Outer Scale Of Turbulence Research Articles

Comparison of mesospheric VHF radar echoes and rocket probe electron concentration measurements

Refractive index irregularities in the equatorial mesosphere have been investigated using both the Jicamarca VHF radar and a rocket‐borne Langmuir probe launched from Punta Lobos, Peru. On February 27, 1983, a single layer of turbulence was observed in the upper mesosphere by both experiments. There is very good agreement between the observed radar echo power and the radar scattering cross section calculated from the rocket data when these are interpreted in the context of isotropic turbulence. The inner and outer scales of turbulence have been calculated from both the radar and the rocket data, and good agreement is found. The radar data show indications of large‐scale vortices in the layer of irregularities. Rocket data show that the inner scale of turbulence in the upper mesosphere is a few tens of meters and that the Jicamarca radar Bragg wavelength (3 m) is well within the viscous subrange of turbulence in this altitude range. The spectral index in the inertial subrange is close to −5/3, changing to about −7 at higher wave numbers. Energy dissipation rate in the layer was calculated to be 0.05 W kg−1, in good agreement with previous estimates.

The dependence of sound extinction on the parameters of thermal turbulence in the atmospheric boundary layer

Scattering coefficients and normalized scattering phase function were calculated using the Von Karman spectrum of temperature fluctuations in the atmospheric boundary layer which enabled one to take into account the outer scale of turbulence. It was shown that the scattering coefficient had a scale of turbulence. At the same time, the scattering cross section was proportional to the cube root of the sound frequency which agreed with the previously obtained results. The normalized scattering phase function was calculated for small scattering angles. It was shown that in the presence of small scale inhomogeneities in the atmosphere the portion of power removed from the original beam increases because the lateral scattering becomes prevailing. The obtained results are in good agreement with the experimental data.

Radar Detection of Turbulence in Precipitation Environments

Abstract Imperfect particle tracer response is incorporated into the relations describing the turbulent air motion contribution to Doppler radar spectrum mean and variance. Tracer effects on radar estimates of the eddy dissipation rate (ϵ) increase with decreasing range, and decreasing turbulence outer scale (λ0). Exclusion of imperfect tracer effects can lead to moderate underestimates (≤30%) of ϵ. These precipitation effects, however, may be dwarfed by the sensitivity of the ϵ estimate to the estimate of maximum eddy size. Nonetheless, good radar estimates of the turbulence severity index (ϵ1/3) can be achieved with only a reasonable estimate of λ0

Estimation of turbulence spectra parameters from log-amplitude fluctuations spectra of an optical wave

From the general formula for the temporal frequency log-amplitude fluctuations spectra the asymptotic expressions are found, valid for the geometric and wave approximations. A comparison of the plane and spherical wave formulas with those valid for the beam wave is made in all frequency regions. The difference between the on-axis and off-axis beam spectra is presented. The possibility of estimating from the log-amplitude scintillations the inner and outer scales of turbulence together with the exponent of turbulence spectra in the subrange is briefly discussed.

Effect of finite bandwidth on far-field performance of modal wavefront—compensative systems

The power spectrum is formulated for the time-dependent expansion coefficients of Zernike polynomials of a wavefront distorted by atmospheric turbulence. This expression is shown to be more general than that derived by Hogge and Butts. The low-frequency asymptotic results are examined along with the effects of the outer scale of turbulence and propagation distance. The integrated power spectra are plotted against the bandwidth for several low-order correcting modes. These graphs are then used to study the combined effect of transverse wind and finite bandwidth on the MTF and irradiance profile at the receiver plane for a phase-compensated Gaussian beam. It is shown that the major effect of low bandwidth is to increase the beam spread and that the phase compensation is slightly less effective along the wind direction.

