1. Introduction
Clear-air turbulence (CAT), either with or without attendant cirrus, frequently occurs in the vicinity of tropopause jet streams (e.g., Ellrod et al. 2015), and presents hazards to commercial aviation because of its difficulty in detection and avoidance. In the current study we examine mechanisms for turbulence onset near the upper-tropospheric–lower-stratospheric (UTLS) jet stream in high-resolution research NWP simulations for two example cases of widespread moderate-or-greater (MOG) turbulence occurring in different synoptic regimes.
Current operational NWP models, including the High-Resolution Rapid Refresh (HRRR) model (Dowell et al. 2022; James et al. 2022) provide valuable information on environmental conditions that are favorable for aviation turbulence, but are unable to explicitly simulate such turbulence. Higher-resolution research NWP models are also unable to directly simulate the turbulence. However, using models with interactive nested grids having horizontal spacings between a few hundred meters and a kilometer is often adequate to establish links between environmental conditions and likely turbulence onset mechanisms (e.g., Sharman et al. 2012; Lane et al. 2012).
The strong vertical shears in the vicinity of jet streams have long been accepted (e.g., Ludlam 1967; Dutton and Panofsky 1970; Klostermeyer and Rüster 1980) as a frequent source of turbulence owing to Kelvin–Helmholtz instability (KHI). The role of distant deep convection located more than several hundreds of kilometers away in enhancing jet stream wind magnitudes, and creating environments more conducive to turbulence, has been illustrated in simulations of observed cases (e.g., Trier et al. 2012; Trier and Sharman 2016). In these cases, and in other research NWP simulations, UTLS convective outflows can significantly enhance the vertical shear and reduce the static stability at altitudes near the synoptic jet, which promotes turbulence through a variety of mechanisms.
In KHI, waves are oriented normal to the environmental vertical shear vector and can overturn, initiating the turbulent cascade of energy to smaller scales. KH waves are sometimes nearly collocated with vertically propagating internal gravity waves (e.g., Trier et al. 2012; Zovko-Rajak and Lane 2014; Sharman and Trier 2019), which can also undergo turbulent breaking immediately below critical levels (e.g., Dörnbrack et al. 1995) where the wave horizontal phase speed c equals the magnitude of the horizontal wind U. Both the onset of KHI, and the altitude of critical levels influencing gravity wave breaking can be significantly influenced by UTLS convective outflows from distant convection (e.g., Trier et al. 2020).
In contrast to KHI, thermal-shear instability (Asai 1970, 1972) leading to turbulence occurs in neutral or statically unstable environments where shallow horizontal convective rolls (HCRs) are oriented parallel to the vertical shear vector. This mechanism has been simulated and linked to (often turbulent) cirrus cloud banding in UTLS outflows extending large distances from parent deep convection in MCSs (Trier et al. 2010), and midlatitude (e.g., Kim et al. 2014; Trier and Sharman 2016) or tropical (Kawashima 2021; Yamazaki and Miura 2021) cyclones. Other factors that can promote turbulence near tropopause jet streams include inertia gravity waves (e.g., Lane et al. 2004; Plougonven and Zhang 2016) and inertial instability (e.g., Knox 1997; Thompson and Schultz 2021), both of which can be either initiated or substantially enhanced by deep convection (e.g., Zhang 2004; Rowe and Hitchman 2015).
Previous modeling studies have examined the influence of model resolution on convectively produced features leading to turbulence, including internal gravity waves (e.g., Lane and Knievel 2005), and on simulation of observed near-cloud turbulence (Lane et al. 2012; Barber et al. 2018). In the current study we examine the effects of resolution on simulated turbulence onset mechanisms near convectively enhanced jet streams at larger horizontal distances from the parent convection.
Section 2 describes our data sources, the horizontal and vertical distribution of turbulence in the two cases, and discusses the differing large-scale weather patterns and vertical structure of the turbulence environments. The experiment design for the simulation of these cases is discussed in section 3. Section 4 presents an overview of simulations and comparison with observations, while effects of model horizontal resolution are analyzed in section 5. Turbulence onset mechanisms in our highest-resolution simulations are described in section 6, and implications for operational forecasts based on input from lower-resolution simulations are discussed in section 7. Results are summarized in section 8.
2. Observed turbulence cases and their synoptic environments
a. Data
This study uses both quantitative in situ observations and qualitative pilot reports (PIREPs) of turbulence intensities. The in situ data are estimates of the energy dissipation rate (EDR) ε1/3 (m2/3 s−1) derived from aircraft vertical motion measurements (Sharman et al. 2014). EDR ranges for severe (≥0.34) and moderate (0.22–0.33) turbulence intensities used in this study follow previously established ones (Sharman et al. 2014) for medium-sized commercial aircraft.
National Weather Service (NWS) soundings launched twice daily at 0000 and 1200 UTC, together with 6-hourly Global Forecast System (GFS) analyses, are used to determine the meteorological conditions in the vicinity of the turbulence. NEXRAD WSR-88D radar mosaics are examined to determine the proximity of the turbulence to organized deep convection.
b. Turbulence cases
The two examined widespread turbulence cases occurred in differing synoptic environments (Fig. 1). Case 1 from 25 October 2019 (Fig. 1a) had a large-amplitude eastward-translating synoptic trough, and much of the turbulence occurs near or slightly east of the trough axis. High amplitude UTLS troughs are a common synoptic feature near which widespread turbulence can occur (e.g., Knox et al. 2016; Trier et al. 2020). This case had 424 MOG turbulence reports above 20 000 ft (Z = 6.1 km MSL, p ≈ 475 hPa) over the eastern 2/3 of the continental United States (CONUS) during 1500–2100 UTC, including 42 that were severe (17 PIREPs, 25 in situ).
Height of the 250-hPa surface (60-m contour interval) and isotach (10 m s−1; gray shading) analysis for (a) 1800 UTC 25 Oct and (b) 1800 UTC 3 Dec 2019, with overlaid MOG turbulence reports from z > 20 000 ft for the indicated time periods. The dashed rectangular regions in (a) and (b) locate the regions of observed turbulence in which the vertical distribution of MOG turbulence reports are presented in Figs. 3a and 3b, respectively.
Citation: Monthly Weather Review 150, 10; 10.1175/MWR-D-22-0077.1
Unlike case 1, the case 2 weather pattern on 3 December 2019 (Fig. 1b) has a weak UTLS synoptic ridge. However, the 468 reports of MOG turbulence during 1700–2300 UTC 3 December 2019, including 41 that were severe (10 PIREPs, 31 in situ), is comparable to the 6-h case 1 reports over a similar sub-CONUS scale (Fig. 1a). Another common aspect of the two cases is the majority of the reported turbulence occurring along the UTLS synoptic jet.
Widespread precipitation with embedded clusters of deep convection occurs across the southern United States in case 1 (Fig. 2a). However, the majority of the concurrent MOG turbulence reports (Fig. 1a) occur within the red ellipse (Fig. 2a) several hundreds of kilometers north of the deep precipitating convection, but close to northern edge of a broader cirrus cloud shield (Fig. 2b). In case 2 there are two concentrated regions of MOG turbulence including one located over California and west central Nevada within the exit region of a jet maximum on the east side of an offshore cutoff low (Fig. 1b). The other is situated downstream from the UTLS ridge axis, and extends from northeastern Utah through southeastern Colorado (Fig. 1b), where cloudiness is less widespread (Fig. 2d). Organized precipitation in case 2 occurs near the California coast (Fig. 2c), though its southwestward extent is likely underestimated due to horizontal range limitations of the land-based NEXRAD radar network.
(left) NEXRAD WSR-88D mosaics of maximum radar reflectivity in a vertical column (MREF) during (a) case 1 at 1800 UTC 25 Oct and (c) case 2 at 0000 UTC 4 Dec 2019. The red ellipses approximate regions containing widespread moderate-or-greater turbulence reports above 20 000 ft for cases 1 and 2, which are displayed in Figs. 1a and 1b, respectively. (right) GOES-16 East true color visible satellite images for cases (b) 1 and (d) 2 at the indicated times.
Citation: Monthly Weather Review 150, 10; 10.1175/MWR-D-22-0077.1
In the remainder of the paper, we focus on the turbulence occurring in the dashed rectangular regions of Fig. 1a for case 1 and Fig. 1b for case 2, which are both far removed from deep precipitating convection (Fig. 2). Despite large differences in the overall 250-hPa geopotential height pattern between these two cases, the rectangular regions of interest both occur within strong geostrophic confluence in the entrance regions of UTLS jet streaks (Fig. 1). However, the vertical distribution of MOG turbulence reports for the 6-h periods within the rectangular regions of Fig. 1 differs between the two cases (Fig. 3).
Vertical distribution of MOG turbulence reports for the indicated 6-h periods over the dashed rectangular regions in (a) Fig. 1a and (b) Fig. 1b.
Citation: Monthly Weather Review 150, 10; 10.1175/MWR-D-22-0077.1
Case 1 MOG turbulence reports decrease monotonically with altitude throughout the UTLS (Fig. 3a). This may be influenced by a large number of aircraft ascending or descending near the commercial aviation hub of Chicago, Illinois, and thus being lower than typical cruising altitudes of 30 000–40 000 ft (9.1–12.2 km). However, simulation results presented later illustrate environmental factors (section 4) and turbulence onset mechanisms (section 6) that are also consistent with this vertical distribution of observed turbulence. Case 2 has a bimodal distribution with a minimum of MOG reports in the upper troposphere between 9.2 and 10.7 km MSL (Fig. 3b), which contrasts with case 1. The differences between the two cases occurs in both PIREPs and in situ data (Fig. 3).
