Radio Science

Near-horizontal line-of-sight millimeter-wave propagation measurements for the determination of outer length scales and anisotropy of turbulent refractive index fluctuations in the lower troposphere



[1] Variances of the normalized power, the angle of arrival, and temporal spectra were obtained from millimeter-wave propagation experiments. For their interpretation the existing theory for monochromatic spherical wave propagation through an anisotropic weakly turbulent medium had been expanded such that anisotropy of the refractive index fluctuations transverse to the propagation path is taken into account. The comparison of experimental and theoretical results provides path-averaged estimates for the outer scale L0 in vertical and horizontal directions within the first few hundred meters above ground. Thus a determination of the atmospheric anisotropy is feasible, and it is found to be related to the atmospheric stability. The values obtained for L0 in vertical and horizontal directions are compatible with those presented in the literature. We found that the size and the anisotropy of L0 depend significantly on atmospheric conditions and vary accordingly widely in time.

1. Introduction

[2] Turbulence is an important aspect of atmospheric motions. It plays a substantial role for the mixing and the transport of atmospheric constituents, as momentum, heat, or pollutants, over many orders of length scales. The bulk of the turbulent kinetic energy is carried at the outer scale of turbulence L0 (similar to the correlation length), where most of the turbulent diffusion and transport takes place.

[3] The intensity and the spatial length scales of turbulence are keys to understand the Earth atmosphere. As the influence of fine-scale turbulence reaches macrostructures of atmospheric flows, its investigation is significant for numerical weather forecasting, formation of clouds and precipitation, and turbulent exchanges and mixing of air masses, as well as for climate modeling (see, e.g., the review paper by Muschinski and Lenschow [2001, and references therein]).

[4] Another important impact of turbulence is its influence on wave propagation [Tatarskii, 1971]. The small-scale atmospheric dynamical processes have practical consequences for all propagation phenomena (light beams, radio waves, sound). Thus wave propagation theory is the basis for remote sensing technologies such as radar, lidar, sodar, and others. Particularly ground-based astronomical observations are severely affected by atmospheric motions, limiting both sensitivity and accuracy. Again the outer scale L0 has a significant influence [e.g., Nightingale and Buscher, 1991; Ziad et al., 1994].

[5] The significance of L0 is generally known, but only few investigations have been carried out in the past (reviews are given by Ziad et al. [1994, 2000]), with the restriction that most of them consider refractivity index fluctuations as being statistically isotropic. In the atmosphere, however, turbulence at large scales is usually anisotropic, so that L0 can be considerably different in horizontal and vertical directions. Estimates of L0 varied from centimeters [Consortini et al., 1970; Lutomirski and Yura, 1969; Gaskill, 1969; Schumann et al., 1995] to a few meters [e.g., Clifford et al., 1971; Nightingale and Buscher, 1991; Krasnenko and Shamanaeva, 1998] and even up to several kilometers [e.g., Colavita et al., 1987], hence providing conflicting evidence for its size. Some investigators suggest a rather large outer scale, close to infinity. However, from in situ measurements made by aircraft [Mahrt, 1985; Schumann et al., 1995; Muschinski and Wode, 1998] and balloon [Bufton, 1973; Dalaudier and Sidi, 1994] it is known that enhanced atmospheric turbulence often exists in thin layers that are only a few meters thick. This would suggest that the turbulent outer scales in these layers were not larger than this, at least in the vertical direction.

[6] The large variability of L0 encountered in the atmosphere is plausible, as it is known that outer scales of turbulence depend significantly on meteorological conditions and therefore vary over many orders of magnitude with altitude, time, and location.

[7] The outer scale values provided by our measurements correspond to the L0 parameter in the Von Karman and the Greenwood-Tarazano models of the refractivity spectrum equations (17) and (18) below), respectively. In both models, this parameter determines the boundary at low spatial wave numbers κ of the inertial subrange of turbulence. Thus the transition between inertial subrange and production range of the turbulent spectrum occurs approximately at κ ≈ 1/L0.

[8] Our experiments were carried out at 212 GHz with two-dimensional angle-of-arrival measurements, with the determination of turbulent outer-scale anisotropy effects being of main interest. The measurements were carried out at two different locations, namely at Bern, Switzerland, and at the astronomical site El Leoncito, Argentina, with near-horizontal test ranges at 10–300 m above ground. Because of the moisture these lowest layers provide the main contributions to the amplitude and phase fluctuations of microwaves propagating through the atmosphere. The two test ranges have lengths of 7.5 and 11 km, respectively. All measurements were carried out in winter during daytime with clear-air conditions. The measuring campaign in Bern occurred during January 1999, when data from 68 ten-minute intervals were collected. For El Leoncito, during September 1999, 64 intervals were obtained.

[9] In order to characterize quantitatively our two-dimensional angle-of-arrival measurements we developed a semianalytical formalism, which is based on the widely discussed smooth perturbation theory [e.g., Tatarskii, 1971; Ishimaru, 1978], in which an anisotropic refractivity spectrum was introduced, as initially suggested by Lee and Harp [1969]. With the derived theoretical expressions and the measured data we estimate the path-averaged outer scales of turbulence L0 in horizontal and vertical directions and describe their anisotropy and their dependence on meteorological conditions and stability. As far as we know, there exist no other investigations, carried out with comparable techniques.

2. A Simple Approach to Wave Propagation in Anisotropic Turbulent Media

[10] In the common theory by Tatarskii [1971] it is assumed that the medium that is crossed by an electromagnetic wave is isotropic. In the troposphere, however, this assumption is not strictly valid as correlation lengths can be substantially different in different directions. There exist only a few attempts to consider the effects of the anisotropic characteristics of the refractivity on electromagnetic wave propagation: Probably one of the first studies was carried out by Consortini et al. [1970], who analyzed mutual deflections of two parallel laser beams propagating through a turbulent atmosphere. Within the geometrical optics approximation, outer scale sizes in horizontal and verical directions between 3 and 6 cm with ratios between 1 and 1.4 were estimated. Similarly, Banakh and Smalikho [1994] investigated analytically random displacements of a laser beam by taking into account anisotropy effects. Several other theoretical papers which include anisotropic turbulent irregularities were also accomplished in the radar framework [Gurvich and Kon, 1992, 1993; Doviak and Zrnic, 1984; Hocking, 1987], with the Born approximation.

[11] For line-of-sight propagation studies, usually the Rytov method is used, as it has several advantages over the algebraic series representation of the Born method [Ishimaru, 1978, p. 349]. In order to interpret our millimeter-wave measurements, semianalytical formulas with a simple model, describing an anisotropic refractive index spectrum, have been derived. The derivation is based on the well-known theory for isotropic turbulence, using the Rytov method (smooth perturbation theory), which is widely applied in line-of-sight propagation problems and well documented in literature [e.g., Tatarskii, 1971; Ishimaru, 1978].

