1. Introduction

The behavior of refractive index gradients in the atmospheric boundary layer (ABL) has long been of interest in electromagnetic propagation studies. Sensitive radars in the VHF, UHF, and lower microwave frequency ranges can measure the backscattered power from refractive index variations in the clear atmosphere, and since the mid-1960s researchers have used such radars to study the morphology of those variations in the convective boundary layer (CBL). Much of that early work is reviewed in Gossard (1990).

The radars used in CBL morphology studies are usually of two types: mechanically scanned systems and vertically pointed, nonscanning systems. The scanning systems are typically employed in the study of kilometer-scale or larger features (Hardy and Ottersten 1969; Konrad 1970; Doviak and Berger 1980). Such features typically remain coherent for as long as 60 min (Doviak and Berger 1980), and thus the one to several minute scan time of those systems is not an issue. Vertically or near vertically pointed systems are either very high resolution FM–CW systems (Richter 1969; Gossard et al. 1982; Eaton et al. 1995), or UHF or microwave profilers (Ecklund et al. 1988; White et al. 1991;Ecklund et al. 1999). Those systems provide information about CBL vertical structure with spatial resolutions from 1 to 100 m. The temporal resolutions are typically 30 s to as fine as 1 s, and give useful insights into the temporal evolution of local features in the vertical profile.

The Turbulent Eddy Profiler (TEP) radar system recently reported in Mead et al. (1998) provides the spatial and temporal resolution of current vertically pointed profiler systems, while adding horizontal spatial coverage through the use of digital beam-forming techniques. TEP forms three-dimensional images consisting of several thousand volumetric pixels of the backscattered power and Doppler velocity within a 25° cone above the radar. The pixel resolution is on the order of 30 m × 30 m × 30 m, and the image formation time is on the order of 1 s. Those features give TEP the unique ability to study the three-dimensional behavior of refractive index structures and velocity vectors in the ABL with a high-temporal resolution.

In August 1996 TEP was deployed at Rock Springs, Pennsylvania, near The Pennsylvania State University (PSU). Also present at the measurement site was an array of sonic anemometers that measured the surface temperature flux, convective velocity scale, and other parameters. In this paper we present radar observations from a highly convective boundary layer encountered during that deployment. The qualitative, three-dimensional structure and evolution of the local reflectivity (η̃) and vertical velocity (w) fields measured by TEP are shown. We then calculate the appropriate correlation distances from the three-dimensional autocorrelation function, quantifying the scales of local structures in a manner similar to previous studies (Deardorff and Willis 1985; Lenschow and Stankov 1986; Mason 1989). The statistical variabilities of η̃ and w measurements are also examined through the analysis of TEP time series data.

We compare many of the presented TEP measurements to CBL predictions of a large eddy simulation (LES). LES has often been applied to studies of the three-dimensional structure of CBL velocity fields (Moeng 1984; Mason 1989; Schmidt and Schumann 1989; Khanna and Brasseur 1998), and recent work (Peltier and Wyngaard 1995; Khanna and Wyngaard 1997) suggests that LES can also be applied to studies of refractive index structures through the calculation of the local refractive index structure–function parameter, C̃²_n. TEP is an appropriate system for LES comparisons due to both its pixel size, which is similar to the 60 m × 60 m × 16 m pixel of LES (Khanna and Brasseur 1998), and its volume-imaging ability that can be compared with the three-dimensional, time-varying fields predicted by LES.

The following section of this paper describes the TEP system and briefly discusses the analysis techniques employed on the acquired data. Section 3 reviews the experimental setup and discusses the meteorological and in situ parameters from the TEP dataset. Section 4 presents qualitative and quantitative measurements from the TEP CBL dataset. Section 5 reviews our method of calculating C̃²_n from LES and presents qualitative LES results. Statistical comparisons of TEP time series measurements and LES appear in section 6 and concluding remarks are found in section 7.