Mutual coherence function of a wave front corrected by zonal adaptive optics

The mutual coherence function (MCF) of a wave propagated through atmospheric turbulence and corrected by zonal adaptive optics is derived for a segmented corrector with either piston only or piston with a local tilt fit. We begin with an intuitive approach which is based on replacing the outer scale of turbulence in the uncorrected MCF with a quantity proportional to the segment size. This approach is especially useful in deriving asymptotes. A rigorous approach provides the full curves, and reduces to a single curve for piston only and a single curve for piston with tilt, after proper parametrization. The effect of amplitude fluctuations is also considered, since scintillation limits the ultimate degree of correction as the segment size goes to zero. Although the analysis is for a segmented corrector, the results apply fairly well to continuous corrector with actuator spacing equal to d/√3, where d is the segment diameter.

Atmospheric turbulence chamber for optical transmission experiment: characterization by thermal method

A turbulence chamber (0.78 x 0.23 x 2.59 m(3)) consisting of ten small electric heater/blowers with an aluminum foil screen and three screens of 2-mm aluminum wire meshes can generate the nearly homogeneous isotropic turbulence within the 0.5 x 0.05 x 2-m(3) region at the 0.11-m height of optical measurements. The temperature structure constant squared C(2)(T) = 52.9 K(2) m(-?) was obtained from the temperature structure function measurements measured by using a differential microthermocouple system. The refractive-index structure constant squared C(2)n<sub/>at the 632.8-nm wavelength was calculated from C(2)(T):C(2)(n) = 3.00 x 10(-11)m(-?). The average wind velocity and temperature were 0.41 m/sec and 53 degrees C, respectively. From the power spectrum of temperature fluctuations, the inner and outer scales of turbulence are determined: l(o) = 5.0 mm and L(0) = 6.5 cm. The measured temperature structure function and power spectrum of temperature fluctuations satisfy the ? and -5/3 power similarity laws in the inertial subrange, respectively.

The effect of the outer scale of turbulence and wavelength on scintillation fading at millimeter wavelengths

Amplitude scintillation measurements made on two-millimeter microwave links at 36 GHz and 110 GHz along a common 4.1 km path across central London are deseribed. The variances of the logarithmic amplitude scintilliations at these two frequencies are compared for different ranges of the outer scale of turbulence. A comparison is made between the experimental results and the theoretical predictions of Tatarski. Good agreement has been found for the two cases when the outer scale of turbulence is greater than and smaller than the Fresnel zone size.

Turbulence-degraded beam quality: improvement obtained with a tilt-correcting aperture

In this paper, we have calculated the degradation in the focal-plane irradiance distribution due to atmospheric turbulence and the potential improvement realizable by employing a wavefront filt-correcting aperture. It has been shown that, when the aperture diameter is of the order of the outer scale of turbulence, virtually no improvement is realized relative to the uncompensated case. For the case when the long-term coherence length is small compared with the diameter, there is a several-decibel improvement over the longterm case; however, for a full phase-compensating aperture (e.g., COAT), peak intensity can be increased an additional several decibels. When the coherence length is not much smaller than the diameter, close to diffraction- limited performance can be expected. Comparisons are also made of the reduction of on-axis intensity with no compensation, tilt-correction, and a full phase-compensating aperture. It is shown that the effective coherence length of the compensated aperture due to the residual amplitude fluctuations is greater than the long-term coherence length by a factor proportional to the square root of the Fresnel number of the aperture. Approximate formulas are also presented for the tilt-corrected MTF for arbitrary aperture irradiance distributions.

Propagation in random media with stationary temporal fluctuations

Born's method is applied to the time‐dependent wave equation. Exact solutions, valid for all wavelengths, are obtained for the first‐order perturbation terms for monochromatic plane and spherical waves propagating in random media with stationary temporal fluctuations. It is shown that the temporal fluctuations of the medium can be neglected if the temporal frequencies of the fluctuations are much less than the frequency of the propagating wave and if the propagation time from the transmitter to receiver is much less than the coherence time of the medium. In the atmosphere for the case where (λ30/l40)L≪ 1, where l0 is the inner scale of turbulence, log‐amplitude and phase covariance functions were calculated for the polarized and depolarized fluctuations of plane waves propagating in slightly different directions. Numerical evaluation revealed that the polarized log‐amplitude fluctuations decorrelate for angular separations on the order of tan−1 [(λ0/L)1/2] while phase fluctuations decorrelate for angular separations on the order of tan−l(L0/L), where L0 is the outer scale of turbulence.

