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however, requires representation of this type of latent heating in terms of the
synoptic-scale field variables, which is much more difficult.
Before considering the problem of condensation heating by cumulus convection,
it is worth indicating briefly how the condensation heating by large-scale forced
uplift can be included in a prediction model. The approximate thermodynamic
energy equation for a pseudoadiabatic process (9.38) states that
L c
c p T
Dq s
Dt
D ln θ
Dt
≈−
(11.16)
The change in q s following the motion is caused primarily by ascent, so that
w∂q s /∂z
Dq s
Dt
for w>0
(11.17)
0
for w<0
and (11.16) can be written in the form
∂t +
ln θ
w ∂ ln θ
∂z
L c
c p T
∂q s
∂z
V
·
+
+
0
(11.18)
for regions where w>0. However, from (9.40)
∂ ln θ e
∂z
∂ ln θ
∂z
L c
c p T
∂q s
∂z
+
so that (11.18) can be written in a form valid for both positive and negative vertical
motion as
∂t +
θ
V
·
+
w e
0
(11.19)
where e is an equivalent static stability defined by
θ∂ ln θ e /∂z
for q
q s
and w>0
e
∂θ/∂z
for q<q s
or w<0
Thus, in the case of condensation due to large-scale forced ascent ( e > 0), the
thermodynamic energy equation has essentially the same form as for adiabatic
motions except that the static stability is replaced by the equivalent static stability.
As a consequence, the local temperature changes induced by the forced ascent will
be smaller than for the case of forced dry ascent with the same lapse rate.
If, however, e < 0, the atmosphere is conditionally unstable and condensation
will occur primarily through cumulus convection. In that case, (11.17) is still
valid, but the vertical velocity must be that of the individual cumulus updrafts, not
the synoptic-scale w. Thus, a simple formulation of the thermodynamic energy
 
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