Airborne Radar Doppler Spectrum Width as a Scale-Dependent Turbulence Metric
1. Introduction
Some of the earliest Doppler weather radars used to investigate stratiform cloud systems (e.g., Boucher et al. 1965) indicated the presence of dynamic turbulent structures, and investigators conjectured about the potentially determinant role they may have on cloud precipitation patterns. In the years since, several generations of Doppler radars have been used to measure dynamic structures within precipitating clouds (e.g., Vali et al. 1998; Houze and Medina 2005; Lothon et al. 2005; Kumjian et al. 2014; Aikins et al. 2016; Bergmaier et al. 2017; Rauber et al. 2017). Of these, millimeter-wavelength (e.g., W-band) cloud radars can resolve the smallest spatial scales, often a few tens of meters, making them well suited for studies investigating microphysical responses to fine-scale turbulence. Many of these radars offer an abbreviated set of measurements to characterize cloud kinematics through the pulse-pair processor (Melnikov and Doviak 2002). Compared to the widely used mean Doppler velocity (first Doppler spectral moment), the spectrum width (second Doppler spectral moment) remains a relatively underutilized data source in airborne cloud radar applications. While spectrum width is most often estimated using the pulse-pair technique, at least one study (Schwartz et al. 2019) compared this field measured by the airborne Hiaper Cloud Radar (HCR) using both the pulse-pair processor and the full Doppler spectrum. Schwartz et al. showed that distribution spectrum widths from the pulse pair was broader than from the Doppler spectrum, though the reason for this is not apparent. Regardless, there remains healthy skepticism of the use for such a noise-dependent and convolved field from airborne radars. For example, spectrum width (i.e., the variation in radial Doppler velocity) is sensitive to fluctuations of air motion and terminal velocity of scatterers, as well as effects related to the aircraft upon which the radar is mounted. Despite this, at least one previous investigation demonstrated the utility of these measurements together with other profiling information providing justification that turbulent dynamics dominate the spectrum width field for the case in their study (Rauber et al. 2017).
Doppler spectrum width measurements from ground-based Doppler radars and lidars can provide a more accurate basis for calculating turbulent kinetic eddy dissipation rate (EDR; ε1/3) and can do so at finer spatial resolutions compared to evaluations using just mean Doppler velocities (Smalikho et al. 2005; Fang et al. 2014a; Wildmann et al. 2019). This latter methodology relies on computing power spectra from windowed time (or spatial) series of mean Doppler velocities and has proven effective at diagnosing turbulent dynamic structures from airborne radars as well, albeit at coarser kilometer resolution (Strauss et al. 2015, hereafter S15). For both methods, estimation of eddy dissipation rates is dependent on well-behaved turbulent energy cascades (preserving eddy scale similarity) following Kolmogorov’s (1941) hypothesis and can be calculated by log-linear (−5/3) regression (Pope 2000). The difference in spatial scales between turbulent motions resolved by variations in the mean Doppler and those resolved by Doppler spectrum width leads to a nontrivial scaling problem between the two (Fang et al. 2014a). However, the turbulent energy cascade bridges both of these scales establishing a relationship between these two measurement techniques. The expected results for isotropic and homogeneous turbulence are two discontinuous statistical estimates of velocity variance taken from 1) the mean Doppler velocities (Robison and Konrad 1974) or 2) the Doppler spectrum width (Fang et al. 2014a), representing resolved and unresolved turbulence, respectively (Rogers and Tripp 1964; Robison and Konrad 1974).
The airborne radar platform, much like scanning Doppler radars (Fang and Doviak 2008), introduces a unique set of uncertainties in the observed Doppler spectrum width and mean Doppler variances (S15). In addition to the spectral broadening effects contributed by the variation in hydrometeor terminal fall velocities along a near-vertical-pointing beam, wind shear, and pulse volume averaging, airborne radars observe further spectral broadening from radar platform motions. Most of this additional broadening results from aircraft motion perpendicular to the beam’s pointing direction, also known as finite beamwidth broadening (Hagen et al. 2021). Moreover, changes in beam-pointing angle due to roll and pitch variations can lead to further positive biases in measured Doppler velocity and spectrum width. These uncertainties were rigorously quantified in S15 where they appear in mean Doppler motions and bias or make uncertain the ensemble variance calculated from windowed mean Doppler velocities. A similar uncertainty analysis or error quantification for individual Doppler spectrum width observations remains to be demonstrated for an airborne cloud radar. Similarly, the utility of available spectrum width information from airborne radars has yet to be fully explored for its characterization of qualitative structural information if not exact quantification.
There exist several methodologies to measure or evaluate turbulence magnitudes from radar and lidar Doppler moments. For airborne cloud radar, the best substantiated metric relies on evaluation of variance and dissipation rates by the inertial dissipation technique (S15), where spatial/temporal series of mean Doppler velocity are transformed into power-frequency (or wavenumber) spectra using Welch’s method. These products are necessarily coarse to resolve both the inertial dissipation range and contain enough spectra to reduce noise—e.g., 2-km output resolution along the aircraft flight track for five overlapped windows of 500-m data segments. Similarly, calculation of the autocorrelation or structure function from nonuniform or sparse Doppler velocity samples can provide insight into the magnitude of fine-scale turbulence (Lothon et al. 2005). For airborne radar, this method has produced cumulative profiles of high vertical resolutions from which variance, dissipation rate, and integral length scale are evaluated. Unfortunately, these coarse (along track) turbulence magnitudes are better suited to evaluating the precipitation response in the more spatially homogeneous marine boundary layer stratus/stratocumulus environment (Lothon et al. 2005) than the orographic winter cloud environment, which exhibits far more horizontally heterogeneous, complex, and sheared eddy structures to be resolved. A final method not yet used by airborne cloud radar utilizes Doppler spectrum width measurements individually or in aggregate as the basic metric for turbulence magnitude retrievals, following ground-based radar and lidar applications (Fang et al. 2014b; Wildmann et al. 2019). This promises to provide the highest resolution and to characterize turbulent motions associated with the finest spatial scales but requires substantial validation by comparison to the structure function and inertial dissipation methods.
The inertial dissipation technique (S15) provides an uncertainty analysis and bias correction derived from mean Doppler velocity measurements from which the variance of vertical air motion is calculated—hereafter we refer to these values as mean Doppler variances. The corresponding Doppler spectrum width characterizes vertical scatterer motion variability of scales smaller than the radar pulse volume—hereafter we refer to the square of the Doppler spectrum width as spectrum width variance.
In this investigation, we demonstrate the utility of measurements of Doppler spectrum width (and spectrum width variance) from an airborne W-band cloud radar as a diagnostic for turbulence magnitude. We compare it to the mean Doppler variance, against flight level in situ observed eddy dissipation rate, and direct measures of vertical wind variance from an in situ gust probe. Close attention is paid to contextualizing the necessary conditions for treating Doppler spectrum width as a metric for turbulent air motion intensity and for use of any subsequent derived turbulence parameters from this airborne Doppler spectrum width field. Section 2 discusses these objectives and methodologies including data handling and uncertainty analyses. Section 3 provides validations and comparisons. Section 4 offers a case study of the utility of this turbulence metric in evaluating cloud structure at finer resolutions compared to other turbulence evaluation methodologies. A summary and conclusions are presented in section 5.
2. Objectives and methodology
The objective of this analysis is to provide the basis for using Doppler spectrum width (and variance) from an airborne W-band cloud radar as a proxy for turbulence magnitude and to determine the general limitations of this metric. This work seeks to demonstrate whether and under what conditions Doppler spectrum width measurements can be used as a diagnostic for turbulence magnitude in relative (qualitative) and absolute (quantitative) senses; what uncertainty is contributed by aircraft, beam angle, and hydrometeor motions; and what atmosphere, flight conditions, and sampling resolutions allow for parameters such as turbulent kinetic energy and EDR to be reasonably derived from airborne measurements of Doppler spectrum width. The need for this study is twofold: 1) measurements of Doppler spectrum width are available at the spatial resolution of individual radar volumes and, for regions of large signal-to-noise ratio, readily capture eddy and other cloud structures. 2) These data, however, remain highly uncertain for lack of strict error quantification as performed for similar turbulence metrics using the inertial dissipation technique (S15). For each of the constituent mean Doppler variance measurements, there is a corresponding spectrum width variance from which something may be said qualitatively or quantitatively (depending on degree of certainty and averaging scale) about variability in air motions or turbulence magnitude. A similar uncertainty analysis for spectrum width variance is not equally possible, due to the difficulty of quantifying broadening due to shear and fall speed away from flight level at the same sampling rate as of the Wyoming Cloud Radar (WCR). Therefore, the main objective of this work is to use coarse-resolution mean Doppler variance as the basis for validating finer-resolution measurements of spectrum width variance. And, to characterize the uncertainty magnitude of radar measured spectrum width and determine for what conditions the measurement maintains a strong correlation with turbulence magnitude.
a. Data acquisition and handling
All data presented herein were collected during the Seeded and Natural Orographic Wintertime clouds: the Idaho Experiment (SNOWIE) field campaign conducted in January through March of 2017. The SNOWIE project focused on winter orographic cloud systems occurring over a region of southwest Idaho in the intermountain western United States. Tessendorf et al. (2019) provide an overview of the campaign and a summary of cloud microphysical and thermodynamic characteristics observed during the 8-week program. Data used in this study were acquired on board the University of Wyoming King Air (UWKA) research platform, a Beechcraft Super King Air 200T modified for research in the lower to midtroposphere (Rodi 2011). The UWKA performed repeated in-cloud passes oriented along the mean horizontal wind direction at flight level. In situ and remote sensors mounted on the UWKA provide detailed observations of cloud thermodynamic, dynamic, and microphysical structure. In this study we use data collected during straight and level flight legs from five research flights which occurred during intensive observation periods (IOPs) 4, 9, 13, 14, and 20 (Tessendorf et al. 2019).
1) In situ measurements
This study focuses on the dynamic characteristics of the observed clouds, in particular the turbulent motions within those clouds. The UWKA’s five-hole gust probe, mounted on an extended nose boom, provides high-rate (25 Hz) measurements of the three-dimensional winds when coupled with measurements from static pressure and filtered GPS/inertial reference system (IRS) for aircraft velocity and attitude (Brown et al. 1983). These wind measurements are calibrated with aircraft maneuvers (Wendisch and Brenguier 2013) representative of the full range of expected flow regimes during the research flights. The accuracy of the wind measurements is 1 m s−1 for the horizontal (u and υ) and about 0.1 m s−1 for the vertical (w) components, with a precision of approximately 0.1 and 0.05 m s−1, respectively. The UWKA also carries a MacCready turbulence meter which characterizes EDR (ε1/3; per aviation unit conventions) by measuring related variations in airspeed (MacCready 1964), serving as a final reference point for in situ comparison.
2) WCR
The WCR is an airborne 95-GHz pulsed Doppler radar able to reveal the fine details of cloud structure and dynamics (Wang et al. 2012). The data presented herein are from fixed, near-zenith and near-nadir antennas, obtained during straight and level flights. For operations during SNOWIE, the radar dwell time (time required to collect one near-vertical radar profile) was approximately 45 ms, resulting in a corresponding along track sampling distance of 4 m at an average true airspeed of approximately 100 m s−1. The half-power beamwidth was ∼0.7° for the zenith antenna and ∼0.5° for the nadir antenna. The utilized radar range sampling was 30 m, resulting in a radar sampling volume at 1-km distance from the UWKA of approximately 10 × 10 × 30 m3. Radar backscattered reflectivity for this analysis was limited to regions with signal-to-noise ratio (SNR) greater than 10 dB to ensure good estimates of Doppler spectrum width (see section 2c). In the SNOWIE clouds, such large signal-to-noise ratios only occurred in ice-dominated clouds. Thus, hydrometeor terminal fall speeds are expected to vary between ∼0.5 and 1.5 m s−1 (Pruppacher and Klett 1997), and upon examination should not include larger variations from any smaller liquid precipitation particles. Supercooled drizzle drops, when observed, exhibited diameters no larger than 500 μm throughout the field campaign. The cases presented included straight and level flight legs in wintertime orographic layer clouds and excluded strong upright convective motions in excess of 5 m s−1; see Cann et al. (2022) for climatological characterization of SNOWIE clouds and Cann et al. (2022) and Grasmick et al. (2021) for characterization of these clouds from IOPs 4 and 14 by example.
b. Radar measurements of Doppler velocity variance
The second Doppler spectral moment is a direct measure of the variance of scatterer velocity along the beam. Airborne cloud radar measurements, such as those from the WCR in this study, that utilize a pulse-pair processor, can provide an estimate of the Doppler spectrum width from a train of pulses (Fig. 1a, lower branch). The spectrum width responds to air and scatterer motions at scales smaller than the individual radar volumes and within its dwell time (Doviak and Zrnić 1993; Rogers and Tripp 1964). Moreover, radar measured Doppler spectrum width is also affected by variation in scatterer terminal fall speed and orientation within the volume, beam-pointing angle changes during the pulse train, motion of the radar platform across the beam, and nonuniform beam filling—all of which will lead to increases in Doppler spectrum width, often referred to as spectral broadening. The difficulty in quantifying this broadening at native pulse-pair resolutions suggests why there has been no previous attempt to rigorously quantify air motion variance uncertainty for individual radar volumes for airborne radars.
Schematic representation of the different pathways for processing Doppler radar data to arrive at the different turbulence measurements. (a) Radar power (P) and phase (δ) returns from a train of n pulses are lagged and correlated to produce Doppler velocities (υD) and spectrum width (συ) values to constitute individual kinematic profiles for each averaged pulse volume. (b) These profiles of Doppler velocities are sequenced into k profiles (along track) corresponding to 2 km of data. These 2-km data sequences were processed in periodograms using five overlapped 500-m windows and applying Welch’s method to estimate power spectral density (PSD). From this PSD, the mean Doppler variance (
Citation: Journal of Atmospheric and Oceanic Technology 40, 12; 10.1175/JTECH-D-23-0056.1
Instead, studies have focused on quantifying the variance of spatial/temporal series of mean Doppler velocities at scales larger than individual radar volumes, retrieved from windowed power spectra following S15, described in more detail in section 2d, and illustrated in Fig. 1b. Uncertainty analysis for this methodology is more easily bracketed than for Doppler spectrum width estimates. Combined with the fine spatial resolution enabled by millimeter-wave airborne radars (e.g., W and Ka bands), the inertial dissipation technique yields both accurate and precise estimates of variance and EDR for the turbulent or inertial subranges down to kilometer resolution (S15).
Both the Doppler spectrum width and power spectra derived from spatial/temporal series of mean Doppler motions can be used to quantify variance of scatterer motions (i.e., mean Doppler variance). However, for mean Doppler power spectra, the variance is separable to uncertainty-bounded air motions following S15. Thus, we use mean Doppler variance as the basis for validation and comparison with the higher-resolution measurements of Doppler spectrum width. This enables a focus toward higher sampling resolutions and smaller contributing spatial scales to better resolve fine turbulent eddy structures. The focus here is to extend the diagnosis of turbulent eddy structure to the sensible limits of the WCR (smaller than the large-eddy scales which start anisotropic energy cascades in the boundary layer; Pinsky et al. 2010). Thus, a goal of this work is to demonstrate that relative changes in Doppler spectrum width from the airborne WCR measurements correspond to changes in air motion turbulence intensities observed in mean Doppler variances and to identify under which cloud conditions this holds.
c. Doppler spectrum width (and spectrum width variance) estimates
The received pulse train from each radar volume is processed to produce the first three moments (μ0–μ2) of the Doppler spectrum. The zeroth moment, or received power (reflectivity), is the average echo power of the pulse train. The first and second moments—i.e., mean velocity and spectrum width (Fig. 1a)—are estimated using the pulse-pair algorithm from zeroth lag and first lag covariances, respectively. The calculation of the spectrum width utilizes the estimated signal power. The WCR has a dedicated channel for measuring the radiometric noise. The mean radiometric noise is then subtracted from the received power and the standard deviation of the radiometric noise is used as a threshold to arrive at the estimate of the signal power. The pulse-pair processing assumes a Gaussian Doppler spectrum and yields larger errors in estimating the second moment, where noisy decorrelated pulse pairs can cause the argument of the logarithm to become very large or negative (as discussed in Doviak and Zrnić 1993). Thus, the Doppler spectrum width is highly sensitive to SNR and unreliable for lower SNRs, which otherwise yield acceptable reflectivity and mean velocity values upon pulse averaging (Fig. 2). Pulse-pair estimated Doppler spectrum width (or squared, spectrum width variance) represents motion within the radar resolution volume and can only resolve eddies of (wavelength) scales smaller than half of the maximum pulse dimension (Doviak and Zrnić 1993; Rogers and Tripp 1964).
Standard deviation of Doppler velocity noise/error as function of SNR. Pulse-to-pulse velocity noise is binned by 1-dB SNR in a histogram from which the shown standard deviation of the Doppler velocity error is calculated. Vertical lines represent different employed thresholds for inclusion in statistical analysis (red) and in plots contained herein (blue).
Citation: Journal of Atmospheric and Oceanic Technology 40, 12; 10.1175/JTECH-D-23-0056.1
Considering source error and uncertainty from target to measurement, we first distinguish radar volumes where radar echo power is insufficient relative to the radar noise level (i.e., volumes containing low SNR) for good spectrum width estimation. Figure 2 shows the standard deviation (error) of the Doppler velocity measurements as a function of signal-to-noise ratio (dB) to demonstrate how lower SNR negatively affects the spectrum width estimates even after signal noise subtraction. Based on this error curve, we employ an SNR threshold of 10 dB (Fig. 2, red vertical line) to discard noisier spectrum width estimates from our analysis.
The hydrometeor fall speed variance within the pulse volume,
The additional broadening contributions in Eq. (1) are relatively small for the data used in this study. Mean wind across the beam at the level of the target,
Rather than trying to actively correct for these broadening effects only crudely characterized here over the legs of interest, we simply acknowledge that the measured spectrum width contains broadening beyond turbulent scatterer motions up to the amount estimated for the individual mechanisms outlined above [Eq. (1)]. Then, we compare those measurements against independent measurements of air motion variance at flight level and against mean Doppler variance to statistically quantify the averaging scales (along-track lengths) over which the measured quantities agree.
d. Mean Doppler variance calculation
The power spectrum density (PSD) of the spatial/temporal series of mean Doppler velocity is estimated by applying Welch’s (1967) method. The PSD estimate is derived from averaging five spectra of 500-m mean Doppler velocity data with 50% overlap. Each 500-m time (spatial) series is first detrended and then passed through a 500-m-wide Hanning window. The resultant 2-km-resolution PSDs provide a similar quantification of the spectrum width variance with a different set of mediating assumptions: isotropic and homogeneous scatterer motions within segments, complete removal of contributions by aliased wavelengths through linear detrending, and ergodicity of the time series. Additionally, these variance estimates include power from the 30–250-m scales, much larger than those from individual radar volumes described above. Given a well-behaved turbulent energy cascade across the inertial subrange, such that a −5/3 slope extends to scales still well within the sample volume, it is expected that strong correlations with spectrum width variance should exist. Where these PSDs often characterize isotropic turbulent energy cascades, it has been demonstrated (S15) that they can be parameterized with a −5/3 log-linear regression over the inertial subrange to estimate eddy dissipation rates at kilometer resolution. The further benefit of examining the variance estimates from these resolved, kilometer-scale PSDs is 1) to confirm that the energy cascades correspond to real turbulent signals and 2) to better characterize uncertainty and systematic biases.
The variance estimated in time series of the scatterers near vertical mean Doppler motions,
Uncertainty contributed by aircraft motion and attitude for the WCR is estimated to be
Based on these bounded, low-resolution air motion variance measurements, the Doppler spectrum width measurements are validated as samples of a coincident distribution.
e. Variance comparison and validation of airborne radar Doppler spectrum width
From the above two measurements of variance—i.e., the spectrum width variance and the mean Doppler variance with bracketed uncertainty—a comparison is performed. Both are measuring coincident regions of cloud, with mean Doppler variance limited to air motions as best as possible. One expects that both metrics should be well correlated if spectrum width variance is dominated by turbulent air motion and not other sources of broadening.
To our knowledge, no such validation of Doppler spectrum width measurements from airborne Doppler radar has been performed, presumably due to the complexity of uncertainty quantification outlined above. We aim to avoid this through a comparison with better substantiated turbulence magnitude measurements and retrievals, to clarify under what meteorological conditions this turbulence magnitude metric holds operationally. The high resolution and near-coincident suite of in situ measurements aboard the UWKA make this problem tractable in ways it may not be for other airborne research platforms.
3. Methodology validation and discussion
Figure 3 shows measurements from the WCR above and below the UWKA from one flight leg during IOP 14. A strong shear layer is present approximately 1.5 km above the flight level at 5 km above mean sea level (MSL). This shear layer results in Kelvin–Helmholtz (KH) waves that are apparent in the reflectivity (Fig. 3a) and vertical Doppler velocity (Fig. 3b). Coincident with the location of the KH waves, there exists a region of elevated Doppler spectrum width (Fig. 3c). Figures 3d and 3e provide zoomed-in images of the region of elevated spectrum width and mean Doppler variance associated with the KH waves. Both methodologies reveal the same general pattern and location of increased velocity variance, but the former can resolve much finer spatial scales.
