Geography Reference
In-Depth Information
must be modified to remove the physical mechanisms responsible for the occur-
rence of the unwanted oscillations, while still preserving the meteorologically
important motions. In modern numerical forecasting, gravity waves are usually
controlled by suitably adjusting the initial data; sound waves, however, are gener-
ally filtered from the dynamical equations.
To help understand how sound and gravity wave noise can be filtered from the
prediction equations, it is useful to refer back to Section 7.3 in which the physical
properties of such waves were discussed. If the pipe shown in Fig. 7.5 is tipped up
in the vertical, it can be used to generate vertically traveling sound waves. If we
now require that pressure be hydrostatic, the pressure at any point along the pipe
is determined solely by the weight of the air above that point. Hence, the vertical
pressure gradient cannot be influenced by adiabatic compression or expansion.
Therefore, vertically propagating sound waves are not among the possible modes
of oscillation in a hydrostatic system. Replacement of the vertical momentum
equation by the hydrostatic approximation is thus sufficient to filter out ordinary
sound waves. This approximation is used in virtually all models for forecasting
medium- and large-scale motions.
A hydrostatically balanced atmosphere can still, however, support a special
class of horizontally propagating acoustic waves. In this type of wave, the ver-
tical velocity is zero (neglecting orographic effects and departures of the basic
state temperature from isothermal conditions). The pressure, horizontal velocity,
and density, however, oscillate with the horizontal structure of the simple acoustic
waves described in Section 7.3.1. These oscillations have maximum amplitude
at the lower boundary and decay away from the boundary with the pressure and
density fields remaining in hydrostatic balance everywhere (see Problem 7.6).
Because these oscillations, known as
Lamb waves
, have maximum pressure per-
turbation amplitude at the ground, they may be filtered simply by requiring that
ω
0 at the lower boundary. This boundary condition is applied most
easily by formulating the equations in isobaric coordinates. In that case the condi-
tion ω
=
Dp/Dt
=
0 at the lower boundary is a natural first approximation for geostrophically
scaled motions over level terrain (see the discussion in Section 3.5).
The set of equations including the above approximations still has solutions in
the form of internal gravity waves. Gravity waves may be important in accounting
for mesoscale variability, and the zonal drag force due to gravity waves must be
included in the overall zonal mean momentum balance. However, gravity waves
are not important for short-range forecasting of synoptic and planetary scale circu-
lations. Due, however, to their high frequency of oscillation, they can create serious
errors in numerical forecasts. One possible solution to this problem is to filter grav-
ity waves from the forecast equations. From the discussion of Section 7.4 it should
be apparent that gravity waves can be solutions only if the dynamical equations
are second order in time. In physical terms this turns out to require that the local
rate of change of the divergence of the horizontal velocity field must be implicitly
=
