Solution of radiative transfer equation for plane parallel atmosphere

The radiative transfer equation describes how electromagnetic radiation (such as light) propagates through a medium. In the case of a plane-parallel atmosphere, the radiative transfer equation can be simplified for a one-dimensional, vertically stratified medium. The equation describes the transfer of radiant energy along the vertical direction through the atmosphere. The equation takes into account absorption, emission, and scattering processes.

Consider a plane parallel atmosphere bounded by 𝜏 = 𝜏∗ (bottom surface) as shown in the figure. Here we are going to derive downward and upward intensity of the radiation as it travels through the plane parallel atmosphere by solving radiative transfer equation.

1. Upward intensity of radiation [ I ( 𝜏 , 𝜇, 𝜙) ]

We know that the radiative transfer equation for plane parallel atmosphere.

Multiplying this equation by e^-𝜏/𝜇 on both sides and rearranging the terms. We have

Integrating on both sides from 𝜏 to 𝜏_*

At the top of the atmosphere, 𝜏 = 0 and upward intensity is thus given by

Downward Intensity of radiation

We know that radiative transfer equation for plane parallel atmosphere is

Integrating above equation from 0 to 𝜏 we have

At the bottom of the atmosphere 𝜏 = 𝜏_*

Radiative Transfer equation for 3 dimension in homogeneous medium

Plane parallel atmosphere assumption is invalid for most atmospheres. In such a situation, the spherical geometry of the atmosphere is used. The general radiative transfer formula is

Let extinction coefficient 𝛽e = 𝜌K𝜆 and omitting subscript 𝜆 for simplicity. We have

The operator d/ds in terms of time and space is given by

where c is velocity of light. Ω is a unit vector specifying the direction of scattering through position vector. In steady state, radiation is independent of time such as illumination from the sun and ∂/ ∂t = vector Ω . vector∇

Now radiative transfer equation can be expressed as

where J gives scattering of direct solar beam, multiple scattering of diffused intensity and emission by the medium.

In cartesian coordinates (x, y, z) we have

In certain cases, the radiative transfer equation may be solved under specific assumptions or approximations. Common techniques include the use of the Eddington approximation or two-stream approximation for simplicity.

Factors like the wavelength/frequency regime (e.g., microwave, infrared, visible) and the participating medium (e.g., gases, aerosols) may affect the radiative transfer equation. Given the conditions of the atmosphere in question, the equation can be modified in accordance.

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