  In this paper we develop and use the two-timing method for a systematic study
of a scalar advection caused by a general oscillating velocity field.
Mathematically, we study and classify the multiplicity of distinguished limits
and asymptotic solutions produced in the two-timing framework. Our calculations
go far beyond the usual ones, performed by the two-timing method. We do not use
any additional assumptions, hence our study can be seen as a test for the
validity and sufficiency of the two-timing hypothesis. Physically, we derive
the averaged equations in their maximum generality (and up to high orders in
small parameters) and obtain qualitatively new results. Our results are: (i)
the dimensionless advection equation contains \emph{two independent
dimensionless small parameters}: the ratio of two time-scales and the spatial
amplitudes of oscillations; (ii) we identify a sequence of \emph{distinguished
limit solutions} which correspond to the successive degenerations of a
\emph{drift velocity}; (iii) for a general oscillating velocity field we derive
the averaged equations for the first \emph{four distinguished limit solutions};
(iv) we show, that \emph{each distinguish limit solution} produces an infinite
number of \emph{parametric solutions} with a Strouhal number as the only large
parameter; those solutions differ from each other by the slow time-scale and
the velocity amplitude; (v) the striking outcome of our calculations is the
inevitable appearance of \emph{vibrodiffusion}, which represents a Lie
derivative of the averaged tensor of quadratic displacements; (vi) our main
methodological result is the introduction of a logical order/classification of
the solutions; we hope that it opens the gate for applications of the same
ideas to more complex systems; (vii) five types of oscillating flows are
presented as examples of different drifts and vibrodiffusion.
