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.

中文翻译：

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倾斜层对流中的不变状态。第 1 部分。沿不变状态之间的动态连接的时间转换

对于固定的普朗特数 Pr=1.07，研究了保持在不同固定温度的两个平行壁之间的倾斜层中的热对流。根据倾斜角度和施加的温差，流动表现出多种自组织的时空对流模式。接近开始时，这些模式已根据主要和次要流动状态的线性稳定性分析进行了解释。在更大的温差下，远远超出开始，实验和模拟显示出复杂的、动态演变的模式，稳定性分析没有描述这些模式，还有待解释。在这里，我们采用动态系统方法。我们构建稳定和不稳定的精确不变状态，包括完全非线性三维 Oberbeck-Boussinesq 方程的平衡和周期轨道。这些不变状态是观察到的对流模式的基础。我们确定状态空间轨迹，从不稳定的层流开始，遵循不稳定不变状态之间的一系列动力学连接，直到动力学接近稳定的吸引子。总之，动态连接的不变状态网络调节共存不变状态之间的时间转换，从而支持在倾斜层对流中观察到的复杂的时间相关动力学。