Nonlinear Dynamics: Coherent Structures

The interaction between shear and double-diffusive convection (DDC) in the diffusive regime (cold fresh water on top of hot salty water) plays an important role in the basal melting of ice shelves in polar regions. This study computes exact coherent structures (ECS) of two-dimensional diffusive-regime DDC with a uniform background shear in a vertically wall-bounded flow layer. The ECS consists of steady-state solutions and periodic orbits. The steady-state solutions include convective rolls with various horizontal wavenumbers, and they are invariant under horizontal translation. All convective roll states undergo the saddle-node bifurcation leading to a stable upper branch and an unstable lower branch, suggesting that they originate from the subcritical bifurcation of conduction base states. Hopf bifurcations appear on the stable upper branch of convective rolls, leading to periodic orbits. Bifurcation diagrams for dimensionless parameters, including the Prandtl number and diffusivity ratio, are established, suggesting subcritical behavior. Chaotic solutions from direct numerical simulations generally visit neighborhoods of these steady or periodic solutions, and these visits leave an imprint on the flow statistics.