copyright © 2012 r. r. dickerson & z.q. li 1 620 lecture10: dilution by entrainment lateral...
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Copyright © 2012 R. R. Dickerson & Z.Q. Li
1
620 Lecture10: 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
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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 © 2013 R. R. Dickerson & Z.Q. Li
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Heat Required to warm the entrained air from T’ to T:
(neglect heat content of vapor and liquid)
Assume that just enough condensate evaporates to saturate the mixture. Let
Copyright © 2012 R. R. Dickerson & Z.Q. Li4
Copyright © 2013 R. R. Dickerson & Z.Q. Li
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Substituting in the values for each individual heat transfer and rearranging:
m
dmww
Tc
LB
Tc
dwLds
p
v
p
sv
)(
With no entrainment (dm = 0) we recover the parcel theory result:
Tc
dwLd
p
sv
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
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If instead of solving for we solved for –dT/dz; we obtain
unstable.ly convective
be tocloud entrainingan for Then
TT and 0 if So
1
)()(1
s
2
2
dz
dm
Tw
RcL
wwcL
TT
dz
dm
mdz
dT
s
p
v
sp
v
s
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An Example from Hess
P = 700 mb; T’ = -1oC at cloud levelT = 0oC; f’ = 67% of f in the cloud.
125.01 km
dz
dm
m
kmC
kmCdz
dT
s
p
/8.5 whereas
/6.6
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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……
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Theory
z
Growth with lateral entrainment
Observed in Sky
The observations suggest downdrafts within cloud which dilute by entrainment of dry ambient air above cloud top.
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Moderate Gale with Heavy Clouds, a Pilot Boat Working its way out to a Waiting Brig
C. W. Eckersberg, 1831
Ny Carlsberg Glyptotek, Copenhagen
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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.
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Life Cycle of a Cumulus Cloud
1. Initial Ascent
Buoyantbubble
Cumulusmass
Spherical cap
Bubble motion
Adiabaticdescent
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Cumulusmass
Turbulentwake
Erosion of cap
Lateral mixingand entrainment
2. Erosion of spherical cap and mixing of ambient and bubble air.
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3. Extension of Cumulus mass
Initial mass
Spherical cap completely eroded and no longer buoyant.
Cloud massevaporating
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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 © 2012 R. R. Dickerson & Z.Q. Li18
Cumuli and Horizontal Winds
z
z
wind
Wake carried downstream
Greatest vertical growth is on down-shear side.
~ Verified by observation ~
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Micro-Pulse Lidar Network (MPLNET)
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Micro-Pulse Lidar Network (MPLNET)
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Micro-Pulse Lidar Network (MPLNET)
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From 500 m resolution WRF run (D-L Zhang)
Micro-Pulse Lidar Network (MPLNET)
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