Radiative Transfer – An Overview

Acknowledgements

• http://cimss.ssec.wisc.edu/goes/comet/radiative_transfer.html• www.weatherchaos.umd.edu/class-presentations/Hudson_I

ntro_rad_trans.ppt• http://timeforchange.org/radiation-wavelength-and-greenho

use-effect

Outline

• What is radiative transfer?

• Why should I care?

• How does it work?

Definition

• Radiative transfer (RT) is the study of how radiation interacts with a medium.

• If you specify the characteristics of the medium and the boundary (external illumination) conditions, RT theory allows you to predict the resulting radiation field.

Just Solve Maxwell’s Equations, Right?

• Yes and no – the RT equation can be derived entirely from Maxwell’s equations.

• But for atmospheric applications, a number of simplifying assumptions can be made that eliminate the need to work from first principles:

1. Far field, time-averaged solutions are sought.

2. Medium is sparsely populated with discrete, randomly-oriented and randomly-placed particles

Statistics

• The RT equation offers a statistical description of the radiation field.

• Exact knowledge of the location of every particle is not necessary.

• Prediction of the fate of each individual photon is impossible, but photons are plentiful, so a statistical description is robust.

• Bulk properties (number density, pressure, temperature) are sufficient to describe the medium.

Neither Scalar nor Vector

• Some atmospheric variables are scalars (single-valued for each point in space). Examples: Temperature, pressure, density.

• Others are vectors (single value and direction associated with each point in space). Example: Wind speed.

• Radiation field has a single value for each direction and each point in space.

Why Bother?

• All atmospheric processes are driven by solar energy. Radiative transfer explains how this energy is distributed throughout the atmosphere.

• The vast majority of our information concerning the atmospheric state is derived from radiation measurements. These measurements cannot be properly interpreted without appreciating the interaction between the atmosphere and the radiation field.

Properties of EM radiation

• Oscillating electric and magnetic fields cause energy to flow through space

• All EM radiation travels at the same speed c: (assuming

vacuum)

• 2 related properties of EM radiation are the wavelength λ and the frequency ν:

smc 8109979.2

c

Electromagnetic spectrum

Energy Packets

• EM radiation is carried in individual “packets” called photons:

• h = Planck’s constant = 6.626 × 10-34 J•s

• All matter emits radiation, due to thermal vibrations

• Hot objects vibrate more energetically, and produce higher-energy photons than cold objects.

• 2 related properties of EM radiation are the wavelength λ and the frequency ν:

hE

Radiance

• Radiance is a radiation variable that describes the brightness at a given point, in a given direction:

• A diffuse (multi-directional) radiation field must be described by the radiance.

• Add up contributions from all wavelengths for the radiance

ddtdAdEddianceSpectralRa

4

Irradiance• Irradiance is a radiation variable that describes

the energy flowing through a specified surface (with a particular orientation) at a given point:

• The irradiance can result from mono-directional beams, or from a collection of diffuse beams.

• Again, the for the total energy, use the integrated quantity (irradiance).

dtdAdEdradianceSpectralIr

3

Blackbodies…

• “Blackbody radiation” represents the maximum radiation that a body at a particular temperature T can emit at each wavelength:

• k = Boltzmann’s constant = 1.381 × 10-23 J/K

4TIrradiance

1)exp(

25

2

kThc

hcBradianceSpectralIr

Sun and Earth’s blackbody intensity

100 MW / m2 / μm (at λ = 0.5 μm) translates to

~ 2.5 × 1023 photons / s / m2 / nm)

…and Non-Blackbodies• The emission by any body can be related to the

blackbody emission through the emissivity ελ:

• ελ represents the efficiency of the emission process for the specified body

• Can vary enormously across the spectrum (ex: snow is black in the IR (ελ ~ 1), but white in the visible (ελ ~ 0).

