1. Introduction

The safe spacing between flight path aircraft is largely determined by the general downwash following each aircraft due to its wingtip vortices. Current methods for vortex detection and estimation of strength, such as various radar, lidar, and massive microphone array configurations, are generally expensive and complex. This remains a largely unsolved problem in spite of large U.S. and European Union (EU) projects, and becomes more urgent with the planned introduction of larger aircraft such as the Airbus A380 (Gerz et al. 2002; Luckner et al. 2004; Perry et al. 1997; Proctor and Switzer 2000).

The most common vortex-detecting systems to date are lidars. Gerz et al. (2002) summarize the state of the art. They state that lidars, especially the combination of a pulsed lidar and several continuous wave (CW) lidars, are capable of identifying and tracking trailing vortices reliably. They point out that to date lidar seems to be the only technology capable of operating reliably under most meteorological conditions but not in fog or heavy rain [although the radar-acoustic method of Rubin (2000) shows promise in these conditions]. Recent results of Frehlich and Sharman (2005) and Köpp et al. (2005) show that vortex positions between CW lidars and pulsed lidars show differences of 9 and 13 m for the vertical and horizontal components, respectively. The accuracy of the lidar-based estimates of vortex circulation is given as 13 m² s⁻¹. Because of the large range of pulsed lidars, they can track the vortices from generation to an advanced state of decay (Frehlich and Sharman 2005). However, range resolution is much higher for the CW lidars with their comparatively short maximum range of a few hundred meters. Therefore, wake vortices can be more easily identified and their internal structure better investigated. However, Gerz et al. (2002) state that an extensive amount of postprocessing is required for lidar data to achieve reliable results.

Other vortex-detecting technologies include microphone arrays (Dougherty et al. 2004), infrasound sensors (Rubin 2005), and radar-acoustic sensors (Rubin 2000). A comprehensive set of sensors including sound detection and ranging (sodar), lidar, and Radio Acoustic Sounding System (RASS), as described in Barker et al. (1997), has been tested for a number of years at New York’s JFK Airport, leading to an integrated set of measurements to feed into the air traffic control Advanced Vortex Spacing System (AVOSS; Dasey and Hinton 1999).

Wake vortex life spans of a few tens of seconds to several minutes have been observed (see, e.g., Dasey and Hinton 1999). An important consideration is the vortex decay rate (Proctor and Switzer 2000; Holzäpfel 2006) and vortex instabilities (Leweke and Williamson 1998). Dasey and Hinton (1999) have summarized the atmospheric processes interacting with vortices. They mention advection, vertical wind shear, small- and large-scale turbulence, as well as stratification. The importance of atmospheric conditions has been widely recognized (see, e.g., Gerz et al. 2002; Spalart 1998), and measurements of sufficient temporal, spatial resolution, and availability are a major obstacle for improvements of vortex decay models (Gerz et al. 2002).

Limited measurements at airports have indicated that sodar, an established tool for acoustic profiling of atmospheric turbulence and winds, can give inexpensive quantitative real-time wind fields from aircraft wing vortices during landing and take off (Burnham and Hallock 1982; Burnham 1997; Burnham and Rudis 1997; E. Mursch-Radlgruber 2000, unpublished manuscript; Mursch-Radlgruber et al. 2004). Bradley et al. (2006) have investigated in detail the data availability for the vortex application, and found that data availability should generally be higher than experienced in more conventional sodar applications. This has some attractiveness, since although heavy rain contaminates data from all current commercially available sodars, fog does not pose a problem. However, although the basic methodology is well established, sodar has not yet been optimized for the vortex measurement task. This requires good-quality nonaveraged profiles of wind speed but only to heights of around 80 m. Also, the results from E. Mursch-Radlgruber (2000, unpublished manuscript) and Mursch-Radlgruber (2004) show wind perturbations from vortices but do not interpret these in terms of vortex strength and geometry. Burnham (1997) describes use of both a vertical-beam sodar and a fan-shaped beam sodar in the AVOSS project, and there is scope for such specializations in sodar design.

Regardless of the manner in which real-time vortex information is conveyed to air traffic coordinators, the essential underlying parameters are the vortex strength (or circulation) and position in space as a function of time. The purpose of this paper is to provide a theoretical framework by which these parameters can be obtained from a modest sodar array, in real time, and with confidence indicators. In the process of developing this methodology, a very simple theory of vortex development is used to derive characteristic time and length scales that might be used as simple guidance for expectations of real vortex behavior. The parameter retrieval framework is applied to the short sequence of observational data already described by E. Mursch-Radlgruber (2000, unpublished manuscript) and Mursch-Radlgruber (2004), and its accuracy and usefulness discussed.

