Geoscience Reference
In-Depth Information
The final term in this equation can now be simplified by recognizing that:
w
u
(
uw
′′ =′
)
(17.24)
u
+ ′
w
x
x
x
with similar equations relevant for ( v
w
) and (w
w
). Consequently Equation
(17.23) can be re-written as:
(
uw
′′
)
(
vw
′′
)
(
ww
′′
)
dw
1
P
(17.25)
=− −
g
+ ∇
u
2
w
+
+
r
dt
z
x
y
z
a
Starting from the equations describing momentum conservation in the ABL at a
point in time in the X and Y directions (Equations (16.36) and (16.37) ), and
following a procedure similar to that just used for the Z direction above gives:
(
uu
′′
)
(
vu
′′
)
(
wu
′′
)
du
1
P
2
=−
fu
+∇−
u
u
+
+
(17.26)
dt
x
x
y
z
r
a
dv
1
P
(
u v
′′
)
(
v v
′′
)
(
w v
′′
)
(17.27)
2
=−
fv
+ ∇
u
v
+
+
dt
y
x
y
z
r
a
There are marked similarities between Equations (17.25), (17.26), and (17.27), the
three equations that describe the evolution of mean flow, and Equations (16.36),
(16.37), and (16.38) that describe instantaneous momentum conservation in the
atmosphere. But differences occur because the effect of turbulent fluctuations must
also be considered when describing the variation in mean quantities, and the
additional terms in the mean flow equation account for contributions from turbulent
flux (as opposed to molecular fl ux) div ergence, e.g., the divergence
(
uw z in
′′
)
the turbulent momentum flux
uw . If there is coherence between fluctuating
components of velocity these give rise to turbulent fluxes that move momentum
from one place to another, and losses/gains in these fluxes, this will cause acceleration/
deceleration in the mean flow. Table 17.1 gives the physical meaning of the three
equations describing momentum conservation for mean flow in a turbulent field.
(
′′
)
Conservation of moisture, heat, and scalars
in the turbulent ABL
Starting from the equations for conservation of moisture, heat and scalar quantities
in the atmosphere, and using an approach analogous to that used in the last section
to derive the equations describing conservation of momentum, equivalent mean
flow equations in the ABL can easily be derived. The form of these equations and
the physical meaning of component terms are given in Tables 17.2, 17.3, and 17.4,
respectively.
 
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