Radiative transfer
Radiative transfer (also called radiation transport) is the physical phenomenon of energy transfer in the form of electromagnetic radiation. The propagation of radiation through a medium is affected by
Definitions
The fundamental quantity that describes a field of radiation is called
In terms of the spectral radiance, , the energy flowing across an area element of area located at in time in the solid angle about the direction in the frequency interval to is
where is the angle that the unit direction vector makes with a normal to the area element. The units of the spectral radiance are seen to be energy/time/area/solid angle/frequency. In MKS units this would be W·m−2·sr−1·Hz−1 (watts per square-metre-steradian-hertz).
The equation of radiative transfer
The equation of radiative transfer simply says that as a beam of radiation travels, it loses energy to absorption, gains energy by emission processes, and redistributes energy by scattering. The differential form of the equation for radiative transfer is:
where is the speed of light, is the emission coefficient, is the scattering opacity, is the absorption opacity, is the mass density and the term represents radiation scattered from other directions onto a surface.
Solutions to the equation of radiative transfer
Solutions to the equation of radiative transfer form an enormous body of work. The differences however, are essentially due to the various forms for the emission and absorption coefficients. If scattering is ignored, then a general steady state solution in terms of the emission and absorption coefficients may be written:
where is the optical depth of the medium between positions and :
Local thermodynamic equilibrium
A particularly useful simplification of the equation of radiative transfer occurs under the conditions of
In this situation, the absorbing/emitting medium consists of massive particles which are locally in equilibrium with each other, and therefore have a definable temperature (Zeroth Law of Thermodynamics). The radiation field is not, however in equilibrium and is being entirely driven by the presence of the massive particles. For a medium in LTE, the emission coefficient and absorption coefficient are functions of temperature and density only, and are related by:
where is the black body spectral radiance at temperature T. The solution to the equation of radiative transfer is then:
Knowing the temperature profile and the density profile of the medium is sufficient to calculate a solution to the equation of radiative transfer.
The Eddington approximation
The Eddington approximation is distinct from the two-stream approximation. The two-stream approximation assumes that the intensity is constant with angle in the upward hemisphere, with a different constant value in the downward hemisphere. The Eddington approximation instead assumes that the intensity is a linear function of , i.e.
where is the normal direction to the slab-like medium. Note that expressing angular integrals in terms of simplifies things because appears in the Jacobian of integrals in spherical coordinates. The Eddington approximation can be used to obtain the spectral radiance in a "plane-parallel" medium (one in which properties only vary in the perpendicular direction) with isotropic frequency-independent scattering.
Extracting the first few moments of the spectral radiance with respect to yields
Thus the Eddington approximation is equivalent to setting . Higher order versions of the Eddington approximation also exist, and consist of more complicated linear relations of the intensity moments. This extra equation can be used as a closure relation for the truncated system of moments.
Note that the first two moments have simple physical meanings. is the isotropic intensity at a point, and is the flux through that point in the direction.
The radiative transfer through an isotropically scattering medium with scattering coefficient at local thermodynamic equilibrium is given by
Integrating over all angles yields
Premultiplying by , and then integrating over all angles gives
Substituting in the closure relation, and differentiating with respect to allows the two above equations to be combined to form the radiative diffusion equation
This equation shows how the effective optical depth in scattering-dominated systems may be significantly different from that given by the scattering opacity if the absorptive opacity is small.
See also
- Beer-Lambert law
- Kirchoff's law of thermal radiation
- List of atmospheric radiative transfer codes
- Optical depth
- Planck's law
- Radiative transfer equation and diffusion theory for photon transport in biological tissue
- Schwarzschild's equation for radiative transfer
- Vector radiative transfer
References
- ^
S. Chandrasekhar (1960). Radiative Transfer. Dover Publications Inc. p. 393. ISBN 978-0-486-60590-6.
- ^
Jacqueline Lenoble (1985). Radiative Transfer in Scattering and Absorbing Atmospheres: Standard Computational Procedures. A. Deepak Publishing. p. 583. ISBN 978-0-12-451451-5.
Further reading
- Ivan Hubeny; ISBN 9780691163291.
- ISBN 978-0-486-60590-6.
- Jacqueline Lenoble (1985). Radiative Transfer in Scattering and Absorbing Atmospheres: Standard Computational Procedures. A. Deepak Publishing. p. 583. ISBN 978-0-12-451451-5.
- Grant Petty (2006). A First Course in Atmospheric Radiation (2nd Ed.). Sundog Publishing (Madison, Wisconsin). ISBN 978-0-9729033-1-8.
- ISBN 978-0-486-40925-2.
- George B. Rybicki; Alan P. Lightman (1985). Radiative Processes in Astrophysics. Wiley-Interscience. ISBN 978-0-471-82759-7.
- G. E. Thomas & K. Stamnes (1999). Radiative Transfer in the Atmosphere and Ocean. ISBN 978-0-521-40124-1.
- ISBN 978-3-527-40503-9.
- ISBN 9780521865562.