Natural Patterns

Lecture 4 : defects

We saw examples of defects in some of the patterns discussed in Lecture 1,
such as the dislocations in sand ripples or in fish skin stripes. The purpose of this
lecture is to introduce the terminology associated with elementary defects displayed
by pattern forming systems.

Defects of stripe patterns

Point defects of stripe patterns are disclinations and dislocations. A brief description of each
type of defect is given below.

Disclinations correspond to either a "bending" of the stripes, as shown in the first sketch of the figure on the right, or to a point where stripes with three different orientations meet, as shown in the second sketch. In both pictures, the solid lines correspond to the crests of the stripe pattern. If one follows a vector (arrow) locally perpendicular to the stripes along the closed path drawn around the core of either defect, one discovers that after one round trip, the orientation of the vector has been reversed. This is a characteristic property of disclinations. Note that the sense of rotation of the vector is different in the two diagrams.

Dislocations correspond to the insertion of an extra period in the stripe pattern. The diagram on the right shows a sketch of a dislocation, where the crests of the stripe pattern appear as solid lines and the troughs as dashed lines. Any closed path around the core of the defect, such as the one indicated on the figure, will be such that the number of periods crossed on one "side" of the defect is larger than the number of periods crossed on the other "side". Note however that a vector locally perpendicular to the stripes is returned to its original orientation after one trip around the closed path.

Line defects of stripe patterns are grain boundaries, which are curves separating regions
of stripes with different orientations.

Defects of hexagonal patterns

Point defects of hexagonal patterns are penta-hepta pairs, which consist of one cell with five nearest neighbors paired to a cell with seven nearest neighbors. If one consider hexagons as resulting from the superposition of three stripe patterns, penta-hepta pairs appear to be due to the presence of two dislocations on two of the stripe patterns, as illustrated in the figure on the right.

Line defects of hexagonal patterns are grain boundaries, which separate regions of hexagons
with different orientations.

Defects of oscillatory patterns

Defects of oscillatory patterns are spiral waves. As shown in the figure on the right, one can label the crests and troughs of the spiral patterns with multiples of p/ 2, and consider that these curves are level curves of some function, called the phase of the pattern. Then one sees that spiral defects are characterized by a phase change of plus or minus 2 p on a closed path around the core of the defect.

Applications

The following web site shows pictures of defects in convection patterns from Guenter Ahler's group
at the University of California Santa Barbara. Can you identify and name each of the defects illustrated
here ?

Defects in Rayleigh-Bénard convection

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