2. Geometry of wing vortices

Aircraft wings change the direction of the airflow, giving a downward component, due to the equivalent to a bound vortex of circulation Γ around each wing. The Helmholtz vortex theorems state that, in the absence of viscous dissipation, this vortex cannot end within the fluid, and so counterrotating vortex tubes of circulation Γ trail back from the end of each wing (see Fig. 1). The downward rate of change of momentum of the air is matched by the lift on the aircraft. The Kutta–Joukowski lift theorem gives lift per unit length as

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where M_ac is the aircraft mass, g is the gravitational acceleration, s is the half-spacing of the trailing vortices, ρ is air density, V_ac is aircraft speed, and Γ is the circulation associated with one of the trailing vortices (Anderson 2001). The vortex spacing, 2s, is about π/4 of the wing span (Dougherty et al. 2004) for an elliptically loaded wing.

The simplest approximation is an inviscid incompressible uniform windless atmosphere and no ground surface. In this case potential flow solutions to the Navier–Stokes equations are applicable, and the tangential velocity due to a simple vortex at distance r from its center is

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This vortex model has frequently been used as a very simple indicator of aircraft vortex behavior (e.g., Greene 1986; Proctor and Switzer 2000). Stewart (1991) found that predicted by potential theory agreed with measured values for many cases.

Other more realistic simple vortex models include the Oseen vortex (Rossow 1999) with

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or

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(Burnham and Hallock 1982), which give a vortex core of width about r = ±r₀ where the velocities are reduced. Other models, such as Corjon and Poinsot (1996, 1997), examine interactions between vortices and the detailed atmospheric stratification, turbulence, wind shear, and advection. This model is extended by Speijker et al. (2000) to include probabilistic wind field models.

When the aircraft is near the ground the model must predict zero flow through the ground surface. This is satisfied by including two image vortices below the two wing vortices of the same strength Γ, as shown in Fig. 2. If a uniform horizontal wind U is present, the four vortices are translated horizontally at the wind speed, the center point of the two wing vortices being at position (x_c, z_c) with respect to an origin at the ground beneath the flight path.

Each vortex is subjected to the velocity field from the other three vortices and to the drift due to the crosswind. The components acting on the center of the starboard vortex are

1. a downward velocity component of Γ/4πs due to the port vortex,

2. a horizontal velocity component of Γ/4πz_c due to the starboard image vortex, and

3. a diagonal velocity component of Γ/4πs² + z²_c due to the port image vortex.

The diagonal component comprises a horizontal component − (Γ/4π)(z_c/s² + z²_c) and a vertical component (Γ/4π)(s/s² + z²_c). Addition gives

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From this

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From (3) and (4)

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giving

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where t is the time elapsed since the vortex half-spacing and height were s₀ and z₀. A similar expression is obtained for z_c. Equation (5a) can be rearranged in the form

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showing that a natural length scale is R and a natural time scale is

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The corresponding natural velocity scale is U_* = R/T = Γ/2πR. Coustols et al. (2004) use a similar time normalization but based on s₀ rather than R, together with a velocity normalization based on V_ac. Note that other authors have suggested characteristic time and length scales based on vortex decay behavior (Gerz et al. 2002; Holzäpfel et al. 2003; Spalart 1998).

The horizontal and vertical velocity components (u, w) at any point (x, y) are the sum of the components from the four vortices:

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where

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Table 1 gives typical values of parameters for a B747-400.

Figure 3a shows the path and shape of the closed streamlines for scaled streamfunction values of 1 and 1.5 at scaled time steps of 1.5, using the parameters defined in Table 1, and for U = 0. In Fig. 3b the streamlines are translated horizontally according to U = 0.5U_*. The streamlines are essentially circles, with centers displaced from the (s, z_c) position. The stronger velocities toward the flight path and toward the ground are evident. Note that this model does not include vortex decay due to entrainment or viscous (turbulent) losses, and wind shear is neglected.

3. Sodar measurements of wing vortices

A sodar emits an acoustic pulse of typically 50-ms duration and then monitors the acoustic echo from turbulent refractive index irregularities. Spectral analysis of successive subsets of the echo time series allow radial wind velocity components to be estimated based on the Doppler shift. Sequential pulse transmissions into off-vertical directions in orthogonal planes allows three-component vector winds to be estimated typically every 5 or 10 m in the vertical to heights of 200–1000 m (depending on the sodar design).

