Statistical topology of streamlines in a time-invariant 2D flow

Two dimensional active flows are relevant in a number of areas in biology, particularly in the system of mucociliary clearance in the airway. In these systems, the flow may have a general direction but also exhibit significant patterns of disorder. The qualitative function of these systems may depend on the topology of the resultant flow patterns. Motivated by this, we ask the following question: what is the probability for a generic streamline in a two-dimensional random flow to be open or closed? To investigate this question, we introduce a framework for studying ensembles of two-dimensional time-invariant flow fields. We are able to establish two separate upper bounds on the trapped area a, one for a general ensemble and one for Gaussian ensembles, by leveraging different insights about the distribution of flow velocities on the closed and open streamlines. We also deduce an exact power series expression for a based on the asymptotic dynamics of flow field trajectories, although the terms in this series remain difficult to compute. Finally, we verify the validity of our bounds using numerical evidence, and comment on the implications of these numerical results on future investigations of this problem.