Re-entrant hexagons in non-Boussinesq convection

Abstract

While non-Boussinesq hexagonal convection patterns are well known to be stable close to threshold (i.e. for Rayleigh numbers $R \approx R_c$), it has often been assumed that they are always unstable to rolls already for slightly higher Rayleigh numbers. Using the {\em incompressible} Navier-Stokes equations for parameters corresponding to water as a working fluid, we perform full numerical stability analyses of hexagons in the strongly nonlinear regime ($\epsilon\equiv R-R_c/R_c=\mathcal{O}(1)$). We find `reentrant' behavior of the hexagons, i.e. as $\epsilon$ is increased they can lose and {\it regain} stability. This can occur for values of $\epsilon$ as low as $\epsilon=0.2$. We identify two factors contributing to the reentrance: i) the hexagons can make contact with a hexagon attractor that has been identified recently in the nonlinear regime even in Boussinesq convection (Assenheimer & Steinberg (1996); Clever & Busse (1996)) and ii) the non-Boussinesq effects increase with $\epsilon$. Using direct simulations for circular containers we show that the reentrant hexagons can prevail even for side-wall conditions that favor convection in the form of the competing stable rolls. For sufficiently strong non-Boussinesq effects hexagons become stable even over the whole $\epsilon$-range considered, $0 \le \epsilon \le 1.5$.