Though the MOG turbulence in the rectangular region for case 1 (Fig. 1a) is far removed from deep precipitating convection (Fig. 2a), NWS soundings from Topeka, KS (TOP) and Davenport, IA (DVN) (locations in Fig. 1a) launched several hours earlier reveal approximately dry adiabatic ice-saturated layers that coincide with the core of the UTLS jet (Fig. 4a). These layers occur from 9.4 to 10.6 km at TOP and through a deeper layer from 9.8 to 12.0 km at DVN, and are consistent with the jet being influenced by moist UTLS outflow from the distant convection. Despite the ability of these layers to support cirrus cloudiness, the majority of the turbulence reports from the region occur below 9.2 km (Fig. 3a), indicating the likely predominance of CAT.
NWS soundings of temperature, dewpoint, and horizontal wind in (a) case 1 at 1200 UTC 25 Oct 2019 from Davenport, IA (DVN), and Topeka, KS (TOP), and (b) case 2 from Grand Junction, CO (GJT), at 1200 UTC 3 Dec 2019 and Denver, CO (DNR), at 0000 UTC 4 Dec 2019. The locations of the soundings are shown relative to moderate-or-greater turbulence reports from 1500 to 2100 UTC 25 Oct (1700–2300 UTC 3 Dec) 2019 for case 1 (case 2) in Fig. 1a (Fig. 1b).
Citation: Monthly Weather Review 150, 10; 10.1175/MWR-D-22-0077.1
In case 2, soundings from Grand Junction (GJT) and Denver, Colorado (DNR) (Fig. 4b), near or within the turbulence cluster of interest (Fig. 1b) also indicate relative humidity enhancements near the UTLS jet maximum that are suggestive of convective outflow effects. However, these soundings lack the deep ice-saturated UTLS layers of the case 1 soundings.
All soundings near the regional MOG turbulence clusters (Fig. 1, dashed rectangles) contain well-defined UTLS jet profiles, but there is variability in the average depth and intensity of the jet among the two cases (Fig. 5a), and at different sounding locations in individual cases (section 4). These differences affect the vertical profiles of gradient Richardson number, Ri = N2/S2, where
Averaged vertical profiles of the (a) horizontal wind speed, (b) square of the vertical shear of the horizontal wind, and (c) dry static stability for the case 1 (red dotted, sounding locations shown in Fig. 1a), case 2 (blue dotted, sounding locations shown in Fig. 1b) soundings, and the average of both cases (black). The horizontal black lines indicate the altitude limits of the 35–40 kft (∼10.7–12.2 km) MOG turbulence report bins in Fig. 3.
Citation: Monthly Weather Review 150, 10; 10.1175/MWR-D-22-0077.1
Stronger vertical shear in the 10.7–12.2 km MSL layer above the jet for case 2 sounding averages than for case 1 (Fig. 5b), despite similar static stability through much of this layer (Fig. 5c), is therefore an environmental factor that could influence the bimodal vertical distribution of MOG turbulence unique to case 2 (Fig. 3). Overall, the static stability is smallest within the UTLS jet maximum (cf. Figs. 5a,c; black curves), and the numerous turbulence reports beneath the jet maximum in both cases (Fig. 3) are consistent with strong vertical shear at these altitudes (Fig. 5b).
3. Numerical model and experiment design
a. Numerical model
Both cases are simulated using version 4.1.3 of WRF-ARW (Skamarock and Klemp 2008). For the current study the model contains 82 vertical levels with a top at 26.3 km MSL. In the lowest 2 km the vertical grid spacing increases approximately linearly from 60 to 240 m and remains roughly constant from 2 to 14 km MSL, before increasing linearly to 1200 m at the model top. A 7-km-deep gravity wave absorbing layer (Klemp et al. 2008) is used at the top of the model to mitigate spurious wave reflection.
The model physical parameterizations include the Noah land surface model (Ek et al. 2003), the RRTMG longwave and shortwave radiation schemes (Mlawer et al. 1997; Iacono et al. 2008), and the MYJ PBL (Janjić 1994, 2001) scheme. The MYJ scheme controls vertical mixing between adjacent vertical layers throughout the model depth and predicts subgrid-scale (SGS) TKE. MYNN (Nakanishi and Niino 2004; Olson et al. 2019) is a more recent Mellor–Yamada-type PBL scheme, which unlike MYJ, has a scale-aware option (e.g., Ito et al. 2015) in WRF. However, use of MYNN resulted in excessive dissipation in the UTLS for a previous turbulence case study, which reduced wave activity compared to MYJ at grid spacings of Δx = 1 km (Muñoz-Esparza et al. 2020). Though older and less optimal for some other applications, this aspect has motivated our use of the MYJ scheme in the current study. A Smagorinsky-type first-order closure is used to determine horizontal mixing in the model (Skamarock and Klemp 2008). For each case simulations use four two-way interactive nested horizontal domains (Fig. 6), which are discussed in the following subsection. In simulations that contain moisture, the Thompson et al. (2008) bulk microphysics scheme is used in each of the four domains and the Tiedtke (1989) cumulus parameterization is used only in the outer domain (d01).
ARW-WRF Model domains used for the two example widespread MOG turbulence cases described in this paper. The horizontal grid spacing used in each of the interactive domains is annotated within the domains. Observed MOG turbulence reports over 6-h periods in d02 are provided in Fig. 1.
Citation: Monthly Weather Review 150, 10; 10.1175/MWR-D-22-0077.1
b. Experiment design
For each of the two cases the horizontal grid spacings are 9, 3, 1, and 0.333 km in domains d01, d02, d03, and d04 (Fig. 6), respectively. Since a major objective of the current study is to examine effects of deep convection on CAT, control simulations (CTRL) are compared to simulations (DRY) that are otherwise identical, except that the bulk cloud microphysics and cumulus parameterizations are deactivated. We also examine the effects of horizontal resolution by successively adding the inner nests (Fig. 6) for different simulations. Here, the two- and one-domain simulations have minimum horizontal grid spacings of Δxmin = 3 and 9 km, respectively, and are expected to produce results broadly representative of high-resolution operational NWP models with and without explicit deep convection, such as the High-Resolution Rapid Refresh (HRRR) (Dowell et al. 2022; James et al. 2022), and the Rapid Refresh (RAP; Benjamin et al. 2016) models, respectively. The four- and three-domain simulations have minimum horizontal grid spacings of Δxmin = 333 m and 1 km, respectively, which are more typical of recent studies with high-resolution research NWP models (e.g., Kim and Chun 2012; Zovko-Rajak and Lane 2014; Zovko-Rajak et al. 2019; Trier and Sharman 2016, 2018; Trier et al. 2020). While unable to resolve scales that directly affect aircraft turbulence, these studies provide insights into possible turbulence onset mechanisms in observed aviation turbulence cases.
Lateral boundary conditions for d01 of the model are from 0.25° NCEP Global Final (FNL) analyses at 6-h frequency. The outer three model domains are initialized using concurrent FNL analyses at 0000 UTC 25 October and 0000 UTC 3 December 2019 for cases 1 (Fig. 6a) and 2 (Fig. 6b), respectively, and these domains are integrated for 24 h in both cases. For case 1, d04 with Δx = 0.333 km, is initialized at 1200 UTC 25 October 2019 and integrated for the remaining 12 h of the simulation for both CTRL and DRY. In case 2, d04 with Δx = 0.333 km, is initialized at 1600 UTC 3 December 2019 and integrated for the remaining 8 h of the CTRL and DRY simulations. For both of these simulated cases, the start and finish times of the highest-resolution domain d04 surround the times during which observed MOG events within the dashed rectangular regions of Figs. 1a and 1b were analyzed (Fig. 3).
4. Overview of simulated cases and comparison with observations
Enhanced model SGS TKE below approximate altitudes of the UTLS jet in d02 of the three-domain CTRL simulations is aligned along the axis of observed severe turbulence reports at surrounding times in both cases 1 (Fig. 7a) and 2 (Fig. 7b). Note that the observed severe turbulence and simulated SGS TKE in the upper Midwest for case 1 (Fig. 7a), and over Colorado in case 2 (Fig. 7b), are located outside of regions of simulated precipitation. The largest TKE values in CTRL are located beneath maximum winds in the UTLS jet for both cases, and are most widespread in case 2 (Fig. 7b).
Model-derived maximum radar reflectivity (MREF) in a vertical column (color shading), simulated winds plotted using the standard meteorological convention [half barb = 5 kt; full barb = 10 kt; pennant = 50 kt; 1 kt = 0.514 m s−1], and TKE (gray shading; 0.5 m2 s−2 contour intervals starting at 0.5 m2 s−2) in domain d02 of the three-domain CTRL simulations for cases (a) 1 and (b) 2. Corresponding (CTRL − DRY) difference winds [barbs plotted as in (a),(b)], (CTRL − DRY) vertical difference of horizontal winds (brown contours with 10 m s−1 intervals, negative values dashed), and (CTRL − DRY) TKE [gray shading as in (a) and (b), negative values in black contours with 0.5 m s−1 intervals] for cases (c) 1 and (d) 2. Observed severe turbulence reports at altitudes of z > 20 000 ft are overlaid for cases (a),(c) 1 and (b),(d) 2. Annotations next to the cross symbols in (a) and (b) indicate horizontal locations of the observed and simulated vertical profiles of wind components plotted in Figs. 8 and 9, respectively.
