Polarimetric Doppler Radar Observations of Kelvin–Helmholtz Waves in a Winter Storm

Jana Lesak Houser School of Meteorology, University of Oklahoma, Norman, Oklahoma

Search for other papers by Jana Lesak Houser in
Current site
Google Scholar
PubMed
Close
and
Howard B. Bluestein School of Meteorology, University of Oklahoma, Norman, Oklahoma

Search for other papers by Howard B. Bluestein in
Current site
Google Scholar
PubMed
Close
Full access

Abstract

Kelvin–Helmholtz waves were observed by the Twin Lakes, Oklahoma (KTLX), Weather Surveillance Radar-1988 Doppler (WSR-88D); the Norman, Oklahoma (KOUN), polarimetric WSR-88D; and the polarimetric Collaborative Adaptive Sensing of the Atmosphere (CASA) radars on 30 November 2006 during a winter storm in central Oklahoma. The life cycle and structure of the waves are analyzed from the radar data, and the nearby atmospheric conditions are examined. The initial perturbations associated with the waves are first evident only in the radars’ radial velocity fields. As the waves mature, perturbations become discernable in the reflectivity factor Z and spectrum width (SW) fields of both radars, and in the differential reflectivity Zdr and, to a lesser extent, the cross-correlation coefficient ρhv fields of KOUN. As the waves break and begin to dissipate, the perturbations subside.

A dual-Doppler analysis is synthesized to examine the kinematic structure of the waves and to relate the polarimetric observations to the kinematics. It is determined that Z and Zdr are enhanced in regions of upward motion (wave crests), and ρhv is reduced in the same vicinity and near the base of the wave circulations. Vertical velocity perturbations transport horizontal momentum upward and downward, inducing horizontal wind perturbations that are approximately 90° out of phase and downstream from their corresponding vertical velocity perturbations. Perturbations in Z, Zdr, and ρhv are observed in the vicinity of wave crests while SW perturbations occur predominately in and just upstream from wave troughs. It is determined that perturbations in the polarimetric variables are a result of the waves modifying local precipitation microphysics. Perturbations in Z and Zdr are hypothesized to be the result of columnar ice crystal generation whereas those in ρhv likely result from the mixing of ice crystals of various shapes and sizes. Perturbations in SW are a result of turbulent motions likely associated with wave breaking and downward advection of a strong shear layer.

Corresponding author address: Jana Houser, University of Oklahoma, 120 David L. Boren Blvd., Suite 5900, Norman, OK 73072. E-mail: jana.b.houser-1@ou.edu

Abstract

Kelvin–Helmholtz waves were observed by the Twin Lakes, Oklahoma (KTLX), Weather Surveillance Radar-1988 Doppler (WSR-88D); the Norman, Oklahoma (KOUN), polarimetric WSR-88D; and the polarimetric Collaborative Adaptive Sensing of the Atmosphere (CASA) radars on 30 November 2006 during a winter storm in central Oklahoma. The life cycle and structure of the waves are analyzed from the radar data, and the nearby atmospheric conditions are examined. The initial perturbations associated with the waves are first evident only in the radars’ radial velocity fields. As the waves mature, perturbations become discernable in the reflectivity factor Z and spectrum width (SW) fields of both radars, and in the differential reflectivity Zdr and, to a lesser extent, the cross-correlation coefficient ρhv fields of KOUN. As the waves break and begin to dissipate, the perturbations subside.

A dual-Doppler analysis is synthesized to examine the kinematic structure of the waves and to relate the polarimetric observations to the kinematics. It is determined that Z and Zdr are enhanced in regions of upward motion (wave crests), and ρhv is reduced in the same vicinity and near the base of the wave circulations. Vertical velocity perturbations transport horizontal momentum upward and downward, inducing horizontal wind perturbations that are approximately 90° out of phase and downstream from their corresponding vertical velocity perturbations. Perturbations in Z, Zdr, and ρhv are observed in the vicinity of wave crests while SW perturbations occur predominately in and just upstream from wave troughs. It is determined that perturbations in the polarimetric variables are a result of the waves modifying local precipitation microphysics. Perturbations in Z and Zdr are hypothesized to be the result of columnar ice crystal generation whereas those in ρhv likely result from the mixing of ice crystals of various shapes and sizes. Perturbations in SW are a result of turbulent motions likely associated with wave breaking and downward advection of a strong shear layer.

Corresponding author address: Jana Houser, University of Oklahoma, 120 David L. Boren Blvd., Suite 5900, Norman, OK 73072. E-mail: jana.b.houser-1@ou.edu

1. Introduction

Kelvin–Helmholtz (K-H) waves are responsible for the vertical transport of heat, momentum, and nonmeteorological aerosols across a stable atmospheric interface, and are a primary source of turbulent mixing and clear air turbulence (Browning and Watkins 1970; Gossard and Richter 1970; Browning 1971; Shapiro 1980). As such, these waves have implications for the momentum and heat budgets of the atmosphere, as well as for the aviation community. Accordingly, the need to understand the structure and environments associated with K-H waves has made them the topic of a great deal of research.

K-H waves have been observed in a variety of capacities over the past several decades. Initially studied and documented by fluid dynamicists Helmholtz (1868) and Lord Kelvin (1871), later studies were done using both visual (Ludlam 1967; Reiss and Corona 1977) and laboratory observations (Drazin 1958; Thorpe 1968, 1971, 1973; Simpson 1969, 1982; Scotti and Corcos 1972; Britter and Simpson 1978; Koop and Browand 1979). As technology advanced, K-H waves were also observed by an assortment of instrumentation including ground-based radars (Hicks and Angell 1968; Browning and Watkins 1970; Mueller and Carbone 1987; Weckwerth and Wakimoto 1992; Chapman and Browning 1997; Petre and Verlinde 2004; Luce et al. 2008) and airborne radars equipped with additional instrumentation (Browning et al. 1973, Hardy et al. 1973; Wakimoto et al. 1992; Wakimoto and Bosart 2001; Friedrich et al. 2008; Geerts and Miao 2010), and were numerically simulated (Fritts 1979; Sykes and Lewellen 1982; Droegemeier and Wilhelmson 1986; Fritts et al. 1996).

With the continual improvement of technology, scientists’ ability to examine these structures and gain a better understanding of the waves’ characteristics and evolution advances. Polarimetric radars are the newest frontier in weather radar observations, and are expected to be implemented into the Weather Surveillance Radar-1988 Doppler (WSR-88D) network over the next several years (Ryzhkov et al. 2005). This study presents polarimetric observations of K-H waves that were observed in stratiform precipitation during a winter storm that occurred in Oklahoma on 30 November 2006. Although K-H waves have been studied extensively and are well understood overall, this paper is the first peer-reviewed publication to document their polarimetric properties as manifest in precipitation. The entire life cycle of the waves is captured in both polarimetric and standard radar fields. Additionally, a dual-Doppler analysis is synthesized when waves are mature and beginning to break, allowing for the polarimetric results to be related to the three-dimensional flow within the waves.

This paper will focus on the waves’ structure, their evolution, and the atmospheric conditions under which they were observed. It is organized in the following manner: instrumentation and methods used to acquire and analyze the data are discussed in section 2. Section 3 provides a brief synoptic overview at the time waves were occurring, introduces the observations that motivated this study, and substantiates the prescribed wave type. The wave characteristics, polarimetric signatures, evolution, and the three-dimensional flow field kinematics are presented in section 4. A discussion of the results is given in section 5, followed by a summary and conclusions in section 6.

2. Instrumentation and methods

a. Overview of observational data

The observational data used in this study came from a variety of sources. Surface observations were retrieved from the Oklahoma Mesoscale Network (Mesonet) (McPherson et al. 2007). Upper-air measurements were obtained from the National Weather Service (NWS) rawinsonde network Norman, Oklahoma, station (OUN) and the National Oceanic and Atmospheric Administration (NOAA) Profiler Network (NPN) Purcell, Oklahoma, profiler. Both horizontal wind observations (Hassle and Hudson 1989; Weber and Wuertz 1990) and radio acoustic sounding system (RASS) temperatures observations (Strauch et al. 1988; May et al. 1990) were utilized from the profiler. Radar data were acquired primarily from the WSR-88D instruments in central Oklahoma: the Twin Lakes, Oklahoma (KTLX), and the test bed polarimetric Norman (KOUN) radars. Data were supplemented by the Collaborative Adaptive Sensing of the Atmosphere (CASA) radars (McLaughlin et al. 2009). The specifications of the various radars are given in Table 1. Dual-Doppler analyses were not generated from the CASA data because data were collected at 1.5° and 3° elevation angles only. All radar data presented herein are processed level-2 data (Crum et al. 1993). Locations of the various instruments are given in Fig. 1a.

Table 1.

Specifications of the KTLX and KOUN WSR-88Ds and the CASA radar network: λ is wavelength, ν is frequency, θ is half-power beamwidth, Rmax is maximum unambiguous range, Vmax is maximum unambiguous velocity, VCP is volume coverage pattern, Z is reflectivity factor (dBZ), Zdr is differential reflectivity (dB), ρhv is cross-correlation coefficient (dimensionless), Vr is radial velocity (m s−1), Kdp is specific differential phase (° km−1), SW is spectrum width (m s−1), Az is azimuth angle (°), El is elevation angle (°), and PPI is plan position indicator.

Table 1.
Fig. 1.
Fig. 1.

(a) Locations of WSR-88D radars, CASA radars, and the Purcell profiler. Range rings are given for the radars with the smaller rings denoting 25 km and the larger ones denoting 50 km. The OUN rawinsonde site is located 4 km southeast of the KOUN radar. (b) Radar locations (black dots) and dual-Doppler analysis domain (thick solid black). The baseline is notated by the thin black line. The angles between KTLX and KOUN to the northwestern and southwestern corners of the grid, which represent the largest and smallest between beam angles within the grid, are given by Θ.

