Modeling the Radar Signature of Raindrops in Aircraft Wake Vortices

Zhongxun Liu Institut Supérieur de l’Aéronautique et de l’Espace (ISAE), University of Toulouse, Toulouse, France, and College of Electronic Science and Engineering, National University of Defense Technology, Changsha, China

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Nicolas Jeannin Department of Electromagntic and Radar, Office National d’Études et de Recherches Aérospatiales (ONERA), Toulouse, France

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Francois Vincent Institut Supérieur de l’Aéronautique et de l’Espace (ISAE), University of Toulouse, Toulouse, France

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Xuesong Wang College of Electronic Science and Engineering, National University of Defense Technology, Changsha, China

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Abstract

The present work is dedicated to the modeling and simulation of the radar signature of raindrops within wake vortices. This is achieved through the computation of the equation of raindrop motion within the wake vortex flow. Based on the inhomogeneous distribution of raindrops within wake vortices, the radar echo model is computed for raindrops in a given resolution cell. Simulated Doppler radar signatures of raindrops within wake vortices are shown to be a potential criterion for identifying wake vortex hazards in air traffic control. The dependence of the radar signature on various parameters, including the radial resolution and antenna elevation angle, is also analyzed.

Corresponding author address: Zhongxun Liu, DEOS, ISAE, 1 Place Emile Blouin, 31056 Toulouse, France. E-mail: zhongxun.liu@isae.fr

Abstract

The present work is dedicated to the modeling and simulation of the radar signature of raindrops within wake vortices. This is achieved through the computation of the equation of raindrop motion within the wake vortex flow. Based on the inhomogeneous distribution of raindrops within wake vortices, the radar echo model is computed for raindrops in a given resolution cell. Simulated Doppler radar signatures of raindrops within wake vortices are shown to be a potential criterion for identifying wake vortex hazards in air traffic control. The dependence of the radar signature on various parameters, including the radial resolution and antenna elevation angle, is also analyzed.

Corresponding author address: Zhongxun Liu, DEOS, ISAE, 1 Place Emile Blouin, 31056 Toulouse, France. E-mail: zhongxun.liu@isae.fr

1. Introduction

When flying in the atmosphere, an aircraft generates two counterrotating vortices of considerable strength lasting for several minutes. These wake vortices are hazardous if encountered by other flying aircraft, especially during the takeoff or landing phases. The current International Civil Aviation Organization (ICAO) safety separation standard is based on the weight category of the aircrafts, without considering the dynamic evolution of the wake vortex. This standard separation is very conservative in most cases but may still permit vortex encounters in particular weather conditions. Erroneous instructions from flight controllers have even caused several accidents, with the aircrafts entering into the wake vortex of the previous aircraft (Gerz et al. 2002; Proctor et al. 2004). With the great growth in air traffic recently, an optimal separation scheme based on monitoring wake vortex behavior in real time can greatly increase aviation safety and airport handling capacity, which are major commercial issues (Bobylev et al. 2010; Barbaresco and Meier 2010).

The candidate technologies for wake vortex monitoring include radar, lidar, and sodar (Rubin 2000; Barbaresco et al. 2007; Holzpfel et al. 2003; Frehlich and Sharman 2005; Zhang et al. 2003; Bradley et al. 2007). Among them, sodar has a too short detection range, lidar is not operational in foggy or rainy weathers and very expensive to purchase and maintain, while radar can detect and locate wake vortex within a relatively long distance under all weather conditions (Barbaresco and Meier 2010; Shariff and Wray 2002) and therefore appears to be a promising choice. In recent years, numerous experiments have been conducted to detect and monitor wake vortices. In Nespor et al. (1994), the vortices of a small fighter aircraft were detected at a range of 2.7 km by using a 1-MW C-band pulsed Doppler radar. The GEC-Marconi Research Center executed wake vortex detection trials by using multifunctional coherent pulse Doppler radar in both the S and X bands (Shephard et al. 1994). In Hanson and Marcotte (1997) and Iannuzzelli and Schemm (1998), the wake of C-130 aircraft was detected by an X-band bistatic radar. In Barbaresco et al. (2007, 2008), aircraft wake vortices at a range of 7 km were effectively detected by an X-band radar in scanning mode, and Doppler radar signatures were observed in rainy weather as well as clear-sky conditions. In Seliga and Mead (2009), a W-band radar was used to detect wake vortices in rainy conditions and the reflectivity RHI plots showed clear evidence of enhanced droplet concentration between the vortices and reduced droplet concentration in columns directly below and toward the outside of each vortex. In Jameson and Kostinski (2010), aircraft wake vortices were found to affect the backscattering properties of raindrops, because the perturbations of wake vortices may generate new periodic clustering in rain capable of increasing the radar coherent backscatter.

Up to now, a significant part of the research into wake vortex radar sensors has been concentrated on field tests. On the other hand, since the 1990s, theoretical studies into the radar scattering mechanism of wake vortices have also been conducted to provide technological guidance for the development of new wake vortex radar sensors. In clear air, the radar reflectivity of the wake vortex is mainly caused by Bragg scattering from the refractive index fluctuations and some fruitful results have been achieved by Marshall and Myers (1996), Myers and Scales (1999), Shariff and Wray (2002), Li et al. (2011), Qu et al. (2010), and Vanhoenacker-Janvier et al. (2012). However, the radar monitoring of wake vortices in clear air for operational application is far from being effective. First, the radar reflectivity of wake vortices in clear air is extremely low and extremely high transmitted powers should be envisaged, leading thus to high infrastructural costs. Second, the characterization of vortex circulation from the radar measurements in clear air has not been made until now. In rainy weather, the radar electromagnetic waves are not severely attenuated by raindrops or fog at short range. Contrarily, the droplets rolled up by the wake vortices are strong scatterers, enhancing the intensity of the scattered signals. Radar could be seen as an efficient instrumentation for monitoring wake vortices and complements other sensors such as lidar that are effective only in clear skies. However, few studies into the radar backscattering of wake vortices in rainy weather have been reported. Thus, the main objective of the present work is to model and simulate the radar signature of raindrops within wake vortices, which may be different from that in still air and can provide potential information for identifying the wake vortex hazard in rainy weather. Actually, the falling raindrops are transported by the wake turbulence flow as inertial particles under the influence of gravity and drag forces, and their trajectories and velocities will be disturbed. This disturbance will change the spatial and velocity distributions of raindrops in wake vortices. For radar observation, the radar signatures will represent the motion of the raindrops and the wake vortex characteristics in each radar cell. This paper is organized as follows, the motion equation of raindrops in wake vortices is analyzed in section 2 and a simplified model for computing the trajectory and velocity of raindrops within wake vortices is described in detail. In section 3, the computation of radar time series from raindrops in wake vortices based on the simulated drops sizes, positions, and velocities is presented and the corresponding signal processing methods are described. In section 4, typical X- and W-band radar signatures from raindrops within wake vortices are obtained and analyzed.

