Transcript Convection
Convection: Parcel Theory
Chapt. 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 1
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 2
Parcel theory equations:
z
d
2
z dt
2
g
(
T
T
'
T
' )
Primes: environmental properties.
You can correct for the effect of moisture on density by using virtual temperature wherever T appears here.
Let B be buoyancy.
or
z
gB with B
T
T
T
Copyright © 2009 R. R. Dickerson & Z.Q. Li 3
Designate updraft velocity then
z
dU dt and
z
U dU dz
gB so
U
dz dt dt
dz U UdU
gBdz U
2
U o
2 2
g z
z B
(
z
)
dz
& Z.Q. Li 4
Alternate formulation using >> d P
g dz
= P’/R’T’ and dP’ = gdz 2
g o z
z B
(
z
)
dz
2
g o p
p
(
T
T
T
)
R T
d P
P
g
2
R
p p
0 (
T
T
)
d P
P
U
2
U o
2 2
R p
0
p
(
T
T
)
d P
P
Copyright © 2009 R. R. Dickerson & Z.Q. Li 5
-RlnP T’
(environment)
Parcel process curve T Hatched area is proportional to the kinetic energy of the parcel. Change in area from level to level represents the change in K.E.
Copyright © 2009 R. R. Dickerson & Z.Q. Li 6
Correcting for Condensed H
2
O
Let
mass of liquid mass of air
; assume μ= total adiabatic liquid water
then
z
g T T
( 1 )
dP
g
( 1 )
dz
Copyright © 2009 R. R. Dickerson & Z.Q. Li 7
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.
Copyright © 2009 R. R. Dickerson 8 & Z.Q. Li