Citation: Journal of the Atmospheric Sciences 68, 8; 10.1175/2011JAS3566.1

b. Polarimetric data

The polarimetric variables archived by KOUN are differential reflectivity Zdr, cross-correlation coefficient ρhv, differential phase Φdp, and specific differential phase Kdp. Differential reflectivity is a measure of the ratio of backscattered power returned from the horizontally polarized channel to that from the vertically polarized one and is therefore a good indicator of hydrometeor shape. For targets with a horizontal dimension larger than the vertical dimension, Zdr > 0 dB; for spherical targets, Zdr ~ 0 dB (Zrnić and Ryzhkov 1999). High values of Zdr are typically found when the dominant hydrometeor type is large, oblate raindrops or horizontally oriented needle-like or platelike ice crystals. Generally, Zdr varies within 0–5 dB for meteorological targets (Ryzhkov et al. 2005). Quantitative measurements of Zdr are also useful in conjunction with other radar parameters (particularly Z and ρhv) in discriminating among hydrometeor types (Ryzhkov and Zrnić 1998a; Straka et al. 2000; Ryzhkov et al. 2005; Park et al. 2009). The cross-correlation coefficient is the correlation between the complex voltages returned from the horizontal and vertical channels. Values for ρhv range from 0 to 1 and are dependent upon the size, orientation, shape, and dielectric constant of the scatterers within a radar resolution volume. For meteorological scatterers, ρhv generally ranges from 0.8 to 1 (Ryzhkov et al. 2005) and is typically greater than 0.95. Wet snow, tumbling hail, and a mixture of precipitation types act to reduce ρhv owing to the different shapes, orientations, sizes, and dielectric constants of the hydrometeors contained in a single sample volume (Zrnić et al. 1993; Zrnić and Ryzhkov 1999).

The differential phase measures the phase lag between the horizontally and vertically polarized waves. When there is a high concentration of large, oblate raindrops, the phase of the horizontally polarized wave increasingly lags the phase of the vertically polarized one. This occurs because the difference in dielectric constant between air and water causes the horizontally polarized wave, which is traveling through more water than the vertically polarized one, to propagate more slowly than the vertically polarized wave. Therefore, the phase of the horizontal wave increasingly lags that of the vertical wave. The specific differential phase is the range derivative of Φdp, which provides information about how Φdp changes with distance. This information is valuable when determining precipitation rates (Chandrasekar et al. 1990; Scarchilli et al. 1993). Because Kdp is not biased by reflectivity, it is effective in correcting for signal attenuation (Bringi et al. 2001; Snyder et al. 2010). For a more detailed explanation of polarimetric variables and KOUN’s specific capabilities, the reader is referred to Zrnić and Ryzhkov (1999), Bringi et al. (2001), and Ryzhkov et al. (2005), among others.

Although polarimetric data are very valuable in providing additional information about hydrometeors, the data are not perfect and they have limitations. Those limitations potentially relevant to this study will now be discussed and arguments will be made to substantiate the validity of the polarimetric observations for this case.

It will be shown subsequently that Kelvin–Helmholtz waves created oscillatory perturbations in Z, Zdr, ρhv, and spectrum width (SW). Ryzhkov and Zrnić (2007) found that errors in Zdr can be introduced by wave depolarization through scattering media with nonzero canting angles, producing unusual-looking radial oscillations of Zdr. Depolarization requires two conditions: 1) pristine ice crystals and 2) an electrical field that is sufficiently strong enough to align the ice crystals vertically along the direction of the electrostatic field. Ryzhkov and Zrnić (2007) have documented this effect in the anvil clouds and upper atmospheric regimes of thunderstorms, where depolarization caused Zdr oscillations in the along-beam direction. The authors feel confident that depolarization is not the cause of the Zdr oscillations observed in this study because the oscillations were not merely parallel to the beam radials, but rather had a persistent cross-radial component. Thus, any depolarization that may have existed would not have explained the presence of the observed Zdr features.

Another source for error in polarimetric data is partial beam filling (Ryzhkov and Zrnić 1998a; Ryzhkov 2007), which occurs when there are significant differences of hydrometeor scattering properties within the horizontal or vertical distance contained in the radar sampling volume. Such errors typically occur along the edges of precipitation where there is a large reflectivity gradient, and in the vicinity of the melting layer. For this case, precipitation was generally light and was not associated with strong horizontal reflectivity gradients. There was, however, a bright band–like feature present, which was associated with a vertical reflectivity gradient. At a range of about 30 km, the partial beam filling associated with the bright band may have negatively biased the vertical placement of the bright band by 100–200 m (Ryzhkov 2007). Oscillatory behavior in Z, Zdr, and ρhv cannot be explained by partial beam filling.

Errors, particularly in Zdr, can occur as a result of ground-clutter filtering algorithms. The filter was off for this case so the data are not affected by such errors. We do not believe the observations of waves presented herein are spurious because oscillations cannot be explained by polarimetric errors, and the atmospheric context and physical locations in which the waves occur match well with where theory predicts waves could form, as will be shown in section 3.

The only source of polarimetric error relevant to the findings of this study is noise biases in ρhv and Zdr from low signal-to-noise ratios (SNRs). Because the precipitation falling during this analysis was snow, reflectivity factors were low, ranging from 0 to 20 dBZ. Despite the proximity of waves to the radar (the range was between 20 and 30 km from KOUN), the SNR still dropped to below 10 dB about halfway into the wave domain. To account for this potential bias, ρhv and Zdr data were corrected according to the method suggested by Schurr et al. (2003):
e1
e2
where , , and α is a ratio of the noise parameters in the horizontal and vertical channels (α = 1.48 for KOUN). These corrections were applied only to the dual-Doppler analyses; the raw PPI files could not be easily corrected.

c. Dual-Doppler analyses

The dual-Doppler analysis was generated from KOUN and KTLX level-2 radar data. The baseline between these radars is 21 km and the location of the dual-Doppler grid domain is illustrated in Fig. 1b. At the time the K-H waves were observed, KTLX was scanning with a stratiform precipitation volume coverage pattern (VCP) that had poor data coverage at elevation angles greater than 8°; consequently, the quality of the dual-Doppler analysis deteriorates above about 4 km. Fortunately, the waves were located below 4 km and were well resolved.

Prior to analysis, the data were manually edited to remove ground clutter and noise and to dealias the radial velocity. Data were then interpolated onto a Cartesian grid using the National Center for Atmospheric Research’s (NCAR)’s Reorder software package, following an exponential analysis scheme, similar to that of Barnes (1964), with a weighting function of
e3
where r is the distance from the radar gate to the grid point; R2 is the sum of the squares of the x, y, and z radii of influence; and β is an attenuation parameter proportional to the coarsest data spacing within the analysis domain. To prevent undersmoothing in data sparse regions, β was chosen based on the most conservative value of maximum data spacing (i.e., where the data are spaced farthest apart) (Trapp and Doswell 2000). The grid was constructed with a domain of 20 × 20 × 10 km3 (x, y, z) and was centered on a point approximately equidistant from both radars. The grid center and domain were specified to ensure the angle of intersection between the two radar beams (between beam angle) would be greater than 30° and less than 150° for all points within the grid, minimizing errors in the u and υ calculations. The horizontal and vertical radii of influence were specified to be 1 km and were chosen based on the horizontal and vertical radar resolution (Trapp and Doswell 2000). Therefore, features with a wavelength less than 2 km were not resolvable. Horizontal and vertical grid spacing was 250 m.

The dual-Doppler analysis was synthesized using NCAR’s Custom Editing and Display of Reduced Information in Cartesian Space (CEDRIC) software. The horizontal wind components and kinematic vertical velocity w were calculated by iteratively invoking the mass conservation equation (Armijo 1969) until the solutions for u, υ, and w converged to a difference of 0.01 between iterations. The data were integrated from the bottom up, with the lower boundary condition specified as w = 0. Upward integration was chosen because the highest radar observations were below the tropopause, and therefore it could not be assumed that w vanished at the top of the grid domain. Errors in vertical velocity are on the order of several meters per second and horizontal velocity errors are even less (Doviak et al. 1976).

3. Synoptic environment and wave classification

a. Synoptic environment

The waves analyzed herein were observed between 1700 and 1830 UTC 30 November 2006. The synoptic environment at 1800 UTC is depicted in Fig. 2. At this time, there was a surface low in northeastern Canada (Fig. 2a). Its trailing cold front, which had passed through central Oklahoma during the morning of 29 November, extended to the southwest through the Texas–Louisiana border. Along the front, a secondary surface low was developing in southwestern Arkansas. At 850 hPa, a closed low associated with an amplified shortwave trough was situated over northeastern Oklahoma (Fig. 2b). This disturbance tilted westward with height, as is evident in the 700-hPa height field (Fig. 2c), and a 500-hPa closed low was located over the Texas Panhandle (Fig. 2d). A large swath of precipitation associated with the front was evident in the national radar reflectivity factor composite (Fig. 3a), extending from the Great Lakes through eastern Texas. A more detailed view of the precipitation in Oklahoma at the time the waves were occurring is given by the regional radar reflectivity factor composite in Fig. 3b. Where the waves were observed, there was a region of light (15–20 dBZ) stratiform precipitation that formed in response to the mid- and upper-level disturbances.

Fig. 2.
Fig. 2.

Synoptic analysis at 1800 UTC 30 Nov 2006. (a) Surface frontal and pressure analysis (NOAA Hydrological Prediction Center). (b) 850-hPa height field (m) and wind barbs from the 1800 UTC 20 km RUC analysis (Benjamin et al. 2004a,b). (c),(d) As in (b), but for (c) 700 and (d) 500 hPa.

Citation: Journal of the Atmospheric Sciences 68, 8; 10.1175/2011JAS3566.1

Fig. 3.
Fig. 3.

Level-3 radar reflectivity factor (dBZ) composites. (a) National composite at 1800 UTC. (b) Regional composite at 1745 UTC. The black box indicates where waves were observed. Both images are on 30 Nov.

