Local and global bifurcations to large-scale oblique patterns in inclined layer convection

TL;DR

This work addresses the emergence of the switching diamond panes SDP in inclined layer convection by computing and continuing invariant solutions of the 3D Oberbeck–Boussinesq equations. Through equivariant bifurcation theory and systematic nonlinear analysis, it uncovers five branches (TW1, TW2, PO1, PO2, PO3) bifurcating simultaneously from the transverse-roll state FP1 at $Ra≈9692.2$ due to broken $D_4$ and $O(2)$ symmetries, and reveals a heteroclinic cycle between PO1 and PO3 that persists into the weakly turbulent regime. It also identifies a global homoclinic bifurcation from ribbons producing PO4 and demonstrates edge states FP3 and PO5 that organize transitions between attractors. Together, these invariant solutions and their bifurcations provide a dynamical-systems framework to interpret SDP’s large-scale oblique structure and the onset of defects far from linear onset.

Abstract

In the inclined layer convection system, thermal convection in a Rayleigh--Bénard cell tilted against gravity, the flow is subject to competing buoyancy and shear forces. For varying inclination angle ($γ$) and Rayleigh number ($Ra$), a variety of spatio-temporal patterns is observed. We investigate the switching diamond panes (SDP) pattern, observed at $(γ, Ra)\simeq(100\degree,10000)$, which exhibits large-scale oblique features and is one of the five complex tertiary patterns at Prandtl number $Pr=1.07$. First, we study the linear instability of the secondary-state transverse convection rolls and the five branches including two travelling waves and three periodic orbits, bifurcating simultaneously from it. These non-generic bifurcations arise from the breaking of the spatial $D_4$ and $O(2)$ symmetries of transverse rolls, and the resulting bifurcated solutions show large-scale diamond-shaped amplitude modulations. Second, we explore a periodic orbit that captures both the large-scale structure and small-scale defects of modulated rolls. Parametric continuation in $Ra$ reveals the global homoclinic bifurcation via which this periodic orbit emerges. Third, the boundary of the basins of attraction of two dynamically relevant periodic orbits has been characterised. Specifically, additional steady and time-periodic solutions are identified on the basin boundary and their bifurcation structures are analysed. Together, using non-linear invariant solutions and their bifurcations, we take a further step toward understanding the emergence and dynamics of SDP far from the onset of convection, where linear methods have not been applied successfully.

Figures (12)

• Figure 1: Schematic of the convection cell where the confined fluid layer is inclined against the gravity by angle $\gamma$. The co-ordinates $x$, $y$ and $z$ represent the streamwise, spanwise and wall-normal directions, respectively. The flow is bounded between two fixed walls in $z$, at $z=-0.5$ where the fluid is heated and at $z=0.5$ where the fluid is cooled. For $\gamma > 90 \degree$, the fluid is heated from above. The velocity ($\boldsymbol u_0$) and temperature ($\mathcal{T}_0$) profile of laminar base solution are shown by orange curve and green line, respectively. All flow field snapshots presented in this paper are temperature field visualised at the midplane $z=0$.
• Figure 2: Snapshots of the midplane temperature field. (a--d) Simulations in the large spatial domain $[L_x, L_y, L_z] = [100, 50, 1]$ at four Rayleigh numbers $Ra=9600$ (a), $9800$ (b), $10000$ (c), and $10200$ (d). From (a) to (d), the flow transitions from almost straight (in $y$) transverse convection rolls to spatio-temporally chaotic switching of large amplitude regions of transverse rolls. The complex pattern in (b) and (c) is called switching diamond panes (SDP). (e--f) Simulations in the small periodic domain $[L_x, L_y, L_z] = [26.6, 12.1, 1]$ at $Ra=9800$ (e, two snapshots) and $Ra=10000$ (f, six snapshots). The time series of $\|\theta\|_2$ at $Ra=9800$ and $10000$ are shown in panels (b) and (f) of figure \ref{['SDP_timeseries9750_10000']}, respectively. The same colour bar is used in all plots.
• Figure 3: (a) Bifurcation diagram of one equilibrium state, two travelling waves, and three periodic orbits. Solid and dashed curves indicate linearly stable and unstable states, respectively. The inset zooms in on the Hopf bifurcation at which five solution branches---TW1, TW2, PO1, PO2, and PO3---bifurcate simultaneously from FP1. The two curves representing the maximum and minimum of $\| \theta \|_2$ for PO3 (shown in orangered) are too close to be distinguished near the bifurcation point. (b) Periods of the three periodic orbits. (c) Time series of $\|\theta\|_2$ for the three periodic orbits at $Ra=10057$. (d--o) Snapshots of the midplane temperature of FP1, TW1, TW2, PO1, PO2, and PO3. In (a--c), the stars and labels indicate the locations of the corresponding states visualised in (d--o). The same colour bar is used in all plots. (q--u) Schematics of five states simultaneously bifurcating from a square lattice (p) in a Hopf bifurcation. The patterns in (q,r) travel in the direction of the orange arrows; in (s,t), they oscillate between opposite edges and vertices of the square; and in (u), they rotate around the origin. The names describing each of the states in (q--u) are those used by Silber1991 and Swift1988. Schematics are inspired by figure 2 of Swift1988 and figure 4 of Silber1991.
• Figure 4: (a) Eigenvector $e_1$ responsible for FP1$\rightarrow$TW1 bifurcation and (b) its quarter-diagonal translation $\tau(L_x/4,L_y/4) e_1$. (c) Linear combination $(e_1 + \tau(L_x/4,L_y/4) e_1)/\sqrt{2}$. (d) Eigenvector $e_2$ responsible for FP1$\rightarrow$TW2 bifurcation and (e) its $xz$-reflection $\pi_{xz} e_2$. (f) Linear combination $(e_2 + \pi_{xz} e_2)/\sqrt{2}$. All eigenvectors are visualised using the temperature field in the $x$--$y$ plane at $z=0$. The same colour bar is used in all plots.
• Figure 5: (a--c) Time series for the long-time dynamics at $Ra=9710$, $9750$, and $9790$. The inset in (a) zooms in on the oscillatory behaviour, corresponding to PO3, between $60000\lesssim t \lesssim 64000$ with very small oscillation amplitude. (d--f) Phase portraits of the long DNS and three periodic orbits. The torus-like dynamics shows a heteroclinic cycle between two saddle orbits PO1 and PO3.