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  • View in gallery

    Topographic maps containing the flight legs during the (a) SNOWIE and (b) CAMPS IOPs. Red lines indicate the flight track for each IOP. Black crosses show the location of rawinsonde launches. White labels indicate locations or ranges mentioned in the text (SPL: Storm Peak Laboratory). Insets depict modeled outer domain (full image), inner domain (solid boundary), and the region shown in the topographic map (dashed boundary).

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    Modeled and observed skew T diagrams at the locations indicated in Fig. 1, for four cases presented in this paper. Solid (dashed) red and green show observed (modeled) temperature and dewpoint profiles. Black (purple) wind barbs to the right of each sounding indicate the corresponding observed (modeled) wind speed (knots; 1 kt = 0.51 m s−1) and direction.

  • View in gallery

    Conceptual diagrams of KH waves developing (a) in the free troposphere, (b) in the mountain’s wake, (c) upstream of a barrier, between layers of blocked and unblocked flow, and (d) at the top of the lofted boundary layer. In this schematic and in all following transects, the mean flow and the shear across the instability layer are from left to right.

  • View in gallery

    Evolution of free-tropospheric KH waves, illustrated from four consecutive transects by the UWKA along a fixed track (shown as IOP-14 in Fig. 1). (a),(c),(e),(g) WCR reflectivity. The flight level is shown as a dashed black line in a white belt (the radar blind zone) in these panels. (b),(d),(f),(h) WCR vertical velocity (plus 1.0 m s−1, as discussed in section 2b); the air vertical velocity measured by the gust probe is shown at flight level, using the same color scale. Lines of constant equivalent potential temperature from the WRF Model are contoured in black in all panels. The dotted boundary in each panel outlines the semilinear turbulent layer (SLTL).

  • View in gallery

    Sounding data from a rawinsonde launched near Crouch, Idaho (black cross in Fig. 1a), at 0000 UTC 19 Feb: (a) the vector magnitude of wind shear (solid) and wind speed (dotted); (b) virtual potential temperature (solid) and equivalent potential temperature (dotted); (c) bulk Richardson number. The orange shaded box shows the level where KH waves were observed. Light-red vertical lines mark layers of potential instability (equivalent potential temperature decreased with height) and dark-red vertical lines mark layers of dry static instability (virtual potential temperature decreased with height).

  • View in gallery

    Transects of turbulence intensity: EDRw calculated from WCR vertical velocity along the flight legs shown in Fig. 1 for (a) SNOWIE IOP-14, (b) SNOWIE IOP-19, (c) CAMPS IOP-05, and (d) CAMPS IOP-01. EDR measured at flight level is also displayed using the same color key, except for (d), where flight-level EDR was not available. Color bar values indicate the locations of “smooth” (blue), “light” (green), “light to moderate” (yellow), “moderate” (orange), “moderate to severe” (red), and “severe” (pink) turbulence categories. Navy blue coloring is reserved for locations where no inertial subrange was present [example in (b)]. Red boxes indicate the region of KH waves.

  • View in gallery

    KH waves within the mountain wake from SNOWIE IOP-19. Panels show cross sections of WCR-derived (a) reflectivity, (b) vertical velocity, (c) horizontal velocity, and (d) wind shear. Lines of constant equivalent potential temperature from the WRF Model are contoured in black. Black arrows in (c) and (d) illustrate the components of the wind in the along-flight and vertical directions. The dotted boundary indicates the region of interest with KH wave activity.

  • View in gallery

    KH waves above blocked flow upwind of the Park Range from CAMPS IOP-05, illustrated using four consecutive flight transects, each separated by 5–15 min. The four panels for each transect display the same WCR-derived variables as those in Fig. 7. The dotted boundary indicates the region of interest with KH wave activity.

  • View in gallery

    As in Fig. 7, but for KH waves at the top of a lofted boundary layer from CAMPS IOP-01.

  • View in gallery

    An extract of Figs. 8a–d, zoomed-in on the KH waves plus (e) the perturbation of the horizontal wind at each level. The dashed lines, overlaid in all panels, mark the locations of two shear layers in (d).

  • View in gallery

    Conceptual diagram of KH wave dynamics showing regions of low-momentum air beneath the waves, high-momentum air above the waves, and the region of intermediate momentum responsible for the braided structure. Phase diagrams indicate the relationship between vertical velocity, horizontal wind speed, pressure, and potential temperature perturbations above and below the layer of KH instability. Above and below the phase relationships are vectors indicating the absolute and wave-relative wind. The green line is dashed because pressure and temperature are inferred from expected dynamic relationships and not directly observed.

  • View in gallery

    Power spectral density of vertical velocity (black) and horizontal velocity (blue) from the box-average FFT from Figs. 10b and 10c. The dashed line shows the slope of turbulent flow in the inertial subrange (−5/3 slope).

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Detailed Dual-Doppler Structure of Kelvin–Helmholtz Waves from an Airborne Profiling Radar over Complex Terrain. Part I: Dynamic Structure

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  • 1 Department of Atmospheric Sciences, University of Wyoming, Laramie, Wyoming
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Abstract

Kelvin–Helmholtz (KH) waves are remarkably common in deep stratiform precipitation systems associated with frontal disturbances, at least in the vicinity of complex terrain, as is evident from transects of vertical velocity and 2D circulation, obtained from a 3-mm airborne Doppler radar, the Wyoming Cloud Radar. The high range resolution of this radar (~40 m) allows detection and depiction of KH waves in fine detail. These waves are observed in a variety of wavelengths, depths, amplitudes, and turbulence intensities. Proximity rawinsonde data confirm that they are triggered in layers where the Richardson number is very small. Complex terrain may locally enhance wind shear, leading to KH instability. In some KH waves, the flow remains mostly laminar, while in other cases it breaks down into turbulence. KH waves are frequently locked to the terrain, and occur at various heights, including within the free troposphere, at the boundary layer top, and close to the surface. They are observed not only upwind of terrain barriers, as has been documented before, but also in the wake of steep terrain, where the waves can be highly turbulent. Vertical-plane dual-Doppler analyses of KH waves reveal the mixing of layers of differential momentum across the high-shear zone. Doppler radar data are used to explore the dynamics of KH waves, including the response of thermodynamic and kinematic variables above, below, and within the instability layer.

© 2020 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Coltin Grasmick, cgrasmic@uwyo.edu

Abstract

Kelvin–Helmholtz (KH) waves are remarkably common in deep stratiform precipitation systems associated with frontal disturbances, at least in the vicinity of complex terrain, as is evident from transects of vertical velocity and 2D circulation, obtained from a 3-mm airborne Doppler radar, the Wyoming Cloud Radar. The high range resolution of this radar (~40 m) allows detection and depiction of KH waves in fine detail. These waves are observed in a variety of wavelengths, depths, amplitudes, and turbulence intensities. Proximity rawinsonde data confirm that they are triggered in layers where the Richardson number is very small. Complex terrain may locally enhance wind shear, leading to KH instability. In some KH waves, the flow remains mostly laminar, while in other cases it breaks down into turbulence. KH waves are frequently locked to the terrain, and occur at various heights, including within the free troposphere, at the boundary layer top, and close to the surface. They are observed not only upwind of terrain barriers, as has been documented before, but also in the wake of steep terrain, where the waves can be highly turbulent. Vertical-plane dual-Doppler analyses of KH waves reveal the mixing of layers of differential momentum across the high-shear zone. Doppler radar data are used to explore the dynamics of KH waves, including the response of thermodynamic and kinematic variables above, below, and within the instability layer.

