Natural convection in a vertical channel. Part 3. Bifurcations of many (additional) unstable periodic orbits and their dynamical relevance

TL;DR

The paper investigates vertical thermal convection as a high-dimensional nonlinear dynamical system by computing a large network of invariant solutions—seven unstable equilibria FP7–FP13 and 26 unstable periodic orbits (UPOs)—in a fixed domain, extending prior work to Ra up to ~6650. Using recurrent flow analysis and a shooting-based Newton solver, the authors perform parametric continuation in Rayleigh number and classify bifurcations (Hopf, pitchfork, saddle–node, period-doubling/halving, global homoclinic/heteroclinic, isolas) across eight symmetry subspaces, revealing complex structures including robust heteroclinic cycles and isola branches. They demonstrate that many unstable orbits capture key spatio-temporal patterns and can reconstruct core attractor statistics, providing a foundation for a periodic-orbit description of transitional turbulence in 3D Navier–Stokes convection. The work also shows that a large domain inherits fingerprints of small-domain invariant solutions, supporting the relevance of periodic orbit theory for understanding and predicting large-domain chaotic convection. Overall, the results offer a detailed, symmetry-resolved map of invariant solutions and underscore the potential of periodic orbit theory to quantify transitional turbulence in high-dimensional flows.

Abstract

Vertical thermal convection system exhibits weak turbulence and spatio-temporally chaotic behaviour. In this system, we report seven equilibria and 26 periodic orbits, all new and linearly unstable. These orbits, together with four previously studied in Zheng et al. (2024b) bring the number of periodic orbit branches computed so far to 30, all solutions to the fully non-linear three-dimensional Navier--Stokes equations. These new invariant solutions capture intricate spatio-temporal flow patterns including straight, oblique, wavy, skewed and distorted convection rolls, as well as bursts and defects in rolls. These interesting and important fluid mechanical processes in a small flow unit are shown to appear locally and instantaneously in a chaotic simulation in a large domain. Most of the solution branches show rich spatial and/or spatio-temporal symmetries. The bifurcation-theoretic organisation of these solutions is discussed; the bifurcation scenarios include Hopf, pitchfork, saddle--node, period-doubling, period-halving, global homoclinic and heteroclinic bifurcations, as well as isolas. These orbits are shown to be able to reconstruct statistically the core part of the attractor, and these results may pave the way to quantitatively describing transitional fluid turbulence using periodic orbit theory.

Figures (28)

• Figure 1: Vertical convection cell with size $[L_x, L_y, L_z] = [1, 8, 9]$. The flow is bounded between two fixed walls at $x=\pm0.5$ at which the flow is heated and cooled respectively. We visualize the flow on the $y$-$z$ plane at $x=0$ (dotted), from left to right as indicated by the eye and arrow. The laminar velocity $\boldsymbol u_0(x) = \sqrt{Ra/Pr} (x/4 - x^3)/6 \:\boldsymbol e_z$ and temperature $\mathcal{T}_0(x) = x$ of this system are traced as an orange curve and a green line, respectively.
• Figure 2: (a) Bifurcation diagram of equilibria and (b-k) flow structures visualized via the midplane temperature field. (b) FP1, (c) FP2 and (d) FP4 have been presented in Zheng2024part2 and are shown with thinner curves in (a). All branches shown are unstable, with the exception of FP1 for $Ra<6056$ and of FP2 for $6056<Ra<6058.5$. Two enlarged diagrams are shown on the right zooming in on the FP2$\rightarrow$FP7$\rightarrow$FP8 and FP9$\rightarrow$FP10$\rightarrow$FP11 bifurcations. (e) FP7 bifurcates from FP2 at $Ra=6279.5$; (f) FP8 bifurcates from FP7 at $Ra=6282.9$. (g) FP9 bifurcates from the unstable base state at $Ra=5941$; (h) FP10 bifurcates from FP9 at $Ra=6360$; (i) FP11 bifurcates from FP10 at $Ra=6369.2$ and undergoes a saddle--node bifurcation at $Ra=6213.5$; (j) FP12 bifurcates from FP9 at $Ra=6184$. (k) FP13 undergoes a saddle--node bifurcation at $Ra=6449$ and both upper and lower branches exist at least until $Ra=6800$. The intersection of the saddle--node bifurcation point of FP11 and FP13 with the nearby branches is due only to projection and does not represent a bifurcation.
• Figure 3: Temperature norms (a) and periods (b) of periodic orbits. Abbreviations PO, RPO and PPO stand for periodic orbit, relative periodic orbit and pre-periodic orbit, respectively. Orbits PO2--PO4 are discussed in detail in Zheng2024part2. In (a), for each orbit, we show two curves, the maximum and minimum of $\lvert\lvert \theta \lvert\lvert_2$ along an orbit. All of RPO5--PPO30 are linearly unstable. The upper limit of (b) is set to $T=700$, even though some orbits are continued to higher period. The bifurcation scenarios include Hopf, pitchfork, saddle--node, period-doubling, period-halving, global homoclinic/heteroclinic bifurcations and isolas. For more clarity, bifurcation diagrams for selected sets of orbits will be shown in figures \ref{['part3_sepa_BD_PO13_15_26_28']}, \ref{['part3_sepa_BD_PO17_18_27']}, \ref{['part3_sepa_BD_PO19_25']}, \ref{['part3_sepa_BD_PO6_10_14_16_29']}, \ref{['part3_sepa_BD_PO5_7_8_9_30']}, \ref{['part3_sepa_BD_PO11_12_20']} and \ref{['part3_sepa_BD_PO21_22_23_24']}. The apparent lack of smoothness in some $\lvert\lvert \theta \lvert\lvert_2$ curves corresponds to the overtaking of one temporal maximum or minimum of $\lvert\lvert \theta \lvert\lvert_2$ by another as $Ra$ is varied.
• Figure 4: Temperature norms (left) and periods (right) of RPO13, RPO15, RPO26 and RPO28. Branch RPO13 bifurcates from and terminates on RPO18 (which is shown more completely in figure \ref{['part3_sepa_BD_PO17_18_27']}) in two period-doubling bifurcations. The bifurcation points are indicated by stars on the right plot. Branches RPO15, RPO26 and RPO28 begin and terminate at saddle--node bifurcations and form isolas.
• Figure 5: Temperature norms (left) and periods (right) of RPO17, RPO18 and RPO27. The RPO17 branch forms an isola. Orbit RPO18 bifurcates from FP2 in a global homoclinic bifurcation at $Ra=6277.96$ and continues to exist up to at least $Ra=6686$. Orbit RPO27 is generated from RPO18 in a pitchfork bifurcation at $Ra=6279.7$ and continues to exist up to at least $Ra=6650$.