the atmosphere: lecture 3 · cpdt gdz convection ii: compressible ideal gas, no condensation...
TRANSCRIPT
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
tropopausetoo cold
surface much too warm
Atmospheric energy balance
Hydrostatic balance
Mass of cylinder M A z
Forces acting:(i) gravitational force Fg −gM −g A z,(ii) pressure force acting at the top face, FT −p A, and(iii) pressure force acting at the bottom face, FB p pA
Fg FT FB 0 → p A −g A z, i.e.,
∂p∂z −g
Pressure and density profiles in a compressible atmosphere
∂p∂z −g
p
RT
∂p∂z − g
RT p
p p0 exp − zHp p0 exp − z
H ; H RTg
hydrostatic balance
perfect gas law
Isothermal atmosphere
p p0 exp −0z dz ′
Hz ′
gas constant for dry air R = 287 J kg-1K-1
More generally, H=H(z) and
Pressure and density profiles in a compressible atmosphere
∂p∂z −g
p
RT
∂p∂z − g
RT p
p p0 exp − zHp p0 exp − z
H ; H RTg
(T=237K)
hydrostatic balance
perfect gas law
Isothermal atmosphere
More generally, H=H(z) and
p p0 exp −0z dz ′
Hz ′
ConvectionI: Incompressible fluid, no condensation
Ts sT
T and ρ are conserved under adiabatic displacement
∂∂z 0 ≡ ∂T
∂z 0
∂∂z 0 ≡ ∂T
∂z 0
stable
unstable
Thermodynamics of dry air
p, T pRT
s sp, T
s cp ln
Cp = 1005 J kg-1K-1
p0 = 1000 hPaκ = R/cp = 2/7 (diatomic ideal gas)
T p 0p
potential temperature
specific entropy
dq cv dT p d 1
cp dT − 1 dp
cp dT − RT dpp
ds dqT cp
dTT − R dp
p cpd
(+ constant)
ds 0 → d 0Adiabatic processes :
θ is conserved under adiabatic displacement
(N. B. θ=T at p =p0= 1000 hPa)
0 d p0p
cpdT − RT
p dp
p0p
cpdT − 1
dpp0p
cpdT g dz
ConvectionII: Compressible ideal gas, no condensation
hydrostatic balance dp −g dz
adiabatic displacement
∂T∂z
−Γ
Γ gcp
9.76 10−3 Km−1
— adiabatic lapse rate
Following displaced parcel
T p 0p
unstable
stable
∂T∂z environment
− Γ
∂T∂z environment
− Γ
dTdz env
−Γ
ddz 0
dTdz parcel
−Γ
∂∂z 0
0 d p0p
cpdT − RT
p dp
p0p
cpdT − 1
dpp0p
cpdT g dz
ConvectionII: Compressible ideal gas, no condensation
hydrostatic balance dp −g dz
adiabatic displacement
∂T∂z
−Γ
Γ gcp
9.76 10−3 Km−1
— adiabatic lapse rate
Following displaced parcel
T p 0p
unstable
stable
∂T∂z environment
− Γ
∂T∂z environment
− Γ
∂∂z 0
dTdz parcel
−Γ
dTdz env
−Γ
ddz 0
Stability of Radiative Equilibrium Profile
-10 K/km
radiativeequilibrium solution
• Radiative equilibrium is unstable in thetroposphere
Effects of convection
Model aircraft observations in an unsaturated convective region (Renno & Williams)
Effects of convection
radiative-convective equilibrium
TRO
PO
SP
HE
RE
STR
ATO
SP
HE
RE
Radiative-Convective Equilibrium
-10 K/km
radiativeequilibrium solution
• Radiative equilibrium is unstable in thetroposphere
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