The scattering volume from which echoes originate at any particular time is defined by the antenna geometry, characterized by a conical beam of angular half-width Δϕ, and the pulse modulation. For simplicity, consider a pulsed system without pulse shaping producing a pulse of constant amplitude and duration τ. The sampling volume is then

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where Δx = (ctΔϕ)/2, Δz = cτ/2, and c is the speed of sound.

Assume that the sodar profiles vertically at x = x_s. For a range gate centered on height z_s, the finite spatial resolution of the sodar causes volume-averaging over the velocity components to give

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Note that we have just assumed that each velocity is weighted according to the size of the incremental volume around it: this corresponds to assuming that the scattering cross section (i.e., turbulence level) is uniform throughout the vortex.

These integrals can be solved analytically, but the resulting expressions are lengthy. A good approximation is to solve the simpler case of a rectangular box shape sampling volume, as shown in Fig. 4, but to allow the box width Δx to vary with range gate height. Consider the starboard vortex. The volume-averaged vertical velocity component is

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where

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The horizontal profile across the vortex at height z_c gives

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and it is clear that w depends on both the horizontal and vertical resolution. The volume-averaged vertical velocity for the four-vortex model comprises four terms like that in (12), but with appropriately chosen volume limits such as those in (13). Figure 5 shows horizontal profiles of scaled four-vortex w as a function of scaled x_s for two vertical resolutions corresponding to 1 and 10 m. The peak vertical velocities are about a factor of 2 higher for the finer-resolution sodar. However, similar peak velocities are recorded with a vertical resolution of 5 m and beam half-width of 2° as for a vertical resolution of 1 m and a beam half-width of 5°. The former is achievable with standard phased-array sodar technology. Figure 6 compares predictions from the four-vortex model with those from a single-vortex model at two different times (or heights). The shape and magnitude of the vertical velocity cross section are captured by the single-vortex model when the two vortices are separated by scaled s > 4. This suggests that, except perhaps at initial vortex generation height, a single-vortex model may adequately describe vortex shape and magnitude. A similar analysis applies to the horizontal velocity component.

In anticipation of the sections that follow, Fig. 7 shows the vertical velocities that would be expected to be recorded from an array of four sodars spaced 25 m apart (at 0, 25, 50, and 75 m from the flight path) on a line perpendicular to the runway, and sampling at every 10-m height range. For the data in Table 1, the sodar spacing is equivalent to a scaled spacing of 1.05 and the range gates are a scaled distance of 0.42 apart. This means that, with x₀ = 0, the center of the starboard vortex initially lies directly over the sodar at 25 m. The first sodar, at x_s = 0 m, always records a downdraft from the edge of the starboard vortex, which progressively weakens as the vortex descends and moves to the right. Volume averaging means that the sodar at 25 m initially records w = 0. As the vortices descend and the starboard vortex moves to the right, this sodar measures more downdraft, until the starboard vortex has moved so far that the second sodar sees a very weak downdraft. The third sodar records an updraft, but ultimately the vortex moves across this sodar. The fourth sodar records similarly, but the vortex center passes between the range gates at 20 and 30 m, so the vortex amplitude is never as large as that recorded by the third sodar.

Figure 8 shows the same sodar configuration, but with a horizontal wind speed of U = 2.5 m s⁻¹. This wind speed is higher than the rate of change of s and so the port vortex is translated across the sodar array. Each sodar generally records a downdraft followed by an updraft, in contrast to the situation in Fig. 7.

The above analysis indicates that this sodar configuration really undersamples the vortices both in the horizontal and the vertical. The vortex natural scale length is R = 24 m, compared with horizontal sodar sampling every 25 m and vertical sampling every 10 m. Valid and useful data can still be obtained at these spacings, but interpretation would be much easier if twice the resolution were used in each direction.

4. Sodar field measurements

E. Mursch-Radlgruber (2000, unpublished manuscript) describes field measurements at the Vienna, Austria, airport. A linear array of four sodars measuring vertical Doppler shifts were spaced 25 m apart at right angles to the flight path on the starboard side, with the first sodar under the flight path. Single-shot profiles of vertical velocity were obtained every 2 s. Additionally, a four-beam sodar was placed near the center of the linear array, giving three-component winds. In both cases the vertical resolution was 10 m and wind data were obtained at 10-, 20-, . . . , 80-m height.

