Convectons in unbalanced natural doubly diffusive convection

Fluids subject to both thermal and compositional variations can undergo doubly diffusive convection when these properties both affect the fluid density and diffuse at different rates. In natural doubly diffusive convection, the gradients of temperature and salinity are aligned with each other and orthogonal to gravity. The resulting buoyancy-driven flows are known to lead to the formation of a variety of patterns, including spatially localized states of convection surrounded by quiescent fluid, which are known as convectons. Localized pattern formation in natural doubly diffusive convection has been studied under a specific balance where the effects of temperature and salinity changes are opposite but of equal intensity on the fluid density. In this case, a steady conduction state exists and convectons bifurcate from it. The aforementioned buoyancy balance underpins our knowledge of this pattern formation but it is an ideal case that can hardly be met experimentally or in nature. This article addresses how localized pattern formation in natural doubly diffusive convection is affected by departures from the balanced case. In particular, the absence of a conduction state leads to the unfolding of the bifurcations to convectons. In thermally dominated regimes, the background flow promotes localized states with convection rolls attached to the end-walls, known as anticonvectons, and the existence of these states is found to be related to the emergence of convectons. The results presented here shed light on the convecton robustness against changes in the buoyancy ratio and extend the scope of our understanding of localized pattern formation in fluid systems.

💡 Research Summary

This paper investigates the formation and bifurcation structure of spatially localized convection patterns—convectons—in natural doubly‑diffusive convection when the classical buoyancy‑ratio balance (N = ‑1) is violated. In the balanced case, temperature and salinity (or any two scalar fields) affect density with equal magnitude but opposite sign, allowing a static conduction state (linear temperature and salinity profiles, zero flow) to exist. Convectons then emerge from this trivial state via transcritical and pitchfork bifurcations, producing families of localized roll structures that exhibit homoclinic snaking. The authors use a two‑dimensional rectangular cavity with no‑slip side walls, fixed temperature and salinity values on the vertical walls, and insulating end walls. The governing equations are the Boussinesq Navier–Stokes system with Pr = 1, Lewis number Le = 5, Rayleigh number Ra as the primary continuation parameter, and buoyancy ratio N as a secondary control parameter. Numerical continuation is performed with a spectral‑element method (Gauss–Lobatto–Legendre nodes, 25 × 25 per element) and Stokes preconditioning.

First, the balanced case (N = ‑1) is reproduced. The conduction state loses stability through a pitchfork bifurcation that generates asymmetric branches, and through a transcritical bifurcation that yields two symmetric branches, denoted L⁺ and L⁻. L⁺ carries an anticlockwise central roll, L⁻ a clockwise one; both evolve into localized convectons with an odd (L⁺) or even (L⁻) number of rolls as Ra is varied. The classic homoclinic snaking scenario is observed: as the solution branch snakes back and forth in parameter space, additional rolls are added one by one, creating a ladder of multi‑convecton states (e.g., L⁺₂, L⁻₂). Domain length influences the ordering of the primary bifurcations, but the overall snaking structure persists.

The core contribution lies in exploring N ≠ ‑1. When N deviates from –1, the conduction state disappears because the thermal and solutal buoyancy contributions no longer cancel. Consequently, the original pitchfork and transcritical bifurcations “unfold,” giving rise to new branches that are not anchored to a trivial state. Two regimes are distinguished:

1. Thermally dominated regime (N > ‑1). Here the thermal buoyancy exceeds the stabilizing solutal contribution, producing a background shear flow that attaches convection rolls to the end walls. These wall‑attached rolls constitute a new family called “anticonvectons.” An anticonvecton features a quiescent core flanked by counter‑rotating rolls attached to the walls; its circulation is opposite to that of the standard convecton. Numerical continuation shows that as N increases, the anticonvecton branch connects smoothly to the L⁺ convecton branch, providing a mechanistic pathway for convecton creation in the absence of a conduction state. The background flow thus replaces the conduction‑to‑convection transition with a flow‑to‑convection transition, and localized structures appear via a secondary bifurcation from the wall‑attached state.

2. Solutally dominated regime (N < ‑1). The solutal buoyancy dominates, raising the critical Rayleigh number for instability. The conduction state persists but becomes less robust; convectons still exist but over a narrower Ra interval, and higher‑order multi‑convecton branches require larger Ra to stabilize. The snaking ladder is compressed, and some branches terminate in saddle‑node bifurcations earlier than in the balanced case.

The authors map out the bifurcation diagrams for three cavity heights (Lz ≈ 4λc, 5λc, 12λc) and demonstrate how the presence of end walls (no‑flux conditions) eliminates translational symmetry, thereby suppressing the pitchfork of revolution seen in periodic domains. This confinement forces the primary eigenmode to vanish at the ends, shaping the spatial structure of the emerging rolls.

Key insights include:

• The buoyancy ratio N is the primary control of convecton robustness; modest departures from –1 already alter the existence and stability ranges.
• Anticonvectons are a novel localized pattern that only appears when thermal buoyancy dominates, highlighting that real‑world systems (where exact N = ‑1 is unlikely) can support qualitatively different localized states.
• The unfolding of bifurcations demonstrates that convectons do not require a static conduction base; they can be generated from a pre‑existing shear flow.
• Multi‑convecton states (bound states of convectons and anticonvectons) persist across a range of N, but their parameter windows shift systematically with N.

The paper concludes by emphasizing the relevance of these findings to laboratory experiments and geophysical contexts (e.g., oceanic thermohaline staircases, planetary interiors) where temperature and compositional gradients coexist but rarely satisfy the ideal N = ‑1 balance. Future work is suggested on three‑dimensional geometries, rotating frames, magnetic fields, and more realistic boundary conditions to further assess the universality of the uncovered mechanisms.