Geography Reference
In-Depth Information
To obtain the zonal-mean angular momentum budget, we must average (10.38) in
longitude. Using the spherical coordinate expansion for the horizontal divergence
as given in Appendix C, we have
∂ (p
s
Mu)
∂λ
1
a cos φ
∂ (p
s
Mv cos φ)
∂φ
∇·
(
p
s
M
V
)
=
+
(10.39)
We also observe that the bracketed term on the right in (10.38) can be rewritten as
p
s
∂x
(p
s
RT )
∂
∂x
(
∂
−
+
RT )
(10.40)
However, with the aid of the hydrostatic equation (10.35) we can write
σ∂
∂σ
∂ (σ)
∂σ
(
−
RT )
=
+
=
Thus, recalling that p
s
does not depend on σ , we obtain
p
s
∂
∂σ
p
s
σ
∂
∂x
∂x
(p
s
RT )
∂
∂x
+
RT
∂p
s
∂x
∂
=
+
(10.41)
Substituting from (10.39) and (10.41) into (10.38) and taking the zonal average
gives
∂
p
s
M
∂t
∂y
p
s
Mv cos φ
1
cos φ
∂
=−
p
s
M
(10.42)
ga cos φ
τ
E
+
∂
∂σ
(a cos φ) σp
s
∂
∂x
−
˙
+
σ
The terms on the right in (10.42) represent the convergence of the horizontal flux of
angular momentum and the convergence of the vertical flux of angular momentum,
respectively.
Integrating (10.42) vertically from the surface of the earth (σ
=
1) to the top of
the atmosphere (σ
=
0), and recalling that
σ
˙
=
0 for σ
=
0, 1wehave
(g cos φ)
−
1
1
1
∂y
p
s
Mv cos φ
dσ
∂
∂t
p
s
Mdσ
∂
g
−
1
=−
(10.43)
0
0
a cos φ
τ
E
p
s
∂h
∂x
−
1
+
σ
=
g
−
1
(x, y, 1) is the height of the lower boundary (σ
where h(x, y)
=
=
1), and
we have assumed that the eddy stress vanishes at σ
=
0.
