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

surface much too warm

tropopause too cold

stratosphere about right

tropospheric lapse rate too large

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

p z

g

pRT

p z

gRT

p

p p0 exp zHp p0 exp z

H; H RT

g

Pressure and density profiles in a compressible atmosphere

hydrostatic balance

perfect gas law

Isothermal atmosphere

p p0 exp 0

z dz

Hz

More generally, H=H(z) and

gas constant for dry air R = 287 J kg-1K-1

p z

g

pRT

p z

gRT

p

p p0 exp zHp p0 exp z

H; H RT

g

Pressure and density profiles in a compressible atmosphere

hydrostatic balance

perfect gas law

Isothermal atmosphere

p p0 exp 0

z dz

Hz

More generally, H=H(z) and

(T=237K)

ConvectionI: Incompressible fluid, no condensation

T

s 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 p

RT Cp = 1005 J kg-1K-1

dQ cv dT p d 1

cp dT 1 dp

cp dT RTdpp

Thermodynamics of dry air

p,T p

RT

s sp,T

Cp = 1005 J kg-1K-1

specific entropy

dQ cv dT p d 1

cp dT 1 dp

cp dT RTdpp

ds dQ

T cp

dTT

R dpp cp

d

Thermodynamics of dry air

p,T p

RT

s sp,T

s cp ln

Cp = 1005 J kg-1K-1

p0 = 1000 hPa κ = R/cp = 2/7 (diatomic ideal gas)

T p 0

p

potential temperature

(+ constant)

specific entropy

dQ cv dT p d 1

cp dT 1 dp

cp dT RTdpp

ds dQ

T cp

dTT

R dpp cp

d

Thermodynamics of dry air

p,T p

RT

s sp,T

s cp ln

Cp = 1005 J kg-1K-1

p0 = 1000 hPa κ = R/cp = 2/7 (diatomic ideal gas)

T p 0

p

potential temperature

Adiabatic processes : ds 0 d 0

θ is conserved under adiabatic displacement

(N. B. θ=T at p =p0= 1000 hPa)

(+ constant)

specific entropy

dQ cv dT p d 1

cp dT 1 dp

cp dT RTdpp

ds dQ

T cp

dTT

R dpp cp

d

0 d p0p

cpdT RT

p dp

p0p

cpdT 1

dp

p0p

cpdT g dz

ConvectionII: Compressible ideal gas, no condensation

adiabatic displacement

T p 0

p

0 d p0p

cpdT RT

p dp

p0p

cpdT 1

dp

p0p

cpdT g dz

ConvectionII: Compressible ideal gas, no condensation

hydrostatic balance

dp g dz

adiabatic displacement

T p 0

p

0 d p0p

cpdT RT

p dp

p0p

cpdT 1

dp

p0p

cpdT g dz

ConvectionII: Compressible ideal gas, no condensation

hydrostatic balance

dp g dz

adiabatic displacement

T z

gcp

9.76 10 3 Km 1

— adiabatic lapse rate

Following displaced parcel

T p 0

p

dTdz parcel

z

0

0 d p0p

cpdT RT

p dp

p0p

cpdT 1

dp

p0p

cpdT g dz

ConvectionII: Compressible ideal gas, no condensation

hydrostatic balance

dp g dz

adiabatic displacement

T z

gcp

9.76 10 3 Km 1

— adiabatic lapse rate

Following displaced parcel

T p 0

p

unstable

stable

T z environment

T z environment

dTdz env

ddz

0

dTdz parcel

z

0

0 d p0p

cpdT RT

p dp

p0p

cpdT 1

dp

p0p

cpdT g dz

ConvectionII: Compressible ideal gas, no condensation

hydrostatic balance

dp g dz

adiabatic displacement

T z

gcp

9.76 10 3 Km 1

— adiabatic lapse rate

Following displaced parcel

T p 0

p

unstable

stable

T z environment

T z environment

z

0

dTdz parcel

dTdz env

ddz

0

Stability of Radiative Equilibrium Profile

• Radiative equilibrium is unstable in thetroposphere

-10 K/km

radiative equilibrium solution

Effects of convection

Model aircraft observations in an unsaturated convective region (Renno & Williams)

Effects of convection

radiative-convective equilibrium

Effects of convection

radiative-convective equilibrium

TR

OP

OS

PH

ER

ES

TR

AT

OS

PH

ER

E

Radiative-Convective Equilibrium

• Radiative equilibrium is unstable in thetroposphere Re-calculate equilibrium subject to the constraint that tropospheric stability is rendered neutral by convection.

-10 K/km

radiative equilibrium solution

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