Physics of Fluctuations of Waves in Turbulent Medium V. I. Tatarskii Zel Technologies & NOAA/PSD firstname.lastname@example.org
AbstractSemi-qualitative description of basic phenomenain wave propagation in a medium withfluctuating parameters is considered. Inparticular, the fluctuations of phase, spatial andtemporal phase differences, angle of arrival,intensity, their correlations and spectra areanalyzed for waves in turbulent atmosphere.The simple consideration, based on descriptivegeometric optics and its diffractive limitations,allows obtaining all basic relations withoutcomplicated mathematics.
Structure of fully developed turbulenceFor very high Reynolds numbers Re = UL/ν, where U is avelocity of flow, L is a scale of flow (for instance, diameter ofjet or pipe, an elevation above a plane boundary of flow), and νis the kinematical viscosity, the structure of flow is turbulent. Thismeans that the values of velocity, density, pressure, temperature, andrefractive index chaotically vary from one point to another.Here, color schematically represents the value of departure ofparameter (say, refractive index) from its mean value. As a rule, thelarger is some inhomogeneity, the more intense it is.
Turbulent flow is usually described statistically. If n is therefractive index, the most important for wave propagation inturbulent medium characteristic is so called structure function Drr , nr nr 2This function is related to the correlation function Brr nrnr , by the formulaDr n 2 r n 2 r 2nrnr ,r
Why the structure function is more useful than the correlationfunction for turbulent medium description?Let us consider two points A and B in a turbulent flow A B
If the scale of inhomogeneity is large in comparison with thedistance r between points A and B (blue inhomogeneity at theprevious slide), it contributes almost the same amount of n in bothpoints, i.e., does not contribute much to the difference n(A) – n(B).If the scale of inhomogeneity is small in comparison with thedistance R between points A and B, such inhomogeneity isrelatively weak and contributes only a small amount to thedifference n(A) – n(B). Thus, the main contribution is caused byinhomogeneities, which scale is of the order of distance R betweenthese points. Because of this, the value of the structure function Dis a measure of the intensity of inhomogeneities having scale R.
For very large Reynolds numbers the structure function D dependsonly on the distance R and is independent of the mutual orientationof points A and B. It was found by A.N. Kolmogorov and A.M.Obukhov in 1941 that the function D has the following shape: Struct. Funct. 100 10 1 0.1 Distance 0.1 1 10 100 1000 C expk 2 l 2 Sk , Dx 2 1 coskxSkdk k 2 4 L 2 2 56 0 There are 3 ranges in this figure: quadratic (left), 2/3 slope (middle) and constant (right). These ranges are separated by the scales l (inner scale, 5 mm in this example) and L (outer scale, 10 m in this example).
Rather often all essential transverse scales of electromagneticproblem, such as radius of the first Fresnel zone, the base of opticalor microwave interferometer, diameter of optical or microwaveaperture are much larger than the inner scale of turbulence l andmuch less than the outer scale L. In such situation it is possible toneglect the effect of these scales and consider the idealized model,for which l = 0 and L = Infinity. In this case, we may use thesimple model Dr C 2 r 23 Struc. Funct. 100 50 10 5 1 0.5 Distance 0.1 1 10 100 1000The model of power structure function (power spectrum) corresponds toconsidering turbulence as a fractal set.
Phase FluctuationsWe start considering of wave propagation in a turbulent mediumwith fluctuations of phase. Fluctuations of Φ are important in suchpractical problems as optical measuring of distances, transmittinghigh-accurate time, accuracy of large-base interferometers.The geometric optics approximation is an adequate tool for thisproblem.We consider the ray intersecting inhomogeneities of refractiveindex. If deviations of refractive index from unity are small, it ispossible to neglect refraction effect and consider the ray as astraight line, because curvature of ray is the second order effect.
After passing a single inhomogeneity having scale Δx and deviationof refractive index from unity equal to Δn, the ray obtains anadditional phase shift equal to ΔΦ = k Δx Δn, where k = 2π/λ is awave number. After passing N inhomogeneities the total phase shiftis equal to k x1 n 1 k x2 n 2 k xN n NThe mean value of Φ is 0, because mean value of Δn is zero. For themean square of Φ we obtain: N 2 k2 xj 2 n j 2 k 2 x l x j n l n j j1 ljBut the second sum vanishes because the fluctuations in differentvolumes are uncorrelated. Thus, N 2 k2 xj 2 n j 2 j1
Which Δx we must choose? We already mentioned that thelarger is inhomogeneity, the stronger is fluctuation Δn. Thus,the main contribution is provided by the largest possibleinhomogeneities, having the size of the order of outer scale L ofturbulence. Thus, we must choose Δx = L. All terms in the lastsum are equal and we may write n j 2 2 k 2 L 2 2 N, 2But the number of inhomogeneities N is equal to the ratio oftotal distance X to L. Thus, 2 k 2 2 LXWe may present the last formula in a little different form, if wesubstitute 2 C 2 L 23where C is the constant, entering in the 2/3 law for refractiveindex structure function.
