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MET 60

Chapter 4:

Radiation & Radiative Transfer

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The layout of chapter 4 is:

• Basics of radiation

• Scattering, Absorption & Emission of radiation

• Radiative Transfer

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Basics of radiation

• Properties of radiation (pp. 113-117)

– wavelength, frequency etc.

– Intensity vs. flux

– Blackbody radiation

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Basics of radiation cont.

• Basic Radiation Laws (pp. 117-120)

– Wien’s Law & Stefan-Boltzman Law

– About the type and amount of radiation emitted

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Scattering, Absorption & Emission of radiation

• Emissivity, absorptivity, transmissivity, reflectivity (p. 120)

– All relate to things that can happen to radiation as it passes through the atmosphere

• The Greenhouse Effect (p.121)

• The physics of scattering (pp. 122-125)

– Type & amount of scattering depends on number, size & shape of particles in the air

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Scattering, Absorption & Emission of radiation contd.

• The physics of absorption

– Lots of details! (pp. 126-130).

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Radiative Transfer

Putting it all together to follow a beam of incident radiation:

• From top of atmosphere to the surface

• And back up

• With interactions along the beam (scattering, absorption etc.)

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Radiative Transfer contd.

With the result being:

• A vertical profile of heating rates due to radiation

• e.g., in the form of the values of

Remember that radiative heating drives the atmosphere!– Vertical distribution (here)

– Horizontal distribution (climatology-related)

zt

T

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Basics of Radiation

• The sun emits radiation (type? amount?)

• Earth intercepts it and also emits its own radiation (type? amount?)

• Radiation is characterized by:– Frequency () … measured in “per sec”– Wavelength () … measured in micrometers (µm) or

microns– Wavenumber (-1)

• Note: all EM radiation travels at the speed of light (c) with c =

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The EM spectrum…

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Solar radiation consists mainly of:

• uv radiation (? – 0.38 µm)

• visible radiation (0.38 – 0.75 µm)

• IR radiation (0.75 - ? µm)

– Near-IR has < 4 µm

– Far-IR has > 4 µm

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Terms…

• Monochromatic intensity of radiation is the amount of energy at wavelength passing through a unit area (normal to area) in unit time

I

• Adding over all wavelengths (all values of ), we get radiance, or intensity: I

• I is also called radiance

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Terms…

• Monochromatic flux density (irradiance) is the rate of energy transfer through a plane surface per unit area due to radiation with wavelength

F

• For, say, a horizontal surface in the atmosphere:

2cos dIF

Monochromatic

radiance

Accounts for radiation arriving in slanted direction

Integrate over ½ sphere

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Note…on confusion!

http://en.wikipedia.org/wiki/Irradiance

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Inverse Square Law…

• Flux density (F) obeys the inverse square law:

F 1/d2

where d = distance from source (sun!)

sun earth mars

149 million km 227.9 million km

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Blackbody Radiation…

A surface that absorbs ALL incident radiation is called a blackbody

All radiation absorbed – none reflected etc.

• Hypothetical but useful concept

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Blackbody Radiation…

• Radiation emitted by a blackbody is given by:

• c1 and c2 are constants

1)(

2

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Tce

cTB

T

Planck Function

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Blackbody Radiation…

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Blackbody Radiation…

• Fig. 4.6 shows how emission varies with for different temperatures

• Choosing T values representative of the sun and of earth gives Fig. 4.7 (upper)

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Wavelength of peak emission?

• Wien’s Displacement Law…

• or

T

1max

T

2897max

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Solar radiation …

Peaks in the visible

Concentrated in uv-vis-IR

Terrestrial radiation …

Peaks in the IR (15-20 m)

All in IR (far IR)

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Maximum intensity of emission?

• Stefan-Boltzmann Law…

So the sun emits much more radiation than earth

since Tsun >> Tearth

4TF

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Example 4.6

Calculate blackbody temperature of earth (Te).

Assume: earth is in radiative equilibrium

energy in = energy out

Assume: albedo = 0.3 (fraction reflected back to space)

Assume: solar constant = 1368 W/m2

= incoming irradiance/flux density @ top of atmosphere

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Incoming energy:

Given by solar constant

spread over area of Earth = area the beam intercepts

area = Re2

Thus incoming = 1368 x (1 – 0.3) x Re2

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Outgoing energy:

Given by Fe = Te4

where we need to find Te

Now this is per unit area, so the total outgoing energy is

Fe = 4Re2Te

4

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Equating:

4Re2Te

4 = 1368 x (1 – 0.3) x Re2

4Te4 = 1368 x (1 – 0.3)

Te = {1368 x 0.7 / 4}¼

Te = 255 K

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Non-blackbody radiation

A blackbody absorbs ALL radiation

A non-blackbody can also reflect and transmit radiation

Example – the atmosphere!

Actually, the gases that make up the atmosphere!

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Definitions:

emissivity

actual radiation emitted / BB radiation

BB has = 1

absorptivity

radiation absorbed / radiation incident

I

B

( )

( )

I absorbed

I incident

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reflectivity

radiation reflected / radiation incident

transmissivity

radiation transmitted / radiation incident

( )

( )

I reflectedR

I incident

( )

( )

I transmittedT

I incident

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incident

reflection

absorption

transmission

absorption

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incident

reflection

absorption

transmission

absorptionscattering

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Kirchoff’s Law

emissivity = absorptivity (at Em)

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An example regarding the greenhouse effect…

1) Pretend the atmosphere can be represented as a single isothermal slab

The slab is transparent to solar radiation (all gets through!)