Spatial–temporal cross sections of (a)–(c) airborne Doppler radar data and (d),(e) subset turbulence metrics for one straight and level flight leg in IOP 14. Along-track, (near) vertical profiles of (a) reflectivity, (b) mean Doppler velocity, and (c) Doppler spectrum width are shown for the entire leg, retrieved from the WCR; blue rectangles indicate the subset area shown in (d) and (e). Subset regions demonstrate scale-dependent measurements of (d) spectrum width variance and (e) mean Doppler variance at 100- and 2000-m resolution, respectively. Mean Doppler variance has been computed as described in section 3a.
Citation: Journal of Atmospheric and Oceanic Technology 40, 12; 10.1175/JTECH-D-23-0056.1
a. Comparison against mean Doppler variance
The uncertainty-bounded mean Doppler variance is calculated from the WCR measurements over a given flight track length following S15. Coincident population distributions of spectrum width variance are plotted as a function of each single corresponding mean Doppler variance for four IOPs and shown in Fig. 4. The spectrum width variances are treated as dependent measurements with convolved random error and other possible systematic biases. Because of the difference in resolution of the two methods, each mean Doppler variance may contain up to 500 spectrum width variance samples. The vertical box and whiskers represent the distribution of spectrum width variances contained within a single mean Doppler variance estimate.
Comparison between mean Doppler variance (x axis) and corresponding distributions of spectrum width variance (y axis). Uncertainty bounds in x follow Eq. (2) above from S15, as described in section 2d. The distribution of corresponding pulse-pair variances in y is characterized by 5th (up-pointing triangle), 25th (-), 50th (small diamond), 75th (-), and 95th (down-pointing triangle) percentiles. The finite-beamwidth broadening minimum (in y) for the up-pointing beam at 100 m s−1 TAS is filled in gray. Pearson’s correlation coefficient (r2) for the line of best fit (black) line and Spearman rank coefficient (ρ) are reported for all points.
Citation: Journal of Atmospheric and Oceanic Technology 40, 12; 10.1175/JTECH-D-23-0056.1
Mean Doppler variance is expected to correlate well with spectrum width variances when measuring spatially coincident weather signals. This correlation should hold if and only if the error and uncertainty sources in spectrum width are small compared to the turbulent air motion signal. To ensure these comparisons occur across similar motion scales, the computation of variance from mean Doppler PSDs are truncated or exclude wavelengths greater than 100 m. This approaches the largest pulse volume dimension (typically 30 m along range) and terminates in noise at nearly the same wavelength. Truncating the mean Doppler variances in this fashion results in a significant deviation from a 1:1 relationship toward larger variances in Doppler spectrum width.
The first clear pattern that emerges is a minimum detectable signal limit in the spectrum width variance. This is a result of spectral broadening due to the variation of aircraft motion across the finite beamwidth (gray shading, Fig. 4). Considering the lower 5th percentiles of the distribution of variances (upward-pointing triangles, Fig. 4), few if any spectrum width variances are less than the up-beam aircraft broadening estimate (∼0.1 m2 s−2). Of the four cases examined, IOP 14 appeared to have the smallest variance from both methodologies (Fig. 4, purple diamonds). Figure 5 separates out the frequency distributions of these paired variances for each IOP, confirming that IOP 14 contains the least variance in both metrics (Fig. 5d). Even for this IOP, some correlation appears between spectrum width variances and the corresponding mean Doppler variance, suggesting that the sensible lower limit for spectrum width as a metric for turbulence is the calculated broadening limit, which may be subtracted in postprocessing.
(a)–(d) Contoured joint frequency distributions of mean Doppler variance (x axis) and median coincident spectrum width variance (y axis) as in Fig. 4, but for mean–median coordinate pairs. Outlier points (red plus symbols) are greater than the top 5th percentile along either axis. Black best-fit line as in Fig. 4 is computed by regression over all cases. The plots in (a), (c), and (d) correspond to the data domains (blue rectangles) shown in Figs. 10, 9, and 3, respectively [in (b), IOP 9, radar data images are not shown].
Citation: Journal of Atmospheric and Oceanic Technology 40, 12; 10.1175/JTECH-D-23-0056.1
Above this threshold, Doppler spectrum width correlates well with air motion variance attributed to real weather signals and therefore is a good metric to characterize turbulence. For both IOPs 9 (Fig. 4, magenta; Fig. 5b) and 14 (Fig. 4, purple; Fig. 5d), low levels of variance from both methods show poorer correlations and differences from the overall best fit line. In both cases, the regions sampled were characterized by relatively quiescent velocities near cloud top. Turbulent eddies from breaking waves (IOP 14) and shear braids (IOP 9) were superimposed on this background flow over narrow layers but were not sufficiently resolved at 2-km resolution. Despite this, the greater spread in these cases still envelope the overall line of best fit despite some apparent bias high and low in IOP 9 and 14, respectively (Figs. 5b,d). Considering data from all four IOPs, the main trend between mean Doppler variance and spectrum width variances appears linear, and certainly monotonic. The r2 value of all 4444 mean Doppler variance–median spectral variance pairs is 0.87, and a rank correlation coefficient (Myers and Well 2003) of 0.85 further demonstrates that the relationship is monotonic at a 0.01 significance level.
There are noticeably different “slopes” across IOPs which characterize the association between mean Doppler variance and spectrum width variance contributing spread between cases at spectrum width values more than 0.5 m2 s−2. Figure 6 shows the averaged energy cascades corresponding to the mean Doppler variances in Fig. 5 following the same layout, separating higher and lower turbulence intensity regions for applicable IOPs. The different “slopes” appear to characterize differences in the mean Doppler PSDs, where the lone PSD containing a flatter turbulent energy cascade below 100-m wavelengths (Fig. 6b, IOP 9) is associated with greater spectrum width variances relative to corresponding mean Doppler variance. The remaining steeper spectra (Figs. 6a,c) demonstrate more horizontal slopes by comparison in Figs. 4 and 5 with especially IOP 13’s points falling mostly below the regression of all mean–median points. The steeper cascades of IOP 4 and 13 also appear to be slightly anisotropic, with inertial energy cascades steeper than −5/3, indicating the presence of more energetic motions at larger wavelengths than would be expected if momentum were mixing down at steady state to viscous scales. Given that the meteorological source in these strong turbulence cases was not the same across the cases—shear-driven wave breaking aloft in IOP 4 and boundary layer turbulence in IOP 13—there is no clear set of meteorological turbulence generation sources which result in these anisotropic cascades that present in variance comparison as more horizontal slopes. Finally, stronger turbulent signals in the mean Doppler PSDs leave more of the energy cascade above the noise level which may otherwise lead to some underestimation of mean Doppler variance in the low turbulence case IOP 14, if not also IOP 9.
Ensemble power spectra from which mean Doppler variances were calculated for each of the four IOPs from Figs. 4 and 5. Dashed 25th- and 75th-percentile lines contain the middle 50% of variation in each wavelength’s power spectral density over the indicated regions, with the median power in solid bold. For panels with two ensemble spectra, black points are from the more turbulent region and red points the more quiescent region. The green line in each panel is a representative −5/3 slope for reference.
Citation: Journal of Atmospheric and Oceanic Technology 40, 12; 10.1175/JTECH-D-23-0056.1
b. Comparison against in situ dynamical measurements
A second set of comparisons is performed to further validate spectrum width against measured in situ turbulence magnitudes from the more turbulent (at flight level) IOP 20. These comparisons rest on the assumption that turbulence magnitude at the aircraft flight level is well correlated to spectrum width measured at the first WCR range gate uncontaminated by receiver noise, 135 m above and below the aircraft for both the near-vertical and near-nadir beam. Two data sources are available for this comparison: 1) the MacReady turbulence meter which provides a direct estimate of EDR and 2) a five-hole gust probe for three-dimensional winds from which variances in vertical wind are calculated. The comparison of Doppler spectrum width against flight level EDR further assumes that the length scales of the turbulent eddies are in the inertial subrange and that the variance of turbulent air motions predominate over spectral broadening by hydrometeor terminal fall speed. If these assumptions are met, EDR should be strongly linearly correlated with Doppler spectrum width (i.e., not spectrum width variance; see Cohn 1995).
A comparison for a single flight leg with moderately strong turbulence at flight level is shown in Fig. 7. Doppler spectrum width for the nearest range gates from the up- and down-pointing beams is plotted as a function of EDR from the MacReady probe. Correlations and resulting linear fits between Doppler spectrum width and in situ-measured EDR are nearly the same for both the up-pointing and down-pointing beams, with only slightly different spectrum width baselines (i.e., the “y intercept” in Fig. 7) resulting from larger zenith beamwidth and larger resulting broadening. This broadening magnitude also explains the consistent difference (in y) between the spectrum width points (Fig. 7, blue/red) and the root of mean Doppler variance points (green). For the green points, the best fit line, if drawn, would pass nearly through the origin since mean Doppler variance is not broadened by the aircraft motion.
Scattered comparison between flight level EDR from the MRI probe (x) and resampled Doppler spectrum width (y). Pearson (r2) and Spearman rank correlation coefficients reported in the legend. Filled green diamonds correspond to equivalent mean Doppler spectrum width (i.e., square root of mean Doppler variance).
Citation: Journal of Atmospheric and Oceanic Technology 40, 12; 10.1175/JTECH-D-23-0056.1
The regressions for both up- and down-beam spectrum width correspond to Pearson correlation coefficients greater than 0.5 in both cases. This comparison demonstrates that at a 1-Hz data rate (∼90-m resolution for the flight leg used in Fig. 7), spectrum width and EDR are reasonably correlated and certainly monotonic (ρ > 0.7). This suggests that spectrum width adequately responds to real air motion turbulence magnitude signals at these resolutions. A final consistency check confirms that these linear correlations persist between MacReady EDR and flight level root of mean Doppler variance, which, in addition to eliminating the broadening bias from aircraft motion in spectrum width, dampen some of the spread in the association at a 2-km calculation resolution. These results indicate (i) if we correct spectrum width values from best estimates of broadening effects (see section 2c) and evaluate the acceptable amounts of error or spread for our use cases, then we can use airborne millimeter-wave radar spectrum width as the basis for retrievals of TKE or EDR. And (ii), for assessing turbulence eddy structure and location, it is often sufficient to utilize spectrum width without correcting for broadening effects, at least in cases where additional broadening does not dominate signal. This can be determined through estimations of values in Eq. (1).