Citation: Journal of the Atmospheric Sciences 68, 8; 10.1175/2011JAS3566.1

Surface winds at 1745 UTC (Fig. 4a) were generally northerly across all of Oklahoma, and wind speeds were between 7.5 and 12.5 m s−1, except in extreme southeastern Oklahoma where they were weaker. Surface temperatures at this time (Fig. 4a) were between −4° and −8°C, except again in the southeast. From the 1200 UTC OUN rawinsonde (Fig. 4b) (6 h earlier), it is seen that the cold surface-layer air was quite shallow, extending to a height of only about 1 km. Above this layer there was a 15°C inversion, about1 km deep, beneath a nearly moist adiabatic layer about 1.25 km thick. Winds veered from north-northeasterly to southeasterly between the surface and the middle of the inversion, decreasing in speed with height. Above this, winds continued to veer until they became southwesterly and speeds increased with height to the tropopause.

Fig. 4.
Fig. 4.

(a) Surface observations [temperature, wind, mean sea level pressure (MSLP)] from the Oklahoma Mesonet at 1745 UTC. Short (long) wind barbs correspond to 2.5 (5) m s−1. Numbers and thin gray contours are 1.5-m temperatures (°C). Thick contours are MSLP (every 3 hPa). The star indicates the location of the OUN rawinsonde release point. (b) OUN rawinsonde from 1200 UTC. Gray line = 0°C. Both images are on 30 Nov.

Citation: Journal of the Atmospheric Sciences 68, 8; 10.1175/2011JAS3566.1

From the hourly averaged Purcell profiler horizontal wind observations (Fig. 5a), it is seen that surface winds remain northeasterly and veer with height to southwesterly at 2 km between 1200 and 1800 UTC. The only noteworthy changes in the low-level wind profile during this period are the winds just above 2 km, which veer to westerly, and the wind speeds between 1 and 2 km, which weaken by about 5–10 m s−1. According to the RASS observations, the thermal profile below 4 km cooled 3°–5°C between 1200 and 1800 UTC (Fig. 5b).

Fig. 5.
Fig. 5.

Purcell profiler data. (a) Hourly horizontal winds (m s−1) with height (m) and pressure (hPa) from 1200 to 1800 UTC 30 Nov. Time increases to the left. (b) RASS temperature (°C) soundings with time (UTC) and height (m). Some missing data have been interpolated.

Citation: Journal of the Atmospheric Sciences 68, 8; 10.1175/2011JAS3566.1

The RASS observations are given to provide an idea of the thermal tendencies of the lower atmosphere during a period in which rawinsonde data are unavailable. The precise RASS temperatures should be viewed with caution. A comparison between the 1200 UTC rawinsonde temperatures (assumed to be correct) and the 1200 UTC hourly averaged RASS temperatures revealed that the RASS temperatures have a warm bias of 1°–4°C in the lowest 4 km. This difference can be accounted for by several factors but occurs primarily because the RASS data are not corrected for vertical velocity. The profiler recorded vertical winds ranging between −1 and 3 m s−1, which correlates to a bias of approximately −1.5° to +5°C. Additionally, vertical gradients in the reflectivity can bias the temperature data (D. Van de Kamp, NOAA, 2011, personal communication). The RASS measures virtual temperature, whereas the rawinsonde measures actual temperature. Given the observed virtual temperatures and the mixing ratios, the difference between true and virtual temperature accounts for about 1°C. The RASS station is also approximately 30 km south of the rawinsonde station and in the presence of a sloped cold front the temperatures likely are not horizontally uniform. RASS temperatures also are averaged over 250-m vertical range bins, which reduces the accuracy of point measurements. Weber et al. (1992) discuss other internal sources of error in RASS temperature observations, which include turbulent fluctuations within the scattering volume, wind variations over the hour in which the temperature is being averaged, and displacement of the acoustic signal by horizontal winds. Despite the shortcomings of the RASS observations, they are still useful for illuminating atmospheric conditions relevant to this study.

b. Wave classification

Between 1200 and 1830 UTC 30 November, wavelike features were identified intermittently in the CASA, KTLX, and KOUN data as areas of undulation in the 0 m s−1 radial velocity Vr line at about 1.4 km AGL, where northerly low-level flow veered to southwesterly aloft, according to the Purcell profiler (Fig. 5a). The result was a zigzag appearance in Vr as illustrated in Figs. 6 and 7. To the authors’ knowledge, such radial velocity features are not documented elsewhere in the literature, which raised the initial question of this study: what are these features? In an effort to answer this question, the thermal and kinematic conditions in the vicinity of the oscillations were examined more closely.

Fig. 6.
Fig. 6.

Radial velocity (m s−1) displays at an elevation angle of 3.3° for (a) KOUN and (b) KTLX at 1741 UTC 30 Nov. Range rings are every 10 km; azimuths are every 30°. Arrows point to wavelike features.

Citation: Journal of the Atmospheric Sciences 68, 8; 10.1175/2011JAS3566.1

Fig. 7.
Fig. 7.

Radial velocity (m s−1) displays from the Cyril CASA radar illustrating the progression of K-H waves with time. Radial velocity is displayed as ±π m s−1, where π corresponds to the Nyquist velocity (15.9 m s−1). Times are (a) 1321 (white lines denote isochrones of the wave indicated by the arrow at the times specified), (b) 1325, (c) 1329, and (d) 1333 UTC.

Citation: Journal of the Atmospheric Sciences 68, 8; 10.1175/2011JAS3566.1

According to the Purcell profiler observations at 1700 UTC (Figs. 5a,b), the base of the cold frontal inversion was located just above 1 km, as was a wind shift from north-northeast to southwest. The wind shift is more obvious in the velocity–azimuth display (VAD) and hodograph generated from KOUN at 1744 UTC (Fig. 8), while the waves were occurring. It is common knowledge that K-H instability can generate waves along a thermally stable interface with strong vertical wind shear (Ludlam 1967; Scorer 1969; Browning 1971; Drazin and Reid 1981) and therefore it was hypothesized that the observed waves were a result of Kelvin–Helmholtz instability.

Fig. 8.
Fig. 8.

(a) Velocity azimuth display (VAD) plotting the u (black) and υ (gray) wind components (m s−1) from the KOUN 7.5° elevation scan at 1744 UTC. (b) Hodograph from the VAD winds shown in (a). Heights are indicated. The lowest VAD wind data have been removed to reflect the true Mesonet winds, rather than the 0 m s−1 radar retrieved velocity.

Citation: Journal of the Atmospheric Sciences 68, 8; 10.1175/2011JAS3566.1

To provide further support of this hypothesis, additional environmental parameters characteristic of K-H instability were examined. Miles and Howard (1964) determined that K-H instability can develop within stably stratified shear flow when the Richardson number Ri < 0.25. Therefore, the bulk Richardson number RiB was calculated from wind data acquired from the Purcell upper-air profiler at 1700 UTC (~15 min prior to wave development). Some RASS temperature data were available at this time; however, data were often missing at critical heights and, as previously established, the RASS temperature observations were positively biased. Therefore, the KOUN sounding temperatures were used and modified to reflect the cooling trend observed in the RASS data (Fig. 5b). The RiB was calculated according to Holton (1992):
e4
where g is the acceleration due to gravity; Δθ is the difference in potential temperature (K) between the top and the bottom of the layer; Θ is the average potential temperature (K) of the layer, here calculated by ; Δz is the depth (m) of the layer; and Δu (υ) is the change in the zonal (meridional) wind component (m s−1) between the top and the bottom of the layer.

The RiB over the depth of the shear layer (between 830 and 1580 m AGL) was 0.61, which does not quite meet the instability criteria for the generation of K-H waves. However, in the layer between 1330 and 1580 m, RiB = 0.14. Some studies have found that, once realized, K-H instability can persist provided that RiB ≤ 1 (Wallace and Hobbs 1977; Weckwerth and Wakimoto 1992). It is therefore hypothesized that K-H waves initially formed within a shallow layer that met the instability criterion. As the waves grew, they likely extracted energy from the deeper shear layer and persisted in that layer because RiB < 1.

According to linear theory, the most unstable wavelength of a K-H wave should lie between 4.4D and 7.5D, where D is the depth of the shear layer (Miles and Howard 1964; Drazin 1958). The average wavelength for the mature waves observed in this case was 5.7 km [determined by calculating the perpendicular distance between consecutive wave crests in Vr over the 45 min (1725–1810 UTC) that multiple wave crests were present and were able to be identified subjectively]. This wavelength would therefore require a shear layer depth between 760 and 1300 m. The observed depth of the shear layer was approximately 1100 m according to the Purcell profiler winds (Fig. 5a) and the VAD generated from KOUN (Fig. 8), which lies within the theoretical range.

Because the wave features observed were within a highly sheared layer and a stable thermal layer, met the RiB < 0.25 criteria, and were observed in a shear layer whose depth was appropriate for the features’ wavelength, it was concluded that these features were Kelvin–Helmholtz waves.

4. Wave structure and evolution

The waves were observed in the Cyril CASA radar from 1240 to 1410 UTC and in the KOUN and KTLX radars from 1715 to 1820 UTC. The reason for the different observation times is unclear. It is not believed that the waves propagated from Cyril, Oklahoma, to KOUN. When waves first appear in the KOUN data, they are in the form of small-amplitude perturbations that appear simultaneously and grow with time, rather than a preexisting wave train entering the domain. It is therefore proposed but unconfirmed that the mesoscale conditions required to generate the waves were not present simultaneously over a large domain, but the necessary conditions were met locally at different times. Even during the times that waves are observed within the KOUN and KTLX data, they are confined to a spatial domain of roughly 30 × 10 km2 at a range of 20 km to the east of KOUN.