2.1. Variances of Angle of Arrival and of Normalized Power

[12] We first consider a plane wave propagating along the x axis of a Cartesian coordinate system, incident upon a turbulent medium at x = 0 with the observation point at distance L. In our experiment, spherical wave propagation is closer to reality: The relevant expressions can easily be obtained by generalizing plane wave propagation. In a general form, the amplitude and phase correlation functions for a plane wave are given by Ishimaru [1978, p. 357]

equation image

where index r stands for Bχ and i stands for BS, with

equation image

Bχ,S(ρ) describes the correlation of the amplitude χ and phase S between two points which are separated by ρ in a transverse plane to the propagation path at distance L; k is 2π/λ, with λ being the wavelength of the electromagnetic wave; κ is the spatial wave number; and Φn(κ) is the three-dimensional refractive index spectrum evaluated at κx = 0 [Lee and Harp, 1969]. As Bχ and BS are functions of the transverse separation ρ it is convenient to express (1) in cylindrical coordinates with κ = κ(cos ϕ, sin ϕ) and ρ = (ρy, ρz) = ρ(cos β, sin β). Adopting the approach to anisotropy given by Lee and Harp [1969], the refractivity spectrum is composed of two orthogonal components Φny(κ,ϕ) and Φnz(κ,ϕ), each of which is separable in κ and ϕ:

equation image

Φny(κ) and Φnz(κ) are three-dimensional spectra which can be derived from measured one-dimensional spectra along the two orthogonal axes y and z transverse to the propagation path. It is assumed that the longitudinal component of the refractivity field has negligible effects, since the length of the test range L is much longer than the correlation length of the medium [e.g., Tatarskii, 1971; Ishimaru, 1978]. The considered model also implies that the anisotropy is elliptical and its main axes are along the y and z axes.

[13] Substituting (2) into (1) and integrating over ϕ yields

equation image

and the structure function Dχ,S(ρ, β) = 2[Bχ,S(0) − Bχ,S(ρ, β)] is given by

equation image

[14] The first terms (equation image respectively) represent the mean characteristics of the refractivity spectrum equation image averaged over the yz plane, and are equal to the corresponding correlation and structure functions for the isotropic case with equation image The second terms (equation image respectively) vanish for isotropy. Their dependence on β (the orientation of ρ) takes care of the anisotropy effects. Instead of expressing Bχ,S(ρ, β) and Dχ,S(ρ, β) as a function of ρ and β it will be also useful to use Cartesian coordinates by replacing cos (2β) = (ρy2 − ρz2)/ρ2.

[15] As Ishimaru [1978] showed, (3) and (4) can be generalized for spherical wave propagation by replacing ρ by γρ and (L − η) by γ(L − η), with γ = η/L (for a plane wave, γ = 1). The amplitude and phase variances σχ2 and σS2 are given by Bχ (ρ = 0) and BS(ρ = 0), respectively, which leads to equation image Therefore the same variances result from anisotropic and isotropic media with equation image The same insensitivity to anisotropy is also found for the normalized variance σp2 = 〈(p − 〈p〉)2〉/〈p2 of the measured power p, an important quantity for experimental comparison. Here σp2 is smaller than the variance σI2 of the normalized intensity I observed by a point receiver. In the case of a finite receiving size, the fluctuations are averaged over the collecting aperture, and the fluctuations are reduced by the factor G [e.g., Tatarskii, 1971, p. 272; Lee and Harp, 1969].

equation image

The aperture averaging factor (for our experimental setup the value of G is 0.92 if L0 > R; compare section 3) G can be expressed as [Tatarskii, 1971, p. 273]

equation image

Σ is the effective aperture area of the receiving system, and bIy, ρz) = BIy, ρz)/BI(0) is the normalized intensity correlation function. The latter can be approximated by Bχy, ρz)/σχ2, provided that σχ2 ≪ 1. The expression Kyz) depends on the geometry of the aperture, which is given for a circular aperture by Tatarskii [1971, p. 273]:

equation image

with R the radius of the effective aperture area. Inserting Bχχ2 and K into (6) and writing it in polar coordinates, it can be seen that the anisotropy term of Bχ (i.e., equation image) has no influence on G, as ∫0 cos (2β) dβ = 0. Therefore the same σp2 results as does in isotropic conditions by simply substituting Φn, iso(κ) = 1/2 [Φny(κ) + Φnz(κ)].

[16] Let us now consider angle-of-arrival fluctuations where an anisotropic refractivity spectrum leads to noticeable effects. Assuming the length of the propagation path to be larger than the Frauenhofer diffraction region of the receiver (far-field condition), Tatarskii [1971, p. 288] derived an expression for the mean-square fluctuation σα2 of the angle of arrival α:

equation image

The ζ-ξ coordinate system lies in the y-z plane with ζ pointing in the direction of ρ. The angle of arrival α is between the x axis and the direction of propagation of the incoming wave, measured in the x-ρ plane. Taking the second derivative of DS along ρ and inserting it into (8) leads to

equation image

[17] For the isotropic case (Φny = Φnz ė Φn) it is clear that σα2 is independent of β, and equation image equals 4Φn(κ). For β = π/4, which corresponds to an angle of arrival in a plane diagonal to the main axes y and z of the anisotropy, both Φny and Φnz are equally weighted, and equation image becomes 2[Φny(κ) + Φnz(κ)]. Hence the same σα2 results as in an isotropic medium with the mean of the spectra Φny(κ) and Φnz(κ). On the other hand, the mean-square fluctuation of the angle of arrival σα2 determined along the yαy2 at β = 0) and z directions (σαz2 at β = π/2) depends considerably on anisotropy, leading to a difference of up to a factor 3. For β = 0, equation image = Φny(κ) + 3Φnz(κ), so that the main contribution is coming from Φnz, and vice versa for β = π/2, with equation image = Φnz(κ) + 3Φny(κ).

[18] The mean-square fluctuation of the angle of arrival is larger in the y direction than in the z direction for Φnz(κ) > Φny(κ) (at all κ: a sufficient but not necessary condition) and vice versa. This conclusion is in contradiction to our previous preliminary interpretation [Lüdi et al., 2000a], which is rectified herewith. With direct measurements of the two components σαy2 and σαz2 an anisotropy coefficient equation image can be defined:

equation image

From (9) and (10) it follows that equation image is limited between 1/3 and 3.

[19] The fact that σαz2 > σαy2 for Φny(κ) > Φnz(κ) (and vice versa) is intuitionally comprehensible: Suppose, for instance, that Φny(κ) > Φnz(κ); thus Φny(κ) extends to lower κ than Φnz (κ), which means that there are longer correlation lengths in the y direction. The phase of the electromagnetic wave is then correlated over longer distances parallel to y, which leads to a smaller phase difference and smaller angular deviations.

[20] This was also concluded by Banakh and Smalikho [1994], who investigated random beam displacements of a laser. They found that large anisotropic inhomogeneities in the refractive index, with scales much larger in the horizontal direction, lead to a dominant contribution in vertical deflections of the beam.

2.2. Temporal Frequency Spectra of Phase Difference, Angle of Arrival, and Normalized Power

[21] From the spectral analysis of both the normalized power Wp(f) and the angle of arrival Wα(f), as a function of the temporal frequency f, information about the characteristics of the refractive index spectrum Φn(κ) is obtained. The spectrum of the power gives only information about the inertial subrange of the spectrum Φn(κ), as the low-frequency part of the refractive index spectrum is filtered by the function Hr. Thus the input range of Φn(κ) has insignificant effects on Wp(f), since equation image On the contrary, the angle-of-arrival spectrum Wα(f) is noticeably affected by large inhomogeneities, as will be discussed below (Figure 2). Therefore also information about large inhomogeneity scales, in particular an estimation of the outer scale L0, is available.

[22] The derivation of temporal frequency spectra for plane and spherical wave propagation, within the Rytov approximation, is well described in earlier publications [e.g., Tatarskii, 1971; Clifford, 1971; Ishimaru, 1978]. In most investigations, Taylor's frozen-in hypothesis is assumed, meaning that the entire random ensemble of inhomogeneities is transported as a whole, ignoring any fluctuations in the transport velocity v and evolution of the inhomogeneities while they are moving [e.g., Tatarskii, 1971, p. 259]. Because the contributions of the wind speed υ|| along the propagation path are insignificant since equation image (fulfilled in most experimental situations), usually only the transverse motion v of inhomogeneities is considered [Tatarskii, 1971, p. 260].

