Lecture 4Lecture 4Atmospheric Radiative Atmospheric Radiative

Transfer; Role of clouds on Transfer; Role of clouds on climateclimate

GEU0136 Climatology

2-Layer Atmosphere2-Layer Atmosphere

Radiative Balances by LayerRadiative Balances by Layer

401(1 )

4 p

ST

4 42 12T T

4 4 41 22sT T T

4 402(1 ) 2

4 p s

ST T

For every layer: Energy In = Energy Out

TOA

L1

L2

Surface

2-Layer B.B. Atmosphere (cont’d)2-Layer B.B. Atmosphere (cont’d)

• Solving energy budgets for all layers simultaneously gives

• Recall from Lecture 3 that a 1 layer B-B atmosphere produces Ts

4 = 2Te4

• In general, an n-layer B-B atmosphere will have Ts

4 = (n+1)Te4

04 44 13 3

p

S e

ST T

Vertical temperature profile for 4-layer atmosphere, with thin graybody layers at top and bottom. Very unrealistic lapse rate!!

Why?

Molecular Absorbers/EmittersMolecular Absorbers/Emitters

• Molecules of gas in the atmosphere interact with photons of electromagnetic radiation

• Different kinds of molecular transitions can absorb/emit very different wavelengths of radiation

• Some molecules are able to interact much more with photons than others

• Different molecular structures produce wavelength-dependent absorptivity/emissivity

Atmospheric AbsorptionAtmospheric Absorption

• Triatomic modelcules have the most absorption bands

• Complete absorption from 5-8 m(H2O) and > 14 m(CO2)

• Little absorption between about 8 m and 11 m (“window”)

Line BroadeningLine Broadening

• Molecular absorption takes place at distinct wavelengths (frequencies, energy levels)

• Actual spectra feature absorption “bands” with broader features.

Why?

① Pressure broadening– Collisions among molecules dissipate

energy as kinetic (Lorentz profile)

② Doppler broadening– Relative motions among molecules

and photons (Doppler profile)

Sun-Earth GeometrySun-Earth Geometry

Sun’s rays

TerminologyTerminology

• Radiance is energy per unit solid angle, usually referred to in a given band of wavelengths

• Flux (or irradiance) is the total energy passing through a plane (integral of radiance)

• = zenith angle• azimuth angle• d solid angle

increment

Solar AbsorptionSolar Absorption

• Absorption depends on path length through the atmosphere, not vertical distance

• dz = ds cos • ds = dz / cos

abs adF k Fds

Beer’s Law (absorption)Beer’s Law (absorption)

Exponential “decay” of radiation as it passes through absorbing gas

abs adF k Fds

(convert from ds to dz)

(define optical depth)

(optical depth is a convenient coord!)

Atmospheric Absorption and Atmospheric Absorption and HeatingHeating

(density of absorbing gas decreases with zH is scale height = RT/g)

(optical depth as a function of height and mixing ratio of absorber)

(differentiate and divide … simple relationship between optical depth and z)

cos

Heating!Local fluxAbsorption

(Heating rate is proportional to flux divergence)

Absorption (Heating) Rate Absorption (Heating) Rate (cont’d)(cont’d)

• Maximum absorption occurs at level of unit optical depth

• Higher in the atmosphere as sun is closer to horizon

Where is max heating? Find out by differentiating previous equation w.r.t. , setting to zero, and solving for

not 0 / = 1

Thermal Absorption and Thermal Absorption and EmissionEmission

• Upwelling terrestrial radiation is absorbed and emitted by each layer

• As with solar radiation, path length ds is the distance of interest, rather than dz

• Also have to consider solid angle d

Infrared Radiative TransferInfrared Radiative Transfer

For radiance of a given frequency passing through a thin layer along a path ds

dI E A

emission

absorption (Beer’s Law)

emissivity Planck

function

Kirchoff’s Law: a =

so =a ds k

Gathering terms:

Planck intensity

Infrared Radiative Transfer Infrared Radiative Transfer (cont’d)(cont’d)

Previous result:

Convert to z:

Define optical depth from surface up:

Rewrite result in coordinate:

IR Radiative Transfer IR Radiative Transfer Schwarzchild’s EquationSchwarzchild’s Equation

Previous result:

Multiply by integrating factor/ne

Interpretation for Interpretation for Schwarzchild’s EquationSchwarzchild’s Equation

• Upwelling radiance at a given level has contributions from the surface and from every other level in between

• Relative contributions are controlled by vertical profiles of temperature and absorbing gases

Radiance at a given optical depth (z) and angle

Emissionfrom sfc

Absorptio

n below

Sum of emissionsfrom each atm level

weighted by absorptivity/emissivity of each layer in between

Simple Form of Schwarzchild’s Simple Form of Schwarzchild’s R.T.E.R.T.E.

