1. Introduction

The rotating air motion arising behind an aircraft can be dangerous for other flying vehicles encountering its wake zone [1]. For safety of airplanes during takeoff and landing, it is important to have real-time information available about the position (with respect to the flight corridor) and intensity of wake vortices generated by other landing or taking-off aircraft [2]. Pulsed coherent Doppler lidars (PCDLs) constitute currently the best technical facilities for experimental studies of aircraft wake vortices. They can also be used for monitoring of the wake situation over an airfield in real time.

A 2-µm PCDL is used most widely for the investigation of aircraft wake vortices [3]. The wake vortex parameters (coordinates of wake vortex axes and wake vortex circulation) are estimated from raw lidar data by different methods, which are described, for example, in [4,5]. The methods and results of extensive lidar studies of aircraft wake vortices in the atmosphere are considered in Chapter 5 of monograph [6].

A necessary condition for lidar measurements of wake vortex parameters is a rather high spatiotemporal resolution of the raw data and a high signal-to-noise ratio, which can be provided by the 2-µm PCDL in the atmospheric boundary layer. In the free atmosphere, where the concentration of aerosol particles is very low either contrails may contribute the required aerosols or smoke generators are installed on the aircraft wings [7].

In comparison with the 2-µm PCDL (the main parameters of this lidar can be found in [8]), the energy of the probing pulse $E_P$ of the 1.5-µm Stream Line PCDL developed by HALO Photonics [9] is 20 times lower [10]. Therefore, the signal-to-noise ratio ($\text{SNR}$) for the Stream Line lidar at the same concentration of aerosol particles in air is few tens (approximately 50) times lower than the $\text{SNR}$ of the 2-µm PCDL. On the other hand, the pulse repetition frequency of the Stream Line lidar is 30 times higher than that of the 2-µm PCDL. This allows achieving the required measurement accuracy even at the low signal-to-noise ratio.

The high $\text{SNR}$ and the relatively long duration of the probing pulse leading to the insignificant (~0.6 MHz) instrumental broadening of the Doppler spectrum for the 2-µm PCDL [8] allows to apply the method of velocity envelopes (VE method) for the estimation of wake vortex parameters from lidar data [6]. However, this method cannot be applied to the processing of data of the Stream Line lidar due to its low SNR. For the Stream Line lidar another method for the determination of wake vortex circulation and positions was proposed [11]. This method employs the array of radial velocities measured by the lidar in order to obtain the required wake vortex information. In what follows, this method is referred to as the method of radial velocities (RV method). In [11], the RV method was tested in field experiment involving a Boeing 737-800 aircraft. However, questions concerning the accuracy of this method and its applicability to other types of PCDL and aircraft remained open.

This paper presents a modified version of the RV method. Operations of the Stream Line lidar and the 2-µm PCDL are simulated numerically to analyze the accuracy of the estimation of wake vortex parameters with this method. Further, results of lidar field experiments on the study of wake vortex axis trajectories and evolution of the wake vortex circulation obtained with the RV method are presented. Finally, the wake vortex parameters estimated with the VE and RV methods are compared employing the 2-μm PCDL.

2. Lidars and measurement strategy

Table 1 summarizes the main parameters of the lidars used in our wake experiments. According to the tabulated data, the probing pulse duration ${\tau }_P$ determined at the FWHM power level for the Stream Line lidar is 2.35 times lower than that for the 2-µm PCDL. The Stream Line lidar obviously allows to measure the radial velocity with significantly better spatial resolution along the optical axis than the 2-µm lidar [8]. Thus, for the rectangular time window [6] with the width $T_W$ = 120 ns, the longitudinal dimension of the probing volume $\Delta z=(c{T}_W/2)/\text{erf}[T_W/(2{\sigma }_P)]$ ($c$ is the speed of light, ${\sigma }_P={\tau }_P/(2\sqrt{\ln 2})$ and $\text{erf}(x)$ is the error function integral) [6] is $\Delta z$ = 30 m for the Stream Line lidar and $\Delta z$ = 65 m for the 2-µm lidar.

Table 1. Parameters of Lidar Experiments

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For wake vortex measurements with a ground-based PCDL, the probing beam scanning is used in a vertical plane intersecting the aircraft flight path. At every scan, the beam intersects the wake behind the aircraft, pointing into the cross section where the two vortices are circulating in opposite directions, until the wake vortex vanishes or leaves the scanning sector. Figure 1 shows the measurement geometry, where the PCDL is installed at a distance $R_r$ from the runway measured along the Y axis perpendicular to the runway. The probing beam scanning at the angular rate ${\omega }_s$ (absolute value) within the prescribed limits of variation of the elevation angle $0°\le \phi \le {\phi }_M$ is carried out in the $Z{Y}^{\prime }$ plane at the azimuth angle $\theta$ to the $Y$ axis. A landing aircraft, while flying over the runway, intersects the scanning plane at the height $Z_A$ at the time $t_A$.

 

Fig. 1 Geometry of measurement by PCDL scanning in the vertical plane.

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The raw data measured by the Stream Line lidar consists of the array of estimates of the correlation functions of the normalized complex lidar signal $\widehat{C}(l{T}_s,R_k,{\phi }_m;n^{\prime })$, where $l=0,1,\mathrm{...},6$, $T_s=B^{-1}$ is the sampling interval, $B$ = 50 MHz is the frequency bandwidth, $R_k=R_0+k\Delta R$ is the distance from lidar, $R_0$ is the minimal measurement range, $k=0,1,\mathrm{...},K-1$, $\Delta R=c{T}_s/2$ = 3 m is the step in range, ${\phi }_m=m\Delta \phi$ is the elevation angle, $m=0,1,\mathrm{...},M$, $\Delta \phi ={\omega }_s\Delta T$ is the scanning angle resolution, $\Delta T=N_a/f_P$ is the duration of the measurement of the array $\widehat{C}(l{T}_s,R_k,{\phi }_m;n^{\prime })$ at fixed $m$ and $n^{\prime }$, $N_a$ is the number of laser pulses used for accumulation of data [12], $f_P$ is the pulse repetition frequency, and $n^{\prime }=1,2,3,\mathrm{...}$ is the scan number.

Let the lidar signal be normalized by the square root of the mean noise power. Then for the estimate of the signal-to-noise ratio $\text{SNR}(R_k,{\phi }_m;n^{\prime })$ defined as a ratio of the mean power of the lidar echo signal to the mean noise power in the 50-MHz receiver passband, we have $\text{S}\widehat{N}\text{R}(R_k,{\phi }_m;n^{\prime })=\widehat{C}(0,R_k,{\phi }_m;n^{\prime })-1$. This estimate is unbiased, that is, $\text{<S}\widehat{N}\text{R}>=\text{SNR}$, where the angular brackets denote ensemble averaging. In the measurements by the Stream Line lidar, the noise is not white. Therefore, first, the normalized correlation function of the noise component of the signal $C_N(l{T}_s)$ ($C_N(0)$ = 1) is determined from the raw data obtained from long distances $R_k$ (in this case, the lidar measures only noise). Then, the equation ${\widehat{C}}_S(l{T}_s,R_k,{\phi }_m;n^{\prime })=[\widehat{C}(l{T}_s,R_k,{\phi }_m;n^{\prime })-C_N(l{T}_s)]/\text{S}\widehat{N}\text{R}(R_k,{\phi }_m;n^{\prime })$ is used to estimate the normalized correlation function of the useful component of the lidar signal ${\widehat{C}}_S(l{T}_s,R_k,{\phi }_m;n^{\prime })$.

Using Eq. (5) from [12], we pass from ${\widehat{C}}_S(l{T}_s,R_k,{\phi }_m;n^{\prime })$ to the array of estimates of Doppler spectra ${\widehat{S}}_D(V_{l^{\prime }},R_k,{\phi }_m;n^{\prime })$, where $V_{l^{\prime }}=(l^{\prime }-L^{\prime }/2)\delta V$, $l^{\prime }=0,1,2,\mathrm{...},L\text{'}-1$, $L\text{'}$ is the number of spectral channels, $\delta V=(\lambda /2)B/L^{\prime }$ is the velocity step, and $\lambda$ is the wavelength. The number of spectral channels $L^{\prime }$ can be arbitrarily large (for example, $L^{\prime }$ = 1024). This is achieved by supplementing the array ${\widehat{C}}_S(l{T}_s,R_k,{\phi }_m;n^{\prime })$ with the necessary number $L^{\prime }-7$ of complex zeros. As a rule, measurements by the Stream Line lidar are conducted at low SNR. This does not allow the spectral moments [4] and velocity envelopes [5] to be estimated from the array ${\widehat{S}}_D(V_{l^{\prime }},R_k,{\phi }_m;n^{\prime })$ with an acceptable accuracy. Therefore, to obtain the information about wake vortices from the Stream Line lidar data, we estimated the radial velocity ${\widehat{V}}_r(R_k,{\phi }_m;n^{\prime })$ from the position of maximum of the Doppler spectrum, i.e. $\max \{{\widehat{S}}_D(V_{l^{\prime }},R_k,{\phi }_m;n^{\prime })\}={\widehat{S}}_D({\widehat{V}}_r,R_k,{\phi }_m;n^{\prime })$.

The wake vortex parameters can be determined from the array ${\widehat{V}}_r(R_k,{\phi }_m;n^{\prime })$ only if all estimates of radial velocities are unbiased. To meet this requirement it is necessary that the lidar echo signal and the number of accumulated pulses $N_a$ are large enough. The focusing of the probing beam to some distance $F$ allows a significant increase of the signal-to-noise ratio in the vicinity of the focus. The optimal parameters of experiment and data processing ($F$, $R_r$, $\theta$, ${\omega }_s$, ${\phi }_M$, $N_a$, and $T_W$) for the Stream Line lidar were found in [11] and are summarized in Table 1. To compare the wake vortex trajectories and wake vortex circulation estimated by the RV and VE methods, we used the raw data of the 2-µm PCDL, whose parameters used during experiment and data processing are also summarized in Table 1. In the VE method, we specified the Gauss time window with the width $T_W$ equal to the pulse duration ${\tau }_P$. In this case, the probing length amounts to $\Delta z$ = 90 m.

For the measurement geometry parameters of the Stream Line lidar from Table 1, the flight height $Z_A$ should not exceed 60 m to enable the observation wake vortices immediately after their formation. The initial vortex separation $b_0$ is given by $b_0=(\pi /4)B_A$, where $B_A$ is the wing span [13]. At this measurement geometry, where the wake vortices interact with the ground surface already during vortex roll-up, the vortex separation always exceeds $b_0$. This is one of the conditions of applicability of the method of radial velocities.

3. RV method

Let the array of lidar estimates of the radial velocity ${\widehat{V}}_r(R_k,{\phi }_m;n^{\prime })$ from the scan number $n^{\prime }=n_0+1$ and to $n^{\prime }=n_0+N$ contain the information about wake vortices. To avoid the influence of the background wind, we obtain the data array

The angular coordinates of the axes of the port vortex (${\widehat{\phi }}_{C1}(n)$) and starboard vortex (${\widehat{\phi }}_{C2}(n)$) are estimated as

The elevation angle ${\phi }_m^{(n)}$ corresponding to the n-th scan is a function of time $t$ measured from the instant as an aircraft intersects the scanning plane: ${\phi }_m^{(n)}\equiv \phi (t)$. The time $t_n^{(i)}$ of intersection of the axis of the $i$-th vortex by the probing beam for the $n$-th scan can be determined from the equality $\phi (t_n^{(i)})={\phi }_{Ci}(n)$. Using the obtained arrays ${\widehat{R}}_{Ci}(n)$ and ${\widehat{\phi }}_{Ci}(n)$, we can represent the coordinates of vortex axis as functions of time in the Cartesian coordinate system:

The circulations of the port vortex ${\widehat{\Gamma }}_1(t_n^{(1)})$ and starboard vortex ${\widehat{\Gamma }}_2(t_n^{(2)})$ can be estimated by fitting of the array of velocities ${\tilde{V}}_r$ measured at the distances ${\widehat{R}}_{Ci}$ to the theoretically calculated values as a result of the least-square minimization of the functionals $\rho ({\Gamma }_1)$ and $\rho ({\Gamma }_2)$:

The algorithm for calculation of ${\overline{V}}_r({\widehat{R}}_{Ci},{\phi }_m|{\Gamma }_1,{\Gamma }_2)$ [11] is based on the theory developed in [6] and can be reduced to the following. It can be shown that the normalized correlation function of the useful component of the complex lidar signal $C_S(l{T}_s,{\widehat{R}}_{Ci},{\phi }_m)=<{\widehat{C}}_S(l{T}_s,{\widehat{R}}_{Ci},{\phi }_m)>$ for the Gaussian temporal profile of the probing pulse can be represented in the form

The following iteration procedure is applied to obtain estimates ${\widehat{\Gamma }}_1$ and ${\widehat{\Gamma }}_2$. When the separation between the vortex axes becomes several times larger than the initial separation $b_0$, the vortices are practically independent. In this case, it is reasonable to take ${\overline{V}}_r={\overline{V}}_r({\widehat{R}}_{C1},{\phi }_m|{\Gamma }_1,0)$ and ${\overline{V}}_r={\overline{V}}_r({\widehat{R}}_{C2},{\phi }_m|0,{\Gamma }_2)$ in Eq. (9) for $\rho ({\Gamma }_1)$ and $\rho ({\Gamma }_2)$, respectively. After minimization of these functionals (Eq. (8)), we have estimates of circulation ${\widehat{\Gamma }}_1^{(1)}$ and ${\widehat{\Gamma }}_2^{(1)}$ in the first iteration. At the second iteration, ${\overline{V}}_r={\overline{V}}_r({\widehat{R}}_{C1},{\phi }_m|{\Gamma }_1,{\widehat{\Gamma }}_2^{(1)})$ and ${\overline{V}}_r={\overline{V}}_r({\widehat{R}}_{C2},{\phi }_m|{\widehat{\Gamma }}_1^{(1)},{\Gamma }_2)$ are specified in Eq. (9) for $\rho ({\Gamma }_1)$ and $\rho ({\Gamma }_2)$, respectively. After minimization of $\rho ({\Gamma }_1)$ and $\rho ({\Gamma }_2)$, we obtain estimates ${\widehat{\Gamma }}_1^{(2)}$ and ${\widehat{\Gamma }}_2^{(2)}$. And so on. The numerical and field experiments have shown that this iteration algorithm converges rapidly. Three iterations are usually enough.

4. Numerical simulation results

The true coordinates and circulation of the $i$-th vortex specified are denoted as $R_{Ci}^{(T)}$, ${\phi }_{Ci}^{(T)}$, and ${\Gamma }_{Ci}^{(T)}$ in the numerical simulation. To study the regular error in estimation of the distance from the Stream Line lidar to the axes of vortices generated by different aircraft types, we calculated the function $D(R_k)$ by Eq. (6) where ${\tilde{V}}_r$ was replaced by ${\overline{V}}_r$. The velocity ${\overline{V}}_r$ was calculated with Eqs. (10)-(12), where ${\widehat{R}}_{Ci}$ was replaced by $R_k$. It was assumed that the circulations of the port vortex and starboard vortex are identical (${\Gamma }_1^{(T)}={\Gamma }_2^{(T)}=\Gamma$), the central point between the vortex axes is at the height $Z_{\text{center}}$ = 30 m, and the line connecting the vortex axes is horizontal.

Figures 2(a, b) depicts $D(R_k)$ calculated for $\Gamma$ = 150 m²/s, $b$ = 15 m, and $r_C$ = 1 m (a) and $\Gamma$ = 250 m²/s, $b$ = 27 m, and $r_C$ = 1.7 m (b), where $b$ is the separation between the vortex axes. For the calculations we used the parameters listed in Table 1 for the Stream Line lidar. Dashed lines show the prescribed positions of the vortex axes $R_{C1}^{(T)}$ and $R_{C2}^{(T)}$. It can be seen from Fig. 2(a) that the positions of the maxima of $D(R_k)$ do not coincide with the dashed lines, that is, ${\widehat{R}}_{C1}>R_{C1}^{(T)}$, while ${\widehat{R}}_{C2}<R_{C2}^{(T)}$. This is explained by the averaging with the low-frequency spatial filter along the optical axis on the radial velocity (Eq. (11)).

 

Fig. 2 Functions $D(R)$ for the Stream Line lidar (a, b) and the 2-µm PCDL (c, d) calculated at $\Gamma$ = 150 m²/s, $b$ = 15 m and $r_C$ = 1 m (a); $\Gamma$ = 250 m²/s, $b$ = 27 m, and $r_C$ = 1.7 m (b, c); $\Gamma$ = 500 m²/s, $b$ = 63 m, and $r_C$ = 3.2 m (d). Dashed lines correspond to the preset model distances from the lidar to the axes of the port vortex and starboard vortex.

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When the longitudinal dimension of the probing volume of the Stream Line lidar $\Delta z$ = 30 m exceeds the separation between the vortex axes $b$ by far, the probing volumes centered at the axes of the port vortex and the starboard vortex overlap, and the RV method does not work sufficiently precise. In Fig. 2(b), $\Delta z\approx b$, the probing volumes do not overlap, and the maxima of $D(R_k)$ roughly coincide with the dashed lines. Our analysis shows that with the Stream Line lidar it is possible to estimate the wake vortex parameters with the acceptable accuracy during the initial vortex evolution, when $b\approx b_0=(\pi /4)B_A$, for aircraft with a wing span $B_A\ge$ 30 m.

Figures 2(c, d) show $D(R_k)$ calculated for the 2-µm PCDL at $\Delta z$ = 65 m. The calculations have been performed for $\Gamma$ = 250 m²/s, $b$ = 27 m, $r_C$ = 1.7 m (c) and $\Gamma$ = 500 m²/s, $b$ = 63 m, $r_C$ = 3.2 m (d); $Z_{\text{center}}$ = 50 m and $\theta =0°$. It can be seen that in Fig. 2(d) maxima $D(R_k)$ meet the dashed lines. Thus, acceptable estimates of wake vortex parameters can be obtained with the 2-µm lidar by the RV method only for big aircraft. Consideration of the azimuth angle of $\theta =37.5°$ yields $b$ = 63 m.

The accuracy of lidar estimates of wake vortex parameters was studied for the given vortex positions and circulations with the aid of closed numerical simulation. The algorithms described in [6] were used for computer simulation of random realizations of PCDL raw data, whose processing yielded arrays of radial velocities. The RV method was then applied to the resulting arrays of radial velocities. In the numerical experiments, the vortex coordinates $\{R_{Ci}^{(T)},{\phi }_{Ci}^{(T)}\}$ and the circulations ${\Gamma }_i^{(T)}$ of the port vortex ($i$ = 1) and the starboard vortex ($i$ = 2) were prescribed and, therefore, known exactly. This allowed us to calculate the errors of their estimation from the simulated lidar data by the equations $E_R=\sqrt{\frac{1}{2}{\displaystyle \sum _{i=1}^2<{\left[{\widehat{R}}_{Ci}-R_{Ci}^{(T)}\right]}^2>}}$, $E_{\phi }=\sqrt{\frac{1}{2}{\displaystyle \sum _{i=1}^2<{\left[{\widehat{\phi }}_{Ci}-{\phi }_{Ci}^{(T)}\right]}^2>}}$, and $E_{\Gamma }=\sqrt{\frac{1}{2}{\displaystyle \sum _{i=1}^2<{\left[{\widehat{\Gamma }}_i-{\Gamma }_i^{(T)}\right]}^2>}}$ based on a statistically significant number of independent estimates of $\{{\widehat{R}}_{Ci},{\widehat{\phi }}_{Ci}\}$ and ${\widehat{\Gamma }}_i$.

To calculate the errors $E_R$, $E_{\phi }$, and $E_{\Gamma }$, it is necessary to specify the signal-to-noise ratio SNR in the numerical simulation. As follows from the experiments with the Stream Line lidar, at the measurement parameters presented in Table 1, $\text{SNR}$ usually ranges from 0.05 to 0.25. Table 2 summarizes the theoretically calculated errors of lidar estimates of wake vortex parameters in this range of SNR. In the simulation, we assumed that the background wind was zero for the measurement parameters and wake vortex parameters corresponding to Fig. 2(b). The results obtained indicate the possibility of measuring the coordinates of axes of aircraft wake vortices and wake vortex circulation by the Stream Line lidar with the rather high accuracy.

Table 2. Theoretical Estimates of Errors in Determination of Wake Vortex Parameters from Data Measured by the Stream Line Lidar

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The closed numerical experiments employing the RV method have been carried out also for the parameters of the 2-µm PCDL in Table 1 and a typical (for this lidar) $\text{SNR}$ = 10 at $\Gamma$ = 500 m²/s, $b$ = 50 m (63 m in scanning plane), and $Z_{\text{center}}$ = 50 m. The calculated errors in this case were $E_R$ = 5.6 m, $E_{\phi }$ = 0.16°, and $E_{\Gamma }$ = 47.5 m²/s, which are higher than the corresponding errors in Table 2. That is, despite the significantly lower $\text{SNR}$ values, the error of estimation of the radial velocity from the Stream Line lidar data appears to be less than the error of estimation of the radial velocity from the data of the 2-µm lidar because in the case of Stream Line lidar we use much more pulses for data accumulation (compare $N_a$ in Table 1). The RV and VE methods for the 2-µm lidar yield comparable errors for the wake vortex axis coordinates, but for circulation the VE method gives a significantly lower error of $E_{\Gamma }$ = 13 m²/s [16].

5. Field experiment results

During summer 2014 we have conducted a series of experiments at the airfield of the Tomsk Airport with the Stream Line lidar. The parameters of these experiments are given in the first column of Table 1. From all the obtained data, we selected only those, in which the probing beam was reflected by the aircraft at the time when the aircraft intersected the scanning plane. This allowed us to determine the vortex generation time $t_A$, the distance from the lidar to the aircraft, and the flight height $Z_A$ with high accuracy.

Figure 3 shows results of lidar experiment conducted on 21 August 2014. Figure 3(a) shows the two-dimensional distribution of lidar estimates of the radial velocity ${\tilde{V}}_r(R_k,{\phi }_m;1)$ obtained from the first scan ($n$ = 1). Figure 3(b) shows the distribution of the radial velocity along the optical axis of the probing beam propagating at the elevation angle ${\phi }_m$ = 4.2° one second after the intersection of the scanning plane by the aircraft. The velocity distribution in Fig. 3(b) corresponds to variation of the radial velocity along the cross section shown by the bold horizontal line in Fig. 3(a). The positions of the minimum and maximum of the radial velocity in this cross section are close to the axes of the starboard vortex and the port vortex, respectively. It is obvious that from the array of radial velocities shown in Fig. 3(a) it is impossible to estimate the aircraft wake vortex parameters, as the short time is not enough for the vortices to form completely.

 

Fig. 3 Radial velocities ${\tilde{V}}_r(R_k,{\phi }_m;1)$ (a), ${\tilde{V}}_r(R_k,\phi ;1)$ at $\phi$ = 4.2° (b), ${\tilde{V}}_r(R_k,{\phi }_m;2)$ (c) and the function $D(R_k;2)$ (d) obtained from the measurements by the Stream Line lidar at the airfield of the Tomsk Airport on 21 August 2014. A landing B737-800 aircraft intersected the probing beam scanning plane at 07:45:24 LT at a height of 35 m.

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Figure 3(c) shows the two-dimensional distribution $\tilde{V}(R_k,{\phi }_m;2)$ obtained for the following scan ($n$ = 2), when the vortices are already further rolled up. The range of elevation angles ${\phi }_m$, within which the summations in Eqs. (6) and (9) were performed, is indicated in Fig. 3(c). The function $D(R_k;2)$ calculated from these data is shown in Fig. 3(d). The distance between the two maxima of this function is equal to 27 m, which coincides with the theoretical estimate of the initial separation between the vortex axes $b_0$ for a B737-800 aircraft, whose wing span is $B_A$ = 34.32 m. The data shown in Fig. 3(c, d) were used to obtain the estimates ${\widehat{R}}_{Ci}$, ${\widehat{\phi }}_{Ci}$, and ${\widehat{\Gamma }}_{Ci}$ by the RV method. For the same case, Fig. 4 depicts the estimates of the Doppler spectra ${\widehat{S}}_D({\tilde{V}}_{l^{\prime }},{\widehat{R}}_{Ci},{\phi }_m;n_0+2)$, where ${\tilde{V}}_{l^{\prime }}=V_{l^{\prime }}-{\widehat{V}}_r({\widehat{R}}_{Ci},{\phi }_m;n_0)$. The results of calculation by Eq. (11) at $R={\widehat{R}}_{Ci}$ and $r_C$ = 1.7 m are shown in this figure as blue curves. It can be seen that the calculated dependences of the radial velocity on the elevation angle correspond well with the Doppler spectra, thus indicating that the estimates of wake vortex parameters by the RV method are acceptable.

 

Fig. 4 Doppler spectra at different elevation angles ${\phi }_m$ and fixed distances ${\widehat{R}}_{C2}$ (a) and ${\widehat{R}}_{C1}$ (b). Blue curves are achieved by Eq. (11).

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The evolution of vortex coordinates and circulations estimated from the lidar data by the RV method during the experiment with В737-800 on 21 August 2014 is shown in Fig. 5. It can be seen from Fig. 5(a) that in the initial period the vortices descend and simultaneously are transported laterally driven by the crosswind. The experimental estimate of the initial descend velocity $\widehat{w}=[Z_A-Z_{Ci}(t_1^{(i)})]/t_1^{(i)}$ = 1.43 m/s is in good agreement with its theoretical estimate by the equation $w_0={\Gamma }_0/(2\pi b_0)$ = 1.47 m/s, where

 

Fig. 5 Trajectories of vortex axes (a) and evolution of vortex circulation (b). The results are obtained by the RV method from the data measured by the Stream Line lidar at the airfield of the Tomsk Airport on 21 August 2014 once the B737-800 aircraft intersected the probing beam scanning plane at 07:45:24 LT.

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Figure 5(b) illustrates the circulation evolution of the port vortex (closed squares) and the starboard vortex (open squares). For the B737-800 aircraft in the considered flight configuration the initial circulation calculated according to Eq. (13) yields ${\Gamma }_0$ = 250 m²/s. One can see that the depicted estimates of the initial circulation (till 20 s) are close to the calculated ${\Gamma }_0$ within the error given in Table 2. With ongoing time the vortex intensity decreases, and the vortices practically vanish approximately 1 min after their formation.

To compare the RV and VE methods for the estimation of aircraft wake vortex parameters, the RV method was applied to raw data of the 2-µm lidar, which allows highly accurate estimations of the vortex parameters with the VE method [6]. Parameters of the lidar and the experiment conducted at the airfield of Munich Airport during 5 April 2011 are summarized in Table 1. Brief description of the Munich measurement campaign and statistical analyses of the 2-mm lidar wake vortex measurements are given in Refs [18, 19]. In order to fulfill the condition of applicability of the RV method for processing of 2-µm lidar data (see Section 3 of this paper), we have chosen the lidar data obtained with the A340-600 aircraft, for which, according to our calculations, $b_0$ = 50 m and ${\Gamma }_0$ = 500 m²/s. Since in this experiment the azimuth angle is $\theta$ = 37.5°, the distance between the vortex axes in the probing beam scanning plane is $b_0/\cos \theta$ = 63 m. Figure 6 exemplifies the array of the radial velocity ${\tilde{V}}_r(R_k,{\phi }_m;4)$ measured by the 2-µm lidar and the corresponding function $D(R_k;4)$. The data presented are obtained for the fourth scan.

 

Fig. 6 Radial velocity ${\tilde{V}}_r(R_k,{\phi }_m;4)$ (a) and function $D(R_k;4)$ (b) obtained from measurements with the 2-µm PCDL at the airfield of Munich Airport on 5 April 2011 once the landing A340-600 intersected the probing beam scanning plane at 07:28:05 LT.

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The VE method [6] consists essentially in obtaining the positive and negative velocity envelopes from the measured Doppler spectra and estimating the aircraft wake vortex parameters from them. In contrast to the RV method, the VE method enables to obtain the results with acceptable accuracy for aircraft of any types under the condition of sufficiently high $\text{SNR}$. Figure 7 shows the results achieved by processing the data with the RV method (squares connected by solid lines) and with the VE method (circles connected by dashed lines) for a duration about 1 min after the A340-600 intersected the scanning plane. The closed squares (open circles) are for the port vortex, while open squares (closed circles) denote the starboard vortex. One can see that the methods yield rather similar results for vortex transport.

 

Fig. 7 Trajectories of vortex axes (a) and evolution of vortex circulation with time (b). The results are obtained by the RV method (squares connected by solid lines) and the VE method (circles connected by dashed lines) from the data measured by the 2-µm PCDL at the airfield of Munich Airport on 5 April 2011 once the A340-600 aircraft intersected the scanning plane at 07:28:05 LT.

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The square roots of mean values of the square difference of the vortex parameters estimated by the RV and VE method are, by the data of Fig. 7, ${\widehat{E}}_Z$ = 1.7 m for the vertical coordinate of the vortex axis, ${\widehat{E}}_Y$ = 5.2 m for the horizontal coordinate of the vortex axis, and ${\widehat{E}}_{\Gamma }$ = 70 m²/s for the circulation. With allowance for the fact that the error of estimation of the circulation by the VE method is 13 m²/s [16], the error of circulation estimation by the RV method is the major contributor to the error ${\widehat{E}}_{\Gamma }$ = 70 m²/s. The experimental error ${\widehat{E}}_{\Gamma }$ is 1.5 times higher than the theoretical error of circulation estimation by the RV method $E_{\Gamma }$ = 47.5 m²/s from the data of the 2-µm lidar. This is possibly connected with the fact that in the numerical simulation we ignored inhomogeneity of the background wind, wind turbulence, and influence of the ground surface on the aircraft vortices.

6. Conclusions

In this paper, we discuss the applicability and accuracy of the method for estimation of aircraft wake vortex parameters from the array of radial velocities measured by the Stream Line lidar and 2-µm PCDL. It is shown that the RV method allows to estimate the position and circulation of aircraft wake vortices from raw data of the Stream Line lidar with an accuracy sufficient for the study of the spatial dynamics and evolution of wake vortices formed behind aircraft flying in the atmospheric surface layer. Owing to the comparatively long duration of the probing pulse, the RV method can be applied to measurement data of the 2-µm PCDL only for the estimation of wake vortex parameters of big aircraft. The error of the estimation of vortex position by the RV method in this case is comparable to the error of the VE method, while the relative error of the vortex circulation estimated by the RV method amounts to about 20%, which is 5 times higher than the relative error of the method of velocity envelopes. The RV method is applicable to data obtained under stationary conditions and weak wind turbulence, when the background wind velocity changes only slightly for the wake vortex lifetime.