Transition to turbulence of convective flows in a cylindrical container

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Abstract
The threshold of the transition to turbulence of low Prandtl number convective flows occurs much closer to the convective threshold in an extended cylindrical cell, than one could infer from a straight roll stability analysis. Convection then involves non local effects together with a closure of the spatial scales. We solve these problems by constructing an explicit analytical solution of the phase fields and of their mean flow fields, which is valid at the dominant orders of pattern distortions. We hence provide an understanding of the low value of the threshold of turbulence at low Prandtl numbers in a cylindrical cell and of the mechanisms that lead to it