Citation: Monthly Weather Review 150, 10; 10.1175/MWR-D-22-0077.1
CTRL − DRY difference fields of SGS TKE and horizontal winds near the UTLS jet, and the difference vertical shear magnitudes immediately beneath the difference winds are shown in Figs. 7c and 7d. Concentrated regions of observed severe turbulence reports coinciding with (CTRL − DRY) southwesterlies in case 1 (Fig. 7c), and with (CTRL − DRY) northwesterlies in case 2 (Fig. 7d) occur along the edges of large anticyclonic circulations emanating from the simulated convection (Figs. 7c,d). The CTRL vertical shear through 2 km depth below the UTLS jet is increased by 15–25 m s−1 (2 km)−1 over that of the DRY simulations in these regions (Figs. 7c,d), demonstrating the strong influence of convectively induced enhancement of the synoptic UTLS jet in providing a favorable environment for turbulence occurring far downstream from the convective source. The broad similarities between the CTRL (Figs. 7a,b) and CTRL − DRY (Figs. 7c,d) SGS TKE fields further reflect the importance of diabatic processes in generating downstream turbulence.
The realism of the simulated vertical structure of the horizontal winds and the effects of distant convection on these winds are assessed by comparing vertical profiles from observed soundings (locations shown in Figs. 7a,b) with simulated ones at corresponding model grid points in CTRL and DRY for cases 1 (Fig. 8) and 2 (Fig. 9). In general, CTRL well simulates the observed profiles in each of the two cases, and has 10–30 m s−1 stronger UTLS jets with greater vertical sharpness in CTRL than in DRY.
Observed (black curves), and simulated vertical profiles of (a),(c) zonal and (b),(d) meridional wind components from corresponding model grid locations in domain d02 of the three-domain CTRL (red curves) and DRY (blue curves) simulations of case 1 at (a),(b) Topeka, KS, and (c),(d) Davenport, IA, at 1200 UTC 25 Oct 2019 (forecast hour t = 12 h). The gray shading represents layers of gradient Richardson number ≤ 0.25 in the observed soundings. The locations of these soundings relative to observed turbulence reports are shown in Figs. 1a and 7a.
Citation: Monthly Weather Review 150, 10; 10.1175/MWR-D-22-0077.1
Observed (black curves), and simulated vertical profiles of (a),(c),(e) zonal and (b),(d),(f) meridional wind components from corresponding model grid locations in domain d02 of the three-domain CTRL (red curves) and DRY (blue curves) simulations of Case 2 for (a),(b) Grand Junction, CO, at 1200 UTC 3 Dec 2019 (forecast hour t = 12 h), and (c),(d) Denver, CO, and (e),(f) Dodge City, KS, at 0000 UTC 4 Dec 2019 (forecast hour t = 24 h). The gray shading represents layers of gradient Richardson number ≤ 0.25 in the observed soundings. The locations of these soundings relative to observed turbulence reports are shown in Figs. 1b and 7b.
Citation: Monthly Weather Review 150, 10; 10.1175/MWR-D-22-0077.1
In case 1, observed layers of Ri ≤ 0.25 are supported both below and above the jet at TOP, despite the much weaker vertical shear components in the small Ri layer above the jet maximum (Figs. 8a,b). Here, small static stability extending above the jet (not shown) supports Ri ≤ 0.25 in this 10.1–10.8 km layer. The UTLS jet is deeper and more westerly at DVN (Fig. 8c), which is located both closer to the centroid of the severe turbulence reports and the entrance to the downstream synoptic jet (Figs. 1a, 7a). Here, the observed jet extends up to the lower stratosphere (∼12 km MSL) where static stability begins increasing with height, and contributes to Ri ≤ 0.25 being restricted to altitudes beneath the jet (Figs. 8c,d).
In case 2, the 1200 UTC 3 December GJT sounding, which is located closest to the ridge axis and upstream convection (Fig. 7b), has an intense, narrow westerly UTLS jet, and supports layers of Ri ≤ 0.25 and turbulence from KHI both below and above the jet (Figs. 9a,b). Farther east, both the DNR (Figs. 9c,d) and DDC (Figs. 9e,f) 0000 UTC 4 December soundings have deeper jet profiles (not unlike DVN from case 1), and layers of Ri ≤ 0.25 are confined to lower altitudes of ∼8–9 km MSL beneath the jet.
Significant grid-resolved vertical velocities (|w| > 1 m s−1) are evident at 8 km MSL in d03 of the four-domain CTRL simulations (Figs. 10a,b), which include feedbacks from d04, thus increasing the “effective” resolution (section 5). Comparisons of Fig. 10a with Fig. 7c and Fig. 10b with Fig. 7d further indicate the strong association of the banded fine-scale vertical motions in the higher-resolution simulations with large enhancements to the environmental vertical shear resulting from the convectively induced anticyclonic UTLS outflow. These SW–NE (Fig. 10a) and NW–SE (Fig. 10b) oriented regions of grid-resolved vertical motions in cases 1 and 2, respectively, are spatially correlated with the observed approximately concurrent in situ MOG turbulence reports. The grid-resolved vertical motions are significantly weaker at 11 km in case 1 (Fig. 10c), which is above the UTLS jet. This is consistent with the small number of corresponding observed MOG reports at higher altitudes in case 1 (Fig. 3a). In contrast, the vertical velocity amplitudes and number of MOG reports above the UTLS jet in case 2 are comparable to those beneath it (cf. Figs. 10b,d), which is consistent with the overall bimodal distribution in case 2 (Fig. 3b).
Vertical velocity (color shading) at (a),(b) 8 km MSL for cases (left) 1 and (right) 2, and at (c),(d) 11 km MSL for cases (left) 1 and (right) 2, in d03 of the four-domain CTRL simulations with overlaid observed in situ reports of moderate (0.22 ≤ EDR < 0.34) and severe (EDR ≥ 0.34) turbulence at (a),(b) z < 30 000 ft and at (c),(d) z ≥ 30 000 ft. The thin black contours in (b) and (d) are surface elevation (1600-, 2400-, and 3200-m values).
Citation: Monthly Weather Review 150, 10; 10.1175/MWR-D-22-0077.1
In case 1 the horizontal location of MOG reports coincides with cirrus banding (Fig. 11a) located along the outer edge of the UTLS outflow. Cirrus banding oriented transverse to the edge of large cloud shields has been widely observed (e.g., Lenz et al. 2009; Knox et al. 2010) and linked to HCRs arising from thermal instability in previous high-resolution simulations (e.g., Trier et al. 2010; Kim et al. 2014; Trier and Sharman 2016). Banded vertical motions occur in a similar location near the simulated cloud edge (Fig. 11b).
(a) GOES-16 true-color visible satellite imagery at 1802 UTC 25 Oct 2019 with superposed observed moderate-or-greater turbulence reports at 1730–1830 UTC from the annotated altitudes (e.g., FL260 = flight level 26 000 ft MSL) and (b) 9.8 km MSL simulated vertical velocity, and approximate visible cloud boundary [qcl (cloud ice + cloud water mixing ratio) ≥ 0.0001 g kg−1] from a portion of d04 (Δx = 333 m, Fig. 6a) of the four-domain case 1 control simulation at 1800 UTC 25 Oct 2019 with overlaid observed 1730–1830 UTC MOG turbulence reports within the indicated altitude range. The dashed inset in (b) indicates the horizontal location of plots presented in Figs. 12a and 12b.
Citation: Monthly Weather Review 150, 10; 10.1175/MWR-D-22-0077.1
Inside the simulated cloud edge of d04 from CTRL, weak bands of vertical motion (|w| ∼ 1 m s−1) originate from an area of negative moist static stability,
Case 1 1800 UTC 25 Oct 2019 vertical velocity, 1-km-deep horizontal wind difference vectors (plotting convention as in Fig. 7), moist static stability (brown contours, negative values dashed), and approximate visible cloud boundary [qcl (cloud ice + cloud water mixing ratio) ≥ 0.0001 g kg−1] from the dashed inset of Fig. 11b at (a) 9.8 and (b) 8.5 km MSL, which is part of d04 (Δx = 333 m, Fig. 6a). Vertical cross sections of potential temperature (3-K contour intervals) and qcl (color shading) along transects (c) AB and (d) CD of (a) and (b), respectively. The bold dashed lines in (c) and (d) indicate the altitude of transects AB and CD in (a) and (b), respectively.
Citation: Monthly Weather Review 150, 10; 10.1175/MWR-D-22-0077.1
The vertical cross section along transect AB (Fig. 12c) shows cloud hydrometeor enhancements in the ∼9.5–10.5 km layer occur near and above incipient wave breaking at ∼8.5 km MSL, where there is more reported MOG turbulence in case 1 (Fig. 3a). Higher amplitude banded vertical motions of |w| ∼ 2–4 m s−1 occur along the edge or outside of the cloud boundary (Fig. 12a), and are located within weak static stability that supports internal gravity waves (section 6). These banded vertical motion features that occur near or immediately outside of the cloud edge (Fig. 12a) also occur above wave breaking, which is evident near 8.5 km MSL along transect CD (Fig. 12d). Along transect CD the 8.5 km MSL vertical motions are approximately normal to the strong vertical shear (Fig. 12b), which implicates KHI as a likely turbulence onset mechanism in case 1. This possibility is investigated further in section 6.
The vertical motion patterns for case 2 contain both a relatively narrow NW–SE-oriented region of intense vertical motions and a broader N–S-oriented region of weaker vertical motions aligned along the elevated terrain of the Rockies (Figs. 10b,d), the latter of which appear to be related to mountain waves (not shown). Examination of d04 in the DRY simulation (Fig. 13a) indicate N–S-oriented vertical motions that strongly resemble the corresponding N–S-oriented vertical motions from CTRL (Fig. 13b). However, the NW–SE band of finer-scale vertical velocities in CTRL (Fig. 13b), which are missing in DRY (Fig. 13a), are best correlated with the NW–SE-oriented envelope of observed MOG turbulence reports in case 2.
