Invariant states in inclined layer convection. Part 1. Temporal transitions along dynamical connections between invariant states

TL;DR

This paper investigates complex convection patterns in inclined layers by constructing invariant states and analyzing their dynamical connections, revealing how these states mediate transitions and sustain intricate flow dynamics.

Contribution

It introduces a dynamical systems framework to identify invariant states and their connections in inclined convection, explaining complex patterns beyond linear stability analysis.

Findings

Invariant states underpin observed convection patterns.

Dynamical connections mediate transitions between states.

Complex time-dependent dynamics are supported by state-space trajectories.

Abstract

Thermal convection in an inclined layer between two parallel walls kept at different fixed temperatures is studied for fixed Prandtl number Pr=1.07. Depending on the angle of inclination and the imposed temperature difference, the flow exhibits a large variety of self-organized spatio-temporal convection patterns. Close to onset, these patterns have been explained in terms of linear stability analysis of primary and secondary flow states. At larger temperature difference, far beyond onset, experiments and simulations show complex, dynamically evolving patterns that are not described by stability analysis and remain to be explained. Here we employ a dynamical systems approach. We construct stable and unstable exact invariant states, including equilibria and periodic orbits of the fully nonlinear three-dimensional Oberbeck-Boussinesq equations. These invariant states underlie the observed convection patterns beyond their onset. We identify state-space trajectories that, starting from the unstable laminar flow, follow a sequence of dynamical connections between unstable invariant states until the dynamics approaches a stable attractor. Together, the network of dynamically connected invariant states mediates temporal transitions between coexisting invariant states and thereby supports the observed complex time-dependent dynamics in inclined layer convection.