© 2020 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Coltin Grasmick, cgrasmic@uwyo.edu

1. Introduction

Recent studies suggest that mountains foster the development of Kelvin–Helmholtz (KH) waves in the overlying boundary layer (BL) and troposphere (Houze and Medina 2005; Geerts and Miao 2010; Barnes et al. 2018). Although they develop in a statically stable environment, KH waves grow in regions of dynamic instability: they are perturbations that expand at the expense of the kinetic energy within the fluid flow. Dynamic instability is associated with layers of high wind shear and, likewise, wind shear is locally enhanced by the underlying terrain. KH instability results when the kinetic energy obtained from the mean horizontal wind exceeds the potential energy needed to exchange vertically stacked parcels in a stable environment. (Markowski and Richardson 2010, chapter 3.5). Put another way, KH instability occurs when dynamic instability exceeds buoyant stability. In effect, the instability can develop during a reduction of static stability, an increase of shear, or a combination of the two (Helmholtz 1868; Kelvin 1871). The exchange (or mixing) of parcels acts to counter the instability by reducing the wind shear and thus the instability. This paper aims to illustrate KH wave kinematics and dynamics, mainly using transects from the Wyoming Cloud Radar (WCR), and classify observed KH waves in terms of the source of instability release.

Atmospheric KH waves have been studied mainly because they are a notorious source for in-cloud and clear-air turbulence, affecting the safety and comfort of airborne travel. Aircraft are most sensitive to turbulent eddies at scales ranging from 100 m to 1 km (Sharman 2016) and the periodic updrafts and downdrafts associated with KH eddies can be on the order of 1 km, becoming smaller as the energy cascades down the inertial subrange (i.e., where turbulent kinetic energy is transferred to increasingly smaller eddies).

Individual case studies in the literature suggest numerous environments in which KH instability can be released: KH waves (or “billows” as they are commonly referred to) have been observed in or near cumulonimbus anvils (Petre and Verlinde 2004; Trier 2016), near the tropopause (Mahalov et al. 2011; Trier et al. 2012), at the base of low-level jets (e.g., Nakanishi et al. 2014; Lin et al. 2019), near the top of density currents associated with a sea breeze (Sha et al. 1991), behind a cold front (Geerts et al. 2006; Friedrich et al. 2008; Samelson and Skyllingstad 2016), at the upper boundaries of other cold-air intrusions (Zhou and Chow 2013), in midlatitude cyclones (Wakimoto et al. 1992), and above mountainous terrain (Geerts and Miao 2010; Medina and Houze 2016).

KH waves (or inferences of KH waves) have been documented extensively in the troposphere and upper atmosphere using a variety of radar and lidar sensing methods. Some of the first radar-observed KH waves were in clear air. Large wind shear is common in regions where stability changes rapidly (e.g., near the top of the BL or troposphere) and the associated large gradient in refractive index, detectible by radars, may reveal embedded waves (e.g., Atlas et al. 1970; Browning and Watkins 1970). The kinematic structure of KH waves can be documented with a Doppler radar when KH waves are within cloud or precipitation (e.g., Luce et al. 2018), or with a Doppler lidar in clear air (e.g., Blumen et al. 2001). Thermodynamic characteristics within KH waves are harder to document, but are occasionally made by instrumented aircraft that penetrate, or pass near, the waves (e.g., Browning et al. 1973; Nielsen 1992; Houze and Medina 2005). Early attempts at describing the formation and kinematics of KH billows, although insightful, lacked the observational systems we have today. Here, we use an airborne profiling Doppler radar with a wavelength of 3 mm and a very high resolution (0.7° beamwidth and 37.5-m range resolution) to uniquely describe the vertical structure KH wave trains, without having to assume steady state for time periods greater than a few seconds (which ground-based profiling systems must do for time-to-distance conversion).

This dataset provides insights in three primary areas. The first two areas are addressed in this paper while the third will be addressed in Part II. First, both the substantial variability and lack of observations over mountainous terrain have made it difficult to ascertain what relationships may exist between mountainous terrain and the KH waves. Of the few documented examples of KH waves near complex terrain, efforts to investigate why and how terrain triggers such waves have remained perfunctory. Barnes et al. (2018) studied three KH wave cases near the Olympic Mountains. Although they pointed out that terrain naturally enhances wind shear, they attributed shear enhancement in their cases to fronts and an upper-level jet. Medina and Houze (2016) identified KH waves in extratropical cyclones passing over the European Alps and the Oregon Cascades. Data from their two cases showed KH waves 1) in highly sheared layers during the passage of a frontal system and 2) between slower low-level flow along the terrain barrier and stronger upper-level flow, likely similar to the blocked flow case discussed below. They concluded that stratified flow, forced over terrain, promotes layers with high wind shear that can result in turbulent eddies such as KH waves. Houze and Medina (2005) establish that the windward slopes are one region of preferred KH wave activity in a study that focuses on blocked flow as a mechanism for enhancing turbulence and precipitation fallout. Our research is devoted to explicitly identifying ways in which terrain enhances shear, leading to KH waves.

Second, literature is lacking a holistic view of KH wave kinematics and dynamics. Early studies documented the general structure of KH waves: radars may see a braided, or “cat’s-eye,” pattern while KH waves are often visible to the eye as vortex rolls, especially in thin cloud layers or along cloud edges (e.g., Atlas et al. 1970; Browning and Watkins 1970; Browning 1971; Reiss and Corona 1977). Such observations coined the term “KH billow” because of the visible plumes of cloud created by the updrafts. With the application of Doppler radars, other studies documented the phase shift, primarily in vertical velocity, that occurs across the level of KH instability (e.g., Klostermeyer and Rüster 1980; Luce et al. 2010). Observing variation in horizontal velocities associated with KH waves is more difficult but has been accomplished with radars scanning in range–height indicator (RHI) mode (Wakimoto et al. 1992; Chapman and Browning 1997; Houser and Bluestein 2011; Barnes et al. 2018) although these slant measurements are partly convoluted with vertical velocity. Kinematic and thermodynamic observations in the vicinity of KH waves are not new, but an observationally based, dynamically consistent description is lacking. As a solution, we design a conceptual diagram with the help of dual-Doppler, high-resolution radar data that matches horizontal and vertical velocity measurements with patterns of reflectivity, turbulence, and wind shear. The airborne dual-Doppler approach here is better suited for unraveling true vertical and horizontal wind velocities. Furthermore, multiple quasi-instantaneous transects provide information about the time evolution of KH waves and, by including multiple cases, what diversity may exist in sizes, turbulence intensity, and evolution.

Third, for KH waves in cloud, the vertical motions and turbulence results in a microphysical response within and downwind of KH waves, including ice initiation and hydrometeor growth. Although not addressed in this paper, we will use airborne observations and an LES framework in Part II, combined with the kinematic information gleaned in Part I, to shed light on the impact of KH waves on cloud-microphysical processes in mixed-phase clouds.

The data and analysis methods are described in section 2. Section 3 describes the four KH wave types. KH wave kinematics are analyzed in section 4. A discussion and a list of conclusions follows in the last two sections.

