A set of filtering approximations originally developed by Boussinesq.

In his attempts to explain the motion of the light in the aether Boussinesq (in 1903) opened a wide perspective of mechanics and thermodynamics. With a theory of heat convection in fluids and of propagation of heat in deforming or vibrating solids he showed that density fluctuations are of minor importance in the conservation of mass. The motion of a fluid initiated by heat results mostly in an excess of buoyancy and is not due to internal waves excited by density variations. In other words, the continuity equation may be reduced to the vanishing of the divergence of the velocity field, and variations of the density can be neglected in the inertial accelerations but not in the buoyancy term. Although used before him, Boussinesq's theoretical approach established a cardinal simplification for a special class of fluids which fundamentally differ from gases and may eliminate acoustic effects.

They result in an equation set applied to almost all oceanic motions except sound waves. The four approximation steps are:

• subtracting a motionless hydrostatically balanced reference state from the equations of motion;
• making the anelastic approximation;
• assuming that the vertical scale of motion is small compared to the scale depth (or height); and
• ignoring the inertial but not the buoyancy effects of variations in the mean density.

Mahrt (1986) addresses the issue of which assumptions properly constitute the Boussinesq approximations:

The derivation of conditions for the validity of the Boussinesq approximations is not as straightforward as many would assume. In the literature, a variety of sets of conditions have been assumed which, if satisfied, allow application of the Boussinesq approximations. The Boussinesq approximation can be divided into two parts. The first group of assumptions allows use of incompressible mass continuity and linearization of the ideal gas law, which are referred to as the shallow motion approximations. Additional restrictions allow neglect of the pressure influence on buoyancy. This more restrictive subclass of shallow motions is equivalent to the full Boussinesq approximations, also referred to as the shallow convection approximations.

The different derivations of the shallow motion approximations share the following conditions:

• the perturbations of variables of state must be small compared to basic state averaged values;
• the motion must be shallow compared to the scale depth of the basic flow; and
• restrictions on the time scale are required.