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