^{1,a)}, Alain Bergeon

^{1,b)}and Edgar Knobloch

^{2,c)}

### Abstract

Natural doubly diffusive convection in a three-dimensional vertical enclosure with square cross-section in the horizontal is studied. Convection is driven by imposed temperature and concentration differences between two opposite vertical walls. These are chosen such that a pure conduction state exists. No-flux boundary conditions are imposed on the remaining four walls, with no-slip boundary conditions on all six walls. Numerical continuation is used to compute branches of spatially localized convection. Such states are referred to as convectons. Two branches of three-dimensional convectons with full symmetry bifurcate simultaneously from the conduction state and undergo homoclinic snaking. Secondary bifurcations on the primary snaking branches generate secondary snaking branches of convectons with reduced symmetry. The results are complemented with direct numerical simulations of the three-dimensional equations.

This work was supported by the Action Thématique de l'Université Paul Sabatier (ATUPS) (C.B.), by CNES under GdR MFA 2799 “Micropesanteur Fondamentale et Appliquée” (C.B. and A.B.), and by the National Science Foundation (Grant Nos. DMS-0908102 and DMS-1211953) (E.K.). E.K. wishes to acknowledge support from the Chaire d'Excellence Pierre de Fermat de la région Midi-Pyrénées (France).

I. INTRODUCTION

II. MATHEMATICAL FORMULATION

III. RESULTS

A. Primary snaking

B. Secondary snaking

IV. TIME-DEPENDENT DYNAMICS

V. DISCUSSION

### Key Topics

- Bifurcations
- 44.0
- Boundary value problems
- 19.0
- Localized states
- 16.0
- Convection
- 15.0
- Fluid equations
- 5.0

## Figures

Sketch of the vertically extended enclosure. The boundary conditions are no-slip for the velocity everywhere, no-flux for the temperature and concentration on all walls except for T = C = 0 at z = 0 (section represented in light gray) and T = C = 1 at z = 1 (section represented in dark gray).

Sketch of the vertically extended enclosure. The boundary conditions are no-slip for the velocity everywhere, no-flux for the temperature and concentration on all walls except for T = C = 0 at z = 0 (section represented in light gray) and T = C = 1 at z = 1 (section represented in dark gray).

Representation of the eigenmodes responsible for the first two instabilities. (a) Marginal eigenmode at the pitchfork bifurcation represented by isosurfaces w = ±W with W chosen appropriately (light indicates w = W > 0 while dark indicates w = −W < 0) and by the isovalues of the streamfunction in the plane y = 1/2 (light indicates clockwise motion while dark indicates counterclockwise motion). (b1) Similar representation of the marginal eigenmode at the transcritical bifurcation (Fig. 3 ). The last two panels show the evolution of the corresponding nonlinear solution along the branch L + using isovalues of the streamfunction in the plane y = 1/2 at Ra ≈ 810 and Ra ≈ 740. (b2) As in (b1) but showing the branch L − corresponding to the eigenfunction .

Representation of the eigenmodes responsible for the first two instabilities. (a) Marginal eigenmode at the pitchfork bifurcation represented by isosurfaces w = ±W with W chosen appropriately (light indicates w = W > 0 while dark indicates w = −W < 0) and by the isovalues of the streamfunction in the plane y = 1/2 (light indicates clockwise motion while dark indicates counterclockwise motion). (b1) Similar representation of the marginal eigenmode at the transcritical bifurcation (Fig. 3 ). The last two panels show the evolution of the corresponding nonlinear solution along the branch L + using isovalues of the streamfunction in the plane y = 1/2 at Ra ≈ 810 and Ra ≈ 740. (b2) As in (b1) but showing the branch L − corresponding to the eigenfunction .

Bifurcation diagram near the primary transcritical bifurcation showing the kinetic energy E as a function of the Rayleigh number Ra along the L + and L − branches. The L − branch bifurcates towards larger Ra but turns around almost immediately in a saddle-node bifurcation that occurs at a very low amplitude. As a result, the two branches appear indistinguishable on the scale of the figure.

Bifurcation diagram near the primary transcritical bifurcation showing the kinetic energy E as a function of the Rayleigh number Ra along the L + and L − branches. The L − branch bifurcates towards larger Ra but turns around almost immediately in a saddle-node bifurcation that occurs at a very low amplitude. As a result, the two branches appear indistinguishable on the scale of the figure.

