Rational derivation of the Boussinesq approximation

This note derives the Boussinesq approximation in a manner consistent with the conservation law of mass. It is shown that the governing equations of a fluid under the approximation can be obtained on the basis of the assumption that the fluid is incompressible, in the sense that the density of the fluid is constant. The equation of motion, in particular, can be formulated with the help of the conservation law of energy. The conditions for the approximation to be valid are also discussed.