The Lagrangian motion of fluid particles in two-dimensional incompressible flow is described by a Hamiltonian system of equations where the streamfunction plays the role of the Hamiltonian function. For time-independent streamfunctions there is no transport across streamlines and no mixing. A theory of mixing in time-dependent Hamiltonian systems, referred to as lobe dynamics, has been developed by Wiggins (1991) and others, beginning with the periodic case and extending to more general time dependence (Haller and Poje, 1998). In Hamiltonian systems mixing is controlled by the geometry of the stable and unstable manifolds associated with the hyperbolic points in the flow. Application of the theory of lobe dynamics generally requires knowledge of the location of the hyperbolic points and of the stable and unstable manifolds. The theory has been applied to comparatively simple flows, where the hyperbolic points and the manifolds are relatively easy to locate (Miller et al., 1996; Poje and Haller, 1999).
Here we present a simple numerical method for locating the stable and unstable manifolds of a complex flow. The method is tested with a simple analytical flow field and then applied to the large-scale quasi-horizontal flow in the lower stratospheric surf zone, where breaking Rossby waves produce complex mixing structures. In the stratosphere the geometry of the manifolds emerges from the flow within a few days. Hyperbolic points and the heteroclinic tangle of stable and unstable manifolds are easily visible in the surf zone. The manifold geometry explains many of the observed mixing properties of the stratosphere. For example, the mixing barrier around the polar vortex is identifiable as a region that is not foliated by the manifolds. In addition to providing theoretical insight into atmospheric transport and dynamics, knowledge of the manifold geometry should be of great use for such tasks as planning aircraft and balloon missions.
Close window or click on previous window to return to the Conference Program.
12th Conference on Atmospheric and Oceanic Fluid Dynamics