Temporal modulation of nonlinear Lagrangian dynamics and finite-time stretching in incompressible flows

Abstract External temporal signals can alter flow deformation and particle transport, yet how changes in signal characteristics are transmitted into Eulerian and Lagrangian responses remains unclear. Here, a temporal input signal modulates the spatial velocity field, and its modulation intensity is quantified by the phase modulation rate (PMR), defined as the standard deviation of the input time derivative. Theoretical derivation shows that signal evolution enters the velocity-gradient tensor, establishing a direct mechanism for input-induced Eulerian deformation. We characterize the resulting response using Eulerian activity, measured by velocity-fluctuation energy and strain-rate magnitude, and finite-time Lagrangian stretching, quantified by the finite-time Lyapunov exponent (FTLE). To examine this framework, a divergence-free two-dimensional multiscale flow is constructed using a phase-modulated stream function and driven by Periodic, Ornstein--Uhlenbeck (OU), Lorenz, and smoothed OU inputs. Across all forcing families, increasing PMR produces a comparatively regular enhancement of Eulerian activity. FTLE statistics show an overall increase with PMR, accompanied by local non-monotonic variations arising from strain--rotation interactions and separation-direction dynamics. The upper-tail FTLE statistics, particularly P95 and P99, show stronger responses to temporal modulation. Refined OU scans confirm the response trends across different baseline correlation times, while Navier--Stokes CFD simulations reproduce the same modulation-dependent progression. These results characterize the relationship between temporal input modulation, Eulerian deformation, and finite-time Lagrangian stretching, with PMR serving as a common measure for comparing responses across the forcing families examined.