A method is presented to compute the spanwise vorticity in polar coordinates from 2D vertical cross sections of high-resolution line-of-sight Doppler wind lidar observations. The method uses the continuity equation to derive the velocity component perpendicular to the observed line-of-sight velocity, which then yields the spanwise vorticity component. The results of the method are tested using a ground-based Doppler lidar, which was deployed during the Terrain-Induced Rotor Experiment (T-REX). The resulting fields can be used to identify and quantify the strength and size of vortices, such as those associated with atmospheric rotors. Furthermore, they may serve to investigate the dynamics and evolution of vortices and to evaluate numerical simulations. A demonstration of the method and comparison with high-resolution numerical simulations reveals that the derived vorticity can explain 66% of the mean-square vorticity fluctuations, has a reasonably skillful magnitude, exhibits no significant bias, and is in qualitative agreement with model-derived vorticity.
The Terrain-Induced Rotor Experiment (T-REX) field campaign was carried out in spring 2006 in Owens Valley, California (Grubišić et al. 2008). The main focus of the experiment was the investigation of the coupled mountain wave rotor system in the lee of the Sierra Nevada. One of the primary objectives is to study the internal dynamics of rotors and their interaction with boundary layer processes near the earth’s surface.
Atmospheric rotors are characterized by strong downslope surface winds in the lee of a mountain range that decelerate rapidly to a weaker, transient, and turbulent flow, often directed back toward the mountain (e.g., Holmboe and Klieforth 1957; Kuettner 1959; Doyle and Durran 2002; Doyle et al. 2009). Based on 3D high-resolution numerical simulations, Doyle and Durran (2007) proposed that the largest hazard for aviation is actually not the whole rotor but smaller vortices, which they called subrotors. These vortices can be produced along a shear line that is generated by boundary layer separation resulting from strong downslope winds and lifted aloft by the lee wave circulation (Doyle et al. 2009). Some of the major challenges for T-REX are to observe these subrotor vortices, to estimate the strength of the horizontal vorticity, and to investigate their characteristics.
In this study, high-resolution scanning Doppler lidar observations taken during T-REX are analyzed, and a method for deriving tangential velocity Vϕ and vorticity fields from lidar line-of-sight velocities Vr is presented (see Fig. 1 for the definition of the coordinate system). During T-REX rotor events, most lidar observations were made in a vertical cross-sectional mode along a baseline 260° from north to 80° from north, which is roughly parallel to the main flow and perpendicular to the valley axis and the Sierra Nevada Ridge. The observations were made by the Institut für Physik der Atmosphäre of the Deutsches Zentrum für Luft- und Raumfahrt (DLR) using a scanning 2-μm coherent Doppler lidar system from Lockheed Martin Coherent Technologies (WindTracer; details available online at http://www.lockheedmartin.com/ssc/coherent/products/windtracer/Specifications.html). The lidar was situated on the alluvial slope of the Owens Valley at 36.79°N, 118.21°W, and 1240 m MSL, which is approximately 2 km west of the town of Independence. The valley is confined by the Sierra Nevada to the west, which includes several peaks above 4000 m MSL and the highest peak in the lower United States, and the Inyo Mountains to the east, with peaks exceeding 3000 m MSL. The width of the valley is approximately 20–25 km. The time required for a full lidar cross section was between 30 and 150 s during the T-REX field campaign, depending on scanning speed. The cross sections used in this note were taken over a time period of 90 s. The resolution of this cross section is 105 m along the lidar beam and 2° elevation.
Doppler lidars only measure the wind component Vr in the direction of the lidar beam, whereas either u and υ or Vr and Vϕ are needed to derive spanwise vorticity fields. Under relatively stationary flow conditions, this can be solved by observations with two (or more) ground-based lidar systems or by making observations from several directions with a lidar mounted on a moving platform (e.g., an aircraft; Reitebuch et al. 2003). Atmospheric rotors during T-REX, however, were highly transient and associated with strong turbulence.
