Mixing and Convection
Chapt 4 page 44.
Isobaric Mixing (p constant) of two samples of moist air:
m1, w1, P1,
q1, T1
?
m2, w2, P2,
q2, T2
m, w, P,
q, T
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?
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Case 1: no condensation
Specific humidity:
Mixing Ratio – since
m1
q
m1  m2
q2
wq
m1
w
m1  m2
So vapor pressure
m2
q1 
m1  m2
e
m2
w1 
m1  m2
w2
m1
e1  m2 e2
m1  m2
m1  m2
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Heat lost by warm sample = heat gained by cold sample
m1 (c p  w1ccp )(T1  T )  m2 (c p  w2ccp )(T  T2 )
or
T
since
m1
m2

T1 
T2
m1  m2
m1  m2
w1  w2  1
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Case 2: condensation and mixing
Question: can condensation occur during the mixing of two
unsaturated samples (isobaric mixing)?
Yes, in the winter when you
see your breath!
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Clausius-Clapeyron
e
es(T)
e2
es
ef
e1
T1
Tf
T2 T
es > ef so isobaric mixing in this case does
NOT result in condensation.
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e
es(T)
e2
ef
es
e1
T1
Tf
T2
T
Isobaric mixing in this case will result in condensation
because es < ef
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How does one determine if
condensation will occur?
1. Determine T & e that would result if no
condensation were to occur.
2. Compare e with es(T):
if e < es(T) - no condensation
if e > es(T) - condensation will occur.
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If Condensation occurs, what is
the final e & T?
• e must be less than that calculated
assuming no condensation because vapor
will be removed.
• T must be greater because latent heat has
been released.
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Latent Heat released during
condensation:
Isobaric Process:
dq = -Lvdw
dq = cpdT
Since w ~ ee/p
- Lvdw = Lv ede/p = cpdT
Or
pc p
de

dT
Lve
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& Z.Q. Li
the equation
of a line!
9
e
Final uncondensed
state
es(T)
(e2 ,T2)
ef
True final state
e’
Isobaric condensation line
(e1 ,T1)
Tf
T’
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T
10
To Determine the Final e & T:
Find the intersection of the isobaric condensation equation
with the Clausius-Clapeyron equation using e &T as “initial
conditions”.
The isobaric condensation equation must be integrated to arrive
at an algebraic form:
es (T ')
 de  
e
pc p
eLv
T'
 dT
so
es (T )  e 
T
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pc p
eLv
(T 'T )
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The Clausius Clapeyron Equation
 Lv  1 1 
es (T ' )  es (T ) exp    
 Rv  T T ' 
Simplifies for T ~ T’ to
 Lv  1 1  
es (T ' )  es (T )1     
 Rv  T T '  
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The simplified form of the Clausius-Clapeyron
equation can be combined with the isobaric
condensation equation to find the final values of
e and T.
But what if conditions don’t allow you
To simplify the equations……?
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Consider Two functions of x: f(x) and g(x)
Assume both are continuous and have
continuous derivatives.
f
g
Find x0 such that
f(x0) = g(x0)
xo
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Since we can not find xo analytically,
how do we proceed?
Expand f and g in a Taylor’s series:
f(x) = f(x*) + f’(x*)(x- x*) + …
g(x) = g(x*) + g’(x*)(x- x*) + …
Neglect higher order terms and solve for x.
Isn’t this what we did for the CC equation?
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f(x) = f(x*) + f’(x*)(x- x*) = g(x*) + g’(x*)(x- x*)
or
g(x )  f (x )
xx 
*
*
g'(x )  f '(x )
*
*
*
or
g(x j )  f (x j )
*
x j 1  x j 
*
*
g'(x j )  f '(x j )
*
*
Newton – Raphson iteration.
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Adiabatic Mixing
• Parcels from different pressure levels are
mixed after being brought together
adiabatically.
• The final state of the combined parcel can
be calculated as shown previously.
• When a column of air is thoroughly mixed,
the specific humidity becomes constant
throughout.
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Specific Humidity of a Mixed Parcel
z2
qmixed
1
  qdz
m z1
Where mass of air per unit area
m
z2
 dz
z1
Using the hydrostatic equation
we can show
qmixed
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& Z.Q. Li
p1
1
=
q dp
ò
Dp p2
18
Likewise,
q mixed
p1
1
=
q dp
ò
Dp p2
Thus for a well mixed layer, q, w and q are
constant throughout. With no condensation,
this must mean that the lapse rate corresponds
to dry adiabatic.
  d
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Convective Condensation Level
CCL – Pressure and temperature at which
condensation occurs in/at top of a well mixed
layer. It can be found by the intersection of
the dry adiabat for the layer with the mixing
ratio isopleth for the layer.
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Lifting Condensation Level
LCL – level at which condensation will occur if
a parcel is lifted from the surface in a dry
adiabatic process with constant w until just
saturated.
Note: LCL = CCL if the layer is well mixed.
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Fair Weather Cumulus Clouds
Fair weather cumulus are form atop buoyant bubbles of air (thermals) that
rise from Earth's surface. As bubbles rise, the water vapor mixing ratio
remains constant but the temperature falls and the relative humidity increases
until it reaches the saturation vapor pressure, 100% RH. Here droplets
condense and clouds form. This occurs at the Lifting Condensation Level,
(LCL) where the flat cloud bases are seen.
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Fair Weather Cumulus
Fair weather cumulus
1 pm EST July 7, 2007,
a smoggy day
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Dickerson & Z.Q. Li
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Boundary Layer Venting
Through Fair Weather Cumulus (Cumulus Humilis)
H2SO4
Cumulus
H2SO4
Cumulus
Inversion
SO2
SO2
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Two Reservoir Model (Taubman et al., JAS, 2004)
H2SO4
cloud
clou
d
SO2
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