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transition period towards the mature stage of the tornado, and the flow becomes
barotropic by the wrap-around mechanism as will be discussed in Sect.
18.7
.
18.6
Comments on Entropy
Next, we will explore more about entropy. For simplicity, it is assumed that the
adiabatic processes are considered to be independent to the diabatic processes, both
are added linearly when both are working. Also, because of the high Reynolds num-
ber for supercell and tornadic cases, starting with the First law of thermodynamics,
d'U D d'Q C d'W where the internal energy U(S,
and external heating d'Q is
expressed by the following relation, using the entropy S and the specific volume
.
D
1=/
/
, and the work d'W by pd
for a dry ideal gas for simplicity as
;
=
:
dU D TdS pd
dS D d'Q
T
(18.26a,b)
Hence, we get
p DU
T D U
S
;
p
D RT
:
(18.27a,b,c)
The entropy change dS is expressed by the temperature change and pressure change
because the adjustment of d'Q is made due to change of temperature and pressure
for the example described in Appendix 4 as, for one mole of gas,
dS D d'Q
=
T D R
=2
dT
=
T Rdp
=
p
(18.28)
where the relations C
V
Rareused.
The internal energy is a function of temperature alone for an ideal gas,
D
3=2
RandC
P
D
5=2
U D c
v
T
;
(18.29)
where
c
v
is the specific heat at constant volume. After mathematical manipulation
from the above (18.26), (18.27), (
18.28
), and (
18.29
), we get
Tp
/
C S
0
;
S D c
p
log
.
(18.30)
where the exponent
c
p
for dry adiabatic processes
and its value is adjusted for moist adiabatic processes. Mathematically, S
0
is the
arbitrarily determined integral constant, but physically it is discussed as to be
determined on the basis of the Third law of thermodynamics, strictly speaking,
different from the convention taken in meteorology.
The potential temperature
is a constant defined as R
=
is conventionally defined in meteorology as the
temperature at a pressure of 1,000 mb after hypothetically moving the particle in
adiabatic process to the pressure level,
/
:
D T
.
p
00
=
p
(18.31)
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