Images of Defects


The research which yielded these images was supported by the Department of Energy.


When appropriate, the radius ratio Gamma, Prandtl number Sigma, and distance from threshold epsilon are provided.



Perhaps the most common and simplest type of defect which occurs in Rayleigh-Benard convection patterns is a dislocation. Sigma = 0.87, epsilon = 0.20.

From K. M. S. Bajaj, N. Mukolobwiez, and G. Ahlers, unpublished.



Dislocations are often created in pairs when the local roll wavelength falls outside of the stability limits of perfect straight-roll patterns. Sigma = 0.87, epsilon = 0.20.

From K. M. S. Bajaj, N. Mukolobwiez, and G. Ahlers, unpublished.



After their birth, the two members of a defect pair typically move away from each other by a combination of climb and glide motion. The direction in which they move seems to be nearly the same as the direction of the perturbation to which the straight rolls have become unstable. In the example shown here this is the direction of the skewed-varicose perturbation. Sigma = 0.87, epsilon = 0.20.

From K. M. S. Bajaj, N. Mukolobwiez, and G. Ahlers, unpublished.



Defects closely related to dislocations also occur in more complicated patterns, like hexagons. Close inspection of the pattern on the left reveals that it contains a penta-hepta defect. The Fourier transform (FT) of the hexagonal pattern is shown on the right. We see that the FT consists of three pairs of points. Sigma = 0.9, epsilon = 0.12.

From E. Bodenschatz, G. Ahlers, and D.S. Cannell, unpublished.



Each pair of points of the FT corresponds to a set of more or less parallel straight rolls, and the hexagonal pattern is a superposition of these three sets. We can take one pair at a time, and back-transform to real space. This enables us to see the nature of each of the roll patterns separately. We find that two of the roll sets contain a dislocation, whereas the third one does not. Sigma = 0.9, epsilon = 0.12.

From E. Bodenschatz, G. Ahlers, and D.S. Cannell, unpublished.



Domain walls are another type of defect often found in disordered patterns. This one was obtained from convection in the presence of rotation. Sigma = 0.9, epsilon = 0.07. The dimensionless rotation rate Omega was 6.2.

From Y. Hu, R.E. Ecke, and G. Ahlers, Phys. Rev. E 55, 6928 (1997).



Foci are an interesting type of defect. Under certain conditions they can emit traveling waves of convection rolls. On the left is a focus formed by a set of concentric rolls in a circular container (in this case the concentric rolls are stabilized by sidewall forcing). On the right is a sidewall focus which forms because, without sidewall forcing, the rolls in a circular container tend to terminate with their axes orthogonal to the wall. Sigma = 0.9. Left: epsilon = 0.04. Right: epsilon = 0.34, Gamma = 41.

From Y. Hu, R.E. Ecke, and G. Ahlers, Phys. Rev. E 48, 4399 (1993).



The defect shown here is known as a convex disclination and was discussed theoretically by Ercolani et al. (J. Nonlin. Sci, to be published). Sigma = 0.33, epsilon = 0.038.

From J. Liu and G. Ahlers, Phys. Rev. E 55, 6950 (1997).



This interesting defect structure seems to be related to the concave disclination found in certain model equations by Ercolani et al. (J. Nonlin. Sci 010, 223 (2000)), but it differs in that it includes a domain-wall structure in the upper center. Gamma = 30, sigma = 0.32, epsilon = 0.07.

From J. Liu and G. Ahlers, Phys. Rev. Lett. 77, 3126 (1996).



This pattern is confined by a stadium-shaped sidewall (two semicircles connected by straight lines). The outermost convection rolls are made to be parallel to the sidewall by heating gently with a heater embedded in the wall. The aspect ratios (half diameter/height) along the major and minor axes are 19.4 and 13.0 respectively. The fluid was ethanol with sigma = 14, and epsilon = 0.80. A single roll pair extends along much of the major diameter, terminating in two convex disclinations.

From W. Meevasana and G. Ahlers, Phys. Rev. E 66, 046308 (2002).



This pattern is confined by an elliptical sidewall. As for the stadium above, the outermost convection rolls are made to be parallel to the sidewall by heating gently with a heater embedded in the wall. The aspect ratios (half diameter/height) along the major and minor axes are 19.4 and 13.0 respectively. The fluid was ethanol with sigma = 14, and epsilon = 0.80. The rolls along the major diameter are not able to maintain the curvature of the sidewall, and instead form an interesting defect structure, known as a protuberance, at two points along the major diameter where the stadium had the two convex disclinations. This structure was envisioned by Ercolani et al. (J. Nonlin. Sci 010, 223 (2000)).

From W. Meevasana and G. Ahlers, Phys. Rev. E 66, 046308 (2002).





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