From a sequence of plan position indicators (PPIs) that increase in elevation angle, and from a vertical cross section through these PPIs generated at a constant range over the azimuthal extent of the waves (Fig. 9), it is seen that the waves are apparent in the radar data at heights between approximately 750 and 1550 m. (Heights were determined from the PPIs by examining the radar beam height at the wave location over various elevation angles and a vertical cross section taken at constant range over consecutive azimuths). The waves were oriented from the west-northwest to the east-southeast, which is approximately orthogonal to the 925–775-hPa (representing the pressures at the lower bound of the wave location and the inversion top, respectively) wind shear vector (18 m s−1 from 210°) and the average wind below the inversion (15 m s−1 from 20°) (refer to Fig. 5a). Wave propagation was determined using the 3° elevation angle CASA data PPIs, which had an update time of about 30 s (Fig. 7), allowing for the wave motion to be defined without aliasing. Waves propagated approximately parallel to the 925–775-hPa shear vector, from the south-southwest to the north-northeast at about 7.8 m s−1, with a period of about 6.5 min. Although the waves did not appear simultaneously in the CASA and KOUN data, it is reasonable to assume the propagation speed and direction did not change because the wind profile remained approximately constant between 1200 and 1700 UTC.

Fig. 9.
Fig. 9.

Sequence of KOUN radial velocity (m s−1) PPIs from 1740 to 1744 UTC illustrating the vertical structure of the radial velocity field associated with K-H waves: (a) 0.5°, (b) 1.5°, (c) 2.4°, (d) 4.3°, (e) 8.7°, and (f) 10°. For (a) and (b) the Nyquist velocity was lower than for the higher elevations, resulting in a slightly different graphical appearance. Velocities are not dealiased for this graphic. (g) Radial velocity (m s−1) constant range (r = 23 km) vertical cross section from 50° to 160° azimuth for the volume shown in (a)–(f). Location of the cross section is indicated by the thin black arc in (d).

Citation: Journal of the Atmospheric Sciences 68, 8; 10.1175/2011JAS3566.1

The polarimetric and dual-Doppler analyses focus on the waves that were observed in a small portion of the KOUN and KTLX radar domains from approximately 1715 to 1820 UTC. These waves were visible from a range of about 25–30 km, between 45° and 165° azimuth in the KOUN display, and from a range of about 20–40 km, between 150° and 210° azimuth in the KTLX display (Fig. 6). Over this period, the entire evolution of the waves is captured, from the initial perturbations, through wave breaking and turbulent mixing, and ultimately dissipation. The previously mentioned zigzag pattern observed in Vr can be explained if parcel momentum is assumed to be largely conserved. According to Fig. 5a, the flow at the base of the observed waves was north-northeasterly; however, the flow at the tops of the waves was southwesterly. Imposing a sinusoidal disturbance in the vertical plane upon such a shear flow would result in southwesterly momentum being transported downward within the troughs of the waves, and north-northeasterly momentum being transported upward within the crests of the waves. Therefore, where wave troughs (crests) are located, Vr would display inbound (outbound) velocities (for a radar situated north of the waves, as was KTLX). The net effect of the sinusoidal disturbance would be to create an undulation between inbound and outbound velocities along the 0 m s−1 radial velocity line at a given height or elevation angle, and this is what is seen in the data from both KOUN and KTLX (Fig. 6).

The evolution of the waves in the Z, Vr, Zdr, ρhv, SW, and Kdp radar fields is presented in Figs. 1015. The waves are first noted in the KOUN Vr field at 1716 UTC (Fig. 10) as two weak perturbations within an approximately 200-m layer (trough to crest depth, determined by the range–height correlation in the PPI radar data), at a height of about 1.4 km. This height lies within the previously mentioned layer over which RiB < 0.25. These initial perturbations are best resolved in the 6.2° elevation scan, but the 4.3° scan was used in Fig. 10 to maintain consistency with the rest of Figs. 1115 and to show that the radial velocity field was not perturbed prior to the onset of the waves. By 1729 UTC, several more Vr perturbations become obvious, now over a depth of approximately 700 m, and perturbations appear in Z and Zdr as elongated zones of enhanced Z and Zdr (Fig. 11). The SW field also exhibits undulations of higher SWs, similar to that of Vr. By 1736 UTC, the waves are well defined and are approaching their peak intensity (Fig. 12). They now perturb a layer approximately 775 m deep and are obvious in the Vr, Z, SW, Zdr, and ρhv fields. The waves do not appear to perturb the Kdp field; throughout this period, Kdp is noisy, having no discernable features. This behavior can be explained because it was snowing at this time, and Kdp is generally about 0° km−1 for dry snow and ice crystals (Ryzhkov and Zrnić 1998b). In locations where Z is enhanced, Zdr is also higher and ρhv is slightly lower, implying a positive correlation between Z and Zdr and a negative correlation between Z and ρhv. The regions between enhanced Z and Zdr and lower ρhv are not oppositely perturbed; the fields remain the same as those in the unperturbed environment. The same relationship between these radar variables was observed for the earlier waves by the Cyril CASA radar (Fig. 16). By 1742 UTC, the depth of the layer over which waves are visible has increased to approximately 800 m. At this time, the waves achieve their peak intensity (Fig. 13). It is hypothesized that waves begin to break at this time because the waves are less prominent in the following radar volume at 1748 UTC (not shown), the depth over which they are visible decreases after this time, the maximum SW increases from the previous scan (1736 UTC), and the perturbations in the other radar fields decline after this point. As waves begin to break, turbulence between the crest and trough of the wave increases, which widens the Doppler spectrum, resulting in larger SW values. By 1801 UTC, the waves are less noticeable in Vr and SW, and the perturbations in Z, Zdr, and ρhv are no longer obvious (Fig. 14). The analysis of the polarimetric perturbations at this time becomes complicated because a break in precipitation coverage enters the wave domain. By 1826 UTC, light precipitation is again falling in the wave domain and there is little evidence of waves in any radar field (Fig. 15). It is suggested from Figs. 1015 that the waves did not merely propagate out of the radar domain during this time, but rather perturbations weakened and dissipated over time. This observation is further supported by animations of the KOUN radar data at various elevation angles (which are unable to be shown in this paper), which make it clear that the wave depths became shallower with time until the perturbations no longer existed.

Fig. 10.
Fig. 10.

KOUN radar display at 4.3° elevation, at 1716 UTC 30 Nov of radar fields just prior to the onset of K-H waves, showing (a) reflectivity (dBZ), (b) radial velocity (m s−1), (c) spectrum width (m s−1), (d) differential reflectivity (dB), (e) cross-correlation coefficient (dimensionless), and (f) specific differential phase (° km−1). Arrows in (d) and (e) point to the bright band. Range rings are every 10 km and azimuth lines every 30°. Gray colors indicate data values that exceed the color bar.

Citation: Journal of the Atmospheric Sciences 68, 8; 10.1175/2011JAS3566.1

Fig. 11.
Fig. 11.

As in Fig. 10, but at 1729 UTC and after the onset of K-H waves. Black lines denote consecutive wave locations according to Vr and are at the same location for all radar fields. Arrows in (d) and (e) point to the bright band.

Citation: Journal of the Atmospheric Sciences 68, 8; 10.1175/2011JAS3566.1

Fig. 12.
Fig. 12.

As in Fig. 11, but at 1736 UTC.

Citation: Journal of the Atmospheric Sciences 68, 8; 10.1175/2011JAS3566.1

Fig. 13.
Fig. 13.

As in Fig. 11, but at 1742 UTC.

Citation: Journal of the Atmospheric Sciences 68, 8; 10.1175/2011JAS3566.1

Fig. 14.
Fig. 14.

As in Fig. 11, but at 1801 UTC.

Citation: Journal of the Atmospheric Sciences 68, 8; 10.1175/2011JAS3566.1

Fig. 15.
Fig. 15.

As in Fig. 11, but at 1826 UTC.

Citation: Journal of the Atmospheric Sciences 68, 8; 10.1175/2011JAS3566.1

Fig. 16.
Fig. 16.

Perturbations from K-H waves in CASA’s Cyril radar display at 3° elevation at 1326 UTC 30 Nov: (a) reflectivity factor (dBZ), (b) radial velocity (m s−1), (c) cross correlation coefficient (dimensionless), and (d) differential reflectivity (dB). Range rings are every 10 km and azimuth lines every 30°. Gray colors indicate data values that exceed the color bar. Spectrum width and Kdp variables were not available. Black boxes indicate the location of waves. Lines denoting waves were not used because they overwhelmed the wave perturbations and made them less visible.

Citation: Journal of the Atmospheric Sciences 68, 8; 10.1175/2011JAS3566.1

To examine further the structure of the K-H waves and how the polarimetric observations are related to the kinematic wind field, a dual-Doppler analysis was synthesized at 1745 UTC, when waves are considered to be mature. The vertical extent of the waves in the dual-Doppler analyses differed from what was seen in the radar PPIs. Waves in the analyses began at a height of approximately 1.25 km and extended to approximately 3.5 km. This discrepancy is addressed at the end of this section. Radar and wind field properties at 1.5 km AGL are shown in Fig. 17. The 1.5-km height was chosen because the wave signals there are the clearest, the oscillation between inbound and outbound velocities is the greatest, and the perturbations are most notable.

Fig. 17.
Fig. 17.

Dual-Doppler analysis depicting the kinematics in a wave train of K-H waves at 1745 UTC 30 Nov at 1.5 km AGL. KTLX is located near (12, 30). KOUN is near (−5, 20) (Fig. 1b). (a) KTLX Vr (m s−1) and dual-Doppler uυ streamlines. The red lines indicate the locations used to calculate the average cross sections in Fig. 19. (b) Kinematic vertical velocity (m s−1) and uυ streamlines. Lines indicate the phase of the horizontal wind perturbations, with the longer one equal to 2π and the shorter one equal to π/2. (c) Horizontal divergence (s−1 × 10−4) and uυ streamlines. (d) Vertical vorticity (s−1 × 10−4) and vertical velocity contours (thick black: w = 0 m s−1; thin black: w = 5 m s−1; dashed: w = −5 m s−1). (e) Spectrum width (m s−1) and vertical velocity contours (m s−1) as in (c).

Citation: Journal of the Atmospheric Sciences 68, 8; 10.1175/2011JAS3566.1

From Fig. 17a it is seen that the 1.5-km AGL horizontal flow is disturbed by the waves. The spatial relationship between vertical velocity and horizontal streamlines is illustrated in Fig. 17b. There is a clear wave signal in w over the northern part of the domain, where vertical motions oscillate between ascending and descending motion. In and downstream (in a wave propagation sense; i.e., to the north-northeast) of downward motion, the flow generally has a southwesterly or westerly component and in and downstream of upward motion it has a northerly or northwesterly component. This suggests that the vertical motions associated with the waves transport northerly (southwesterly) momentum upward (downward) in the vicinity of outbound (inbound) radial velocities, as suggested previously. The flow is most southwesterly (northerly) between regions of downward (upward) and upward (downward) motion just downstream (relative to wave motion) from the maximum negative (positive) w. Toward the southern portion of the domain, there is less structure to the w field, implying a more turbulent flow field. The incoherence in w could be an indication that waves in this region were breaking.

Vertical velocities are generally on the order of 5–10 m s−1 (Fig. 17b), which is similar to what Geerts and Miao (2010) also found for K-H waves in a winter storm. It should be noted that the exact values of w should be viewed with caution because the kinematically derived w is contaminated both through errors introduced in the u and υ calculations and through vertical integration. In regions where there is divergence, the average w (wave) is −2.2 m s−1, and where there is convergence wave is 3.04 m s−1. Thus, at this height, which is near the base of the wave according to the dual-Doppler analyses (refer to the end of section 4 for a discussion of the wave layer in the dual-Doppler analyses), horizontal convergence is associated with wave crests (w > 0), and divergence with troughs (w < 0), which would be expected.

The oscillations in the horizontal wind are also associated with periodic fluctuations of horizontal divergence δ (Fig. 17c) and vertical vorticity ζ (Fig. 17d), suggesting that the K-H waves are also associated with patterns in δ and ζ. Positive (cyclonic) ζ was collocated with wave crests and upward motion, while negative (anticyclonic) ζ was collocated with wave troughs and downward motion (Figs. 17b,d). The pattern of vertical vorticity is consistent with the tilting and advection of the environmental vertical wind shear by the wave-relative wind. An illustration of the environmental kinematics in the presence of vertical wave perturbations and how vorticity is tilted and advected over the vertical perturbations is given in Fig. 18a. Most of the contribution to tilting is from the (∂w/∂y)(∂u/∂z) term, since there is little variation in w in the x direction where the wave signal is strongest (refer to Fig. 17b for structure of w field). Although the 925–775-hPa shear vector was south-southwesterly, the vertical shear just above and below 2 km (the average height of wave circulation centers) was northwesterly (8 m s−1 from 315°), which was along the wave axis. According to Markowski and Richardson (2010), the vertical vorticity equation with respect to a moving feature linearized around a mean base-state flow—assuming the base-state environment has no vertical motion or vertical vorticity, only vertical shear, and neglecting friction, the Coriolis force, and products of perturbations—is
e5
where ζ′ is perturbation vertical vorticity, is the base-state horizontal velocity vector, c is the wave propagation vector, ∂vh/∂z is the horizontal shear vector, and w′ is perturbation vertical velocity. According to (5), vertical vorticity perturbations are the result of either horizontal advection of vertical vorticity or tilting of horizontal winds by vertical velocity gradients. Without advection, vorticity perturbations should straddle the regions of upward or downward vertical velocity. For this case, (5) implies that ζ′ > 0 (ζ′ < 0) would be to the right of the shear vector in a region upward (downward) motion, as illustrated in Fig. 18a. The wave relative wind (, where is approximated as the average horizontal wind over the depth of the wave layer) is approximately 2.5 m s−1 from 220°, according to wind data from the Purcell profiler (Fig. 5a). Therefore, wave-relative advection would displace the vertical vorticity downstream relative to the wave motion, causing the vorticity centers to be collocated with regions of upward and downward motion, with cyclonic vorticity collocated with upward motion and anticyclonic vorticity collocated with downward motion, as shown in Fig. 18a. According to Fig. 17d, this is indeed what is observed.
Fig. 18.
Fig. 18.

Conceptual models of the production of vertical vorticity in K-H waves and the structure of the waves. (a) The tilting of horizontal vorticity (vortex line shown as solid line with arrows) associated with vertical wind shear (Vs layer) into the vertical by gradients in vertical velocity (upward and downward motion indicated by open arrows). Combined with wave-relative horizontal advection, the vorticity centers move downstream, to be collocated with updrafts and downdrafts. Based on (5). (b) Model of the K-H wave structure and the relationships among polarimetric parameters, the 3D wind field, and hypothetical thermal perturbations. The wave to the far left shows only the velocity perturbations, the wave in the middle shows the velocity and polarimetric perturbations together, and the wave to the far right shows polarimetric perturbations with thermal perturbations. The thick line denotes the theoretical boundary between cold dense air below and warm less dense air above, essentially outlining the shape of the waves. Large open arrows denote the flow field. The dashed ovals identify regions of horizontal wind perturbations due to downward transport of momentum. The dotted ovals denote regions of high spectrum width and horizontal wind perturbations due to upward transport of momentum. The solid circles denote regions of positive Z and Zdr perturbations, while the thick dashed ovals identify regions of negative ρhv perturbations.

Citation: Journal of the Atmospheric Sciences 68, 8; 10.1175/2011JAS3566.1

The horizontal flow undulations become increasingly amplified and less coherent toward the southern portion of the domain, as do δ and ζ, suggesting that the waves in the southern portion are more mature, with stronger perturbations and vertical motion than those to the north. While it is possible that some of the variability in δ and ζ might be due to increasing error toward the edge of the dual-Doppler domain, the errors resulting from the smaller between-beam angle near the bottom edge of the domain are nearly the same as those for the top edge where the between-beam angle is large. Since similar behavior of δ, ζ, and w is not seen along the top edge, the variability along the bottom edge is likely not spurious. High values of spectrum width are apparent in and just upstream (in a wave-relative sense) from regions where w < 0 (Fig. 17e). It is expected that SW should be largest here, where vortex roll-up and wave breaking occurs, resulting in a wider velocity spectrum and consequently higher SW.

To determine the relationships between the polarimetric fields and the wind field, average north–south vertical cross sections of Z, Zdr, ρhv, and Vr with overlays of the υw wind vectors were generated roughly perpendicular to the waves’ axes (Fig. 19). The averages were computed by taking a series of cross sections along a constant x at locations denoted by red lines in Fig. 17a, shifting the data in the y direction to compensate for the horizontal wave slope, and calculating the average field values at common y grid points. From Figs. 19a and 19b it is seen that the positive Z and Zdr perturbations appear to be nearly collocated with positive w perturbations. The relationship between ρhv and w is similar but not as well behaved. It is seen from Fig. 19c that two of the three well-defined circulations have lower values of ρhv (~0.87) where w > 0. The ρhv perturbation associated with the middle circulation is toward the base of the circulation, where upward motion is beginning.

Fig. 19.
Fig. 19.

Average north–south vertical cross sections and Doppler-derived υw vectors (m s−1) through a series of K-H waves. The locations of the cross sections contributing to the average are noted by red lines in Fig. 17a. Locations where arrows are present but the other plotted variable is not indicate that the value exceeds the range of the color bar. (a) Reflectivity factor (dBZ). (b) Differential reflectivity (dB). (c) Cross-correlation coefficient (unitless). Circles denote upward vertical velocities and encompass the same regions. Arrows denote low ρhv perturbations. (d) Radial velocity (m s−1). Circles identify circulations associated with the waves. Black dots denote the centers of circulations.

Citation: Journal of the Atmospheric Sciences 68, 8; 10.1175/2011JAS3566.1

In summary, perturbations in the horizontal winds associated with upward and downward transport of horizontal momentum are located just downstream from their associated vertical velocity maxima or minima. High Z and Zdr perturbations are collocated with upward motion, while deficits of ρhv occur toward the base of the upward branch of the circulation, and SW perturbations are located just upstream from downward motion. The relationships among the wind field and the polarimetric parameters as described above are summarized in Fig. 18b.

From the radial–velocity cross section (Fig. 19d), it is seen that the circulations associated with the K-H waves are collocated with a shear zone separating low-level northerly flow from southerly flow aloft (KTLX is located to the north of the cross section). The average shear across this interface was 13 m s−1 over 1500 m, or 0.0087 s−1. The circulations are associated with flow perturbations above and below them, have an average vertical depth of about 2 km, and are centered at slightly different heights, as noted in previous studies (Simpson 1969; Droegemeier and Wilhelmson 1985; Weckwerth and Wakimoto 1992), between 1.75 and 2.25 km. As previously mentioned, according to the raw Vr data at the time the dual-Doppler analysis was synthesized, the waves perturbed the wind field over a layer only about 800 m deep, between about 750 and 1550 m. The difference in wave depth and vertical placement between what was observed directly by the radar and what was resolved by the analyses can be attributed to the following possibilities. 1) The entire wave depth may not have been detectable in Vr using the qualitative method described. 2) The vertical radius of influence in the dual-Doppler analysis was 1 km; therefore the wave signal may have been spuriously stretched in the vertical, making it appear that the waves were deeper and higher than they truly were.

5. Discussion

When the waves first formed, a feature that resembled a bright band was apparent within a range of approximately 20–27 km in the Zdr and ρhv fields (Ryzhkov and Zrnić 1998b) (Figs. 10d,e). However, the reflectivity field (Fig. 10a) does not exhibit a classic brightband signature (Martner et al. 1993). There are breaks in the higher reflectivity factor ring corresponding to the range of the feature, which may be attributed to an actual gap in the precipitation falling within the radar domain since reflectivity factors fall to about 0 dBZ. Additionally, reflectivity factors in the bright band usually exceed 30–35 dBZ and they do not do so in this case.

A typical bright band indicates the existence of melting and water-coated hydrometeors where snow transitions to rain; however, precipitation reaching the surface at this time was recorded as snow. According to the RASS observations (Fig. 5b), a thin region of the inversion was at or just above freezing. It is therefore hypothesized that hydrometeors falling through the bright band began to melt but never became purely liquid and the liquid water that had been coating the hydrometeor refroze when the precipitation fell into the subfreezing region beneath the bright band.

