Uj
I could digress here and spend 1020 pages discuss-
=0
(13a)
ing what the averaging denoted by that straight over-
xj
bar means. Ideally, it denotes an ensemble average
(Lumley and Panofsky 1964, p. 6)--the average formed
uj
= 0.
from data collected during repeated, identical experi-
(13b)
xj
ments. In the atmospheric sciences, however, we do
not have the luxury of ensemble averaging. To create
The first term on the right-hand side of eq 12 is a con-
averages, we must make an ergodic hypothesis (Lum-
sequence of another of the Boussinesq approxi-
ley and Panofsky 1964, p. 35)--that conditions are near
mations--that fluctuations in fluid density are much
enough to steady state that a time average is equivalent
less than the mean density and, therefore, only impor-
to an ensemble average. Lumley and Panofsky (1964,
tant when they multiply g.
p. 35 ff.), Haugen et al. (1971), Wyngaard (1973) and
In horizontally homogeneous conditions, the x (or x1)
Andreas (1988), among others, have considered what
and y (or x2) derivatives in eq 12 produce zeroes, except
the appropriate averaging time is for ABL statistics. In
in P/ x and P/ y, which are imposed synoptic-scale
light of these studies, henceforth in this report I will
forcings. Near an impermeable surface, such as sea ice
use upper-case variables to mean averages obtained
or the ocean, the average vertical velocity, W (or U3),
from an hour's worth of instantaneous values. Like-
must be zero. Lastly, the viscous term in eq 12 is
wise, the straight overbar, in uiuj for example, will
always small except within a few millimeters of the
mean a turbulence statistic obtained by averaging for 1
surface. Hence, in an ABL above a horizontally homo-
hour. Notice that, because these are time averages not
geneous surface, the three equations implicit in eq 12
volume averages, an averaged quantity may still de-
become
pend on position within the fluid.
To write eq 4, 7 and 8, we have already made some
U equation
approximations. To simplify these further, we will make
additional approximations. These are the Boussinesq
U
uw 1 P
=-
-
+ fV
approximations (Busch 1973; Businger 1982; Garratt
(14a)
ρ0 x
t
z
1992, p. 20 f.), which I summarize here.
The Boussinesq Approximations
cp is the specific heat of air at constant pressure; and Dw are constants through-
out the fluid.
2. The flow speeds are low enough that the air behaves as an incompressible fluid.
3. Turbulent fluctuations in fluid properties are much smaller than the corresponding
averages; that is θ/Θ<<1, q/Q<<1, ρ/ρ0<<1 and p/P<<1.
4. p/P can be neglected in comparison to θ/Θ, q/Q and ρ/ρ0.
5. The heat generated by viscous stresses can be neglected. In other words, term VI
in eq 4 does not need to appear as a source term in eq 7.
6. Turbulent density fluctuations, ρ, are significant only when they multiply g.
Now back to the Navier-Stokes equation, eq 4. Sub-
V equation
stitute in it eq 9a, 9d and 9e, then average. The result is
V
vw 1 P
=-
-
- fU
(14b)
ρ0 y
t
z
1 P
Ui
Ui
+ Uj
+
uiuj =
ρ0 xi
t
xj
xj
W equation
1 P
g=-
2
.
Ui
(14c)
- g δi3 - 2Ω εijk ηj Uk + ν
ρ0 z
.
(12)
xj xj
The W equation, which results from the assumption that
The second and third terms on the left-hand side derive
vertical accelerations are much less than pressure gra-
from the assumption that the atmosphere is incompress-
dient forces (Pielke 1984, p. 30 ff.), shows that, on
ible
average, the ABL is in hydrostatic balance.
3
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