In addition to the DLR lidar, Arizona State University (ASU) operated a similar Doppler lidar during the T-REX field campaign. Both lidars performed coplanar lidar observations with the goal of deriving the horizontal and vertical velocity through a combination of both datasets through dual Doppler analysis (Drechsel et al. 2009; Hill et al. 2010). Results from dual Doppler can also be used to derive spanwise vorticity, but the combination of two different datasets inevitably leads to errors resulting from the time difference of the observations (up to about 30–150 s with the scan strategy during T-REX). Furthermore, the volume of the observations is never completely the same because of the observation geometry. Thus, it is concluded that, in events with strong turbulence and transient flow, the time sequencing and differences of the observational volume likely lead to significantly larger errors than the method presented in this note. For example, relatively small vortices propagate with the mean flow, which can lead to a displacement of up to 600 m (nearly six range gates) in 30 s for a wind speed of 20 m s−1.
This note presents an alternative method that exploits the continuity equation to derive Vϕ and the spanwise vorticity from vertical slice line-of-sight observations of a single lidar (section 2). This approach has the advantage that the time difference between two adjacent observations is in the range of only 0.2–1 s. The errors of lidar Vr observations are small compared to the velocity fluctuations associated with turbulent flows as, for example, in rotors. Errors of coherent 2-μm Doppler lidars are often estimated to be on the order of 0.1 m s−1 under favorable conditions and may increase by a factor of approximately 2–3 at a lower signal-to-noise ratio (Grund et al. 2001; Chai et al. 2004).
The main errors for the derived Vϕ and vorticity fields are expected to occur because of flow divergence perpendicular to the plane of the lidar cross section. For this reason, the errors of the calculation resulting from divergent lateral flow are investigated with tests using 2D and 3D high-resolution numerical simulations with the atmospheric portion of the Coupled Ocean–Atmosphere Mesoscale Prediction System (COAMPS) in section 3. Furthermore, the application of the calculation to T-REX observations is also shown in section 3. A discussion of the results is presented in section 4.
2. Tangential velocity and vorticity calculations derived from Vr observations
The tangential velocity Vϕ (i.e., velocity component perpendicular to Vr) can be calculated for the grid points in each quadrant through the use of the 2D continuity equation in polar coordinates and the application of a boundary condition such that the vertical velocity is zero at the surface. It could also be calculated for the whole hemicycle starting at ϕ = 0°, but errors of the calculation accumulate with every elevation increment. It follows that it is more accurate to begin the calculation from both sides (i.e., at ϕ = 0° and ϕ = 180°, respectively).
Assuming that there is no divergence perpendicular to the plane of the lidar observations (r–ϕ plane), the continuity equation in polar coordinates (r, ϕ) is
where r is the radial distance from the lidar; ϕ is the elevation angle; and Vr and Vϕ are the radial and tangential velocities, respectively. At the point (i, k), ∂Vr /∂r can be approximated as
where Δr is the radial distance between two range gates, i is the index of the range gate, and k is the index of the elevation step. Similarly, ∂Vϕ /∂ϕ at the point (i, k) can be approximated as
Equation (4) can be solved stepwise by starting at ϕ = 0°, where Vϕ(i,k−1) = 0, and incrementing to ϕ = 90°. Likewise, Vϕ can be calculated from ϕ = 180° to ϕ = 90°. However, in the case of divergence in the y direction, this leads to errors of ∂Vr /∂r resulting in errors of Vϕ that propagate from one elevation increment to the next. This aliasing effect causes “rings” of high/low Vϕ and large gradients of ∂Vϕ/ ∂ϕ. To mitigate the aliasing, a smoothing parameter S is introduced to average Vϕ at the grid point (i, k − 1), resulting in
Tests with model and lidar fields show that the best results are achieved with S in the range of 0.2–0.33 (see section 3). No significant improvements are achieved with a smoothing algorithm that includes the second level of range gates (not shown). This smoothing algorithm is expected to reduce errors that arise as a result of the numerical approximations in Eqs. (2) and (3) and as a result of the small-scale variability of the flow. However, the smoothing procedure does not remove errors arising from large-scale lateral flow divergence.
Without the along-beam smoothing, the method produces spurious structures of Vϕ in some locations, as mentioned previously. This is understandable, because errors resulting from lateral divergence in one range gate will propagate to higher elevations at constant Δr because of the iterative algorithm. However, divergence resulting from small-scale flow variability is usually compensated by convergence nearby. Thus, the spurious structures can be effectively mitigated through the smoothing procedure, which is confirmed by the visual inspection of the results that show realistic structures and no obvious rings after smoothing. The improvements through such an approach are further supported by the tests with output fields of the numerical model.
As a consequence of larger-scale lateral flow divergence, there is a difference of Vϕ at ϕ = 90° for the calculations beginning with ϕ = 0° and ϕ = 180°. Therefore, a correction is performed by assuming a constant error of [Vr(i+1,k) − Vr(i−1,k)], which is equal to the difference of the two Vϕ calculations at ϕ = 90° divided by the number of elevation steps (further discussed in section 3).
After calculating Vϕ, the vorticity field can be computed by using the definition of the spanwise vorticity component η in polar coordinates,
For the vorticity calculation, ∂Vϕ/ ∂r and ∂Vr /∂ϕ at the point (i, k) were approximated using centered finite differencing, similar to that in Eq. (2).
3. Evaluation of the velocity and vorticity calculations using model fields and T-REX observations
An example of T-REX lidar observations is shown in Fig. 2. A more detailed discussion of the rotor event on 16 April 2006 can be found in Doyle et al. (2009). The observations were taken during one of the strongest rotor events of T-REX and show strong downslope winds about 8–10 km upstream of the lidar. A region with weaker flow partly directed backward toward the mountain range can be seen between the lidar and 5 km upstream of the lidar. Strong westerlies are apparent above this region with weaker flow. A distinct vortex with a diameter of about 1 km that formed at the shear layer at about 3 km MSL is visible 2 km upstream of the lidar.
A vertical slice from the 3D high-resolution numerical simulations using an eddy-resolving version of the COAMPS model was used to optimize the smoothing parameter S to decide if the along-beam smoothing and the correction for differences at ϕ = 90° should be applied and to estimate the uncertainty of the Vϕ and vorticity calculation. The grid spacing of the simulations is 60 m in the horizontal and 50 m in the vertical (see Doyle et al. 2009). The baseline of the cross section is the same as for the most common scan pattern during T-REX (roughly parallel to the main flow). There are obviously differences between the simulations (Fig. 2f) and the lidar observations (Fig. 2a), but the simulations appear to be a suitable test case because the strength and size of the vortices have a similar magnitude and structure. More details of the simulations, similarities, and differences to T-REX observations are discussed in Doyle et al. (2009).
The model output fields were averaged to a polar grid that is similar to the lidar observations with a radial resolution of 105 m and 2.0054° for ϕ. In the vicinity of the lidar, where the resolution of the polar grid is higher than that the model grid, the angular tolerance for the averaging was increased by a factor of 4 for the first four range gates and by a factor of 2 for range gates 6–10. After averaging the model winds to a polar grid, Vr and Vϕ were calculated (Figs. 2f,g). Additionally, Vϕ was calculated from model Vr using the continuity equation, as discussed in section 2 (Vϕ–CONT; Fig. 2h).
The spanwise vorticity was calculated from a vertical slice of the model test fields using three different methods: 1) Cartesian coordinates and then averaged to polar grid (not shown in Fig. 2); 2) polar coordinates using Eq. (6) with approximations described by Eqs. (7) and (8) (Fig. 2i); and 3) polar coordinates using Vϕ calculated with the continuity equation (Fig. 2j; VORT–CONT). The calculations for Figs. 2h were made using S = 0.33 with a correction for differences of Vϕ at ϕ = 90°.
The standard deviation and mean of these fields are shown in Table 1. All values of Tables 1, 2, and 3 were only calculated for the domain where corrected Vϕ–CONT is available (Fig. 2j) to make the values comparable. Table 2 shows the mean-square differences of Vϕ calculated using the continuity equation to the averaged model Vϕ for a range of S from zero (no smoothing) to 0.33. The setting of S = 0.33 implies that the three range gates used for the averaging are nearly equally weighted. In addition, these fields are normalized with the mean-square values of Table 1. Table 3 shows the vorticity differences in a similar manner as the Vϕ differences displayed in Table 2.
The Vϕ and vorticity fields are calculated with simplified numerical approximations of the derivatives, which inevitably leads to numerical errors proportional to the grid increment. For this reason, the vorticity calculated in polar coordinates has smaller fluctuations than the vorticity calculated in Cartesian coordinates (Table 1). The calculation of Vϕ using continuity and the smoothing parameter S leads to a further decrease of fluctuations that can be resolved by the method. However, small differences of the vorticity fields calculated in Cartesian coordinates and the ones in polar coordinates (not shown) indicate that these numerical errors are smaller than the ones resulting from divergence perpendicular to the lidar scan.
Figure 2h shows that the calculation of Vϕ using continuity leads to reasonably accurate results in the region with fairly uniform flow (e.g., upstream 3 km and above 4 km MSL), but that lateral divergence can lead to errors of Vϕ: for example, in the turbulent region above the lidar, where the model fields show reversed flow in contrast to calculated Vϕ. Despite these errors of Vϕ–CONT, the VORT–CONT in the same area is in reasonable agreement with the vorticity fields.
Overall, the error of Vϕ–CONT is 6% of the total Vϕ fluctuations for S > 0.1 with the correction for differences at ϕ = 90° and 7% for S ≥ 0.2 without correction (Table 2). Both smoothing and the correction show a clear improvement of Vϕ–CONT. According to the results, S should be in the range of 0.2–0.33 with lowest errors at S = 0.33. The total fluctuations of Vϕ–CONT (Table 1) are realistic given the fact that the test field is highly turbulent.
The smoothing is assumed to reduce errors that primarily arise because of divergence perpendicular to the plane of the lidar scans associated with turbulent fluctuations. As a consequence, the optimal setting of S is expected to be dependent on the ratio of the dominant turbulence length scale and the resolution of the observations. The optimal S parameter is defined as the value of S that leads to the smallest square difference of Vϕ–CONT and VORT–CONT to model Vϕ and vorticity, respectively. This dependency on S is demonstrated for the 3D simulation fields shown in Fig. 3, where the optimal S parameter is shown for different along-beam resolutions.
Further tests using a 2D simulation (Fig. 3) show that the smoothing parameter also improves the results in the absence of divergence perpendicular to the cross section. Otherwise, the optimal S parameter would be zero or close to zero for 2D model fields. This is interpreted as a reduction of numerical errors by the smoothing, which also explains why the optimal S parameter shows a clearly lower dependency on the along-beam resolution for the 2D fields. A direct comparison of the errors using 2D and 3D output fields is not possible because the flow field itself is significantly different in 2D and 3D simulations.
The error resulting from the vorticity calculation is less than 37% for S ≥ 0.2 with the correction and less than 40% without correction. The smallest error of VORT–CONT is reached using S = 0.33. With this setting, the error of the calculation is 34%, which implies that the vorticity derived using the continuity equation can explain 66% of the vorticity fluctuations. The positive effect of the correction actually does not appear to be that large for the vorticity field compared to the improvement of Vϕ–CONT. This is understandable, because only the gradients and not the absolute values of Vϕ are important for the vorticity. A qualitative comparison of the results shown in Fig. 2 indicates reasonable agreement of the vorticity fields: the magnitude of the fluctuations of VORT–CONT is realistic (Table 1), and the maxima are of similar magnitude (∼0.2 s−1).
The application of the method to T-REX observations is shown in Figs. 2a–e. The qualitative interpretation of using different smoothing parameters for vorticity calculations with T-REX observations confirmed that 0.2 < S < 0.33 is a reasonable choice for the parameter (not shown). Because the calculation of Vϕ needs a continuous field, five isolated missing values of Vr were interpolated from neighboring values for the calculation of Vϕ and vorticity.
Despite differences of Vϕ that arise as a result of the correction, the results for vorticity are very similar with and without the correction for differences at ϕ = 90° (Fig. 2). Because observations at all elevation angles are needed to perform a correction, the coverage of corrected VORT–CONT is lower than that of the uncorrected field. In the field displayed in Fig. 2, this is clearly a limitation, because the strongest vortex appearing at a height of 2 km AGL 1.7 km upstream of the lidar is not completely covered in the corrected field. The vorticity calculation has been tested with several of the lidar scans during T-REX and shows reasonable results in all cases. These tests confirm that the calculation leads to reasonable values for vorticity. The Vϕ–CONT field is improved further with the inclusion of the correction factor.
We have demonstrated that realistic vorticity fields can be derived from vertical slice lidar Vr scans with a relatively simple and computationally inexpensive method that calculates Vϕ solving the 2D continuity equation numerically. An along-beam smoothing of Vϕ is essential to retain realistic results. A further correction for differences of Vϕ at ϕ = 90° with a constant error term improves the fields of Vϕ but does not appear essential for the calculation of vorticity because only errors of the along-beam gradient of Vϕ lead to vorticity errors and not errors of the absolute Vϕ.