This formula has the form 2 MC 2 k 2 L 53 XWe inserted in the last formula some unknown numericalcoefficient M, because all previous reasoning was performed onlywith the accuracy of indefinite coefficient. The value of M can beobtained only by more rigorous theory.
Phase differences fluctuationsFluctuations of spatial or temporal phase differences are importantfor many practical problems: measurements of angle of wavearrival, resolution of images, interferometry.Let us consider two parallel rays in a turbulent medium separated atdistance ρ. The most Effective inhomogeneity Large inhomogeneity provides the same phase shift to both ρ rays and does not The most contribute to phase effective difference inhomogeneity Contribution to phase difference from these inhomogeneities is small
Let us consider contribution to the phase difference provided byinhomogeneities of different scales. If the size of someinhomogeneity is small in comparison with the distance ρ, it maycontribute only to phase shift along a single ray. Thus, suchinhomogeneity contributes to a phase difference. The mostimportant contribution will be provided by the largestinhomogeneities of such type, i.e., by inhomogeneities of sizeabout ρ. The inhomogeneities having the size much larger than ρ,are stronger, but they provide the same contribution to the phaseshifts for both rays. Thus, contribution to the phase differencefrom such inhomogeneities will be small. Therefore, the mostimportant contribution to the phase difference are due to theinhomogeneities of the size about ρ.According to the 2/3 law, Dr C 2 r 23 the deviation of n from unity is of the order ofn j ~ j Cr 13 , where j 0, 2 j 1, i j 0 for i j
Now we can calculate the total contribution of essentialinhomogeneities to the phase difference. We must choose r = ρ inthe last formula and obtain N j1 kn j N j1 k j C13 Ck43 N j1 jThe mean value of ΔΦ = 0, and for the mean square of ΔΦ we have 2 Ck43 j1 jl j l C 2 k 2 83 j1 2 C 2 k 2 83 N 2 N N N j The number of essential inhomogeneities N = X/ρ, where X is the total distance from the source of wave to the receiver and ρ is the longitudinal scale of essential inhomogeneities, which for isotropic turbulence is equal to its transverse scale. Thus, 2 KC 2 k 2 53 X where some unknown numerical coefficient K was introduced.
Let us find such transverse distance ρ, for which the variance 2 C 2 k 2 53 X 1. This value is called “radius ofcoherence.” We find 0 1 65 1 35 C 2 k 2 X 35 C k 65 X Coherence radius plays an important role in the problem of resolution of telescopes and other optical devices. Only if 0 D where D is diameter of aperture, there exists a possibility of coherent summation of waves in the focal plane. If 0 D, different parts of aperture send incoherent waves to the focal plane, and it is impossible to achieve diffraction limit of the lens resolution. The important parameter is a ratio of coherence radius to the radius of the first Fresnel zone. For this ratio it is easy to find 0 1 X C 2 k 76 X 116 610
Angle of arrival fluctuationsThe angle of arrival is related to the phase difference. If we measurethe phase difference by interferometer having the base ρ and the Phase shift angle between the wave vector of incident wave and the normal to ρ the base of interferometer is γ, the γ phase shift δ = kρ sin γ appears. Wave front Thus, for small γ, k Thus, fluctuations of angle of arrival and phase differences are determined by the formula 2 2 KC 2 13 X k 2 2
Dependence of on ρ is shown in the following plot in the 2 semi-logarithmic scale 2 KC2 X 2.5 2 1.5 1 0.5 0.1 0.2 0.5 1 2 5 10 Decreasing of 2 with increasing ρ is caused by the effect of averaging fluctuations over the interferometer base (or aperture of telescope).
Formula for shows unlimited increasing of fluctuations of γ 2while ρ tends to zero. This result is incorrect, because the 2/3 lawis valid only for r > l. If ρ becomes less than l, the 2/3 lawchanges for Dr C 2 l 23 r2 l2 If we repeat the derivation of 2 for this case, we obtain thefollowing result for the case ρ < l: 2 KC 2 l 13 X We may suggest the interpolating formula, working for all ρ: 2 KC 2 X l 2 2 16
The plot of this function is presented in the following Figure: 2 l2 3 KC2 X 1 0.8 0.6 0.4 0.2 0.01 0.1 1 10 100 l Variance of angle of arrival fluctuations in entire range of ρ. The case ρ < l corresponds to the aperture less than inner scale of turbulence.
