Transcript Lecture 7, Dry Parcel Theory

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:
d z
(T  T ' )
z  2  g
dt
T'
2
or
z  gB
with
T T
B
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.
Copyright © 2009 R. R. Dickerson
& Z.Q. Li
3
dz
U
dt
Designate updraft velocity
then
dU
z 
dt
so
dz
dt 
U
and
dU
z  U
 gB
dz

UdU  gBdz
z
U 2  U o2  2 g  B( z )dz
zo Dickerson
Copyright © 2009 R. R.
& Z.Q. Li
4
dP
   g
dz
Alternate formulation using >>
 = P’/R’T’ and dP’ = - gdz
(T  T ) RT dP
2 g  B ( z )dz  2 g 

T
Pg
z
p
z
p
o
o
dP 
 2 R  (T  T )
P
p0
p
dP 
 2 R  (T  T )
P
p0
p
U
2
 Uo
2
Copyright © 2009 R. R. Dickerson
& Z.Q. Li
5
T’ (environment)
-RlnP
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 H2O
Let
mass of liquid

;
mass of air
assume μ= total adiabatic liquid water
dP
T

then z  g   (1   )   
 g (1   )
dz
T

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 Copyright
theory
U => expected upper limit.8
© 2009 R. R. Dickerson
& Z.Q. Li