Radiative transfer

Radiative transfer is the physical phenomenon of energy transfer in the form of electromagnetic radiation. The propagation of radiation through a medium is affected by absorption, emission and scattering processes. The equation of radiative transfer describes these interactions mathematically. Equations of radiative transfer have application in wide variety of subjects including optics, astrophysics, atmospheric science, and remote sensing. Analytic solutions to the radiative transfer equation (RTE) exist for simple cases but for more realistic media with complex multiple scattering effects numerical methods are required.

The present article is largely focused on the condition of radiative equilibrium. .

Definitions

The fundamental quantity which describes a field of radiation is the spectral intensity. If we think of a very small area element in the radiation field, there will be radiation energy flowing through that area element. The flow can be completely characterized by the amount of energy flowing per unit time per unit solid angle, the direction of the flow, and the wavelength interval being considered (polarization will be ignored for the moment).

In terms of the spectral intensity, $I\_\backslash nu$, the energy flowing across an area element of area $da\backslash ,$ located at $\backslash mathbf\{r\}$ in time $dt\backslash ,$ in the solid angle $d\backslash Omega$ about the direction $\backslash hat\{\backslash mathbf\{n\}\}$ in the frequency interval $\backslash nu\backslash ,$ to $\backslash nu+d\backslash nu\backslash ,$ is

:$dE\_\backslash nu\; =\; I\_\backslash nu(\backslash mathbf\{r\},\backslash hat\{\backslash mathbf\{n\}\},t)\; \backslash cos\backslash theta\; \backslash \; d\backslash nu\; \backslash ,\; da\; \backslash ,\; d\backslash Omega\; \backslash ,\; dt$

where $\backslash theta$ is the angle that the unit direction vector $\backslash hat\{\backslash mathbf\{n\}\}$ makes with a normal to the area element. The units of the spectral intensity 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, and redistributes energy by scattering. The differential form of the equation for radiative transfer is:

:$\backslash frac\{1\}\{c\}\backslash frac\{\backslash partial\}\{\backslash partial\; t\}I\_\backslash nu\; +\; \backslash hat\{\backslash Omega\}\; \backslash cdot\; \backslash nabla\; I\_\backslash nu\; +\; (k\_\{\backslash nu,\; s\}+k\_\{\backslash nu,\; a\})\; I\_\backslash nu\; =\; j\_\backslash nu\; +\; \backslash frac\{1\}\{4\backslash pi\; c\}k\_\{\backslash nu,\; s\}\; \backslash int\_\backslash Omega\; I\_\backslash nu\; d\backslash Omega$

where $j\_\backslash nu$ is the emission coefficient, $k\_\{\backslash nu,\; s\}$ is the scattering cross section, and $k\_\{\backslash nu,\; a\}$ is the absorption cross section.

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 solution in terms of the emission and absorption coefficients may be written:

:$I\_\backslash nu(s)=I\_\backslash nu(s\_0)e^\{-\backslash tau(s\_0,s)\}+\backslash int\_\{s\_0\}^s\; j\_\backslash nu(s\text{'})\; e^\{-\backslash tau(s\text{'},s)\}\backslash ,ds\text{'}$

where $\backslash tau(s\_1,s\_2)$ is the optical depth of the atmosphere between $s\_1$ and $s\_2$:

:$\backslash tau(s\_1,s\_2)\; \backslash \; \backslash stackrel\{\backslash mathrm\{def\}\}\{=\}\backslash \; \backslash int\_\{s\_1\}^\{s\_2\}\; \backslash alpha\_\backslash nu(s)\backslash ,ds$

Local thermodynamic equilibrium

A particularly useful simplification of the equation of radiative transfer occurs under the conditions of local thermodynamic equilibrium (LTE). In this situation, the atmosphere consists of massive particles which are in equilibrium with each other, and therefore have a definable temperature. The radiation field is not, however in equilibrium and is being entirely driven by the presence of the massive particles. For an atmosphere in LTE, the emission coefficient and absorption coefficient are functions of temperature and density only, and are related by:

:$\backslash frac\{j\_\backslash nu\}\{\backslash alpha\_\backslash nu\}=B\_\backslash nu(T)$

where $B\_\backslash nu(T)$ is the black body intensity at temperature T. The solution to the equation of radiative transfer is then:

:$I\_\backslash nu(s)=I\_\backslash nu(s\_0)e^\{-\backslash tau(s\_0,s)\}+\backslash int\_\{s\_0\}^s\; B\_\backslash nu(T(s\text{'}))\backslash alpha\_\backslash nu(s\text{'})\; e^\{-\backslash tau(s\text{'},s)\}\backslash ,ds\text{'}$

Knowing the temperature profile and the density profile of the atmospheric components will be enough to calculate a solution to the equation of radiative transfer.

The Eddington approximation

The Eddington approximation is a special case of the two stream approximation. It can be used to obtain the intensity in a plane-parallel atmosphere with isotropic frequency-independent scattering. It assumes that the intensity is a linear function of $\backslash mu=\backslash cos\backslash theta$. i.e.

:$I\_\backslash nu(\backslash mu,z)=a(z)+\backslash mu\; b(z)$ where $z$ is the normal direction to the slab atmosphere. Note that expressing angular integrals in terms of $\backslash mu$ simplifies things because $d\backslash mu=-\backslash sin\backslash theta\; d\backslash theta$ appears in the Jacobian of integrals in spherical coordinates.

Extracting the first few moments of the intensity with respect to $\backslash mu$ yields

:$J\_\backslash nu=\backslash frac\{1\}\{2\}\backslash int^1\_\{-1\}I\_\backslash nu\; d\backslash mu\; =\; a$ :$H\_\backslash nu=\backslash frac\{1\}\{2\}\backslash int^1\_\{-1\}\backslash mu\; I\_\backslash nu\; d\backslash mu\; =\; \backslash frac\{b\}\{3\}$ :$K\_\backslash nu=\backslash frac\{1\}\{2\}\backslash int^1\_\{-1\}\backslash mu^2\; I\_\backslash nu\; d\backslash mu\; =\; \backslash frac\{a\}\{3\}$

Thus the Eddington approximation is equivalent to setting $K\_\backslash nu=1/3J\_\backslash nu$. 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. $J\_\backslash nu$ is the isotropic intensity at a point, and $H\_\backslash nu$ is the flux through that point in the $z$ direction.

The radiative transfer through an isotropically scattering atmosphere at local thermal equilibrium is given by :$\backslash mu\; \backslash frac\{dI\_\backslash nu\}\{dz\}=-\; \backslash alpha\_\backslash nu\; (I\_\backslash nu-B\_\backslash nu)\; +\; \backslash sigma\_\{\backslash nu\}(J\_\backslash nu\; -I\_\backslash nu)$

Integrating over all angles yields :$\backslash frac\{dH\_\backslash nu\}\{dz\}=\backslash alpha\_\backslash nu\; (B\_\backslash nu-J\_\backslash nu)$ Premultiplying by $\backslash mu$, and then integrating over all angles gives :$\backslash frac\{dK\_\backslash nu\}\{dz\}=-(\backslash alpha\_\backslash nu+\backslash sigma\_\backslash nu)H\_\backslash nu$

Substituting in the closure relation, and differentiating with respect to $z$ allows the two above equations to be combined to form the radiative diffusion equation :$\backslash frac\{d^2J\_\backslash nu\}\{dz^2\}=3\backslash alpha\_\backslash nu(\backslash alpha\_\backslash nu+\backslash sigma\_\backslash nu)(J\_\backslash nu-B\_\backslash nu)$

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

Absorption (electromagnetic radiation)

Atomic line spectra

Beer-Lambert law

Emission

List of atmospheric radiative transfer codes

Scattering

Radiative transfer equation and diffusion theory for photon transport in biological tissue

Further reading

References

Category:Radiometry Category:Electromagnetic radiation Category:atmospheric radiation