Simulated 8 km MSL vertical motions and SGS TKE (green contours with 0.5 m2 s−2 intervals) in d04 (Δx = 333 m) of four-domain (a) DRY and (b) CTRL simulations at 2030 UTC 3 Dec 2019 (forecast hour t = 20.5 h). The overlaid observed PIREPs and in situ turbulence reports are from altitudes of z < 30 000 ft. EDR threshold values for the in situ turbulence reports are as in Fig. 10.
Citation: Monthly Weather Review 150, 10; 10.1175/MWR-D-22-0077.1
5. Sensitivity of simulations to horizontal resolution
Both horizontal and vertical resolution in numerical models can be important factors for anticipating and forecasting widespread turbulence events (e.g., Trier and Sharman 2016). The effects of horizontal resolution are demonstrated in Fig. 14 for case 1 at two different times. SGS TKE in the single domain CTRL simulation, with horizontal grid spacing of Δxmin = 9 km, is located in southwesterly flow ahead of the 1600 UTC trough axis at 8.5 km MSL (Fig. 14a), where much MOG turbulence is reported during 1500–2100 UTC (cf. Fig. 3a). SGS TKE increases at the finer grid spacing of Δxmin = 3 km in the two-domain simulation (Fig. 14c) but decreases as grid-scale vertical motions begin to successively appear in higher-resolution four- (Fig. 14g) and three-domain (Fig. 14e) simulations.
Plots of 8.5 km MSL vertical velocity, horizontal winds (plotted as in Fig. 7), and subgrid-scale TKE (black contours with 0.5 m2 s−2 intervals starting at 0.5 m2 s−2) from case 1 CTRL simulations at (left) 1600 UTC (forecast hour t = 16 h) and (right) 1900 UTC 25 Oct 2019 (forecast hour t = 19 h) using (a),(b) 1 domain; (c),(d) 2 domains; (e),(f), 3 domains; and (g),(h) 4 domains with minimum horizontal grid spacings of Δx = 9, 3, 1, and 0.333 km, respectively. The total area displayed comprises the dashed rectangle in Fig. 6a, and the dashed rectangular regions in (f) and (h) are enlarged to better display vertical velocity at z = 8.5 km MSL in Figs. 16a and 16c, respectively.
Citation: Monthly Weather Review 150, 10; 10.1175/MWR-D-22-0077.1
SGS TKE increases by 1900 UTC in the lower-resolution simulations (Figs. 14b,d), whereas the amplitude and coverage of the grid-scale vertical motions increase in the higher-resolution simulations (Figs. 14f,h). By this time, the locations of these features correspond well to the mean locations of the observed 1500–2100 UTC MOG turbulence reports in the dashed rectangular region of Fig. 1a. Here, the grid-resolved vertical motions in the highest-resolution four-domain simulation with Δxmin = 333 m (Fig. 14h) account for most of the vertical mixing and stabilization that was being accomplished prior to 1600 UTC (not shown) by the PBL scheme (section 3).
Neither the one- nor two-domain simulations have sufficiently fine horizontal grid spacings to produce significant gridscale vertical velocities (Figs. 14a–d). However, the two-domain simulation with Δxmin = 3 km has larger SGS TKE than in the one-domain simulation with Δxmin = 9 km (Figs. 14b,d). This result could be influenced by subtle differences in the UTLS convective outflow in the two-domain simulation with explicit deep convection from that of corresponding UTLS outflow owing to the parameterized convection in the single-domain simulation, which is an important topic for future research.
For case 2 we examine effects of the horizontal resolution at the higher altitude of 11.5 km MSL (Fig. 15) where, in contrast to case 1 (Fig. 3a), observations indicate many reports of MOG turbulence above the UTLS jet (Fig. 3b). Here, evidence of the terrain-related N–S-oriented region of vertical motions discussed in the previous section is found in the single-domain simulation at 1930 UTC 3 December 2019 (Fig. 15a), though these simulated vertical motions are significantly under-resolved at this resolution. This area of vertical motions does become qualitatively similar at Δxmin = 3 km (Fig. 15c) to analogous N–S-oriented regions of vertical motions in the higher-resolution simulations (Figs. 15e,g).
Plots of 11.5 km MSL vertical velocity, horizontal winds (plotted as in Fig. 7), and subgrid-scale TKE (black contours with 0.5 m2 s−2 intervals starting at 0.5 m2 s−2) from case 2 CTRL simulations at (left) 1930 UTC (forecast hour t = 19.5 h) and (right) 2100 UTC 3 Dec 2019 (forecast hour t = 21 h) using (a),(b) one domain; (c),(d) two domains; (e),(f), three domains; and (g),(h) four domains with minimum horizontal grid spacings of Δx = 9, 3, 1, and 0.333 km, respectively. The total area displayed comprises the dashed rectangle in Fig. 6b, and the dashed rectangular regions in (e) and (g) are enlarged to better display vertical velocity at z = 11.5 km MSL in Figs. 16b and 16d, respectively.
Citation: Monthly Weather Review 150, 10; 10.1175/MWR-D-22-0077.1
The NW–SE-oriented finer-scale vertical motions in the three- and four-domain simulations (Figs. 15f,h) develop at a faster rate than in case 1, but like in case 1, remain completely unresolved in the one- and two-domain simulations (Figs. 15b,d). Also similar to case 1, their spatial coverage in three- and four-domain simulations at 2100 UTC (Figs. 15f,h) roughly coincides with the patterns of SGS TKE in the lower-resolution one- and two-domain simulations (Figs. 15b,d).
Differences in the structure of the vertical motion fields between simulations with Δxmin = 1 km and 333 m are highlighted in Figs. 16a,c and in Figs. 16b,d as close-up views from the dashed boxes in cases 1 (Figs. 14f,h) and 2 (Figs. 15e,g), respectively. Both cases exhibit a banded vertical motion pattern having a characteristic horizontal wavelength of roughly λ ≈ 10 km. In the higher-resolution four-domain simulations, there are also smaller-scale banded and cellular structures (Figs. 16c,d), which are absent in the coarser-grid three-domain simulations (Figs. 16a,b). The finest-scale vertical motions in the four-domain simulations may interfere with the λ ≈ 10-km-scale motions in some locations, thus making the latter appear less regular. Another characteristic of both cases is the greater horizontal extent of significant vertical motions (i.e., |w| ≥ 0.5 m s−1) in the four-domain simulations (Figs. 16c,d).
Enlarged views of vertical velocity and subgrid-scale TKE (black contours with 0.5 m2 s−2 intervals starting at 0.5 m2 s−2) over the dashed rectangular insets of Figs. 14f and 14h for the case 1 CTRL simulations at 8.5 km MSL using (a) three domains with minimum horizontal grid spacing of Δx = 1 km and (b) four domains with minimum horizontal grid spacing of Δx = 333 m, respectively, and over the dashed rectangular insets of Figs. 15e and 15g for the case 2 CTRL simulations at 11.5 km MSL using (b) three domains with minimum horizontal grid spacing of Δx = 1 km and (d) four domains with minimum horizontal grid spacing of Δx = 333 m, respectively. The yellow transects labeled SW–NE and W–E in (c) and (d) indicate horizontal locations of the vertical cross sections displayed in Figs. 18a and 18b, respectively.
Citation: Monthly Weather Review 150, 10; 10.1175/MWR-D-22-0077.1
Figure 17 provides a more quantitative overall characterization of the model horizontal resolution dependence in the two cases over the physical areas of inner domains d04 (Figs. 6a,b). Included are probability density functions (PDFs) of vertical velocity in the four-domain (Δxmin = 333 m) CTRL simulations at different times (Figs. 17a,b), energy spectra of vertical velocity at the peak of turbulence activity in the CTRL simulations with different minimum horizontal grid spacing (Figs. 17c,d), and the corresponding PDFs of the SGS TKE from the PBL scheme in these simulations (Figs. 17e,f). In both cases, the PDFs of vertical velocity widen over time with peaks at 1900 UTC 25 October 2019 for case 1 and 2100 UTC 3 December 2019 for case 2.
(a),(b) Hourly evolution of probability density functions (PDFs) of vertical velocity in domain d04. (c),(d) Energy spectra of vertical velocity (Ew) for domains d02, d03, and d04, corresponding to the peak time of turbulence activity. (e),(f) PDFs of subgrid-scale (SGS) TKE from the PBL scheme across domains d02, d03, and d04. Case 1 (case 2) utilizes fields interpolated to a constant height every 500 m in the altitude range of 8.0 ≤ z ≤ 9.0 km (9.0 ≤ z ≤ 10.0 km). The vertical lines in (c) and (d) indicate the effective resolution keff for the color-coded simulations.
