Transport of passive scalars in a chain of vortices in a shear layer is studied using a model motivated by the quasigeostrophic equation, and a discrete map model. Surrounding the vortices there is a stochastic layer where particles alternate chaotically between being trapped in the vortices, and moving following the shear flow. Transport in the stochastic layer is asymmetric: Mixing between the vortices and the up-stream flow is, in general, different from mixing between the vortices and the down-stream flow. We use the Melnikov method to study this asymmetry, and to construct a generalized separatrix map model for asymmetric transport. The statistics of the passive scalar is non-Gaussian. In particular, there is anomalous advection, and anomalous (non-Brownian) diffusion. Thus, transport in this system cannot be described by an advection-diffusion equation with an effective diffusivity. The probability density function (PDF) of particle displacements&dgr;x,P(&dgr;x,t),is asymmetric and broader than Gaussian. At large times,Prelaxes to a self-similar limit distribution of the formt−&ggr;/2f(X/t&ggr;/2),whereX≡&dgr;x−⟨&dgr;x⟩,fis a scaling function, and &ggr; is the anomalous diffusion exponent. As a result, the moments scale as⟨Xn⟩∼tn&ggr;/2.We present a systematic study of the dependence of the mean, the variance, the skewness, and the flatness, on the parameters controlling the asymmetry of the flow. The PDFs of the duration of flight (motion following the shear flow) events, and vortex trapping events, exhibit algebraic decay. In some cases, the flights correspond to Le´vy flights. The results of the model are compared with recent experiments on chaotic advection and Le´vy flights in a rotating annulus. ©1998 American Institute of Physics.