2. Radar measurements

TEP (Mead et al. 1998) is a 915-MHz volume-imaging radar system that uses digital beam-forming techniques to obtain high-resolution measurements in the ABL. The vertical (range) resolution is 30 m and the horizontal (azimuthal) resolution is 4.5°. At a height of 380 m, a pixel has a horizontal extent of 30 m. Figure 1 shows TEP deployed at the Rock Springs site near the PSU campus at University Park, Pennsylvania.

In normal operation a vertically pointed horn antenna illuminates a 25° conical volume of the ABL above a 60 element receiver array. Each element of the array receives the backscattered signal from the full field of view. Echoes from each of the 50 range bins are digitized, coherently averaged, and output at a rate of 100 Hz. This dictates the range of resolvable Doppler velocities of ±8.2 m s⁻¹. The receiver array is focused in postprocessing by combining the outputs of every receiver with appropriate phase delays. A focused beamwidth of 4.5° is obtained by processing the outputs of 60 receivers. After focusing, backscattered power and mean velocity are estimated from the first two moments of the Doppler spectrum. Figure 2 shows typical TEP power and Doppler data products for altitudes between 900 and 1200 m, averaged over a 5-s interval. The area of high intensity in the backscattered power corresponds to a coherent, downward velocity feature.

TEP estimates the velocity vector and local volume backscattered power coefficient, which we label as η̃, in each of its pixels and the estimation techniques are discussed in Mead et al. (1998). To calculate η̃, an absolute calibration of the system is necessary. For this deployment, TEP is calibrated using a cross-dipole target on a tethered balloon flown above the receiver array.

The roots of (1) and (2) lie in the hypotheses of Kolmogorov (1941) concerning the structure of turbulence in the inertial subrange. Also central to (1) and (2) is the notion of the ensemble average. These equations are purely statistical relations that emerge only after averaging over a sufficiently large number of realizations of the flow. In practice, to obtain η and C²_n, we average over time, invoking the ergodic hypothesis (Tennekes and Lumley 1972) that in a statistically steady flow a time average converges to the ensemble average.

For many radar measurements, (1) can be used to a good approximation in obtaining C²_n from measured values of η. Equation (1) is not a good approximation, however, for systems such as TEP with 10-m-scale pixels and short averaging times. In such systems, the η values fluctuate considerably in time and space; the expected values of η and squared two-point difference in the refractive index, however, vary only on the timescale of the mean-flow evolution, and on the spatial scale of the mean-flow structure. The faster, smaller-scale variations in backscattered power and in refractive-index structure are often erroneously interpreted as variations in η and in C²_n.

A separate issue regarding (4) is the assumption that the dominant scattering mechanism is Bragg scattering, or scattering from spatial fluctuations in the index of refraction at a characteristic scale of λ/2. One consideration in estimating η̃ is the effect of non-refractive-index scatterers in the TEP volume. Recent results (Wilson et al. 1994; Ecklund et al. 1996) suggest that biological targets such as birds and insects can contribute substantially to the backscattered power at 915 MHz. Both Wilson et al. (1994) and Ecklund et al. (1996) find that in a typical CBL over land, insects tend to be the dominant scattering mechanism in the lower half of the CBL. To mitigate that problem, we follow the approach of (Angevine et al. 1994), and statistically filter the data in each pixel, discarding data values that fall outside of three standard deviations from the local 30-s running intensity mean. Similarly, velocity measurements that fall outside of five standard deviations are discarded.

3. Description of the experiment

Figure 3 illustrates the August 1996 site plan of the Rock Springs, Pennsylvania, site. TEP is situated between two fields, approximately 30 m from a road. A horizontal array of anemometers is located approximately 300 m from TEP. In addition, a nearby operations building contains a variety of meteorological sensors for collecting ground level winds, humidity, temperature, and other parameters.

The ground station parameters for the afternoon data are shown in Table 1. Ground level winds are in the range of 1–2 m s⁻¹ from the west, or near the direction of acceptable winds as illustrated in Fig. 3. The temperature rises from 1200 eastern daylight savings time EDT) values near 24° to over 26°C in the late afternoon and there is little or no cloud cover.

