasma - which forms similar branching patterns, which can be observed in discharges through plasma - as seen in "plasma balls".

I will also argue that the reason why the nervous system, the circulatory system, and the lungs all take the form they do is because of fractal drainage patterns - and that the same is true of the forms of trees, leaf veins, ferns, filter-feeding equipment, drainpipes and memory busses in computers.

It might seem that a tree is formed by a branching instability at the growing tips. However, looking at more fundamental causes the reason the genetic program of the tree causes that growth pattern is so the tree can channel liquid, gasses, and act as a better light trap. Material is channeled from many, widely dispersed locations towards the body of the tree. The same is true of the tree roots - their purpose is to drain their environment of nutrients, in order to help the tree reproduce.

Much the same rationale applies to the circulatory system - though there the fluid flows in both directions through a branching structure. It flows from the heart towards the capiliaries through arteries, and back from the capiliaries to the heart through veins.

The nervous system mirrors the circulatory system, in this respect - again with separate pathways for sensory and motor pathways. The respiratory system much the same - although this time inhalaled air flows (from the mouth to the lung alveolae) along exactly the same paths that it takes when exhaling (from the lungs out of the mouth). The fractal dimension of the respiratory system is very nearly three - it is very effective a space-filling structure.

History

Some of the first attempts to study drainage networks were made in the 1930s by an American engineer named Robert Horton. He developed scaling laws for stream networks, which expressed how the number of branches of each size varied.

For more information on the history of drainage patterns - and the mechanisms responsible for their production - I would recommend chapters 5 and 6 of Philip Ball's book, The Self-Made Tapestry.

Fractal drainage systems compared to diffusion-limited aggregation

While diffusion-limited aggregation typically produces static structures, fractal drainage systems are dynamic - and constantly changing.

Diffusion-limited aggregation generally involves growth governed by accumulations of randomly-moving particles fractal drainage systems are generally formed by the flow of fluids (or gasses) moving in channels.

Information flow

Very often, the key to understanding nature's trees is to look at them from the point of view of what flows from inwards, from the periphery and channels itself down to the root.

Sometimes (as in the case of water-erosion) the flow from the streams towards the oceans is obvious. However sometimes nothing at all seems to be flowing in that direction. This may seem to be true of cracks, lightning, tree growth and so on. However, normally if you look closeley enough, you can usually find something that moves in the opposite direction. Very often you can obly see the resulting snowflake - and the water vapor responsible for its formation remains invisible.

While both Diffusion Limited Aggregation and Fractal Drainage Systems are characterised by tree-shaped structures that look like they have been grown using a branching root, very often the key to understanding them is to look at flow in from the periphery that channels itself down to the root.

Artistic usage

Drainage patterns have been employed for artistic purposes for some time:

Encaustic painting often involves melting wax with an iron.
The drainage patterns are formed by lifting the iron from the surface.
Paint butterflies are the patterns formed by folding a piece of paper in half after appying paint. The resulting symmetrical pattern often contains drainage patterns.
Paint rollers have been used to produce decorative effects for decades.

As an example of the effect encaustic painting produces, here is an image left by a lifted iron:

As you can see - despite the fact the the fluid does not flow very far - the resulting drainage patterns can be quite impressive.

Cellular Automata Model

To go on to read about the cellular automata model, click here.

tim@tt1.org | http://fractaldrainage.com/