Bifurcation diagrams representing the kinetic energy E as a function of the Rayleigh number Ra along the (a) L + branch and (b) L − branch. The solutions at each saddle-node are represented in Figs. 5(a) and 5(b) . The L + branch consists of solutions with an odd number of convection rolls. The oscillations in the branch are associated with the nucleation of new rolls at either side of the structure (left edge of the snaking region) and their growth towards full amplitude (right edge). The repeated nucleation of rolls ends when five rolls are present and the container is full after which the energy E increases monotonically with Ra. The L − branch consists of solutions with an even number of rolls but instead of increasing monotonically after the container is full the L − branch turns back towards smaller energies and the solution splits into a two-pulse state with a defect in the center of the domain before leaving the snaking region.

Bifurcation diagrams representing the kinetic energy E as a function of the Rayleigh number Ra along the (a) L + branch and (b) L − branch. The solutions at each saddle-node are represented in Figs. 5(a) and 5(b) . The L + branch consists of solutions with an odd number of convection rolls. The oscillations in the branch are associated with the nucleation of new rolls at either side of the structure (left edge of the snaking region) and their growth towards full amplitude (right edge). The repeated nucleation of rolls ends when five rolls are present and the container is full after which the energy E increases monotonically with Ra. The L − branch consists of solutions with an even number of rolls but instead of increasing monotonically after the container is full the L − branch turns back towards smaller energies and the solution splits into a two-pulse state with a defect in the center of the domain before leaving the snaking region.

Solutions at successive saddle-nodes in the bifurcation diagrams in Fig. 4 . (a) L + branch, (b) L − branch. The solutions are ordered from left to right in terms of increasing distance from the primary bifurcation at Ra t ≈ 850.86. The solutions are shown in terms of surfaces of constant vertical velocity, u = ±U, with U chosen appropriately (light indicates u = −U < 0 while dark indicates u = U > 0). The last snapshot in (a) is taken at Ra ≈ 841 and that in (b) at Ra ≈ 840. In each roll, the flow is upwards near the z = 1 wall and downwards near z = 0.

Solutions at successive saddle-nodes in the bifurcation diagrams in Fig. 4 . (a) L + branch, (b) L − branch. The solutions are ordered from left to right in terms of increasing distance from the primary bifurcation at Ra t ≈ 850.86. The solutions are shown in terms of surfaces of constant vertical velocity, u = ±U, with U chosen appropriately (light indicates u = −U < 0 while dark indicates u = U > 0). The last snapshot in (a) is taken at Ra ≈ 841 and that in (b) at Ra ≈ 840. In each roll, the flow is upwards near the z = 1 wall and downwards near z = 0.

Representation of the marginal eigenmode generating the branches (a) (S c -symmetric), (b) (S △-symmetric), (c) (S △-symmetric), and (d) (S c -symmetric) using a pair of equal and opposite isovalues of the vertical velocity (left panels) and of the z component of the vorticity (right panels). To reveal the twisting nature of the eigenmodes, the coordinate axes have been rotated with respect to earlier figures.

Representation of the marginal eigenmode generating the branches (a) (S c -symmetric), (b) (S △-symmetric), (c) (S △-symmetric), and (d) (S c -symmetric) using a pair of equal and opposite isovalues of the vertical velocity (left panels) and of the z component of the vorticity (right panels). To reveal the twisting nature of the eigenmodes, the coordinate axes have been rotated with respect to earlier figures.

Secondary snaking on the primary branches of localized states in terms of the kinetic energy E as a function of the Rayleigh number Ra. (a) L +, (b) L −. Secondary bifurcations occur on the subcritical parts of the primary branches and lead to the formation of localized twisted solutions represented by the secondary branches , , , , , and . Snapshots of these solutions are available in Figs. 8 and 9 .

Secondary snaking on the primary branches of localized states in terms of the kinetic energy E as a function of the Rayleigh number Ra. (a) L +, (b) L −. Secondary bifurcations occur on the subcritical parts of the primary branches and lead to the formation of localized twisted solutions represented by the secondary branches , , , , , and . Snapshots of these solutions are available in Figs. 8 and 9 .