The naming convention for the data segments presented in this paper is outlined in Table 2. The 40–50-s gaps occurring between 10-min data segments are due to the data storage scheme used. For brevity, the segments are referred to by the letter of their label (i.e., N, O, P) throughout this paper.

The anemometer array was operated on 22 August beginning at 13 09 EDT for approximately 1 h. It measured the surface temperature flux 〈wθ〉_s and the friction velocity u∗, from which the Monin–Obukhov length scale L is calculated (Tennekes and Lumley 1972, p. 100). The measured geostrophic wind is taken from the TEP dataset as the average mean wind above z_i, and the measured mean boundary layer depth z_i is found from the maximum power return as suggested by (Wyngaard and LeMone 1980) and as implemented in (Angevine et al. 1994), among others. The measurement of z_i allows the calculation of the convective velocity scale w∗ (Stull 1988, p. 118). The moisture flux was not measured. Table 3 summarizes the results obtained with the anemometer array.

The LES results provided for comparisons are also summarized in Table 3. The simulations were generated using the code of (Moeng 1984), with a 128 × 128 × 128 grid in a 5 km × 5 km × 2 km domain, and originally appeared in Khanna and Brasseur (1998). These results are not specifically matched to the conditions of 22 August 1996. In particular, the temperature jump at the top of the mixed layer on that day is not known, making it difficult to perform an LES with an exact match of the conditions. The LES set used, however, is typical of a CBL state, and the differences in features and statistics are scaled by the traditional mixed-layer scales of Deardorff (1972), z_i and w∗.

4. Morphological observations

5. LES CBL features

a. LES C̃²_n calculation

The LES results included in this paper originally appeared in Khanna and Brasseur (1998). New for this paper, however, are the LES C̃²_n values. The method by which they were obtained is summarized in this section.

We calculate the refractive-index fluctuation n from the expressions presented by Wesely (1976). These expressions have the general form:

naθbq,(7)

where θ and q are the temperature and water vapor mixing ratio fluctuations, respectively, and the coefficients a and b depend on the mean values of temperature, pressure, and water vapor mixing ratio. For radio waves the effect of water vapor fluctuations on n considerably exceeds that of temperature, so for simplicity we evaluate the temperature coefficient a for a dry atmosphere. The resulting expression is

n⁻⁷θq(8)

for θ in kelvins and q in g kg⁻¹. Therefore, C̃²_n is

C̃²_n⁻¹⁴C̃²_θC̃_θqC̃²_q(9)

where C̃²_θ and C̃²_q are the local structure–function parameters of temperature and humidity, respectively, and C̃_θq is the local joint structure–function parameter.

We calculate the local structure–function parameters as

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The local dissipation rate of kinetic energy ϵ̃ is given by

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where the effective scales Δs and l of the grid volume and the subgrid-scale turbulence, respectively, are

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Here χ̃_θ², χ̃_q², and χ̃_θq are the local rates of destruction of temperature variance, mixing ratio variance, and temperature–humidity covariance,

View Expanded

where q^r and θ^r are the resolvable-scale water vapor mixing ratio and temperture, respectively, and K is the subgrid-scale eddy diffusivity:

Klsle(13)

Moeng (1984) and Peltier and Wyngaard (1995) discuss those relations in detail.

The water vapor mixing ratio fields are generated by superposition of fields of top–down and bottom–up passive, conservative scalars calculated through large eddy simulation (Moeng and Wyngaard 1984). Denoting the fluctuating top–down and bottom–up scalar fields by c_t and c_b, the q field is taken as

qxtc₁c_txtc₂c_bxt(14)

The constants c₁ and c₂ are chosen such that the mixing ratio flux at the surface (c₁〈wc_b〉|₀) and at the mixed-layer top (c₂〈wc_t〉|_zi) were 0.05 (gm of vapor) (kg of air)⁻¹ m s⁻¹. The resulting mixing ratio flux was nearly uniform throughout the mixed-layer depth. With (10) through (14), we calculate C̃²_n from (9).

As discussed by Peltier and Wyngaard (1995), LES does not model the local molecular destruction rates ϵ̃ and χ̃; it models the rates of inertial transfer of energy and scalar variance from resolvable scales to the subgrid scales. Because of the nature of the subgrid-scale variance budgets, these transfer rates can differ locally from the corresponding destruction rates, although their expected values are the same. We expect the extent to which the two quantities correlate increases with the scale of the local averaging volume, much as we expect the correlation of the TEP backscattered power (η̃) and C̃²_n to increase with increasing pixel size (appendix A). In the remainder of this paper, we examine the impact of those differences on LES C̃²_n by comparing the LES predictions to TEP measured values.

6. Statistical comparisons

Although the TEP field of view is inadequate for capturing the largest C̃²_n features, we can construct larger effective volumes by examining TEP time series data and invoking Taylor’s hypothesis. In this section, we examine the statistics of w and C̃²_n from volumes constructed from time series data. We also examine the measured horizontal correlation distance of C̃²_n structures.

One advantage of the TEP system over other vertically pointed systems in the analysis of time series data is the simultaneous measurement of pixels in the crosswind dimension. Figure 10 illustrates the time series data segments. The crosswind dimension includes multiple independent pixels and thus improves the time series statistics.

The TEP measurements within three-dimensional effective volumes are another opportunity for LES comparisons, and for each of the results presented below we also provide corresponding LES results. In each case we use four segments of TEP data to facilitate that comparison; with a mean wind speed near 2 m s⁻¹, four TEP data segments would form an equivalent distance of approximately 5 km, equivalent to the horizontal, streamwise dimension in the LES domain.

b. Statistics of C̃²_n

Peltier and Wyngaard (1995) present the results of their LES structure–function parameter calculations in terms of a “variability index,” or normalized variance F_n:

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Here F_n is calculated as the variance of C̃²_n over slices of constant altitude, similar to the 〈w²〉 calculation in the previous section. At each height, the variance is normalized by the squared mean of C̃²_n at that altitude.

In analyzing TEP F_n measurements, two factors need be considered. First, it is important to quantify the effects of fading on a variance calculated from the radar data. Each radar power estimate is subject to Rayleigh fading statistics that limit the accuracy of that estimate;a single estimate has a normalized variance of unity (Ulaby et al. 1982, p. 480). The variance due to fading is reduced by a factor of N_i by averaging N_i independent samples. For each estimate, we average two 64-point FFTs, each spanning 0.64 s. The tails of the measured spectrum typically fill more than 8 FFT bins, implying that each spectrum represents roughly 8 independent samples. Those 8 samples along with the factor of 2 from incoherently averaging two spectra, reduce the normalized error in the power estimate due to fading to 6% of the estimated value (±0.27 dB). That error is small compared to the variance obtained from TEP data, presented below.

Second, the pixel efficiency should be considered in TEP C̃²_n estimates. We define the pixel efficiency ϵ as the ratio of the power received from within a single pixel to the total power received. Here, the pixels are defined by the 6-dB contour of the radar beam. With that definition,

P_mϵP_aϵP_s(16)

where P_m is the measured return power in each pixel, P_a is the actual power that would be measured by the sensor having a perfect pixel efficiency, and P_s is the power received from pixels outside of the main beam. As discussed in Mead et al. (1998), the TEP antenna pattern, or any real antenna pattern, has a ϵ less than 1. With the TEP receiver array, however, phase errors between the many elements reduce the pixel efficiency to values near 70% (Mead et al. 1998).

For statistical comparisons, we define F_a as the normalized variance of the actual power return from each pixel:

View Expanded

It is shown in appendix B that the actual normalized variance F_a, is

View Expanded

for a beam efficiency of 70%, where F_m is defined for P_m similarly to F_a in (17). Equation (18) corrects each of the measured values presented in this paper.

The results of the TEP F_n calculations are presented in Fig. 12, where segments Q through T are shown by the triangles. Here F_n is only calculated above 0.4z_i, as results below that altitude are highly intermittent due to occasional clutter sources. The F_n values seem to increase with increasing height up to approximately 0.75z_i, near the edge of the high C̃²_n region surrounding z_i. Above 0.75z_i F_n begins to decrease, reaching a minimum at z_i. The increase above z_i is due to the decreasing denominator in (15) as well as the changing thickness of the high C̃²_n region.

Segments Q through T are shown because they agree quite closely to the LES F_n, the stars in Fig. 12. The shape of the two curves agrees quite well, while the differences in value near 1.0z_i could be due to the F_n correction, as is discussed in the appendix. With other TEP datasets, however, the LES F_n agrees less well. The error bars in Fig. 12 show the minimum and maximum F_n from other combinations of four adjacent TEP data segments. The general shape presented by the error bars, however, seems similar to segments Q through T.

7. Summary

This paper shows a unique dataset from the CBL obtained with the TEP radar system. We present qualitative and quantitative measurements of the CBL and examine C̃²_n and w statistics through time series measurements.

Measured vertical profiles of C̃²_n show a bright band near the capping inversion layer, z_i. The vertical profiles also show intermittent plumes of enhanced C̃²_n in the mid-boundary layer. In the horizontal TEP images, we find 100-m-scale coherent structures of enhanced C̃²_n that correspond to converging horizontal winds and coherent downdrafts in w. The presented feature has a vertical extent of 0.05z_i.

The three-dimensional autocorrelation function is calculated within the TEP field of view and we have presented the vertical and streamwise correlation distances, λ_η, for the fluctuating component of C̃²_n. In the streamwise direction, TEP measures the full scales of structures in the lower boundary layer, but does not capture the full scales of structures near z_i.

We compare LES C̃²_n predictions to the radar measurements and find a similar vertical profile, although missing the plumes of high C̃²_n in the lower boundary layer that occur in the measured data. The streamwise, horizontal λ_η is calculated from the LES C̃²_n set mapped into cones of TEP-like dimensions and the LES results agree well with the measured data.

To study larger-scale features, we construct effective volumes from the measured time series data. Those volumes differ from those formed with traditional, vertically profiling instruments, as the TEP volumes add a crosswind dimension improving the statistics of our estimates. We compare the measured variability of C̃²_n and w in those volumes to LES.

TEP and LES curves of w variance reproduce well-known results in the boundary layer. Above z_i, however, the measured and simulated values diverge, a result seen in other recently published remotely sensed data. That divergence may well be due to atmospheric motions above the capping inversion layer that are not well modeled by LES.

The measurements of C̃²_n variance show a vertical profile that is consistent with the qualitative results. The LES C̃²_n variance agrees well with the some, but not all of the measured data. Some of the differences between measured and simulated values may be due to the effects of pixel efficiency, as discussed in appendix B.

We have also compared correlation distances from TEP time series data with LES. The two agree well in the region surrounding the capping inversion layer, but the radar measures a much lower correlation distance below 0.75z_i than predicted by LES. That difference is perhaps due to the lack of narrow plumes in the LES C̃²_n predictions and will be a subject of further study.

In summary, we have found many similarities in C̃²_n from TEP measurements and LES predictions, suggesting that LES predictions are quite good, not only in their statistical behavior, but also in their local, instantaneous behavior. The differences we have found occur in the mid- to lower boundary layer and require further study. The results here suggest, however, that LES may well become a useful tool in the simulation of electromagnetic propagation in the ABL.

Finally, this study introduces some of the unique capabilities of the TEP instrument. The ability to image structures throughout the boundary layer with a high time resolution is applied here to LES CBL comparisons of C̃²_n and velocity vectors, but in the future may be applied to many other problems of interest.