Transcript Lecture CP6, Wet Parcel Theory

Continuing to build a cloud model:
Consider a wet air parcel
Parcel boundary
As the parcel moves assume no mixing with environment.
Pressure inside = pressure outside
Copyright © 2014 R. R. Dickerson
& Z.Q. Li
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We have already considered a dry parcel, now consider
a parcel just prior to saturation (the book leaves out
several steps):
mass of parcel = md + mv = mp
Suppose we expand the parcel reversibly and
adiabatically and condense out some mass of liquid =
mL, keeping total mass constant.
or
mp = md + mv’ + mL
mL + mv‘ = mv
Where mv’ is the new (lower for rising motion) mass
of vapor in the parcel.
2
mL = mv - mv’
Let the mass ratio of liquid to dry air be c.
m
Let c  L
md
 w  w for this example.
Since for this small change, w is the sum of water
vapor and liquid; at w’, the parcel is saturated.
c  w  ws
Consider the initial state to be just saturated:
dc  dws
This is Eq. 2.37 in Rogers and Yau.
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& Z.Q. Li
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Define c to be the adiabatic liquid water content,
and dc is the increase in adiabatic liquid water
mixing ratio. c increases from the LCL (sub C)
where it is zero to….
c = ws(Tc,pc) - ws(T,p)
at any other level.
Define the parcel total water mixing ratio QT:
QT = ws + c
QT (the book uses just Q) is conserved in a closed
system meaning no precip.
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& Z.Q. Li
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Consider an adiabatic displacement of a saturated parcel.
Assume a reversible, closed process with total mass
conserved. From R&Y 2.4 at const. Temp. Lv = T(fv - fw).
The specific (per kg) entropy of cloudy air and vapor will be:
j = j d + wsj v + cj w
Lv
jv = jw +
T
Lv ws
j = j d + (ws + c )j w +
T
æ Lv ws ö
dj = dj d + (ws + c )j w + d ç
÷
è T ø
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& Z.Q. Li
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The system is closed so we can consider an
isentropic process:
df  0
df w
dT
 cw T
where cw  specific heat of liquid
 Lv ws 
0  dfd  QT dfw  d 

 T 
Remember for dry air
df d
dT
dP
 c p T  R' P
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& Z.Q. Li
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Adding together the entropy changes with
temperature for dry and wet air with the entropy of
evaporation, setting total entropy change to zero:
æ Lv w s ö
0 = c p d lnT + cwQT d(lnT ) - Rd(ln Pd ) + d ç
÷
è T ø
æ Lv w s ö
0 = (c p + QT cw )d(lnT ) - Rd(ln Pd ) + d ç
÷
è T ø
æ Lv w s ö
0 = Ad(lnT ) - Bd(ln Pd ) + d ç
÷
è T ø
æ Lv w s ö
A
B
0 = d(lnT - ln Pd ) + d ç
÷
è T ø
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& Z.Q. Li
With further rearrangement:
 Lv ws 
d ln T  ( A)d ln Pd  d 
0
 AT 
B

B A
d
d ln TP
exp
 R ' /(c p  QT cw )
d
TP
   0
Lv ws
AT


Lv ws
  constant
exp 
 T (c  Q c ) 
 p T w 
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& Z.Q. Li
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Wet, Equivalent Potential Temperature, qq
and Equivalent Potential Temperature, qe.
θq is the temperature a parcel of air would reach if all
of the latent heat were converted to sensible heat by a
reversible adiabatic expansion to w = 0 followed by a
dry adiabatic compression to 1000 hPa.
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& Z.Q. Li
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Because the previously derived quantity is a constant,
we may define the wet-equivalent temperature as:
q q  1000
R / A
 R /(c p  QT cw )
d
 TP
 1000 

so q q  T 
 P 
 d 
 R /(c p  QT cw )