[23] With the frozen-in approximation the time dependence of fluctuations can be obtained from the time-independent case simply by replacing ρ by ρ-vτ. Based on the latter, Tatarskii [1971] derived the time correlation function RδSp(τ) of the phase difference for a plane wave

equation image

For clarity we introduced the upper index p, indicating that (11) applies for a plane wave. For a spherical wave, however, the amount of shift depends also on the distance η of the inhomogeneity from the source, and the motion of an inhomogeneity is observed with a velocity v(L/η) [Clifford, 1971]. Therefore ρ must be replaced by ρ -vτL/η, and (11) must be adjusted accordingly:

equation image

with BSs the phase correlation function for a spherical wave and γ = η/L.

[24] Finally, the spectrum WδS(f) of the phase difference is given by

equation image

Substituting (12) into (13) involves an integration over τ, which, for an arbitrary orientation of the anisotropic inhomogenities to the transverse wind, seems to be analytically unsolvable. However, if we choose one axis of the y-z coordinate system along the wind v, the integration is achievable. For this we define v parallel to the y axis (vz = 0). Φny(κ) is thus parallel to v (and Φnz(κ) is perpendicular to the wind direction). This restraint is unproblematic, as several measurements show that one of the anisotropic inhomogeneity axes is usually oriented along the direction of the mean wind velocity [e.g., Pospelov, 1996].

[25] The integration over τ with vz = 0 yields (compare Appendix A)

equation image

[26] With Φny(κ) = Φnz(κ) (isotropy) and ρ parallel to vz = 0), (14) coincides with the expression given by Clifford [1971].

[27] For numerical computation of WδS(f) the following form is more convenient:

equation image

The mean-square fluctuations of the angle of arrival can be approximated by dividing the phase structure function DS(ρ = 2R) by (2Rk)2 [Tatarskii, 1971]. Applying this relation finally yields the angle-of-arrival spectrum:

equation image

where α is the measured angle of arrival in the x-ρ plane.

[28] In order to obtain numerical results the refractivity spectrum Φn(κ) must be known. A frequently used spectrum in atmospheric research is the Von Karman spectrum [Ishimaru, 1978]

equation image

which takes into account the finite size of the outer scale L0, where Cn2 is the refractive structure constant. Another occasionally used refractivity spectrum, which considers a finite L0, is the Greenwood-Tarazano model [e.g., Buscher et al., 1995]:

equation image

We will use both the Von Karman (VK) and the Greenwood-Tarazano (G-T) models throughout the subsequent text. In the scale region (1/κ) smaller than L0, the spectra follow the Kolmogorov spectrum Φn(κ) = 0.033Cn2κ−11/3. In the inhomogeneity regime with scale lengths larger than L0 (production range or input range), energy is introduced into turbulence by shear, convection, or both. The spectrum Φn(κ ≲ 1/L0) over the production range is basically constant for the VK model, and in the case of the G-T model the rise toward smaller κ is reduced, as Φn(κ ≲ 1/L0) is approximately proportional to κ−1.8. In this range the spectrum may depend on how turbulence is created, and therefore no universal description may be possible. However, the VK spectrum and the G-T spectrum serve as approximations for the production range.

2.2.1. Isotropic medium

[29] Figure 1 shows numerically computed spectra fWδS(f) for an isotropic refractivity spectrum Φn(κ) by inserting (17) and (18), respectively, into (14). We have chosen the fWδS(f) representation, as the differences at the lower frequencies f become more prominent. The computations were carried out for a spherical wave with parameters L0 = 20 m, L = 7.5 km, ρ = 0.27 m, and λ = 1.42 mm, relevant to our experiments (section 3). The solid curves were computed with the VK model, and the dashed curves were computed with the G-T model. The two curves next to the parallel symbol show the phase difference spectrum for the wind parallel to ρ, those labeled with the perpendicular symbol represent WδS(f) for a perpendicular wind direction, and finally, those marked by the diagonal symbol correspond to a diagonal situation (ρy = ρz). Obviously, the lower end of the phase difference spectrum depends markedly on the wind direction with respect to the direction of the measurements (ρ) whereas the influence of the shape of Φn(κ) (G-T or VK) is small: the VK model yielding slightly higher phase differences at low frequencies than the G-T model. At high frequencies the spectra WδS(f) become independent of the wind direction and follow a power law, WδS(f) ∝ f−8/3, i.e., fWδS(f) ∝ f−5/3.

Figure 1.

Modified phase difference spectrum fWδS(f), computed from (14) with an isotropic refractivity spectrum Φn(κ). The solid curves are calculated with Φn(κ) given by VK, and the dashed curves are calculated with that given by the G-T model. Curves with the parallel symbol represent fWδS(f) measured parallel, curves with the perpendicular symbol represent that measured perpendicular, and curves with the diagonal symbol represent that measured diagonal to the wind. The computations are for a spherical wave with L = 7.5 km, ρ = 0.27 m, λ = 1.42 mm, and L0 = 20 m.

2.2.2. Anisotropic medium

[30] The anisotropic characteristics of the medium are described by Φny(κ) and Φnz(κ). Because mainly large-scale inhomogeneities are anisotropic and as small scales, which are well below the production range, are believed to be isotropic, we characterize anisotropy solely by two different scales L0y and L0z:

equation image

This retains the assumption of local isotropy in the inertial subrange of turbulence (scales less than L0). The scales L0y and L0z must be interpreted as projections of the outer scale of turbulence on the y and z directions.

[31] Figure 2 shows computed spectra fWδS(f) with (19) inserted in (14) for different combinations of outer scale of 30 and 5 m, respectively. The parameters λ, L, and ρ are the same as in Figure 1. Anisotropy affects only the low-frequency part of WδS(f) as expected (Figure 2). It is further seen that irrespective of the orientation of the anisotropy, WδS(f) is always higher in the low-frequency part, when the wind is perpendicular to ρ. Another remarkable point is that if L0z dominates L0y, then ∫0WδSy(f) df is larger than ∫0WδSz(f) df, and vice versa. This corresponds to the results for the mean-square fluctuation of the angle of arrival for which σαy2 > σαz2 when L0z > L0y. The graphs in Figures 1 and 2 are also representative for angle-of-arrival spectra as they differ only by a numerical constant.

Figure 2.

Modified phase difference spectrum fWδS(f) with an anisotropic refractivity spectrum from (14) (by VK), computed for a spherical wave (with L = 7.5 km, ρ = 0.27 m, and λ = 1.42 mm). The curves show WδS(f) for parallel and perpendicular wind to ρ with variable L0. The parallel and perpendicular signs mean parallel (y) and perpendicular (z) to the wind.

[32] Let us now consider the spectrum of the received power fluctuations. Because the normalized variance σp2 of the power is found to be basically unaffected by anisotropy, it is evident that the substitution Φn,iso(κ) = 1/2 [Φny(κ) + Φnz(κ)] applies also for the spectrum. The temporal frequency spectrum of the normalized power Wp(f) is thus given by Tatarskii [1971, p. 277]

equation image

[33] Finally, we must mention here that the formulas for the spectra do not take into account aperture averaging. It causes a suppression of scales which are small compared to the aperture radius, whereas the contributions from large-scale inhomogeneities in the low-frequency part of the spectra are unaffected. As we are not considering here the very high frequency part (dissipation range), we neglect the influence of the finite antenna size. However, for completeness we indicate that the given formulas for the frequency spectra are easily supplemented by multiplying by the appropriate averaging factor given by, for example, Tatarskii [1971, p. 277].

