20
1. Introduction
not even approximately propagate along the characteristics. There is no relation
between the characteristics and the paths of the gravity-waves because some of
the physical processes essential for gravity-wave propagation are mathematically
represented by undifferentiated functions of the unknown variables, and as such
exert no influence on the shape of the characteristics.
1.2.2 Filtered Equations
The Euler equations support sound waves, but sound waves have no direct influence on many types of atmospheric and oceanic motion . Analytic simplicity can
often be achieved by approximating the Euler equations with alternative sets of
filtered governing equations that do not support sound waves. As will be discussed
in Chapter 7, eliminating the sound waves may also allow the resulting system of
equations to be numerically integrated using a much larger time step than that
which would be required for a similar numerical integration of the original Euler
equations. These sets of filtered equations are not hyperbolic systems, but they
support gravity waves that closely approximate the gravity-wave solutions to the
full Euler equations. If the latitudinal variation of the Coriolis parameter is included, the filtered equations also support Rossby waves. The Coriolis parameter
will, however, be neglected in the following discussion in order to present the
essential ideas in the simplest context.
According to (1.35), the pressure perturbations in a perfect gas arise from variations in density and entropy. Variations in entropy play no fundamental role in
the physics of sound wave propagation. Indeed, for the general dass of fluids
described by an equation of state of the form
p == p(p , S),
the speed of sound is given by the square root of (apjap)s (Batchelor 1967,
p. 166). In order to filter sound waves from the governing equations, it is therefore
necessary to sever the link between density perturbations and pressure perturbations. This can be accomplished through any one of a family of related approximations that neglect terms involving the time variation of the density in the mass
continuity equation (1.32).
One approximation that will filter sound waves is obtained by assuming that
the flow is incompressible, in which case
'I/ ·v=O,
(1.51)
and mass conservation is replaced by volume conservation. The approximation
of (1.32) by (1.51) is widely referred to as the Boussinesq approximation. Unfortunately, the term " Boussinesq approximation" has been used in two different
senses. In some disciplines , the Boussinesq approximation refers only to the approximation of mass conservation by volume conservation . In the atmospheric
and oceanic sciences, the Boussinesq approximation is generally understood to
1. Introduction
not even approximately propagate along the characteristics. There is no relation
between the characteristics and the paths of the gravity-waves because some of
the physical processes essential for gravity-wave propagation are mathematically
represented by undifferentiated functions of the unknown variables, and as such
exert no influence on the shape of the characteristics.
1.2.2 Filtered Equations
The Euler equations support sound waves, but sound waves have no direct influence on many types of atmospheric and oceanic motion . Analytic simplicity can
often be achieved by approximating the Euler equations with alternative sets of
filtered governing equations that do not support sound waves. As will be discussed
in Chapter 7, eliminating the sound waves may also allow the resulting system of
equations to be numerically integrated using a much larger time step than that
which would be required for a similar numerical integration of the original Euler
equations. These sets of filtered equations are not hyperbolic systems, but they
support gravity waves that closely approximate the gravity-wave solutions to the
full Euler equations. If the latitudinal variation of the Coriolis parameter is included, the filtered equations also support Rossby waves. The Coriolis parameter
will, however, be neglected in the following discussion in order to present the
essential ideas in the simplest context.
According to (1.35), the pressure perturbations in a perfect gas arise from variations in density and entropy. Variations in entropy play no fundamental role in
the physics of sound wave propagation. Indeed, for the general dass of fluids
described by an equation of state of the form
p == p(p , S),
the speed of sound is given by the square root of (apjap)s (Batchelor 1967,
p. 166). In order to filter sound waves from the governing equations, it is therefore
necessary to sever the link between density perturbations and pressure perturbations. This can be accomplished through any one of a family of related approximations that neglect terms involving the time variation of the density in the mass
continuity equation (1.32).
One approximation that will filter sound waves is obtained by assuming that
the flow is incompressible, in which case
'I/ ·v=O,
(1.51)
and mass conservation is replaced by volume conservation. The approximation
of (1.32) by (1.51) is widely referred to as the Boussinesq approximation. Unfortunately, the term " Boussinesq approximation" has been used in two different
senses. In some disciplines , the Boussinesq approximation refers only to the approximation of mass conservation by volume conservation . In the atmospheric
and oceanic sciences, the Boussinesq approximation is generally understood to