Passive tracer patchiness and particle trajectory stability in incompressible two-dimensional flows

Particle motion is considered in incompressible two-dimensional flows consisting of a steady background gyre on which an unsteady wave-like perturbation is superimposed. A dynamical systems point of view that exploits the action--angle formalism is adopted. It is argued and demonstrated numerically that for a large class of problems one expects to observe a mixed phase space, i.e., the occurrence of ``regular islands'' in an otherwise ``chaotic sea.'' This leads to patchiness in the evolution of passive tracer distributions. Also, it is argued and demonstrated numerically that particle trajectory stability is largely controlled by the background flow: trajectory instability, quantified by various measures of the ``degree of chaos,'' increases on average with increasing $|\mathrm{d}\omega/\mathrm{d}I|$, where $\omega (I)$ is the angular frequency of the trajectory in the background flow and $I$ is the action.