[34] Expressions for the mean-square angle-of-arrival fluctuations σα2(β) and the variance of the normalized power σp2 as well as the spectra Wα(f) and Wp(f) are derived within the model framework, proposed by Lee and Harp [1969]. It was found that anisotropy has a notable influence on the angle of arrival, while power fluctuations are basically unaffected. In section 3 the derived formulas will be applied to line-of-sight propagation measurements in order to estimate the correlation lengths of the atmospheric refractivity in transverse directions to the propagation path, thus allowing us to determine anisotropy.

3. Experiments and Their Interpretation

3.1. Test Ranges and Measurement Equipment

[35] In order to test the influence of weather conditions on turbulence, two test ranges with considerably different climates have been used. The test range at Bern (Switzerland) is slightly tilted over a length of 7.5 km, crossing a mixture of urban, suburban, rural, and forestal terrain [Lüdi and Magun, 1999]. The transmitter is placed on a telecommunication tower 50 m above ground at a height of 990 m above sea level (asl). The receiving system was at 580 m asl, 10 m above ground. Most parts of the subjacent ground were slightly covered by snow (∼3–10 cm). The test range at El Leoncito (Argentina) spans a length of 11 km at heights from 2960 (transmitter) to 2550 asl (receiver) and crosses desert-like “pampa” landscape. Both test ranges are some 10 to 300 m above ground. Next to the transmitter (half-power beam width ≈1°) and the receiver, weather stations measured meteorological data.

[36] The Solar Submillimeter Telescope (SST) [e.g., Kaufmann et al., 1997] was used for measuring the incident power and the angle of arrival in azimuth and elevation. SST is a multibeam and multi frequency system with four antenna beams at 212 GHz and two at 405 GHz. The main reflector has a diameter of 1.5 m designed to provide half-power beam widths of ∼4 and ∼2 arc min at 212 and 405 GHz, respectively. Three beams at 212 GHz are intersecting near their half-power values for the angle-of-arrival determination.

3.2. Principle of Measurement

[37] To determine the angle of arrival, we use the method that has been successfully applied for the precise location of solar bursts at 48 GHz with the multibeam system at Itapetinga, Brazil [e.g., Herrmann et al., 1992]. For our measurement at 212 GHz the transmitting source was positioned on the central axis of the three-beam cluster of SST. Because we detect the incident signal with each of the three beams near their half-power values, the effective aperture radius, which is relevant for analyzing the data, is much smaller than the geometrical aperture. The radius (R = 0.27 m) of the effective antenna aperture area was derived from the measured directivity in the source direction [Lüdi et al., 2000b; Lüdi and Magun, 1999]. The angle of arrival that corresponds to the apparent angular position of the transmitter (ϑss) can then be derived from the simultaneously measured receiver output voltages that are proportional to the power beam patterns Piss) of the individual antenna beams (i) pointing into slightly different directions. The angle of arrival can be reconstructed with the help of the measured far-field beam patterns [Lüdi and Magun, 1999; Lüdi et al., 2000b] to an accuracy of better than 1 arc sec at each instant of time (every 2 ms).

[38] The received power fluctuates because of focusing/defocusing effects and diffractional spreading introduced by atmospheric inhomogeneities as well as by changes of the angle of arrival. Focusing effects and diffractional spreading cause equal power fluctuations in all three beams, whereas angle of arrival variations change the directivity and thus the received signal. The measurement of the angle allows us to determine the power fluctuations in the incoming wave, unaffected by angle-of-arrival effects. It was found that the reconstructed power, used for further analysis, typically fluctuates by a factor of ∼2 less than the power in single beams. For the statistical analysis presented here, measurement periods of 10 min were used, as the assumption of stationarity is valid only within the limited period of time of a few minutes. On the other hand, the period must be long enough in order to take into account all turbulent scales. A reasonable compromise of these two restraints is approximately 10 min [Vanhoenacker et al., 1991].

[39] Before analyzing the normalized power and the angle-of-arrival time series, the raw data were processed in two ways. From the 10-min samples we eliminated linear drifts, which may be introduced by the instrument and by a slight nonstationarity of the atmosphere due to weather changes. These drifts were usually very small or vanishing. In a second step, we filtered out harmonic signals which appeared in the power spectrum as sharp spikes, most likely caused by instrumental effects. Such resonances were small and contained very little power. The statistical convergence of the time series was thereafter checked by comparing the variances of power, azimuth, and elevation over different time periods from 1 to 10 min. Only those 10-min samples were considered for analysis for which the variances changed by less than 5% for averaging times above 7 min. The preprocessed data were then used to determine variances as well as temporal frequency spectra, which then were compared with the theoretical expressions of section 2 to estimate characteristic parameters of the atmosphere. The necessary theoretical quantities (Wα(f), Wp(f), σp2, and σα2(β)) for comparison with the measurements were computed numerically with the assumption of spherical wave propagation because the transmitter is located within the turbulent medium and most of the random medium between the transmitter and the receiver is located in the far zone of the transmitter [Clifford et al., 1971; Clifford, 1971; Ishimaru, 1978]. This was also confirmed by the fact that the agreement between measurements and model was better than that for the plane wave assumption.

[40] For the representation of our results we will assume further on that y and z directions point into horizontal and vertical directions, respectively, and that the wind direction is horizontal. Therefore the indices z and y are replaced by v and h, respectively.

3.3. Temporal Frequency Spectra

[41] Typical measured frequency spectra are presented in Figure 3. The left panel shows the measured angle-of-arrival spectra (fWα(f)) for both azimuth (thin solid line) and elevation (thin dashed line). The right panel displays the spectrum fWp(f) of the normalized measured power. All spectra are fitted with theoretical curves derived with (16) and (20), respectively (assuming VK). For the fit of the measured angle-of-arrival spectrum the wind direction must be known. As the horizontal wind component dominates the vertical one by typically a factor of more than 10, the spectrum of the angle of arrival in horizontal direction (azimuth) is fitted with the assumption of ρ parallel to the wind. Similarly, the measured spectrum in vertical direction (elevation) is fitted with ρ perpendicular to the wind. In Figure 3 the computed curves in azimuth and elevation (thick solid and thick dashed curves) fit the measured spectra reasonably well with outer scales of L0h = 80 m and L0v = 4 m in horizontal and vertical directions,respectively.

Figure 3.

Typical spectra fWα(f) for the (left) angle of arrival and (right) normalized power fWp(f). The measured spectra (19 January 1999, 1540–1550 UT) are fitted by theoretical curves (VK model). The measured horizontal (solid curves) and vertical (dashed curves) angle-of-arrival spectra are fitted with L0h = 80 m and L0v = 4 m.

[42] In the high-frequency part of the spectra, which reflects the inertial subrange of Φn(κ), the two angle-of-arrival spectra coincide and follow a power law with an index of −5/3, also in agreement with theory (at the highest frequencies the spectra have a smaller slope because of noise contamination). Similarly well fitted is also fWp(f) by ∝f−5/3. In order to substantiate the agreement between the model and the measured high-frequency index, a statistical evaluation was made. As no statistically significant difference was found of the high-frequency behavior between the two locations of measurements (Bern or El Leoncito) and the different quantities (power, azimuth, or elevation), all data were used to obtain a larger statistical database (393 samples). The spectra Wα,p(f) in log-log representation above the production range (∼1 Hz) and below the part with noise contamination (∼10 Hz) were fitted with a straight line (least squares fit).

