Geography Reference
In-Depth Information
Du/Dt
−
fv
+
∂/∂x
=
X
(10.1)
Dv/Dt
+
fu
+
∂/∂y
=
Y
(10.2)
H
−
1
RT
∂/∂z
=
(10.3)
ρ
−
1
0
∂u/∂x
+
∂v/∂y
+
∂ (ρ
0
w) /∂z
=
0
(10.4)
+
κT
H
w
J
c
p
DT Dt
=
(10.5)
where
D
Dt
≡
∂
∂t
+
∂
∂x
+
∂
∂y
+
∂
∂z
u
v
w
and X and Y designate the zonal and meridional components of drag due to small-
scale eddies.
For convenience in this and the following chapters we have dropped the aster-
isk notation used in Chapter 8 to distinguish the log-pressure coordinate from
geometric height. Thus, z here designates the log-pressure variable defined in Sec-
tion (8.4.1).
Analysis of the zonally averaged circulation involves study of the interaction of
longitudinally varying disturbances (referred to as eddies, and denoted by primed
variables) with the longitudinally averaged flow (referred to as the mean flow
an
d denoted by overbars). Thus, any variable A is expanded in the form A
=
A
. This sort of average is an
Eulerian mean
, as it is evaluated at fixed latitude,
height, and time. Eulerian mean equations are obtained by taking zonal averages of
(10.1)-(10.5). Such averaging is facilitated by using (10.4) to expand the material
derivative for any variable A in flux form as follows:
A
+
ρ
0
∂
A
A
∂z
(ρ
0
w)
DA
Dt
=
∂
∂z
∂
ρ
0
∂t
+
V
·∇
+
w
+
∇·
(ρ
0
V
)
+
(10.6)
∂
∂t
(ρ
0
A)
∂
∂x
(ρ
0
Au)
∂
∂y
(ρ
0
Av)
∂
∂z
(ρ
0
Aw)
=
+
+
+
where we recall that ρ
0
=
ρ
0
(z).
Applying the zonal averaging operator then gives
ρ
0
Av
A
v
ρ
0
Aw
A
w
(10.7)
∂t
ρ
0
A
+
DA
Dt
=
∂
∂
∂y
∂
∂z
ρ
0
+
+
+
