Aerosol Size Distribution
Atmospheric aerosols are polydisperse . Although their sizes can vary in the
0.01—100 pm range of diameters, the number of small particles is limited by the coagulation process and the number of large particles by gravitational sedimentation. Between these limits the number of particles varies with size.
The manner in which the particle population is spread over the range of sizes is defined by the size distribution function. The size distribution of the atmospheric aerosol is one of its core physical parameters. It determines how the various properties like mass and number density, or optical scattering, are distributed as a function of particle radius. Particle size distributions are necessary inputs for models used to predict the attenuation and scatter of radiation between the transmitter and receiver in different applications (optical communication, satellite image restoration, weapons-based electro-optical systems, etc.).
The number n(r) of particles per unit interval of radius and per unit volume is given by
The differential quantity dN(r) expresses the number of particles having a radius between r and r + dr, per unit volume, according to the distribution function n(r).
Because of the many orders of magnitude present in atmospheric aerosol concentrations and radii, a logarithmic size distribution function is often used:
The much used distribution function is the power law first presented by Junge [24,40]. Junge’s model is
Or, in a nonlogarithmic form,
where C is a normalizing constant to adjust the total number of particles per unit volume, and v is the shaping parameter. Most measured size distributions can best be fit by values of v in the range 3 < v < 5 for hazy and clear atmospheric conditions, and for aerosols whose radii lie between 0.1 and 10 pm . According to the power-law size distribution, the number of particles decreases monotonically with an increase in radius. In practice, there is an accumulation in the small particle range. Actual particle size distributions may differ considerably from a strict power-law form.
The modified power-law distribution was given by McClatchey et al.  and was modified in Reference 42 by use of the gamma probability density function (PDF) to
where a is the total number density, and а, в, and b are shaping parameters.
The total particle concentration given by the integral over all particle radii according to Reference 39 is, for this distribution:
The mode radius for this distribution is given by [6,39]
The value of the distribution at the mode radius is [6,39]
Because it has four adjustable constants, (1.15) can be fitted to various aerosol models. The gamma PDF is usually employed to model haze, fog, and cloud particle size distributions.
The tropospheric aerosols above the boundary layer are assumed to have the same composition, but their size distribution is modified by eliminating the large particle component because of the higher elevation.
To utilize the extensive measurements, a series of aerosol models for different environmental conditions and seasons were constructed . The models have been divided into four altitude regimes, as described above. Many experiments were carried out over the last three decades to test and modify the present aerosol models. The new data available show that the distribution varies dramatically with altitude, often within meters. Large variations exist in the data from different locations .