Geography Reference
In-Depth Information
where E is the evaporation rate (kg m
−
2
s
−
1
) and z
m
is the top of the moist layer
(z
m
≈
2 km over much of the equatorial oceans). Substituting into (11.24) from
the approximate continuity equation for q,
∇·
(ρq
V
)
+
∂ (ρqw) /∂z
≈
0
(11.25)
we obtain
P
=
(ρwq)
z
m
+
E
(11.26)
Using (11.26) we can relate the vertically averaged heating rate to the synoptic-
scale variables w(z
m
) and q(z
m
).
We still, however, need to determine distribution of the heating in the vertical.
The most common approach is to use an empirically determined vertical distribu-
tion based on observations. In that case, (11.18) can be written as
∂
∂t
+
ln θ
ρc
p
T
η
(
z
)
(
ρwq
)
z
m
+
E
w
∂ ln θ
∂z
L
c
V
·∇
+
=
(11.27)
where η(z)
=
0 for z<z
c
and z>z
T
and η(z) for z
c
≤
z
≤
z
T
is a weighting
function that must satisfy
z
T
η
(
z
)
dz
=
1
z
c
Recalling that the diabatic heating must be approximately balanced by adiabatic
cooling as indicated in (11.22), we see from (11.27) that η(z) will have a vertical
structure similar to that of the large-scale vertical mass flux, ρw . Observations
indicate that for many tropical synoptic-scale disturbances, η(z) reaches its max-
imum at about the 400-hPa level, consistent with the divergence pattern shown in
Fig. 11.5.
The above formulation is designed to model average tropical conditions. In
reality the vertical distribution of diabatic heating is determined by the local dis-
tribution of cloud heights. Thus, the cloud height distribution is apparently a key
parameter in cumulus parameterization. A cumulus parameterization scheme in
which this distribution is determined in terms of the large-scale variables was
developed by Arakawa and Schubert (1974). A number of other schemes have
been suggested in the past decade. Discussion of these is beyond the scope of this
text.
11.4
EQUATORIAL WAVE THEORY
Equatorial waves are an important class of eastward and westward propagat-
ing disturbances in the atmosphere and in the ocean that are trapped about the
equator (i.e., they decay away from the equatorial region). Diabatic heating by
