Geography Reference
In-Depth Information
Here we have used the fact that ∂ ( )/∂x
0, as quantities with the overbar are
independent of x. We have also used the fact that for any variables a and b
=
a
)
b
b
a
b
which follows from that fact
s th
at a and b are independent of x and a
=
ab
+
a
b
a
b
ab
=
(a
+
+
=
ab
+
+
=
ab
+
b
=
0
so that, for example, ab
ab
0.
The terms involving (v, w) on the right in (10.7) can be rewritten in advective
form with the aid of the zonal mean of (10.4):
∂v
∂y
=
=
ρ
−
1
0
+
∂(ρ
0
w)/∂z
=
0
(10.8)
Applying the chain rule of differentiation to the mean terms on the right in (10.7)
and substituting from (10.8) we can rewrite (10.7) as
ρ
0
A
v
ρ
0
A
w
Dt
ρ
0
A
+
ρ
0
DA
D
∂
∂y
∂
∂z
Dt
=
+
(10.9)
where
D
Dt
≡
∂
∂t
+
∂
∂y
+
∂
∂z
v
w
(10.10)
is the rate of change following the mean meridional motion (v, w).
10.2.1
The Conventional Eulerian Mean
Applying the averaging scheme of (10.9) to (10.1) and (10.5) we obtain the
zonal-mean zonal momentum and thermodynamic energy equations for quasi-
geostrophic motions on the midlatitude β plane:
∂
u
v
/∂y
∂u/∂t
−
f
0
v
=−
+
X
(10.11)
∂
v
T
/∂y
N
2
HR
−
1
w
∂T/∂t
+
=−
+
J/c
p
(10.12)
where N is the buoyancy frequency defined by
κT
0
H
+
R
H
dT
0
dz
N
2
≡
In (10.11) and (10.12), consistent with quasi-geostrophic scaling, we have neglected
advection by the ageostrophic mean meridional circulation and vertical eddy flux
divergences. It is easily confirmed that for quasi-geostrophic scales these terms
are small compared to the retained terms (see Problem 10.4). We have included
the zonally averaged turbulent drag in (10.11) because stresses due to unresolved
eddies (such as gravity waves) may be important not only in the boundary layer,
but also in the upper troposphere and in the middle atmosphere.
