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The mesochronic velocity is the average of the velocity field alongtrajectories generated by the same velocity field over a time interval offinite duration. In this paper we classify initial conditions of trajectoriesevolving in incompressible vector fields according to the character of motionof material around the trajectory. In particular, we provide calculations thatcan be used to determine the number of expanding directions and the presence ofrotation from the characteristic polynomial of the Jacobian matrix ofmesochronic velocity. In doing so, we show that (a) the mesochronic velocitycan be used to characterize dynamical deformation of three-dimensional volumes,(b) the resulting mesochronic analysis is a finite-time extension of theOkubo--Weiss--Chong analysis of incompressible velocity fields, (c) thetwo-dimensional mesochronic analysis from Mezic et al. \emph{A New MixingDiagnostic and Gulf Oil Spill Movement}, Science 330, (2010), 486-489, extendsto three-dimensional state spaces. Theoretical considerations are furthersupported by numerical computations performed for a dynamical system arising influid mechanics, the unsteady Arnold--Beltrami--Childress (ABC) flow.

Mesochronic classification of trajectories in incompressible 3D vector fields over finite times