Figure 9a shows typical horizontal velocity profiles from the four-beam sodar. The difficulty in analyzing these is that the mean horizontal wind needs to be subtracted at each height in order to isolate the vortex perturbation. Consequently, and because deployment of vertical-only sodars will be simpler operationally, our analysis below concentrates on vertical velocity profiles. Figure 9b shows a typical dataset from the four vertical-pointing sodars during one aircraft landing. This shows some features in common with Figs. 7 and 8, but requires considerable interpretation.

5. Real-time estimation of vortex strength and position

The sodar characteristics Δx, Δz are known and at each 2-s time interval w values are measured at four x_s locations and eight heights z_s. Figures 7 and 8 suggest that an “event” lasts around 5T to 7T, or about 70–100 s, which is consistent with Fig. 9. This means there are about 4 × 8 × (35–50) = 1100–1600 observations per event. There are five unknowns, x_c, z_c, s, Γ, and U if only vertical profilers are used. But all of these, except possibly U, are changing throughout the landing event.

One of the major challenges in vortex prediction is a sufficient input of meteorological data to adequately model the interaction between vortices and the atmosphere (Gerz et al. 2002). To provide a useful dataset from sodar observations, it is important that the procedure adopted to retrieve the required parameters x_c, z_c, and Γ not make assumptions regarding the dynamical interactions between the vortices and the atmosphere. The simple four-vortex model described above is useful for indicating likely behavior and characteristic scales, and for first interpretations of recorded data. However, this model, or more complex models, should not be used in the parameter estimation algorithm if they involve temporal development. On the other hand, both the single-vortex model and the four-vortex model can be used to provide a “snapshot” of the gross vortex system shape, without invoking the dynamical interactions driving the vortex path. Coustols et al. (2004) describe a trial using a similar approach with a single-vortex model.

By fitting the observations to either model independently at each time step, no assumptions regarding dynamical interactions with the atmosphere are included, other than that the vortex has a shape predicted by potential flow. This approach also means that the technique is capable of real-time estimation and display of vortex position and strength.

We use the well-known Levenberg–Marquardt method of fitting a function, which is nonlinear in its parameters, to observations in a least squares sense. The code used is a modified version of the MathWorks nonlinear least squares routine, with some bugs corrected (see http://w3eos.whoi.edu/12.747/resources/lsq/nlleasqr.m). Here we wish to minimize

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where the four instruments are sited at x_s,i (i = 1, 2, 3, 4) and the range gates are centered on heights z_s,h (h = 1, 2, . . . , 8). There are N = 32 observations at each time step. The nonlinear function is, from (12),

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and there are M = 3 parameters (x_c + s, z_c, and Γ) for the single-vortex (V = 1) model, and M = 4 parameters (x_c, z_c, Γ, and s) for the four-vortex (V = 4) model. The sampling volume limits and rotation sense are

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Before performing parameter estimations on real data, we first apply the fitting method to synthetic w data based on Fig. 7 but with added Gaussian random noise, having zero mean and σ_w standard deviation, at each of the 32 observation points. For this test the single-vortex model is used. Figure 10 shows the RMS position error for σ_w = 0, 0.2, 0.4, 0.5, and 0.6 m s⁻¹. The position uncertainty in this simulation is 1–2 m. Figure 11 shows the theoretical and estimated path of the vortex and demonstrates that the position error occurs in both horizontal and vertical directions. Gaps in the data sequence are due to nonconvergence of the iterative least squares process in some instances. When nonconvergence occurs, that particular time step is ignored. We have not explored whether application of other constraints would remove these nonconvergence cases. The RMS uncertainty in retrieved Γ values over this same range of vortex movement is 24 m² s⁻¹. Figure 12 shows the estimated Γ values and the parameter uncertainties predicted from the nonlinear least squares fit. Note that the fitted model is a single-vortex model without any vortex interactions and treating each time step as totally independent.

These simulations provide some confidence in the proposed parameter estimation scheme. In particular, there is a relatively smooth progression in all estimated parameters in the expected manner.

6. Results of estimation procedure

A 45-min time series of vertical velocities from the four sodars was analyzed. The record was visually searched for sequences of around 1 min that appeared to contain vortices. In this way the start and end times for 20 such sequences were identified. Each of these sequences was then used for vortex parameter estimation with both the single-vortex and the four-vortex model functions.

