Transcript The Atmosphere: Lecture 3

The Atmosphere:
Part 3: Unsaturated convection
•
Composition / Structure
•
Radiative transfer
• Vertical and latitudinal heat transport
•
•
Atmospheric circulation
Climate modeling
Suggested further reading:
Hartmann, Global Physical Climatology (Academic Press, 1994)
Full calculation of radiative equilibrium
stratosphere
about right
tropospheric lapse
rate too large
tropopause
too cold
surface much too
warm
Atmospheric energy balance
Hydrostatic balance
Mass of cylinder
M  A z
Forces acting:
(i) gravitational force F g gM gA z,
(ii) pressure force acting at the top face, F T p A, and
(iii) pressure force acting at the bottom face, F B 
p p
A
F g F T F B 0  p A gA z, i.e.,

p
g

z
Pressure and density profiles in a compressible atmosphere
gas constant for dry air R = 287 J kg-1K-1
hydrostatic balance
perfect gas law

p
g

z

p
RT

p
g
 p
RT

z
Isothermal
atmosphere
z
p p
0 exp
p p
zH; H  RT
0 exp
g
H
More generally, H=H(z) and
dz 
0 H
z 
p p 0 exp 
z
Pressure and density profiles in a compressible atmosphere
hydrostatic balance
perfect gas law

p
g

z

p
RT

p
g
 p
RT

z
Isothermal
atmosphere
z
p p
0 exp
p p
zH; H  RT
0 exp
g
H
More generally, H=H(z) and
dz 
0 H
z 
p p 0 exp 
z
(T=237K)
Convection
I: Incompressible fluid, no condensation
 T
s s
T
T and ρ are conserved
under adiabatic displacement


T 0
0  

z

z
stable


T 0
0  

z

z
unstable
Thermodynamics of dry air

p, T
p
RT
1
dQ c v dT p d 
1 dp
c p dT  
dp
c p dT RT p
Cp = 1005 J kg-1K-1
Thermodynamics of dry air

p, T
specific entropy
s s
p, T
p
RT
1
dQ c v dT p d 
Cp = 1005 J kg-1K-1
1 dp
c p dT  
dp
c p dT RT p
ds 
dQ
T
c p dT
R
T
dp
p
c p d
Thermodynamics of dry air

p, T
s s
p, T
Cp = 1005 J kg-1K-1
p0 = 1000 hPa
κ = R/cp = 2/7 (diatomic ideal
gas)
dp
c p dT RT p
potential
temperature
 T
1
dQ c v dT p d 
1 dp
c p dT  
specific entropy
p0
p
p
RT

ds 
dQ
T
c p dT
R
T
s c p ln 
dp
p
(+ constant)
c p d
Thermodynamics of dry air

p, T
p
RT
s s
p, T
dp
c p dT RT p
potential
temperature
 T
Cp = 1005 J kg-1K-1
p0 = 1000 hPa
κ = R/cp = 2/7 (diatomic ideal
gas)
1 dp
c p dT  
specific entropy
p0
p
1
dQ c v dT p d 
ds 

dQ
T
c p dT
R
T
s c p ln 
Adiabatic processes :
dp
p
c p d
(+ constant)
ds 0  d 0
θ is conserved under adiabatic displacement
(N. B. θ=T at p =p0= 1000 hPa)
Convection
II: Compressible ideal gas, no condensation
 T
p0
p
adiabatic displacement
p0
p
p
 p0
p0
p
0 d 



c p dT  RT
p dp
1 dp
c p dT  
c p dT g dz

Convection
II: Compressible ideal gas, no condensation
 T
p0
p

adiabatic displacement
p0
p
p
 p0
p0
p
0 d 



c p dT  RT
p dp
1 dp
c p dT  
c p dT g dz
hydrostatic balance
dp gdz
Convection
II: Compressible ideal gas, no condensation
 T
p0
p

adiabatic displacement
p0
p
p
 p0
p0
p
0 d 
dT
dz
parcel




c p dT  RT
p dp
1 dp
c p dT  
c p dT g dz
Following displaced parcel

T 

z


0

z
g
 c 9. 76 10 3 Km 1
p
— adiabatic lapse rate
hydrostatic balance
dp gdz
Convection
II: Compressible ideal gas, no condensation
p0
p
 T

adiabatic displacement
dT
dz

env
d 0
dz
dT
dz
parcel
p0
p
p
 p0
p0
p
0 d 


c p dT  RT
p dp

1 dp
c p dT  

c p dT g dz
hydrostatic balance
Following displaced parcel

T 

z


0

z
g
 c 9. 76 10 3 Km 1
p
— adiabatic lapse rate

T

z
environm ent

T

z
environm ent

unstable

stable
dp gdz
Convection
II: Compressible ideal gas, no condensation
p0
p
 T

adiabatic displacement
dT
dz
parcel

p0
p
p
 p0
p0
p
0 d 

c p dT  RT
p dp

1 dp
c p dT  

c p dT g dz
hydrostatic balance
Following displaced parcel
dT
dz

T 

z

env
d 0
dz


0

z
g
 c 9. 76 10 3 Km 1
p
— adiabatic lapse rate

T

z
environm ent

T

z
environm ent

unstable

stable
dp gdz
Stability of Radiative Equilibrium Profile
radiative
equilibrium
solution
-10 K/km
• Radiative equilibrium is unstable in the
troposphere
Effects of convection
Model aircraft observations
in an unsaturated convective
region (Renno & Williams)
Effects of convection
radiative-convective
equilibrium
Effects of convection
TROPOSPHERE
STRATOSPHERE
radiative-convective
equilibrium
Radiative-Convective Equilibrium
radiative
equilibrium
solution
-10 K/km
• Radiative equilibrium is unstable in the
troposphere
Re-calculate equilibrium subject to the
constraint that tropospheric stability is
rendered neutral by convection.
Radiative-convective equilibrium
(unsaturated)
Better, but:
• surface still too
warm
• tropopause still too
cold
Moist convection
Above a thin boundary layer, most
atmospheric convection involves
phase change of water:
condensation releases latent heat