Turbulence and Dispersion of Contaminants in the Planetary Boundary Layer 65
The oscillation period in Equations 3.116a and b depends upon the ∂u ˉ /∂y and
∂v ˉ /∂x values (see Equations 3.108, 3.109a and b, and 3.112) and the Coriolis parameter. It can be seen that when ∂u ˉ /∂y and ∂v ˉ /∂x are both equal to zero, the present
solution gives the inertial oscillation period 2π/f c ≈ 17 h (Stull, 1988). It may also
be noted that with increasing wind speed, and thus increasing spatial derivates of
the wind speed components, the oscillation period decreases. This means that variance associated to meandering shifts to higher frequencies in power spectra.
An interesting result is obtained by performing a scale analysis of Equations
3.116a and b. The term
+
2
2
/
(
)
D p
q is approximately zero, − +
+
1
2
11
1
((
)/ )
pr qr a r b
is two orders of magnitude lower than −
− +
2
1
12
1
((
)/ )
pr qr a r b , also r 2 is two orders
of magnitude lower than r 1 , a 1 /b 1 and c 1 /b 1 are also negligible with respect to the
remaining terms, −pr 2 + a 1 r 2 is approximately zero, and
≈
1
( / ) 1
q b
. Accounting for
these simplifi cations, Equations 3.116a and b become:
−
= 1
( )
cos( )
pt
u t r e
qt
(3.118a)
−
= 1
( )
sin( )
pt
v t r e
qt
(3.118b)
These can be written in an analytical function form as
− −
=
(
)
1
( )
pt iqt
t
e
u
α
(3.119)
Solutions (118a circles) and (118b triangles) are plotted in Figure 3.5, where they
are labeled ( )
S t
u
and ( )
S
v t , respectively. It clearly appears that these simplifi ed
–1.0
0
1000
2000
3000
4000
Time (s)
5000
6000
7000
–0.5
0.0
Velocity (m/s)
0.5
1.0
u s (t)
v s (t)
u(t)
v(t)
FIGURE 3.5 Time series of calculated velocities with Equations 3.116a and b and the corresponding simplifi ed Equations 3.118a and b.
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