Advection equation analysed by two-timing method
  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. 
V.A. Vladimirov
physics.flu-dyn math-ph math.MP