Citation: Monthly Weather Review 150, 10; 10.1175/MWR-D-22-0077.1
The one-dimensional energy spectra, which are derived from percentiles over azimuthal rings of constant wavenumber for the 90th percentile computed from horizontal two-dimensional spectra, show increases in resolved vertical kinetic energy as the finest grid spacings are refined from Δxmin = 3 km to 333 m. Figures 17c and 17d clearly depict how the spectrum of resolved motions is progressively expanded to higher wavenumbers that allow finer-scale waves to be explicitly resolved. The two-domain simulations with Δxmin = 3 km do not provide sufficiently fine effective resolution to start capturing motions on horizontal scales ≤13 km, as implicit diffusion from the model advection scheme strongly damps energy at these scales. The three-domain simulations with Δxmin = 1 km, while capable of starting to resolve these scales, exhibit differing behaviors in the two cases. In case 1 (Fig. 17c), the three-domain simulation (d03/3dom) has excess energy for scales larger than its effective horizontal resolution (keff ≈ 1.4 × 10−4 m−1, approximated by 1/7Δx) indicated by the turquoise vertical line. Note that this energy content exceeds that for the same scales in the four-domain simulation with Δxmin = 333 m (d04/4dom in Fig. 17c), consistent with Muñoz-Esparza et al. (2020), who reported that horizontal grid spacings of 1 km resulted in over estimates of UTLS wave energy owing to gray-zone effects (e.g., Ching et al. 2014) and PBL scheme limitations.
In contrast, the three-domain simulation (d03/3dom) in case 2 displays a slightly lower energy content around the dominant wavenumber (Fig. 17d). This corresponds to the suppression of the waves developing over northwestern Colorado (Fig. 15f), as opposed to the spurious amplification of waves in case 1 (Fig. 14f). Case 2 retains large SGS TKE over northwestern Colorado which prevents fine-scale features from fully amplifying (Fig. 15f). This result is corroborated by the PDFs of SGS TKE in Figs. 17e and 17f, which indicate considerably larger values in the d03/3dom simulation for case 2 than case 1.
The foregoing demonstrates that horizontal grid spacings in the gray-zone of UTLS wave dynamics (i.e., only permitting the onset of these structures) can lead to either amplification or suppression of vertical motions, owing to the lack of a proper parameterization for SGS effects at these grid spacings. Therefore, caution must be exercised when interpreting vertical motion amplitudes from simulations with Δxmin ≈ 1 km for these clear-air turbulence scenarios. It is worth noting that when a finer-scale Δx = 333-m nest is included and the simulations use two-way nesting, the resulting vertical velocity field on the Δx = 1 km grid (d03/4dom) more closely resembles that on the Δx = 333 m grid (d04/4dom) than that on the Δx = 1 km grid of the three-domain simulation (d03/3dom). Quantification through energy spectra (Figs. 17c,d) shows energy distributions in d03/4dom that nearly match those of d04/4dom, with only a minimal reduction of energy at high wavenumbers, which we attribute to additional truncation errors in the integration of domain d03 that uses a time step twice as large as for d04. Nevertheless, the constant feedback from the nested domain is found to counter the implicit filtering from the advection scheme in the numerical model, leading to a solution consistent with the highest-resolution domain and one that is less affected by gray-zone effects.
6. Turbulence onset mechanisms in high-resolution simulations
Vertical cross sections of potential temperature and vertical velocity from the four-domain CTRL simulations, during times at which fine-scale wavelike structures are well-developed (Figs. 16c,d), reveal considerably different vertical structure in cases 1 (Fig. 18a) and 2 (Fig. 18b). Case 1, taken along transect SW–NE in Fig. 16c, exhibits deep tropospheric waves with approximate horizontal wavelengths of λ ≈ 9 km, which undergo intermittent breaking between 7.9 and 8.5 km MSL, as evidenced by overturning isentropes in this layer (Fig. 18a). This altitude (∼26 000–28 000 ft) of wave breaking is consistent with the earlier noted predominance of observed MOG turbulence reports at lower than average commercial aviation cruising altitudes (Figs. 3a, 11a). Vertically propagating internal gravity waves, examples of which are indicated by the bold dashed lines in Fig. 18a, occupy the remainder of the troposphere above the wave breaking layer, and may be secondary waves generated by the wave breaking region, as found in simulations of mountain waves (e.g., Satomura and Sato 1999) and convective gravity waves (e.g., Lane and Sharman 2006).
Vertical cross sections of vertical velocity and potential temperature (black contours at 3 K intervals) from four-domain CTRL simulations (with Δx = 333 m) along transects (a) SW–NE in Fig. 16c for case 1 and (b) W–E in Fig. 16d for case 2. The bold dashed lines in (a) indicate phase lines of internal gravity waves referred to in the text. The bold black line in each panel indicates surface elevation.
Citation: Monthly Weather Review 150, 10; 10.1175/MWR-D-22-0077.1
In contrast, the vertical structure in the case 2 cross section (Fig. 18b), which is taken along transect WE of Fig. 16d, exhibits waves of λ ≈ 12 km that overturn in opposite directions within nearly adjacent UTLS layers. The uppermost layer of wave overturning near the tropopause excites secondary deep stratospheric internal gravity waves, which are evanescent (i.e., decaying away from their source) and are, themselves, unlikely to be associated with aviation-scale turbulence.
The structural differences between these two cases are consistent with case differences in vertical structure of the environmental flow and static stability (Figs. 19a,b). Here, the vertical profiles are constructed from horizontal averages of vertical cross sections from the same locations as those in Fig. 18, but are instead from the two-domain CTRL simulations, which cannot resolve the waves, and therefore better represent their environments.
Vertical profiles of horizontal wind (solid red curves), dry static stability (red dotted curves), and gradient Richardson number < 0.25 (gray shading) from the two-domain CTRL simulations (with Δx = 3 km) horizontally averaged along transects (a) SW–NE in Fig. 18a for case 1 and (b) W–E in Fig. 18b for case 2. Corresponding horizontally averaged vertical profiles of the Scorer parameter squared for (c) case 1 and (d) case 2. The blue vertical lines in (c) and (d) represent the square of the horizontal wavenumber of the waves shown in Figs. 18a and 18b, respectively. The dashed horizontal lines in (c) and (d) indicate the locations of critical levels where the horizontal phase speeds of waves c equal the component of the horizontal flow U along the vertical cross sections, which are the solid red curves shown in (a) and (b), respectively.
Citation: Monthly Weather Review 150, 10; 10.1175/MWR-D-22-0077.1
The most striking difference between the two cases is the more intense and much sharper UTLS jet in case 2 (Fig. 19b) compared to case 1 (Fig. 19a). The two layers of Ri < 0.25 in case 2 (Fig. 19b) occur within layers of strong vertical shear located immediately above and below the jet, which are consistent with the KH-like waves overturning in opposite directions (Fig. 18b). Note that the simulated overturning waves (Fig. 18b) correspond to ≥500-m-deep layers of Ri < 0.25 in the 1200 UTC GJT sounding (Figs. 9a,b) located at altitudes similar to those simulated (Fig. 19b).
Unlike for case 2, vertical shear large enough to result in Ri < 0.25 occurs only in a ∼0.9 km layer beneath the broader and weaker simulated UTLS jet in the case 1 vertical cross section (Fig. 19a). In case 2, small static stability of N2 ≤ 0.0001 s−2 extends a sufficient distance above the narrow simulated jet to support the upper layer of Ri < 0.25 despite weaker estimated mean vertical shear of ∼0.03 s−1 in this layer compared to ∼0.05 s−1 in the deeper layer of Ri < 0.25 beneath the jet (Fig. 19b). The layer of 0 < Ri < 0.25 beneath the jet in case 1 (Fig. 19a) coincides with the layer of overturning isentropes in the higher-resolution simulation (Fig. 18a). The longest wavelength for two-dimensional KHI in a flow with linear shear is λmax = πΔz/Ri (Scorer 1969). Using Δz = 900 m and taking Ri as 0.25 through the depth of this layer yields λmax = 11.3 km, which exceeds the λ ≈ 9 km horizontal wavelength estimated from Fig. 18a.
Estimates of the ground-relative c obtained from animations of the isentropes in the vertical cross sections of Fig. 18 using model output at 3-min frequency are c = 44 and 61 m s−1 for cases 1 and 2, respectively. The resulting vertical profiles of l2 (Figs. 19c,d) each indicate two critical levels (U = c), including one above and the other below the UTLS jet in each case (cf. Figs. 19a,b). However, these profiles also indicate several important differences between the cases that influence the structure of the waves and their vertical propagation.
The l2 profile for case 1 (Fig. 19c) has conditions suitable for vertical propagation of waves, with l2 > k2 in a layer from z = 10.3 to a critical level at 13.2 km MSL. Breaking KH waves beneath this layer (Fig. 18a) could be a source for secondary gravity waves. These internal gravity waves are expected to experience some trapping between two thin evanescent (l2 < k2) layers centered near 9.5 and 10.1 km MSL (Fig. 19c). However, the thinness of these layers is likely to permit some wave leakage to reach 10.3 km MSL, above which vertical propagation is supported. This interpretation is consistent with the nearly vertical orientation of the potential temperature troughs between 9 and 10.3 km MSL (Fig. 18a), followed by a more substantial vertical tilt (characteristic of vertically propagating waves) above, shown by the bold dashed phase lines. The amplification of the isentrope pattern in the vertical cross section near and slightly above 13 km MSL (Fig. 18a) suggests evidence of a critical level, and could represent an additional possible mechanism for turbulence onset. However, there were few aircraft reports of MOG turbulence near the 13.2 km MSL (∼43 000 ft) altitude of the critical level (Fig. 3a) to substantiate this. Elsewhere, the l2 profile (Fig. 19c) supports stronger vertical trapping and decay of downward propagating waves beneath 6.2 km MSL (Fig. 18a).
In contrast to case 1, the case 2 l2 profile (Fig. 19d) representing conditions in the corresponding vertical cross section (Fig. 18b) lacks a deep upper-tropospheric layer amenable to vertical wave propagation. Here, the lack of vertical tilt and slow decay with height of the lower-stratospheric waves, which were likely generated by KHI near the tropopause (Figs. 18b, 19b), is explained by the deep layer of l2 ≈ k2 between ∼13 and 20 km (Fig. 19d).
