Models for mixed ensemble of hydrometeors and their use in calculating the total random cross section of a resolution volume filled by random scatterers

American Journal of Applied Sciences, Dec, 2008 by Ayman Al- Lawama, Salaheddin Malkawi, Abdullah Al- Odienat, Felix Yanovsky

INTRODUCTION

Hydrometeors, such as water drops and ice crystals, scatter incident electromagnetic waves. Polarization of backscattered field depends on the shape, size, orientation and type of the particles. While models of backscattering from water drops and snowflakes separately are well developed, the mixed ensemble of hydrometeors requires more accurate consideration. This problem arises very sharply in the context of weather aviation maintenance (1). Particularly, as the new methods of aircraft icing that have been proposed recently (2), (3) are based on the difference in behavior of the polarization of the radar signal in case of water clouds and ice clouds. This reason motivated the work in this research, which deals with the simulation of backscattering and calculation of polarimetric variables in case of pure water, crystal clouds and their mixed ensemble.

MATERIALS AND METHODS

The problem of simulation of scattering from hydrometeors is rather difficult because of different parameters of scatterers and their distributions on shape, size and orientation, which should be taken into account. Key point in this research is placed on the possibility of polarimetric weather radar to determine the type of hydrometeors in case of homogeneous medium (scatterers of one type) and in case of a mixture of different types of scatterers in ensemble of hydrometeors.

The possibility to determine a percentage of different types of hydrometeor is also a subject of consideration in this research. This is important for flight safety and can be used particularly, in modern aircrafts for remote sensing of in-flight icing.

The microstructure of precipitation can be characterized by physical and statistical properties of the individual particles. A hydrometeor can be modeled by ellipsoid without consideration of any possible mutual interaction of scatterers (4).

The average drop size distribution N (D) in rain can be described by the gamma-distribution:

N(D) = [N.sub.0][D.sup.[mu]] exp(-[[3.67 + [mu]]/[D.sup.0]]D) (1)

where, [mu] is a spread parameter and [D.sub.0] is a median drop diameter.

If [mu] = Zero (Marshall-Palmer case), [N.sub.0] = 8000 and if [mu] [not equal to] 0, [N.sub.0] is derived from the Marshall-Palmer distribution by keeping the total volume of amount of water per [m.sup.3] constant for a given D. If [mu] is integer, this leads to the following formula (5):

[N.sub.0][approximately equal to][[264.59*[(3.67 + [mu]).sup.[mu] + 4]]/[[D.sub.0.sup.[mu]]([mu] + 3)!]] (2)

The normalized raindrop diameter distribution n(D), which can be used as a probability density function, is derived using expression (1) at [D.sub.min] = zero, [D.sub.max] = [infinity] as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The case of rain is the simplest one. It is much more difficult to find an adequate hydrometeor size distribution for clouds. In case of liquid droplet cloud without rain, the gamma-distribution is used for the description of the drop size distribution in cloud as follows (6):

n(r) = [[N.sub.0]/[[GAMMA]([alpha] + 1)[[beta].sup.[alpha] + 1]]][r.sup.[alpha]] exp[-r/[beta]] (4)

Other form of the same distribution is the normalized gamma distribution, which can be written as:

f(r) = [1/[[GAMMA]([alpha] + 1)[[beta].sup.[alpha] + 1]]][r.sup.[alpha]] exp[-r/[beta]] (5)

The original gamma-distribution was introduced for the description of atmospheric particle spectrum in the following form (6):

n(r) = c[r.sup.[alpha]][e.sup.-br] (6)

For clouds of a given type, averaged over large space and carried out for many cases, the drop size distribution is usually well described by Hrgian-Mazin distribution that is a special case of the formula (6) when [alpha] = 2 (Hrgian-Mazin case) (6):

n(r) = [cr.sup.2][e.sup.-br] (7)

Median drop diameter is taken in the range from 30 to 100 microns. Such values of this parameter in combination with negative cloud temperature and super cooled liquid water content of more than 0.2 g [m.sup.-3] can lead to significant aircraft icing. Maximum diameter of droplet in such cloud type does not exceed 1 mm. The shape of such droplets is almost ideal sphere.

In contrast to water droplets, ice crystals are characterized by an extremely wide variety of shapes. A crystal shape depends on the conditions under which they are formed. However, the three main types of ice crystals that are usually formed in clouds at temperature from 0 to -35[degrees]C, namely: columnar crystals, needles and plates. Their percentage varies depending on cloud type, temperature and humidity of air, as well as on some other reasons. The size distribution of ice crystals is defined by the expression:

N(L) = 1000.([L.sup.-23]) (8)

with L as a characteristic size, e.g. length for columnar crystals and needles and diameter for ice plates. The shape of columnar crystals and needles was modeled by spheroid with the model being approximated by the relation d(L), where L is the length and d is the diameter of the column or needle. The case of crystals of lamellar form the model was approximated by the relation h(d), where h is the thickness and d is the diameter of the crystals.