Advection equation analysed by two-timing method

TL;DR

This paper systematically analyzes the advection equation with oscillating velocity fields using the two-timing method, classifying multiple asymptotic solutions, introducing pseudo-diffusion, and providing a framework for complex oscillatory systems.

Contribution

It introduces a classification of infinite distinguished limits and solutions for the advection equation, including pseudo-diffusion, without additional assumptions, advancing the methodological understanding.

Findings

Identified an infinite sequence of distinguished limits.

Derived averaged and oscillatory equations up to fourth order.

Discovered pseudo-diffusion as a Lie derivative of quadratic displacements.

Abstract

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.