[43] From the histogram of Figure 4 it is seen that the power law index clusters around a value of −2.74, which is in good agreement with the theoretical value of −8/3 (−2.67). It is also obvious that the indices are normally distributed with a standard deviation of 0.52. This considerable width of the distribution means that many of the measured high-frequency power law indices deviate notably from the theoretical value of −8/3, which was also noted by other authors. Rao et al. [1999] obtained mainly values which are smaller than −8/3 and found power law index variations between approximately −7 and −2. On the other hand, Buscher et al. [1995] measured rather larger indices (mean value of −2.55) with variations between −2 and −3. As argued by Rao et al. [1999], indices below the theoretical value may be explained by an increased influence of the dissipation range (large inner scale), which causes a stronger decrease of the spectrum at high frequencies. Indices that are larger than −8/3 may be the outcome of measurements during strong atmospheric attenuation, which resulted in a decreased sensitivity and hence in increased noise at high frequencies. It is also possible that these spectra are measured during turbulence that is not fully developed, where a well-developed inertial subrange was not present. Other reasons for the index deviations from −8/3 may be due to a real departure from Kolmogorov statistics in the turbulence [Nightingale and Buscher, 1991]. It is important to mention that in our measurements, in spite of the considerable variability, the high-frequency indices are very well correlated for the three simultaneously measured spectra and the differences between the spectra for azimuth and elevation are insignificant at higher f (generally, different by less than a factor of 1.5). The correlation coefficients between azimuth and power and between elevation and power are both 0.7, and between azimuth and elevation it is 0.9. From this, it is seen that even though Φn(κ) at low κ may be unequal in different directions (because of an anisotropic L0), Φn(κ) at high κ seems to be basically equal in all directions, giving evidence for the validity of the assumption of a locally isotropic refractivity spectrum.

Figure 4.

Statistical distribution of the measured high-frequency index in the inertial subrange of all spectra (total number of samples is 393). The distribution is well fitted by a normal probability density function with a mean value of −2.74 (dashed line) and a standard deviation of 0.52. The theoretically expected value is −2.67 (dotted line) and thus in good agreement.

[44] The frequency spectra also serve to estimate the transverse wind speed, as all spectra Wp(f) and Wα(f) are shifted in frequency by the transverse drift velocity v. Most suitable for this purpose is Wp(f). The peak frequency f0 (compare Figure 3, right panel) has a theoretical dependence of equation image (spherical wave propagation). From the example of Figure 3 with f0 ≃ 0.34 Hz, a transverse wind speed of approximately 1 m/s is estimated. This value, representing an average over the whole test range, is in agreement with the in situ measured transverse wind speeds of ∼1.5 m/s and ∼0.2 m/s next to the transmitter and receiver, respectively. For plane wave propagation, equation image and hence the estimated v, would be twice as large, what is usually not between the two in situ values. Therefore the assumption of spherical wave propagation in our experiment is justified.

3.4. Outer Scale of Turbulence L0

3.4.1. Methods to estimate the outer scale

[45] In section 3.3 we showed an example of a measured angle-of-arrival spectrum with fitted theoretical curves. As already mentioned, from these fits an estimation of the outer scale of turbulence is feasible by varying L0h and L0v until the theoretical curves fit the measured spectra best. The fitting procedure is pragmatic: We computed many curves with different outer scales L0h and L0v, which were then fitted to the measured spectra (least squares). Because the angle-of-arrival spectrum in azimuth is primarily governed by L0v, and since L0h has a dominant influence on elevation, the spectra could be fitted separately with only one free parameter L0, which represents the outer scale orthogonal to the measuring direction.The applicability of the described procedure is limited: As angle-of-arrival spectra with L0 ≅ 100 m and L0 → ∞ do not differ much, this technique is restricted to cases where L0 ≤ 100 m, with its accuracy decreasing for increasing outer scales. Another method derives the outer scale of turbulence from the variance of the angle of arrival σα2, where we take advantage of the fact that σα2 depends significantly on the outer scale of turbulence (Figure 5).The variance of the normalized power σp2 also depends on L0, but basically only if equation image.

Figure 5.

Relative comparison of the theoretical L0 dependence on the mean-square fluctuations, σα2 and σp2, of the angle of arrival and of the normalized power, respectively, computed with an isotropic Φn(κ) given by VK and G-T. L0 has a significant influence on σα2, whereas σp2 is only affected if equation image.

[46] The determination of the two scales L0h and L0v and Cn2 can be done by comparing the theoretical σαh2, σαv2, and σp2 (equations (9) and (5)) with the corresponding measured variances. The refractivity structure constant Cn2 will be investigated in a separate paper.

[47] Similar to the outer-scale estimation from spectral fits, the estimation from variances is also limited, as the dependence of σα2 on L0 reduces with increasing L0: Variance σα2 for L0 = 100 m and L0 = 1000 m (Figure 5) differs merely by 10%, so that also here only outer scales below ∼100 m can be reliably determined. The two methods for estimating L0 have their advantages and disadvantages. The variance method needs very accurate measurements in both the angle of arrival and the normalized power. In this regard, an important criterion in our case is that we have to know the antenna diagrams very accurately, which is ensured here [Lüdi and Magun, 1999; Lüdi et al., 2000b]. Additionally, for the determination of the normalized power fluctuations a stable transmitter power is also essential in order to avoid non atmospheric fluctuations in the analyzed signal (≲2%). A further disadvantage of this method is that it can only be applied if Cn2 is basically constant along the propagation path, as the contributions from turbulence to σα2 are more weighted close to the receiver.

[48] In the spectral method only the general shape of Wα(f) is used, and systematic errors (e.g., systematic offsets due to improper calibration of the power) are considerably less critical. On the other hand, a disadvantage of the spectral method is that the wind direction must be known and then, depending on the wind, fits in different directions are carried out with different computed curves. A further penalty is the fact that the spectral fit uses Taylor's hypothesis; namely, it assumes that the wind is constant during the measuring period and along the path. Lee and Harp [1969] showed that nonuniform winds (different directions or different speeds) along the path lead to different time-lagged correlation functions and thus to different frequency spectra. Finally, it must be noted that the theoretical formalism for the temporal frequency spectra is less accurate than that for the variances.

[49] From a theoretical standpoint we may consider the variance method to be more accurate and easier to use for estimation of L0. It has, however, the disadvantage that the normalized power is also used and is therefore affected by the stability of the transmitter. Also, the necessity of the assumption that Cn2 is basically constant along the propagation path is a significant drawback. Hence both methods have their advantages and disadvantages, but, overall, they give close results (see section 3.4.2).

3.4.2. Results

[50] A comparison of the estimated outer scales L0 in horizontal and vertical directions with the two different methods is shown in Figure 6. The data were measured at Bern, and the VK model was used. A similar agreement was also found for El Leoncito and for the G-T model as well.

Figure 6.

Comparison between the outer scale estimated by the variance method L0σ and that measured by the spectral fit method L0W.

[51] The outer scales determined from the variance and spectral fits are denoted L0σ and L0W, respectively. They correlate reasonably well and are of the same order. This agreement is not self-evident, as for the variance method a constant Cn2 along the propagation path should be assumed. Because the propagation paths at both locations were only nearly horizontal, the latter could not be taken for granted.A significantly enhanced Cn2 close to the receiver (lower altitude) would lead to overestimated sizes of L0. Since the agreement between the two methods is reasonably good, it seems that Cn2 varied insignificantly along the path.

[52] For both outer-scale estimations (by spectral fits and by variance adjustments) it is found that L0 determined with the G-T model is slightly larger (by less than a factor of 2) than that with the VK model, confirming the results of Buscher et al. [1995] and Ziad et al. [1994, 2000]. Because we do not know which one of the two models is a better approximation and since the differences of the outer scales from the two models are not substantial, we take the mean values of L0 (VK and G-T) for further analysis.

