Under conditions of reduced gravity, surface tension gradients become a dominant force in driving convective flows. In thin layers they lead typically to the classic hexagonal convection patterns. Here we are interested in the possible types of spatio-temporal chaos that can arise from instabilities of such patterns.
In buoyancy-driven convection and other pattern-forming systems spatio-temporal
chaos has been investigated in great detail. In most of these systems it arises from a
stripe-type platform, e.g. convection rolls. As the case of spiral-defect chaos illustrates
vividly, the chaotic state can strongly depend on the underlying destabilized pattern.
Therefore we expect qualitatively different dynamics if the destabilized convection pattern
is hexagonal rather than stripe-like.
To destabilize the hexagonal convection pattern we consider convection in a
rotating system. Rotation is known to have a strong impact on convection patterns due to
the Coriolis force. A particularly interesting case is rotating buoyancy-driven
Boussinesq convection where due to the K¨uppers-Lortz instability the usually observed steady roll
pattern is transformed into a pattern of ever-changing patches of rolls in di.erent
orientations. Within the framework of the full .uid equations for Marangoni convection a
detailed stability analysis of hexagonal convection and simulations of the evolution
ensuing from the instabilities is a formidable undertaking. We will discuss results for two
order-parameter models [1,2] and for coupled Ginzburg-Landau equations [3,4].
As a minimal model for the description of hexagonal patterns in the presence of
rotation we discuss an extension of the Swift-Hohenberg equation [1],
Here ø is a typical physical quantity like the temperature of the fluid in the
mid-plane and ˆez is the unit vector perpendicular to the (x, y)-plane. The
rotation enters (1) via the term proportional to ã, which breaks the chiral symmetry of the system. Linear
stability analysis of the weakly nonlinear hexagon patterns described by (1) reveals long- and
short-wave instabilities, which can be steady and oscillatory. The oscillatory
instabilities can induce a transition to hexagon patterns that are periodically modulated in space and
time. Further instabilities can lead to disordered patterns as shown in .g.1a. Strikingly, in
this regime the pattern still retains a residual six-fold symmetry reflected in 6 broad peaks in
the spatial
Fourier spectrum. Fig.1b shows a space-time plot of the angle dependence of this
Fourier spectrum (radically integrated). Most notably, the spectrum exhibits an
effective precession in time, which on longer time scales displays intermittent behavior.
For poorly conducting top and bottom boundaries Marangoni convection arises at
long wavelengths. For the case of a nondeformable fluid surface we have
systematically derived the long-wave equation [2]
As in (1), the rotation can drastically reduce the stability range of the hexagon patterns.
Fig.1c shows a case in which above a certain reduced Marangoni number µc ¡Ö 0.05
the steady hexagons become unstable at all wavenumbers. In this regime chaotic
dynamics may be expected in analogy to the results found for the Swift-Hohenberg equation (1)
[1]. Rotation can induce a Hopf bifurcation to oscillating hexagons. Using coupled
Ginzburg- Landau equations for the hexagon patterns [3] we have shown that independent of
the values of the coefficients in these equations oscillating hexagons with the critical
wavenumber are linearly stable. However, larger perturbations induce a transition to persistent
defect chaos [4], a state that has been studied in great detail in the single complex
Ginzburg-Landau equation but has so far not really been accessible in clean experiments.
Riecke, H., Echebarria, B., Sain, F., Instabilities and Spatio-Temporal Chaos of Hexagonal Convection Patterns in the Presence of Rotation, Proceedings of the Fifth Microgravity Fluid Physics and Transport Phenomena Conference, NASA Glenn Research Center, Cleveland, OH, CP-2000-210470, pp. 839-855, August 9, 2000.