The following constraints were used in the Levenberg–Marquardt method.

1. Equal weighting was applied to each observation.

2. The maximum fractional change in each parameter between iterations was 0.2.

3. Fitting was abandoned if

   • a) convergence did not occur within 40 iterations or

   • b) Γ < 50 m² s⁻¹ or

   • c) Γ > 400 m² s⁻¹ (for the case studies discussed here).

4. Fitting was completed when

   • a) the fractional change in χ² < 10⁻⁴ or

   • a) the fractional change in all parameters < 0.01.

5. Parameter estimates from the (j − 1)th time step were used as initial guesses for the jth time step.

The quality, and sometimes convergence, of the fitting process had some sensitivity to the initial parameter guesses. Generally, convergence occurred to similar parameter estimates when initial values were 20 < x_c < 40 m, 50 < z_c < 90 m, 100 < Γ < 400 m² s⁻¹, and 15 < s < 30 m. For the results discussed below, initial guesses of x_c = 20 m, z_c = 65 m, Γ = 150 m² s⁻¹, and s = 20 m were used.

At each time step the parameters (x_c + s, z_c, and Γ), or (x_c, z_c, Γ, and s), their uncertainties (σ_x+s, σ_z, σ_Γ) or (σ_x, σ_z, σ_Γ, and σ_s), and the RMS of residuals, χ, were recorded. For each entire event, the RMS values of each of these quantities were also recorded.

It was found that the single-vortex model produced uncertainties in parameters about twice as large as those for the four-vortex model. The overall RMS values for χ for each event were typically 30%–40% larger for the single-vortex model. The following discussion therefore concentrates on the four-vortex model.

The mean event RMS residual vertical velocity error was 0.83 m s⁻¹. For comparison, RMS vertical velocities were calculated from nonevent times, giving 0.48 m s⁻¹. This “natural” variation is partly due to actual vertical velocity fluctuations, and partly due to statistical errors in velocity retrievals caused by background noise in the sodar signal. Sodar Doppler spectra are usually averaged over typically 40 profiles, but in this application velocities are estimated from each separate spectrum, giving relatively large random errors in the vertical velocities. The least squares model is contributing about 0.34 m s⁻¹ RMS velocity error. Considering that volume-averaged vertical velocities due to vortices will often be greater than 3 m s⁻¹, the model is contributing of order 10% error to these peak values.

Figure 13 shows a typical time series of estimated Γ values for an event. Parameter estimation uncertainties are shown as error bars. The low values of Γ at the start of the event may be due to incorrectly identifying the event start time. In this example there is little development of Γ with time. In contrast, Fig. 14 shows estimated Γ values from an event in which there appears to be significant enhancement of vortex strength through the event duration. Beyond the first 20 s or so, this enhancement is not attributable to possible incorrect identification of the event start time. Also, the variations are larger than the parameter uncertainties and therefore are statistically significant. Since the parameter estimation described is capable of putting significance levels against such changes, the method has the potential of both indicating development in real time (profile by profile) and also indicating quality of the data.

Figure 15 shows an example of the evolution of vortex center height z_c and vortex half-spacing s, together with estimation uncertainties. Although z_c initially rapidly decreases and s eventually increases, the interrelationship between the two parameters does not follow the smooth form predicted by the simple four-vortex theoretical model described above. In fact the “hump” in z_c at around 24 s appears to be statistically significant and is not accompanied by a corresponding drop in s. We have not investigated the independence of s and x_c estimates, nor have we made the (reasonable) assumption that U is constant throughout an event and therefore that x_c should be linearly varying with time. Another example is shown in Fig. 16. In this case z_c falls rapidly initially and then more slowly, whereas s increases slowly initially and then more rapidly, as expected. The overall rate of vortex descent is about −0.7 m s⁻¹, although we do not actually know the type of aircraft involved.

The mean values of uncertainties are given in Table 2.

7. Discussion and conclusions

Sodars have previously been shown to be capable of obtaining high-resolution real-time measurements of transient atmospheric structures and, in some limited instances, of aircraft wake vortices. One configuration, a linear array of sodars placed perpendicular to the flight path, has been described E. Mursch-Radlgruber (2000, unpublished manuscript) and Mursch-Radlgruber (2004). The temporal resolution (2 s) of this configuration is very good in comparison with other vortex measurement approaches, and provides adequate sampling rates to characterize vortex developments in real time. However, the spatial sampling of the vortex flow has necessarily been restricted by the number of available sodars so that the horizontal sampling interval (25 m) is quite large compared to the characteristic scale of a typical aircraft wing vortex. The vertical sampling interval of 10 m is adequate, although more samples in this direction would also no doubt enhance vortex visualization.

In this paper we have developed a simplistic potential flow four-vortex model which allows us to put in context these spatial and temporal sampling scales. The model allows for the ground surface and produces an analytic expression for vortex development. However, interactions with atmospheric buoyancy, wind shear, and turbulence are ignored.

This simple model provides a “snapshot” in time of the basic vortex flow which includes ground effect and combination of the wing vortex pair. This wind pattern, described by four parameters (strength, vortex-pair center horizontal and vertical position, and vortex spacing) is then used as a model for the vertical wind field observed at each sodar sampling time, with no coupling assumed between parameters at one sampling time and the next. The vertical wind is used because the naturally occurring vertical motion near the flat terrain of an airport is much smaller than the vertical wind perturbations produced by vortices. Although the use of the model applied independently at each time step makes minimal assumptions about vortex interaction with the atmospheric structure, some of the resulting parameter uncertainties will be invariably due to “model error.” In particular, the four-vortex model described assumes symmetry (both wing vortices are identical in strength and have centers at the same height) whereas it is known that wind shear can enhance one vortex and diminish the other and also tilt the vortex centerline.

One of the features of a sodar is its generally large sampling volume. For observations of the spatially compact vortex structures this needs to be taken into account. The four-vortex model is used as a basis for volume averaging the flow according to reasonable sodar beam patterns. In practice, analytic expressions for conical acoustic beams can be obtained, but with substantive algebra and the resulting expressions are so lengthy as to obscure any physical interpretation. Given the approximate nature of the vortex model, we have chosen to use a simple box for the sodar sampling volume as giving sufficient accuracy. Box dimensions are allowed to increase with height in the manner of the actual conical beam. This approach has allowed investigation of the limitations imposed by sodar beam geometries of various resolutions. It is found that both horizontal and vertical sodar resolution affects peak vertical velocities measured, and there is little point in enhancing just the vertical resolution if the beamwidth remains relatively wide. The simple potential flow vortex predicts a nonphysical singularity at the vortex center, and in practice the peak velocities may not be as high or as sharply delineated as shown in Figs. 5 and 6. For example, Hahn (2002) finds that the core is about ±0.07 s or about ±2.5 m wide, which agrees with lidar measurements (Coustols et al. 2004). Consequently, finer spatial resolution than, say, 2 m in the vertical, will probably not yield increased information about the vortex structure.

As a case study example of the parameter-fitting methodology, a limited case study of 20 successive aircraft landings is analyzed. It is found that the parameter values, and convergence, are relatively insensitive to initial parameter guesses. The retrieved parameters generally developed more or less smoothly with time during any one landing event. The previous parameters were used to initialize the next time step, but the insensitivity to initial parameter guesses, and the scope for parameters to change by a factor of 1.2⁴⁰ = 1500 during the iterative process meant that the convergence was merely speeded up by this choice of parameters, rather than constrained by it.

The position and spacing of the vortex pair was determined to about 5 m, an uncertainty that is adequate for air traffic purposes. The uncertainty in vortex strength was around 25%, again acceptable for air traffic safety.

Examination of the residuals (fitted vertical velocity minus measured vertical velocity) showed that the model error was less than the random measurement error of the sodar. However, there may be systematic features of this model error, which remain to be studied.

The method described appears to provide a potentially useful technique for obtaining real-time visualizations of vortex behavior. However, the work in this paper is exploratory rather than exhaustive, and much remains to be done. In particular, the method needs to be tested on a much larger dataset, spanning a wide range of atmospheric conditions, and where more is known of the atmospheric structure and of the aircraft type and flight speed. Also, more intensive investigation of parameter behavior and parameter errors is planned for a subsequent publication. Finally, other models, allowing for nonsymmetric vortices or more complex rotational structures as described by Stumpf (2005) should be tested using a similar approach to that described above.

The sodars used in this study (E. Mursch-Radlgruber 2000, unpublished manuscript) are physically small (about 300 mm in each dimension, with acoustic baffles) and inexpensive. Given that lidars have a range advantage and better spatial resolution transverse to their beam, but sodars have advantages of simplicity, low maintenance, and performance in fog, the optimum configuration may be a combination of the two technologies.