Geoscience Reference
In-Depth Information
Summary of equations of atmospheric flow
In addition to the several conservation equations introduced above, the suite of
basic equations describing the atmosphere also includes the ideal gas law for moist
air here given in terms of virtual temperature (see Equation (2.11) and associated
text), i.e., in the form:
(16.51)
PRT
=
adv
r
where
R
d
is the gas constant for air (287 J kg K) and
T
v
=
T
(1 + 0.61
q
) is the virtual
temperature. Table 16.1 summarizes the resulting set of equations used to describe
the movement and evolution of atmospheric variables.
Important points in this chapter
Prognostic equations
the set of equations that describe atmospheric flow at
a point in time, which are often called prognostic equations, are just the local
conservation equations for each atmospheric variable, plus the ideal gas law.
:
●
Time rate of change in fluids
there are two ways in which a property of a mov-
ing fluid measured at a point can change, either in response to mechanisms
acting within the fluid itself (e.g., forces, internal diffusion, or source/sink pro-
cesses), or because the property is not constant with distance inside the fluid
as it moves past the point. Their sum is the
total rate of change
with time.
:
●
Momentum conservation
the prognostic equations for kinematic velocity
are given by applying momentum conservation equations along three axes,
with the total rate of change of momentum equated to the sum of 'forces'
associated with pressure gradients, molecular diffusion of momentum, and
axis-specific (Coriolis and gravity) forces.
:
●
Mass conservation
the continuity equation for air mass is given by equating
the local rate of change in air density to the difference between incoming and
outgoing air flow along all three axes. In the ABL, changes in air density
equilibrate quickly (at about the speed of sound) and are often neglected.
:
●
Conservation of other variables
the prognostic equations for other varia-
bles (e.g., moisture, temperature, CO
2
) are given by equating their total rate
of change to the relevant mechanisms by means of which they might be
changed (e.g., atmospheric sources/sinks, phase changes, radiation
divergence).
:
●