In sections 2 and 4 we noted spatiotemporal differences in environmental conditions from NWS soundings within individual cases, particularly for case 2 (Fig. 9), and mentioned that individual soundings could not easily explain the regional vertical distributions of MOG turbulence (Fig. 3). The simulated waves (Fig. 18b) and analysis of their supporting environmental conditions (Figs. 19b,d) are representative of region A in case 2 located in northwestern Colorado (Fig. 20a). A nearly simultaneous region of fine-scale wavelike vertical velocity features is simulated in domain 4 over eastern Colorado and is denoted by region B (Fig. 20a). Region B coincides with nearly synchronous (within 90 min) MOG turbulence reports from altitudes both above and below the UTLS jet.
(a) The 10 km MSL vertical velocity and horizontal winds (plotted as in Fig. 7) from domain d04 of the four-domain CTRL simulation (Δxmin = 333 m) with the locations and annotated altitudes of approximately synchronous observed MOG turbulence reports (from within 90 min), (b) vertical cross section taken along transect NW–SE in (a) of vertical velocity and potential temperature (3-K contour intervals), and (c) vertical profile of along-cross-section wind component (solid red), dry static stability (dotted red), and gradient Richardson number < 0.25 (shaded) averaged along the cross-section location in (b) but for the two-domain simulation (Δxmin = 3 km) at 1930 UTC 3 Dec 2019 (forecast hour t = 19.5 h).
Citation: Monthly Weather Review 150, 10; 10.1175/MWR-D-22-0077.1
Vertical cross section NW–SE (Fig. 20b) taken from within region B (Fig. 20a) reveals overturning isentropes between 8.4 km (∼26 000 ft) and 9 km (∼29 000 ft) MSL that coincide vertically with a simulated ≈600-m-deep layer of environmental Ri < 0.25 in domain 2 (Fig. 20c). These waves have a somewhat shorter horizontal wavelength of λ ≈ 7 km than those found in the previous vertical cross sections (Figs. 18a,b). The longest wavelength for two-dimensional KHI is λmax = πΔz/Ri = 7.5 km, using Δz = 600 m and taking Ri as 0.25 through the depth of this layer, which is close to the λ ≈ 7 km estimated horizontal wavelength from Fig. 20b. Simulated vertically propagating gravity waves occur above the breaking KH waves and the isentropes are highly amplified (Fig. 20b), though not yet breaking, in a vertical shear layer (Fig. 20c) from ∼11.2 to 11.6 km MSL (∼36 500–38 000 ft). Together, the KH waves and the amplified gravity waves located a few km above provide a possible explanation for the altitudes of the observed turbulence over eastern Colorado in region B (Fig. 20a). Though somewhat weaker, the structure of the vertical motions in Fig. 20b bears greater resemblance to those in case 1 (Fig. 18a) than those from region A in case 2 (Fig. 18b). Both case 1 (Fig. 19a) and region B of case 2 (Fig. 20c) have deeper UTLS jets than region A of case 2 (Fig. 19b), and possess stronger static stability immediately above the UTLS jet maximum, which hinders the development of KHI.
7. Forecast implications
Given the small horizontal grid spacings identified in the previous sections as being necessary for resolving possible turbulence onset mechanisms for the two discussed cases, it is of interest to investigate the veracity of operational turbulence forecasts for aviation, which use larger horizontal grid spacings. These operational turbulence forecasts are typically based on output from operational NWP models to infer turbulence potential. Over the United States, the highest-resolution operational NWP model at this writing is NOAA’s HRRR system (Dowell et al. 2022; James et al. 2022). The HRRR has a horizontal grid spacing of 3 km, 50 vertical levels, a model top at 20 hPa (∼25 km MSL), is convection-permitting, is updated hourly, and provides 18- or 48-h forecasts depending on the initialization time.
One aviation turbulence forecast model that is available operationally and uses HRRR model output is the Graphical Turbulence Guidance (GTG) (http://aviationweather.gov/adds). The technique and its verification statistics are provided in Sharman et al. (2006), Sharman and Pearson (2017), Muñoz-Esparza and Sharman (2018), and Kim et al. (2018). In the GTG system several turbulence indices or diagnostics are computed, with each examining the NWP model output in slightly different ways, and the final forecast is developed as the ensemble mean of the diagnostics used. Alternatively, or in addition, a probabilistic forecast can be obtained as the percentage agreement of diagnostics that exceed a certain threshold (e.g., Kim et al. 2018). The turbulence diagnostics are related to strong spatial gradients derived from the NWP model-resolved synoptic or mesoscale atmospheric scales, and assumes a downscale energy cascade from these large-scale disturbances to small-scale eddies (e.g., Cho and Lindborg 2001; Tung and Orlando 2003) that are responsible for aircraft-scale turbulence. The set of turbulence diagnostics used is developed by statistical comparisons to PIREPs and in situ EDR data to identify the best performing set overall. The final deterministic GTG contains gridded forecasts of EDR, on the same domain as the input NWP grid. The output is provided at 1000-ft (∼300-m) vertical intervals between the surface and flight level (FL) 450 (45 000 ft). Probabilistic forecasts can also be provided on this domain.
Figures 21 and 22 summarize GTG output for the two cases. In case 1 (Fig. 21), at FL270 (∼344 hPa) and FL360 (∼227 hPa), both the deterministic GTG ensemble mean EDR and the probabilistic [P(EDR ≥ 0.3) = fraction of indices agreeing that the EDR is at least 0.3 m2/3 s−1] compare well with the in situ and PIREP observations, and are consistent with the patterns obtained from the WRF simulations (Figs. 7a and 10a,c). Similarly, for case 2 (Fig. 22) at FL270 (∼344 hPa) and FL360 (∼227 hPa), there is good overall agreement with the observations and with the results of the WRF simulations (Figs. 7b and 10b,d). The forecasted maximum EDR is 0.48 m2/3 s−1 with a P(EDR ≥ 0.3) of ∼0.9 at FL270 and similar values occur at FL360. Consistent with observations in both cases, the forecasted elevated turbulence regions are concentrated in narrow bands, with little forecasted outside these bands.
(a),(b) HRRR/GTG deterministic mean (EDR; m2/3 s−1) and (c),(d) probabilistic [P(EDR > 0.3 m2/3 s−1)] 6-h forecasts valid at 1800 UTC 25 Oct 2019 computed from an ensemble of 11 turbulence diagnostics at (a),(c) FL270 (∼344 hPa) and at (b),(d) FL360 (∼227 hPa). Thin black lines are contours of geopotential height on the pressure surface of the flight level (FL), and the gray shadings are HRRR MREF values with a 10-dBZ contour interval. Colored circles are turbulence observations within ±1 h of valid time and within 2000 ft of flight level, which follow the color convention of Fig. 1, but also include “smooth” (EDR < 0.05 m2/3 s−1) observations (small magenta dots).
Citation: Monthly Weather Review 150, 10; 10.1175/MWR-D-22-0077.1
As in Fig. 21, but for 6-h forecasts valid at 2000 UTC 3 Dec 2019.
Citation: Monthly Weather Review 150, 10; 10.1175/MWR-D-22-0077.1
Like for WRF, these results are a manifestation of the HRRR’s ability to accurately represent the convection in these two cases, which is evident from comparing the coverage of MREF ≥ 10 dBZ in the HRRR (gray shadings in Figs. 21 and 22) with the available observations (Figs. 2a,c). The UTLS outflows resulting from the simulated convection enhance the large scale synoptically driven vertical shear to values expected to lead to turbulence, even though possible direct turbulence onset mechanisms (e.g., KHI, internal gravity wave breaking) are not resolved. These results should correspond to other cases provided that the HRRR or other NWP models driving the turbulence forecast system can correctly locate the timing and position of the deep convection (even when it is parameterized). In this situation the enhanced vertical shear that accompanies the convection can allow operational turbulence forecast systems such as GTG to correctly locate mesoscale regions of enhanced turbulence.
8. Summary
Environmental conditions and possible onset mechanisms for two cases of widespread moderate-to-severe clear-air turbulence outbreaks were examined using observations and high-resolution NWP simulations. The turbulence regions inferred from the simulations were well supported by turbulence observations from commercial aircraft. We illustrated that even though such events can occur in distinctly different synoptic patterns, a common environmental factor was a convectively enhanced UTLS synoptic jet that significantly influenced turbulence from hundreds to more than 1000 km downstream from active deep convection.
Favorable conditions for KHI occurred on the vertical flanks of the convectively enhanced UTLS jet in full-physics simulations, in contrast to simulations in which cloud microphysics and cumulus parameterizations were deactivated. Beneath the UTLS jet maximum, gradient Richardson number Ri < 0.25 supporting KHI is dominated by strong vertical shear. Above the UTLS jet maximum, Ri < 0.25 results from a combination of increasing vertical shear and small static stability (often minimized near the jet maximum) extending upward beneath the tropopause. Vertically propagating internal gravity waves often occupied large depths of the troposphere as well, and reached a critical level in the lower stratosphere above the jet, thereby providing an additional possible mechanism for turbulence from gravity wave breaking located up to several kilometers above the wave breaking beneath the jet. The occurrence of these different turbulence onset mechanisms and their simulated altitudes were strongly influenced by the intensity and vertical sharpness of the convectively enhanced UTLS jet.
