Damping filter method for obtaining spatially localized solutions

Intermittent direction reversals of moving spatially localized turbulence observed in two-dimensional Kolmogorov flow

We have found that in two-dimensional Kolmogorov flow a spatially localized turbulent state (SLT) exists stably and travels with a constant speed on average switching the moving direction randomly and intermittently for moderate values of the control parameters: Reynolds number and the flow rate. We define the coarse-grained position and velocity of an SLT and separate the motion of the SLT from its internal turbulent dynamics by introducing a co-moving frame. The switching process of an SLT represented by the coarse-grained velocity seems to be a random telegraph signal. Focusing on the asymmetry of the internal turbulence we introduce two coarse-grained variables characterizing the internal dynamics. These quantities follow the switching process reasonably. This suggests that the twin attracting invariant sets each of which corresponds to a one-way traveling SLT are embedded in the attractor of the moving SLT and the connection of the two sets is too complicated to be represented by a few degrees of freedom but the motion of an SLT is correlated with the internal turbulent dynamics.

Chaotic self-sustaining structure embedded in the turbulent-laminar interface

An interface structure between turbulence and laminar flow is investigated in two-dimensional channel flow. This spatially localized structure not only sustains itself but also converts the laminar state into turbulence actively. A filtered simulation technique is introduced to understand the invading process as an inhomogeneity-induced self-sustaining coherent structure, which consists of a meandering jet on bulk-region and near-wall vortex pairs. A phenomenological model, called the ejection-jet cycle, reveals the relationship between the spatial inner structure of the interface and the invading speed. This model gives insight on the inner-outer interaction in wall-turbulence.

Solitary solutions including spatially localized chaos and their interactions in two-dimensional Kolmogorov flow

Two-dimensional Kolmogorov flow in wide periodic boxes is numerically investigated. It is shown that the total flow rate in the direction perpendicular to the force controls the characteristics of the flow, especially the existence of spatially localized solitary solutions such as traveling waves, periodic solutions, and chaotic solutions, which can behave as elementary components of the flow. We propose a procedure to construct approximate solutions consisting of solitary solutions. It is confirmed by direct numerical simulations that these solutions are stable and represent interactions between elementary components such as collisions, coexistence, and collapse of chaos.

Unstable spiral waves and local Euclidean symmetry in a model of cardiac tissue

This paper investigates the properties of unstable single-spiral wave solutions arising in the Karma model of two-dimensional cardiac tissue. In particular, we discuss how such solutions can be computed numerically on domains of arbitrary shape and study how their stability, rotational frequency, and spatial drift depend on the size of the domain as well as the position of the spiral core with respect to the boundaries. We also discuss how the breaking of local Euclidean symmetry due to finite size effects as well as the spatial discretization of the model is reflected in the structure and dynamics of spiral waves. This analysis allows identification of a self-sustaining process responsible for maintaining the state of spiral chaos featuring multiple interacting spirals.