copyright © 2009 r. r. dickerson & z.q. li 1 convection: parcel theory chapt. 4 page 48....
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Copyright © 2009 R. R. Dickerson & Z.Q. Li
1
Convection: Parcel TheoryChapt. 4 page 48.
• Vertical motion of parcels of air.
• Buoyant convection leads to the formation of cumulus clouds (though other clouds have convective features).
• Parcel theory is an elementary treatment, but useful for qualitative understanding.
Copyright © 2009 R. R. Dickerson & Z.Q. Li
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Parcel Theory Assumptions
• Parcel and environment are in dynamic equilibrium: P = P’.
• Parcel maintains its identity; no mixing with environment.
• Atmosphere in hydrostatic equilibrium.
• No compensating motion by atmosphere as parcel moves.
Copyright © 2009 R. R. Dickerson & Z.Q. Li
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Parcel theory equations:
'
)'(2
2
T
TTg
dt
zdz
Primes: environmental properties.You can correct for the effectof moisture on density by usingvirtual temperature whereverT appears here.Let B be buoyancy.
or
gBz
with
T
TTB
Copyright © 2009 R. R. Dickerson & Z.Q. Li
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Designate updraft velocitydt
dzU
then
dt
dUz so
U
dzdt
and
gBdzUdUgBdz
dUUz
z
z
o
o
dzzBgUU )(222
Copyright © 2009 R. R. Dickerson & Z.Q. Li
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Alternate formulation using >> gdz
Pd
p
p
o
p
p
p
p
z
z
P
PdTTRUU
P
PdTTR
gP
PdTR
T
TTgdzzBg
oo
0
0
)(2
)(2
)(2)(2
22
= P’/R’T’ and dP’ = - gdz
Copyright © 2009 R. R. Dickerson & Z.Q. Li
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T
-RlnP
T’
Parcel process curve
(environment)
Hatched area is proportional to the kinetic energy ofthe parcel. Change in area from level to level representsthe change in K.E.
Copyright © 2009 R. R. Dickerson & Z.Q. Li
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Correcting for Condensed H2O
Let ;
assume μ= total adiabatic liquid water
mass of liquid
mass of air
)1()1(
g
dz
dP
T
Tgzthen
Copyright © 2009 R. R. Dickerson & Z.Q. Li
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Parcel Theory is Useful for:
• Estimating cloud base and top heights• Estimating stability• Estimating vertical motion U
But we’ve neglected• Drag• Mixing• Compensating motion by environment• Effects of condensed water
Therefore: parcel theory U => expected upper limit.