Reentrant Hexagons in non-Boussinesq Convection

While non-Boussinesq hexagonal convection patterns are well known to be stable close to threshold (i.e. for Rayleigh numbers R ≈ Rc), it has often been assumed that they are always unstable to rolls already for slightly higher Rayleigh numbers. Using the 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 (ǫ ≡ R − Rc/Rc = O(1)). We find ‘reentrant’ behavior of the hexagons, i.e. as ǫ is increased they can lose and regain stability. This can occur for values of ǫ as low as ǫ = 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 ǫ. 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 ǫ-range considered, 0 ≤ ǫ ≤ 1.5.