Figure 8 similarly compares variances computed from the vertical component of the in situ gust probe measurement and nearest-gate spectrum width variance. In this case, in situ vertical velocity measurements are available at 25 Hz, corresponding to a spatial resolution of 4 m at a true airspeed of 100 m s−1. Flight level vertical wind variance is calculated over a region of 100-m (blue), 500-m (magenta), and 1-km (green) windows along the vertical velocity time series. The 100-m sampling resolution represents scales similar to the separation distance between the aircraft and nearest radar range gate. At this resolution, the blue circles in Fig. 8 demonstrate large scatter and relatively weak linear correlation with spectrum width variance averaged over that same 100 m (r2 = 0.31). To investigate the influence of resolution on this comparison, variances of vertical wind are calculated over longer regions and compared with mean spectrum width variances over that same region. At both 500 and 1000 m, which are significantly longer than the spatial separation between the aircraft and the nearest radar range gate, correlations between in situ and radar measured variances increase to between 0.62 and 0.76. Stronger correlations suggest that spectrum width variance responses in nearest radar gates come from real, turbulent air motions confirmed at flight level, especially at longer averaging lengths. The best-fit slopes at these longer scales also approach 1:1, indicating that there is nearly the same power density in the motion scales of the inertial cascade resolved between flight level gust probe variance and nearest radar gate spectrum width variance, especially at windowing lengths 500 m and larger (or sampling rates less than 0.2 Hz at 100 m s−1 TAS).
Scattered comparison between variances calculated from 100-m (blue), 500-m (magenta), and 1-km (green) windowed vertical wind time series (x) and corresponding Doppler spectrum width (y). Pearson correlation coefficients (r2) reported for each sampling resolution in legend and accompanying line of best fit in thick color. The thin black line is 1:1.
Citation: Journal of Atmospheric and Oceanic Technology 40, 12; 10.1175/JTECH-D-23-0056.1
4. Case study results and analysis
To illustrate the utility of Doppler spectrum width as a turbulence metric from airborne radar measurements we present a few cases to demonstrate its ability to discriminate turbulent eddies, outline its operative considerations, and discuss limitations to interpretation in practice. The general set of cloud conditions sampled in these cases are precipitating wintertime orographic layer clouds as described above in section 2a. In the following straight and level flight legs, distinct cloud regions of moderate to strong turbulent wave trains and eddies are visually identified from the analysis domains of reflectivity, mean Doppler velocity, and spectrum width, to demonstrate how this spectrum width allows one to locate regions of enhanced mixing, entrainment, or turbulent overturning.
For the vertical cross sections of Doppler spectrum width, we employ a more conservative SNR threshold of 30 dB to mute noisier radar volumes and demonstrate how little (∼10%–20% around cloud top, Fig. 9c) of the observed cloud falls outside this more conservative limit. This coincidentally allows for a narrowing of focus toward the eddy structures best resolved by the Doppler spectrum width and embedded within the layer clouds.
IOP 13 illustrates an example of a boundary layer turbulence feature. The panels—(a) equivalent radar reflectivity (dBZe), (b) Doppler velocity (m s−1), and (c) spectrum width (m s−1)—characterize the near-surface turbulent layer. Blue boxes indicate the sample domain and data sources used in Figs. 4 and 5.
Citation: Journal of Atmospheric and Oceanic Technology 40, 12; 10.1175/JTECH-D-23-0056.1
Of the two presented cases, IOP 13 (Fig. 9) has more intense radar echoes near the turbulent feature of interest, a near-surface (1.5–3 km MSL) layer of boundary layer turbulence. By comparison, IOP 4 (Fig. 10) contains weaker echoes capping the turbulent wave feature. Strong echoes (Ze ≥ 5 dBZe) and correspondingly high SNR (>30 dB) ensures that pulse-to-pulse noise does not contaminate the spectrum width field, apparent by comparing IOP 13 near ground (Fig. 9c) against IOP 4 near cloud top (Fig. 10c, directly below muted light green region). Combined with elevated reflectivity signatures and highly variable mean Doppler velocities (±4 m s−1), the spectrum width allows us to distinguish not only the location of layers of turbulence (Fig. 9c, nearest 1.5 km to terrain), but also the fine detail of wave crests (Fig. 10c, near cloud top) and other turbulent eddy structures. For instance, the enhanced reflectivity and spectrum width wave crests near cloud top in IOP 4 correspond to regions of turbulent eddies between otherwise smoother and larger wavelength (>1 km) mean vertical velocity couplets above and below (Figs. 10a,c).
As in Fig. 6, but for a decaying wave train from IOP 4.
Citation: Journal of Atmospheric and Oceanic Technology 40, 12; 10.1175/JTECH-D-23-0056.1
The feature in IOP 4 appears to be associated with decaying or breaking shear waves. This characterization is consistent with higher values of reflectivity contained below the wave crests (Fig. 10a) and in the elevated, overturning spectrum width vortices (along the SNR threshold interface, Fig. 10c). It is impressive, given the relatively weak echoes (0–5 dBZe), that such structures appear above the conservative 30-dB SNR threshold. Despite this, it is difficult to resolve the full picture of turbulent motions in this region of relatively weak reflectivity—suggesting also that while structure may still be apparent, averaging for more accurate turbulence magnitudes may be impossible near highly heterogeneous entraining cloud interfaces given the uncertainties introduced by low reflectivity/low SNR volumes. In precipitating, mixed phase layer clouds this consideration typically only impacts unglaciated cloud tops or where reflectivity remains lower than −10 dBZe. In purely liquid stratus as well as convective clouds, these cloud boundary considerations may limit the certainty of averaging operations in such regions due to the weak signal and corresponding measurement uncertainty.
