Geoscience Reference
In-Depth Information
2
Ψ
[
v
=
(7.3)
2
and
Ψ
=−
2
,
[]
ω
(7.4)
2
which have been constructed so that Eq. 7.2 is satisfied. A variation on this idea
is commonly used to quantify the Hadley circulation. Spherical coordinates
(Appendix B) are used because the space scale of the circulation system is large,
and the constant
g
is added so the units of are kg/s, that is, a mass flux. The
Stokes stream function
is defined by
g
2
Ψ
2
[]
v
=
(7.5)
2
πφ
a
cos
p
and
g
2
Ψ
,
[]
ω
=−
(7.6)
πφ
φ
2
2
2
a
cos
which satisfy the divergence equation
2
1
2
(
cos
φ
[ ])
[]
v
ω
v
d
$
v
=
+
=
0
.
(7.7)
a
cos
φ
2
φ
2
p
Because vertical velocities are small in the atmosphere and, therefore, diffi-
cult to observe accurately, the Stokes stream function is calculated from obser-
vations of zonally averaged meridional velocity with the assumption (boundary
condition) that the stream function is zero at the tropopause. Integrating Eq.
7.5 from the tropopause
down
to some level
p
at a fixed latitude gives
p
Ψ
(,)
φ
p
p
2
πφ
φ
a
cos
2
πφ
φ
a
cos
#
#
#
[(,)]
vpdp
=
d
ψ
Ψ
( ,)
φ
p
=
[(,)]
vpdp
.
(7.8)
&
g
g
p
0
p
TROP
TROP
The integral on the right-hand side of Eq. 7.8 can be calculated numerically
from an observed meridional velocity field.
(,
p
φΨ is a mass transport (kg/s),
equal to the rate at which air is transported meridionally in the region between
pressure level
p
and the tropopause by the zonal mean circulation.
Figure 7.3
shows observed monthly mean values of the Stokes stream
function. During December, January, February, and March, the mean meridi-
onal circulation consists of a single Hadley cell centered near 15°N with its
up-branch in the Southern Hemisphere tropics. During May, June, July, Au-
gust, and September the pattern is reversed, and the Hadley cell is centered
in the Southern Hemisphere (the winter hemisphere) with its up-branch in
the Northern Hemisphere (the summer hemisphere). The pattern is symmetric
about the equator—with two Hadley cells of approximately equal strength—
only during three months (April, October, and November) and in the annual
mean (
Fig. 7.2)
.
The following features of the climate system are directly related to the Had-
ley circulation:
