Atmospheric Convection
A Primer
Homogeneous Compressible Gases
Buoyancy and Entropy
Specific Volume:
α = 1ρ
Specific Entropy:
s
α = α ( p, s)
 ∂α   ∂T 
Maxwell : 

 =
∂
∂s
p

p 
s
 ∂T
 ∂α 
=
 δ s = 
 ∂s  p
 ∂p
B=g
(δα ) p
(δα ) p g  ∂T 
α
=

α  ∂p

 δ s
s
 ∂T 
 δ s = − 
 δ s ≡ Γδ s
 ∂z  s
s
The adiabatic lapse rate:
First Law of Thermodynamics :
dsrev
dT
dα
Q =T
= cv
+p
dt
dt
dt
dp
dT d (α p )
= cv
+
−α
dt
dt
dt
dT
dp
= ( cv + R )
−α
dt
dt
dT
dp
= cp
−α
dt
dt
Adiabatic : c p dT − α dp = 0
Hydrostatic : c p dT + gdz = 0
→
(
dT
dz )
s
=− g
cp
≡ −Γ d
g
Γ= c
p
1K
Γ
=
Earth’s atmosphere:
100 m
Model Aircraft Measurements
(Renno and Williams, 1995)
Image courtesy of American Meteorological Society.
Prandtl’s (1925) problem
Q
∞ ∫0 Qdz
=F
F ~ L2t −3
→w~
( Fz )
1
3
1
 3
F2
B~ z


Above a thin boundary layer, most atmospheric
convection involves phase change of water:
Moist Convection
Moist Convection
• Significant heating owing to phase
changes of water
• Redistribution of water vapor – most
important greenhouse gas
• Significant contributor to stratiform
cloudiness – albedo and longwave
trapping
Water Variables
Mass concentration of water vapor (specific humidity):
q≡
MH O
2
M air
Vapor pressure (partial pressure of water vapor):
Saturation vapor pressure:
C-C:
e*
17.67(T −273)
T +30
e* = 6.112 hPa e
Relative Humidity:
H ≡ e e*
e
The Saturation Specific
Humidity
Ideal Gas Law:
R*T
p=ρ
m
R*T
e = ρv
mv
mv e
ρ
v
q= ρ =
m p
mv e*
q*=
m p
Phase Equilibria
Bringing Air to Saturation
e = qp  m m 

v
e* = e * (T )
1. Increase q (or p)
2. Decrease e* (T)
When Saturation Occurs…
• Heterogeneous Nucleation
• Supersaturations very small in atmosphere
• Drop size distribution sensitive to size
distribution of cloud condensation nuclei
ICE NUCLEATION PROBLEMATIC
Percentage of clouds with ice particle
concentrations above detectable level
15
16 16
80
7 1 2 1 1 1
8
23
60
25
40
12
10
25
14
20
33
15
20
0
0
4
3
100
1
1
-5
-10
-15
Cloud top temperature (oC)
Figure by MIT OCW.
-20
-25
Precipitation Formation:
• Stochastic coalescence (sensitive to drop
size distributions)
• Bergeron-Findeisen Process
• Strongly nonlinear function of cloud water
concentration
• Time scale of precipitation formation ~1030 minutes
Stability
No simple criterion based on entropy:
T 
 p
sd = c p ln   − Rd ln  
 p0 
 T0 
α = α ( sd , p )
T 
 p 
q
+L
− qR ln (H
s = c ln   − R ln 
p T 
d  p  vT
v
 0
 0
α = α ( s, p, qt )
)
Virtual Temperature and Density
Temperature
Assume all condensed water falls at
terminal velocity
Va + Vc
α=
Md + Mv + Mc
pV = nR *T
R *T
Va =
p
 Md Mv 
+

,
 md mv 
1
md ≡
1
Md
Rd T
→ Va =
p
where
ε≡
mv
Mi
∑i m
i
 M + Mv
 d
ε

md
,


≅ 0.622
Rd ≡ R *
md
(
)

