Near-onset dynamics in natural doubly diffusive convection

Abstract

Doubly diffusive convection is considered in a vertical slot where horizontal temperature and solutal variations provide competing effects to the fluid density while allowing the existence of a conduction state. In this configuration, the linear stability of the conductive state is known, but the convection patterns arising from the primary instability have only been studied for specific parameter values. We have extended this by determining the nature of the primary bifurcation for all values of the Lewis and Prandtl numbers using a weakly nonlinear analysis. The resulting convection branches are extended using numerical continuation and we find large-amplitude steady convection states can coexist with the stable conduction state for sub- and supercritical primary bifurcations. The stability of the convection states is investigated and attracting travelling waves and periodic orbits are identified using time stepping when these steady states are unstable.

Toward Convectons in the Supercritical Regime: Homoclinic Snaking in Natural Doubly Diffusive Convection

“…Previous studies of localization in this system [6,11,12,34] were restricted to parameter values that provided both subcriticality and bistability. Here, we characterize how the homoclinic snaking that was found in the subcritical regime [12] changes as the system is driven toward supercriticality by decreasing the Prandtl number [8]. The key results we obtained are the discovery of convectons in the supercritical regime and the systematic characterization of bifurcation diagrams during this transition.…”

“…In the next section, we detail the mathematical formulation of the system. In section 3, we review the weakly nonlinear analysis of the system previously derived in [8] and extend the analysis into vertically extended domains, which provides insight into the origin of secondary branches of convectons. In section 4, we consider the nonlinear behavior of the system and describe the successive stages that the snaking secondary branches undergo as the Prandtl number is decreased until the supercritical regime is reached.…”

“…We consider a domain of nondimensional vertical extent L z = 12\lambda c , where \lambda c \approx 2.48 is the wavelength of the critical eigenmode from the primary bifurcation of the conduction state for all values of the Prandtl and Lewis numbers [8,24,51]. The nondimensional boundary conditions read…”

“…Details of the system when N \not = - 1, which leads to the conduction state being replaced by a base flow, are considered in [46]. Numerical continuation of branches in the doubly diffusive convection system was carried out using a spectral element numerical method based on a Gauss-Lobatto-Legendre discretization and supplemented by Stokes preconditioning [3,10,45].…”

Double-diffusive transport in multicomponent vertical convection

“…In the motivating example of convection at a tidewater glacier face, the buoyancy perturbations in the boundary layer are significantly greater than the buoyancy differences in the ambient, so we do not expect detrainment from the wall and layering due to stratification. The flow we consider is turbulent due to the strong buoyancy forcing, and we are far from the marginal stability curves identified for this problem [24,25].…”