We develop a framework to study the role of variability in transport across a streamline of a reference flow. Two complementary schemes are presented: a graphical approach for individual cases, and an analytical approach for general properties. The s
patially nonlinear interaction of dynamic variability and the reference flow results in flux variability. The characteristic time-scale of the dynamic variability and the length-scale of the flux variability in a unit of flight-time govern the spatio-temporal interaction that leads to transport. The non-dimensional ratio of the two characteristic scales is shown to be a a critical parameter. The pseudo-lobe sequence along the reference streamline describes spatial coherency and temporal evolution of transport. For finite-time transport from an initial time up to the present, the characteristic length-scale of the flux variability regulates the width of the pseudo-lobes. The phase speed of pseudo-lobe propagation averages the reference flow and the flux variability. In contrast, for definite transport over a fixed time interval and spatial segment, the characteristic time-scale of the dynamic variability regulates the width of the pseudo-lobes. Generation of the pseudo-lobe sequence appears to be synchronous with the dynamic variability, although it propagates with the reference flow. In either case, the critical characteristic ratio is found to be one, corresponding to a resonance of the flux variability with the reference flow. Using a kinematic model, we demonstrate the framework for two types of transport in a blocked flow of the mid-latitude atmosphere: across the meandering jet axis and between the jet and recirculating cell.
Hamiltonian dynamical systems possessing equilibria of ${saddle} times {centre} times...times {centre}$ stability type display emph{reaction-type dynamics} for energies close to the energy of such equilibria; entrance and exit from certain regions of
the phase space is only possible via narrow emph{bottlenecks} created by the influence of the equilibrium points. In this paper we provide a thorough pedagogical description of the phase space structures that are responsible for controlling transport in these problems. Of central importance is the existence of a emph{Normally Hyperbolic Invariant Manifold (NHIM)}, whose emph{stable and unstable manifolds} have sufficient dimensionality to act as separatrices, partitioning energy surfaces into regions of qualitatively distinct behavior. This NHIM forms the natural (dynamical) equator of a (spherical) emph{dividing surface} which locally divides an energy surface into two components (`reactants and `products), one on either side of the bottleneck. This dividing surface has all the desired properties sought for in emph{transition state theory} where reaction rates are computed from the flux through a dividing surface. In fact, the dividing surface that we construct is crossed exactly once by reactive trajectories, and not crossed by nonreactive trajectories, and related to these properties, minimizes the flux upon variation of the dividing surface. We discuss three presentations of the energy surface and the phase space structures contained in it for 2-degree-of-freedom (DoF) systems in the threedimensional space $R^3$, and two schematic models which capture many of the essential features of the dynamics for $n$-DoF systems. In addition, we elucidate the structure of the NHIM.
A quantum version of transition state theory based on a quantum normal form (QNF) expansion about a saddle-centre-...-centre equilibrium point is presented. A general algorithm is provided which allows one to explictly compute QNF to any desired orde
r. This leads to an efficient procedure to compute quantum reaction rates and the associated Gamov-Siegert resonances. In the classical limit the QNF reduces to the classical normal form which leads to the recently developed phase space realisation of Wigners transition state theory. It is shown that the phase space structures that govern the classical reaction d ynamicsform a skeleton for the quantum scattering and resonance wavefunctions which can also be computed from the QNF. Several examples are worked out explicitly to illustrate the efficiency of the procedure presented.
