Dilution by Entrainment
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Transcript Dilution by Entrainment
620 Dilution by Entrainment
Lateral entrainment: mix cooler, drier air through cloud’s
lateral boundaries. The cloud is warmer
than surroundings and actively growing.
Effects:
Reduce T – T’
Reduce w
Copyright © 2012 R. R. Dickerson
& Z.Q. Li
1
Cloudy Air of mass m consists of dry air,
water vapor, and condensed water.
Assume that as cloudy air ascends a
distance dz, a mass dm of environmental
air is entrained. Condensed water in the
cloud will evaporate in response to the
entrainment of drier air.
Primes will denote properties of ambient
(environmental) air.
Copyright © 2012 R. R. Dickerson
& Z.Q. Li
2
Heat Required to warm the entrained air from T’ to T:
d 'Q1 = c p (T - T ¢)dm
(neglect heat content of vapor and liquid)
Assume that just enough condensate evaporates to
saturate the mixture.
Let
d ¢Q2 = heat required for this evaporation
d ¢Q2 = Lv (ws - w¢)dm
Where w' is the mixing ratio of environmental air,
less than ws.
Copyright © 2013 R. R. Dickerson
& Z.Q. Li
3
Condensation occurs during ascent, releasing
latent heat : d ¢Q3 = -mLv dws
Cloudy air loses d ¢Q1 + d ¢Q2 but gains d ¢Q3
Applying the first law to mass m, with a = RT/P
dP
d ¢Q1 + d ¢Q2 - d ¢Q3 = m(c p dT - RT
)
P
Compare to
dq
dT
dP
cp
= cp
-R
q
T
P
Copyright © 2012 R. R. Dickerson & Z.Q. Li
4
Substituting in the values for each individual heat transfer and
rearranging:
dm
Lv dws
Lv
B
( ws w)
m
c pT
c pT
d
With no entrainment (dm = 0) we recover the parcel
theory result:
d
Lv dws
c pT
So for a bouyant parcel with entrainment, we see that the
magnitude of d/ is larger than the pure parcel result.
Temperature falls off at a faster rate: buoyancy is impaired.
Copyright © 2013 R. R. Dickerson
& Z.Q. Li
5
If instead of solving for we solved for –dT/dz; we obtain
Lv
(T T ) c ( ws w)
dT
1 dm
p
s
L2v ws
dz
m dz
1
2
c
R
T
p
So if
dm
0
dz
and
T T
Then s for an entraining cloud to be
convective ly unstable.
Copyright © 2013 R. R. Dickerson
& Z.Q. Li
6
An Example from Hess
o
P = 700 mb; T’ = -1 C at cloud level
o
T = 0 C; f’ = 67% of f in the cloud.
1 dm
0.25 km1
m dz
dT
p 6.6 C / km
dz
whereas s 5.8 C / km
Copyright © 2013 R. R. Dickerson
& Z.Q. Li
7
Aircraft observations show T ~ T’ in many
clouds. It is possible to integrate to find
m(z) for specified p and f. Results show
the cloud mass may easily double or triple
in a few km of ascent. Lab measurements
of man-made buoyant plumes bear out the
theory.
But there’s a problem……
Copyright © 2013 R. R. Dickerson
& Z.Q. Li
8
Theory
Observed in Sky
z
Growth with lateral
entrainment
The observations suggest downdrafts within cloud which
dilute by entrainment of dry ambient air above cloud top.
Copyright © 2013 R. R. Dickerson
& Z.Q. Li
9
Copyright © 2013 R. R. Dickerson
& Z.Q. Li
10
Copyright © 2013 R. R. Dickerson
& Z.Q. Li
11
Moderate Gale with Heavy Clouds, a Pilot Boat
Working its way out to a Waiting Brig
C. W. Eckersberg, 1831
Ny Carlsberg Glyptotek, Copenhagen
Copyright © 2013 R. R. Dickerson
& Z.Q. Li
12
Bubble Theory
Observations of cumuli indicate towers grow
for a while, lose their impetus and are
succeeded by new ones. This phenomena
led Scorer (1958) to propose what is known
as the “Bubble Theory” of convection.
Copyright © 2013 R. R. Dickerson
& Z.Q. Li
13
Life Cycle of a Cumulus Cloud
1. Initial Ascent
Bubble motion
Adiabatic
descent
Spherical cap
Cumulus
mass
Buoyant
bubble
Copyright © 2013 R. R. Dickerson
& Z.Q. Li
14
2. Erosion of spherical cap and mixing of
ambient and bubble air.
Erosion of cap
Turbulent
wake
Cumulus
mass
Lateral mixing
and entrainment
Copyright © 2013 R. R. Dickerson
& Z.Q. Li
15
3. Extension of Cumulus mass
Spherical cap completely
eroded and no longer buoyant.
Cloud mass
evaporating
Initial mass
Copyright © 2013 R. R. Dickerson
& Z.Q. Li
16
Net Result of Bubble Cycle
The bubble has enriched the ambient air above
the original cloud (moistened the environment).
Thus the next bubble can penetrate further than
the first. Successive bubbles extend the cloud
further in the vertical direction.
Copyright © 2013 R. R. Dickerson
& Z.Q. Li
17
Cumuli and Horizontal Winds
Wake carried
downstream
z
wind
Greatest vertical
growth is on
down-shear side.
z
~ Verified by observation ~
Copyright © 2012 R. R. Dickerson & Z.Q. Li
18
Copyright © 2013 R. R. Dickerson
& Z.Q. Li
19
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