Theoretical analysis of particle number density in steady aeolian saltation

- The particle number density in steady aeolian saltation is deduced.
- Two different types of PDF of vertical lift-off velocity are analyzed.
- Vertical ejection velocity distribution affects the particle number density profile.
- Height parameters from two types of PDF are unequal to mean saltation height.

Particle number density or particle concentration in aeolian saltation is one important input parameter to calculate the sand flux, kinetic energy and mid-air collision probability in the aeolian saltation and particle concentration is also related to the wind erosion capacity, hence, in the present paper, the vertical distribution of particle number density in steady aeolian saltation is analyzed based on two different types of probability density functions of vertical lift-off velocity of saltating particles: one is the PDF (probability density function) of vertical velocity of lift-off particles in the three-dimensional space defined as a type-A PDF which considers the number of particles in various velocity bins per unit volume; and the other is the PDF of vertical velocity of lift-off particles ejected from the sand bed surface in a period of time as a type-B PDF which considers the number flux of particles in various velocity bins per unit surface area. These two types of PDFs are from two different perspectives (i.e., volume- and surface-based perspectives, respectively), and can be deduced from each other. The half-normal and exponential distributions are recommended for the type-A PDF, and the corresponding type-B PDF is expressed by Rayleigh and Gamma(2) distributions. The PDF distribution pattern of vertical velocity of lift-off particles has an important influence on the vertical profile of particle number density. If the type-A PDF of vertical velocity of ejected particles is a half-normal distribution, the particle number density decays exponentially with height. If the type-A PDF is an exponential distribution, the particle number density also decreases with height. If the type-A PDF is Gamma(3) and Rayleigh distributions, the particle number density first increases, then decreases with height. The type-A and type-B height parameters, which are calculated according to the mean vertical lift-off velocity from the type-A and type-B PDFs, respectively, are not simply considered as the actual mean saltation height, although if the type-A PDF of vertical lift-off velocity is half-normal PDF, the mean dispersion height of particle concentration equals the mean saltation height. The vertical distribution of particle number density can determine the dislodgement rate on bed surface which further affects the surface bedforms.