streamlines’ dynamics in the vicinity of a resonance. Scattering on resonance and capture into resonance are discussed in Subsects. 4.1 and 4.2, respectively. We show that both capture and scattering destroy the adiabatic invariance and we obtain expressions for jumps of adiabatic invariants during a single passage. As the magnitude of the jumps is very sensitive to initial conditions, in the case of multiple passages through resonance these phenomena can be treated as random processes and we derive their statistical properties. In Sect. 5 we apply the technique developed in the previous sections to a particular problem of mixing between two counter-rotating cylinders. In Sect. 6 we discuss the long time dynamics of streamlines and present results of numerical simulations. We show that multiple passages through resonance lead to the destruction of adiabatic invariance and chaotic advection and estimate characteristic time of mixing. And finally, we show that mixing can be described using a single diffusion-type PDE, in which the diffusion coefficient is provided not by the molecular diffusion, but by the averaged influence of the resonance processes. Thus, we call the resulting diffusion the [*adiabatic diffusion*]{}.

Main equations. The unperturbed and the averaged systems.
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In the first part 