Refractive Index Structure Constant Research Articles

Volume scattering by tropospheric turbulence at midlatitudes: Frequency validity range

A theoretical propagation model is developed to quantify the probability density function of the power received on tropospheric transhorizon radio links at midlatitudes. Experimental radiosonde data are used to build a new random model of the refractive index structure constant. This model takes into account the turbulence strength for both long‐term and altitude variations. A method is developed to evaluate the received power probability density function by numerical integration through the common volume allowing the consideration of antenna radiation patterns and ground effects. After numerous comparisons between our new theoretical propagation model and the semiempirical one, NBS 101, we have set up an upper limit for the validity of the volume scattering theory. The limit is found near 3 GHz for beyond line of sight radio links in the case of a turbulence described by the Kolmogorov's spectrum.

A deterministic temperature model for stratospheric optical turbulence

A deterministic model for the refractive index structure constant, Cn2, is presented for the nighttime, subtropical stratosphere from 17 to 30 km. The model stems from a theoretical formulation of Tatarski and contains the product of an unknown parameter and quantities calculated from meteorological data. Using high resolution thermosonde data for Cn2, altitude profiles for the fitting parameter are calculated and fitted. These exhibit a great similarity in magnitude and trend and an average fitted curve is presented for the fitting parameter. A sample model Cn2 profile is presented and this exhibits the correct magnitude and trend and shows a degree of stratification.

Higher-order statistics of laser-irradiance fluctuations due to turbulence

Higher-order coefficients (up to eighth order) that give information concerning the contribution from the tails of non-Gaussian distributions are utilized to compare model distributions (K, log-normal, Rice–Nakagami, and a universal statistical model distribution) with the measured data of 0.6328-μm laser-irradiance fluctuations propagated through a laboratory-simulated turbulence. Expressions for the coefficients of the K distribution are derived in this paper. The comparison is made using a plot of the coefficients as a function of σ1 = (1.23Cn2k7/6L11/6)1/2, Cn2 being the refractive-index structure constant. Different values of the exponent α (or p) in Γn ~ σ1α (or Γn ~ Lp) computed from the measured data show that the exponent parameters increase with the order of n of the coefficients Γn. The parameters α or p can be useful for accurate modeling of the complex process.

Turbulence and waves in the optically clear planetary boundary layer resolved by dual‐Doppler radars

Dual Doppler radars provide a unique method to resolve the spatial structure of planetary boundary layer (PBL) air motions. Both the wave and turbulent fluctuations of air in the convectively unstable PBL are revealed from synthesis of the Doppler velocities of echoes (from the atmosphere's intrinsic scatterers) returned to two spaced (40 km) radars. The smaller turbulent scales follow closely the two‐thirds law‐of‐similarity theory, whereas the observed 4‐km wave is compared with those from theoretical models. Power spectra derived from Fourier space analysis of Doppler‐synthesized wind are compared to those obtained from time‐space conversion of Fourier time analysis applied to wind measurements on a 500‐m tower. It is shown that turbulent eddies move with the mean wind and maintain an identity for nearly 60 min. Larger wind variance is observed in the cross‐wind component than in the along‐wind component. Error variances are in agreement with estimates from theoretical studies. Comparisons of refractive index structure constants Cn2 deduced by radar and aircraft are within 1 dB.

Reflection and scattering of VHF radar signals from atmospheric refractivity structures

Very high frequency (VHF) radar echoes received with a quasi‐vertical antenna beam from the troposphere and stratosphere are caused by reflection and scattering. The essential difference between these mechanisms is described by different atmospheric refractivity structures, viz., stable laminae and turbulent irregularities. Several methods for discriminating between reflection and scattering are described and tested experimentally. These involve comparing estimates of the reflection coefficient and the turbulence refractive index structure constant using acceptable models. They also involve the wavelength and volume dependence, as well as the angular distribution of the radar echoes. Doppler spectra, phase and amplitude distributions, temporal and spatial correlation functions, and the dependence between radar echo power and coherence are examined. Diffuse reflection from refractivity laminae, corrugated by turbulence, appears to be an appropriate model for many observations. Finally, the implications of this process for VHF radar investigations of the atmosphere are outlined.

Refractive-index structure constant dependence on height

The variance of the angle of arrival of an optical beam passing through a turbulent atmosphere was measured, utilizing a telescope and a TV measuring system. The results were compared with predictions obtained using existing models for the dependence of the refractive-index structure constant on the height above the ground. It was found that the best fitting model, for lines of sight near the ground (10m≤h≤ 100 m), is Tatarskii’s model for convective conditions including buoyancy.

Atmospheric propagation of partially coherent radiation*

An extended, Rayleigh-Sommerfeld integral method is used to derive expressions for the mutual coherence function and radiation intensity derived from a planar, partially coherent source propagating through the atmosphere. The derived results reduce to previous results for (i) coherent radiation propagation in the atmosphere and (ii) the relations relating the far-field intensity angular distribution and the source coherence for a partially coherent source in vacuo. A mathematical description of the predicted results in terms of the vacuum distribution and scattering functions (related to the Fourier-transformed two-source mutual coherence function) is permitted by this development. Analytical results are calculated for a homogeneous atmosphere and a source coherence that simulates a laser-illuminated rough surface. The effective far-field range is determined by the source size, wavelength, and source coherence length. The phase of the calculated mutual coherence function is determined by the field-point separation for off-axial propagation directions. Numerical results for the amplitude and phase coherence lengths are calculated and illustrated as a function of the source size, source coherence length, propagation angle, range, and refractive-index structure constant.

Refractive index variance and its height distribution in different air masses

With the advent of radars capable of detecting backscatter from turbulent inhomogeneities in the clear air, there is renewed interest in the refractive character of the atmosphere. The radio refractive index structure constant and its spatial and temporal distribution are of particular importance. The general refractive properties of air masses are discussed in this paper. Based on certain assumptions, the distribution of optical and radio refractive index structure constants are then calculated and the results are compared with the limited observational data available.

Turbulence-Induced Phase Errors in Synthetic-Aperture Radars

Radar waves propagating through the turbulent troposphere acquire phase fluctuations which appear as aperture phase errors in synthetic-aperture radar systems. The phase error description follows directly from Tatarski's theory of wave propagation through turbulence; for most situations of interest, the phase error is Gaussian with stationary first increments. The structure function of the phase error is a function of the radar parameters and of the vertical profile Cnm(h) of the microwave refractive index structure constant of the turbulence. A closed form expression for best attainable resolution can then be given as a function of these same parameters, via the results of a phase error analysis by Brown and Riordan. Published data on Cnm(h) and its optical-region counterpart are reviewed. The data from a recent microwave propagation experiment by Thompson and Janes are then analyzed and are consistent with the order-of-magnitude estimates of Cnm: Some unsolved problems associated with applying the results to synthetic-aperture systems are discussed.

Measurement of the average structural characteristic of the atmospheric refractive index

Measurements of the square of the structure constant of the atmospheric refractive index averaged over a 25-km propagation path are presented. The daily behavior of this parameter C <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</inf> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> for 4 days in July, 1969, are shown. The measured values are in the range from 0.5 to 4×10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-17</sup> cm <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-2/3</sup> .

Measurements of Laser-Beam Scintillation in the Atmosphere

Measurements have been made of the scintillation of a laser beam after propagating over an 8-km path near the ground. The measurements were made with collection apertures ranging from 1 mm to 1 m in diameter. The probability distribution of the scintillation was found to be log-normal for all collector diameters. The log-normal variance decreased smoothly for diameters from 1 mm to about 10 cm, but showed no decrease between 10 cm and 1 m. A hypothesis is offered which explains those results which are anomalous in terms of present theories. Measurements of the log-normal variance were made over extensive periods on three days, during winter, spring, and summer. From these the atmosphere’s refractive-index structure constant, CN2, was computed. Values of CN2 were found to be fairly constant over the full range of measurements, falling between 3 and 6×10−15m−23, except near sunrise and sunset, when values of about 1×10−15m−23 were more typical.