2. Observational campaigns, instrumentation, and data processing

a. The SNOWIE and CAMPS campaigns

Data for this study were drawn from two field campaigns, both set in the intramountain, western United States. The first and most recent project, Seeded and Natural Orographic Wintertime Clouds: The Idaho Experiment (SNOWIE), was launched in order to address questions pertaining to both natural and seeded orographic clouds (Tessendorf et al. 2019). SNOWIE took place between 7 January and 17 March 2017 in the Payette basin northeast of Boise, Idaho (Fig. 1a). Tessendorf et al. (2019) describe the complete SNOWIE experimental design, project domain, and observed cloud and precipitation properties. The experimental design did not consider KH waves; their prevalence in all stratiform precipitation systems sampled during SNOWIE came as a surprise, and led to the questions that motivated this study.

Fig. 1.
Fig. 1.

Topographic maps containing the flight legs during the (a) SNOWIE and (b) CAMPS IOPs. Red lines indicate the flight track for each IOP. Black crosses show the location of rawinsonde launches. White labels indicate locations or ranges mentioned in the text (SPL: Storm Peak Laboratory). Insets depict modeled outer domain (full image), inner domain (solid boundary), and the region shown in the topographic map (dashed boundary).

Citation: Journal of the Atmospheric Sciences 77, 5; 10.1175/JAS-D-19-0108.1

The ubiquity of KH waves in SNOWIE drew our attention to a 2011 winter storm study that deployed the same instruments over the Park Range near the town of Steamboat Springs, Colorado (Fig. 1b). Known as the Colorado Airborne Multi-Phase Cloud Study (CAMPS), the project aimed to investigate the influence of complex terrain on clouds and precipitation in midlatitude cyclones (Dorsi et al. 2015). The site was prime for the CAMPS objectives, as well as for this project, due to the prominent north–south ridge of the Park Range and the extensive upstream (westbound) fetch without substantial topography.

b. The Wyoming Cloud Radar

The primary data platform for this project, and for both field campaigns, was the University of Wyoming King Air (UWKA), a Beechcraft King Air 200T aircraft outfitted with in situ and remote sensing instrumentation for studying atmospheric dynamics and cloud microphysics in the lower troposphere (Wang et al. 2012). The key instrument aboard the UWKA for this study was the WCR, a W-band (95 GHz, 3-mm wavelength) pulsed Doppler radar that collected finescale observations of equivalent radar reflectivity and radial velocity (University of Wyoming Flight Center 1995). The WCR creates a profile every 0.045 s (about 4.5 m) that, with a beamwidth of 0.7°, provides a total along-track resolution of about 17 m at 1-km range. Three fixed antennas were used, one pointing up (zenith), one pointing down (nadir), and one pointing 30° forward of nadir (down–forward beam). The up and down beams provided reflectivity and vertical velocity between the ground and cloud top (except for a 250-m-deep blind zone centered at flight level). The combination of the down and down–forward beam allowed for the estimation of the wind speed along the flight track (section 2c).

The WCR measures hydrometeor vertical velocity, the sum of air vertical velocity and hydrometeor fall speed. To estimate air vertical velocity, an estimated fall speed of the scatterers (generally unrimed snow particles in these cases), was subtracted from the WCR vertical velocity in all cross sections shown herein. A comparison between flight-level gust-probe air vertical velocity and close-gate up and down WCR vertical velocity (not shown) indicated that this fall speed was about 1.0 m s−1 in both campaigns, at least above the freezing level. (This correction does not apply for rain of course; in all but one of the cases presented here, snow falls down to ground level.)

c. Dual-Doppler synthesis

The down and down–forward beams of the WCR were combined to synthesize a 2D wind field in a plane below the UWKA and aligned with aircraft heading. The dual-Doppler technique used here (developed in Damiani and Haimov 2006), has been used in many other studies to reveal finescale dynamics of atmospheric phenomena including lake-effect snowbands (e.g., Bergmaier and Geerts 2016), orographic flow (Geerts et al. 2015), BL separation (French et al. 2015), BL turbulence (Geerts et al. 2011), drylines (Miao and Geerts 2007), cold fronts (Geerts et al. 2006), horizontal convective rolls (Yang and Geerts 2006), and cumulus thermals (Damiani et al. 2006). While Geerts and Miao (2010) used the WCR to document reflectivity and vertical velocity structures of KH billows, their KH waves occurred above flight level, so they could not synthesize the dual-Doppler winds.

The dual-Doppler synthesis assumes that the same volume of scatterers is sampled by both the down–forward and down beams. This assumption is most valid when the UWKA flight path is parallel with the wind direction so that the measured horizontal wind is close to the actual full horizontal wind component. When a substantial crosswind is present, this assumption is less reliable, as the aircraft must “crab” into the wind to maintain its flight track and the two beams are no longer aligned (Bergmaier and Geerts 2016). Cross-track advection of the parcel increases the uncertainty. However, the flight tracks presented herein were predominantly aligned with the mean wind direction, increasing the quality of the dual-Doppler calculations. The largest deviation occurred in the CAMPS IOP-01 case, where the flight track was 12° off from the mean wind direction. Flying with the wind (or rather, the wind shear) is also important for observing KH wave structure because the waves align with the direction of the wind vector (Damiani and Haimov 2006).

During WCR operation, a parcel of air passes through the down–forward beam first. A few seconds later (depending on aircraft speed, parcel speed, and range from the UWKA) the parcel of air is resampled by the down beam. Doppler velocities from the two beams provide two resolved components of the 3D wind: vertical velocity and horizontal velocity along the flight track. The cross-track wind is unresolved by the WCR but is estimated from a nearby sounding. Winds from the sounding profile are further used to correct vertical velocities when they are “contaminated” by horizontal wind. Contamination occurs from aircraft attitude fluctuations during which the up and down WCR beams tilt away from the zenith and nadir directions. The dual-Doppler synthesis occurs after the horizontal and vertical wind components are interpolated onto a grid. For the transects in this project, the wind was binned into 30-m sections in the horizontal (along track) and vertical dimensions, thereby smoothing the original WCR resolution, but still maintaining the resolution necessary for velocity fluctuations in KH waves. For more information on beam geometry and error estimation for the dual-Doppler synthesis, the reader is referred to Damiani and Haimov (2006).

Once the dual-Doppler synthesis is complete, other variables, such as shear and vorticity, can be derived from the along-track and vertical wind (u, w). The vertical shear of the horizontal wind (hereafter “wind shear”) is of particular interest in the vicinity of KH waves and is therefore included in the transects shown below.

d. Case-specific WRF simulations

To supplement the UWKA observations with a dynamically consistent depiction of the sampled winter storm, the Advanced Research version of the Weather Research and Forecasting (WRF) Model (WRF-ARW) (Skamarock et al. 2008) was used to simulate each of the four cases examined here. The model domain of each case consisted of an outer and inner domains of 2.7- and 0.9-km grid spacing, respectively, driven by hourly ERA5. The ERA5 dataset was assimilated as part of ECMWF’s Integrated Forecast System and contains 137 hybrid sigma–pressure levels (not including available surface-level data) with the top level at 0.01 hPa and a horizontal resolution of 31 km (ECMWF 2017). The outer and inner domains for the SNOWIE and CAMPS campaigns are shown in Fig. 1. The modeling framework for each of the simulations are listed in Table 1. The two SNOWIE cases and the two CAMPS cases were run with nearly identical initializations: both utilized the same physics parameterizations, driving dataset, and horizontal resolution. The only differences between the two simulations were the WRF version, the domain, and the vertical resolution. The CAMPS cases were run with more than twice the vertical resolution of the SNOWIE cases (201 vs 81 vertical levels) to test whether increasing the number of grid points in a column would improve the model’s ability to represent locations of dynamic instability. For comparison, one of the CAMPS cases (IOP-05) was simulated using the lower vertical resolution as a control. The results of this analysis are briefly examined in the discussion section.