Finally recall, on the other hand, that the coarse-resolution mean Doppler variances from the S15 method represent best estimates of actual air motion variances—after bias correction and uncertainty estimation—more useful for applications where precise statistical characterizations of turbulence magnitude are needed for prediction with relaxed resolution constraints to describe eddy structure and location (Fig. 3e). For example, evaluating turbulence effects on hydrometeor growth by riming or aggregation over a synoptic frontal boundary does not require as fine a resolution as describing the wake structure of a wind turbine enveloped in freezing fog or blowing snow. The higher-resolution (Fig. 3d, 100-m) spectrum width variances may prove sufficient for quantitative estimates of one-dimensional turbulence magnitude for validation against high-resolution convection-permitting or large-eddy models. Here, higher spatial resolutions of turbulence information are necessary without sacrificing accuracy but will require fitting correct scaling parameters by comparison to turbulence measurements from flux towers or in situ gust probes. Finally, the finer resolutions (Fig. 3d) may allow for examination of the link between increased turbulence and microphysical response in aggregate—that is, for specific layers, eddy structures, etc., which are well resolved by spectrum width and from which statistical or spatial comparisons can be demarcated and tested for inference.
5. Summary and conclusions
In this study measurements of pulse-pair spectrum width from an airborne W-band Doppler cloud radar were validated against several independent turbulence air motion measurements. Median values of spectrum width were compared against much lower-resolution retrievals of the mean Doppler variance (S15) from four research flights in orographic wintertime layer clouds from SNOWIE. The two measurements demonstrated strong agreement (r2 = 0.85). This agreement appeared to hold in aggregate for all radar profiles with spectrum width above the MDS of finite beamwidth broadening (0.36 and 0.26 m s−1 for up and down beam, respectively).
Further comparisons to in situ turbulence measurement confirmed that this agreement persisted even at higher resolutions. First, and perhaps most conclusively, spectrum width demonstrated a significantly monotonic relationship with EDR as measured by the MacReady turbulence meter at its native 1 Hz (∼90-m) sampling rate (r2 > 0.5, ρ > 0.7). Second, spectrum width variance demonstrated reasonable agreement with vertical velocity variance from a gust probe at sampling resolutions of 500 m and greater (r2 > 0.6). These results suggest that even at 100-m sampling resolutions, spectrum width from the WCR can serve as a good metric for quantifying the air motion turbulence magnitude and that spectrum width measurements at these sampling rates respond to in situ turbulence magnitudes sufficiently to serve as the basis for retrievals of TKE or EDR if scaling is addressed. It was demonstrated that even at 100-m resolution, spectrum width provided better discrimination of turbulent eddies than coarser mean Doppler variance measurements following S15. Furthermore, even for the native range and along-track resolution of Doppler spectrum width from the WCR (10 × 10 × 30 m3 at 1-km range, oversampled to ∼5-m along-track spatial resolution), the random pulse-pair estimation error did not obscure apparent eddy structures. This suggests the utility of pulse-pair spectrum width toward diagnosing turbulent regions of cloud and allows for examining the spatial turbulent structure closer to the scales of the turbulence cascade where turbulent mixing is coupled to microphysical process rates.
Finally, these results are limited by the threshold that turbulent air motion signal must overcome the combined effects of Doppler spectrum broadening. In this study, spectral broadening was dominated by two main sources: 1) that which results from the motion of the platform perpendicular to the finite beamwidth of the radar and 2) that which results from variations of the terminal fall speed of the scatterers. For the configuration of the WCR and Wyoming King Air, the former is on the order of 0.3 m s−1 and remains fairly constant. The latter, on the other hand, strongly depends on the conditions being sampled by the radar. In this study, fall speed variations were expected to contribute less than 0.2 m s−1 to spectral broadening. However, for cloud conditions containing rain drops, graupel, and/or hail, the terminal fall speed distribution of particles is expected to broaden by a factor of 2 or 3 times (or more) compared to mixed planar ice or aggregated flakes, introducing a noticeable dependence of spectrum width response to the fall streaks and convective cores where such hydrometeors are found in abundance. For this reason, foremost among others, we do not expect spectrum width to respond predominantly to air motion turbulence in conditions of upright convection and rain and suggest caution in applying these results beyond mixed phase layer clouds and/or ice-only clouds.
In summary, we expect pulse-pair spectrum width to be a sufficient metric for turbulent air motion intensity for the following conditions:
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airborne cloud radars with short (<100 ms) dwell times,
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reflectivity SNR greater than 10 dB,
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regions in which hydrometeor terminal fall speed variations do not dominate the signal, such as cases without graupel, hail, or large drops and away from any bright bands, and
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turbulence intensities leading to signal greater than the aircraft motion–induced finite beamwidth broadening.
In future research we will explore subtracting aircraft motion–induced finite beamwidth broadening from the measured pulse-pair spectrum width.
Acknowledgments.
The authors would like to acknowledge the National Science Foundation Seeded and Natural Orographic Wintertime clouds: the Idaho Experiment grant (Awards 1547101 and 2016077) in funding this research. The authors thank Lukas Strauss for insightful correspondence expanding on the inertial dissipation variance bias constants, the entire SNOWIE principal investigator team for steering feedback and timely encouragement, and Dave Leon and Dave Kingsmill for corroborating radar subject matter expert opinion.
Data availability statement.
The L1 WCR data used for this analysis are available from the UWKA facility at https://doi.org/10.15786/M2CD4J (University of Wyoming Research Flight Center 2017b). The flight level aircraft data are available similarly at https://doi.org/10.15786/M2MW9F (University of Wyoming Research Flight Center 2017a). The hydrometeor size spectra data used to evaluate terminal fall speed broadening are available from the SNOWIE EOL/UCAR Data Archive at https://doi.org/10.5065/D6GT5KXK (French and Majewski 2017). The sounding data used to evaluate shear effects are available from the SNOWIE EOL/UCAR Data Archive at https://doi.org/10.5065/D6J38R93 (Blestrud 2021).
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