Antenna aperture effects on measurements of propagation through turbulence

The first Born approximation is applied to propagation of a spherical wave through a layer of random irregularities to an aperture antenna. The effect of the receiving aperture on measurements of the variance of in-phase and quadrature components of the received signal, and the variances of their spatial derivatives is determined. In the weak scattering limit, these quantities are related to the scintillation index, mean-square phase fluctuations, mean-square angle-of-arrival fluctuations, and fading rates. Physical situations corresponding to the generic problem considered here include satellite communication through ionospheric fluctuations, solar wind observations, observations through irregularities caused by instabilities in barium release clouds, etc. Numerical results are presented which show the aperture effect for irregularities with power law spectra of the form <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S(K) \propto (1 + K^{2}L_{0}^{2})^{-N/2}</tex> for values of the spectral index <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</tex> ranging from 3 to 5 and for many ratios of the three lengths, turbulent outer scale <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L_{0}</tex> , aperture radius, and Fresnel length. These results are presented in convenient form for estimating the aperture effect in many cases of interest. Aperture effects (aperture smoothing) generally dominate the behavior of the spatial derivatives, while a nearfield effect is shown which causes enhancement of the variance of the in-phase fluctuations over the value predicted ignoring the receiving aperture. It is suggested that this effect may be useful to explain some aspects of observed gigahertz scintillation.

Beam wander in a turbulent medium: An application of Ehrenfest’s theorem

Beam wander of a finite optical beam propagating in a turbulent medium is investigated theoretically. Using the optical analog of Ehrenfest’s theorem, it is shown that the centroid of a finite beam propagates as a paraxial ray in a certain effective refractive index that depends on the irradiance profile of the beam. Ray statistics in the effective refractive index are studied for arbitrary irradiance profiles and new results are obtained for the variance of spot displacement and beam angle of arrival. These results are then applied to the particular cases of focused and collimated gaussian beams in atmospheric turbulence with a modified Von Karman power spectrum to yield the functional dependence of spot dancing and angle-of-arrival statistics on the inner and outer scales of turbulence and on the Fresnel number for focused gaussian beams.

Observations of Turbulence in the Atmosphere of Venus using Mariner 10 Radio Occultation Measurements

A technique is developed for estimating the outer scale of turbulence in a planetary atmosphere using dual-frequency radio occultation measurements. This technique is based on the frequency dependence of the temporal frequency spectra for the log-amplitude fluctuations and is particularly useful when probing the upper atmosphere where the transverse velocity of the line-of-sight path is decelerating very rapidly. It is applied to the region of strong turbulence located in the vicinity of 60 km on the dayside of the atmosphere of Venus using the Mariner 10 S/X radio occultation measurements. It is found that, contrary to earlier findings from Mariner 5, the outer scale of turbulence is at least 5 km. It appears that the outer scale of turbulence is as large as the vertical extent of the region of strong turbulence. Estimates of the structure constants for refractive index and temperature fluctuations indicate that the turbulence is stronger than that measured near the earth's tropopause.

Self-induced irradiance fluctuations of a laser beam in an absorbing, turbulent medium

The log-irradiance variance of an intense laser beam that propagates in an absorbing medium with a turbulent velocity is determined. The irradiance fluctuations result from the random refractive-index variations created by turbulent convection of the energy absorbed from the beam. The index of refraction, assumed to depend only upon temperature, is found by linearizing the heat-transport equation. Linearization is shown to be valid because of the finite extent of the induced temperature gradient. The results are valid in the geometrical-optics approximation with the restriction that l0/5.91 ≪ a ≪ L0, where l0 and L0 are the inner and outer scales of turbulence, respectively, and a is the laser-beam radius.