[53] Figure 6 shows that we often measured outer scales of only a few meters, even in the horizontal direction, which may be thought to be rather small for the boundary layer in the first hundreds of meters above ground. However, these small values are found with two different methods, and both of them are more accurate for small L0. It must also be remarked that errors in the angle-of-arrival spectrum, due to instrumental drifts or weather changes, would increase the low-frequency part of the spectrum, which would lead to an overestimate of L0. In these terms, the estimated L0 could rather be interpreted as being an upper limit of the real outer scale.

[54] For further discussion, we consider only the estimates of L0 derived from the variance method which were in the same order of magnitude as L0 determined from spectral fits. As we are also interested in comparing L0 for vertical and horizontal directions, the variance method is preferable, as both directions are treated the same way, unlike in the spectral fit procedure (see discussion in section 3.4.1). We selected 68 and 64 ten-minute samples, obtained at Bern and El Leoncito, respectively, during clear-air conditions and determined the outer scales L0 in horizontal and vertical directions (Table 1).

Table 1. Distribution and Mean Values of the Outer Scales L0v and L0h in Vertical and Horizontal Directionsa
  0.5–2 m2–10 m10–100 m>100 mequation imageL0Median L0
  • a.

    equation image specifies the mean value, and 〈L0〉 is deduced from logarithmic means of the outer scales. equation image and 〈L0〉 are computed by taking into account only the estimates for L0 < 100 m. The median L0 are determined from all data.

BernL0h29%24%29%18%14 m5 m8 m
BernL0v64%30%3%3%1.9 m1.1 m1.3 m
El LeoncitoL0h7%54%28%11%16 m8 m10 m
El LeoncitoL0v25%44%19%12%8 m4 m4.5 m

[55] The measurements at Bern yielded a wide range of outer scale sizes, ranging from less than 1 m to more than 100 m. Vertical scales were almost always significantly smaller than horizontal ones. The wide spread of L0 values is mainly due to the highly changeable weather and can be explained by meteorological conditions (see section 3.5). The size of L0 is of particular interest for astronomical observations because it limits the spatial resolution: A small outer scale is favored [e.g., Nightingale and Buscher, 1991; Ziad et al., 1994]. At the astronomical site, El Leoncito, L0 was found to be less variable, mainly because the weather conditions were much more constant than those at Bern. Obviously, the outer scales at El Leoncito are clustered between 1 and 10 m, and we can roughly confirm that L0 is log normally distributed, as found by Martin et al. [1998]. The mean value of the logarithmic distribution of L0h is 0.9, corresponding to 〈L0h〉 ≅ 8 m, close to the median value of ∼10 m. The mean value of the horizontal outer scale equation image of 16 m is noticeably larger. (The determination of 〈L0〉 and equation image is computed by taking into account only the estimates for L0 < 100 m. The median L0 are determined from all data.) Similarly, in the vertical direction, equation image and the median of L0v is 4.5 m. Also, at El Leoncito the L0 distributions show larger outer scales in the horizontal than in the vertical direction. However, it is obvious from Table 1 that the differences between horizontal and vertical outer scales are considerably smaller than those at Bern. This is due to the fact that the ratio of the horizontal to vertical length scale depends on the stability of the atmosphere, as will be discussed in section 3.5.

3.4.3. Discussion and comparison with other investigations

[56] Several measurements of outer scales were carried out in the past using different methods (a more complete review is given by, for example, Ziad et al. [1994, 2000]. It is generally believed that the outer scale increases for an increased height above ground within the boundary layer [e.g., Krasnenko and Shamanaeva, 1998]. From horizontal line-of-sight propagation measurements, several L0 estimates are available at different heights for comparison. Most of the horizontal line-of-sight phase front measurements (phase difference or angle of arrival) were carried out in the horizontal plane and hence reveal inhomogeneity sizes mainly in the vertical direction. Close to the ground, in the surface layer, it is expected that the outer scale is comparable to the height above ground level [Muschinkski and Roth, 1993], which was also observed by several authors: At a height of 1.6 m, Clifford et al. [1971] measured optical phase variations and temperature spectra and determined L0 ∼ 1–2 m. R. J. Lataitis et al. (unpublished manuscript, 1990) estimated the outer scale from one angle-of-arrival measurement in the millimeter-wavelength region (142 GHz) at 3.7 m above ground as being approximately 6 m. Some experiments from similar ground levels, however, are also reported that obtained significantly smaller L0: Lutomirski and Yura [1969] discuss measured phase structure functions, carried out with a laser at approximately 1 m above ground, measuring L0 ≅ 10–50 cm. Even smaller are the estimations by Gaskill [1969] from measurements with an interferometric laser technique along horizontal paths at heights of ∼4–9 m above ground. He estimated correlation lengths of the refractive index varying over approximately 1 order of magnitude, between 10 and 100 mm, only. The fact that L0 is not constant with time was also observed by the following authors: Bester et al. [1992] report measurements of laser fringe motion power spectra with an interferometer, carried out near the ground, indicating outer scales between 5 and 20 m. Some 50 m above ground, and thus at heights similar to those at which our measurements were carried out, Cole et al. [1978] obtained L0 varying between 1 and 50 m: values which are compatible with our L0 distribution.

[57] Because the outer scale is of particular interest for astronomers, several investigations about L0 were also undertaken in that community, by observing a star in a near-vertical direction. In this case, outer scales must be interpreted in terms of an average over all atmospheric layers. Because vertical propagation path measurements are primary affected by horizontal correlation lengths of the turbulence, it can be assumed that the estimated outer scales are larger.

[58] Nightingale and Buscher [1991] and Buscher et al. [1995] measured phase difference spectra with optical interferometers and noted many characteristic features that would be expected from Kolmogorov turbulence with an infinite outer scale. However, other data showed evidence for an outer scale of turbulence as short as 2 m (La Palma [Nightingale and Buscher, 1991]) and approximately 30 m (Mount Wilson [Buscher et al., 1995]). As a possible explanation for the observed L0 with large sizes, Buscher et al. [1995] argued that by overlaying several spectra with a finite outer scale, but with different winds, the average of these spectra can look like a spectrum with L0 → ∞. This statement also supports the fact that, as we mentioned in section 3.4.2, estimated L0 rather represent an upper limit of the real outer scale. Even though our test ranges were nearly horizontal (2° and 3° in elevation), a similar assumption could also be applied in our case, as noticeable differences of wind speeds between receiver and transmitter were sometimes measured. The large variability of the outer scale was also observed by the GSM team (University of Nice) [e.g., Agabi et al., 1995; Martin et al., 1998; Ziad et al., 1994, 2000], who measured L0 at different astronomical sites. They report about temporal changes of the outer scales between ∼1 and 300 m, mentioning that L0 may change over more than 1 order of magnitude in less than a few minutes.

[59] Thus our values obtained for L0 in both directions, vertical and horizontal, fit well into the large range of those presented in literature. Additionally, we confirm the large variability of outer scales with time. As generally expected, the correlation length in the horizontal direction is usually larger than that in the vertical direction. The scales at El Leoncito in the vertical and horizontal directions are not that much different and are in both directions quite small. This may be interesting for that astronomical site, as a small L0 improves seeing conditions.

