  The aim of this paper is to study and classify the multiplicity of
distinguished limits and asymptotic solutions for the advection equation with a
general oscillating velocity field with the systematic use of the two-timing
method. Our results are:
  (i) the dimensionless advection equation contains two independent small
parameters, which represent the ratio of two characteristic time-scales and the
spatial amplitudes of oscillations; the scaling of the variables and parameters
contains Strouhal number;
  (ii) an infinite sequence of distinguished limits has been identified; this
sequence corresponds to the successive degenerations of a drift velocity;
  (iii) we have derived the averaged and oscillatory equations for the first
four distinguished limits; derivations are performed up to the forth orders in
small parameters;
  (v) we have shown, that each distinguish limit solution generates an infinite
number of parametric solutions; these solutions differ from each other by the
slow time-scale and the amplitude of prescribed velocity;
  (vi) we have discovered the inevitable appearance of pseudo-diffusion, which
appears as a Lie derivative of the averaged tensor of quadratic displacements;
we have clarified the meaning of pseudo-diffusion using a simple example;
  (vii) our main methodological result is the introduction of a logical order
into the area and classification of an infinite number of asymptotic solutions;
we hope that it can help to study the similar problems for more complex
systems;
  (viii) since in our calculations we do not employ any additional assumptions,
our study can be used as a test for the validity of the two-timing hypothesis;
  (ix) the averaged equations for five different types of oscillating velocity
fields have been considered as the examples of different drifts and
pseudo-diffusion.