Snapshots of the secondary branches of solutions shown in Fig. 7 . The snapshots are taken at each saddle-node, beginning at the bifurcation point, and ending at a point on the right of the bifurcation diagram. The same representation as in Fig. 5 is employed. (a) Solutions on the branch (S c symmetric, last snapshot at Ra ≈ 837). (b) Solutions on the branch (S △ symmetric, last snapshot at Ra ≈ 845). (c) Solutions on the branch (S △ symmetric, last snapshot at Ra ≈ 862). (d) Solutions on the branch (S c symmetric, last snapshot at Ra ≈ 864).

Snapshots of the secondary branches of solutions shown in Fig. 7 . The snapshots are taken at each saddle-node, beginning at the bifurcation point, and ending at a point on the right of the bifurcation diagram. The same representation as in Fig. 5 is employed. (a) Solutions on the branch (S c symmetric, last snapshot at Ra ≈ 837). (b) Solutions on the branch (S △ symmetric, last snapshot at Ra ≈ 845). (c) Solutions on the branch (S △ symmetric, last snapshot at Ra ≈ 862). (d) Solutions on the branch (S c symmetric, last snapshot at Ra ≈ 864).

As in Fig. 8 but for (a) and (b). Last snapshots are taken at Ra ≈ 836 and Ra ≈ 832, respectively.

As in Fig. 8 but for (a) and (b). Last snapshots are taken at Ra ≈ 836 and Ra ≈ 832, respectively.

Slices at x = X (top) and x = −X (bottom) of the rightmost states in Figs. 8(c), 8(d), 9(a), and 9(b) , showing isovalues of the vertical velocity u, with dark/white indicating positive/negative values. (a) . (b) . (c) . (d) . In each case, X is chosen to capture the outermost rolls.

Slices at x = X (top) and x = −X (bottom) of the rightmost states in Figs. 8(c), 8(d), 9(a), and 9(b) , showing isovalues of the vertical velocity u, with dark/white indicating positive/negative values. (a) . (b) . (c) . (d) . In each case, X is chosen to capture the outermost rolls.

(a) Time evolution at Ra = 900 of the maximum vertical velocity u max from a small perturbation of the conduction state proportional to the eigenvector responsible from the transcritical bifurcation at Ra = 850.86. The dashed line shows the corresponding result with imposed S y symmetry. (b) Solutions at several different times. (c) Solutions at t = 150 and t = 250 from the direct numerical simulation of Eqs. (2)–(5) with imposed S y symmetry. All snapshots show isovalues of u as in Fig. 5 .

(a) Time evolution at Ra = 900 of the maximum vertical velocity u max from a small perturbation of the conduction state proportional to the eigenvector responsible from the transcritical bifurcation at Ra = 850.86. The dashed line shows the corresponding result with imposed S y symmetry. (b) Solutions at several different times. (c) Solutions at t = 150 and t = 250 from the direct numerical simulation of Eqs. (2)–(5) with imposed S y symmetry. All snapshots show isovalues of u as in Fig. 5 .

(a) Time evolution at Ra = 810 of the maximum vertical velocity u max starting from a solution at Ra ≈ 804 close to the second saddle-node of L +. The dashed line shows the corresponding result with imposed S y symmetry. The lower panel shows the evolution of the asymmetry with respect to S y using the L 2 norm , where f y ≡ (u, v, w, T, C) − S y (u, v, w, T, C), measuring the difference between the solution and its image under S y . When the symmetry S y is not imposed, the solution decays to the conduction state. This is not the case when S y imposed. Panels (b) and (c) show snapshots of the solution in these two cases. All snapshots show isovalues of u as in Fig. 5 .

(a) Time evolution at Ra = 810 of the maximum vertical velocity u max starting from a solution at Ra ≈ 804 close to the second saddle-node of L +. The dashed line shows the corresponding result with imposed S y symmetry. The lower panel shows the evolution of the asymmetry with respect to S y using the L 2 norm , where f y ≡ (u, v, w, T, C) − S y (u, v, w, T, C), measuring the difference between the solution and its image under S y . When the symmetry S y is not imposed, the solution decays to the conduction state. This is not the case when S y imposed. Panels (b) and (c) show snapshots of the solution in these two cases. All snapshots show isovalues of u as in Fig. 5 .

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