  The motions of a passive scalar $\hat{a}$ in a general high-frequency
oscillating flow are studied. Our aim is threefold: (i) to obtain different
classes of general solutions; (ii) to identify, classify, and develop related
asymptotic procedures; and (iii) to study the notion of drift motion and the
limits of its applicability. The used mathematical approach combines a version
of the two-timing method, the Eulerian averaging procedure, and several novel
elements. Our main results are: (i) the scaling procedure produces two
independent dimensionless scaling parameters: inverse frequency $1/\omega$ and
displacement amplitude $\delta$; (ii) we propose the \emph{inspection
procedure} that allows to find the natural functional forms of asymptotic
solutions for $1/\omega\to 0, \delta\to 0$ and leads to the key notions of
\emph{critical, subcritical, and supercritical asymptotic families} of
solutions; (iii) we solve the asymptotic problems for an arbitrary given
oscillating flow and any initial data for $\hat{a}$; (iv) these solutions show
that there are at least three different drift velocities which correspond to
different asymptotic paths on the plane $(1/\omega,\delta)$; each velocity has
dimensionless magnitude ${O}(1)$; (v) the obtained solutions also show that the
averaged motion of a scalar represents a pure drift for the zeroth and first
approximations and a drift combined with \emph{pseudo-diffusion} for the second
approximation; (vi) we have shown how the changing of a time-scale produces new
classes of solutions; (vii) we develop the two-timing theories of a drift based
on both the \emph{GLM}-theory and the dynamical systems approach; (viii)
examples illustrating different options of drifts and pseudo-diffusion are
presented.
