A generic model of double-diffusive convection: extended and localized patterns

Two-dimensional double-diffusive convection in a horizontal layer of fluid heated from below exhibits a variety of dynamical behaviours. The system has two competing gradients that drive motion in the fluid: the temperature gradient and the solute gradient. With low solute gradient, the first bifurcation for the resting (trivial) state as the temperature gradient is increased is a pitchfork bifurcation leading to steady convection. With larger solute gradient, the bifurcation changes to a Hopf bifurcation leading to oscillatory convection. In double-diffusive convection with idealised boundary conditions, these two forms of convection set in with the same horizontal wavelength. The point where the pitchfork and Hopf bifurcation coincide is called the Takens–Bogdanov point. This dissertation concentrates on understanding the bifurcation behaviour close to this Takens–Bogdanov point in domains that are large compared to the wavelength of the pattern of convection. A new partial differential equation (PDE) model that replicates the linear behaviour of double-diffusive convection is presented. The model has a variety of nonlinear terms, which allows considerable flexibility in its behaviour, and is the first Swift–Hohenberg-type model that has a Takens–Bogdanov primary bifurcation. Compared to the full PDEs for double-diffusive convection, the model is simple, which helps to investigate the nonlinear behaviour numerically and analytically, especially in large domains. From solving the model numerically, different patterns have been obtained: extended and localized patterns. Extended solutions such as steady states (SS), travelling waves (TW), and standing waves (SW) convection have been found in small and large domains. In large domains, and at different parameter values, localized patterns have also been found. Localized steady states (LSS) are found in the subcritical regime of the pitchfork bifurcation and localized travelling waves (LTW) are found in the subcritical regime of the Hopf bifurcation. In both cases, the trivial state and a large-amplitude stable pattern coexist. Previously, LSS and LTW have been found in numerical and experimental investigations of thermosolutal and binary convection. Our work helps explain the origins of these states and allows detailed investigation of their properties. The model also exhibits new types of patterns; LSS with modulated waves (MW) background in the parameter region where small-amplitude SW and large-amplitude SS are both stable, and LTW with SS background in the region where small-amplitude SS and large-amplitude TW are both stable. We make use of the analysis of the Takens–Bogdanov normal form done by Dangelmayr and Knobloch (1987) in order to find parameter values in the model PDE where the pitchfork or Hopf bifurcations are supercritical or subcritical. This analysis does not predict large-amplitude stable extended patterns in the subcritical cases; we rely on global stability of the PDE model to ensure that these stable large-amplitude states exist. The two last states we found (LSS with MW background and LTW with SS background) are new and have not been observed before in systems of double-diffusive convection. The new model is sufficiently general that it could be used to investigate other convection problems such as magnetoconvection and rotating convection.