For years people have been fascinated with the regular natural patterns they saw around them - orderly rows of clouds, waves on the surface of the ocean extending for miles and miles, the washboard pattern of sand ripples on river bottoms and in deserts, not to mention countless examples from biological systems, such as the ordered growth of some bacterial colonies and colored markings on animals. Yet it was not until the 20th century that the problem of pattern formation was first tackled by physicists.

At the beginning of this century Lord Rayleigh realized that pattern formation was the result of an instability undergone by a driven system, that is a system which is supplied energy by external forces, like the flow of water over the sandy river bottom or the circulation of air in the atmosphere which produces cloud streets. Rayleigh's insight came from considering experiments done on thermal convection. Convection patterns arise from the circulation of hot and cold fluid, and this motion organizes itself into regular rolls.

The results of Rayleigh's qualitative analysis were confirmed many years later by the detailed solutions to the full equations of motion that describe the fluid. Physicists went further and looked at equations describing convection in binary fluids, that is a mixture of two fluids like water and alcohol, or water with a salinity gradient, such as any ocean (Dead Sea, which has a particularly strong salt concentration is an ideal example). To their surprise they discovered that while convection in pure water forms stationary rolls, in a binary fluid the rolls move! The system proved to be richer than that. It was observed that binary fluids confined in narrow channels were capable of forming localized states of convection which were very stable.(1) These localized states were composed of four or five short moving rolls surrounded by a quiescent fluid. They were like ice cubes floating in water, but unlike ice cubes which coexist with water only at a unique temperature, zero degrees Celsius, these localized pulses were stable over a range of temperatures! While their existence came as a surprise, these pulses are a prototype for other structures observed in nature, but not understood yet, such as a solitary wave that travels for hundreds of miles in the ocean completely unchanged. Much of the pioneering experimental work on localized pulses was done in the last ten years by the group here in Santa Barbara.(2) When I joined the group my first project was to see if localized states of convection, which were observed in narrow channels, what physicists like to call 1-dimensional geometry, were also stable in wide containers, 2-dimensional geometry. The theorists know the equations that describe this system, and others like it, but solving them at present is an insurmountable task. In some sense we "solve" them experimentally, by seeing which solutions are physically possible.

The set-up I work with is extremely simple, yet fine tuning it to the level demanded by the standards of this field took me well over a year. The convection patterns are imaged by means of a shadowgraph technique.

When convection starts, the rolls are traveling. In their passage through the container they interact with the rolls reflected by the circular sidewalls. This interaction produces a pattern of roughly circular concentric rolls. As convection grows in strength, the circular symmetry is broken and the pattern becomes strongly enhanced along a diameter of the container. After a while this line of convection may die and another one, roughly perpendicular to the first, may develop, and so on. Eventually the bright line of convection will collapse to form a localized pulse of convection just like the one observed in narrow rectangular channels.

The pulses formed spontaneously in my cell, just like in 1-dimensional systems, but unlike those systems, these pulses were not stable, that is they didn't persist indefinitely.(3) Over a period of many hours they either died away completely or grew to fill the container. Little is known of the solutions of the equations of the fluid that describe these pulses. It is a challenge to the theorists to explain their existence and the lack of stability.

While the later stages of convection are too complex to analyze, we discovered that we could characterize and study quantitatively the so-called linear regime, when the strength of convection is not too large. We found that the geometry of the system plays a crucial role in the development of the convection pattern. The rolls have a unique size determined by the physical conditions. Since the size of the rolls cannot change, the pattern will change its underlying symmetries to adjust itself to the geometry. As I change the diameter of the convection cell in increments of the roll size, the pattern switches from having purely concentric rolls, to one whose rotation symmetry is broken so that something like a "drum mode" is seen. This means that upflowing fluid in one part of the container corresponds to downflowing fluid on the opposite side of the container, or light stripes become dark as you follow them along a circle at a particular radius. We call this the "m=1 mode," meaning that the strength of convection goes as cosQ, where Q is the angular position along a circle in the container. Actually, I discovered that the state can be best described by a superposition of different m modes (i.e. the strength of convection can be expressed as A0 + A1cosQ + A2cos2Q + A3cos3Q + ...). The strongest (biggest Am) modes are m=0 and m=2 when the size of the container is equal to an odd number of convection rolls, and they are m=1 and m=3 when the container is an even number of convection rolls in diameter. In the container that's not an integer number of rolls long, I would expect a competition between odd and even modes, though I have not yet investigated this region. A neat mathematical trick allows me to separate these modes from each other and study their properties, such as the spatial and temporal growth rates, independently. The dynamics of these modes, their selection as a function of the size of the system and their evolution are the subject of my current research.

**For more information, please refer to the following papers**

(1) R. Heinrichs, G. Ahlers and D. S. Cannell, *Physical
Review* **A 35**, 2761 (1987).

(2) Eberhard Bodenschatz, David S. Cannell, John
R. de Bruyn, Robert Ecke, Yu-Chou Hu, Kristina Lerman and Guenter Ahlers,
"Experiments in three systems with non-variational aspects", *Physica***D
61**, 77 (1992).

(3) Kristina Lerman, Eberhard Bodenschatz, David S.
Cannell and Guenter Ahlers, "Transient
Localized States in 2d Binary Liquid Convection", *Physical Review
Letters* **70**, 3572, (1993).

This work was done with **Guenter
Ahlers** and **David Cannell** at the Physics Department and CNLS,
UCSB.

Last modified: September 29, 1994 Kristina Lerman, lerman@isi.edu