The slab is opaque to terrestrial radiation (none gets through!)

Everything is in Em.

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z=0

“top”

incoming = F unitsoutgoing = F units for balance

F units emitted downwards

Surface receives 2F units

Surface must emit 2F units for balance

F

F

F

2F

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Now use the 2F units of radiation emitted by the surface to compute Te via Stefan-Boltzman.

F = 1368 W/m2 modified by albedo

result: Te = 303 K

Greenhouse effect delivers 48 K of “warming” (single slab model)

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2) Pretend the atmosphere can be represented as two isothermal slabs…or three etc. – see text

per p.122, Te = 335 K etc.

Note:

include more layers → steeper lapse rate in lower atmosphere

eventually … > d … unstable atmosphere “predicted”

→ use a Radiative-convective model instead…”convective adjustment”

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Physics of Scattering, Absorption & Emission

Need to understand physics of these processes to come up with expressions for how much radiation is scattered etc. from a beam.

Scattering

Consider a “tube” of incoming radiation – Fig. 4.10.

Radiation may be scattered by:

– gas molecules (tiny)

– aerosol particles (small – tiny).

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Physics of Scattering, Absorption & Emission

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Scattering contd.

Scattering amount depends on:

1) Incident radiation intensity (I)

2) Amount of scattering gases/aerosols

3) Ability of these to scatter (size, shape etc.)

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Scattering contd.

For an incident intensity of I, an amount dI is lost by scattering, with

N = number of particles (gas, aerosol) per unit volume. = c/s area of each particleds = path length (see diagram)

K = (scattering or absorption) efficiency factor (large “K”)

Note:

K(total extinction) = K(scattering) + K(absorption)

.dI I K N ds

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Scattering contd.

For a gas, we write:

r = is the mass of the absorbing gas per unit mass of air

= air density

k = mass absorption coefficient (m2kg-1)(small “k”)

.dI I rk ds

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Scattering contd.

For a column (“tube”) of air from height z to the top of the atmosphere, we can integrate:

This represents the amount of absorbing material in the column down to height z.

Called the optical depth or optical thickness ( ).

Large much extinction in the column.

Note that is wavelength-dependent.

.z

rk dz

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Scattering is very complicated.

Scattering particles have a wide range of sizes and shapes (and distributions).

Start by looking at a sphere of radius r.

How does this scatter?

Extinction is given by Eq. 4.16 – need to know K - provided by theory (which we will not do!!)

???

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First…Fig. 4.11

y-axis: r = scattering radius (m)

x-axis: = wavelength (m)

Plotted is:

Fig. 4.11 shows us the different regimes of scattering that occur as a function of:

- wavelength of radiation (solar vs. terrestrial)

- size of scattering particle

2 rx

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Results of theory…

With small particles (x << 1), we get Rayleigh Scattering

And theory gives:

Particles scatter radiation forward and backward equally!

Fig. 4.12a.

4K

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As particle size increases, we get more forward scattering…Fig. 4.12 b,c.

For larger particles with x > 1, we get Mie Scattering.

In this case, values of K are oscillatory - Fig. 4.13.

Note: an index of refraction has entered

m = mr + imi

mr = (speed of light in vacuum) / (speed of light through particle)mi = absorption (mi = 0 no absorption; mi = 1 complete absorption)

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Example 4.9…the sky is blue because…

Blue light is scattered 3.45 times more efficiently than red light!

ALSO…p.124 2nd column

…tells us that to understand satellite imaging and retrievals, as well as weather radar etc., we need to apply the ideas in this section.

4( ) 0.64

3.45( ) 0.47

K blue m

K red m

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Absorption by non-gaseous particles

Not much information BUT read last sentence of p.126

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Absorption - and emission - by gas molecules

Energy arrives, is emitted and absorbed in discrete amounts called photons

Having energy

E = h

And c = E = hc/

h = Planck’s constant

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Atomic energy states

An atom has electrons in orbit around the nucleus

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For the electron to jump into a higher orbit (higher energy level), a discrete amount of energy must be absorbed

So only discrete orbits are allowed

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When this discrete amount of energy (E) is absorbed, a spectrum of absorption versus wavelength shows a spike.

Finite absorption at certain wavelengths.

Zero absorption otherwise (transparency).

wavelength (m)

absorption

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→ line spectrum

Each species → different line spectrum

All overlap & combine in the atmosphere

Adding molecules → additional complications

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Energy of a molecule, E is:

E = Eo + Ev + Er + Et

Energy due to

electron orbits

in atoms Energy due to vibration of molecule

Energy due to rotation of molecule

translational energy

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As a result, the spectrum is more complicated.

Adding E of energy (e.g., incident from the sun) can result in changes to the rotational state of the molecule, ditto vibrational, ditto electron states etc.

→ complex absorption spectrum (one for each species)

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Examples:

http://www2.ess.ucla.edu/~schauble/molecular_vibrations.htm

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Examples:

Atmospheric

Absorption spectra

for the main gases

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Examples:

http://en.wikipedia.org/wiki/Electromagnetic_spectroscopy