The range at which the Zdr and ρhv feature appears corresponds to a height of 1.5–2 km. According to the RASS, the freezing layer at 1700 UTC extends from about 2 to about 2.6 km. The height discrepancy can be understood by considering the following. 1) RASS observations are of virtual temperature, which are approximately 1°C higher than the air temperature. By preserving the lapse rate just above the inversion and shifting the temperature profile to the left by 1°C, the upper boundary of the melting layer would decrease by 200 m. 2) Nonuniform beam filling across the bright band at a range of 20–30 km introduces errors to the upper and lower vertical extent of the bright band of 100–200 m (Ryzhkov 2007). 3) Because the RASS measures the average temperature over 250 m, the temperature profile is spread out in the vertical. This could account for an error in the 0°C height of about 100 m given that the environmental lapse rate was 5°C km−1. 4) From Fig. 5b, it is apparent that the top of the melting level decreased with time, probably owing to increasing cold air advection over a deeper layer, which causes heights to fall. Putting this into spatial context, freezing layer heights will be lower to the north where the average temperature of the air column would be lower. Thus, the physical distance between KOUN and the RASS could account for some of the difference in the freezing layer heights.

Between 1716 and 1729 UTC, periodic high Z and Zdr and lower ρhv values become apparent beneath the brightband feature, creating a zigzag pattern similar to what is seen in Vr (Fig. 11), particularly in Zdr. As time proceeds, the zigzag shape is amplified in Z and Zdr (Fig. 12), and the original bright band disappears (Fig. 13). By the time the waves dissipate, Z and Zdr (ρhv) values are no longer high (low) anywhere (Fig. 15). The appearance of wavelike features in all radar variables implies that the K-H waves modified precipitation microphysics, which is an important finding of this study.

The removal of the brightband feature is attributed mostly to cold air advection occurring over the lower 2.5 km of the troposphere between 1700 and 1900 UTC (Fig. 5b). It is possible that some of the change in the thermal profile may have been enhanced by the turbulent motions in the waves; the temperature difference ΔT between the base and the top of the inversion decreased from 12° to 9°C between 1700 and 1900 UTC. Thus, turbulent motions could have helped to mix the inversion to the point of eliminating the layer that was above freezing. If this were the case, these waves have the potential to modify the local thermodynamic environment and may have implications for winter storm precipitation type. Unfortunately, temperature data are not available where the waves were occurring (the Purcell profiler was about 40 km to the southwest) to determine the relative amount of cooling attributable to large-scale cooling compared to the actual cooling observed. Therefore, the degree to which the waves modulated the thermal profile is unknown.

Previous studies have concluded that K-H waves are associated with vertical motion and temperature perturbations (Browning 1971; Hardy et al. 1973; Browning et al. 1973). Because the waves were initially generated in close proximity to the bright band, it is tempting to deduce that vertical motions caused local variations in the lower boundary of the melting level by advecting warm air downward and cold air upward, perturbing the bright band and resulting in the undulating Z, Zdr, and ρhv values noted. As mentioned, the Z, Zdr, and ρhv perturbations were located beneath the bright band. Therefore, this argument would favor higher Z and Zdr in the troughs below the bright band owing to the downward displacement of the melting layer. However, the observations depict enhanced Z and Zdr in closer proximity to wave crests (Figs. 19a,b). Additionally, reflectivity values associated with the bright band are typically on the order of 35 dBZ, while in this case the wavelike features in Z were only around 20 dBZ.

Because the observations are not consistent with perturbations of the bright band, it is likely that other microphysical processes contributed to the Z, Zdr, and ρhv perturbations. The observations from the Cyril radar (Fig. 16) provide further evidence that these perturbations are not associated with the bright band because both prior to and at the time waves were occurring here, there was no evidence of a bright band. As mentioned, Z in the wave perturbations was somewhat low (15–20 dBZ), but in the same locations Zdr was quite high, ranging from 2 to 3 dB. Such a correlation is a characteristic polarimetric signature of horizontally oriented columnar or needle ice crystals (Ryzhkov et al. 2005; Hogan et al. 2002). These crystals typically form when temperatures range from −4° to −10°C (Fletcher 1962). The RASS temperatures at the inversion base are about −10°C, and about −8°C at 1.5 km (Fig. 5b), where the upward motion associated with the waves begins, according to the dual-Doppler analyses. Thus, the thermodynamic environment was favorable for the growth of such crystals. The near collocation of these perturbations with upward motion indicates that development of these crystals was enhanced in the upward-directed branch of the K-H wave circulation. The atmosphere within the inversion was nearly saturated with respect to liquid water (Fig. 4b) and was supersaturated with respect to ice (observed vapor pressure e = 352 Pa; saturated vapor pressure with respect to ice ei = 336 Pa for T = −7°C). The upward motion likely fostered ice crystal growth by enhancing riming, supplying moisture and supercooled droplets (some of which may have been circulated into the updraft region from the melting layer above), opposing the fall speed of the crystals, and prolonging the dwell time of the crystals, thus allowing them to grow for a longer time. Additionally, considering the mixing that occurs with the waves, it is possible that the updraft temperature was slightly warmer than the −8°C RASS observation. If temperatures were approximately −5°C, the Hallett–Mossop ice multiplication process may have caused rapid generation of needle crystals (Hallett and Mossop 1974). This process occurs when riming hydrometeors splinter, creating a large number of small crystals, which grow by deposition and riming, rapidly increasing the quantity of ice crystals. This process was likely favored in the updraft rather than the downdraft regions because the downdrafts, which advected warmer air from above, were probably too warm for the ice multiplication process, and the dwell time of hydrometeors in the downdraft is shorter.

The lower ρhv values at the base of the circulations can be explained by considering the variety of ice crystal shapes over the depth of the inversion. The crystals associated with the warmer temperatures above were likely plates and aggregates since the air temperature was about 0°C, while those below the inversion were likely sheaths or columns, since the temperature was about −10°C (Rogers and Yau 1996). As the platelike crystals and aggregates were circulated into the colder region below, the increased variety of crystal shapes, orientations, and sizes likely contributed to low ρhv. The same reasoning would also support a region of lower ρhv above the circulations. However, uncorrected ρhv above the circulations is already low above the circulations, owing to low Z, which inhibits the identification of lower ρhv due to the circulation process. (The corrected ρhv at the top of the domain in Fig. 19c should be viewed with caution.)

The observations of enhanced Z associated with upward motion might suggest an increase in precipitation accumulation. However, while wave features are obvious in Z at intermediate radar elevations (~2.5°–10°), waves are not apparent in the Z field of the lowest radar tilt (0.5°) (0.5° Vr shown in Fig. 9a). Therefore, it is unlikely that these waves affect surface precipitation accumulation. It is also observed that the wave crests are associated with positive Z and Zdr perturbations, but the troughs are not associated with negative perturbations. In the troughs, Z and Zdr are the same as those that are not influenced by the waves, which agrees with what Zhang et al. (2007) found.

To examine how radar variables evolve over the time waves exist, average and maximum radar parameters associated with the waves at a constant elevation angle and crest–trough wave depth over the life cycle of the waves are presented in Fig. 20. The spectrum width initially is somewhat large even prior to wave development as a result of the preexisting shear layer that initially generated the K-H instability. The wave depth increases until 1742 UTC, when wave-breaking begins, evinced by the sudden increase in the maximum SW and decrease in wave depth (Fig. 20a). The downward trend in SW and wave depth continues until the waves dissipate, at which time the average and maximum SW are lower than they were a priori to wave development, suggesting that the waves slightly reduced the local wind shear.

Fig. 20.
Fig. 20.

Plots of wave and radar parameter characteristics over time for the 4.3° elevation scan. (a) Radar-observed wave layer depth (m) (left abscissa), and average and maximum spectrum width (m s−1) (right abscissa) over time (ordinate). (b) Maximum and average differential reflectivity (dB) (left abscissa), and minimum and average cross-correlation coefficient (right abscissa) over time (ordinate). Fourth-order polynomial trend line (thick black line) is fit to the minimum ρhv data. The averages for the radar fields were determined from the raw radar data by identifying wave locations and examining the radar field values over a range of 5 km (20 range gates) for five consecutive azimuths, yielding a total of 100 samples. Maximum (minimum) values were determined by finding the highest maximum (lowest minimum) value that was not considered to be an outlier in the averaging dataset. The wave life cycle is noted across the bottom.

Citation: Journal of the Atmospheric Sciences 68, 8; 10.1175/2011JAS3566.1

Of particular interest are the maximum Zdr values, which often exceed 3.5 dB, and exceed 4 dB in two scans; the minimum ρhv values, which are minimized at less than 0.8; and the maximum SW, which surpasses 5 m s−1. (These values come from the raw PPI data; therefore, Zdr and ρhv have not been corrected for low SNR biases). In general, the maximum Zdr increases as the wave is amplifying, oscillates a little during the mature phase and the beginning of the dissipation period, and then decreases. Conversely, the minimum ρhv values exhibit a decreasing trend during wave amplification and maturation but then rise during the breaking and dissipating phases. The extent to which the Z, Zdr, and ρhv relationships found herein are applicable to other winter storms and to nonwinter precipitation scenarios is unknown and provides motivation for future study.

6. Conclusions

Kelvin–Helmholtz waves formed along a shear interface associated with a cold-frontal boundary separating shallow, cold surface air from warmer air aloft. The waves were observed as undular perturbations in Vr, SW, Z, Zdr, and ρhv. They were associated with periodic fluctuations in horizontal divergence and vertical vorticity with regions of cyclonic (anticyclonic) vorticity collocated with positive (negative) vertical velocity. Vertical motion was about 90° out of phase with the horizontal flow perturbations; positive (negative) w was located generally just downstream of southwesterly (northerly) winds for waves where this relationship could be determined. The dual-Doppler analysis also reveals that the K-H rolls were three-dimensional features, with complex flow patterns and organization and a complex relationship between the horizontal and vertical motion in the vicinity of mature, breaking waves.