Fine-scale vertical motions associated with these possible turbulence onset mechanisms of KHI and internal gravity wave breaking were present in similar horizontal locations in simulations with minimum horizontal grid spacings of Δxmin = 1 and 0.333 km. However, energy spectra revealed the amplitudes of the wavelike vertical motions to be unreliable in the Δxmin = 1 km simulations, containing too much energy for scales larger than the model effective resolution in one case (case 1) and too little in the other case (case 2). This shortcoming of the Δxmin = 1 km simulations is conjectured to result from a combination of gray-zone effects at this resolution and limitations of SGS PBL parameterizations that also influence vertical mixing in the free troposphere. Here, the effective resolution is similar to the horizontal wavelength of the banded vertical motion features. Nevertheless, such simulations could inform forecasters on possible locations of turbulence and likely mechanisms of onset. At Δxmin = 0.333 km banded vertical motion features with horizontal wavelengths of λ ≈ 7–12 km are well resolved and are spatially well correlated with observed MOG turbulence reports.
The fine-scale vertical motions were entirely unresolved in coarser simulations with Δxmin = 9 and 3 km, which are representative of high-resolution operational NWP models currently in use. However, these simulations did produce mesoscale regions of subgrid-scale TKE that coincided with the location of both fine-scale vertical motions in the Δxmin = 1 and 0.333 km simulations, and observed reports of moderate-or-greater turbulence. This is attributed to the intensity and broad horizontal scale of the simulated UTLS convective outflows, which can strongly enhance vertical shear layers and influence static stability in the model, resulting in production of SGS TKE.
Even though current turbulence forecast systems use relatively coarse NWP model input, the Graphical Turbulence Guidance (GTG) turbulence forecast system (Sharman et al. 2006; Sharman and Pearson 2017), which uses output from the HRRR operational NWP model, demonstrated success in capturing regions of moderate-or-greater turbulence in both examined cases. It is anticipated that similar success would likely occur in other cases where the parent NWP model used to drive turbulence forecast systems accurately represents organized deep convection and their UTLS outflows.
Acknowledgments.
The authors thank David Ahijevych (NCAR) for providing Python scripts to plot satellite data in Fig. 11a, and Richard Rotunno (NCAR) for his helpful internal review of the paper. The comments and suggestions of three anonymous reviewers are also appreciated. This research is in response to requirements and funding by the Federal Aviation Administration (FAA). The views expressed are those of the authors and do not necessarily represent the official policy or position of the FAA. The National Center for Atmospheric Research is sponsored by the National Science Foundation.
Data availability statement.
All simulations reported in this study will be made available from the corresponding author upon request. The NWS radiosonde data were obtained from the University of Wyoming (http://weather.uwyo.edu/upperair/). The NOAA GOES satellite data examined in this study were accessed from https://registry.opendata.aws/noaa-goes/. The PIREPs used here can be obtained from NOAA’s Family of Services (https://weather.gov/noaaport/), and the in situ EDR data are available through NOAA’s MADIS data at https://madis-data.noaa.gov/madisPublic1/data/archive.
REFERENCES
Asai, T., 1970: Stability of a plane parallel flow with variable vertical shear and unstable stratification. J. Meteor. Soc. Japan, 48, 129–139, https://doi.org/10.2151/jmsj1965.48.2_129.
Asai, T., 1972: Thermal instability of a shear flow turning the direction with height. J. Meteor. Soc. Japan, 50, 525–532, https://doi.org/10.2151/jmsj1965.50.6_525.
Barber, K. A., G. L. Mullendore, and M. J. Alexander, 2018: Out-of-cloud convective turbulence: Estimation method and impacts of model resolution. J. Appl. Meteor. Climatol., 57, 121–136, https://doi.org/10.1175/JAMC-D-17-0174.1.
Benjamin, S. G., and Coauthors, 2016: A North American hourly assimilation and model forecast cycle: The Rapid Refresh. Mon. Wea. Rev., 144, 1669–1694, https://doi.org/10.1175/MWR-D-15-0242.1.
Ching, J., R. Rotunno, M. LeMone, A. Martilli, B. Kosovic, P. A. Jimenez, and J. Dudhia, 2014: Convectively induced secondary circulations in fine-grid mesoscale numerical weather prediction models. Mon. Wea. Rev., 142, 3284–3302, https://doi.org/10.1175/MWR-D-13-00318.1.
Cho, J. Y. N., and E. Lindborg, 2001: Horizontal velocity structure functions in the upper troposphere and lower stratosphere 1. Observations. J. Geophys. Res., 106, 10 223–10 232, https://doi.org/10.1029/2000JD900814.
Dörnbrack, A., T. Gerz, and U. Schumann, 1995: Turbulence breaking of overturning gravity waves below a critical level. Appl. Sci. Res., 54, 163–176, https://doi.org/10.1007/BF00849114.
Dowell, D. C., and Coauthors, 2022: The High-Resolution Rapid Refresh: An hourly updating convection-allowing forecast model. Part I: Motivation and system description. Wea. Forecasting, 37, 1371–1395, https://doi.org/10.1175/WAF-D-21-0151.1.
Durran, D. R., and J. B. Klemp, 1982: On the effects of moisture on the Brunt–Väisälä frequency. J. Atmos. Sci., 39, 2152–2158, https://doi.org/10.1175/1520-0469(1982)039<2152:OTEOMO>2.0.CO;2.
Dutton, J., and H. A. Panofsky, 1970: Clear air turbulence: A mystery may be unfolding. Science, 167, 937–944, https://doi.org/10.1126/science.167.3920.937.
Ek, M. B., K. E. Mitchell, Y. Lin, E. Rogers, P. Grummann, V. Koren, G. Gayno, and J. D. Tarpley, 2003: Implementation of Noah land-surface model advances in the NCEP operational mesoscale Eta model. J. Geophys. Res., 108, 8851, https://doi.org/10.1029/2002JD003296.
Ellrod, G. P., J. A. Knox, P. F. Lester, and L. J. Ehernberger, 2015: Clear air turbulence. Encyclopedia of Atmospheric Science, 2nd ed. G. R. North, J. Pyle, and F. Zhang, Eds., Academic Press, 177–186.
Iacono, M. J., J. S. Delamere, E. J. Mlawer, M. W. Shephard, S. A. Clough, and W. D. Collins, 2008: Radiative forcing by long-lived greenhouse gases: Calculations with the AER radiative transfer models. J. Geophys. Res., 113, D13103, https://doi.org/10.1029/2008JD009944.
Ito, J., H. Niino, M. Nakanishi, and C. H. Moeng, 2015: An extension of the Mellor–Yamada model to the terra incognita zone for dry convective mixed layers in the free convection regime. Bound.-Layer Meteor., 157, 23–43, https://doi.org/10.1007/s10546-015-0045-5.
James, E. P., and Coauthors, 2022: The High-Resolution Rapid Refresh (HRRR): An hourly updating convection-allowing forecast model. Part II: Forecast performance. Wea. Forecasting, 37, 1397–1417, https://doi.org/10.1175/WAF-D-21-0130.1.
Janjić, Z. I., 1994: The step-mountain eta coordinate model: Further development of the convection, viscous sublayer, and turbulence closure schemes. Mon. Wea. Rev., 122, 927–945, https://doi.org/10.1175/1520-0493(1994)122<0927:TSMECM>2.0.CO;2.
Janjić, Z. I., 2001: Nonsingular implementation of the Mellor-Yamada level 2.5 scheme in the NCEP Meso Model. NCEP Office Note 437, NCEP, 61 pp.
Kawashima, M., 2021: A numerical study of cirrus bands and low static stability layers associated with tropical cyclone outflow. J. Atmos. Sci., 78, 3691–3716, https://doi.org/10.1175/JAS-D-21-0047.1.
Kim, J.-H., and H.-Y. Chun, 2012: A numerical investigation of convectively induced turbulence above deep convection. J. Appl. Meteor. Climatol., 51, 1180–1200, https://doi.org/10.1175/JAMC-D-11-0140.1.
Kim, J.-H., H.-Y. Chun, R. D. Sharman, and S. B. Trier, 2014: The role of vertical shear on aviation turbulence within cirrus bands of a simulated western Pacific cyclone. Mon. Wea. Rev., 142, 2794–2813, https://doi.org/10.1175/MWR-D-14-00008.1.
Kim, J.-H., R. Sharman, M. Strahan, J. W. Scheck, C. Bartholomew, J. C. H. Cheung, P. Buchanan, and N. Gait, 2018: Improvements in non-convective aviation turbulence prediction for the World Area Forecast System (WAFS). Bull. Amer. Meteor. Soc., 99, 2295–2311, https://doi.org/10.1175/BAMS-D-17-0117.1.
Klemp, J. B., J. Dudhia, and A. D. Hassiotis, 2008: An upper gravity-wave absorbing layer for NWP applications. Mon. Wea. Rev., 136, 3987–4004, https://doi.org/10.1175/2008MWR2596.1.
Klostermeyer, J., and R. Rüster, 1980: Radar observation and model computation of jet stream-generated Kelvin–Helmholtz instability. J. Geophys. Res., 85, 2841–2846, https://doi.org/10.1029/JC085iC05p02841.
Knox, J. A., 1997: Possible mechanisms of clear-air turbulence in strongly anticyclonic flows. Mon. Wea. Rev., 125, 1251–1259, https://doi.org/10.1175/1520-0493(1997)125<1251:PMOCAT>2.0.CO;2.
Knox, J. A., A. S. Bachmeier, W. M. Carter, J. E. Tarantino, L. C. Paulik, E. N. Wilson, G. S. Bechdol, and M. J. Mays, 2010: Transverse cirrus bands in weather systems: A grand tour of an enduring enigma. Weather, 65, 35–41, https://doi.org/10.1002/wea.417.
