## 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**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**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, *C̃*^{2}_{n}

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 *C̃*^{2}_{n}

## 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,**η̃,*

*C*

^{2}

_{n}

*η*

*C*

^{2}

_{n}

*λ*

^{−1/3}

*λ*is the radar wavelength. The

*C*

^{2}

_{n}

*n*

**x**

*t*

*n*

**x**

**r**

*t*

^{2}

*C*

^{2}

_{n}

*r*

^{2/3}

*r*= |

**r**|, where

*n*(

**x**,

*t*) is the index of refraction at a point

**x**and time

*t,*and the brackets, 〈 · 〉, denote an ensemble average.

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 *C*^{2}_{n}

For many radar measurements, (1) can be used to a good approximation in obtaining *C*^{2}_{n}*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 *C*^{2}_{n}

*C̃*

^{2}

_{n}

*r.*Its expected value is the traditional

*C*

^{2}

_{n}

*C̃*

^{2}

_{n}

*C*

^{2}

_{n}

*η̃,*

*η̃*

*C̃*

^{2}

_{n}

*λ*

^{−1/3}

*η̃*

*C̃*

^{2}

_{n}

*η̃*

*C̃*

^{2}

_{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*

## 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

### a. Qualitative observations

A single frame of the basic TEP data products is shown in Fig. 2; we extract temporal information from a sequence of such images. Figure 4 shows plots of the backscattered power in a single, vertically pointed beam with respect to time, similar to the output of a boundary layer wind profiler. The data in the top panel of Fig. 4 is from segments M to Q, and the data in the lower panel is from segments R to V. The segments are separated by black bars that represent the downtime between data segments.

An obvious feature of the plots in Fig. 4 is the expected bright band of high *C̃*^{2}_{n}*z*_{i}. Segment V follows with a collapse of *z*_{i} due to a deep entrainment process. The downdraft in Fig. 2 is obtained from segment V near time *t* = 100 min.

The spatial and temporal evolution of structures is observable in time sequences of horizontal slices through the field of view. Figure 5 shows a series of images of *C̃*^{2}_{n}*z* = 0.82*z*_{i}, beginning at 1538:11 EDT. A region of high reflectivity propagates through the image from left to right with the mean wind. Each frame represents a 1.28-s average, and the time difference between frames is 5.12 s. Horizontal velocity vector estimates are overlaid on each image and converge on the area of high reflectivity. Frames of the vertical velocity, *w,* are shown in Fig. 6 corresponding to the backscatter in the Fig. 5 frames with the same horizontal velocity vectors overlaid. From these it is evident that the regions of high *C̃*^{2}_{n}*w* seen in Fig. 6.

This small downdraft feature, with a vertical extent (not shown) on the order of 60 m (0.05*z*_{i}), remains coherent as it advects through the TEP field of view, a time interval in excess of 40 s. It moves along the direction of the mean wind at a speed of approximately 6 m s^{−1}, over twice the measured mean wind speed at that altitude.

### b. Scales of *C̃*^{2}_{n} structures

*C̃*

^{2}

_{n}

*w*structures in the CBL are well studied. Several authors (Deardorff and Willis 1985; Lenschow and Stankov 1986; Mason 1989) use the correlation distance

*λ*

_{w}from the

*w*horizontal autocorrelation function,

*R*

_{ww}, given by

*R*

_{ww}

*w*

*x*

*w*

*x*

*r*

*w*(

*x*) at a point

*x,*where

*r*is a horizontal separation distance. The correlation distance,

*λ*

_{w}is the point at which

*R*

_{ww}falls below

*e*

^{−1}times its peak value. In Deardorff and Willis (1985) and Mason (1989),

*λ*

_{w}is seen to grow with height to near the mid-boundary layer, remain constant into the upper boundary layer, and decrease as it approaches

*z*

_{i}.

In a similar manner, we can quantify the scales of *C̃*^{2}_{n}*C̃*^{2}_{n}*λ*_{η}, the correlation distance for the fluctuating *C̃*^{2}_{n}*R*_{ηη}, in both vertical and horizontal dimensions.

*R*

_{ηη}as

*R*

_{ηη}

*η̃*′(

*x*)

*η̃*′(

*x*+

*r*)

*η̃*′(

*x*)

*x,*and

*η̃*′(

*x*+

*r*)

*r.*Here

*η̃*′

*R*

_{ηη}is computed from the inverse Fourier transform of the three-dimensional power spectral density of

*η̃*′

Each *η̃**R*_{ηη} is calculated every 300 s in order to ensure each calculation is independent; in a time of 300 s, a structure moving at the mean wind speed of approximately 2 m s^{−1} should move completely through the TEP 25° field of view.

The averaged measured vertical correlation distance, *λ*_{ηυ}, is small, near 0.04*z*_{i} (45 m), on the same order as the structures in the previous section. Within each TEP cone, we examine the height dependence of *λ*_{ηυ} by correlating 300-m sections of the vertical profile. The *λ*_{ηυ} calculated from each of those sections remains roughly constant at approximately 0.04*z*_{i}.

For horizontal autocorrelations, Fig. 7 shows *λ*_{ηh} along the mean wind direction calculated from all of the TEP segments listed in Table 2. Three curves are shown. The straight, dashed line represents the limits imposed by the finite field of view, and represents *λ*_{ηh} for a perfectly correlated (uniform) scene. The dashed and dotted line is *λ*_{ηh} obtained from the direct calculation of *R*_{ηη}; near the upper altitudes, it is quite close to the perfectly correlated scene, suggesting that the field of view is nearly filled with large-scale structures. The solid line is *λ*_{ηh} calculated after subtracting the mean received power in each time sample at each altitude. This subtraction is a high-pass filter, removing structures in *η̃*′*λ*_{ηh} shows several interesting characteristics. First, it agrees well with the direct *λ*_{ηh} calculation below 0.5*z*_{i}, suggesting that TEP is capturing the full spectrum of *C̃*^{2}_{n}*z*_{i} where the direct and high-pass curves diverge, the slope of the two curves differs as well. The total curve changes to a slope similar to the perfectly correlated scene, while the high-pass curve keeps a roughly constant slope throughout the vertical profile.

Here *λ*_{ηh} quantifies the *C̃*^{2}_{n}*C̃*^{2}_{n}*λ*_{ηh} data of Fig. 7 below 0.5*z*_{i}. Above 0.5*z*_{i}, Fig. 4 suggests that the *C̃*^{2}_{n}*z*_{i}, where they reach a maximum. Near *z*_{i} we would not expect the radar field of view to contain completely the large *C̃*^{2}_{n}*λ*_{ηh} approaches the TEP limit in the regions near *z*_{i}.

## 5. LES CBL features

### a. LES *C̃*^{2}_{n} calculation

*C̃*

^{2}

_{n}

The LES results included in this paper originally appeared in Khanna and Brasseur (1998). New for this paper, however, are the LES *C̃*^{2}_{n}

*n*from the expressions presented by Wesely (1976). These expressions have the general form:

*n*

*aθ*

*bq,*

*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*

^{−7}

*θ*

*q*

*θ*in kelvins and

*q*in g kg

^{−1}. Therefore,

*C̃*

^{2}

_{n}

*C̃*

^{2}

_{n}

^{−14}

*C̃*

^{2}

_{θ}

*C̃*

_{θq}

*C̃*

^{2}

_{q}

*C̃*

^{2}

_{θ}

*C̃*

^{2}

_{q}

*C̃*

_{θq}is the local joint structure–function parameter.

*ϵ̃*

*s*and

*l*of the grid volume and the subgrid-scale turbulence, respectively, are

*χ̃*

_{θ2}

*χ̃*

_{q}

^{2}

*χ̃*

_{θq}

*q*

^{r}and

*θ*

^{r}are the resolvable-scale water vapor mixing ratio and temperture, respectively, and

*K*is the subgrid-scale eddy diffusivity:

*K*

*l*

*s*

*l*

*e*

*c*

_{t}and

*c*

_{b}, the

*q*field is taken as

*q*

**x**

*t*

*c*

_{1}

*c*

_{t}

**x**

*t*

*c*

_{2}

*c*

_{b}

**x**

*t*

*c*

_{1}and

*c*

_{2}are chosen such that the mixing ratio flux at the surface (

*c*

_{1}〈

*wc*

_{b}〉|

_{0}) and at the mixed-layer top

*c*

_{2}〈

*wc*

_{t}〉|

_{zi})

^{−1}m s

^{−1}. The resulting mixing ratio flux was nearly uniform throughout the mixed-layer depth. With (10) through (14), we calculate

*C̃*

^{2}

_{n}

As discussed by Peltier and Wyngaard (1995), LES does not model the local molecular destruction rates *ϵ̃**χ̃**η̃**C̃*^{2}_{n}*C̃*^{2}_{n}

### b. LES *C̃*^{2}_{n} morphology

*C̃*

^{2}

_{n}

Typical instantaneous vertical structure of *C̃*^{2}_{n}*C̃*^{2}_{n}*C̃*^{2}_{n}*x* axis of Fig. 8 is a distance of 5 km, which is roughly equivalent to the 50-min plots in Fig. 4. The TEP and LES vertical beam plots look quite similar in their contrast, especially in the high *C̃*^{2}_{n}

Figures 5 and 6 show TEP measurements of 100-m-scale structures in *C̃*^{2}_{n}*w.* Previous work shows similar qualitative features from LES (Pollard et al. 1998). We can quantify the scales of the fluctuating LES *C̃*^{2}_{n}*λ*_{η}. We use (4) to estimate *η̃**C̃*^{2}_{n}*η̃*′*η̃**λ*_{η} is found from the 1/*e* point of the three-dimensional *R*_{ηη}, computed from (6), where *λ*_{ηh} is the horizontal, streamwise correlation length.

Figure 9 shows the average *λ*_{ηh} from the LES *C̃*^{2}_{n}*λ*_{ηh} for three cases. In each case, *λ*_{ηh} is estimated along the mean wind direction. Figure 9 shows curves for the limits of the LES results mapped into the TEP cone dimensions, *λ*_{ηh} from the direct *R*_{ηη}, and *λ*_{ηh} from the high-pass filtered *R*_{ηη}, calculated with the mean removed in each field of view. The *λ*_{ηh} from the direct *R*_{ηη} curve is much closer to the cone limit than the radar curve at lower altitudes, but shows comparable structure near *z*_{i}. The high-pass *λ*_{ηh} curve, however, compares very well with the radar case, both in value and in slope. Thus, while the curves in Figs. 7 and 9 show the limits of the TEP field of view in imaging *C̃*^{2}_{n}*λ*_{ηh} of the high-pass filtered *C̃*^{2}_{n}*C̃*^{2}_{n}

## 6. Statistical comparisons

Although the TEP field of view is inadequate for capturing the largest *C̃*^{2}_{n}*w* and *C̃*^{2}_{n}*C̃*^{2}_{n}

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.

### a. Vertical velocity statistics

The statistics of the vertical velocity *w* in the CBL are well studied, and provide a first-order comparison of LES with radar measurements. The second moment 〈*w*^{2}〉 is calculated from the variance of *w* in each pixel over horizontal slices of constant normalized height. The mean of *w* at each height is very close to zero for both TEP and LES, as expected.

Figure 11 shows a comparison of the vertical velocity variances from TEP and LES. Here the radar data are taken from segments R to U, although all combinations of data segments show a similar behavior. The values shown are scaled by the respective convective velocity scales *w*∗, shown in Table 3. Below *z*_{i} the results compare well and are similar to those found in other LES studies (Moeng 1984; Mason 1989; Schmidt and Schumann 1989), and laboratory (Willis and Deardorff 1974;Deardorff and Willis 1985), and radar (Kropfli and Hildebrand 1980) measurements.

Above 1.0*z*_{i}, however, the LES predictions diverge from the TEP measurements. The observation of increased vertical velocity variance above *z*_{i} is seen in other profiler data in (Angevine et al. 1994), and also in lidar data (Frehlich et al. 1998). One explanation for the difference between measurement and simulation is that there is atmospheric motion above 1.0*z*_{i} that is not well modeled by LES.

### b. Statistics of *C̃*^{2}_{n}

*C̃*

^{2}

_{n}

*F*

_{n}:

*F*

_{n}is calculated as the variance of

*C̃*

^{2}

_{n}

*w*

^{2}〉 calculation in the previous section. At each height, the variance is normalized by the squared mean of

*C̃*

^{2}

_{n}

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.

*C̃*

^{2}

_{n}

*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*

_{m}

*ϵP*

_{a}

*ϵ*

*P*

_{s}

*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).

*F*

_{a}as the normalized variance of the actual power return from each pixel:

*F*

_{a}, is

*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 0.4*z*_{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 0.75*z*_{i}, near the edge of the high *C̃*^{2}_{n}*z*_{i}. Above 0.75*z*_{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 *C̃*^{2}_{n}

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 1.0*z*_{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.

### c. *C̃*^{2}_{n} autocorrelations

*C̃*

^{2}

_{n}

As in section 4.2, we can calculate the temporal (TEP) or spatial (LES) autocorrelation functions to examine the scales of *C̃*^{2}_{n}*R*_{ηη} from the inverse Fourier transform of the three-dimensional spectral density of *η̃*′*λ*_{ηh}, the horizontal correlation distance. The TEP values are presented as a normalized distance by multiplying the observed correlation lag time in the temporal dimension by the mean wind. Both TEP and LES calculations are in the streamwise direction.

Figure 13 shows the TEP and LES *λ*_{ηh} vertical profiles. The radar data are taken from segments R to U, but all combinations of four adjacent segments show very similar behavior. As in the previous section, only results above 0.4*z*_{i} are shown. The TEP curve shows a small *λ*_{ηh} with a fairly constant slope up through 0.75*z*_{i}. Above 0.75*z*_{i} the TEP *λ*_{ηh} curve behaves as expected, with the largest correlation distance occurring near *z*_{i}, the area of largest *C̃*^{2}_{n}

Near *z*_{i}, the LES *λ*_{ηh} agrees well with the measurements. However, the LES values in the mid-boundary layer are much larger than the TEP measured *λ*_{ηh}. One explanation for that difference is due to the lack of narrow plumes of *C̃*^{2}_{n}*C̃*^{2}_{n}*C̃*^{2}_{n}*C̃*^{2}_{n}*λ*_{ηh} measurements in the mid-boundary layer. LES, on the other hand, may predict coherent structures with small values of *C̃*^{2}_{n}

## 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 *C̃*^{2}_{n}*w* statistics through time series measurements.

Measured vertical profiles of *C̃*^{2}_{n}*z*_{i}. The vertical profiles also show intermittent plumes of enhanced *C̃*^{2}_{n}*C̃*^{2}_{n}*w.* The presented feature has a vertical extent of 0.05*z*_{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 *C̃*^{2}_{n}*z*_{i}.

We compare LES *C̃*^{2}_{n}*C̃*^{2}_{n}*λ*_{η} is calculated from the LES *C̃*^{2}_{n}

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 *C̃*^{2}_{n}*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 *C̃*^{2}_{n}*C̃*^{2}_{n}

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 0.75*z*_{i} than predicted by LES. That difference is perhaps due to the lack of narrow plumes in the LES *C̃*^{2}_{n}

In summary, we have found many similarities in *C̃*^{2}_{n}

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 *C̃*^{2}_{n}

## Acknowledgments

This work was supported by the U. S. Army Research Office under Grant DAAL03-92-G-0110 at UMass and Grant DAAL03-92-G-0117 at PSU. The authors would like to thank Chenning Tong for collecting and analyzing the anemometer data, Dick Thompson for his extensive help at the Rock Springs site, Minfei Leng for his assistance in the TEP deployment at Rock Springs, and Geoff Hopcraft for his assistance in the TEP system development. The authors also would like to thank Jim Mead and Keith Wilson for helpful discussions, and the three reviewers for their helpful comments.

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## APPENDIX A

### A Discussion of *C̃*^{2}_{n} and *η̃*

*C̃*

^{2}

_{n}

*η̃*

The roots of (1) and (2) lie in the hypotheses of Kolmogorov (1941) concerning the structure of turbulence in the inertial subrange of wavenumbers, that is, turbulence at spatial scales small compared to those of the energy-containing range, but large compared to those of the dissipative range. The extent of this inertial subrange is proportional to *R*^{3/4}_{t}*R*_{t} is a Reynolds number of the energy-containing turbulence. Here *R*_{t} is typically so large in the atmospheric boundary layer that the inertial subrange there is at least a decade wide, and often wider.

As discussed in section 2, also central to (1) and (2) is the notion of the ensemble average. Equations (1) and (2) are purely statistical relations that emerge only after sufficient averaging. We typically use a time average in place of an ensemble average and assume ergodicity (Tennekes and Lumley 1972).

Short-term averages of backscattered power and of squared, two-point differences in refractive index vary considerably in time and in space. Their expected values can, of course, depend on time and on position in a flow, but due to the smoothing effects of the ensemble averaging operator any temporal or spatial variations in expected values are also smooth; they vary on the timescale of mean flow evolution and on the spatial scale of mean flow structure. The faster, smaller-scale variations in backscattered power and in refractive-index structure should not be interpreted as variations in *η* and *C*^{2}_{n}

*n*depends on fluctuating temperature and fluctuating water vapor mixing ratio

*q*(section 5). For simplicity in this discussion let us take the refractive-index fluctuations to be dominated by the water vapor contribution, so

*n*=

*cq*with

*c*a constant (section 5). It follows that

*C*

^{2}

_{n}

*c*

^{2}

*C*

^{2}

_{q}

*C*

^{2}

_{q}

*ϵ*

^{−1/3}

*χ*

_{q}

*ϵ*is the dissipation rate of turbulent kinetic energy per unit mass and

*χ*

_{q}is the molecular destruction rate of 〈

*q*

^{2}〉. The

*C*

^{2}

_{q}

*ϵ,*and

*χ*

_{q}are ensemble-mean quantities.

During the years following Kolmogorov’s (1941) hypotheses, turbulence researchers discovered that the small-scale properties of turbulence in large *R*_{t} flows are quite nonuniformly distributed in space at any given time and quite intermittent in time at a given point in space. Thus, short time averages of statistics of finescale properties, such as the dissipation rate of turbulence energy and molecular destruction rate of squared water vapor fluctuations, can have large fluctuation levels. Kolmogorov (1962) and Obukhov (1962) introduced the notion of finescale properties averaged over a local volume of space of characteristic dimension *r,* and reinterpreted Kolmogorov’s original (1941) hypotheses in terms of these *local* variables, as we will call them.

*local*structure–function parameters. The local version of (A1) for water vapor is

*C̃*

^{2}

_{q}

*ϵ̃*

^{−1/3}

*χ̃*

_{q}

*C̃*

^{2}

_{q}

*C̃*

^{2}

_{q}

*C*

^{2}

_{q}

*η̃,*

*η.*

*η̃*

*C̃*

^{2}

_{n}

*C̃*

^{2}

_{n}

*λ*

^{−1/3}

*η̃*

*C̃*

^{2}

_{n}

*η̃,*

*C̃*

^{2}

_{n}

*η̃*

*C̃*

^{2}

_{n}

*λ*

^{−1/3}, does not directly follow, however. Thus, the Kolmogorov–Obukhov hypotheses do not directly imply that the local backscattered power coefficient “tracks” (is perfectly correlated with) the local structure function parameter.

*η̃*

*C̃*

^{2}

_{n}

*β*

_{1}(

*η̃*

*C̃*

^{2}

_{n}

*η̃*

*C̃*

^{2}

_{n}

*β*

_{2}(

*η̃*)

*β*

_{1}

*η̃*

*C̃*

^{2}

_{n}

*δ*

*η̃*

*C̃*

^{2}

_{n}

*λ*

^{−1/3}

*η̃*

*C̃*

^{2}

_{n}

*η̃*

*C̃*

^{2}

_{n}

We do not expect, in general, that *η̃**C̃*^{2}_{n}*η̃**C̃*^{2}_{n}*η̃**C̃*^{2}_{n}*C̃*^{2}_{n}

## APPENDIX B

### Pixel Efficiency

*ϵ*is used to form (16) above,

*P*

_{m}

*ϵP*

_{a}

*ϵ*

*P*

_{s}

*P*

_{m}is the measured return power in each pixel,

*P*

_{a}is the actual power that would be measured by a sensor with a perfect pixel efficiency, and

*P*

_{s}is the power received from outside of the main beam. Using the variability index of (15) for

*σ*

^{2}

_{a}

*P*

^{2}

_{a}

*P*

_{a}〉

^{2}yields

*P*

_{m}

*P*

_{s}

*P*

_{a}

*P*

_{m}

*P*

_{s}〉/〈

*P*

_{m}〉〈

*P*

_{s}〉 is unity if

*P*

_{m}and

*P*

_{s}are independent, and for simplicity that assumption is used, implying that the third term of (B4) can be ignored.

*F*

_{s}term 0.18 times the

*F*

_{m}term, implying that even if

*F*

_{s}≈

*F*

_{m}, the error in assuming

*F*

_{s}is sampled from a much broader scene it should be less than

*F*

_{m}, and the relationship in (B8) should be a good approximation, albeit perhaps somewhat low because the term examined in (B5) is ignored. That term, ignored by assuming that

*P*

_{m}and

*P*

_{s}are independent, may explain the slightly lower values of the TEP

*F*

_{n}when compared to the LES values.

Snapshots of backscattered intensity and radial velocity from the TEP system. The horizontal axes represent the angle off of zenith, and the averaging time of the images is 5 s. The area of high-relative intensity is seen to correspond to a coherent downdraft feature in radial velocity.

Citation: Journal of the Atmospheric Sciences 57, 14; 10.1175/1520-0469(2000)057<2281:LSOTCB>2.0.CO;2

Snapshots of backscattered intensity and radial velocity from the TEP system. The horizontal axes represent the angle off of zenith, and the averaging time of the images is 5 s. The area of high-relative intensity is seen to correspond to a coherent downdraft feature in radial velocity.

Citation: Journal of the Atmospheric Sciences 57, 14; 10.1175/1520-0469(2000)057<2281:LSOTCB>2.0.CO;2

Snapshots of backscattered intensity and radial velocity from the TEP system. The horizontal axes represent the angle off of zenith, and the averaging time of the images is 5 s. The area of high-relative intensity is seen to correspond to a coherent downdraft feature in radial velocity.

Citation: Journal of the Atmospheric Sciences 57, 14; 10.1175/1520-0469(2000)057<2281:LSOTCB>2.0.CO;2

The layout of the experiment site at Rock Springs, PA, six miles from the PSU campus. TEP and the anemometer array are separated by a distance of approximately 300 m. The local clutter environment includes fields of beans and corn and the area is flat except for a ridge to the south where the local topography increases by more than 300 ft.

The layout of the experiment site at Rock Springs, PA, six miles from the PSU campus. TEP and the anemometer array are separated by a distance of approximately 300 m. The local clutter environment includes fields of beans and corn and the area is flat except for a ridge to the south where the local topography increases by more than 300 ft.

The layout of the experiment site at Rock Springs, PA, six miles from the PSU campus. TEP and the anemometer array are separated by a distance of approximately 300 m. The local clutter environment includes fields of beans and corn and the area is flat except for a ridge to the south where the local topography increases by more than 300 ft.

Time–height plots of the TEP measured backscattered power, or *C̃*^{2}_{n}

Time–height plots of the TEP measured backscattered power, or *C̃*^{2}_{n}

Time–height plots of the TEP measured backscattered power, or *C̃*^{2}_{n}

A sequence of TEP *C̃*^{2}_{n}*z* = 0.82*z*_{i} with horizontal velocity vectors overlaid. Each image is a 1.28-s average of data and the temporal spacing between images is 5.12 s.

A sequence of TEP *C̃*^{2}_{n}*z* = 0.82*z*_{i} with horizontal velocity vectors overlaid. Each image is a 1.28-s average of data and the temporal spacing between images is 5.12 s.

A sequence of TEP *C̃*^{2}_{n}*z* = 0.82*z*_{i} with horizontal velocity vectors overlaid. Each image is a 1.28-s average of data and the temporal spacing between images is 5.12 s.

A sequence of TEP vertical velocity (*w*) images from an altitude of *z* = 0.82*z*_{i} with horizontal velocity vectors overlaid. These *w* images correspond to the *C̃*^{2}_{n}*w.* As in Fig. 5, the averaging time here is 1.28 s.

A sequence of TEP vertical velocity (*w*) images from an altitude of *z* = 0.82*z*_{i} with horizontal velocity vectors overlaid. These *w* images correspond to the *C̃*^{2}_{n}*w.* As in Fig. 5, the averaging time here is 1.28 s.

A sequence of TEP vertical velocity (*w*) images from an altitude of *z* = 0.82*z*_{i} with horizontal velocity vectors overlaid. These *w* images correspond to the *C̃*^{2}_{n}*w.* As in Fig. 5, the averaging time here is 1.28 s.

The correlation length, *λ*_{ηh}, for the average spatial autocorrelation function for the fluctuating component of *C̃*^{2}_{n}*λ*_{ηh} is calculated directly from the measured *C̃*^{2}_{n}*λ*_{ηh} is calculated after removing the mean value at each altitude in each measurement cone.

The correlation length, *λ*_{ηh}, for the average spatial autocorrelation function for the fluctuating component of *C̃*^{2}_{n}*λ*_{ηh} is calculated directly from the measured *C̃*^{2}_{n}*λ*_{ηh} is calculated after removing the mean value at each altitude in each measurement cone.

The correlation length, *λ*_{ηh}, for the average spatial autocorrelation function for the fluctuating component of *C̃*^{2}_{n}*λ*_{ηh} is calculated directly from the measured *C̃*^{2}_{n}*λ*_{ηh} is calculated after removing the mean value at each altitude in each measurement cone.

A profile of the *C̃*^{2}_{n}

A profile of the *C̃*^{2}_{n}

A profile of the *C̃*^{2}_{n}

Similar to Fig. 7 for LES data mapped into a volume equivalent to the TEP field of view. As in Fig. 7, the TEP limit represents the limits of the TEP field of view for a perfectly correlated scene, the total *λ*_{ηh} is calculated directly from the autocorrelation function for the fluctuating component of *C̃*^{2}_{n}*λ*_{ηh} is calculated after removing the mean *C̃*^{2}_{n}

Similar to Fig. 7 for LES data mapped into a volume equivalent to the TEP field of view. As in Fig. 7, the TEP limit represents the limits of the TEP field of view for a perfectly correlated scene, the total *λ*_{ηh} is calculated directly from the autocorrelation function for the fluctuating component of *C̃*^{2}_{n}*λ*_{ηh} is calculated after removing the mean *C̃*^{2}_{n}

Similar to Fig. 7 for LES data mapped into a volume equivalent to the TEP field of view. As in Fig. 7, the TEP limit represents the limits of the TEP field of view for a perfectly correlated scene, the total *λ*_{ηh} is calculated directly from the autocorrelation function for the fluctuating component of *C̃*^{2}_{n}*λ*_{ηh} is calculated after removing the mean *C̃*^{2}_{n}

The method used in forming three-dimensional volumes from crosswind slices through the TEP field of view. Crosswind slices are stacked in time, forming a three-dimensional dataset.

The method used in forming three-dimensional volumes from crosswind slices through the TEP field of view. Crosswind slices are stacked in time, forming a three-dimensional dataset.

The method used in forming three-dimensional volumes from crosswind slices through the TEP field of view. Crosswind slices are stacked in time, forming a three-dimensional dataset.

A comparison of the variance of the vertical velocity, normalized by the convective velocity scale *w*∗. The TEP data segments used in this comparison are segments R–U.

A comparison of the variance of the vertical velocity, normalized by the convective velocity scale *w*∗. The TEP data segments used in this comparison are segments R–U.

A comparison of the variance of the vertical velocity, normalized by the convective velocity scale *w*∗. The TEP data segments used in this comparison are segments R–U.

A comparison of *F*_{n}, the variability index, for LES *C̃*^{2}_{n}*C̃*^{2}_{n}*F*_{n} from other sets of four TEP data segments.

A comparison of *F*_{n}, the variability index, for LES *C̃*^{2}_{n}*C̃*^{2}_{n}*F*_{n} from other sets of four TEP data segments.

A comparison of *F*_{n}, the variability index, for LES *C̃*^{2}_{n}*C̃*^{2}_{n}*F*_{n} from other sets of four TEP data segments.

The correlation distance *λ*_{ηh} for *C̃*^{2}_{n}^{−1} mean wind speed to produce a correlation distance.

The correlation distance *λ*_{ηh} for *C̃*^{2}_{n}^{−1} mean wind speed to produce a correlation distance.

The correlation distance *λ*_{ηh} for *C̃*^{2}_{n}^{−1} mean wind speed to produce a correlation distance.

Ground parameters measured on the afternoon of 22 August 1996.

TEP datasets collected on 22 August 1996.

LES parameters and TEP measured meteorological conditions for 22 August 1996.