Temporal correlation and spectrum of angle fluctuations Let us consider a temporal fluctuations of the angle of arrival. The angle of arrival in some moment t is determined by the pair of rays and inhomogeneities located at these rays. At the moment t + τ all inhomogeneities will be shifted to a new position and instead of them a new inhomogeneities will cover our two rays.1 t2 3 t+τ 4 At the moment t these inhomogeneities were located at the positions 3 and 4, opposite to a wind direction.
Let us consider the temporal correlation function of the angle ofarrival γ. We have 1 2 3 4 1 2 3 4 1 , 2 , 1 2 k k k 2 2We can use the algebraic identity a bc d 1 a d 2 b c 2 a c 2 b d 2 2 and present the correlation function in the form 1 4 2 2 3 2 1 3 2 2 4 2 1 2 2k 2 2 But we already determined the variance of the phase differences for an arbitrary separation between two rays: KC 2 Xk 2 2 2 k 2 2 2 2 l 2 16
For the rays 1 and 4 we must instead of ρ substitute ρ +Vτ, where Vis the transverse component of wind. For the rays 2 and 3 we mustsubstitute instead of ρ the value Vτ – ρ, and for pairs 1, 3 and 2, 4we substitute instead of ρ the value Vτ. The formula forautocorrelation function of γ takes the form 2 V 2 V 2 V 2 1 2 B KC 2X 16 16 2 16 2 V l 2 2 V l 2 2 V 2 l 2This function for KC^2X=1, V=500 cm/s, ρ = 5 cm, and l = 0.5 cm isshown in the following Figure:
B 1 0.8 0.6 0.4 0.2 ,s 0.1 0.2 0.3 0.4 0.5The auto-correlation function of the angle of arrival fluctuations.
The spectrum of γ is determined by the formula Q cosB d 0 Q 1 23 0.001 6 1. 10 9 1. 10 1 ,s 0.001 0.1 10 1000Zeroes in the spectrum are caused by the presence of difference inthe definition of γ (for the case of interferometer). In the real spectrathese zeroes will be filled in because of wind fluctuations. Thestraight line in the spectrum corresponds to 23 dependence.
Intensity fluctuationsIn geometric optics, the product of intensity by the cross-section ofbeam (ray tube) is constant. Intensity is determined by the cross-section. The small cross-section The large intensityThe large cross-sectionThe small intensityThe shape of a beam is determined by distribution of refractiveindex in space.
Inhomogeneities of refractive index play role of random lensesfocusing or defocusing light. Initial Final intensity intensity Less intensity Negative lens Positive lensIt is known that for a spherical lens the focal distance F is equal toratio of the curvature radius ® to (n-1), where n is the refractiveindex of lens material. For turbulent inhomogeneities the curvatureradius is of the order of scale R of inhomogeneity. Thus, F R nR 1
It follows from 2/3 law that nR 1 C R 13 Thus, F R 23 C Let us calculate the intensity change after passing a single inhomogeneity. δR α α R F X R We have F CR 13 and l X CR 13 X
Thus, the change of the intensity is determined by the relation IR 2 I IR R 2 , or I I R RUsing the obtained formula for δR we obtain I CX I R 23 It is clear from this formula that the smaller is the scale of inhomogeneity, the larger is the changing of intensity. Thus, the most important contribution to the change of intensity is provided by the smallest possible inhomogeneities of the order of inner scale l of turbulence. Thus, we must set R = l in the last formula. We also introduce a random number ξ, which accounts that the sign of fluctuation of refractive index is random. Thus, for the contribution of a single j-th inhomogeneity to intensity fluctuation we obtain I j CX j I l 23
The total change of intensity is determined by the sum N I CX I l 23 j j1 For the mean square of relative fluctuations of intensity we obtain N N I j m l 43 2 C2X2 C2X2 N I l 43 j1 m1 The last step is to substitute N = X/l. The result is I 2 C2X3 73 I l This formula was obtained by geometric optics approach and is valid if the geometric optics is true for this problem.
The effect of diffraction at inhomogeneities leads to spreading ofall rays. At the distance X from the inhomogeneity the sharpboundary of ray tube spreads to the size X X l X X If X l it is possible to neglect a diffraction and use the geometric optics result. But if X l, diffraction compensates the focusing effect, and the inhomogeneity of the scale l does not produce change of intensity. Thus, the minimal scale of inhomogeneities, which still may cause the intensity fluctuations, isl X . Thus, in this case we must replace the inner scale ofturbulence l in the formula for intensity fluctuations for X .
