The Atmosphere: Lecture 3 · cpdT gdz Convection II: Compressible ideal gas, no condensation...
Transcript of The Atmosphere: Lecture 3 · cpdT gdz Convection II: Compressible ideal gas, no condensation...
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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)
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Full calculation of radiative equilibrium
stratosphere about right
tropospheric lapse rate too large
tropopausetoo cold
surface much too warm
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Atmospheric energy balance
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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
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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
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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 ′
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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
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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)
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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
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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
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Stability of Radiative Equilibrium Profile
-10 K/km
radiativeequilibrium solution
• Radiative equilibrium is unstable in thetroposphere
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Effects of convection
Model aircraft observations in an unsaturated convective region (Renno & Williams)
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Effects of convection
radiative-convective equilibrium
TRO
PO
SP
HE
RE
STR
ATO
SP
HE
RE
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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.
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Radiative-convective equilibrium(unsaturated)
Better, but:
• surface still too warm
• tropopause still too cold
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Moist convection
Above a thin boundary layer, most atmospheric convection involves phase change of water: condensation releases latent heat