Knox, J. A., A. W. Black, J. A. Rackley, E. N. Wilson, J. S. Grant, S. P. Phelps, D. S. Nevius, and C. B. Dunn, 2016: Automated turbulence forecasting strategies. Aviation Turbulence: Processes, Detection, Prediction, R. Sharman and T. Lane, Eds., Springer, 243–260.
Lane, T. P., and J. C. Knievel, 2005: Some effects of model resolution on simulated gravity waves generated by deep, mesoscale convection. J. Atmos. Sci., 62, 3408–3419, https://doi.org/10.1175/JAS3513.1.
Lane, T. P., and R. D. Sharman, 2006: Gravity wave breaking, secondary wave generation, and mixing above deep convection in a three-dimensional cloud model. Geophys. Res. Lett., 33, L23813, https://doi.org/10.1029/2006GL027988.
Lane, T. P., J. D. Doyle, R. Plougonven, M. A. Shapiro, and R. D. Sharman, 2004: Observations and numerical simulations of inertia-gravity waves and shearing instabilities in the vicinity of a jet stream. J. Atmos. Sci., 61, 2692–2706, https://doi.org/10.1175/JAS3305.1.
Lane, T. P., R. D. Sharman, S. B. Trier, R. G. Fovell, and J. K. Williams, 2012: Recent advances in the understanding of near-cloud turbulence. Bull. Amer. Meteor. Soc., 93, 499–515, https://doi.org/10.1175/BAMS-D-11-00062.1.
Lenz, A., K. M. Bedka, W. F. Feltz, and S. A. Ackerman, 2009: Convectively induced transverse band signatures in satellite imagery. Wea. Forecasting, 24, 1362–1373, https://doi.org/10.1175/2009WAF2222285.1.
Ludlam, F. H., 1967: Characteristics of billow clouds and their relation to clear-air turbulence. Quart. J. Roy. Meteor. Soc., 93, 419–435, https://doi.org/10.1002/qj.49709339803.
Mlawer, E. J., S. J. Taubman, P. D. Brown, M. J. Iacono, and S. A. Clough, 1997: Radiative transfer for inhomogeneous atmospheres: RRTM, a validated correlated-k model for the longwave. J. Geophys. Res., 102, 16 663–16 682, https://doi.org/10.1029/97JD00237.
Muñoz-Esparza, D., and R. D. Sharman, 2018: An improved algorithm for low-level turbulence forecasting. J. Appl. Meteor. Climatol., 57, 1249–1263, https://doi.org/10.1175/JAMC-D-17-0337.1.
Muñoz-Esparza, D., R. D. Sharman, and S. B. Trier, 2020: On the consequences of PBL scheme diffusion on UTLS wave and turbulence representation in high-resolution NWP models. Mon. Wea. Rev., 148, 4247–4265, https://doi.org/10.1175/MWR-D-20-0102.1.
Nakanishi, M., and H. Niino, 2004: An improved Mellor-Yamada level 3 model with condensation physics. Bound.-Layer Meteor., 112, 1–31, https://doi.org/10.1023/B:BOUN.0000020164.04146.98.
Nappo, C. J., 2002: An Introduction to Atmospheric Gravity Waves. Academic Press, 276 pp.
Olson, J. B., and Coauthors, 2019: Improving wind energy forecasting through numerical weather prediction model development. Bull. Amer. Meteor. Soc., 100, 2201–2220, https://doi.org/10.1175/BAMS-D-18-0040.1.
Plougonven, R., and F. Zhang, 2016: Gravity waves generated by jets and fronts and their relevance for clear-air turbulence. Aviation Turbulence: Processes, Detection, Prediction, R. Sharman and T. Lane, Eds., Springer, 385–406.
Rowe, S. M., and M. H. Hitchman, 2015: On the role of inertial instability on stratosphere–troposphere exchange near midlatitude cyclones. J. Atmos. Sci., 72, 2131–2151, https://doi.org/10.1175/JAS-D-14-0210.1.
Satomura, S., and K. Sato, 1999: Secondary generation of gravity waves associated with the breaking of mountain waves. J. Atmos. Sci., 56, 3847–3858, https://doi.org/10.1175/1520-0469(1999)056<3847:SGOGWA>2.0.CO;2.
Scorer, R. S., 1969: Billow mechanics. Radio Sci., 4, 1299–1308, https://doi.org/10.1029/RS004i012p01299.
Sharman, R. D., and J. Pearson, 2017: Prediction of energy dissipation rates for aviation turbulence. Part I: Forecasting nonconvective turbulence. J. Appl. Meteor. Climatol., 56, 317–337, https://doi.org/10.1175/JAMC-D-16-0205.1.
Sharman, R. D., and S. B. Trier, 2019: Influences of gravity waves on Convectively-Induced Turbulence (CIT): A review. Pure Appl. Geophys., 176, 1923–1958, https://doi.org/10.1007/s00024-018-1849-2.
Sharman, R. D., C. Tebaldi, G. Wiener, and J. Wolff, 2006: An integrated approach to mid- and upper-level turbulence forecasting. Wea. Forecasting, 21, 268–287, https://doi.org/10.1175/WAF924.1.
Sharman, R. D., S. B. Trier, T. P. Lane, and J. D. Doyle, 2012: Sources and dynamics of turbulence in the upper troposphere and lower stratosphere: A review. Geophys. Res. Lett., 39, L12803, https://doi.org/10.1029/2012GL051996.
Sharman, R. D., L. B. Cornman, G. Meymaris, J. Pearson, and T. Farrar, 2014: Description and derived climatologies of automated in situ eddy-dissipation-rate reports of atmospheric turbulence. J. Appl. Meteor. Climatol., 53, 1416–1432, https://doi.org/10.1175/JAMC-D-13-0329.1.
Skamarock, W. C., and J. B. Klemp, 2008: A time-split nonhydrostatic atmospheric model for weather research and forecasting applications. J. Comput. Phys., 227, 3465–3485, https://doi.org/10.1016/j.jcp.2007.01.037.
Thompson, C. F., and D. M. Schultz, 2021: The release of inertial instability near an idealized zonal jet. Geophys. Res. Lett., 48, e2021GL092649, https://doi.org/10.1029/2021GL092649.
Thompson, G., P. R. Field, R. M. Rasmussen, and W. D. Hall, 2008: Explicit forecasts of winter precipitation using an improved bulk microphysics scheme. Part II: Implementation of a new snow parameterization. Mon. Wea. Rev., 136, 5095–5115, https://doi.org/10.1175/2008MWR2387.1.
Tiedtke, M., 1989: A comprehensive mass flux scheme for cumulus parameterization in large-scale models. Mon. Wea. Rev., 117, 1779–1800, https://doi.org/10.1175/1520-0493(1989)117<1779:ACMFSF>2.0.CO;2.
Trier, S. B., and R. D. Sharman, 2016: Mechanisms influencing cirrus banding and aviation turbulence near a convectively enhanced upper-level jet stream. Mon. Wea. Rev., 144, 3003–3027, https://doi.org/10.1175/MWR-D-16-0094.1.
Trier, S. B., and R. D. Sharman, 2018: Trapped gravity waves and their association with turbulence in a large thunderstorm anvil during PECAN. Mon. Wea. Rev., 146, 3031–3052, https://doi.org/10.1175/MWR-D-18-0152.1.
Trier, S. B., R. D. Sharman, R. G. Fovell, and R. G. Frehlich, 2010: Numerical simulation of radial cloud bands within the upper-level outflow of an observed mesoscale convective system. J. Atmos. Sci., 67, 2990–2999, https://doi.org/10.1175/2010JAS3531.1.
Trier, S. B., R. D. Sharman, and T. P. Lane, 2012: Influences of moist convection on a cold-season outbreak of Clear-Air Turbulence (CAT). Mon. Wea. Rev., 140, 2477–2496, https://doi.org/10.1175/MWR-D-11-00353.1.
Trier, S. B., R. D. Sharman, D. Muñoz-Esparza, and T. P. Lane, 2020: Environment and mechanisms of severe turbulence in a midlatitude cyclone. J. Atmos. Sci., 77, 3869–3889, https://doi.org/10.1175/JAS-D-20-0095.1.
Tung, K. K., and W. W. Orlando, 2003: The k3 and k5/3 energy spectrum of atmospheric turbulence: Quasigeostrophic two-level model simulation. J. Atmos. Sci., 60, 824–835, https://doi.org/10.1175/1520-0469(2003)060<0824:TKAKES>2.0.CO;2.
Yamazaki, K., and H. Miura, 2021: On the formation mechanism of cirrus banding: Radiosonde observations, numerical simulations, and stability analyses. J. Atmos. Sci., 78, 3477–3502, https://doi.org/10.1175/JAS-D-20-0356.1.
Zhang, F., 2004: Generation of mesoscale gravity waves in upper-tropospheric jet-front systems. J. Atmos. Sci., 61, 440–457, https://doi.org/10.1175/1520-0469(2004)061<0440:GOMGWI>2.0.CO;2.
Zovko-Rajak, D., and T. P. Lane, 2014: The generation of near-cloud turbulence in idealized simulations. J. Atmos. Sci., 71, 2430–2451, https://doi.org/10.1175/JAS-D-13-0346.1.
Zovko-Rajak, D., T. P. Lane, R. D. Sharman, and S. B. Trier, 2019: The role of gravity wave breaking in a case of upper-level near-cloud turbulence. Mon. Wea. Rev., 147, 4567–4588, https://doi.org/10.1175/MWR-D-18-0445.1.