Distinguished hyperbolic trajectories in time-dependent fluid flows: analytical and computational approach for velocity fields defined as data sets

In this paper we develop analytical and numerical methods for finding special hyperbolic trajectories that govern geometry of Lagrangian structures in time-dependent vector fields. The vector fields (or velocity fields) may have arbitrary time dependence and be realized only as data sets over finite time intervals, where space and time are discretized. While the notion of a hyperbolic trajectory is central to dynamical systems theory, much of the theoretical developments for Lagrangian transport proceed under the assumption that such a special hyperbolic trajectory exists. This brings in new mathematical issues that must be addressed in order for Lagrangian transport theory to be applicable in practice, i.e. how to determine whether or not such a trajectory exists and, if it does exist, how to identify it in a sequence of instantaneous velocity fields. We address these issues by developing the notion of a distinguished hyperbolic trajectory (DHT). We develop an existence criteria for certain classes of DHTs in general time-dependent velocity fields, based on the time evolution of Eulerian structures that are observed in individual instantaneous fields over the entire time interval of the data set. We demonstrate the concept of DHTs in inhomogeneous (or "forced") time-dependent linear systems and develop a theory and analytical formula for computing DHTs. Throughout this work the notion of linearization is very important. This is not surprising since hyperbolicity is a "linearized" notion. To extend the analytical formula to more general nonlinear time-dependent velocity fields, we develop a series of coordinate transforms including a type of linearization that is not typically used in dynamical systems theory. We refer to it as Eulerian linearization, which is related to the frame independence of DHTs, as opposed to the Lagrangian linearization, which is typical in dynamical systems theory, which is used in the computation of Lyapunov exponents. We present the numerical implementation of our method which can be applied to the velocity field C, given as a data set. The main innovation of our method is that it provides an approximation to the DHT for the entire time-interval of the data set. This offers a great advantage over the conventional methods that require certain regions to converge to the DHT in the appropriate direction of time and hence much of the data at the beginning and end of the time interval is lost.