2. Motion of raindrops within wake vortices

Starting with the well-known characteristics of the raindrops in terms of size distribution and terminal falling velocity in the absence of turbulence, and considering a generic flow model for the vortex, a model for analyzing the concentration of raindrops in the vicinity of the vortex is presented in this section. This model will be used later to compute the radar signature of the raindrops within wake vortices.

a. Parameterization of raindrops in still air

Drop-size distributions (DSDs) have been widely used by radar meteorologists, as they can directly be related to the radar reflectivity (Marshall and Palmer 1948). A series of models for describing this kind of distribution in different climatic regions and seasons were summarized in Owolawi (2010). For a given rain rate R (mm h−1), a suitable model to describe the size distribution of the rainfall in Europe is given by (Atlas and Ulbrich 1974; Jiang et al. 1997; Owolawi 2010)
e1
where N0 = 64 500R−0.5 (m−3 mm−3) and Λ = 7.09R−0.27 (mm−1), and where N(D) (m−3 mm−1) represents the number of raindrops with the diameter D per unit volume per unit diameter class interval. If the raindrops are divided into several diameter classes, the number of raindrops with the diameters between D − ΔD/2 and D + ΔD/2 in a unit volume can be approximately computed by ΔDN(D). Figure 1a presents a raindrop-size distribution for the given rain rates of 2, 5, and 10 mm h−1. In still air, a falling raindrop reaches its terminal fall velocity VT with the equilibrium between the inertial force and the drag force acting on it (Sauvageot 1992). A widely used exponential expression between VT and the diameter D is given by (Atlas et al. 1973; Bringi and Chandrasekar 2004; Fang 2003)
e2
where α1 = 9.65 m s−1, α2 = 10.3 m s−1, α3 = 0.6 m s−1, and (ρ0/ρ)0.4 is a density ratio correction factor adjusting the deviation of the terminal fall velocity due to the air density change with the fall altitude. Figure 1b presents the terminal fall velocity of raindrops for different altitude levels.
Fig. 1.
Fig. 1.

Characteristics of raindrops in still air. (a) Raindrops size distributions for R = 2, 5, and 10 mm h−1; the diameter interval is 0.1 mm. (b) Raindrop terminal falling velocities at altitudes of 50, 1000, and 4000 m.

Citation: Journal of Atmospheric and Oceanic Technology 30, 3; 10.1175/JTECH-D-11-00220.1

b. Wake vortex velocity profiles

In the atmosphere, the flow induced by the aircraft wake vortex introduces fluctuations of temperature, pressure, humidity, and velocity field inside and outside the vortex core. As shown in Fig. 2, the aircraft vortex wake may be described by four specified zones and the vortex velocity profile in each zone changes with time and space (Ginevsky and Zhelannikov 2009, chapter 1). In the wake formation zone, small vortices emerge from the vortex sheet at the wingtips and at the edges of the landing flaps. After rolling up, the wake enters into its stable phase, maintaining a relatively static velocity profile that may last for several minutes. In the following analysis, we focus on the fully rolled-up wake vortex in the stable phase.

Fig. 2.
Fig. 2.

Breakdown of the aircraft vortex wake in accordance with its evolution phases (Ginevsky and Zhelannikov 2009).

Citation: Journal of Atmospheric and Oceanic Technology 30, 3; 10.1175/JTECH-D-11-00220.1

In its stable phase, the aircraft wake vortex is represented by superposing two coherently counterrotating vortices with equal intensity and axis-symmetric velocity distributions, as shown in Fig. 3. According to the Biot–Savart law, the local flow velocity at the point P can be expressed as
e3
where VL, VR are the velocity components induced by the left and right vortex separately. Due to the mutual interference between the two vortices, the positions of the two vortex cores will descend with a downwash velocity Vd. In the following analysis of interaction between wake vortices and raindrops, Vd is not taken into account for the preliminary research.
Fig. 3.
Fig. 3.

The velocity field of two counterrotating vortices (Li et al. 2011).

Citation: Journal of Atmospheric and Oceanic Technology 30, 3; 10.1175/JTECH-D-11-00220.1

For each single vortex, the widely used Hallock–Burnham model is considered to depict its tangential velocity distribution. The tangential velocity fields from this model are expressed as (Hinton and Tatnall 1997; Gerz et al. 2002)
e4
where Vθ(r) (m s−1) is the tangential velocity at the distance r (m) from the vortex core, rc (m) is the vortex core radius, and Γ0 (m2 s−1) is the initial circulation that describes the vortex strength and is determined mainly by the aircraft lift (Holzpfel et al. 2003). In the case of an elliptical distribution of the lift, the circulation is expressed as (Winckelmans et al. 2005)
e5
where M (kg) is the weight of the airplane, U (m s−1) is the flying speed, g (m s−2) is the gravitational acceleration, ρ is the air density, b0 is the separation between two vortices, and b (m) is the aircraft’s wingspan. During the takeoff and landing phases near the airport runway, U (m s−1) is relatively small; thus, the circulation of wake vortices is larger than that in the cruise phase. Figure 4 illustrates the two-dimensional velocity distribution of the wake vortices generated by a A340 airplane, whose maximum landing weight, landing velocity, and wingspan are 259 000 kg, 290 km h−1, and 60.30 m, respectively, with each white arrow representing the velocity of the local vortex flow.
Fig. 4.
Fig. 4.

Two-dimensional velocity distribution of the wake vortices generated by an A340 airplane.

Citation: Journal of Atmospheric and Oceanic Technology 30, 3; 10.1175/JTECH-D-11-00220.1

c. Motion equation of raindrops within wake vortices

To compute the trajectory of the raindrops in the neighborhood of the wake vortex, some basic assumptions are made. First, it is assumed that the raindrops are spherical and not deformable. Second, the interaction between raindrops such as collision or coalescence, as well as the effects of wind and atmospheric turbulence on the raindrops, are not considered. Third, the raindrops could possibly decrease the vorticity and could therefore speed up the decay process of the vortex; however, considering that the mass loading of raindrops for nonexceptional rain rates is far below 1, it can be assumed that this effect is very small and negligible (Dowell et al. 2005). Fourth, no significant evaporation or condensation of the raindrops occurs in the vortex flow. Hence, the problem is simplified as a point particle dynamic problem.

