**We**
have found that traveling-wave convection in a binary mixture is an
interesting system for studying spatiotemporal disorder. When TW convection
is first initiated, the patterns are extemely disordered, consisting of many
small domains of traveling waves, small, unstable spirals, and sources of
waves. As the time passes, the pattern resolves itself into a few domains of
uniform traveling waves, separated by well defined domain boundaries.

**The**
two pictures below show the pattern before and after the coarsening
transition. Frame (a) is taken immediately after the onset of convection,
when the pattern is most disordered, and frame (b) is taken and after about
2 days, after the pattern has ordered itself and formed the multi-domain
state.

**The**
most straightforward method of characterizing order in a pattern is probably
to take the spatial Fourier transform and look at the wave vector spectrum,
or to evaluate the autocorrelation function. The Fourier transforms of the
patterns in (a) and (b) are shown in frames (c) and (d). Surprisingly, the
difference in the Fourier transforms is rather subtle, even though the
patterns are very different.

**Our**
goal was to find a better way to characterized the level of disorder. The
key was to calculate the complex amplitude of the pattern. Basically, since
the pattern is composed of traveling-waves, the intensity of the pattern at
any given point oscillates in time as waves pass by. Therefore, you could
describe the pattern by specifying the phase and amplitude of oscillation of
each pixel. It turns out that the amplitude is rather boring, and most of
the information about the pattern is contained in the phase. (See our other
web page about traveling-wave states). It turns out that a good way to
characterize the level of disorder is to look at the phase field. The phase
field of the ordered pattern is smooth, but the phase field of the
disordered pattern has many singularities (called phase defects).

**A**
phase defect is a point in the complex amplitude where the contours of
Re(A)=0 and Im(A)=0 cross, and the phase of the field is
undefined. Below are close-up pictures of typical phase defects which were
found in TW convection patterns. The pictures represent the phase in a
rainbow palette, and contours of Re(A)=0 and Im(A)=0 are
marked by black and white lines.

**
DISLOCATION
**

**In**
the dislocation, a single phase defect marks the termination of a roll. A
few dislocation are visible within the large domains of traveling waves in
the ordered pattern, above.

**
ZIPPER
BOUNDARY
**

**In**
this zipper boundary, waves are moving down on the left side, and up on the
right side. Along the boundary, a row of phase defects is observed.

**
PERPENDICULAR
BOUNDARY
**

**In**
the perpendicular boundary, waves are moving down from the top and to the
left along the bottom of the pattern. Again, there is a row of phase defects
along the boundary.

**
CROSS
ROLL
PATTERN
**

**The**
cross roll patch is a very complicated pattern element, in which two mutually
perpendicular standing waves alternate in time. This pattern manifests itself
as a lattice of phase defects.

**Now**
that the phase defects can be identified in the patterns, we can use them to
characterize the level of disorder.

**
PHASE
CONTOURS IN THE
DISORDERED
PATTERN
**

**In**
this image, cyan and yellow lines represent contours of Re(A)=0 and Im(A)=0.
In the disordered pattern, the phase contours form a tangled mess, because
the pattern mostly consists of nonuniform cross-roll patches. Disorder is
marked by a large neutral population of phase defects (at the crossings of
the contour lines).

**
PHASE
CONTOURS IN THE
ORDERED
PATTERN
**

**In**
the ordered patterns, the contour lines cross themselves the minimum number
of times possible to satisfy the boundary conditions. Order is marked by a
small number of defects, all having the same topological charge.

**The**
picture below shows the array of defects at four times during the evolution
of the pattern. Defects with positive and negative topological charge are
marked by circles and triangles (a little small, so they are hard to see),
and the trajectories of the defects are marked by grey trails. As the pattern
orders, the number of defects decreases dramatically, and the defects arrange
themselves into rows.

**The**
disordered pattern contains a high density of defects, where as the ordered
pattern contains very few defects. The coarsening of the pattern can
therefore be described in terms of the number of defects in the pattern.
These are shown below.

**The**
ordering of the pattern is marked by a decrease in the total number of
defects, and is preceded by an accumulation of net defect charge.