Just prior to wave formation, a radar brightband feature was apparent in the KOUN data. Shortly after the waves were observed, the bright band disappeared, but perturbations of relatively high Z and Zdr values and somewhat lower ρhv values persisted until waves began to break. The bright band was not evident in the CASA Cyril data, which provides evidence that the observed perturbations are associated with processes independent of the melting layer.

The vertical motions induced by the waves transport horizontal momentum vertically, creating a zigzag pattern in the radial velocity display, and likely affect Z, Zdr, and ρhv by altering hydrometeor microphysics and mixing different crystal types. Regions of enhanced Z and Zdr, were located in the vicinity of upward motion and likely result from ice crystal generation through the Hallett–Mossop ice multiplication process. Perturbations of low ρhv are found in upward motion and toward the base of the circulations, most likely resulting from decreased homogeneity of the ice crystal shapes and sizes due to vertical mixing of ice crystal type. High values of SW were associated with sinking motion in the wave-breaking region. Spectrum width increased as waves developed and rose rapidly when wave breaking began as a result of the turbulent motions generated within the wave roll-up and breaking regions. After the waves dissipated, SW was reduced to a value lower than it had been at the onset of the waves.

This study helps to elucidate the microphysical structure of K-H waves in winter precipitation, illustrates the evolution of K-H waves as seen in polarimetric radar fields, and expands upon the current understanding of the three-dimensional structure of flow within these waves. It provides evidence that K-H waves are capable of modulating precipitation microphysics, which is reflected in quantities measured by the polarimetric variables, and shows that these perturbations are related to the kinematic flow within the waves.

Acknowledgments

This work has been made possible through the guidance, insight and contributions of many people. Terry Schuur (NSSL) furnished the KOUN radar data. Jerry Straka (OU) provided a translator to convert the KOUN data to a usable format. Curtis Alexander (NOAA) and Jim Marquis (Pennsylvania State University) provided helpful advice concerning the dual-Doppler analysis. Tom Galarneau (State University at Albany, State University of New York) provided gridded RUC and GFS analysis files. Jeff Snyder (OU) provided technical support and contributed Matlab code. Jerry Brotzge (Center for Analysis and Prediction of Storms) provided assistance with the CASA data and Yanting Wang provided a translator for the CASA data. The authors would also like to thank Matt Kumjian, Alexander Ryzhkov, and Doug Van de Kamp (NOAA) for their insightful comments and discussion and two anonymous reviewers for their contributions. Funding for this research came from the National Science Foundation Engineering Research Centers Program under NSF award ATM-0313747 and from Grant ATM-0637148.

REFERENCES

  • Armijo, L., 1969: A theory for the determination of wind and precipitation velocities with Doppler radars. J. Atmos. Sci., 26, 570573.

    • Search Google Scholar
    • Export Citation
  • Barnes, S. U., 1964: A technique for maximizing details in numerical weather map analysis. J. Appl. Meteor., 3, 396409.

  • Benjamin, S. G., G. A. Grell, J. M. Brown, T. G. Smirnova, and R. Bleck, 2004a: Mesoscale weather prediction with the RUC hybrid isentropic–terrain-following coordinate model. Mon. Wea. Rev., 132, 473494.

    • Search Google Scholar
    • Export Citation
  • Benjamin, S. G., and Coauthors, 2004b: An hourly assimilation–forecast cycle: The RUC. Mon. Wea. Rev., 132, 495518.

  • Bringi, V. N., T. D. Keenan, and V. Chandrasekar, 2001: Correcting C-band radar reflectivity and differential reflectivity data for rain attenuation: A self-consistent method with constraints. IEEE Trans. Geosci. Remote Sens., 39, 19061915.

    • Search Google Scholar
    • Export Citation
  • Britter, R. E., and J. E. Simpson, 1978: Experiments on the dynamics of a gravity current head. J. Fluid Mech., 88, 223240.

  • Browning, K. A., 1971: Structure of the atmosphere in the vicinity of large amplitude Kelvin–Helmholtz billows. Quart. J. Roy. Meteor. Soc., 97, 283299.

    • Search Google Scholar
    • Export Citation
  • Browning, K. A., and C. D. Watkins, 1970: Observations of clear air turbulence by high power radar. Nature, 277, 260263.

  • Browning, K. A., G. W. Bryant, J. R. Starr, and D. N. Axford, 1973: Air motion within Kelvin–Helmholtz billows determined from simultaneous Doppler radar and aircraft measurements. Quart. J. Roy. Meteor. Soc., 99, 608618.

    • Search Google Scholar
    • Export Citation
  • Chandrasekar, V., V. N. Bringi, N. Balakrishnan, and D. S. Zrnić, 1990: Error structure of multiparameter radar and surface measurements of rainfall. Part III: Specific differential phase. J. Atmos. Oceanic Technol., 7, 621629.

    • Search Google Scholar
    • Export Citation
  • Chapman, D., and K. A. Browning, 1997: Radar observations of wind-shear splitting within evolving atmospheric Kelvin–Helmholtz billows. Quart. J. Roy. Meteor. Soc., 123, 14331439.

    • Search Google Scholar
    • Export Citation
  • Crum, T. D., R. L. Alberty, and D. W. Burgess, 1993: Recording, archiving, and using WSR-88D data. Bull. Amer. Meteor. Soc., 74, 645653.

    • Search Google Scholar
    • Export Citation
  • Doviak, R. J., P. S. Ray, R. G. Strauch, and L. J. Miller, 1976: Error estimation in wind fields derived from dual-Doppler radar measurement. J. Appl. Meteor., 15, 868878.

    • Search Google Scholar
    • Export Citation
  • Drazin, P. G., 1958: The stability of a shear layer in an unbounded heterogeneous inviscid fluid. J. Fluid Mech., 4, 214224.

  • Drazin, P. G., and W. H. Reid, 1981: Kelvin-Helmholtz instability. Hydrodynamic Stability, Cambridge University Press, 14–22.

  • Droegemeier, K. K., and R. B. Wilhelmson, 1985: Three-dimensional numerical modeling of convection produced by interacting thunderstorm outflows. Part I: Control simulation and low-level moisture variations. J. Atmos. Sci., 44, 23812403.

    • Search Google Scholar
    • Export Citation
  • Droegemeier, K. K., and R. B. Wilhelmson, 1986: Kelvin–Helmholtz instability in a numerically simulated thunderstorm outflow. Bull. Amer. Meteor. Soc., 67, 416417.

    • Search Google Scholar
    • Export Citation
  • Fletcher, N. H., 1962: The Physics of Rain Clouds. Cambridge University Press, 383 pp.

  • Friedrich, K., D. E. Kingsmill, C. Flamant, H. V. Murphey, and R. M. Wakimoto, 2008: Kinematic and moisture characteristics of a nonprecipitating cold front observed during IHOP. Part II: Along front structures. Mon. Wea. Rev., 136, 37963821.

    • Search Google Scholar
    • Export Citation
  • Fritts, D. C., 1979: The excitation of radiating waves and Kelvin–Helmholtz instabilities by the gravity wave–critical level interaction. J. Atmos. Sci., 36, 1223.

    • Search Google Scholar
    • Export Citation
  • Fritts, D. C., T. L. Palmer, O. Andreassen, and I. Lie, 1996: Evolution and breakdown of Kelvin–Helmholtz billows in stratified compressible flows. Part I: Comparison of two- and three-dimensional flows. J. Atmos. Sci., 53, 31733191.

    • Search Google Scholar
    • Export Citation
  • Geerts, B., and Q. Miao, 2010: Vertically pointing airborne Doppler radar observations of Kelvin–Helmholtz billows. Mon. Wea. Rev., 138, 982986.

    • Search Google Scholar
    • Export Citation
  • Gossard, E. E., and J. H. Richter, 1970: The shape of internal waves of finite amplitude from high-resolution radar sounding of the lower atmosphere. J. Atmos. Sci., 27, 971973.

    • Search Google Scholar
    • Export Citation
  • Hallett, J., and S. C. Mossop, 1974: Production of secondary ice particles in the riming process. Nature, 249, 2628.

  • Hardy, K. R., R. J. Reed, and G. K. Mather, 1973: Observation of Kelvin–Helmholtz billows and their mesoscale environment by radar, instrumented aircraft, and a dense radiosonde network. Quart. J. Roy. Meteor. Soc., 99, 279293.

    • Search Google Scholar
    • Export Citation
  • Hassle, N., and E. Hudson, 1989: The wind profiler for the NOAA Demonstration Network: Instruments and observing methods. Papers presented at the Fourth WMO Technical Conference on Instruments and Methods of Observations (TE-CIMO-IV), Brussels, WMO/TD Rep. 35, 261–266.

    • Search Google Scholar
    • Export Citation
  • Helmholtz, H. L. F., 1868: Über discontinuierliche Flüssigkeits-Bewegungen (On the discontinuous movements of fluids). Monatsber. König. Preuss. Akad. Wiss. Berlin, 23, 1–215.

    • Search Google Scholar
    • Export Citation
  • Hicks, J. J., and J. K. Angell, 1968: Radar observations of breaking gravitational waves in the visually clear atmosphere. J. Appl. Meteor., 7, 114121.

    • Search Google Scholar
    • Export Citation
  • Hogan, R. J., P. R. Field, A. J. Illingworth, R. J. Cotton, and T. W. Choularton, 2002: Properties of embedded convection in warm-frontal mixed-phase cloud from aircraft and polarimetric radar. Quart. J. Roy. Meteor. Soc., 128, 451476.

    • Search Google Scholar
    • Export Citation
  • Holton, J. R., 1992: An Introduction to Dynamic Meteorology. 3rd ed. Academic Press, 511 pp.

  • Kelvin, L., 1871: Hydrokinetic solutions and observations. Philos. Mag., 42, 362377.

  • Koop, C. G., and F. K. Browand, 1979: Instability and turbulence in stratified fluid with shear. J. Fluid Mech., 93, 135159.

  • Luce, H., G. Hassenpflug, M. Yamamoto, and S. Fukao, 2008: High-resolution observations with MU radar of a KH instability triggered by an inertia–gravity wave in the upper part of a jet stream. J. Atmos. Sci., 65, 17111718.