As shown in Fig. 5, the axis of the wake vortex is assumed to be perpendicular to the wake vortex flow between the two boundaries l and l′. When a raindrop enters into the wake vortex flow at P, its movement is governed by
e6
where t is the time, a is the acceleration of the raindrop, Fd is the fluid drag force acting on the raindrop, mp is the mass of the raindrop, and g is the downward gravitational acceleration, which is taken as negative. For a raindrop moving with velocity vp in the fluid velocity field u[zp(t)], if its diameter D ranges from 0.5 to 4 mm, the drag force Fd can be approximately considered in the Newton regime (Lovejoy and Schertzer 2008) and given by
e7
where zp(t) denotes the raindrop’s position, δv is the relative velocity between the vortex flow and the raindrop, and Cd is the fluid drag coefficient. For a raindrop with diameter D and density ρw, the impact of air density variations in the vortex flow on Cd can be neglected because the raindrops are relatively dense (Dowell et al. 2005); thus, Cd is derived by the equilibrium equation of its weight and the drag force when falling at its terminal falling velocity in still air:
e8
Substituting Eqs. (7) and (8) into Eq. (6), the motion equation of raindrops within the wake vortices can be further expressed as
e9
The instantaneous position and velocity of raindrops can be obtained from the above equation. However, its nonlinearity does not enable its analytical computation. A fourth-order four-variable Runge–Kutta algorithm is used to compute the equation of motion.
Fig. 5.
Fig. 5.

Interaction between raindrops and wake vortices.

Citation: Journal of Atmospheric and Oceanic Technology 30, 3; 10.1175/JTECH-D-11-00220.1

d. Trajectories of raindrops within wake vortices

For a raindrop falling from point P as shown in Fig. 5, P′ and P″ show its trajectories within wake vortices and in still air, respectively. In still air, the raindrop falls down along the trajectory that is vertical to the ground. In the presence of wake vortices, the trajectory of a raindrop depends on the diameter of the raindrop and the location where it enters into the wake vortex flow. In Fig. 6, the trajectories of four groups of raindrops with different diameters (0.5, 1, 2, and 4 mm) are illustrated. The vertical position of the vortices is 150 m above the ground. Initially, the raindrops are released on a horizontal plane 315.7 m above the ground at the terminal falling velocity. The background color indicates the velocity of the two-dimensional wake vortex flow. The white arrow denotes both the position and velocity of the raindrops at each time step. The trajectories of raindrops with smaller diameters seem to be more largely changed by the vortex flow. The fall duration is defined as the total time the fastest-moving raindrop takes from the initial position to the ground. The fall duration of the 0.5-mm group of raindrops is longer than the fall duration of other groups of raindrops, as their terminal falling velocity is much lower than those of larger drops. In some part of the region under the vortex flow, there are no raindrops falling down. In the region between the two vortex cores, some of raindrops are preferentially concentrated due to the influence of the vortex velocity field; the velocities of the raindrops here are also considerably increased.

Fig. 6.
Fig. 6.

The trajectory of raindrops in wake vortices: (a) D = 0.5 mm, duration = 106.6 s; (b) D = 1.0 mm, duration = 62.2 s; (c) D = 2.0 mm, duration = 41.0 s; and (d) D = 4.0 mm, duration = 31.8 s.

Citation: Journal of Atmospheric and Oceanic Technology 30, 3; 10.1175/JTECH-D-11-00220.1

3. Radar signature of raindrops in wake vortices

The signal arriving from the scattering volume that contains raindrops can be used to retrieve parametric information on the raindrop through the interpretation of the Doppler information, as is already widely used for weather monitoring purposes. In the present case, the heterogeneous distribution and velocities of the raindrops perturbed by the vortex flow should induce a distinct signature from that obtained from volumes with raindrops in still air. The radar signature of the raindrops in the vicinity of the wake vortices is modeled and analyzed in the remainder of this section.

a. Microwave properties of raindrops

Considering a spherical raindrop of diameter D (m) and an incident electromagnetic wave of wavelength λ (m), the radar cross section (RCS) of the raindrop is closely related to its radio-electric size α = πD/λ. Here, α is generally smaller than 1 for a wide range of radar bands (S, C, and X bands), because the sizes of raindrop scatterers are smaller than the wavelengths. In this case, the radar cross section of a raindrop is well described by the Rayleigh approximation (Gunn and East 1954):
e10
where |K|2 is a coefficient related to the dielectric constant of water and m is the complex refractive index of the raindrops relative to the air background, which can be expressed as m = nik. In addition, is the ordinary refractive index with ɛr the constant dielectric and k the absorption coefficient of the raindrops. In the Rayleigh approximation region, the temperature change does not significantly affect the scattering. Practically, |K|2 could be considered to be a constant for radar application at temperatures found in the atmosphere and centimeter wavelengths (Sauvageot 1992). In the following simulation, |K|2 = 0.93 is adopted. For higher-frequency bands, if α is comparable to 1, the backscattering cross section of a raindrop is usually given by Mie formulas:
e11
where an and bn are the Mie coefficients obtained from Bessel and Hankel functions with arguments α and m. Detailed expressions for this computation can be found in Sauvageot (1992). In Fig. 7, the backscattering cross-section function of a raindrop’s diameter is computed by Rayleigh and Mie formulas for 10- and 94-GHz incident waves, respectively.
Fig. 7.
Fig. 7.

Backscattering cross section of a raindrop as a function of its diameter.

Citation: Journal of Atmospheric and Oceanic Technology 30, 3; 10.1175/JTECH-D-11-00220.1

b. Radar signal time series

Combining the results of the previous section, it is at this stage possible to simulate the radar echo from the raindrops in the wake vortex region in any configuration. For airport applications, the runway wake vortex radar detection is of great significance (Barbaresco et al. 2011) and we concentrate our efforts on radar monitoring of wake vortices near the airport runways. As shown in Fig. 8, the XOZ plane denotes the ground; and the airport runway is assumed to be perpendicular to the XOY plane, the radar is pointing perpendicularly to the rotation axis of wake vortex, with an elevation angle α0. The distance between the radar and the airport runway is R0, and the height of the wake vortex core is h0. Above the wake vortex region, the raindrops are assumed to be distributed as shown in Eq. (1). In the area impacted by the vortex flow in stable stage, the new size distribution of the raindrops is obtained by numerical computation of the motion via Eq. (9). In the following analysis, we consider the radar echo model of raindrops in a given resolution volume, which is defined by the radar range resolution and the beamwidths of the radar antennas.

Fig. 8.
Fig. 8.

The geometry of radar configuration.

Citation: Journal of Atmospheric and Oceanic Technology 30, 3; 10.1175/JTECH-D-11-00220.1

Supposing that the Doppler radar transmits a series of Np pulses with unitary normalized amplitudes, then
e12
where fc is the carrier frequency of the signal, T is the pulse repetition interval, and τ is the pulse width. Here, rect(·) is a rectangular function equal to 1 for 0 ≤ (·) ≤ 1 and to 0 otherwise and μ(t) is a pulse modulation function that can be set as a linear frequency modulation, or a phase-coded modulation. In this case, the simplest situation with μ(t) = 1 is considered. In Fig. 8, the radar antenna beam points to the direction of (α0, φ0), where φ0 is set to be 0. Considering the kth raindrop in the given resolution volume at time t = 0, the beam is positioned at P with the spherical coordinates (rk, αk, φk), where rk is the radial distance from the raindrop to the radar, αk is the elevation angle between OP and the XOZ plane, and φk is the azimuthal angle between OX and the projection of OP on the XOZ plane. The backscattered baseband signal of the kth raindrop from the nth pulse can be expressed as
e13
where c is the speed of light, υk is the radial velocity of the raindrop moving toward the radar, and Ak is the amplitude of the signal, which is derived from the radar equation with the following analytical form:
e14
where Pt is the transmitted power, G is the antenna gain, λ is the wavelength, L is the total loss of the radar system, σk is the radar cross section of the raindrop, and wa and wr are the angular and radial weighting functions, respectively, depending on the location of the raindrop in the radar scattering volume (Cheong et al. 2008; Capsoni and D’Amico 1998; Capsoni et al. 2001). Thus, the composite signal time series reflected from the transmitted pulses can be approximated by the superimposition of the baseband signals backscattered from all the raindrops in the radar scattering volume:
e15
where Nr represents the number of raindrops in the given resolution volume and ns is a centered complex Gaussian white noise. In our analysis, the multiple scattering of raindrops is ignored and the ground clutter competing with the raindrops’ signatures is not taken into account, as it can be easily removed (Richards et al. 2010).