Tests with an eddy-resolving model output from a 3D simulation reveal that the vorticity derived using the continuity equation can explain 66% of the vorticity fluctuations. In addition, the fluctuations of vorticity have a realistic magnitude, and the calculation does not show a significant bias. The derived vorticity and model vorticity are qualitatively in good agreement. The Vϕ field calculated from Vr using the continuity equation has an accuracy of 94%, but errors are larger in regions of strong turbulence.
The method is based on the assumption of 2D continuity, whereas the flow also has fluctuations in the third dimension perpendicular to the vertical slice. Thus, the main errors in the derived tangential velocity and vorticity result from small-scale flow variability and large-scale divergence in the third dimension. The first contributing factor should be mitigated through the along-beam smoothing. This is shown by tests with the velocity fields from the model simulations where the third velocity component is known. The errors resulting from large-scale flow divergence are attempted to be reduced through a correction based on the differences of Vϕ at ϕ = 90° after calculating the tangential velocity incrementally from the surface on both sides of the lidar up to ϕ = 90°. Without a large-scale flow divergence, the calculations from both sides should lead to the same tangential velocity at ϕ = 90°. Thus, the Vϕ field was linearly relaxed to minimize the difference with this correction. Some of the remaining differences in the Vϕ field indicate that the correction does not fully remove these errors. However, as illustrated by the model simulations, this large-scale flow divergence primarily affects the tangential velocity, and the derived vorticity is not substantially impacted. Instrumental noise associated with coherent Doppler lidars is estimated to be in a range of 0.1–0.3 m s−1, according to Grund et al. (2001) and Chai et al. (2004); thus, it is comparably smaller than the velocity gradients of several meters per second between neighboring range gates. Errors resulting from experimental noise can also be expected to be reduced by the along-beam smoothing, because these errors are spatially uncorrelated.
In summary, the proposed method is useful to quantify the strength and size of vortices, such as those associated with atmospheric rotors, which pose a severe hazard to aviation. One limitation of the method is that only the spanwise vorticity component is derived. However, because there is no observing system that can observe the full 3D flow field within a time range of minutes, it is seen as a useful approach for topographic flow problems and other flows when the main wind component is along the plane of vertical slice lidar scans.
The along-beam smoothing is performed through a weighted moving average of the neighboring range gates, whereby different weighting parameters for the average are investigated. The optimal smoothing parameter is expected to be dependent on eddy size, resolution of the observations, and accuracy of the observing system. This smoothing may need to be adapted when the method is applied for different observing systems or if the vortices of interest are significantly different, but the general approach can be assumed to be valid for various applications where the strength of vorticity needs to be quantified by vertical cross sections of line-of-sight observations from Doppler lidar or also Doppler radar instruments. In recent years, several airports around the world (e.g., Hong Kong International Airport; Shun and Chan 2008) installed operational wind lidar systems for aircraft safety, and the presented method is also seen as a possibility to provide quantitative information on vortices and potential aviation hazards in real time.
The efforts of the National Center for Atmospheric Research (NCAR) Field Project Support (FPS) and the T-REX staff are greatly appreciated. The primary sponsor of T-REX is the U.S. National Science Foundation (NSF); furthermore, the DLR lidar observations were sponsored by funds from the Austrian Science Foundation (FWF) and internal funds of DLR. COAMPS is a registered trademark of the Naval Research Laboratory (NRL). The third author acknowledges support through ONR PE 0601153N. The Doppler lidar operated by DLR was leased from Lockheed Martin Coherent Technologies, and the technical support by Mark Vercauteren and Keith Barr is acknowledged. The assistance of Andreas Wieser (Forschungszentrum Karlsruhe) for DLR lidar observations, Matlab software from Ron Calhoun (ASU) and Christian Kühnlein (DLR) that provided the basis for the analysis of DLR lidar data, and help from Qingfang Jiang (NRL) with the formulation of the vorticity equation in polar coordinates are highly appreciated.
Corresponding author address: Dr. Martin Weissmann, DLR Oberpfaffenhofen, D-82230 Wessling, Germany. Email: firstname.lastname@example.org
This article included in the Terrain-Induced Rotor Experiment (T-Rex) special collection.