),(..),(

TEmissionBB

TianceEmittedRad

UV-VISIBLE SOLAR SPECTRUM

Radiation in a Medium

• Radiation incident upon a medium can be:

1. Absorbed (photon is converted to another form of energy). Efficiency of absorption is described by the absorptivity aλ:

)()(

dianceIncidentRadianceAbsorbedRaa

Radiation in a Medium

2. Reflected (photon leaves the medium in the direction opposite the incident beam). Efficiency of reflection is described by the reflectivity rλ:

)()(Re

dianceIncidentRaianceflectedRadr

Radiation in a Medium

3. Transmitted (photon leaves the medium in the same direction as the incident beam). Efficiency of transmission is described by the transmissivity tλ:

• From conservation of energy,

)()(

dianceIncidentRadRadianceTransmittet

1 tra

Radiation at a Surface

• Radiation incident upon an opaque surface is described the same way, except that transmission is impossible, leaving:

(again from conservation of energy).

1 ra

Earth Radiation Budget

1 ra

Kirchhoff’s Law

• The efficiency of the emission and absorption processes are related through Kirchhoff’s Law, which states that:

under conditions of “local thermodynamic equilibrium” (LTE).

• LTE conditions prevail when the absorbers collide frequently enough to effectively re-distribute the absorbed energy, so that the subsequent emission is related to the bulk T of the medium.

(again from conservation of energy).

a

Please Don’tMisunderstand…

• Kirchhoff’s Law does not imply that emission and absorption are equal at all wavelengths (under LTE conditions). It only implies that emission and absorption are equally efficient at all wavelengths (under LTE conditions).

• For example, high-energy solar photons may be strongly absorbed (aλ ~ 1) in the atmosphere, implying that the atmosphere emits the maximum possible amount at that λ (ελ ~ 1)… but if T is low, that may produce a negligible amount of energy.

Sources and Sinks

Sources Sinks

Emission Absorption

--------Scattering -------

• Scattering redirects the radiation beam, without adding or removing energy. So the sink for one direction becomes the source for another.

Scattering is All Relative

Rayleigh Scattering

• Describes the case for which the scatterer is much smaller than the wavelength of the incident radiation.

• Probability of scattering is proportional to λ-4.

• Scattering pattern depends upon the scattering angle Θ as (1 + cos2Θ), so it varies by a factor of 2 between minimum (at Θ = 90º) and maximum (at Θ = 0º or Θ = 180º).

Mie Scattering

• General expression for scattering by dielectric spheres.

• Needed to describe scattering by a scatterer that is the same size or larger than the wavelength of the incident radiation.

• Probability of scattering is proportional to λ-1 (roughly).

• Scattering pattern is greatly elongated towards forward scattering.

Phase diagrams for aerosols

Optical Thickness

• Radiative unit of length• The probability P that a photon will pass

through a specified medium without interaction with it (absorption or scattering) is given by:

where τ = the “optical thickness” of the medium.

)exp( P

Radiative Transfer Equation

• Iλ = spectral radiance

• Jλ = “source function”

• Jλ accounts for the possibility of photons added to a particular line of sight, due to emission and/or scattering

• Infrared RT and UV/visible RT are nearly distinct specialties, due to the differences in the physics involved.

IJ

ddI

UV/vis Radiative Transfer

• Atmospheric emission is negligible; Jλ is entirely due to scattering.

• Absorption events change the electronic states of the molecules.

• Absorption features tend to be broad.

Infrared Radiative Transfer

• Atmospheric scattering (by molecules) is negligible; Jλ is entirely due to emission.

• Absorption events change the vibrational-rotational states of the molecules.

• Absorption features tend to be a forest of narrow, closely-spaced absorption lines.

Greenhouse Effect

Greenhouse Effect

• Without our atmosphere, the surface temperature of Earth would be ~ 255 K. The natural greenhouse effect makes our planet inhabitable for humans.

• Water vapor is the strongest greenhouse gas.• We enhance this natural effect when we inject

additional gases that absorb in the infrared (CO2, CH4, N2O, CFCs, …).

Greenhouse Effect

• Without our atmosphere, the surface temperature of Earth would be ~ 255 K*. The natural greenhouse effect makes our planet inhabitable for humans.

• Water vapor is the strongest greenhouse gas.• We enhance this natural effect when we inject

additional gases that absorb in the infrared (CO2, CH4, N2O, CFCs, …).

Greenhouse Effect

Radiative forcing of long-lived greenhouse gases, relative to 1750 (From http://www.esrl.noaa.gov/gmd/aggi/, by Hoffman, 2007)

Volcanic Stratospheric Aerosol Cooling

http://www.giss.nasa.gov/data/strataer/

Thanks to Makiko Sato:

THE END