Based on the above equation, the radar echo of a given resolution volume can be computed for each transmitted pulse. It is worth pointing out that the coordinates of each raindrop are changing for different pulses due to their motion in the wake vortex flow, the mathematical relationship between its coordinates, and its position in two-dimensional wake vortex region, which can be obtained through the computation of the motion equation. At each instant when the radar transmits a pulse, the positions and velocities of the raindrops are updated according to the motion equation by numerical simulation. A general description of the procedure to compute the radar signal time series of raindrops in wake vortices is summarized in Fig. 9.

Fig. 9.
Fig. 9.

The procedure of generating a radar signal time series.

Citation: Journal of Atmospheric and Oceanic Technology 30, 3; 10.1175/JTECH-D-11-00220.1

c. Radar signal processing

The radar reflectivity and Doppler spectrum characteristics are of great importance in the radar observation of meteorological targets. In radar meteorology, the reflectivity of raindrops is defined as the sum of the backscattering cross sections of each individual raindrop over the unit volume:
e16
which implies that the raindrops are homogeneously distributed in the given radar resolution volume. However, the spatial distribution of raindrops moving in the wake vortex flow is no longer homogeneous. Therefore, we introduce the equivalent radar cross section συ, to represent the radar reflectivity of raindrops in the given resolution volume. One of the advantages of calculating συ is that, for the radar observation of wake vortex in rainy weather conditions, the RCS contribution of the raindrops due to the Rayleigh scattering could then be compared with the RCS of the wake vortex due to the Bragg scattering induced by the refractivity gradient within the flow, as studied in (Myers and Scales 1999; Shariff and Wray 2002; Li et al. 2009, 2011). Assuming that the angular and radial weighting functions are equivalent to 1, συ can be approximately derived from the radar equations as
e17
where r0 denotes the distance from the radar to the center of the radar resolution volume, H is the constant defined in Eq. (14), and Pr is the average received power, which can be obtained easily by analyzing the Doppler spectrum of the recorded or simulated radar time series Sr(n) and summing the power spectrum over all the Doppler spectrum bins as
e18
where Np is the number of coherent integration pulses, Δfd is the Doppler frequency resolution, and Y(m) denotes the energy contribution to the Doppler spectrum from each Doppler frequency bin. Under the conditions of short radar time series, a number of high-resolution spectral estimation methods, including the maximum entropy method and parametric methods, can be utilized to estimate the Doppler spectrum of the Sr(n) as well (Barbaresco and Meier 2010). Actually, the Doppler velocity is proportional to the Doppler frequency via the relationship υd = 2λfd; thus, the Doppler spectrum is a power-weighted distribution of raindrop radial velocities within the radar resolution volume (Yu et al. 2007). In the wake vortex region, the raindrops move in various directions, which are determined by the combination of drag forces and gravity, and the Doppler spectrum will be broadened correspondingly. The width of the Doppler spectrum is quite significant in identifying the possible wake vortex warnings, the estimation of its location and strength, as well as the prediction of wake vortex evolutions.

4. Simulation and discussion

In this section, typical X- and W-band radars are assumed to be used for monitoring aircraft wake vortices in rainy weather. From the model of interactions between raindrops and wake vortices, considering a given radar configuration, the radar signatures of raindrops within wake vortices are here illustrated by numerical simulation. The main objectives of the simulation are to estimate the Doppler velocity spectrum of raindrops within a wake vortex and to evaluate the radar detectability of the wake vortex in rainy weather. The simulation results are expected to help us develop a deeper knowledge of wake vortex radar sensors.

a. Initial position of raindrops

The first step in the simulation is to generate the initial distribution of raindrops in wake vortices. As shown in Fig. 8, a region centered around the vortex pair is considered to be the wake vortex region, with measurements of 250 m × 150 m × 1 m along the X, Y, and Z axes, respectively. In the region above the wake vortex, the raindrop-size distribution is assumed to be given by Eq. (1). The raindrops with diameters between 0.5 and 4 mm are considered in the simulation. The diameter class interval is of 0.1 mm. For each diameter class, the raindrops are initially released in a box above the wake vortex region, whose measurements along the X, Y, and Z axes are 250 m × 5 m × 1 m, respectively. At each time step, some of the raindrops will enter into the wake vortex region, and their positions and velocities are updated from the numerical computation of the motion Eq. (9). The procedure is continued until the smallest raindrops reach the ground or pass through the wake vortex region. To illustrate the initial position of raindrops, the 2D wake vortex region is divided into M × N independent grids whose size is 2 m × 2 m, and in each grid, the number of raindrops for each diameter class is counted. Figure 10 illustrates the number of raindrops in 2D wake vortices for the diameter classes of 0.5, 1, 2, and 4 mm, where the color bar indicates the number of raindrops in each grid. The rain rate is assumed to be 5 mm h−1 and the simulated wake vortex has the characteristics of a stable stage wake vortex generated by an A340. It is interesting to note the enhanced concentration of raindrops between the two vortices and the reduced concentration of raindrops in the columns below the two vortices. Indeed, this heterogeneous concentration of raindrops is one of the elements that may enable our detection of the vortices.

Fig. 10.
Fig. 10.

Illustration of the number of raindrops in 2D wake vortices where the diameter = (a) 0.5, (b) 1.0, (c) 2.0, and (d) 4.0 mm.

Citation: Journal of Atmospheric and Oceanic Technology 30, 3; 10.1175/JTECH-D-11-00220.1

b. X-band radar Doppler signatures

The X-band radar is an interesting sensor for monitoring wake vortex due to its high temporal and spatial resolution and its low cost. In the vicinity of the airport, radar waves at X band are not severely attenuated in rainy weather due to the short detection range of wake vortices. In the simulation, the rain rate is set to be 5 mm h−1 and the input radar parameters are listed in Table 1, referring to the X-band radar parameters for monitoring wake vortices described in Barbaresco and Meier (2010). To learn more about the Doppler signature of raindrops within wake vortices, different radar geometry configurations are considered as shown in Fig. 11.

Table 1.

Input parameters of the X-band radar simulation.

Table 1.
Fig. 11.
Fig. 11.

Different radar geometry configurations for X-band simulations: (a) two vortex cores in two adjacent cells; (b) two vortex cores in two interval cells; (c) elevation angles = 3°, 5°, and 7°; and (d) range resolution = 100 m.

Citation: Journal of Atmospheric and Oceanic Technology 30, 3; 10.1175/JTECH-D-11-00220.1

In Fig. 11a, the concerned radar scattering volume is divided into six continuous range cells where the two vortex cores are located in two adjacent cells, and the range resolution is 40 m. The elevation angle of the radar beam is 5°. Figure 12 shows the spectrum of Doppler radar echoes obtained from the six radar cells under consideration. One can notice that the Doppler spectrum is wider for radar cells 02–05, and especially so for cells 03 and 04 compared with cells 01 and 06, which are farther from the vortex cores. Actually, this phenomenon is caused by the richness of the radial velocities of the scattered raindrops in the given radar cell, as well as the inhomogeneous concentration of raindrops in the wake vortex region. In Barbaresco (2012), the Doppler spectrum of a wake vortex in rain was also reported to be extended periodically while the X-band radar worked in scanning mode. This extended Doppler spectrum indication is a possible signature that denotes the existence of a wake vortex.

Fig. 12.
Fig. 12.

The X-band radar Doppler velocity spectrum of raindrops with vortex cores in two adjacent cells: cells (a) 01, (b) 02, (c) 03, (d) 04, (e) 05, and (f) 06.

Citation: Journal of Atmospheric and Oceanic Technology 30, 3; 10.1175/JTECH-D-11-00220.1

In Fig. 11b, the two vortex cores are located in two-interval radar cells and the radar echoes from five successive radar cells are simulated. In Fig. 13, the Doppler velocity spectra from the raindrops in the three radar cells around the vortices are illustrated. The elevation angle of the antenna is configurable to be 3°, 5°, and 7°, as shown in Fig. 11c. It is easily found that the width of the Doppler velocity spectrum is dependent on the elevation angle. At an elevation angle of 3°, the radar antenna points to the raindrops below the wake vortex cores where the raindrops tend to fall vertically to the ground, as shown in Fig. 6; thus, the richness of the radial velocities of raindrops is less than those at 5° and 7° elevation.

Fig. 13.
Fig. 13.

The X-band radar Doppler velocity spectrum of raindrops with vortex cores in two interval radar cells: cells (a) 02, (b) 03, and (c) 04.

Citation: Journal of Atmospheric and Oceanic Technology 30, 3; 10.1175/JTECH-D-11-00220.1

In Fig. 11d, two vortex cores are located in the same radar resolution volume while the range resolution is set to 100 m. The corresponding Doppler velocity spectrum is illustrated in Fig. 14. In radar cell 02, the Doppler velocity spectrum is almost symmetrically distributed and the width is also dependent upon the elevation angle. In radar cells 01 and 03, the Doppler velocity spectrum of raindrops is not extended and is similar to the Doppler spectrum of raindrops falling in still air, because in these two radar cells, the raindrop trajectories and velocities have not been changed significantly and at elevation angles of 3°, 5°, and 7° the radar radial velocity of the raindrops falling in still air is very small. Actually, the raindrops size distribution in each pulse resolution volume depends on the division of the radar cells, which are defined by the pulse width and the radar antenna beamwidth.

Fig. 14.
Fig. 14.

The X-band radar Doppler velocity spectrum of raindrops with vortex cores in the same radar cell with 100-m range resolution: cells (a) 01, (b) 02, and (c) 03.

Citation: Journal of Atmospheric and Oceanic Technology 30, 3; 10.1175/JTECH-D-11-00220.1

c. W-band radar Doppler signatures

Other recent radar trials attempting to detect wake vortex in rainy weather have been reported upon in Seliga and Mead (2009). The parameters of the W-band radar used for the experiments are illustrated in Table 2. Considering the proposed methodology, this measurement configuration is taken as a reference to simulate the radar signature of the raindrops in wake vortices. The altitude of the A340 wake vortices is assumed to be 200 m above the airport runway and the radar is deployed 1000 m away from the runway. The W-band radar is working in vertical scanning mode, with a scan rate of 5° s−1. The rain rate is set to be 2 mm h−1. The received radar time series at each radar cell are processed to obtain the average power and the Doppler velocity field. In Fig. 15, the average received power and the corresponding Doppler velocity are illustrated separately. The simulated results are very similar to the W-Band radar observations in Seliga and Mead (2009). In the scanned radar reflectivity map, there are two columns with reduced reflectivity from raindrops due to the presence of a pair of wake vortices. Between these two columns, there are several radar cells with enhanced radar reflectivity due to the enhanced concentration of raindrops within wake vortices. The difference in the radar reflectivity level is about 6 dB (the absolute value is not comparable, as the number of simulated raindrops is far lower than the actual one to get reasonable computing times). For the retrieved Doppler velocity map, both the negative and positive peak Doppler velocities are obtained in radar cells around the vortex cores asymmetrically. This kind of information on the radial velocity of raindrops within wake vortices is critical to estimating the vortex circulation for air traffic control applications, as related to the danger associated with encountering a wake vortex. It must be noted that due to the low dimensions of the radar cells and the low number of simulated drops the simulated Doppler information is not relevant in the radar cells with an extremely low reflectivity (i.e., under the vortex cores).

Table 2.

Input parameters of the W-band radar simulation.

Table 2.
Fig. 15.
Fig. 15.

The W-band radar signature of raindrops in wake vortices: (a) average received power and (b) retrieved Doppler velocity field.

Citation: Journal of Atmospheric and Oceanic Technology 30, 3; 10.1175/JTECH-D-11-00220.1

5. Conclusions

This paper has presented a methodology for simulating the radar signature of wake vortices during rainy conditions. The theoretical model for the computation of raindrop trajectories within wake vortices, as well as the interaction between raindrops and electromagnetic waves, has been presented. Typical X- and W-band radar parameters have been used to compute the radar echoes considering all the raindrops within the observation volume. In X-band simulation, the particular shape of the Doppler spectrum of the raindrops within wake vortices can provide potential information that may be helpful for identifying wake vortex hazards in air traffic control operations. In W-band simulations, the enhanced concentration of raindrops between the two vortex cores due to the influence of vortex flow is clearly observed; to some extent, this phenomenon is consistent with the experimental results in Seliga and Mead (2009). The simulation results will later be validated by radar data observations of wake vortices taken during rainy weather. At this stage, the relevance of some of the simplifications and assumptions will be evaluated and if needed, the effects of crosswinds, wind shear, and atmospheric turbulence on raindrop motion or collisions between raindrops will be taken into account. The simulated radar signature of wake vortices will then be used to evaluate different algorithms to detect vortices from the radar signal and also to retrieve some vortex parameters such as the circulation or the vortex core separation and radius. It will hence enable to assess the most relevant radar configurations for the observation of wake vortices in rainy weather.

Acknowledgments

The authors thank Mr. Florent Christophe (ONERA/DEMR) and Mr. Frederic Barbaresco (THALES AIR OPERATIONS) for their helpful supervisions of this work. The authors strongly acknowledge the anonymous referees for their help and suggestions.

REFERENCES

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    • Search Google Scholar
    • Export Citation
  • Atlas, D., Srivastava R. C. , and Sekhon R. S. , 1973: Doppler radar characteristics of precipitation at vertical incidence. Rev. Geophys.,11, 1–35.

  • Barbaresco, F., 2012: Radar/lidar sensors for wind & wake-vortex monitoring on airport: First results of SESAR P12.2.2 XP0 trials campaign at Paris CDG airport. Wake Turbulence in Current Operations and Beyond: WakeNet3–Europe Fourth Major Workshop, Langen, Germany, European Cockpit Association and DFS Deutsche Flugsicherung GmbH.

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    • Search Google Scholar
    • Export Citation
  • Barbaresco, F., Jeantet A. , and Meier U. , 2007: Wake vortex detection & monitoring by x-band Doppler radar: Paris Orly radar campaign results. Proc. Int. Conf. on Radar Systems, Edinburgh, United Kingdom, Institution of Engineering and Technology.

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    • Search Google Scholar
    • Export Citation
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    • Search Google Scholar
    • Export Citation
  • Cheong, B. L., Palmer R. D. , and Xue M. , 2008: A time series weather radar simulator based on high-resolution atmospheric models. J. Atmos. Oceanic Technol.,25, 230–243.

  • Dowell, D. C., Alexander C. R. , Wurman J. M. , and Wicker L. J. , 2005: Centrifuging of hydrometeors and debris in tornadoes: Radar-reflectivity patterns and wind-measurement errors. Mon. Wea. Rev.,133, 1501–1524.

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  • Gerz, T., Holzpfel F. , and Darracq D. , 2002: Commercial aircraft wake vortices. Prog. Aerosp. Sci.,38, 181–208, doi:10.1016/S0376-0421(02)00004-0.

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    • Search Google Scholar
    • Export Citation
  • Hanson, J. M., and Marcotte F. J. , 1997: Aircraft wake vortex detection using continuous-wave radar. Johns Hopkins APL Tech. Dig., 18, 348357.

    • Search Google Scholar
    • Export Citation
  • Hinton, D. A., and Tatnall C. R. , 1997: A candidate wake vortex strength definition for application to the NASA Aircraft Vortex Spacing System (AVOSS). NASA Tech. Memo. 110343, 32 pp.

  • Holzpfel, F., Gerz T. , Kpp F. , Stumpf E. , Harris M. , Young R. I. , and Dolfi-Bouteyre A. , 2003: Strategies for circulation evaluation of aircraft wake vortices measured by lidar. J. Atmos. Oceanic Technol.,20, 1183–1195.

  • Iannuzzelli, R. J., and Schemm C. E. , 1998: Aircraft wake detection using bistatic radar: Analysis of experimental results. Johns Hopkins APL Tech. Dig., 19, 299314.

    • Search Google Scholar
    • Export Citation
  • Jameson, A. R., and Kostinski A. B. , 2010: Partially coherent backscatter in radar observations of precipitation. J. Atmos. Sci.,67, 1928–1946.

  • Jiang, H., Sano M. , and Sekine M. , 1997: Weibull raindrop-size distribution and its application to rain attenuation. Microwaves, Antennas Propag., 144, 197200, doi:10.1049/ip-map:19971193.

    • Search Google Scholar
    • Export Citation
  • Li, J., Wang X. , Wang T. , and Liu Z. , 2009: Study on the scattering characteristics of stable-stage wake vortices. Proc. Int. Radar Conf., Paris, France, IEEE.

  • Li, J., Wang X. , and Wang T. , 2011: Modeling the dielectric constant distribution of wake vortices. Aerosp. Electron. Syst.,47, 820–831, doi:10.1109/TAES.2011.5751228.

  • Lovejoy, S. and Schertzer D. , 2008: Turbulence, raindrops and the I1/2 number density law. New J. Phys.,10, 075017, doi:10.1088/1367-2630/10/7/075017.

  • Marshall, J. S., and Palmer W. M. , 1948: The distribution of raindrops with size. J. Meteor., 5, 165166.

  • Marshall, R. E., and Myers T. , 1996: Wingtip generated wake vortices as radar target. IEEE Aerosp. Electron. Syst. Mag., 11, 2730, doi:10.1109/62.544796.

    • Search Google Scholar
    • Export Citation
  • Myers, T. J., and Scales W. A. , 1999: Determination of aircraft wake vortex radar cross section due to coherent bragg scatter from mixed atmospheric water vapor. Radio Sci.,34, 103–111.

  • Nespor, J. D., Hudson B. , Stegall R. L. , and Freedman J. E. , 1994: Doppler radar detection of vortex hazard indicators. NASA Tech. Memo. 19950006799, 38 pp.

  • Owolawi, P. A., 2010: Characteristics of rain at microwave and millimetric bands for terrestrial and satellite links attenuation in South Africa and surrounding islands. Ph.D. thesis, University of KwaZulu-Natal.

  • Proctor, F. H., Hamilton D. W. , Rutishauser D. K. , and Switzer G. F. , 2004: Meteorology and wake vortex influence on American Airlines FL-587 accident. NASA Tech. Memo. NASA/TM-2004-213018, 60 pp.

  • Qu, L., Li J. , Wang T. , Zhao Z. , and Wang X. , 2010: Simulation on the evolution and rcs of aircraft wake vortices. Int. Conf. on Computer Application and System Modeling, Vol. 2, Taiyuan, China, IEEE, V2-376 –V2-379, doi:10.1109/ICCASM.2010.5620751.

  • Richards, M. A., Scheer A. , and Holm W. A. , 2010: Principles of Modern Radar: Basic Principles. SciTech Publishing, 924 pp.

  • Rubin, W. L., 2000: Radar-acoustic detection of aircraft wake vortices. J. Atmos. Oceanic Technol.,17, 1058–1065.

  • Sauvageot, H., 1992: Radar Meteorology. Artech House, 366 pp.

  • Seliga, T. A., and Mead J. B. , 2009: Meter-scale observations of aircraft wake vortices in precipitation using a high resolution solid-state W-band radar. Preprints, 34th Conf. on Radar Meteorology, Williamsburg, VA, Amer. Meteor. Soc., P10.25. [Available online at http://ams.confex.com/ams/pdfpapers/155796.pdf.]

  • Shariff, K. and Wray A. , 2002: Analysis of the radar reflectivity of aircraft vortex wakes. J. Fluid Mech.,463, 121–161.

  • Shephard, D., Kyte A. , and Segura C. , 1994: Radar wake vortex measurements at F and I band. Colloquium on Radar and Microwave Imaging, London, United Kingdom, IEEE, 7/1–7/5.

  • Vanhoenacker-Janvier, D., Kahina D. , and Barbaresco F. , 2012: Model for the calculation of the radar cross section of wake vortices of take-off and landing airplanes. European Radar Conference (EuRAD 2012), Amsterdam, The Netherlands, European GAAS Association and IEEE.

  • Winckelmans, G., Duquesne T. , Treve V. , Desenfans O. , and Bricteux L. , 2005: Summary description of the models used in the Vortex Forecast System (VFS)—VFS version with added improvements done after completion of the Transport Canada funded project. Catholic University of Louvain, 18 pp.

  • Yu, T.-Y., Wang Y. , Shapiro A. , and Yeary M. B. , Zrnić D. S. , and Doviak R. J. , 2007: Characterization of tornado spectral signatures using higher-order spectra. J. Atmos. Oceanic Technol.,24, 1997–2013.

  • Zhang, Y., Wang F. Y. , and Hardin J. C. , 2003: Spectral characteristics of wake vortex sound during roll-up. NASA Tech. Memo. 20040013294, 30 pp.

Save
  • Atlas, D., and Ulbrich C. W. , 1974: The physical basis for attenuation-rainfall relationships and the measurement of rainfall parameters by combined attenuation and rader methods. J. Rech. Atmos., 8, 275298.

    • Search Google Scholar
    • Export Citation
  • Atlas, D., Srivastava R. C. , and Sekhon R. S. , 1973: Doppler radar characteristics of precipitation at vertical incidence. Rev. Geophys.,11, 1–35.

  • Barbaresco, F., 2012: Radar/lidar sensors for wind & wake-vortex monitoring on airport: First results of SESAR P12.2.2 XP0 trials campaign at Paris CDG airport. Wake Turbulence in Current Operations and Beyond: WakeNet3–Europe Fourth Major Workshop, Langen, Germany, European Cockpit Association and DFS Deutsche Flugsicherung GmbH.

  • Barbaresco, F., and Meier U. , 2010: Radar monitoring of a wake vortex: Electromagnetic reflection of wake turbulence in clear air. C. R. Phys., 11, 5467.

    • Search Google Scholar
    • Export Citation
  • Barbaresco, F., Jeantet A. , and Meier U. , 2007: Wake vortex detection & monitoring by x-band Doppler radar: Paris Orly radar campaign results. Proc. Int. Conf. on Radar Systems, Edinburgh, United Kingdom, Institution of Engineering and Technology.

  • Barbaresco, F., Wasselin J. P. , Jeantet A. , and Meier U. , 2008: Wake vortex profiling by Doppler X-band radar: Orly trials at initial take-off & ILS interception critical areas. Proc. RADAR ’08, Rome, Italy, IEEE, doi:10.1109/RADAR.2008.4721113.

  • Barbaresco, F., Juge P. , Klein M. , Ricci Y. , Schneider J. , and Moneuse J. , 2011: Optimising runway throughput through wake vortex detection, prediction and decision support tools. 2011 Tyrrhenian Int. Workshop on Digital Communications—Enhanced Surveillance of Aircraft and Vehicles, Capri, Italy, IEEE, 27–32.

  • Bobylev, A. V., Vyshinsky V. V. , Soudakov G. G. , and Yaroshevsky V. A. , 2010: Aircraft vortex wake and flight safety problems. J. Aircr., 47, 663674.

    • Search Google Scholar
    • Export Citation
  • Bradley, S., Mursch-Radlgruber E. , and von Hünerbein S. , 2007: Sodar measurements of wing vortex strength and position. J. Atmos. Oceanic Technol.,24, 141–155.

  • Bringi, V. N., and Chandrasekar V. , 2004: Polarimetric Doppler Weather Radar: Principles and Applications. Cambridge University Press, 636 pp.

  • Capsoni, C., and D’Amico M. , 1998: A physically based radar simulator. J. Atmos. Oceanic Technol., 15, 593598.

  • Capsoni, C., D’Amico M. , and Nebuloni R. , 2001: A multiparameter polarimetric radar simulator. J. Atmos. Oceanic Technol., 18, 17991809.

    • Search Google Scholar
    • Export Citation
  • Cheong, B. L., Palmer R. D. , and Xue M. , 2008: A time series weather radar simulator based on high-resolution atmospheric models. J. Atmos. Oceanic Technol.,25, 230–243.

  • Dowell, D. C., Alexander C. R. , Wurman J. M. , and Wicker L. J. , 2005: Centrifuging of hydrometeors and debris in tornadoes: Radar-reflectivity patterns and wind-measurement errors. Mon. Wea. Rev.,133, 1501–1524.

  • Fang, F., 2003: Raindrop size distribution retrieval and evaluation using an S-band radar profiler. M.S. thesis, Dept. of Electrical and Computer Engineering, University of Central Florida, 180 pp.

  • Frehlich, R. and Sharman R. , 2005: Maximum likelihood estimates of vortex parameters from simulated coherent doppler lidar data. J. Atmos. Oceanic Technol.,22, 117–130.

  • Gerz, T., Holzpfel F. , and Darracq D. , 2002: Commercial aircraft wake vortices. Prog. Aerosp. Sci.,38, 181–208, doi:10.1016/S0376-0421(02)00004-0.

  • Ginevsky, A., and Zhelannikov A. , 2009: Vortex Wakes of Aircrafts, Foundations of Engineering Mechanics. Foundations of Engineering Mechanics, Springer-Verlag, 154 pp.

  • Gunn, R. L. S., and East T. W. R. , 1954: The microwave properties of precipitation particles. Quart. J. Roy. Meteor. Soc., 80, 522545.

    • Search Google Scholar
    • Export Citation
  • Hanson, J. M., and Marcotte F. J. , 1997: Aircraft wake vortex detection using continuous-wave radar. Johns Hopkins APL Tech. Dig., 18, 348357.

    • Search Google Scholar
    • Export Citation
  • Hinton, D. A., and Tatnall C. R. , 1997: A candidate wake vortex strength definition for application to the NASA Aircraft Vortex Spacing System (AVOSS). NASA Tech. Memo. 110343, 32 pp.

  • Holzpfel, F., Gerz T. , Kpp F. , Stumpf E. , Harris M. , Young R. I. , and Dolfi-Bouteyre A. , 2003: Strategies for circulation evaluation of aircraft wake vortices measured by lidar. J. Atmos. Oceanic Technol.,20, 1183–1195.

  • Iannuzzelli, R. J., and Schemm C. E. , 1998: Aircraft wake detection using bistatic radar: Analysis of experimental results. Johns Hopkins APL Tech. Dig., 19, 299314.

    • Search Google Scholar
    • Export Citation
  • Jameson, A. R., and Kostinski A. B. , 2010: Partially coherent backscatter in radar observations of precipitation. J. Atmos. Sci.,67, 1928–1946.

  • Jiang, H., Sano M. , and Sekine M. , 1997: Weibull raindrop-size distribution and its application to rain attenuation. Microwaves, Antennas Propag., 144, 197200, doi:10.1049/ip-map:19971193.

    • Search Google Scholar
    • Export Citation
  • Li, J., Wang X. , Wang T. , and Liu Z. , 2009: Study on the scattering characteristics of stable-stage wake vortices. Proc. Int. Radar Conf., Paris, France, IEEE.

  • Li, J., Wang X. , and Wang T. , 2011: Modeling the dielectric constant distribution of wake vortices. Aerosp. Electron. Syst.,47, 820–831, doi:10.1109/TAES.2011.5751228.

  • Lovejoy, S. and Schertzer D. , 2008: Turbulence, raindrops and the I1/2 number density law. New J. Phys.,10, 075017, doi:10.1088/1367-2630/10/7/075017.

  • Marshall, J. S., and Palmer W. M. , 1948: The distribution of raindrops with size. J. Meteor., 5, 165166.

  • Marshall, R. E., and Myers T. , 1996: Wingtip generated wake vortices as radar target. IEEE Aerosp. Electron. Syst. Mag., 11, 2730, doi:10.1109/62.544796.

    • Search Google Scholar
    • Export Citation
  • Myers, T. J., and Scales W. A. , 1999: Determination of aircraft wake vortex radar cross section due to coherent bragg scatter from mixed atmospheric water vapor. Radio Sci.,34, 103–111.

  • Nespor, J. D., Hudson B. , Stegall R. L. , and Freedman J. E. , 1994: Doppler radar detection of vortex hazard indicators. NASA Tech. Memo. 19950006799, 38 pp.

  • Owolawi, P. A., 2010: Characteristics of rain at microwave and millimetric bands for terrestrial and satellite links attenuation in South Africa and surrounding islands. Ph.D. thesis, University of KwaZulu-Natal.

  • Proctor, F. H., Hamilton D. W. , Rutishauser D. K. , and Switzer G. F. , 2004: Meteorology and wake vortex influence on American Airlines FL-587 accident. NASA Tech. Memo. NASA/TM-2004-213018, 60 pp.

  • Qu, L., Li J. , Wang T. , Zhao Z. , and Wang X. , 2010: Simulation on the evolution and rcs of aircraft wake vortices. Int. Conf. on Computer Application and System Modeling, Vol. 2, Taiyuan, China, IEEE, V2-376 –V2-379, doi:10.1109/ICCASM.2010.5620751.

  • Richards, M. A., Scheer A. , and Holm W. A. , 2010: Principles of Modern Radar: Basic Principles. SciTech Publishing, 924 pp.

  • Rubin, W. L., 2000: Radar-acoustic detection of aircraft wake vortices. J. Atmos. Oceanic Technol.,17, 1058–1065.

  • Sauvageot, H., 1992: Radar Meteorology. Artech House, 366 pp.

  • Seliga, T. A., and Mead J. B. , 2009: Meter-scale observations of aircraft wake vortices in precipitation using a high resolution solid-state W-band radar. Preprints, 34th Conf. on Radar Meteorology, Williamsburg, VA, Amer. Meteor. Soc., P10.25. [Available online at http://ams.confex.com/ams/pdfpapers/155796.pdf.]

  • Shariff, K. and Wray A. , 2002: Analysis of the radar reflectivity of aircraft vortex wakes. J. Fluid Mech.,463, 121–161.

  • Shephard, D., Kyte A. , and Segura C. , 1994: Radar wake vortex measurements at F and I band. Colloquium on Radar and Microwave Imaging, London, United Kingdom, IEEE, 7/1–7/5.

  • Vanhoenacker-Janvier, D., Kahina D. , and Barbaresco F. , 2012: Model for the calculation of the radar cross section of wake vortices of take-off and landing airplanes. European Radar Conference (EuRAD 2012), Amsterdam, The Netherlands, European GAAS Association and IEEE.

  • Winckelmans, G., Duquesne T. , Treve V. , Desenfans O. , and Bricteux L. , 2005: Summary description of the models used in the Vortex Forecast System (VFS)—VFS version with added improvements done after completion of the Transport Canada funded project. Catholic University of Louvain, 18 pp.

  • Yu, T.-Y., Wang Y. , Shapiro A. , and Yeary M. B. , Zrnić D. S. , and Doviak R. J. , 2007: Characterization of tornado spectral signatures using higher-order spectra. J. Atmos. Oceanic Technol.,24, 1997–2013.

  • Zhang, Y., Wang F. Y. , and Hardin J. C. , 2003: Spectral characteristics of wake vortex sound during roll-up. NASA Tech. Memo. 20040013294, 30 pp.

  • Fig. 1.

    Characteristics of raindrops in still air. (a) Raindrops size distributions for R = 2, 5, and 10 mm h−1; the diameter interval is 0.1 mm. (b) Raindrop terminal falling velocities at altitudes of 50, 1000, and 4000 m.

  • Fig. 2.

    Breakdown of the aircraft vortex wake in accordance with its evolution phases (Ginevsky and Zhelannikov 2009).

  • Fig. 3.

    The velocity field of two counterrotating vortices (Li et al. 2011).

  • Fig. 4.

    Two-dimensional velocity distribution of the wake vortices generated by an A340 airplane.

  • Fig. 5.

    Interaction between raindrops and wake vortices.

  • Fig. 6.

    The trajectory of raindrops in wake vortices: (a) D = 0.5 mm, duration = 106.6 s; (b) D = 1.0 mm, duration = 62.2 s; (c) D = 2.0 mm, duration = 41.0 s; and (d) D = 4.0 mm, duration = 31.8 s.

  • Fig. 7.

    Backscattering cross section of a raindrop as a function of its diameter.

  • Fig. 8.

    The geometry of radar configuration.

  • Fig. 9.

    The procedure of generating a radar signal time series.

  • Fig. 10.

    Illustration of the number of raindrops in 2D wake vortices where the diameter = (a) 0.5, (b) 1.0, (c) 2.0, and (d) 4.0 mm.

  • Fig. 11.

    Different radar geometry configurations for X-band simulations: (a) two vortex cores in two adjacent cells; (b) two vortex cores in two interval cells; (c) elevation angles = 3°, 5°, and 7°; and (d) range resolution = 100 m.

  • Fig. 12.

    The X-band radar Doppler velocity spectrum of raindrops with vortex cores in two adjacent cells: cells (a) 01, (b) 02, (c) 03, (d) 04, (e) 05, and (f) 06.

  • Fig. 13.

    The X-band radar Doppler velocity spectrum of raindrops with vortex cores in two interval radar cells: cells (a) 02, (b) 03, and (c) 04.

  • Fig. 14.

    The X-band radar Doppler velocity spectrum of raindrops with vortex cores in the same radar cell with 100-m range resolution: cells (a) 01, (b) 02, and (c) 03.

  • Fig. 15.

    The W-band radar signature of raindrops in wake vortices: (a) average received power and (b) retrieved Doppler velocity field.

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