Observations of Small-Scale Turbulence in the Atmosphere of Venus by Mariner 5

Information regarding small-scale turbulence in the Venus atmosphere is important and desirable because it contributes to understanding of the atmosphere's circulation. It is demonstrated that the radio occultation data of a flyby spacecraft such as Mariner 5 can provide valuable information on turbulence in the Venus atmosphere. Unlike previous studies of the Mariner 5 data, this paper is based on the frequency spectrum rather than the variance of the log-amplitude fluctuations. The excellent agreement between the processed and previously derived theoretical spectra furnishes strong evidence that the Mariner 5 fluctuations are primarily turbulence-induced. It is seen that, above 35 km, turbulence is strongest in the vicinity of 45 and 60 km, and that the outer scale of turbulence is of the order of 100 m. Comparison with the results obtained from the Venera missions is also discussed.

Role of the outer scale of turbulence in atmospheric degradation of optical images

The atmospheric degradation of optical images is investigated by assuming the refractive-index structure function to be independent of the distance between two correlation points when such distance is larger than the outer scale (Von Karman model). It is found that, in most practical cases, the atmospheric turbulence puts a limit to the resolution as predicted by the 2/3-power law, however, only when the turbulence is sufficiently strong. The parameter distinguishing strong from weak turbulence turns out to be the limit value of the wave structure function Dw (∞) for infinite distance. Weak turbulence, with Dw (∞) ≲ 20, does not put a limit to the resolution of an optical system.

Spectral broadening of an acoustic pulse propagating through turbulence

The following analysis develops the equation for the scattered acoustic power and spectral broadening of a plane acoustic pulse propagating through atmospheric turbulence that is moving with a uniform wind transverse to the beam. Assuming weak single scattering, the resulting propagation broadening is proportional to the wind velocity divided by the outer scale of turbulence and is independent of the strength of turbulence and total propagation range. Further, the broadening for a medium with variable outer scale and wind shear (e.g., a vertical propagation path) is determined primarily by the highest ratio of transverse wind to outer scale encountered by the pulse.

Remote sensing of the turbulence characteristics of a planetary atmosphere by radio occultation of a space probe

The purpose of this paper is to analyze the effects of small‐scale turbulence on radio waves propagating through a planetary atmosphere. The analysis provides a technique for inferring the turbulence characteristics of a planetary atmosphere from the radio signals received from a spacecraft as it is occulted by the planet. The planetary turbulence is assumed to be localized and smoothly varying with the structure constant varying exponentially with altitude. Rytov's method is used to derive the variance of log‐amplitude and phase fluctuations of a wave propagating through the atmosphere. When the distance between the spacecraft and the planetary atmosphere exceeds L02/λ, where L0 is the outer scale of turbulence, the effects of inhomogeneity and finite size of the turbulent medium become important. A recent study by Gurvich [1969] pointed out that the structure constant for the Venusian atmosphere inferred from the Mariner‐5 occultation data was significantly lower than that inferred from the Venera‐4 data. By applying the analysis developed in this paper it is shown that the effects of inhomogeneity and finite size of the turbulence, which were excluded in Gurvich's analysis, account for this difference. More important, however, is the fact that the new techniques are available for use in future radio‐occultation missions.

Optical beam spread in a turbulent medium: effect of the outer scale of turbulence*

Optical beam spread in a turbulent medium: effect of the outer scale of turbulence*

Beam Spreading in a Turbulent Medium

The Kolmogorov spectrum with its associated structure functions is generally used to study wave propagation in turbulent media. It is known, however, that this spectrum does not accurately represent the real atmosphere outside the inertial range. In this paper, the broadening of a focused gaussian beam is calculated, using instead a modified von Karman spectrum better to describe the medium. The Born approximation is also used, because it follows naturally from the use of this spectrum. The dependence of beam spreading on the outer scale of turbulence is obtained and the results generally reduce to those corresponding to the Kolmogorov spectrum and the Rytov approximation, which in fact do not depend on the outer scale of turbulence. The results can be applied to arbitrary source configurations or extended to include the calculation of other field quantities of interest.