The resulting formula has the form I 2 C2X3 C 2 X 116 C 2 k 76 X 116 I X 73 76Here, k = 2π/λ is the wave number.It is possible to write a simple interpolation formula whichprovides transition from geometric optics case to diffraction case: I 2 C2X3 I l 143 X 73 More rigorous theory provides some numerical coefficients in above formulae and another type of transition from geometric optics range to diffraction range of distances.
2 I 10 1 0.1 0.01 0.001 X, m 10 20 50 100 200 500 1000Dependence of intensity fluctuations on distance for λ = 0.63 μ,l = 5 mm, C2 10 16 m23 . Transition from geometric optics regimeto diffraction regime takes place at the distance X = 40 m.
Saturated Intensity FluctuationsComparison of the experimental values of II 2 2 with Ithe theoretically predicted value 2 C 2 k 76 X 116 shows that there I0is a good agreement between them if 2 1. But in the region I0where I0 1 the experimental value of I 2 2 does not increasewhile 2 increases and remains approximately constant. I0
The region 2 1 is called the region of strong or saturated I0fluctuations. It starts at the distance X 0 C 1211 k 711 where thetheoretically predicted value of I0 becomes unity. 2Let us consider the ratio of coherence radius to the radius of thefirst Fresnel zone. This value was found above: 0 1 X C 2 k 76 X 116 610It is clear from this formula that in the region of weak fluctuations,i.e., 2 1 , the radius of coherence is large in comparison with I0the radius of the firs Fresnel zone, while it is small in comparisonwith X in the region of strong fluctuations. This is clear fromthe following plot.
The radius of coherence 0 (red) decreases and the radius of thefirst Fresnel zone X (blue) increases while the distance Xincreases. At some distance X 0 C 1211 k 711 the coherenceradius becomes less than the radius of the first Fresnel zone. 0, X , cm 2 1.75 1.5 1.25 1 0.75 0.5 Weak fluctuations Saturated (strong) fluctuations 0.25 X, m 500 1000 1500 2000
Previously we found that the most important for the intensitychanges inhomogeneities have size of the order of X . Suchinhomogeneity caused focusing or defocusing of a beam. Thisfocusing (defocusing) is possible because the inhomogeneity actssimilarly to a lens, which coherently summarizes all wave field atits surface. Such situation is possible only if the field, incident at alens, is coherent, i.e., if 0 X . But in the region 0 X ,where the coherence radius is small in comparison with the scaleof lens, different parts of lens transmit (radiate) incoherent waves.Because of this, such lens is unable to focus / defocus radiation. Wave front
It is convenient to call the wave field for which 0 X asdegenerated wave field. The highly degenerated field ( 0 X )can not be focused and for such field it is impossible to obtain asharp image in the focal plane.If we return to the intensity fluctuations, we may conclude thatonly the inhomogeneities located in the initial part X X 0 of thepropagation path may produce intensity fluctuations. Allinhomogeneities, which are located in the region of strongfluctuations, can not focus or defocus the wave and because of thisthey have no (or have very little) influence on I . But these 2inhomogeneities continue to contribute to decreasing of radius ofcoherence.This qualitative picture explains the phenomenon of strongfluctuations, but the corresponding rigorous theory is rathercomplicated and is based either on the theory of random Markovfields or diagram technique.
For more detailed information concerning the discussed problem, itis useful to refer to the following publications (the simplest are listedprior to more complicated).1. Tatarskii V.I. Review of Scintillation Phenomena. In Wave propagation inRandom Media (Scintillation). Edited by V.I. Tatarskii, A. Ishimaru, V.U.Zavorotny. Copublished by SPIE Press and IOP, 1993.2. S.M. Rytov, Yu.A. Kravtsov, V.I. Tatarskii. Principles of StatisticalRadiophysics. vol. 4. Wave Propagation Through Random Media. Springer-Verlag, 1989.3. V.I. Tatarskii. The effects of the turbulent atmosphere on wave propagation.Translated from the Russian by the Israel Program for Scientific Translations,Jerusalem, 1971. Available from the U.S. Dept. of Comm., Nat. Tech. Inf. Serv.,Springfield, VA, 221514. V.I. Tatarskii and V.U. Zavorotniy. Strong Fluctuations in Light Propagation in aRandomly Inhomogeneous Medium. In Progress in Optics, vol. XVIII, edited byE. Wolf, North-Holland Publishing Company, Amsterdam - New York - Oxford,1980.
This presentation may be downloaded from the website http://home.comcast.net/~v.tatarskii/vit.htm