Lv ws

exp 
 T (c  Q c ) 
 p T w 


Lv ws

 exp 
 T (c  Q c ) 
 p T w 
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& Z.Q. Li
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Equivalent Potential Temperature qe
If one assumes the latent heat goes only to heat dry air
and not H2O, this is called a pseudo adiabatic process.
Set QT = 0, then one obtains the equation for the
equivalent potential temperature.
æL w ö
æ 1000 öR / cp
qe = T ç
· exp çç w s ÷÷
÷
è Pd ø
è Tc p ø
æL w ö
q e » q · exp çç w s ÷÷
è Tc p ø
Copyright © 2014 R. R. Dickerson
Note the AMS Glossary of Meteorology defines these two temps as the same.
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Pseudoadiabatic Process
Consider a saturated parcel of air.
Expand parcel from T, p, wo …
…to T+dT, p+dp, wo+dwo
(note: dT, dp, dwo are all negative)
This releases latent heat = – Lvdwo
Assume this all goes to heating dry air, and not into
the water vapor, liquid, or solid (rainout) .
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In contrast to the reversible adiabatic process, we assume that
all condensation products fall out of parcel immediately.
-Lv dwo = dq¢ = c p dT - a d dpd
dpd = d( p - es ) @ dp
ad @ a =
'
RT
p
dp
-Lv dwo = c p dT - R 'T
p
dT R' dp Lv
or
=
dwo
T
c p p c pT
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& Z.Q. Li
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 p
Remember t hat
  
q  po 
ln T  ln q  k ln p  k ln po
T
R '/ c p
; po  1000 hPa
dT dq
dp
or

k
(k  R' / c p )
T
q
p
dT R' dp Lv
using


dwo
T
c p p c pT
dq
Lv
we get

dwo for a pseudoadia batic process.
q
c pT
A nice clean answer!
Since dw0 < 0, q increases for a pseudoadiabatic process.
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& Z.Q. Li
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dwo
 wo 
 d
 (see Hess)
T
T 
Lv  wo 
dq

d

q
cp  T 
Integrate
from the condensation level
where T = Tc, q  original qo
to a level where ws ~ 0
0


Lv 
dq
 wo 

q q   c p w / T d  T 
  o c
qe
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& Z.Q. Li
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Use of Equivalent Potential Temperature qe
 Lv ws 

q e  q exp 

c
T
 p c
qe is the temperature that a parcel of air would have
if all of its latent heat were converted to sensible
heat in a pseudoadiabatic expansion to low
pressure, followed by a dry adiabatic compression
to 1000 hPa. qe is conserved in both adiabatic and
pseudoadiabatic processes. See Poulida et al., JGR,
1996.
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& Z.Q. Li
Adiabatic Equivalent Temperature Tea
Adiabatic equivalent temperature (also known as
pseudoequivalent temperature): The temperature
that an air parcel would have after undergoing the
following process: dry-adiabatic expansion until
saturated; pseudoadiabatic expansion until all
moisture is precipitated out; dry-adiabatic
compression to the initial pressure. Glossary of
Met., 2000.
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& Z.Q. Li
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Adiabatic Equivalent Temperature Tea
Instead of compressing to 1000 hPa, we go instead
to the initial pressure.
 Lv ws 

Tea  T exp 
c T 
 p c 
 1000 

i.e., q e  Tea 
 p 
k
 1000 

while q  T 
p 

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k
& Z.Q. Li
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Note:
3

c p ~ 10 J/kg K
L v ~ 2.5x10 6 J/kg
Lv
~ 2.5x103 K
cp
Since T is in the range of 200-300 K and
wo is generally < 20 x 10-3
Lv wo

~ 10wo ~ 0.2 or less.
c pT
Lv wo
Thus, Tea ~ T 
 Tep
cp
Generally Tea and Tep (equivalent potential temp) are
o
within 5 C.
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& Z.Q. Li
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Additional Temperature Definitions
Wet Bulb Potential Temperature qw
Defined graphically by following pseudo/saturated
adiabats to 1000 hPa from Pe, Tc. This temp is
conserved in most atmos. processes.
Adiabatic Wet Bulb Temperature Twa (or Tsw)
Follow pseudo/saturated adiabats from Pe, Tc to
initial pressure.
o
|Tw- Twa| ~ 0.5 or less.
http://glossary.ametsoc.org/wiki/
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& Z.Q. Li
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Conservative Properties of Air Parcels
Variable
dry adiabatic
saturated/pseudo adiabatic
q
C
NC
qe
C
C
qw
C
C
Td
NC
NC
Tw
NC
NC
w
C
NC
T* (Tv)
NC
NC
Te
NC
NC
Tc
C
NC
f
NC
C
C
C
qq
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& Z.Q. Li
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Remember Thermodynamic
Diagrams (lecture 4)
A true thermodynamic diagram has Area a Energy
T-f gram
Emagram
isotherms
Dry adiabats
RlnP
lnP
T
T
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& Z.Q. Li
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In the U.S. a popular meteorological thermodynamic
diagram is the Skew T – LogP diagram:
y = -RlnP
x = T + klnP
k is adjusted to make the angle between isotherms and
o
dry adiabats nearly 90 .
See Hess for more complete information.
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& Z.Q. Li
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Copyright © 2010 R. R. Dickerson
& Z.Q. Li
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