Table 1.

Domain and parameterization choices for the SNOWIE and CAMPS WRF simulations.

Table 1.

The main function of these numerical simulations was to overlay model output, such as contours of equivalent potential temperature θe, onto WCR observations, in order to provide thermodynamic context. This was accomplished by creating model cross sections along the flight tracks at the time of flight. Model output variables were interpolated to the WCR transects using the four closest grid points at each level. Contours of θe are used as indicators of moist isentropic flow, that is, air trajectories, if we can assume that these contours are steady state, locked to the terrain. WCR-derived wind shear also provides some indication of moist isentropic flow (Geerts and Christian 2017).

Other studies have used numerical models to describe KH waves using either large-eddy simulations (e.g., Sauer et al. 2016) or high-resolution numerical weather prediction models such as WRF (e.g., Mahalov et al. 2011; Conrick et al. 2018). However, in this study, we do not attempt to resolve the KH waves themselves (900-m grid spacing is insufficient for the small wavelengths we observe), only the environment in which they form.

Model simulations were validated using rawinsondes launched near the UWKA transects (rawinsonde locations shown in Fig. 1). In three of four profiles, the environment is saturated, or nearly saturated, from the surface to 300 mb (Figs. 2a,b,d) and the model is in close agreement. During CAMPS IOP-05 (Fig. 2c), observations indicated rapid drying near 520 mb (about 5.1 km MSL; 1 mb = 1 hPa) but the model maintains a saturated layer up to 450 mb. This is the only significant disagreement between observations and simulations; however, our analysis is restrained to the lowest 4.5 km for this case, where differences between the model and observations are small.

Fig. 2.
Fig. 2.

Modeled and observed skew T diagrams at the locations indicated in Fig. 1, for four cases presented in this paper. Solid (dashed) red and green show observed (modeled) temperature and dewpoint profiles. Black (purple) wind barbs to the right of each sounding indicate the corresponding observed (modeled) wind speed (knots; 1 kt = 0.51 m s−1) and direction.

Citation: Journal of the Atmospheric Sciences 77, 5; 10.1175/JAS-D-19-0108.1

3. KH wave types

Upon examining WCR dual-Doppler transects of numerous KH wave occurrences from the SNOWIE and CAMPS field campaigns, we found that the release of terrain-driven KH instability is generally associated with a local increase in shear. We distinguish four different situations in which this occurs, all near complex terrain (Fig. 3). To be concise, we describe each situation and provide one example in the following sections. This list of orographic forcing mechanisms is not comprehensive, but it covers the dominant mechanisms observed during these campaigns and can be used, or built upon, in future case studies.

Fig. 3.
Fig. 3.

Conceptual diagrams of KH waves developing (a) in the free troposphere, (b) in the mountain’s wake, (c) upstream of a barrier, between layers of blocked and unblocked flow, and (d) at the top of the lofted boundary layer. In this schematic and in all following transects, the mean flow and the shear across the instability layer are from left to right.

Citation: Journal of the Atmospheric Sciences 77, 5; 10.1175/JAS-D-19-0108.1

a. Free-tropospheric KH waves

1) WCR observations

In SNOWIE IOP-14 (18 February 2017), a sequence of KH waves were encountered above flight level, between 4.5 and 5.5 km MSL (Fig. 4). The KH waves were present during multiple passes of the UWKA and their lifespan was over 70 min. The colloquial “billow” appearance was present in the WCR reflectivity while alternating regions of upward and downward motion were evident in the vertical velocity measurements. The billows were most apparent in legs 2 and 3, between 10 and 30 km near 5 km MSL (Fig. 4c–f). The WCR vertical velocity measurements showed that the entire KH wave train was colocated with a layer of high turbulence; the larger, more coherent KH wave motion developed amid a layer of smaller, irregular vertical motions that was visible in all legs (Figs. 4b,d,f,h) and annotated with a dotted oval. This semilinear turbulent layer, likely a result of the dynamic mixing, was present even before the distinct billows developed (Fig. 4b) and was useful for identifying regions of KH wave activity. Below flight level, the model contains some potential instability where θe decreases with height, although there was no evidence in the WCR reflectivity or vertical velocity that this instability was being released. Above flight level, where the KH waves were located, θe is stratified and the environment is statically stable. Because the KH waves were above flight level, it was not possible to perform dual-Doppler synthesis in that region.

Fig. 4.
Fig. 4.

Evolution of free-tropospheric KH waves, illustrated from four consecutive transects by the UWKA along a fixed track (shown as IOP-14 in Fig. 1). (a),(c),(e),(g) WCR reflectivity. The flight level is shown as a dashed black line in a white belt (the radar blind zone) in these panels. (b),(d),(f),(h) WCR vertical velocity (plus 1.0 m s−1, as discussed in section 2b); the air vertical velocity measured by the gust probe is shown at flight level, using the same color scale. Lines of constant equivalent potential temperature from the WRF Model are contoured in black in all panels. The dotted boundary in each panel outlines the semilinear turbulent layer (SLTL).

Citation: Journal of the Atmospheric Sciences 77, 5; 10.1175/JAS-D-19-0108.1

We identify these as “free troposphere” KH waves, to emphasize that the bulk of the shear does not appear to be produced by a specific mountain ridge and that the wave train developed well above the underlying terrain and BL (Fig. 3a). Instead, shear was already present within the larger-scale flow above the BL (shown in the sounding analysis next) and probably related to the larger-scale terrain [the gradual slope from the Snake River plain (left) to the Sawtooth Range (some 50 km east of the eastern end of the transect shown here (Fig. 1a)]. Even so, the KH wave train appeared to be mostly tied to underlying terrain, with waves forming on the upstream western (left) side and decaying on the right. The wind speed at the level of the wave train was 18 m s−1, implying that air parcels traveled considerable distances from one transect to the next (27 km between legs 1 and 2 and between legs 3 and 4, and 13 km between legs 2 and 3) providing evidence of a stationary wave train. This suggests that the terrain perturbed, or enhanced, the shear layer in a manner that allowed the waves to develop. The waves may have been triggered by a standing orographic gravity wave, evident in the alternating blue and pink hues in the vertical velocity panels with a wavelength of 12–14 km. The model contours of equivalent potential temperature (black contours in Fig. 4) showed similar gravity wave undulations. As the individual KH waves advected downstream and decayed, new waves formed upwind and allowed the wave train to remain stationary. Observations showed new waves forming upwind of the terrain, at least until the third transect.

Other studies have suggested some interaction between vertically propagating gravity waves and KH waves. Scinocca and Peltier (1993) uncovered a mechanism for producing KH waves in which a well-mixed layer, resulting from breaking gravity waves, leads to layers of enhanced shear. In a case study in Peru, Kelley et al. (2005) provided evidence that KH instability is enhanced in the presence of orographic gravity waves using a 50-MHz radar and radiosondes. In general, the vertical gradients of wind speed and stability found in layers promoting KH waves can also act as critical layers for gravity waves, produce wave breaking, reflect gravity waves, or have other nonlinear interactions (Doyle et al. 2016).