    • Search Google Scholar
    • Export Citation
  • Ludlam, F. H., 1967: Characteristics of billow clouds and their relation to clear air turbulence. Quart. J. Roy. Meteor. Soc., 93, 419435.

    • Search Google Scholar
    • Export Citation
  • Markowski, P. M., and Y. P. Richardson, 2010: Mesoscale Meteorology in Midlatitudes. Wiley-Blackwell, 430 pp.

  • Martner, B. E., J. B. Snider, R. J. Zamora, G. P. Byrd, T. A. Niziol, and P. I. Joe, 1993: A remote-sensing view of a freezing-rain storm. Mon. Wea. Rev., 121, 25622577.

    • Search Google Scholar
    • Export Citation
  • May, P. T., R. G. Strauch, K. P. Moran, and W. L. Ecklund, 1990: Temperature sounding by RASS with wind profiler radars: A preliminary study. IEEE Trans. Geosci. Remote Sens., 28, 1928.

    • Search Google Scholar
    • Export Citation
  • McLaughlin, D., and Coauthors, 2009: Short-wavelength technology and the potential for distributed networks of small radar systems. Bull. Amer. Meteor. Soc., 90, 17971817.

    • Search Google Scholar
    • Export Citation
  • McPherson, R. A., and Coauthors, 2007: Statewide monitoring of the mesoscale environment: A technical update on the Oklahoma Mesonet. J. Atmos. Oceanic Technol., 24, 301321.

    • Search Google Scholar
    • Export Citation
  • Miles, J. W., and L. N. Howard, 1964: Note on heterogeneous shear flow. J. Fluid Mech., 20, 331336.

  • Mueller, C. K., and R. E. Carbone, 1987: Dynamics of a thunderstorm outflow. J. Atmos. Sci., 44, 18791895.

  • Park, H. S., A. V. Ryzhkov, D. S. Zrnić, and K. E. Kim, 2009: The hydrometeor classification algorithm for the polarimetric WSR-88D: Description and application to an MCS. Wea. Forecasting, 24, 730748.

    • Search Google Scholar
    • Export Citation
  • Petre, J. M., and J. Verlinde, 2004: Cloud radar observations of Kelvin–Helmholtz instability in a Florida anvil. Mon. Wea. Rev., 132, 25202523.

    • Search Google Scholar
    • Export Citation
  • Reiss, N. M., and T. J. Corona, 1977: An investigation of a Kelvin–Helmholtz billow cloud. Bull. Amer. Meteor. Soc., 58, 159162.

  • Rogers, R. R., and M. K. Yau, 1996: A Short Course in Cloud Physics. 3rd ed. Butterworth-Heinemann, 290 pp.

  • Ryzhkov, A. V., 2007: The impact of beam broadening on the quality of radar polarimetric data. J. Atmos. Oceanic Technol., 24, 729744.

    • Search Google Scholar
    • Export Citation
  • Ryzhkov, A. V., and D. S. Zrnić, 1998a: Discrimination between rain and snow with a polarimetric radar. J. Appl. Meteor., 37, 12281240.

    • Search Google Scholar
    • Export Citation
  • Ryzhkov, A. V., and D. S. Zrnić, 1998b: Beamwidth effects on the differential phase measurements of rain. J. Atmos. Oceanic Technol., 15, 624634.

    • Search Google Scholar
    • Export Citation
  • Ryzhkov, A. V., and D. S. Zrnić, 2007: Depolarization in ice crystals and its effect on radar polarimetric measurements. J. Atmos. Oceanic Technol., 24, 12561267.

    • Search Google Scholar
    • Export Citation
  • Ryzhkov, A. V., T. J. Schuur, D. W. Burgess, P. L. Heinselman, S. E. Giangrande, and D. S. Zrnić, 2005: The Joint Polarization Experiment: Polarimetric rainfall measurements and hydrometeor classification. Bull. Amer. Meteor. Soc., 86, 809824.

    • Search Google Scholar
    • Export Citation
  • Scarchilli, G., E. Goroucci, V. Chandrasekar, and T. A. Seliga, 1993: Rainfall estimation using polarimetric techniques at C-band frequencies. J. Appl. Meteor., 32, 11501160.

    • Search Google Scholar
    • Export Citation
  • Schurr, T., A. Ryzhkov, and P. Heinselman, 2003: Observations and classification of echoes with the polarimetric WSR-88D radar. NOAA/National Severe Storms Laboratory Rep., 46 pp. [Available online at http://arrc.ou.edu/~guzhang/Polarimetry/img/class/Schuur2003.pdf.]

    • Search Google Scholar
    • Export Citation
  • Scorer, R. S., 1969: Billow mechanics. Radio Sci., 4, 12991308.

  • Scotti, R. S., and G. M. Corcos, 1972: An experiment of the stability of small disturbances in a stratified free shear layer. J. Fluid Mech., 52, 499528.

    • Search Google Scholar
    • Export Citation
  • Shapiro, M. A., 1980: Turbulent mixing within tropopause folds as a mechanism for the exchange of chemical constituents between the stratosphere and troposphere. J. Atmos. Sci., 37, 9941004.

    • Search Google Scholar
    • Export Citation
  • Simpson, J. E., 1969: A comparison between laboratory and atmospheric density currents. Quart. J. Roy. Meteor. Soc., 95, 758765.

  • Simpson, J. E., 1982: Gravity currents in the laboratory, atmosphere and ocean. Annu. Rev. Fluid Mech., 14, 213234.

  • Snyder, J. C., H. B. Bluestein, G. Zhang, and S. J. Frasier, 2010: Attenuation correction and hydrometeor classification of high-resolution, X-band, dual-polarized mobile radar measurements in severe convective storms. J. Atmos. Oceanic Technol., 27, 19792001.

    • Search Google Scholar
    • Export Citation
  • Straka, J. M., D. S. Zrnić, and A. V. Ryzhkov, 2000: Bulk hydrometeor classification and quantification using polarimetric radar data: Synthesis of relations. J. Appl. Meteor., 39, 13411372.

    • Search Google Scholar
    • Export Citation
  • Strauch, A. G., K. P. Moran, P. T. May, A. J. Bedard, and W. L. Ecklund, 1988: RASS temperature sounding techniques. NOAA Tech. Memo. ERL WPL-158, 12 pp. [Available from National Technical Information Service, 5285 Port Royal Rd., Springfield, VA 22161.]

    • Search Google Scholar
    • Export Citation
  • Sykes, R. I., and W. S. Lewellen, 1982: A numerical study of breaking Kelvin–Helmholtz billows using a Reynolds-stress turbulence closure model. J. Atmos. Sci., 39, 15061520.

    • Search Google Scholar
    • Export Citation
  • Thorpe, S. A., 1968: A method of producing a shear flow in a stratified fluid. J. Fluid Mech., 32, 693704.

  • Thorpe, S. A., 1971: Experiments on the instability of stratified shear flows: Miscible fluids. J. Fluid Mech., 46, 299319.

  • Thorpe, S. A., 1973: Experiments on instability and turbulence in a stratified shear flow. J. Fluid Mech., 61, 731751.

  • Trapp, R. J., and C. A. Doswell III, 2000: Radar data objective analysis. J. Atmos. Oceanic Technol., 17, 105120.

  • Wakimoto, R. M., and B. L. Bosart, 2001: Airborne radar observations of a warm front during FASTEX. Mon. Wea. Rev., 129, 254274.

  • Wakimoto, R. M., W. Blier, and C. Liu, 1992: The frontal structure of an explosive oceanic cyclone: Airborne radar observations of ERICA IOP 4. Mon. Wea. Rev., 120, 11351155.

    • Search Google Scholar
    • Export Citation
  • Wallace, J. M., and P. V. Hobbs, 1977: Atmospheric Science: An Introductory Survey. Academic Press, 467 pp.

  • Weber, B. L., and D. B. Wuertz, 1990: Comparison of rawinsonde and wind profiler radar measurements. J. Atmos. Oceanic Technol., 7, 157174.

    • Search Google Scholar
    • Export Citation
  • Weber, B. L., D. B. Wuertz, D. C. Welsh, and R. McPeek, 1992: Quality controls for profiler measurements of winds and RASS temperatures. J. Atmos. Oceanic Technol., 10, 452464.

    • Search Google Scholar
    • Export Citation
  • Weckwerth, T. M., and R. M. Wakimoto, 1992: The initiation and organization of convective cells atop a cold-air outflow boundary. Mon. Wea. Rev., 120, 21692187.

    • Search Google Scholar
    • Export Citation
  • Zhang, P., A. Ryzhkov, and D. Zrnić, 2007: Kelvin–Helmholtz waves observed by a polarimetric prototype of the WSR-88D radar. Extended Abstracts, 33rd Conf. on Radar Meteorology, Cairns, Australia, Amer. Meteor. Soc., P10.4. [Available online at http://ams.confex.com/ams/33Radar/techprogram/paper_122956.htm.]

    • Search Google Scholar
    • Export Citation
  • Zrnić, D. S., and A. V. Ryzhkov, 1999: Polarimetry for weather surveillance radars. Bull. Amer. Meteor. Soc., 80, 389406.

  • Zrnić, D. S., N. Balakrishnan, C. L. Ziegler, V. N. Bringi, K. Aydin, and T. Matejka, 1993: Polarimetric signatures in the stratiform region of a mesoscale convective system. J. Appl. Meteor., 32, 678693.

    • Search Google Scholar
    • Export Citation
Save
  • Armijo, L., 1969: A theory for the determination of wind and precipitation velocities with Doppler radars. J. Atmos. Sci., 26, 570573.

    • Search Google Scholar
    • Export Citation
  • Barnes, S. U., 1964: A technique for maximizing details in numerical weather map analysis. J. Appl. Meteor., 3, 396409.

  • Benjamin, S. G., G. A. Grell, J. M. Brown, T. G. Smirnova, and R. Bleck, 2004a: Mesoscale weather prediction with the RUC hybrid isentropic–terrain-following coordinate model. Mon. Wea. Rev., 132, 473494.

    • Search Google Scholar
    • Export Citation