Exact Coherent Structures of Sheared Double-Diffusive Convection

TL;DR

The paper computes exact coherent structures (ECS) for sheared double-diffusive convection (SDDC) in the diffusive regime within a wall-bounded domain, focusing on 2D equilibria and periodic orbits that arise under uniform background shear. Using a ChFlow-DDC spectral framework, it identifies coexisting tilted-roll equilibria (E1–E3) and mixed modes (C12, C24) connected through subcritical saddle-node bifurcations, with Hopf bifurcations giving rise to periodic orbits (PO1, PO2). Extending to 3D reveals transverse and longitudinal rolls (E2-3D, E4-3D) that can bifurcate subcritically from their 2D counterparts, producing elongated structures and enhanced mixing. DNS shows chaotic trajectories that visit neighborhoods of these ECS, indicating that the ECS provide a skeleton for the turbulent dynamics in SDDC and delineate the shear-influenced transport regime characterized by $\gamma$ being smaller than the DDC-dominated limit but larger than the purely conductive limit. The work advances understanding of nonlinear coherent states in coupled shear–diffusion systems and lays groundwork for exploring higher-Rayleigh-number regimes and multi-layer configurations in ocean-relevant contexts.

Abstract

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 heat and mass transport of polar region oceans. This study computes exact coherent structures (ECS) of diffusive-regime DDC with a uniform background shear in a vertically wall-bounded flow layer. We focus on the shear-influenced regime and present two-dimensional (2D) ECS consisting of steady-state solutions and periodic orbits. The steady-state solutions include tilted convective rolls with various horizontal wavenumbers, and they are invariant under horizontal translation. All tilted convective roll states undergo 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 tilted convective rolls, leading to periodic orbits. Bifurcation diagrams for dimensionless parameters, including the Rayleigh number, the Prandtl number, and the diffusivity ratio, are established, suggesting subcritical behavior. Increasing shear strength stabilizes the 2D tilted convective roll, while these tilted convective rolls continue to exist in the limit of zero density ratio corresponding to sheared Rayleigh-Bénard convection. Extension to the three-dimensional (3D) domain leads to 3D streamwise-elongated roll solutions; one of them originates from a subcritical bifurcation of the corresponding 2D roll solution. 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.

Figures (22)

• Figure 1: Illustration of the flow configuration with the computational domain as $L_x=2\pi$, $L_z=1$ (with vertical coordinates $z\in [-0.5,0.5]$), and varying spanwise size $L_y$. Laminar base state velocity is $\boldsymbol{U}_b=z\hat{\mathbf{e}}_x / \sqrt{Ri}$, where $Ri$ represents the Richardson number. The conduction base state temperature and salinity profiles are $T_b=S_b=-z$.
• Figure 2: Direct numerical simulation of SDDC in a vertically wall-bounded flow layer with parameters $(Ra,Ri,Pr,\tau,\Lambda)=(10^6,1,10,0.01,2)$. (a) Time evolution of $Nu$ and $Sh$. (b) Time history of horizontally-averaged total density $\langle \rho\rangle_h$ is plotted. (c) Distributions of the instantaneous total density field ($\rho$) at three different times illustrate initial development ($t=300$) and density stratification with two layers ($t=6000$) and one layer ($t=30000$) as the final state. (d) Time and horizontally averaged scalar profiles of total temperature $\overline{\langle T\rangle}_{h}$ and salinity $\overline{\langle S\rangle}_{h}$. Dashed gray lines correspond to mean profiles averaged over $t\in [3000,8000]$ displaying two layers, while solid black lines are averaged over $t\in [25000,32000]$ associated with the one-layer state. These time-averaging domains correspond to two red boxes in panel (b). See supplementary movie 1.
• Figure 3: Statistics of DNS data for SDDC. The Rayleigh number $Ra$ versus the time-averaged (a) Nusselt number $\overline{Nu}$, (b) Sherwood number $\overline{Sh}$, and (c) total flux ratio $\overline{\gamma}=\Lambda \tau \overline{Sh}/\overline{Nu}$. The black solid circles represent the laminar (trivial) flow regime, while the green solid circles represent the non-trivial flow regime. The white and gray zones in panels (a, b) indicate the steady and unsteady flow behavior, respectively. The blue zone in panel (c) indicates where tilted rolls (TR) are observed.
• Figure 4: Power spectral density (PSD) of $Nu$ for different $Ra$. Green and red boxes of dashed lines indicate the interested zones of frequencies ($f_{1,2}$), representing differently dominant flow structures with and without dominant tilted rolls, respectively, near the onset of the shear-influenced regime.
• Figure 5: Fluctuations of (a) streamwise velocity $u$, (b) vertical velocity $w$, (c) temperature $\theta$, and (d) salinity $s$, illustrating one pair of counter-rotating convection rolls (E1) for SDDC at parameters $(Ra, Ri, Pr, \tau, \Lambda)=(10^5, 1, 7, 0.01, 2)$.