Joseph Boussinesq and his approximation: a contemporary view

A hundred years ago , in his 1903 volume II of the monograph devoted to ‘Théorie Analytique de la Chaleur’, Joseph Valentin Boussinesq observes that: “The v ariations of density can be ignored except were they are multiplied by the acceleration of gravity in equation of motion for the vertical component of the velocity vector .” A spectacular consequence of this Boussinesq observation (called, in 1916, by Rayleigh, the ‘Boussinesq approximation’) is the possibility to work with a quasi-incompressible system of coupled dynamic, (Navier) and thermal (Fourier) equations where buoyancy is the main driving force. After a few words on the life of Boussinesq and on his observation, the applicability of this approximation is briefly discussed for various thermal, geophysical, astrophysical and magnetohydrodynamic problems in the framework of ‘Boussinesquian fluid dynamics’. An important part of our contemporary view is devoted to a logical (100 years later) justification of this Boussinesq approximation for a perfect gas and an ideal liquid in the framework of an asymptotic modelling of the full fluid dynamics (Euler and Navier–Stokes–Fourier) equations with especially careful attention given to the validity of this approximation.