2) Sounding analysis

This case provided an excellent comparison with other cases that are unequivocally terrain-driven. A nearby rawinsonde, launched at 0000 UTC 19 February (Blestrud 2018), indicated a local maximum in wind shear magnitude near 5.0 km MSL that coincided with the height of the KH waves (Fig. 5a). The shear vector was westerly, consistent with the orientation of the KH billows (Fig. 4) (hodograph not shown). In addition, profiles of virtual potential temperature suggested rather weak stability (Fig. 5b). These variables were used to calculate the bulk Richardson number (Ri), as follows:
Ri=(gθυ¯)ΔθυΔz(ΔU)2+(ΔV)2.
The static stability term in the numerator is composed of the gravitational constant g, average virtual temperature Tυ¯ of the layer, and a virtual potential temperature difference ∆θυ across the layer height ∆z. The denominator corresponds to dynamic stability and incorporates wind shear in both the horizontal components of u and υ, over the same depth. Theory (e.g., Markowski and Richardson 2010) predicts that dynamic instability is released (i.e., overcomes the static stability) when Ri locally falls below 0.25 in stratified, sheared flow. The sounding data used in the calculation had an average resolution of 4 m (∆z = 4 m) and were smoothed using a moving average about 100 m thick. Ri fell below the critical value of 0.25 in the layer of interest (Fig. 5c). There were a few shallow, critical layers below, mostly near the surface, far from the region of interest, and likely associated with BL turbulence encountered by the balloon. (This BL turbulence can be seen in the WCR vertical velocity transects, although contaminated by the transition from snow to rain with a much higher fall speed.)
Fig. 5.
Fig. 5.

Sounding data from a rawinsonde launched near Crouch, Idaho (black cross in Fig. 1a), at 0000 UTC 19 Feb: (a) the vector magnitude of wind shear (solid) and wind speed (dotted); (b) virtual potential temperature (solid) and equivalent potential temperature (dotted); (c) bulk Richardson number. The orange shaded box shows the level where KH waves were observed. Light-red vertical lines mark layers of potential instability (equivalent potential temperature decreased with height) and dark-red vertical lines mark layers of dry static instability (virtual potential temperature decreased with height).

Citation: Journal of the Atmospheric Sciences 77, 5; 10.1175/JAS-D-19-0108.1

3) Turbulence analysis

This analysis would not be complete without a quantitative measurement of the turbulence in the vicinity of the KH waves in a way that infers its potential effects on aircraft, as well as potential microphysical effects within cloud (discussed in Part II). The energy (or eddy) dissipation rate (EDR) is commonly measured by commercial aircraft because of its close relation to the standard deviation of vertical acceleration that is also related to the turbulence felt by aircraft passengers and crew (Cornman 2016). Therefore, we calculated EDR from WCR vertical velocity, following the technique of Strauss et al. (2015) who also calculated EDR from the WCR vertical velocity (EDRw) to study atmospheric turbulence generated by orographic wave breaking and rotor circulations. The method adapts the inertial dissipation technique (e.g., Champagne 1978), which calculates EDR from the turbulent energy spectrum in the inertial subrange:
EDRw={2πTAS[Sυr(f)f5/3¯αw]3/2}1/3.
In the EDRw calculation [Eq. (2)], TAS refers to the true airspeed of the UWKA, Sυr(f) is the power spectral density of WCR vertical velocity within the inertial subrange, f is frequency (corresponding to each Sυr), and αw is the Kolmogorov constant for the vertical component (αw = 0.707). An EDRw cross section for leg 3 in SNOWIE IOP-14 is displayed in Fig. 6a with the color bar discretized with EDR thresholds for “smooth,” “light,” “light–moderate,” “moderate,” “moderate–severe,” and “severe” turbulence. Occasionally, the vertical velocity segment that underwent the Fourier transform did not have an identifiable inertial subrange (i.e., −5/3 slope). Such cases appeared most commonly in regions of laminar flow and are differentiated in EDR panels with purple (e.g., Fig. 6b).
Fig. 6.
Fig. 6.

Transects of turbulence intensity: EDRw calculated from WCR vertical velocity along the flight legs shown in Fig. 1 for (a) SNOWIE IOP-14, (b) SNOWIE IOP-19, (c) CAMPS IOP-05, and (d) CAMPS IOP-01. EDR measured at flight level is also displayed using the same color key, except for (d), where flight-level EDR was not available. Color bar values indicate the locations of “smooth” (blue), “light” (green), “light to moderate” (yellow), “moderate” (orange), “moderate to severe” (red), and “severe” (pink) turbulence categories. Navy blue coloring is reserved for locations where no inertial subrange was present [example in (b)]. Red boxes indicate the region of KH waves.

Citation: Journal of the Atmospheric Sciences 77, 5; 10.1175/JAS-D-19-0108.1

The largest EDRw values on leg 3 were located near the surface but remained in the “light” turbulence category. Another local maximum was apparent within the KH waves near 5 km MSL but this also remained in the “light” category. Even so, the turbulence associated with the KH waves extended over 40 km horizontally and over a kilometer in height. Onboard the UWKA, the MacCready turbulence meter (MacCready 1964) directly measured EDR. These measurements are displayed at flight level using the same color bar. Generally, the flight-level EDR was slightly larger than the EDR values derived from WCR vertical velocity, but show the same trend, lending credence to the derivation.

b. Mountain wake

The second type of KH instability prevalent throughout the two campaigns manifested as regularly spaced waves that were near, or even in contact with the surface and often downwind of a prominent peak. In these cases, the increase in wind shear responsible for the wave train development is between unimpeded flow above the peak and a region of stagnant flow downwind and below the height of the peak (Fig. 3b). This appears similar to the LES by Sauer et al. (2016) wherein cases prescribed with nondimensional mountain height greater than unity developed a trapped lee wave above a stagnant region with reverse flow. It seems likely that a similar mechanism is at work in this case, contributing to the turbulence in wake of the mountain. Furthermore, the well-mixed BL advected up to the peak may separate in its wake and produce a region of increased shear (e.g., Chow and Street 2009; French et al. 2015). The localized extreme shear in this region of “wake turbulence” can produce a series of regularly spaced, billow-like, updrafts and downdrafts that may decay into turbulence. This wake turbulence was observed in many cross-mountain transects in SNOWIE and CAMPS. Such an event was observed by the UWKA at 1338 UTC 4 March 2017 during SNOWIE IOP-19 (Fig. 7), along a transect shown in Fig. 1a.

Fig. 7.
Fig. 7.

KH waves within the mountain wake from SNOWIE IOP-19. Panels show cross sections of WCR-derived (a) reflectivity, (b) vertical velocity, (c) horizontal velocity, and (d) wind shear. Lines of constant equivalent potential temperature from the WRF Model are contoured in black. Black arrows in (c) and (d) illustrate the components of the wind in the along-flight and vertical directions. The dotted boundary indicates the region of interest with KH wave activity.

Citation: Journal of the Atmospheric Sciences 77, 5; 10.1175/JAS-D-19-0108.1

In this single transect, the billows were apparent in both WCR reflectivity and vertical velocity (Figs. 7a,b). They began near the secondary peak at 13 km along-track distance and 2.3 km MSL, increasing in height to near 3 km MSL by around 17 km. Along-track wind (Fig. 7c) and vertical wind shear (Fig. 7d) were obtained from WCR dual-Doppler synthesis of the nadir and down–forward beams (section 2). Within the area of the waves, plumes of lower-momentum air were drawn upward in updrafts while higher-momentum air descended in downdrafts and were mixed within the eddies (blue arrows in Figs. 7b,c). Flow above the mountain peak contained small magnitudes of both positive and negative shear layers that aligned with the modeled virtual potential temperature contours. However, shear in the wake of the peak, directly above the surface, was much larger and displayed high turbulence. The large wind shear (resulting in dynamic instability) was subsequently mixed out downstream of the waves where flow became less turbulent. Derived EDRw (Fig. 6b) was largest near the surface, downstream of the dominant peak in the mountain wake.

The EDRw cross section has a second layer of elevated values spanning ~21 km at a height of about 4 km MSL. Interestingly, the two regions of elevated EDRw were conspicuously colocated with the largest reflectivity values (Fig. 7a). It is possible the larger EDRw was in response to the higher fall speeds of larger hydrometeors but the circumstance is redolent of underlying microphysical effects.

c. Blocked flow

When the air approaching a mountain is relatively stable so that its inertia cannot overcome the potential energy needed to raise over the mountain, the flow into the mountain becomes blocked, with very little low-level cross-mountain flow. Mathematically, blocked flow occurs when the Froude number (i.e., the wind speed divided by the Brunt–Väisälä frequency and mountain height) is less then unity. Blocked flow is another mechanism to increase wind shear, in this case, in the layer between the low-level blocked flow that decelerates and/or deflects, and the upper-level unblocked flow (Fig. 3c). Thus, the region for potential KH instability develops upstream of the blocking terrain. This mechanism has been illustrated over the Cascade Mountains (Houze and Medina 2005) and the Sierra Nevada (Medina and Houze 2015).

On 9 January 2011, during CAMPS IOP-05, the UWKA flew a series of nine transects, oriented southwest to northeast across the Park Range (Fig. 1b), which raises steeply above lower terrain to the west. Four of these transects are displayed in Fig. 8. As with the SNOWIE cases, dual-Doppler data were corrected with a nearby sounding from Steamboat Springs (Holdridge et al. 2010). These four sequential transects showed numerous terrain-induced flows. Our focus here is on the remarkably pristine KH waves in the range 0 < x < 20 km at the height of 3–3.5 km MSL seen best in the kinematic observations. As described earlier, the instability developed within a layer of high shear (Figs. 8d,h,l,p). Below, wind is blocked by the Park Range, with WCR-derived cross-mountain wind speeds between 0 and 5 m s−1. Above this layer, the wind was much stronger, between 10 and 15 m s−1 (Figs. 8c,g,k,o). In this case, it appears that blocking was being enhanced by the intrusion of a postfrontal, cold air mass below 3 km MSL, indicated by the (model derived) θe contours.

Fig. 8.
Fig. 8.

KH waves above blocked flow upwind of the Park Range from CAMPS IOP-05, illustrated using four consecutive flight transects, each separated by 5–15 min. The four panels for each transect display the same WCR-derived variables as those in Fig. 7. The dotted boundary indicates the region of interest with KH wave activity.

Citation: Journal of the Atmospheric Sciences 77, 5; 10.1175/JAS-D-19-0108.1

The KH waves in these transects are easily spotted in the cross sections of WCR-derived wind shear, where there are obvious braided structures approximately 0.2–0.3 km thick. Others (e.g., Dutton and Panofsky 1970; Medina and Houze 2016; Barnes et al. 2018) have also observed these braided structures within KH waves. The multiple passes allowed an investigation into the wave train evolution. On average, the flight legs were about 13 min apart. The KH waves first appeared in the fourth transect across the mountain (leg 1). The previous flight leg (not shown) did not show any wave structure in any of the cross-section variables. From leg 1 to leg 2, the waves strengthened: the vertical velocity magnitudes increased, the wind shear “braids” became more coherent, and distinct plumes of low and high momentum air (from the layers below and above the waves in Fig. 8g) were drawn upward and downward, respectively. By leg 3, all wave structures nearly vanished. Only a slight wave pattern was observable in wind shear (Fig. 8l). In leg 4, a KH wave train reappeared, initially weak and unorganized. These four panels span a period of about 45 min during which the KH waves developed, strengthened, transitioned into incoherent turbulence, and then reorganized. This 45-min cycle seems at least semiregular in this case because the KH waves go through another period of strengthening and dissipation in the next four legs (not shown). The transient nature of the KH waves in this section illustrates their effectiveness in mixing out the momentum of different layers thereby reducing the shear and dynamic instability (e.g., Medina and Houze 2016; Stull 1988, chapter 5.5). A forcing mechanism (such as continued blocking of the low-level flow) must once again increase shear to the critical value before the KH waves can redevelop.

According to EDRw (Fig. 6c), the airflow during this IOP was more turbulent than the previous cases at most levels. A majority of the transect contained light turbulence with some areas of light–moderate turbulence downstream of Mount Werner (the top of the Park Range in this transect). A large region of light to moderate turbulence was colocated with plunging flow downstream of the relatively low terrain (Emerald Mountain) to the SW (best seen in Figs. 8m–p), followed by what appeared to be a hydraulic jump in the Yampa valley. In addition, there was a low-level rotor (vorticity pointing out of the page) in the Yampa valley just east of the hydraulic jump, enhancing the shear near mountaintop level. Kingsmill et al. (2016) suggested that this rotor could be a result of low-level blocking by the Park Range, associated with a trapped lee wave in the wake of Emerald Mountain, or with a hydraulic jump in the lee slope of Emerald Mountain (see Fig. 12 in Kingsmill et al. 2016). At the location of KH waves (0 < x < 14 km, 3.5 km MSL), a layer of enhanced EDRw was present. Regions above and below this layer were less turbulent (lower EDRw).

d. Lofted BL

In section 3a, we demonstrated a case where linear, vertically propagating gravity waves may trigger KH waves by acting on preexisting shear. Nonlinear effects of gravity waves (e.g., downslope windstorms) and interactions with the BL produce more substantial shear. In this case, when unblocked slower flow within a well-mixed turbulent BL is lifted to a maximum height above the high terrain, it is in closer proximity to high-speed flow aloft, thereby increasing shear. This is the fourth mechanism illustrated herein (Fig. 3d).

The WCR cross section in Fig. 9, during CAMPS (Fig. 1b) on 15 December 2010 (IOP-01), shows turbulent, low-momentum flow within the BL below 1.2 km AGL and much faster winds aloft (Figs. 9c,d). Again, there was a shallow plunging current in the lee of Emerald Mountain. In this case the nonlinear wave updraft (reminiscent of a hydraulic jump) was strongest at low levels but widens with height over a great depth. A downstream rotor was present again, against the western slope of the Park Range, with vorticity pointing into the page. KH billows emerge from the shear layer over the Park Range at 3.8 km MSL as low-momentum air is carried in close proximity to more laminar strong westerly flow above the rotor. The laminar nature of the free-tropospheric flow was apparent from the striations in the wind shear (Fig. 9d) (Geerts and Christian 2017) and the lack of small-scale vertical velocity variations (Fig. 9b). The layer (striation) of positive shear starting at 3.6 km MSL descended into the Yampa valley before it ascended toward the Park Range (Fig. 9a). As the BL top (close to the 308-K equivalent potential temperature contour) moved upward and neared the high terrain of the Park Range, the shear layer reached values close to 0.1 s−1 before KH waves developed, destroying the shear layer.

