# Boussinesq approximation

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A **Boussinesq approximation** is a set of filtering approximations originally developed by french mathematician Joseph Valentin Boussinesq (1842 – 1929). This approximation is often applied in fluid dynamics to water waves and to buoyancy driven flow (natural convection).

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According to Sander [1998]:

"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."

The approximations 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.

The term “Boussinesq approximation” is not always used identically with the above series of approximation steps, e.g. it may or may not include the assumption of incompressibility.

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.

## Further Reading

- Physical Oceanography Index
- E. A. Spiegel and G. Veronis. On the Boussinesq approximation for a compressible fluid. Astrophys. J., 131:442–447, 1960.
- J. M. Mihaljan. A rigorous exposition of the Boussinesq approximations applicable to a thin layer of fluid. Astrophys. J., 136:1126–1133, 1962.
- H. P. Greenspan. The Theory of Rotating Fluids. Cambridge University Press, N.Y., 1969.
- O. M. Phillips. The Dynamics of the Upper Ocean. Cambridge University Press, New York, 1977. pp. 15-20
- L. Mahrt. On the shallow motion approximations. JAS, 43:1036–1044, 1986.
- R. Zeytounian. Asymptotic Modeling of Atmospheric Flows. Springer-Verlag, New York, 1990. pp. 142-176
- Peter Muller. Ertel’s potential vorticity theorem in physical oceanography. Rev. Geophys., 33:67–97, 1995.
- P. Thunis and R. Bornstein. Hierarchy of mesoscale flow assumptions and equations. JAS, 53:380–397, 1996.
- J. Sander. Dynamic equations and turbulent closures in geophysics. Continuum Mech. Thermodyn., 10:1–28, 1998.

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