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I've found lots of information on the formation of Benard Cells and convection currents but very little information about what happens to the self-organized structures when the energy gradient being applied begins to get smaller.

Do the complex structures persist as the gradient is diminished?

Do they break and then form slightly less complex structures?

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The dimensionless Rayleigh number characterizes buoyancy driven convection. When the Rayleigh number is below a critical value, heat transfer is primarily by conduction (e.g. no Benard convection cells). When the Rayleigh number is above the critical value, heat transfer is primarily by convection (e.g. Benard convection cells spontaneously form and persist). The Rayleigh number is linearly proportional to the temperature difference, and can be written in terms of the temperature gradient.

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As far as I remember, the RB convection system very far away from the onset has not been analytically studied to a great extent. There has been quite a bit of numerical work studying pattern formation and structure generation (finally leading to chaotic turbulence). Far from the onset, I suppose one might see ergodicity breaking with periodic orbits and intermittency. I think an accepted route to turbulence is via quasiperiodicity. Reducing the thermal gradient will change the existing structures, but it is not clear to me how this will occur (though intuitively, I would expect some sort of hysterisis to be present and so the changes would be expected to be sudden and discontinuous). Close to the onset though, we can say a little more about the loss of order as you tune down the thermal gradient. The convective transition is first order (in the so called Brazovskii class of fluctuation induced discontinuous transitions) and is supercritical (into rolls, while subcritical into more complex structures like hexagons)and so the broken symmetry phase will discontinuously disorder as you tune below the critical point.

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