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equation in terms of only the synoptic-scale variables is not possible. We can
still, however, simplify the thermodynamic energy equation to some extent. We
recall from Section 11.2 [see Eq. (11.10)] that due to the smallness of temperature
fluctuations in the tropics, the adiabatic cooling and diabatic heating terms must
approximately balance. Thus, (11.16) becomes approximately
w
∂ ln θ
∂z
L
c
c
p
T
Dq
s
Dt
≈−
(11.20)
The synoptic-scale vertical velocity w that appears in (11.20) is the average of very
large vertical motions in the active convection cells and small vertical motions in
the envi
ro
nment. Thus, if we let w
be the vertical velocity in the convective cells
and the w vertical velocity in the environment, we have
aw
+
w
=
(1
−
a)w
(11.21)
where a is the fractional area occupied by the convection. With the aid of (11.17),
we can than write (11.20) in the form
w
∂ ln θ
∂z
L
c
∂q
s
∂z
c
p
T
aw
≈−
(11.22)
The problem is then to express the condensation heating term on the right in (11.22)
in terms of synoptic-scale field variables.
This problem of
parameterizing
the cumulus convective heating is one of the
most challenging areas in tropical meteorology. A simple approach that has been
used successfully in some theoretical studies
4
is based on the fact that, because
the storage of water in the clouds is rather small, the total vertically integrated
heating rate due to condensation must be approximately proportional to the net
precipitation rate:
z
T
ρaw
∂q
s
/∂z
dz
−
=
P
(11.23)
z
c
where z
c
and z
T
represent the cloud base and cloud top heights, respectively, and
P is the precipitation rate (kg m
−
2
s
−
1
).
Because relatively little moisture goes into changing the atmospheric vapor
mixing ratio, the net precipitation rate must approximately equal the moisture
convergence into an atmospheric column plus surface evaporation
z
m
∇·
ρq
V
dz
P
=−
+
E
(11.24)
0
4
See, for example, Stevens and Lindzen (1978).
