SEGURIDAD ACCESO CONDICIONAL IPTV Y SISTEMAS DRM

viously, the actual attenuation for a specific propagation path through the atmosphere depends heavily on local humidity and environmental conditions, in particular for long horizontal paths at low altitude.

Depolarisation

Especially in the case of multiple scattering, depolarisation of the incident light can occur [121]. The depolarisation factor depends on the anisotropy of the scatterer, that is, on the deviation from the spherical form. In fact, depolarisation measurements of back- scattered light can be used to determine the physical composition of cloud constituents, such as the relative ratio of water vapour or ice crystals in a cloud. However, quantitative measurements over horizontal propagation paths in the lower clear atmosphere [124,125] indicate that the polarisation of a propagating wave is only minimally affected, often below the sensitivity of the apparatus.

3.3 Kolmogorov theory of turbulence

Turbulence of a viscous fluid is fundamentally a nonlinear process and described by the Navier-Stokes equations. Because of mathematical difficulties in solving these equations, Kolmogorov developed a statistical approach of turbulence [126], that relies on certain simplifications, but still allows to deduce important implications for wave propagation in random media. A comprehensive treatment of the topic can be found, for example, in [127].

Fluid mechanics distinguishes two types of motion in a viscous medium: laminar and turbulent flow. While the associated velocity field is continuous in laminar flow, it loses these characteristics in turbulent flow, and dynamic mixing with random subflows (called

turbulent eddies) occurs. The state of motion is described by the dimensionless Reynolds number Re = v l/ν, where v and l are a characteristic velocity and a characteristic dimension of the flow, respectively, and ν is the kinematic viscosity. The transition from laminar to turbulent conditions takes place at a critical Reynolds number, which depends on the exact flow configuration and must be determined empirically (common values range from 1 to 103). Under atmospheric conditions typically prevailing close to the ground (l ∼ 1 m, v ∼ 1−5 m/s), the Reynolds number reaches easily large values on the order Re∼105_{, which means that the motion of the air is highly turbulent. The}

source of energy in atmospheric turbulence is either wind shear (i.e., a wind gradient) or convection. The wind velocity increases until the critical Reynolds number is exceeded. At that point, local unstable air masses are created (large eddies), that break up into smaller eddies because of inertial forces. As the eddies become smaller and smaller, the relative energy dissipated by viscous forces increases until it matches the supplied kinetic energy: the eddies disappear, and the remaining energy is dissipated as heat. Thus, a continuum of eddies from a macroscale L₀ (outer scale of turbulence) to a microscale l₀

(inner scale of turbulence) is formed. The outer scale L₀ denotes the scale size below which turbulence properties are independent of the parent flow. In the surface layer up to 100 m,L₀grows roughly linearly with the height above groundhand is approximately of the same order ash, whereas l₀ is typically 1−10 mm [127].

According to Kolmogorov’s work [126], the turbulence on scales between l₀ and L₀

(called theinertial subrange) can be described by statistical means under the assumption of statistical homogeneity and isotropy of the random velocity field. This means that the mean value of wind velocity is constant over the considered region, and that correlations between random fluctuations from one point to another depend only on the absolute value of the vector connecting the two observation points. If such correlations of a property x between two different points R₁ and R₂ are locally homogeneous, they can be described by structure functions D_x, defined as

D_x(R₁,R₂)≡D_x(R) =[x(R₁)−x(R₁ +R)]2,

where the brackets denote an ensemble average andR:=R₂−R₁. The longitudinal structure function of wind velocity (parallel to the vector R) is found to satisfy the power laws D_RR(R) =[v(R₁)−v(R₂)]2= C2 vR2/3 : l0 R L0 C2 vl0−4/3R2 : R l0, (3.7) Here, C_v2 is the velocity structure constant, that is dependent on the average energy dissipation rate. Since only eddies of scale sizes smaller thanL0 are assumed statistically

homogeneous and isotropic, by definition no general prediction ofD_RR exists forR > L₀. For instance, in altidudes above ∼100 m, eddies of greater size than L₀ are often much larger in horizontal dimension than in vertical dimension because of stratification. Hence, the turbulence is generally nonisotropic on that scale. Likewise, a temperature structure functionD_T(R) exists, that obeys the same power laws as the velocity structure function

D_RR(R), but has a different structure constant C_T2.

Optical wave propagation in a transparent medium is governed by the index of re- fraction, which is sensitive to small-scale temperature fluctuations. At any point R, the index of refraction of the atmosphere can be written as the sum of its mean value

n₀ =n(R) and the random deviation n₁(R) from the mean value

n(R) = n₀+n₁(R)

1 + 7.76·10−5(1 + 7.52·10−3λ−2)p(R)

T(R) (3.8)

1 + 8·10−5 p(R)

T(R), (3.9)

where λ is wavelength in μm, p is pressure in mbar, and T is temperature in Kelvin. The wavelength dependence is small for optical frequencies, so expression (3.8) is a