Bifurcation diagram based on Ref. 24 showing the L 2 norm of stationary localized solutions u(x) of Eq. (1) in a periodic domain of period 20π as a function of r when ν = 2. The localized structures are homoclinic orbits to the trivial state u = 0 passing near a periodic state and lie on a pair of snaking branches.
Sketch of the horizontally extended domain. The boundary conditions on the square cross-section are no-slip, with no-flux conditions for the temperature and concentration on the z-walls, T = C = 0 at y = 0 (section represented in light gray) and T = C = 1 at y = 1 (section represented in dark gray). The boundary conditions on the x-walls may be either periodic or no-slip/no-flux.
Bifurcation diagram for CCBC showing the vicinity of the primary bifurcation from the conduction state, Ra ≈ 1719, in terms of the kinetic energy E as a function of the Rayleigh number Ra. The bifurcation is transcritical and generates two branches of solutions. Of these, the subcritical branch L 0 contains localized states while the supercritical branch consists of spatially extended states. The left (Ra ≈ 1649) and right (Ra ≈ 1771) panels show, respectively, the isovalues of the vertical velocity w in the central cross-section on the subcritical and supercritical parts of the L 0 branch, dark (light) shading indicating positive (negative) velocity.
Bifurcation diagrams for CCBC showing the kinetic energy E as a function of the Rayleigh number Ra for (a) the L 0 branch and (b) the secondary branches and that bifurcate from L 0 at Ra ≈ 1224 and Ra ≈ 1226, respectively.
Snapshots of localized states on the L 0 branch. From bottom to top: representation of the marginal eigenmode and of the solution at Ra ≈ 752, followed by the solutions at each of the three saddle-nodes, ending with the solution at Ra ≈ 851 along the upper part of the branch. The flow is represented using two isovalues of the y velocity, v = ±V, with V chosen appropriately (light indicates v = −V < 0 while dark indicates v = V > 0). A different value of V is used for the marginal eigenmode in the lowest panel.
Snapshots of the localized states lying on (a) the branch and (b) the branch. From bottom to top: representation of the solution at the branching point on L 0 followed by the solutions at each subsequent saddle-node along the branch and ending with a solution taken on the upper part of the snaking branch at Ra ≈ 1380. The flow is represented by two isovalues of the z velocity, w = ±W, with W chosen appropriately (light indicates w = −W < 0 while dark indicates w = W > 0).
An solution at Ra = 1364 in terms of the three velocity fields u, v, and w. In each panel the representation uses equal and opposite contour values, the light (dark) color indicating negative (positive) velocity. For this solution the maximum values of u, v, and w are u max ≈ 0.40, v max ≈ 0.23, and w max ≈ 0.62.
(a) Bifurcation diagram representing the kinetic energy E as a function of the Rayleigh number Ra along branch L 0 (dashed line) for a closed container, and H x and (solid lines) for a periodic container. The branch H x bifurcates from the conduction solution E = 0 and consists of solutions invariant in the x direction. The branch of localized states bifurcates from H x and is similar to the branch L 0 in the closed container, but unlike L 0 it terminates on H x close to its saddle-node instead of extending towards higher Rayleigh numbers. (b) From bottom to top: snapshot along H x at Ra ≈ 1485 (lower part of ), at Ra ≈ 1462 (upper part of H x ) and along at Ra ≈ 685. Left slices represent isovalues of the vertical velocity w in the plane x = L/2 while the right snapshots represent isosurfaces of opposite values of the y-velocity (dark: v > 0, light: v < 0).
(a) Bifurcation diagram representing the kinetic energy E as a function of the Rayleigh number Ra for the branches of twisted localized states with no-slip ( , dashed lines) and periodic ( , full lines) boundary conditions. (b) Bifurcation diagram for the twisted localized states with PBC: emerge from a branch of homogeneous localized states and terminate on a branch of periodic twisted states P x . The latter bifurcates from the branch H x of 2D states connecting P x and .
Periodic solution P x at Ra ≈ 1600 consisting of a zigzag pattern of twisted rolls, visualized using two opposite values of the z component of the velocity w. (a) A three-dimensional rendering. (b) Top view showing the twist from a better angle.
Same as Fig. 1 but showing collapsed snaking in a 40π domain when ν = 3.75. The localized structures are fronts connecting two distinct homogeneous states and lie on a branch that grows vertically as the nontrivial state u ≠ 0 invades the domain.
Sketch of the bifurcation scenario for periodic boundary conditions in the extended direction. A typical integral quantity such as the kinetic energy is represented against the Rayleigh number. The bifurcation diagram has been stretched for convenience and is not to scale. Representative solutions along H x , P x , and are shown in Figs. 8(b), 10, and 6 , respectively, while those along resemble those in Fig. 5 .
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