3.5. Anisotropy and Stability of the Atmosphere

[60] As already mentioned in 1969 by Lee and Harp [1969], angle-of-arrival measurements in two dimensions allow the determination of anisotropic inhomogeneities. It is generally assumed and also seen in section 3.4 that horizontal length scales are usually larger than vertical ones. This anisotropy at large scales, where most of the energy and energy input are allocated, depends on the stability of the atmosphere and the height of measurement [Panofsky and Dutton, 1984, p. 217]. The nocturnal boundary layer is usually stable, and the atmosphere is stratified. During daytime, if a convective mixed layer in the atmosphere is present, the stratification is significantly reduced, and it cannot be taken for granted any longer that the horizontal length scale dominates over the vertical one. The spatial correlation and the mixing processes are mainly governed by the wind shear and the temperature gradient, which both influence the anisotropy and the stability of the atmosphere. The latter can be determined from the bulk Richardson number Rb using meteorological measurements at two different heights. Rb is an approximation of the Richardson number Ri, whereby wind and temperature gradients are approximated by their respective differences at discrete heights [e.g., Stull, 1988, p. 177]:

equation image

where g is the gravity acceleration and Δθ, Δvx, and Δvy are the differences of the potential temperature and of the horizontal wind components at heights z1 and z2, with Δ z = z1z2. At both locations (Bern and El Leoncito) the meteorological data were measured next to the transmitter and receiver. Our estimations of height gradients are based on the assumption that these measurements are representative for the air masses at this level. Because of the large separation of the weather stations (7.5 and 11 km, respectively) this assumption is certainly not always valid, and a more qualitative than quantitative analysis of the atmospheric conditions is possible. Also, the large vertical separation (410 m) of the weather stations makes it evident that particularly the estimated Rb must be considered qualitatively, as any small-scale gradients are smoothed out. Because the atmospheric stability depends considerably on the location and climate, the measurements at Bern and El Leoncito are discussed separately.

3.5.1. Bern

[61] The measurement campaign in Bern took place during winter (13–25 January 1999), and its results are summarized in Figure 7. For each day the average measured values of the meteorological parameters and outer scales are shown: Figure 7e presents the measured anisotropy coefficient equation image (equation (10)), Figure 7d presents the estimated outer scales, Figures 7a and 7b present the measured wind and temperature differences, and Figure 7c presents the bulk Richardson number Rb according to (21). The atmosphere was almost always stably stratified, even though the measurements were carried out during daytime. On 13–14 January the potential temperature next to the transmitter was always higher than that next to the receiver. During 19–24 January, even the absolute temperature was higher at larger heights, indicating a strong inversion (Figure 7). In a pronounced thermally stratified atmosphere it is intuitionally understandable that the spatial correlation length is larger along the atmospheric layers than perpendicular to them, i.e., L0h > L0v, and hence it is clear that mainly equation image was measured. However, equation image was never less than 0.3, as required by the theoretical model.

Figure 7.

Daily mean values of Bern in January 1999: the measured (a) wind and (b) temperature differences |Δv| and ΔT between the different heights of the weather stations, (c) the bulk Richardson number Rb, (d) the estimated outer scales in the horizontal and vertical directions L0h and L0v, and (e) the anisotropy coefficient equation image.

[62] In Figure 7 it is seen that we observed a high degree of anisotropy when Rb was large (very stable atmosphere) whereas equation image approached 1 for small positive Rb (around approximately neutral conditions). During two days, from 13 to 14 January, the atmosphere was nearly neutral, and the ratio L0h/L0v was moderate, varying from ∼0.9 to 30 with an average of 5. Within the period of 19–22 January a strong inversion was present, leading to a pronounced stratification of the atmosphere. In addition, these days were characterized by weak wind (1–2 m/s). Hence the anisotropy was accordingly pronounced, and L0h/L0v occasionally exceeded 100, with an average between 15 and 30. L0h was, most of the time, between 10 and 100 m and sometimes above 100 m, while L0v was mainly well below 10 m with an average around ∼0.5–1 m. A comparably strong thermal inversion was also present on 24 January. However, because of the increased wind speed, Rb was smaller, which explains the reduced anisotropy (equation image compare Figure 7), and the ratio L0h/L0v was always below 10 and occasionally close to 1. On 25 January the inversion was significantly relaxed, the wind was strong, and hence the anisotropy small.

[63] Besides the general correlation between stability and anisotropy, we can also confirm that if the turbulent energy is mainly generated by mechanical production, the eddies are small [Blackadar, 1998, p. 123]. The latter is seen, as the outer scales tend to decrease for increased wind speed, and vice versa. This is much more apparent for L0h than it is for L0v. The vertical outer scale L0v did not increase during 20–22 January probably because of the strong thermal stratification, limiting L0v to less than a few meters. Very small outer scales in the vertical direction (meters or even less) are quite typical for atmospheric inversion conditions. The latter was also observed by in situ measurements in the stably stratified atmosphere [e.g., Dalaudier and Sidi, 1994; Schumann et al., 1995; Muschinski and Wode, 1998], which also support the existence of substantial anisotropies in the meter and submeter scales of velocity, temperature, and humidity fields in the quiet, stably stratified atmosphere, similar to our findings.

3.5.2. El Leoncito

[64] During the period of measurements (12–17 September 1999) the weather conditions were much more constant at El Leoncito than at Bern, also indicated by much smaller variations of L0, as discussed in section 3.4.2 (Table 1). This is not surprising as the surrounding area is a flat and uniform desert-like “pampa” landscape where humidity is usually very low (during the days of measurements, relative humidity was 10–20%). The wind was mainly weak, and all days were almost cloudless, so that most of the generated turbulence was due to buoyancy. From the temperature differences of the two meteorological stations it was evident that the supply of solar energy, early in the morning, transformed the stable boundary layer with thermal inversion into a convective layer, also called the mixed layer. Therefore, during midday, equation image often exceeded 1, and L0v dominated L0h. We frequently observed that equation image most likely became larger than 1, right after sunrise, until approximately the temperature maximum was reached (usually, 1–2 hours after local noon), followed by a decrease below 1.

[65] We focus on the two days of 16–17 September. These two days are characterized by extremely light wind, less than 0.5 m/s at 10 m height. These days were also quite vernal, with warm weather (up to 22°C, the warmest days during the measuring period). Such atmospheres are governed only by buoyancy, so that the temperature gradient is sufficient to characterize the atmospheric stability.

[66] Figure 8 presents the measured anisotropy coefficient equation image and the estimated ratio L0v/L0h as a function of ΔTz, with ΔT being the temperature difference measured at the two different heights separated by Δz. The correlation is obviously very good. Particularly, equation image follows ΔTz steadily but stays within the limits of 1/3 to 3, as predicted by the theory. The correlation between L0v/L0h and ΔTz shows more scatter; however, the correlation is still reasonably good. It should be remembered that equation image is directly measured and depends only on the angle of arrival. In contrast, the estimated ratio L0v/L0h depends on the theoretical model used, which may not reflect the real world. In addition, the estimation of L0 also needs the variance of the normalized power, which may also be slightly corrupted by small errors, so that equation image must be judged to be more accurate than L0v/L0h. It is seen that equation image occurs around neutral atmospheric conditions, approximately at −0.01 ≲ ΔTz ≲ 0 (ΔTz = −0.0098 K/m corresponds to zero difference of the potential temperature). In the range ΔTz ≲ −0.01, where the atmosphere is convective, the vertical scales are larger than the horizontal ones (Figure 8).

Figure 8.

equation image versus ΔTz at El Leoncito. The temperature difference ΔT was measured at the two different heights z1 and z2 next to the transmitter and receiver. Measurements are shown where the wind was very calm, and thus the only main driving force of turbulence was buoyancy (i.e., ΔT). With increasing buoyancy the highly elongated inhomogeneities in the horizontal direction become more isotropic in neutral conditions (−0.01 ≲ ΔTz ≲ 0) up to even larger vertical than horizontal length scales (in the range ΔTz ≲ −0.01).

