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Turbulent and other nonlinear flows are highly complex and time dependent, but are not fully random. To capture this spatiotemporal coherence, we introduce the idea of a linear neighborhood, defined as a region in an arbitrary flow field where the velocity gradient varies slowly in space over a finite time. Thus, by definition, the flow in a linear neighborhood can be approximated arbitrarily well by only a subset of the fluid-element trajectories inside it. This slow spatiotemporal variation also allows short-time prediction of the flow. We demonstrate that these linear neighborhoods are computable in real data using experimental measurements from a quasi-two-dimensional turbulent flow and find support for our theoretical arguments. We also show that our kinematically defined linear neighborhoods have an additional dynamical significance, in that the scale-to-scale spectral energy flux that is a hallmark of turbulent flows behaves differently inside the neighborhoods. Our results add additional support to the conjecture that turbulent flows locally tend to transport energy and momentum in space or in scale but not both simultaneously.

Local linearity, coherent structures, and scale-to-scale coupling in turbulent flow