Va + Vc
Rd T
qc ρ a 
q
=
α=
1 − qt +
1+


ε
Md + Mv + Mc
p
 1− qc ρc 
Rd T
≅
1 − qt + q
ε
p
Mv + Mc
Mv
qt ≡
,
q≡
M
M
(
)
Density temperature:
(
q
Tρ ≡ T 1 − qt +
α=
Rd Tρ
p
ε
)
Trick:
Define a saturation entropy, s* :
s* ≡ s (T , p,q*)
α = α ( s*, p, qt )
We can add an arbitrary function of qt to s* such that
α ≅ α ( s*', p )
Stability Assessment using Tephigrams:
100
Pressure (mb)
200
300
400
500
600
700
800
900
1000
-40
-30
-20
-10
0
10
Temperature (C)
20
30
40
50
Stability Assessment using Tephigrams:
Convective Available Potential Energy
(CAPE):
CAPEi ≡ ∫ p (α p − α e )dp
pi
n
(
)
= ∫ p Rd Tρ − Tρ d ln ( p )
pi
p
e
Other Stability Diagrams:
“Air-Mass” Showers:
This image is copyrighted by Oxford, NY: Oxford University Press, 2005. Book title is Divine Wind: The History and Science of
Hurricanes. ISBN: 9780195149418. Used with permission.
Image courtesy of American Meteorological Society.
Precipitation Effects:
Buoyancy Reversal:
Summary of Differences Between Dry and
Moist Convection:
• Possibility of metastable states
• Strong asymmetry between cloudy and
clear regions
• Typically, only thin layers near surface are
unstable to upward displacements
• Mixing can cause buoyancy reversal
• Large potential for evaporatively cooled
downdrafts
• Separation of buoyancy from displacement
can lead to propagating convection
• Buoyancy of unsaturated downdrafts
depends on supply of precipitation
• Entropy produced mostly through
mixing, not dissipation
• Internal waves can co-exist with
unstable convection
Tropical Soundings
Annual Mean Kapingamoronga:
Radiative-Moist Convective
Equilibrium
Precipitating Convection favors Widely
Spaced Clouds (Bjerknes, 1938)
This image is copyrighted by Oxford, NY: Oxford University Press, 2005.
Book title is Divine Wind: The History and Science of Hurricanes. ISBN: 9780195149418. Used with permission.
Properties:
• Convective updrafts widely spaced
• Surface enthalpy flux equal to vertically
integrated radiative cooling
c pT ∂θ
= −Q
M
•
θ ∂z
• Precipitation = Evaporation = Radiative
Cooling
• Radiation and convection highly interactive
5
4.5
Pre cipita tion (mm/da y)
4
3.5
3
2.5
2
1.5
1
0.5
0
1
2
3
4
5
6
Time (days)
7
8
9
10
This image is copyrighted by Academic Press, NY, 1999. It is in a chapter entitled "Quasi-equilibrium Thinking" (author is Kerry Emanuel) that
appears in the book entitled General Circulation Model Development: Past, Present and Future. Edited by D. Randall. ISBN: 9780125780100.
Used with permission.
This image is copyrighted by Academic Press, NY, 1999. It is in a chapter entitled "Quasi-equilibrium Thinking" (author is Kerry Emanuel) that
appears in the book entitled General Circulation Model Development: Past, Present and Future. Edited by D. Randall. ISBN: 9780125780100.
Used with permission.
This image is copyrighted by Academic Press, NY, 1999. It is in a chapter entitled "Quasi-equilibrium Thinking" (author is Kerry Emanuel) that
appears in the book entitled General Circulation Model Development: Past, Present and Future. Edited by D. Randall. ISBN: 9780125780100.
Used with permission.
This image is copyrighted by Academic Press, NY, 1999. It is in a chapter entitled "Quasi-equilibrium Thinking" (author is Kerry Emanuel) that
appears in the book entitled General Circulation Model Development: Past, Present and Future. Edited by D. Randall. ISBN: 9780125780100.
Used with permission.
Recovery from mid-level specific humidity
perturbation
Specific humidity perturbation (g/Kg), from -5.479 to 2.645
-100
-200
-300
P (mb)
-400
-500
-600
-700
-800
-900
-1000
1
2
3
4
5
6
Time (days)
7
8
9
10
Tv Perturbation, from -6.87 to 0.848
-100
-200
-300
P (m b)
-400
-500
-600
-700
-800
-900
-1000
1
2
3
4
5
6
Time (days)
7
8
9
10
Robe and Emanuel, J. Atmos. Sci., 1996
Image courtesy of American Meteorological Society.
Islam et al. Predictability Experiments
Robe and Emanuel, J. Atmos. Sci., 2001
Image courtesy of American Meteorological Society.
Fovell and Ogura, J. Atmos. Sci., 1988, 1989
Image courtesy of American Meteorological Society.
Rotunno, Klemp and Weisman, J. Atmos.
Sci., 1988
Image courtesy of American Meteorological Society.
Speculative Regime Diagram
2-D Shear-Parallel
Convection
E, zb
2-D Shear-Perpendicular
Convection
3-D Convection
/DU/
Figure by MIT OCW.
Non-equilibrium Convection
Desert
Low qe
Gulf of Mexico
High qe
Figure by MIT OCW.
Klemp and Wilhelmson, J. Atmos. Sci., 1978
Image courtesy of American Meteorological Society.