Fig. 9.
Fig. 9.

As in Fig. 7, but for KH waves at the top of a lofted boundary layer from CAMPS IOP-01.

Citation: Journal of the Atmospheric Sciences 77, 5; 10.1175/JAS-D-19-0108.1

The billows are visible as plumes of enhanced reflectivity in WCR reflectivity (Fig. 9a) and adjacent updrafts and downdrafts in WCR vertical velocity (Fig. 9b) above the Park Range near 4 km MSL. Similar to previous cases, KH instability was locked to the terrain because the instability was continuously created as long as the shear layer between the BL and free troposphere was dynamically unstable. In effect, KH waves aged as they move downstream; smaller, pristine updrafts and downdrafts become larger, more turbulent masses with time until the shear is reduced (Figs. 9b,d). The IOP01 flight consisted of a set of parallel east–west flight transects shifted north–south along the Park Range. The set (and each transect identified by two end points) was repeated three times but the intermediate transects introduced a delay, just over an hour, before a specific transect was repeated. For the leg shown in Fig. 9, the identical transect 1 and 2 h later (not shown) still displayed regular updrafts and downdrafts but they were less coherent. It is also noteworthy that the plunging flow and hydraulic jump features were no longer present in the second and third transect, implying a role in shear enhancement (i.e., the most coherent billows occurred when the hydraulic jump was present). Because a hydraulic jump produces a transition from supercritical to subcritical flow (i.e., faster shallow flow to slower deeper flow), it is possible they may lead to larger shear over the subcritical region.

WCR-derived turbulence (EDRw) was rather elevated in layers throughout the transect (Fig. 6d). Upstream, there was a clear distinction between the turbulent boundary layer, below 3.6 km MSL, and the more laminar flow in the free troposphere above 3.6 km MSL (dashed line). As shear increased and KH waves developed, this boundary became less clear.

4. KH wave kinematics from dual-Doppler observations

One of the better examples of a KH wave train is the one observed on leg 1 during CAMPS IOP05, so we zoom in on this train (Fig. 10). The billows are not obvious in WCR reflectivity (Fig. 10a); however, the wave structure is very clear in WCR vertical velocity (Fig. 10b), dual-Doppler wind speeds (Fig. 10c), and dual-Doppler shear (Fig. 10d). Dual-Doppler error, estimated from the least squares residual norm during Doppler synthesis, has an average value of less than 1 m s−1 within this cross section. This value increases to 1–2 m s−1 for the shear layer. As described in section 3a, vertical velocity measurements depict a semilinear turbulent layer within the center of the billows, that is, the level of instability release. Above and below this layer are regularly spaced updrafts and downdrafts that were not tilted and whose magnitude decreases with height away from the instability source (similar to exponential decay in an evanescent wave). The lower-level updrafts are phase shifted relative to the upper-level ones. The KH wave train is embedded in a broader gravity wave, as is evident from the vertical velocity and the height variation of the wave train.

Fig. 10.
Fig. 10.

An extract of Figs. 8a–d, zoomed-in on the KH waves plus (e) the perturbation of the horizontal wind at each level. The dashed lines, overlaid in all panels, mark the locations of two shear layers in (d).

Citation: Journal of the Atmospheric Sciences 77, 5; 10.1175/JAS-D-19-0108.1

Dual-Doppler wind in Fig. 10c displays stunning, although subtle details about KH wave kinematics. The layers of high shear (dashed lines in Fig. 10d) bounded the linear turbulent layer. Turbulent mixing within a breaking billow wave or “roll” creates an intermediate wind speed having a momentum that was between that of the upper and lower levels. As an effect, the “parcel” of intermediate momentum produces two shear layers: one at the boundary with the low-momentum air and one at the boundary with the high-momentum air. This provides an explanation for the two-layer braid observed in shear (e.g., Chapman and Browning 1997; Medina and Houze 2016).

Expectedly, updrafts below the turbulent layer transport slower wind speeds (cold shading) upward while downdrafts above this layer transport faster wind speeds (warm shading) downward (Fig. 10c). Above the turbulent layer, the maximum wind speeds lay between the upper-level updrafts and downdrafts (i.e., horizontal wind speed quadrature lags vertical velocity) from flow acceleration over wave crests. Observations by Wakimoto et al. (1992), Chapman and Browning (1997), Houser and Bluestein (2011), and Barnes et al. (2018) also show the greatest wind speeds over crests. Maximum wind speeds below the turbulent layer follow a downdraft (i.e., vertical velocity quadrature lags horizontal wind speed).

The KH wave kinematics are combined into a conceptual diagram in Fig. 11. The phase relationships between vertical velocity, wind speed, potential temperature, and pressure above and below the dynamically unstable layer are shown, in addition to the region of turbulent mixing and intermediate wind speed (gray shading). Although the potential temperature and pressure perturbations are not observed with the current dataset, they are inferred from the horizontal and vertical wind speeds. Above the billows, the potential temperature perturbations quadrature lead vertical velocity perturbations as downward velocities transport larger potential temperature downward in a stable environment. The wind speed and pressure perturbations can be understood by applying the Bernoulli energy conservation principle to a laminar parcel moving from left to right through the wave train. According to Bernoulli’s principle, the total energy of the fluid (the sum of the pressure energy, kinetic energy, and potential energy) must remain constant. The wind speed is greatest over wave crests where the flow is constricted and, because potential energy remains nearly constant, colocated with a minimum in pressure. Beneath the billows, the phase relationships are shifted and flipped. This results from the lower layer’s wind speed relative to the billows. The phase relationships in Fig. 11 are not wave relative; rather, the relationships are depicted as a snapshot in time to match observations from the WCR (i.e., Eulerian framework). In a framework relative to the waves (i.e., Lagrangian framework), the framework moves with the KH waves that are advecting at a speed between that of the upper- and lower-layer wind speeds. This is supported by Fig. 10e in which the KH waves appear in horizontal velocity only after subtracting the mean wind. Consequently, the upper-layer wind speeds are slower in a wave-relative reference frame but the direction remains the same (from left to right). However, because the individual waves move faster than the lower-layer wind speed, flow moves from right to left beneath the waves and the low-level phase relationships can only be understood in a wave-relative framework. Although negative (moving from right to left), the largest wind speed magnitude is beneath the trough of the KH wave, which is consistent with Bernoulli’s principle.

Fig. 11.
Fig. 11.

Conceptual diagram of KH wave dynamics showing regions of low-momentum air beneath the waves, high-momentum air above the waves, and the region of intermediate momentum responsible for the braided structure. Phase diagrams indicate the relationship between vertical velocity, horizontal wind speed, pressure, and potential temperature perturbations above and below the layer of KH instability. Above and below the phase relationships are vectors indicating the absolute and wave-relative wind. The green line is dashed because pressure and temperature are inferred from expected dynamic relationships and not directly observed.

