The Quasi-Boussinesq Equation Set
Derivation by assuming small perturbations about a vertically-stratified atmosphere
As discussed earlier, the Euler equations of compressible fluid motion describe many types of fluid motions, not all of which are of relevance to meteorology. For instance the sound emitted from trees when a strong wind blows through their branches is a form of fluid motion, yet it is intuitively reasonable to suppose that it is does not have an important impact on meteorological flows -- at least in the lower atmosphere. For some purposes therefore, the compressibility of air is an unnecessary complication yet there are other situations (e.g. airflow through a deep convective storm) where the fractional change of density following an air parcel is considerable. We need, therefore, an equation set that will filter sound waves yet still represent the volume expansion of air as it ascends or descends a great distance in the atmosphere.
Another aspect of atmospheric flows which can be used to simplify the Euler equations is the fact that the basic thermodynamic variables are strongly stratified in the vertical at all times. The height variation of pressure and density is never far from exponential and horizontal variations are very small relative to their mean values. The quasi-Boussinesq system (as we shall refer to it) exploits these properties to effectively linearize the pressure gradient term as the following analysis shows. Assume that the pressure, density and entropy of dry air can be expressed as :
and
respectively where the subscript ‘0’ refers to basic state variables and the departure from these is denoted by primes with:
Now the basic state satisfies the hydrostatic relation:
so that the vertical component of the pressure gradient force can be expanded thus: