Modeling of Wake Vortex Radar Detection in Clear Air Using Large-Eddy Simulation

a. Governing equations

The computational box size is set to 4b₀, which corresponds to a few hundred meters for a heavy aircraft (i.e., b₀ on the order of 50 m). Regardless of the vortex wake simulated, the computational domain is too small to consider the Coriolis term (Rossby number ≪ 1) in the momentum equation; therefore, it has been neglected.

b. Numerical method

To solve those equations, the same pseudospectral Navier–Stokes solver as in De Visscher et al. (2013) is used here with phase shift partial dealiasing (Canuto et al. 2006; Jeanmart 2002). The governing equations are thus solved in a periodic domain. The main advantage of spectral solvers is that they are free of dispersion and diffusion errors. The time integration is carried out using Williamson’s (1980) third-order, low-storage Runge–Kutta scheme. The SGS model used for the simulations is the regularized variational multiscale model (Jeanmart and Winckelmans 2007): it dissipates only at the smallest resolved scales of the LES, thus allowing the dynamics of the large and medium scales (the most important ones) to be preserved. The temperature and humidity SGS fluxes are obtained assuming unity SGS Prandtl and Schmidt numbers as in (Sagaut 2006).

c. Numerical setup

In the present simulations, atmospheric conditions encountered during approach are considered, that is, a very weakly/weakly stratified atmosphere combined with atmospheric turbulence ranging from weak to strong. Note that very strong unstable conditions may also be encountered in the lower layers of the atmosphere, yet the periodic setup used here does not allow us to simulate such unstable conditions.

To simulate the wake vortices in realistic conditions, background turbulent fields for each case have been generated first and converged statistically in the computational domain. Those fields were converged using the Alvelius (1999) forcing term, which is characterized by a random phase and an amplitude computed according to a prescribed injection spectrum (Vanhoenacker-Janvier et al. 2014). The integral of this spectrum corresponds to the amount of energy injected per unit time dE_f/dt = ε_f, and thus also to the dissipation rate when the stratified turbulence is statistically converged.

Once the stratified turbulence has been produced, the two counterrotating vortices (aligned with the x direction) are superimposed over the background turbulent velocity field. A low-order algebraic profile [also called Burnham–Hallock (B-H) in the aircraft community] is used as the initial profile of the vortices. The effective core size r_c (i.e., the radius of maximum induced velocity) is set to r_c/b₀ = 0.05, which is typical of what is used when modeling aircraft wake vortices with that model. The Reynolds number based on the vortex circulation is very high: Re_Γ = Γ₀/ν ≃ 10⁷.

The box size L is 4b₀, with a uniform grid size Δ of resolution equal to b₀/Δ = 64; r_c/b₀ = 0.05 has been chosen for the core size ratio, because such a resolution ensures that the most relevant dynamics of the vortices are well captured (r_c/Δ > 3) (De Visscher 2012). For each case, the simulation was run for two additional characteristic times t₀, after reaching the statistical turbulent equilibrium of the vortices with the background turbulence [=time of maximum eddy dissipation rate (EDR); Fig. 1].

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Time evolution of the EDR for the three investigated cases. Red symbols highlight the intervals during which the simulations represent a real vortex system as obtained after roll-up.

Citation: Journal of Atmospheric and Oceanic Technology 36, 10; 10.1175/JTECH-D-18-0127.1

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The three cases used in the radar simulator are summarized in Table 1. The dimensional/physical parameters were here obtained by assuming a heavy aircraft (Γ₀ = 400 m² s⁻¹; b₀ = 50 m), an initial height z₀ of 500 m, and typical atmospheric conditions; thus, V₀ = 1.27 m s⁻¹ and t₀ = 39.3 s.

Table 1.

Simulation parameters of the three simulated cases used for the radar simulator. The subscripts g and 0 denote atmospheric factors at ground level (z = 0) and at the release height z₀ of the two vortices, respectively.

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d. Results

Once the vortices are introduced in the simulation, the vortex pair starts to descend at a velocity V₀ = Γ₀/(2πb₀). The turbulence surrounding the vortices is affected by two main phenomena. First, the background turbulence is stretched by the vortex system, generating intense transverse vortical structures. Second, due to the stratification, baroclinic vortical structures are generated at the border of the Rankine oval. This extra source term of vorticity arises from the difference in density between the warmer fluid carried by the vortices inside the Rankine oval and the cooler fluid outside. Subsequently, small-scale instabilities develop inside those structures, leading to a fully turbulent vortex system. The turbulence and the associated EDR increase until reaching a statistical equilibrium with the vortex system (Fig. 1). From that moment on, the flow becomes representative of a real vortex system as obtained after roll-up. This physical evolution is highlighted in red on the graphs. After reaching this equilibrium, the circulation of both vortices decreases smoothly (Fig. 2a), with the vortex system sustaining its own turbulence. Therefore, the descent velocity as well as the maximum tangential velocity (Fig. 2b) of the vortices also decreases. This equilibrium is reached at time t* = t/t₀ = 2.5 for cases 1 and 3 and t* = 4 for case 2.

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Time evolution of: (a) ${\overline{\mathrm{\Gamma }}}_{\text{tot}}^{*}$, (b) the maximum tangential velocity of the WV normalized by the initial descent velocity, (c) ${\overline{z}}_c^{*}$, and (d) the ratio between ${\overline{\mathrm{\Gamma }}}_{\text{tot}}/\left(2\pi {\overline{b}}_c\right)$ and the WV descent velocity ${\overline{V}}_c$. Each panel contains graphs for the three investigated cases. The reported values are averaged between the left and right wake vortices. Red symbols highlight the intervals during which the simulations represent a real vortex system as obtained after roll-up.

Citation: Journal of Atmospheric and Oceanic Technology 36, 10; 10.1175/JTECH-D-18-0127.1

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In the LES, the total circulation of each vortex ${\overline{\mathrm{\Gamma }}}_{\text{tot}}\left(t\right)$ is obtained by finding first, in each y–z slice, the local WV center. This localization is performed by locating the minimum of pressure in each slice. Those y–z planes are the planes perpendicular to the initial vortex tube direction. Afterward, the circulation profile Γ(x_i, r, t) is computed in each slice (with r measured from the local WV center). Then we keep the maximum of the circulation Γ_tot(x_i, t) in each slice. Last, these maximum values are averaged over all of the slices (256 slices) to get ${\overline{\mathrm{\Gamma }}}_{\text{tot}}\left(t\right)$.

We see that the total circulation has already decayed greatly, as shown in Fig. 2a, which depicts the time evolution of the dimensionless vortex circulation ${\overline{\mathrm{\Gamma }}}_{\text{tot}}^{*}={\overline{\mathrm{\Gamma }}}_{\text{tot}}\left(t\right)/{\mathrm{\Gamma }}_0$. The intensification of the turbulence surrounding the vortices in case 3 is stronger than in case 2 with a weaker turbulent background; there, even though the atmosphere is equally stratified, the vortex tubes remain stronger and coherent for a longer time.

The stratification also creates baroclinic vortical structures. Those structures quickly interact with the turbulent vortices, enhancing the decay of the vortices compared to the less stratified case at same level of turbulence, that is, case 1 (Fig. 2a). This baroclinic vorticity also has an opposite sign relative to the primary vortices and thus tends to slow down the vortices as seen in Fig. 2c, which depicts the dimensionless vertical position of the vortex tubes: ${\overline{z}}_c^{*}=z_c\left(t\right)/z_0$.

For nonstratified or weakly stratified atmosphere, the circulation is proportional to the product of the descent velocity V_c and the distance between the WV cores b_c. Figure 2d illustrates this ratio: for case 1 the circulation calculated from the WV velocity field is equal to that calculated from V_c and b_c, whereas for the more stratified cases 2 and 3 the ratio deviates significantly from unity.

Figure 5b shows the humidity field combined with the wind velocity (w, u) of case 1 at time t* = 3.75 in a vertical cross section of the computational box. The black crosses represent the center of the vortex tubes. These were located by identifying the location of the minimum pressure. During its descent, the vortex system carries the lower humidity, which is situated at a high altitude, downward. Such a pattern in the humidity field (as well as in the temperature and pressure fields) may backscatter a characteristic return signal to the radar, which could be used to detect the vortex pair and possibly estimate its intensity as well.

a. Geometry and parameters of the radar used for the simulation

The main parameters of the radar are listed in Table 2. Figure 3 illustrates the structure of the radar signal: during the dwell time, the radar emits 512 pulses with a pulse repetition time of 50 μs. The radar receives the pulses scattered by a given radar cell located at a distance of cτ/2, where τ is the time necessary for an electromagnetic wave to travel from the radar to the scatterer and back and c is the speed of light. A pulse received at time mT + τ, where m is the consecutive number of the pulses in the burst, corresponds to the pulse emitted at time mT and scattered back onto the refractive index inhomogeneities. For the sake of simplicity, the pulses in Fig. 3 are given in the form of a voltage that modulates the carrier frequency of the radar; the carrier frequency is not represented. It is assumed that the modulation of the pulses provides the needed range resolution.

Table 2.

Parameters of the simulated X-band radar.

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Schematic representation of the train of emitted (black) and received (gray) pulses.

Citation: Journal of Atmospheric and Oceanic Technology 36, 10; 10.1175/JTECH-D-18-0127.1

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Figure 4 illustrates the geometry used for the simulation: the radar is located at the side of the glide with the distance r to the runway axis equal to 1100 m. Two geometrical setups are used in the simulator. For setup A, the size of the radar resolution volume in range (Δr_rc) is 60 m and the elevation angle of the antenna beam is fixed at 5.3° to monitor the WV at the altitude of the glide (h₁ = 100 m). For setup B, the range resolution is 6 m and the radar performs a scan of the LES box in elevation (with a step of 0.27°) to track the WV descent.

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Geometric configuration of the radar measurements simulation: the radar is located at the side of the glide with the distance r to the runway axis; Δr_rc is the radar cell size in range; the radar beam is pointed perpendicular to the glide at the elevation angle α; h₁ is the altitude at which the center of the LES box is placed; φ_a and θ_a are the antenna beam widths at 3 dB in azimuth and elevation, respectively.

Citation: Journal of Atmospheric and Oceanic Technology 36, 10; 10.1175/JTECH-D-18-0127.1

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b. Calculation of the radar signal

The discrete nature of the output of the LES imposes the transformation of the integral into a sum over the radar cell volume for which the product $W\left(\mathbf{r}\right)f_{\theta \varphi }^2\left(\mathbf{r}\right)$ is significant. The numerical calculation of the scattering integral necessitates the value of the refractive index irregularities n′(r, t) with a resolution better than the Bragg wavelength. For classical simulators using 2D models for the wake vortices, the resolution could possibly be adapted to the wavelength of the radar. The resolution of the present 3D LES suffices to capture the most relevant dynamics of the medium to large scales of the vortices and the associated turbulence, yet is not fine enough to capture the Bragg wavelength. This one is smaller than the LES resolution by a factor of 50, making a full calculation of the scattering integral impossible. Naturally, it is not possible to increase the LES resolution by a factor of 50, which would amount to 125 000 additional grid points.

Because the LES output data are 3D snapshots of meteorological parameters at discrete time values, time t in Eq. (16) is replaced by the discrete values of kT, where T is the pulse repetition time of the radar.

Taylor’s hypothesis of the “frozen turbulence” (Taylor 1938) is used: because of the relatively short dwell time (25.6 ms), it is assumed that the scattering pattern within the LES cell remains constant and is advected in time by the local velocity field; $C_n^2$ is assumed to be constant over the dwell time. If the dwell time is greater than the LES time step, proper use of Eq. (16) necessitates the interpolation of the values of n and the velocity field for the time corresponding to the radar pulses (Muschinski et al. 1999).

The Muschinski et al. (1999) model uses two main hypotheses:

1. First, the LES subgrid turbulence obeys the Kolmogorov scaling; that is, it belongs to the inertial subrange of the turbulence spectrum, at the Bragg scale.

2. Second, the initial phases φ_0p of the LES grid cell contributing to the sum over the radar cell are statistically independent.

c. Calculation of n and $C_n^2$

The $C_n^2$ calculated at each LES grid node takes the six closest nodes (at a distance Δ equal to the size of the LES grid cell) and averages the square difference between values of n at plus and minus the LES resolution, taken two by two along the three axes. So, in this case, |δ| = 2Δ.

An example of the distribution of $C_n^2$ in a vertical cross section of the LES box is provided in Fig. 5a for case 1. The humidity field combined with the wind velocity field is shown in Fig. 5b. The wake vortex is clearly visible, with high values of $C_n^2$ at the limit of the core and at the limit of the oval surrounding the vortices.

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(a) The $C_n^2$ distribution and (b) the humidity distribution combined with the wind velocity streamlines. Both are in a vertical cross section of the LES box for case 1 at t* = 3.75. To obtain the streamlines in (b), υ and w components of the velocity are averaged along the x axis and the WV descent velocity V₀ is added to the w component to show the Rankine oval formed by the WV velocity field. The black times signs in (b) correspond to the WV core centers averaged along the x axis.

Citation: Journal of Atmospheric and Oceanic Technology 36, 10; 10.1175/JTECH-D-18-0127.1

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d. Calculation of the phase

The calculation of the time series of the scattered signal is based on the assumption that all Bragg’s scatterers behave like point scatterers with an infinite lifetime. However, the refractive index irregularity pattern at the Bragg length scale λ/2 is reorganized after a time interval on the order of λ/(εΔ)^1/3 within the LES grid cell, where (εΔ)^1/3 is the characteristic turbulence velocity. Muschinski et al. (1999) estimated the upper boundary of this time, during which the Bragg scatterers pattern with a scale Δ remains coherent. This time is called the “phase correlation time scale” and is estimated as the time interval after which the RMS phase difference between two scatterers separated by a distance Δ is equal to π:

${\tau }_{\phi }\left(\mathrm{\Delta }\right)=0.16{\displaystyle \frac{\lambda }{{\left(\epsilon \mathrm{\Delta }\right)}^{1/3}}}.$(20)

Referring to the three LES cases (Table 1), with a spatial resolution of Δ ≃ 78 cm and a wavelength of 3.16 cm, the phase correlation time scale τ_φ corresponding to the forcing background turbulence is shown in Table 3 for the various cases.

Table 3.

Estimation of the phase correlation time scale τ_ϕ, the inner scale l₀, and the broadening of the Doppler spectrum Δσ_υ from the subgrid turbulence contribution.

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As discussed before, the developed and turbulent WV have much higher EDR values than those used as the forcing background turbulence. For all cases, the mean value of the EDR increases to 3.9 × 10⁻³–8 × 10⁻³ m² s⁻³, which corresponds to τ_φ changing in the range from 35 to 27 ms. Proper use of Eq. (19) necessitates a random change of the phases of Bragg scatterers after time intervals of τ_φ.

e. Estimation of the subgrid turbulence contribution to the Doppler spectrum width

Table 3 shows the broadening of the Doppler spectrum Δσ_υ caused by the subgrid turbulence. These estimates are given for the background forcing turbulence as well as for the strong increase of the EDR caused by the developed turbulent WV, which results in an increase of Δσ_υ for all cases.

a. Geometrical setup A

Figure 9 shows the details of geometrical setup A: the radar cell is aligned with the WV core center; the size of the radar cell in range is approximately 60 m; the half-power width of the antenna beam is 3.1°. A decision was made to position the WV core at the center of the radar cell here so as to be able to compare the WV circulation estimated from the Doppler spectrum to the exact value as measured in LES (section 6). At each LES time step, the LES box is placed at the same altitude as defined in section 3a. This compensation for the WV descent provides equivalent conditions for the estimation of the WV circulation.

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Alignment of the radar cell relative to the wake vortex core. The black dashed lines indicate the borders of the LES box, red curves show the positions of the wake vortex cores, and gray delineates the borders of one radar cell.

Citation: Journal of Atmospheric and Oceanic Technology 36, 10; 10.1175/JTECH-D-18-0127.1

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Figure 10 represents a distribution of the local velocity (intensity) and $C_n^2$ in horizontal (Figs. 10c,d) and vertical cross sections of the LES (Figs. 10a,b). Each distribution is an average of 80 adjacent LES layers taken from the center of the LES, in the z and x directions for the horizontal and vertical cross sections, respectively. The borders of the radar cell centered at the initial position of the left WV are indicated by dashed lines.

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Distribution of (b),(d) $C_n^2$ and (a),(c) the intensity of the velocity for case 1 at t* = 3.75. The orders of the radar cell are indicated by the dashed lines.

Citation: Journal of Atmospheric and Oceanic Technology 36, 10; 10.1175/JTECH-D-18-0127.1

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Figure 11 shows an example of the Doppler spectrum of the simulated signal. The periodogram gives an estimate of the power spectral density based on a limited number of samples determined by a window (rectangular, Bartlett, Hamming, etc.). However, this estimator is not consistent because its variance does not converge to zero when the number of samples becomes large. Using a single realization of the radar scattered signal gives a very spiky spectrum (Blackman and Tukey 1958). To get statistically stable results, various Doppler spectra can be averaged for an ensemble of scattered signal realizations. Each realization is obtained with a randomly generated initial phase φ_0p entered in Eq. (19). The generation of an ensemble of scattered signal time series with Eq. (16) is computationally expensive. To speed up the simulation, the approach introduced by Muschinski et al. (1999) is used:

$S\left({\omega }_n\right)={\displaystyle \sum _{p=1}^N}{a}_p^2{Z}_{np}{Z}_{np}^{*},$(28)

where $a_p={\left[{\left(C_n^2\right)}_p{V}_{p}g\left(\eta k_B\right)\right]}^{1/2}$; ω_p = k_Bpυ_p; ω_n = 2πn/T_d for n = …, −2, −1, 0, 1, 2, …; and Z_np are the Fourier amplitudes

$Z_{np}{Z}_{np}^{*}=\{\begin{array}{cc}{\displaystyle \frac{2}{{\left[\left({\omega }_p-{\omega }_n\right)T_d\right]}^2}}\left\{1-\hspace{0.17em}\cos \left[\left({\omega }_p-{\omega }_n\right)T_d\right]\right\},& {\omega }_p\ne {\omega }_n\\ 1,& {\omega }_p={\omega }_n\end{array}.$(29)

The two Doppler spectrum estimators are compared: the Doppler spectrum calculated directly with the closed form Eq. (28) is compared with the ensemble average of the Doppler spectra calculated over 100 independent time series of the radar signal. These time series are generated with Eq. (16) using randomly generated initial phases φ_0p in Eq. (19) to simulate 100 realizations of time series. The Doppler spectra are estimated for each realization and averaged. Figure 11 clearly shows that the ensemble-averaged Doppler spectrum is not distinguishable from the spectrum calculated with Eq. (28) and has a reduced variance with respect to a single realization. In the following sections, the closed-form Doppler spectrum is used.

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Doppler spectrum of the scattered signal (case 1). The ensemble average is calculated over 100 realizations. The closed form represents the Doppler spectrum estimated with Eq. (28). All Doppler spectra are normalized to the area under their graphs.

Citation: Journal of Atmospheric and Oceanic Technology 36, 10; 10.1175/JTECH-D-18-0127.1

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b. Geometrical setup B

Geometrical setup B is similar to setup A but the altitude of the LES box tracks the altitude of the descending WV. The radar performs a scan of the LES box in elevation with a step of 0.27°. It is assumed that the position of the LES box does not change during each scan. The range resolution is 6 m. For each radar resolution volume, the Doppler spectrum is estimated with Eq. (28).

Figure 12 illustrates the distribution of the Doppler spectra width among radar cells in one elevation scan. The position of the WV cores is estimated by the local maxima of the Doppler spectrum width (Fig. 13). There are two causes of the discontinuities in estimated coordinates (y, z): 1) a relatively large radar range resolution (6 m) and 2) the fact that the radar cell with the maximum Doppler spectrum width does not necessarily correspond to the actual location of the WV core. To reduce the variance of the estimated WV coordinates, the width of the Doppler spectra in a given radar cell is obtained as an average over 25 closest radar cells in elevation and range (in an area of 5 × 5 of the radar resolution volumes centered at a given radar cell).

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Distribution of the Doppler spectra width in one elevation scan for case 1 at t* = 3.75. The white times signs show the position of the WV cores as measured in the LES; the red open circles show those as estimated by the radar.

Citation: Journal of Atmospheric and Oceanic Technology 36, 10; 10.1175/JTECH-D-18-0127.1

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Time evolution of (a) y and (b) z coordinates of the WV cores for case 1. Dashed lines show LES data, and solid lines are estimated from the Doppler spectrum width distribution.

Citation: Journal of Atmospheric and Oceanic Technology 36, 10; 10.1175/JTECH-D-18-0127.1

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a. Setup A

However, the Rubin model assumes that the wake vortex circulation Γ_tot is linked to V_max by Γ_tot = 2πr_cV_max and thus a top-hat wake vortex model (i.e., uniform vorticity within r_c and zero outside). In the B-H vortex model used as the initial condition in this study, Γ(r) varies smoothly from 0 to Γ_tot and is equal to Γ_tot/2 at r_c. Although the circulation distribution of the developed turbulent WV no longer follows the simple B-H model, it is found that Γ(r_c) is also roughly equal to Γ_tot/2, which holds fairly true for the representative time intervals; see the red parts of Fig. 14: Γ(r_c) deviates from Γ_tot/2 by approximately ±12% for all cases. To take this variation into account, it is proposed to modify Eq. (30) by using Γ_tot/2 ≃ 2πr_cV_max. The coefficient k was evaluated using r_c = 0.05b₀ = 2.5 m. The actual change of r_c over time with respect to the initial spacing between WV cores b₀ is shown in Fig. 15: r_c deviates from the initial value by from +10% to −20% depending on the LES case. As compared with Rubin (2000), the interval of integration in Eq. (30) is changed to [−V_max, −V_min] and [V_min, V_max] because the Doppler spectrum is not symmetric: the negative and positive parts of the spectrum represent different parts of the wake vortex system.

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Time evolution of the ratio between ${\overline{\mathrm{\Gamma }}}_{\text{tot}}/2$ and $\overline{\mathrm{\Gamma }}\left(r_c\right)=2\pi {\overline{r}}_c{\overline{V}}_{\max }$, as measured from the LES data.

Citation: Journal of Atmospheric and Oceanic Technology 36, 10; 10.1175/JTECH-D-18-0127.1

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Time evolution of the ratio ${\overline{r}}_c^{*}={\overline{r}}_c/b_0$, as measured from the LES data.

Citation: Journal of Atmospheric and Oceanic Technology 36, 10; 10.1175/JTECH-D-18-0127.1

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To compare the circulation retrieved from a given radar cell to the circulation measured from the LES velocity field, the radar cell center is aligned with the vortex core center. The LES reference Γ(x_i, r_max, t) is sought at each x_i, where r_max is equal to one-half of the radar cell width in range. Then it is averaged over x_i (within the radar resolution volume) to obtain Γ_tot(t). The total circulation shown in Fig. 16 is the average of the left and right wake vortex circulations, normalized by Γ₀. The wake vortices decay in time, dissipating their energy by interacting with each other and with the surrounding turbulent background. Consequently, the WV circulation decreases in time as shown in Fig. 16 for the considered LES cases.

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Temporal evolution of the normalized circulation of the wake vortices retrieved from the radar simulator (solid) and obtained directly from the LES (dashed).

Citation: Journal of Atmospheric and Oceanic Technology 36, 10; 10.1175/JTECH-D-18-0127.1

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The radar estimates of the circulation at each LES time step are obtained from the radar cells centered at the initial positions of the left and right WV cores (at t* = 0). Then the average between the left and right WV estimates is calculated. The results provided in Fig. 16 show a good agreement with the circulation calculated directly from the LES field. Note that in the LES t* = 0 corresponds to the time of the injection of the wake vortices into the initial stratified turbulence. The moment at which the LES reaches a statistical equilibrium—the state at which the energy dissipation rate reaches its maximum—indicates the time after which the LES provides a physically meaningful simulation of fully developed turbulent wake vortices in the atmosphere. Those moments differ among the LES cases, and consequently, the estimation of Γ_tot starts accordingly (the red parts in the previous figures, lasting for a total time of 2t₀).

The limits of integration (V_min) in Eq. (30) significantly influence the accuracy of the estimation of Γ_tot. The lower limits V_min are chosen so as to exclude the influence of the Doppler spectrum broadening caused by the turbulence induced by the WV: 1.15 m s⁻¹ for cases 1 and 3 and 1 m s⁻¹ for case 2. The values for V_min are estimated by minimizing the error between the estimate Γ_radar and the reference circulation from the LES Γ_tot. However, in real systems, V_min should be estimated from the radar signal backscattered by the turbulence. For all cases, a constant upper limit V_max = 12.8 m s⁻¹ is used. Nevertheless, the value of V_max at each time step can be accurately estimated from a total width of the Doppler spectrum (Fig. 17). The evolution of the Doppler spectrum is shown in Fig. 17 for case 1: the total width of the Doppler spectrum decreases with time; the width of the central part represents the velocity variance related to the turbulence.

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Time evolution of the Doppler spectrum for case 1. The black curves correspond to the −26.5-dB level (following approximately V_max); the white curves correspond to the −3.5-dB level.

Citation: Journal of Atmospheric and Oceanic Technology 36, 10; 10.1175/JTECH-D-18-0127.1

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The influence of V_min on the accuracy of the circulation estimation (with fixed V_max) is shown in the right panel of Fig. 18 in the form of a normalized root-mean-square error:

$\text{NRMSE}={\left\{{\displaystyle \frac{{\displaystyle \sum _{i=1}^N}{\left[{\mathrm{\Gamma }}_{\text{radar}}\left(t_i\right)-{\mathrm{\Gamma }}_{\text{tot}}\left(t_i\right)\right]}^2}{{\displaystyle \sum _{i=1}^N}{\mathrm{\Gamma }}_{\text{tot}}{\left(t_i\right)}^2}}\right\}}^{1/2}.$(31)

It first follows a type of hyperbolic law due to the presence of the 1/V_min factor in the k coefficient of Eq. (30). It then departs form the hyperbolic behavior and eventually reaches a minimum error for some optimal V_min value. These are the values mentioned above. After reaching the minimum, the curve reflects because of the quadratic form of the NRMSE definition.

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Influence of (left) V_min and (right) the misalignment of the radar resolution volume relative to the WV core on the NRMSE (case 1).

Citation: Journal of Atmospheric and Oceanic Technology 36, 10; 10.1175/JTECH-D-18-0127.1

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To evaluate the influence of the misalignment of the radar resolution volume center relative to the WV core, the location of the radar cell center is changed in range from −45 to 90 m. For each shift in range, the circulation within a given radar cell is computed and Eq. (31) is used to calculate the error of the estimation (the right panel of Fig. 18). It can be seen that this error is minimal as long as the shift of the radar cell center is within ±15 m relative to the center of the WV core. Deviations in the local minima of the NRMSE from the initial positions of the WV cores (range shifts of 0 and 50 m for the left and right WV, respectively) are due to the fact that the wake vortices do not remain symmetrical and centered at their initial positions as they evolve in the turbulent stratified atmosphere.

b. Setup B

We note that De Visscher et al. (2013) showed that the influence of the stratification can be accurately modeled. For cases with significant stratification, the model they proposed could potentially be used for estimating the WV circulation from the measurement of the WV descent velocity.

For geometrical setup B, the coordinates of the WV cores are estimated as explained in section 5b. Then the distance between the WV cores is computed and the descent velocity is estimated using three successive measurements for each WV core. Figure 19 shows the results of the circulation estimation obtained using Eq. (32). The parameters ${\overline{V}}_c$ and ${\overline{b}}_c$ of the descending WV are estimated with a reasonable accuracy (with some outliers due to the discontinuities in the estimation of the WV coordinates). As seen from Fig. 19, the method provides accurate estimates of ${\overline{\mathrm{\Gamma }}}_{\text{tot}}$ for low stratification only (case 1). For higher stratifications, a significant deviation of the estimated value is observed, as expected.

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Circulation estimated from the descent velocity of the WV.

Citation: Journal of Atmospheric and Oceanic Technology 36, 10; 10.1175/JTECH-D-18-0127.1

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c. Impact of the noise on the WV detection

Figure 20 provides an example of a distribution of the LES cells’ RCS, calculated as η_pV_p, where η_p is the reflectivity (m⁻¹) calculated for the pth LES node having a volume V_p:

${\eta }_p=0.38{\lambda }^{-1/3}{\left(C_n^2\right)}_{p}g\left(\eta k_B\right).$(33)

This distribution is given in a cross section of LES case 1. Two regions A (containing the left WV core) and B (outside the WV) are considered to illustrate the difference between the RCS calculated for the volumes with high and low EDR: the median values of ε^1/3 is 0.185 m^2/3 s⁻¹ for region A, and 0.075 m^2/3 s⁻¹ for region B. The regions are extended in the x-axis direction to form cubes having dimensions of 60 × 60 × 60 m³, which roughly correspond to the size of the radar resolution volume used for geometrical setup A. The RCS of these volumes is −81.9 dB for A and −85.4 dB for B. The difference of 3.5 dB comes from the decrease of the attenuation function g(ηk_B) by a factor of approximately 2 corresponding to the ε^1/3 decrease from 0.185 to 0.075 m^2/3 s⁻¹ (Fig. 7).

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Distribution of the RCS of the LES cells having a volume of approximately 0.5 m³.

Citation: Journal of Atmospheric and Oceanic Technology 36, 10; 10.1175/JTECH-D-18-0127.1

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Low RCS of the WV limits the radar detection on long distances. Figure 21 shows the SNR estimated for one pulse for different combinations of emitted peak power P_tx and range r. The value of −85 dB is taken for the WV RCS and the noise level of the receiver is −140 dB. The other parameters of the radar are given in Table 2. For the distance of 1050 m and the emitted peak power of 600 W, the single-pulse SNR is approximately −18 dB. However, it is estimated that 7–10 dB of the single-pulse SNR is needed for the accurate retrieval of the WV circulation from the Doppler spectrum (geometrical configuration A). For this SNR, the side parts of the Doppler spectrum (Fig. 11) are above the noise floor. So, for the emitted power of 600 W, the WV circulation can be retrieved at the distance of approximately 250 m. To increase the range, the emitted power should be increased correspondingly.

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Dependence of the SNR (estimated for a single pulse) on the distance between the radar and the WV and on the emitted power estimated for the WV RCS of −85 dB. The lowest limit of the emitted peak power range is 600 W.

Citation: Journal of Atmospheric and Oceanic Technology 36, 10; 10.1175/JTECH-D-18-0127.1

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