3.5.3. Comparison with other studies

[67] Even though height gradients and thus atmospheric stability could be determined rather qualitatively, it is obvious that the size of the outer scale and its anisotropy depend on meteorological conditions. Our results are in good agreement with those obtained by other authors who made in situ measurements. Panofsky and Dutton [1984, p. 217] estimated the ratio of the horizontal to vertical integral scale (similar to the outer scale) of wind fluctuations and showed that for an increasing positive Richardson number Ri, much larger scales can be expected in the horizontal direction. For neutral conditions (Ri ∼ 0), horizontal and vertical scales are similar, and for unstable air (Ri < 0), eddies are vertically elongated.

[68] Some investigators analyzed variances of turbulence-induced fluctuating atmospheric parameters. As they depend on the outer scale, an estimation of the anisotropy is also assessable, if variances are measured simultaneously in different directions. Deardorff [1970] and Caughey and Palmer [1979] measured the variance of the three velocity components which must be equal if isotropy prevails. If the variance of one component is larger, then the outer scale in this direction is larger. In the middle of the unstable boundary layer (0.3 ≲ z/zi ≲ 0.7, where zi is the height of the boundary layer top), usually larger RMS vertical wind velocity fluctuations (σw2) are measured than are measured in the horizontal (σu,v2) direction [Deardorff, 1970; Caughey and Palmer, 1979], indicating an increased outer scale in the vertical direction during convective conditions.

[69] A few investigations were carried out also considering scalar quantities such as temperature, which are better comparable to our measurements. Antonia and Chambers [1978] measured the three components of the fluctuating temperature gradient vector in a weakly unstable atmosphere and found a slightly larger RMS value in the vertical direction, indicating a larger correlation length in this direction. A similar conclusion can also be drawn by Moulsley et al. [1981], who found indications for vertically oriented temperature inhomogeneities in convective atmospheric layers.

4. Conclusions

[70] A simple theory to investigate large-scale anisotropy effects is described and applied to millimeter-wave line-of-sight propagation experiments. We demonstrate the applicability of two-dimensional angle-of-arrival measurements for the characterization of anisotropic inhomogenities and reveal their dependence on atmospheric conditions and stability. The dimension of L0 in different directions is mainly important for the characterization of atmospheric diffusion processes as well as for estimating astronomical seeing conditions.

[71] Semianalytical expressions for the mean-square angle-of-arrival fluctuations σα2 and the variance of the normalized power σp2, as well as the spectra Wα(f) and Wp(f) for monochromatic spherical wave propagation with arbitrary wind directions, are derived within the model framework, proposed by Lee and Harp [1969]. From this, it is seen that anisotropy has a significant influence on the angle of arrival (up to a factor 3 for σα2) while power fluctuations are basically unaffected.

[72] The measurements at 212 GHz were made with the new Solar Submillimeter Telescope, which features a multibeam antenna to allow the determination of the angle of arrival in two dimensions with an acuracy of better than 1 arc sec at each instant of time (2 ms). Our measuring setup allows, additionally, the determination of the power fluctuations irrespective of angle-of-arrival fluctuations.

[73] Good agreement between measured and theoretical spectra of the power and the angle of arrival in both horizontal and vertical directions was found. The comparison of the experimental and theoretical variances and spectra yielded path-averaged outer scales L0 in the vertical and horizontal directions over the lowest few hundred meters above ground. The finite outer scale L0 for the VK model is slightly larger than that for the G-T model (by less than a factor of 2), confirming the results of Buscher et al. [1995] and Ziad et al. [1994, 2000]. The investigated layers are most important for millimeter-wave observations as moisture from this part gives the main contribution to signal fluctuations.

[74] The outer scale L0 varies strongly with time as it depends considerably on meteorological conditions. The latter varied considerably at Bern, so that a wide range of L0 from less than 1 m to more than 100 m was measured. Small outer scales were generally measured for large wind speeds while L0 increased for weaker wind. At El Leoncito the weather conditions were much more constant, which therefore resulted in smaller variations of L0, clustering between ∼8 and 10 and between ∼4 and 5 m in the horizontal and vertical directions, respectively. These scales are quite small and hence advantageous for that astronomical site, as large scales limit spatial resolution of ground-based observations.

[75] The anisotropy depends on meteorological conditions: The estimated large-scale inhomogeneities were usually larger in the horizontal than in the vertical direction, as generally expected and intuitively understandable. If the atmosphere is strongly stratified and very stable, the horizontal scale can even exceed the vertical one by more than 2 orders of magnitude. On the other hand, if the atmosphere is unstable and convective, the dominance of L0 in the horizontal direction cannot be taken for granted any longer and vertical length scales are larger than horizontal ones, which is in good agreement with in situ investigations [e.g., Panofsky and Dutton, 1984, p. 217]. The observed anisotropy, however, seems hardly to affect the inertial subrange of turbulence, as the frequency spectra Wα(f) of the angle of arrival in different directions coincide at high frequency, giving evidence for the validity of the assumption of a locally isotropic refractivity spectrum.

[76] From our investigation it can be concluded that the described technique and analysis have great potential in the remote characterization of outer scales in two dimensions. Further investigation during other seasons than winter is required to characterize the anisotropy and the size of L0 for atmospheres with enhanced convective conditions. Also, in future this kind of measurement should be joined by, for example, radiosonde measurements for profiling the atmosphere, allowing a more accurate estimation of atmospheric conditions and stability. In principle, similar investigations are conceivable with other instruments where a determination of the angle of arrival (or phase difference) in two dimensions with arc second accuracy is possible.

Appendix A

[77] In order to obtain (14), two integrals equation image have to be evaluated:

equation image

Using Graf's addition theorem for Bessel functions [Abramowitz and Stegun, 1970, p. 363, equation 9.1.79] and noting that Jk = (−1)kJk yields

equation image

Using Neumann's addition formula [Abramowitz and Stegun, 1970, p. 363, equation 9.1.75] and rearranging the terms leads to

equation image

Substitution of (24) into (23) and the latter into (22) and integrating yields

equation image

where T2k(z) = cos [2k arccos (z)]. With equation image [Abramowitz and Stegun, 1970, p. 361, equation 9.1.42], (25) finally becomes

equation image

[78] The second integral equation image is given by

equation image

Applying again Graf's addition formula and noting that Jk = (−1)kJk, the expression equation image of (27) becomes

equation image

[79] Using Neumann's addition formula, rearranging (analog to (24)), substituting into (27), and integrating yields

equation image

and rearranging equation image yields

equation image

It is seen that equation image is proportional to equation image of (25); hence equation image finally becomes

equation image


[80] The authors appreciate the assistance by Axel Murk (IAP, University of Bern) in operating and maintaining the technical equipment during the El Leoncito campaign. We wish also to express our thanks to the SST team of CRAAE (São Paulo, Brazil), especially Joaquim Costa, Guigue de Castro, and Pierre Kaufmann (PI of SST) for making SST operative. Further thanks go to Christian Mätzler (IAP, University of Bern), Werner Eugster (GIUB, University of Bern), and Guergana Guerova (IAP, University of Bern) for many interesting and helpful discussions. We are also most grateful for comments and useful suggestions by Andreas Muschinski (NOAA) and another (anonymous) referee which helped to improve the paper. We acknowledge FAPESP (Brazil) for funding and supporting the SST project as well as Swiss-SNF (contract 2000-055687.98) and CONICET (Argentina) for their support.