Many Faces of Boussinesq Approximations

TL;DR

This paper systematically analyzes the Boussinesq approximation equations using asymptotic theory, revealing an infinite variety of models classified by parameters that affect their accuracy and applicability.

Contribution

It introduces a classification of Boussinesq models based on two parameters, q and k, detailing their impact on model quality and scale variations.

Findings

Infinite set of asymptotic models classified by parameters q and k

Higher q improves approximation quality but reduces applicability

Scales of velocity, time, and viscosity can be varied independently

Abstract

The \emph{equations of Boussinesq approximation} (EBA) for an incompressible and inhomogeneous in density fluid are analyzed from a viewpoint of the asymptotic theory. A systematic scaling shows that there is an infinite number of related asymptotic models. We have divided them into three classes: `poor', `reasonable' and `good' Boussinesq approximations. Each model can be characterized by two parameters $q$ and $k$, where $q =1, 2, 3, \dots$ and $k=0, \pm 1, \pm 2,\dots$. Parameter $q$ is related to the `quality' of approximation, while $k$ gives us an infinite set of possible scales of velocity, time, viscosity, \emph{etc.} Increasing $q$ improves the quality of a model, but narrows the limits of its applicability. Parameter $k$ allows us to vary the scales of time, velocity and viscosity and gives us the possibility to consider any initial and boundary conditions. In general, we discover and classify a rich variety of possibilities and restrictions, which are hidden behind the routine use of the Boussinesq approximation. The paper is devoted to the multiplicity of scalings and related restrictions. We do not study any particular solutions and particular failures of EBA.