Integrate Schwarzchild across thermal IR and across all angles and make simplifying assumptions to obtain simpler expressions for upwelling and downwelling radiative fluxes

upwelling:

downwelling:

blackbody emission(temperature dependence)

transmission functions(emissivity and radiances)

IR Fluxes and HeatingIR Fluxes and Heating

OLR and downward IR at surface depend on temperature profile and transmission functions

Net flux(z):

Heating rate:

TOA OLR

IR at sfc

Transmission Functions and Transmission Functions and HeatingHeating

• Think of upwelling and downwelling IR as weighted averages of T4

• The change in transmission function with height is the weighting function

• Downwelling IR at surface comes from lower troposphere

• Upwelling IR at TOA comes from mid-upper troposphere

• This is the very basis for the so-called “greenhouse effect”

Vertical profiles of atmospheric LW transmission functions and temperature

Cloud Radiative PropertiesCloud Radiative Properties

Cloud Radiative Properties:Cloud Radiative Properties:Dependence on Liquid Water PathDependence on Liquid Water Path

• Recall a + r + = 1

• Thick clouds reflect and absorb more than thin (duh!)

• Generally reflect more than absorb, but less true at low solar zenith angles

Cloud Radiative Properties:Cloud Radiative Properties:Dependence on Drop SizeDependence on Drop Size

• Small droplets make brighter clouds

• Larger droplets absorb more

• Dependence on liquid water path at all droplet sizes too

Cloud Radiative PropertiesCloud Radiative PropertiesLongwave EmissivityLongwave Emissivity

• Clouds are very good LW absorbers. • Clouds with LWC > 20 g/m2 are almost blackbodies!

Radiative-Convective ModelsRadiative-Convective Models(a recipe)(a recipe)

• Consider a 1-D atmosphere

• Specify solar radiation at the top, emissivity of each layer

• Calculate radiative equilibrium temperature for each layer

• Check for static stability

• If layers are unstable, mix them! – (e.g. if > d, set both T’s to mass-

weighted mean of the layer pair)

• Add clouds and absorbing gases to taste

Tn

T1 1

n

T3 3

…

Manabe and Strickler (1964)

Radiative-Convective Radiative-Convective EquilibriumEquilibrium

• Pure radiative equilibrium is way too hot at surface

• Adjusting to d still too steep

• Adjusting to observed 6.5 K km-

1 produces fairly reasonable profile:– Sfc temp (still hot)– Tropopause (OK)– Stratosphere (OK)

Radiative-Convective Radiative-Convective EquilibriumEquilibrium

Effect of Different AbsorbersEffect of Different Absorbers• Water vapor

alone … atmosphere is cooler

• H2O + CO2 … almost 10 K warmer

• H2O + CO2 + O3 … stratosphere appears!

Radiative-Convective Radiative-Convective EquilibriumEquilibrium

Radiative Heating RatesRadiative Heating Rates• L indicates longwave

cooling• S indicates heating

(by solar absorption)• NET combines all• Heating and cooling

nearly balance in stratosphere

• Troposphere cools strongly (~ 1.5 K/day)

• How is this cooling balanced?– In the R.C.M?– In the real world?

Radiative-Convective Radiative-Convective EquilibriumEquilibriumEffects of CloudsEffects of Clouds

• Clouds absorb LW

• Clouds reflect SW

• Which effect “wins?”

• Depends on emitting T

• For low clouds, T4 ~ Ts4

, so SW effect is greater

• For high clouds, T4 << Ts

4 so LW effect “wins”

• High clouds warm

• Low clouds cool

Details are sensitive to optical properties and distributions of clouds, but remember the basic conclusions

Observed Mean Cloud FractionObserved Mean Cloud Fraction

• High clouds mostly due to tropical convection (Amazon, Congo, Indonesia, W. Pacific)

• Low clouds (stratocumulus) over eastern parts of subtropical ocean basins – Cold SST– Subsiding air– Strong inversion

high clouds( < 440 mb)

low clouds( > 680 mb)

all clouds

Annual Mean Cloud ForcingAnnual Mean Cloud Forcing• “Cloud forcing” is

defined as the difference between a “clearsky” and “all sky” measurement

• At the surface, (a) is all warming, and (b) is all cooling

• Net effect of clouds is to cool the surface, but changes can go either way

OLR

solar abs

Rnet

Global Mean Cloud Radiative Global Mean Cloud Radiative ForcingForcing

• Clouds increase planetary albedo from 15% to 30%

• This reduces absorbed solar by 48 W m-2

• Reduced solar is offset by 31 W m-2 of LW warming (greenhouse)

• So total cloud forcing is –17 W m-2

• Clouds cool the climate. How might this number change if cloudiness increased?

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