1. Mixing and Convection R&Y Chapt 4 page 44. Salby Chapt. 5&7.
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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2. Copyright © 2013 R. R. Dickerson & Z.Q. Li
Case 1: no condensation
Specific humidity:
Mixing Ratio – since
So vapor pressure
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3. Copyright © 2013 R. R. Dickerson & Z.Q. Li
Heat lost by warm sample = heat gained by cold sample or
since
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4. 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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5. Copyright © 2013 R. R. Dickerson & Z.Q. Li
Clausius-Clapeyron e
es(T) e2 Tf ef es e1 T1 T2 T
es > ef so isobaric mixing in this case does NOT result in condensation.
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6. Copyright © 2013 R. R. Dickerson & Z.Q. Li
es(T) e2 Tf ef es e1 T1 T2 T
Isobaric mixing in this case will result in condensation because es < ef
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7. 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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8. If condensation occurs, what are 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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9. Copyright © 2013 R. R. Dickerson & Z.Q. Li
Latent Heat released during
condensation: dq = -Lvdw
Isobaric Process: dq = cpdT
Since w ~ ee/p
- Lvdw = Lv e de/p = cpdT Or
the equation
of a line!
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10. Copyright © 2013 R. R. Dickerson & Z.Q. LiTf ef
Final uncondensed
state
es(T)
(e2 ,T2) e’ T’
True final state
Isobaric condensation line
(e1 ,T1) T
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11. 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: so
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12. Copyright © 2013 R. R. Dickerson & Z.Q. Li
The Clausius Clapeyron Equation
Simplifies for T ~ T’ to
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13. Copyright © 2013 R. R. Dickerson & Z.Q. Li
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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14. Solving Transcendental Equations
Consider Two functions of x: f(x) and g(x)
Assume both are continuous and have
continuous derivatives. f g xo
Find x0 such that
f(x0) = g(x0)
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15. Copyright © 2013 R. R. Dickerson & Z.Q. Li
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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16. Copyright © 2013 R. R. Dickerson & Z.Q. Li
f(x) = f(x*) + f’(x*)(x- x*) = g(x*) + g’(x*)(x- x*) or
Newton – Raphson iteration.
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17. Copyright © 2013 R. R. Dickerson & Z.Q. Li
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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18. Specific Humidity of a Mixed Parcel
Where mass of air per unit area
Using the hydrostatic equation
we can show
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Likewise,
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.
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20. 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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21. 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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22. 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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24. Boundary Layer Venting Through Fair Weather Cumulus (Cumulus Humilis)
H2SO4
H2SO4
Cumulus
Cumulus
Inversion
SO2
SO2
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25. Two Reservoir Model (Taubman et al., JAS, 2004)
H2SO4
cloud
cloud
SO2
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