Geoscience Reference
In-Depth Information
After applying the customary time averaging over a suitable time period, one obtains
from Equation (2.25) for the mean flux components of water vapor
F
v
x
=
ρ
(
u q
+
u
q
)
F
v
y
=
ρ
(
v
q
+
v
q
)
(2.26)
+
w
q
)
F
v
z
=
ρ
(
w
q
The first terms on the right of these three equations represent the advective transport of
water vapor by the mean motion of the air. The second terms are the components of the
advective vapor transport by the turbulence; they are also often called the Reynolds fluxes,
and statistically speaking, they are covariances. The estimation and parameterization of
these flux components is one of the core problems of hydrology.
Conservation equation of water vapor
The standard procedure for a more thorough analysis of the water vapor transport consists
of combining the expressions for the fluxes, Equations (2.26), with the principle of mass
conservation (1.8) applied to water vapor. This is accomplished by substituting
ρ
v
for
ρ
and
F
v
for (
ρ
v
) in Equation (1.8); since in this derivation, the bulk air itself is of less concern, it
can be as
s
umed to have a constant density, which allows use of Equation (1.9) for the mean
v
elocity
v
. Thus, one obtains the conservation equation for the mean specific humidity,
q
(see Brutsaert, 1982), as follows
∂
∂
z
w
q
∂
x
u
q
+
∂
y
v
q
+
∂
q
∂
t
+
u
∂
q
∂
x
+
v
∂
q
∂
y
+
w
∂
q
∂
∂
∂
z
=−
(2.27)
in which, it should be noted again, the molecular diffusion term is neglected. In princi-
ple, it should be possible to solve Equation (2.27) with appropriate boundary conditions to
describe water vapor transport in the atmosphere. However, this equation presents several
difficulties, which make its solution extremely difficult. First, since the fluxes in (2.26)
are intrinsically dependent on the velocity of the air and the turbulence, it is necessary to
consider the dynamics of the flow and to include the conservation equations for momen-
tum and temperature in the solution process as well. A second and more fundamental
difficulty is that this conservation equation for the mean specific humidity contains not
only
q
as a dependent variable, which is the first moment, but also the covariances of
q
with the velocity fluctuations
u
,v
and
w
, which are second moments. This means that
Equation (2.27) has more than one unknown; this fact is an instance of the notorious closure
problem of turbulence and it indicates that, without additional relationships, this equation
cannot be solved mathematically.
Fortunately, it is possible to simplify the general problem, as formulated with the
above fluxes, considerably and still obtain meaningful results. This is accomplished,
first, by assuming that the atmosphere nearest the surface can be considered as a steady
boundary layer above a quasi-homogeneous surface (Section 2.4), and, second, by the
application of similarity assumptions to alleviate the turbulence closure problem by
appropriate parameterization (Section 2.5).
