Anisotropic scattering of light in the spherical geometry of the Earth's atmosphere

If light hits on particles that are the same size or larger than the wavelength of the light, the light might be scattered. Many particles in the Earth's atmosphere have this property, like aerosol particles, water droplets in haze or clouds, ice particles, but also air molecules.

The phase function is a measure of the anisotropy of the scattering. It provides a factor for each direction with which the incoming intensity has to be multiplied to give the outgoing intensity. Hence, for isotropic scattering, the phase function is 1 for all directions.

Here are some phase functions of tropospheric haze particles, aerosol particles and air molecules:

The dashed curve represents the phase function for a wavelength of the scattered light of 325nm, the dotted curve for light of 400nm and the solid curve for light of 700nm. The dash-dotted curve in the right figure represents the phase function of air molecules. As is apparent, especially for large particles and small wavelengths, the scattering process can be very anisotropic. However, the anisotropy is not large enough to merritt the asumption of the Eddington-delta approximation, ie to approximate the phase function by a forward scattering delta-distribution plus a linearly anisotropic term. This is only the case for even larger particles, like cloud droplets or ice particles in clouds.

The problem of anisotropic scattering in a plane geometry of the Earth's atmosphere has been solved some time ago. However, when the sun is below, say, 75 or 80 degrees, ie when the sun is close to the horizon or even below the horizon, in other words, during sun-rise or sun-set, the spherical aspect of the Earth's geometry plays a vital role. It is these situations, anisotropic scattering in a spherical geometry, ie anisotropic scattering during sun-rise or sun-set, that I am most interested in.

To solve the problem of anisostropic scattering in spherical geometry, I developed a new mathematical equation describing the physics of this process accurately in this context. This new equation is

It is the numerical solution of this equation that is displayed in the sunset simulations and the UVB radiation results underneath an ozone hole.

Details of the derivation of the equation and its numerical solution are found in:

Balluch, M., 1996:"A new numerical model to compute photolysis rates and solar heating with anisotropic scattering in spherical geometry". Annales Geophysicae, 14, 80-97.