Citation: Journal of the Atmospheric Sciences 77, 5; 10.1175/JAS-D-19-0108.1

These fundamental relationships also apply to wave trains triggered by terrain—when KH instability is triggered, the individual KH waves still form in the shear layer and travel at a phase speed between the upper-level and lower-level wind speeds. KH wave trains triggered by terrain are unique in that they are typically stationary (i.e., locked to the terrain) and do not advect with the mean wind. Because new waves would form upstream where the instability first develops and decay downstream as the instability is mixed out, the group speed opposes the phase speed. Unfortunately, the time lag between consecutive WCR transects is too large to experimentally verify this. The shortest time between wave encounters was 5 min, which occurred between legs 1 and 2 in Fig. 8. Considering the dual-Doppler measured intermediate speed of 7 m s−1, individual waves would have advected 2.1 km downstream. The consistency between legs and shift in the position of the vertical velocity couplets advocate for the theory’s plausibility but we would need even shorter periods between observations for confirmation.

The dual-Doppler synthesis presents a unique opportunity to compare the depth of the shear layer with the wavelength of the KH waves. Early studies of KH waves suggest theoretical relationships between shear depth h and wavelength λ. Miles and Howard (1964) proposed the range of 4.4 < λ < 7.5 h while Turner (1973) provided the smaller range of 6.3 < λ < 7.5 h, where h represents the thickness of the unstable shear layer.

To objectively retrieve the dominant wavelength for the KH waves shown in Fig. 9, a spectral analysis was performed (Fig. 12). The power spectral density plots of WCR vertical velocity and the dual-Doppler horizontal wind speed panel were created by averaging the frequencies for all the individual levels in Figs. 10b and 10c. First, data from each level were linearly detrended before applying the Fourier transform. Next, the power spectrum was smoothed using a Tukey filter with a width of 5 points and frequencies were converted to wavelengths using the UWKA air speed. Between 1.0 and 1.4 km, there is a peak in energy within vertical velocity w and horizontal kinetic energy u representing the KH waves. The energy near 10 km is related to the larger-scale gravity wave evident within this segment. From Fig. 10d, the shear layer appears to be about 210 m thick. The range of observed wavelengths (1.0–1.4 km) falls at the lower end of Turner’s prediction (1.3 < λ< 1.6 km) but well within the prediction by Miles and Howard (0.9 < λ < 1.6 km). At wavelengths shorter than 0.5 km (0.16 s−1), the w′ energy distribution follows the inertial subrange (−5/3 slope), implying turbulent flow, at an energy level considerably below that contained in the KH waves.

Fig. 12.
Fig. 12.

Power spectral density of vertical velocity (black) and horizontal velocity (blue) from the box-average FFT from Figs. 10b and 10c. The dashed line shows the slope of turbulent flow in the inertial subrange (−5/3 slope).

Citation: Journal of the Atmospheric Sciences 77, 5; 10.1175/JAS-D-19-0108.1

5. Discussion

This study presents four cases of KH waves observed by an airborne profiling radar, the WCR. The cases were chosen to illustrate different ways in which the KH waves are triggered in relationship with the underlying terrain. In all cases, topography increases the shear within a layer inducing dynamic instability. The detailed 2D wind field (u, w) and shear pattern associated with the KH waves is described for each of the four cases. Vertical velocity measurements are also used to calculate the EDR along the flight path for a quantitative turbulence measurement. This confirms that KH waves are usually colocated with a local maximum in turbulence (a local maximum was not visible in our last case likely due to large background levels of boundary layer turbulence). The cases explored here contain values that are relatively benign; all show EDR values that peak in the light to moderate turbulence category.

In addition, the output from WRF simulations (900-m resolution) is used to describe the environment in which the KH waves are observed, and to test WRF’s capability of identifying terrain-driven dynamic instability. The simulations predict regions of potential KH instability poorly, which is not a surprise since the expression for Ri contains vertical gradients, and thus is sensitive to small changes in wind shear and static stability as the flow crosses the mountain. Ri is sensitive to the model’s vertical mixing (PBL) scheme, to the accuracy of vertical gradients in upstream boundary conditions, and to the details of the flow over upstream mountain ranges, which may alter the stratification. The model’s difficulty in producing critical layers may be attributed to resolving complex terrain-induced flows. Conrick et al. (2018) also used the WRF Model to analyze critical layers based on the Richardson number albeit with more success. A likely explanation may exist in the different scales of essential features being simulated. The wind shear in the simulations by Conrick et al. (2018) was produced by synoptic features (terrain-modified shallow frontal passage and a low-level jet) while the terrain-driven features we simulate are smaller scale. Only the blocked flow case produces dynamic instability (i.e., 0 < Ri < 0.25) in the actual KH wave region and the blocked flow (and postfrontal air mass) is deep enough to be resolved by the model.

This study shows radar reflectivity as well, in part to demonstrate reflectivity enhancement in the region of the KH waves, suggesting that the waves impact cloud microphysics. This agrees with Houser and Bluestein (2011) and Barnes et al. (2018), who provide evidence via polarimetric variables that KH waves impact microphysical processes in liquid, ice, and mixed-phase clouds. While progress has been made, there are still questions regarding the total effect of in-cloud KH waves: whether they generate extra cloud water that remains suspended in cloud, or remove water by enhancing precipitation efficiency. These questions will be addressed in Part II.

6. Conclusions

The introduction stated two goals: 1) to examine the relationship of tropospheric KH waves to the underlying terrain; and 2) to observationally describe the KH wave train flow and associated turbulence patterns, using very high-resolution vertical transects. The conclusions drawn from the analysis, in response to these two goals, are as follows:

  1. How does complex terrain enhance shear and generate KH waves?
    • KH waves can occur well above the underlying terrain and the BL, in the free troposphere, and may be triggered by vertically propagating gravity waves in layers of preexisting shear.
    • Flow stagnation downstream of a mountain crest enhances shear that can lead to KH instability. WCR observations suggest this occurs very frequently and that KH waves are highly turbulent.
    • Blocked flow enhances shear between low-level deterred flow and upper-level unblocked flow creating a layer, upstream of a mountain crest, prime for KH waves.
    • Low-momentum, well-mixed BL air lifted by terrain into close proximity of higher-momentum air aloft, increases the shear locally over the mountain crest.
  2. What are the key kinematic features of KH waves?
    • As previous research has shown, KH waves develop in stratified flow containing a layer of high shear where the Richardson number drops below 0.25. The KH waves deform this shear layer into a braided pattern. The resulting vertical velocity couplets, which are strongest just above and below the instability layer, increase turbulence on scales that affect aircraft.
    • Observations depict a horizontal layer of turbulence from which KH waves emerge. This “linear turbulent layer” is likely a result of dynamic overturning in the layer of high shear and an incipient feature of developing KH waves.
    • Pristine KH waves influence the dynamics of their environment in three regions (Fig. 11). Within the waves (at the level of KH instability), overturning billows mix air parcels of different momentum, producing an intermediate speed at which the billows advect downstream. This region contains small-scale, erratic turbulence. Above the waves, flow is faster than the advection speed of the individual waves; the flow moves through regularly spaced updrafts and downdrafts producing perturbations in horizontal wind speed, pressure, and temperature. Below the waves, flow is slower than the waves so that, relative to individual waves, the flow moves in the opposite direction. This results in dynamic perturbations that are flipped relative to the upper-level perturbations.

Acknowledgments

This study was funded by the National Science Foundation Grant AGS-1547101. Phillip Bergmaier and Sam Haimov assisted with the analysis and display of WCR data and Lulin Xue provided the SNOWIE WRF simulations.

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