**3.4. Turbulence**

frequencies above approximately 10 GHz are adversely impacted by rain and little impacted by fog. This is because of the closer match of RF wavelengths to the radius of raindrops, both being larger than the moisture droplets in fog [14]. The rain scattering coefficient can be

> ( a

can be calculated using equation following:

R

1.33(πa3)V a λ

) (19)

(21)

) (22)

(20)

calculated using Stroke Law [25]:

180 Contemporary Issues in Wireless Communications

*a*: is the radius of raindrop, (

The raindrop distribution

*R*: is the rainfall rate (cm/s),

*V*a: is the limit speed precipitation.

Limiting speed of raindrop [25] is also given as:

ρ =1 g / cm<sup>3</sup> ).

η

g= 980

= 1.8 \* 10 -<sup>4</sup>

wavelengths readers can refer to references [26,27].

Na : is the rain drop distribution, (cm-3).

: is the scattering efficiency.

Where:

Qscat

Where:

Where:

: is water density, (

: is viscosity of air,

: is gravitational constant,

ρ

g

η

βrain scat= πa2NaQscat

cm).

Na

> Na=

> > V a <sup>=</sup> <sup>2</sup>a2 ρg

cm / sec

g / cm.sec.

τ=exp (-

The rain attenuation can be calculated by using Beer's law as:

9η

2 .

βrain scatL

For more details about several weather conditions and the corresponding visibility at various

Clear air turbulence phenomena affect the propagation of optical beam by both spatial and temporal random fluctuations of refractive index due to temperature, pressure, and wind variations along the optical propagation path [28,29]. Atmospheric turbulence primary causes phase shifts of the propagating optical signals resulting in distortions in the wave front. These distortions, referred to as optical aberrations, also cause intensity distortions, referred to as scintillation. Moisture, aerosols, temperature and pressure changes produce refractive index variations in the air by causing random variations in density. These variations are referred to as eddies and have a lens effect on light passing through them. When a plane wave passes through these eddies, parts of it are refracted randomly causing a distorted wave front with the combined effects of variation of intensity across the wave front and warping of the isophase surface [30]. The refractive index can be described by the following relationship [31]:

$$n \cdot 1 \approx 79 \times \frac{p}{T} \tag{23}$$

Where:

P: is the atmospheric pressure in mbar .

T: is the temperature in Kelvin K .

If the size of the turbulence eddies are larger than the beam diameter, the whole laser beam bends, as shown in Fig. 9. If the sizes of the turbulence eddies are smaller than the beam diameter and so the laser beam bends, they become distorted as in Fig. 10. Small variations in the arrival time of various components of the beam wave front produce constructive and destructive interference and result in temporal fluctuations in the laser beam intensity at the receiver see Fig. 10.

**Figure 9.** Laser beam Wander Due to turbulence cells that are larger than the beam diameter.

#### *3.4.1. Refractive index structure*

Refractive index structure parameter Cn 2 is the most significant parameter that determines the turbulence strength. Clearly, Cn 2 depends on the geographical location, altitude, and time of

**Figure 10.** Scintillation or fluctuations in beam intensity at the receiver due to turbulence cells that is smaller than the beam diameter.

day. Close to ground, there is the largest gradient of temperature associated with the largest values of atmospheric pressure (and air density). Therefore, one should expect larger values Cn 2 at sea level. As the altitude increases, the temperature gradient decreases and so the air density with the result of smaller values of Cn 2 [8].

In applications that envision a horizontal path even over a reasonably long distance, one can assume Cn <sup>2</sup> to be practically constant. Typical value of Cn <sup>2</sup> for a weak turbulence at ground level can be as little as 10-17m-2/3, while for a strong turbulence it can be up to 10-13m-2/3 or larger. However, a number of parametric models have been formulated to describe the Cn 2 profile and among those, one of the more used models is the Hufnagel-Valley [32] given by:

$$\begin{aligned} C\_n^2(h) &= 0.00594 (\nu/27)^2 \{10^{-5} h\}^{10} \exp\left(\cdot \frac{h}{1000}\right) + \\ &2.7 \times 10^{-16} \exp\left(\cdot \frac{h}{1500}\right) + A\_o \exp\left(\cdot \frac{h}{100}\right) \end{aligned} \tag{24}$$

Where:

h: is the altitude in m].

v: is the wind speed at high altitude m / s .

A0: is the turbulence strength at the ground level, Ao= 1.7 × 10-14m-2/3.

The most important variable in its change is the wind and altitude. Turbulence has three main effects ; scintillation, beam wander and beam spreading.
