1. Introduction
Our study analyzed atmospheric bores and other convergent boundaries observed from 15 May to 25 June 2002 within a 1600–0600 LST (2200–1200 UTC) time window using data collected during the International H20 Project (IHOP_2002) over the southern Great Plains of the United States (Weckwerth et al. 2004). A majority of the thunderstorms within this region occur at night (Wallace 1975; Carbone et al. 2002), a subset of which is organized convective systems that provide a significant portion of the region’s rainfall (Fritsch et al. 1986) and the region’s extreme rainfall events (Schumacher and Johnson 2006). IHOP_2002 data are ideal for studying bores in the nocturnal environment because the measurements include a dense Oklahoma Mesonet surface network, special soundings launched from five sites every 3 h, and the National Center for Atmospheric Research (NCAR) Multiple Antenna Profiler (MAPR). The main objective of our study is to use IHOP_2002 data to explain the frequency and behavior of the bores with the characteristics of their environment.
Common progenitors of atmospheric bores are synoptic-scale cold fronts and convective outflows. Tepper (1950), in one of the first studies to document a cold front generating an atmospheric bore, used the term “pressure jump line” to describe the mechanism driving a squall line. Such pressure jump lines are now recognized to be atmospheric bores (Smith 1988; Rottman and Simpson 1989, hereafter RS89; Houze 2004; Knupp 2006, hereafter K06; Coleman and Knupp 2011, hereafter CK11). Over the southern Great Plains, atmospheric bores are also generated from convective outflows (e.g., Wilson and Roberts 2006, hereafter WR06; Whiteman et al. 2006; K06; Koch et al. 2008a,b, hereafter K08a, K08b, respectively; Tanamachi et al. 2008; Marsham et al. 2011).
Historically, case studies identify fine lines in radar reflectivity as provisional bores whose dynamic and thermodynamic structure is then studied with surface observations and prebore environmental soundings (e.g., Locatelli et al. 1998; K06; K08a,b; Tanamachi et al. 2008; CK11; Marsham et al. 2011). These case studies of bores generally found good agreement with predictions from hydraulic and linear wave theory. However, it is difficult to draw generalizations from case studies and such an approach does not provide information on the frequency of such events. Our study moves beyond the individual case study approach and systematically examines the relationship between multiple convectively generated outflows, bores, and their nocturnal environment. When appropriate, we employ statistical models to make generalizations.
As in WR06, we analyze bores that are evident in radar fine lines (RFLs) and surface wind data from IHOP_2002. Our studies use the data with a differing purpose. WR06 uses surface winds to discriminate elevated from surface-based boundaries that initiate convection, while we incorporate surface thermodynamic meteorological measurements to describe evolutions unobservable by radar and wind measurements alone. Our method assisted the identification of a nonundular bore. This term refers to bores that display a prominent single rise in pressure and one wind shift with little indication of a secondary fluctuation. By our definition, this also includes rarefaction waves (White and Helfrich 2012).
Using the methods described in section 3, we characterized 152 RFLs in IHOP_2002 data. In addition, our study related the generation and evolution of the bores to their nocturnal environment by applying hydraulic theory to density currents and linear theory to waves that evolve from the bores. Our focus on bores is of practical relevance to understanding how organized nocturnal convection evolves, including how bores can initiate (e.g., WR06) and maintain (e.g., Parker 2008; French and Parker 2010; Blake et al. 2017) convection. Our insight into the frequency of bores provides a building block for future investigations into the relationship between bores and nocturnal convective systems.
2. Theory applied to atmospheric bores
The pair of Fr and H indicates one of four flow regimes. When the flow is supercritical (regime a in Fig. 1) or subcritical (regime d in Fig. 1), a density current will not generate a bore. When the flow is partially or completely blocked (regime b or c in Fig. 1), a semipermanent deepening of the inversion layer occurs ahead of the density current, hereafter called a bore; in water, this is called a hydraulic jump. The bore attains a height determined by Fr and H4 and, without considering the loss of energy due to wave radiation or turbulence, continues to expand horizontally for as long as the environmental flow remains partially or completely blocked (Carbone et al. 1990; Wakimoto and Kingsmill 1995; Koch and Clark 1999). We will utilize hydraulic theory to determine which flow regimes are likely observed on any night during IHOP_2002.
The flow regimes in a two-layer flow (approximated to a one-layer flow containing an upper layer of infinite depth) over a streamlined obstacle. Diagram lifted from RS89 as adapted from Baines and Davies (1980).
Citation: Monthly Weather Review 145, 10; 10.1175/MWR-D-16-0415.1
While hydraulic theory has been used to explain when the inversion fluid becomes blocked (partially or completely) and the depth of the bore response (K06; K08a,b), hydraulic theory does not account for how a bore interacts with surroundings that contain varying shear and stratification. RS89 avoids this complication by restricting stratification to an infinitely thin layer between two neutral, quiescent fluids, which does not allow vertical wave propagation. In the atmosphere, there are large layers of varying stratification imbedded within varying profiles of shear, allowing (or restricting) the vertically propagation of waves. Linear theory describes a wave within this medium as an internal buoyancy wave (B95). In the case of an atmospheric bore, the bore consists of a wave packet—a spectrum of gravity waves each characterized by a horizontal and vertical wavelength. Linear wave theory is appropriate to gauge when a linear wave mode5 becomes trapped within a horizontal wave duct and which modes are untrapped and propagate vertically away from the disturbance. Untrapped waves can quickly diminish in amplitude with time from imperfect reflection within a duct (Scorer 1949; Lindzen and Tung 1976, hereafter LT76; Lindzen and Rosenthal 1976).
We shall assume that the atmosphere is well represented by a two layer system partitioned into a layer characterized by a constant positive Scorer parameter 
Large changes in m2 have been observed in the presence of nocturnal low-level jets (NLLJs) (K91). In atmospheric flows, changes in the sign of m2 have been connected to the sign change in the curvature, described by Koch and Clark (1999), K08b, Marsham et al. (2011), and CK11. This study will examine the vertical wavenumber in environments with a NLLJ and discuss how its curvature plays a role in trapping wave modes.
3. Data analysis
The observations utilized for this study were obtained during the IHOP_2002 experiment (Weckwerth et al. 2004) conducted over the Great Plains (Fig. 2) and include the following:
A radar mosaic (available online at http://catalog.eol.ucar.edu/ihop/) constructed at 15-min intervals from the operational S-band WSR-88D network (Hardy and Gage 1990) and the National Center for Atmospheric Research’s S-Pol Ka radar (Keeler et al. 2000);
The Automated Surface Observing System (ASOS) stations providing T, RH, p, wind speed, and direction every 1 min (ASOS Program Office Staff 1998), and the Oklahoma Mesonet stations providing T, Td, p, 10-m wind speed, and direction every 5 min (Brock et al. 1995);
Radiosonde soundings launched by the Atmospheric Radiation Measurement (ARM) Program (Stokes and Schwartz 1994) every 3 h from 26 May to 25 June from the ARM central facility (Lamont, Oklahoma) and from 26 May to 15 June at four auxiliary sites (Hillsboro, Kansas; Morris, Oklahoma; Purcell, Oklahoma; and Vici, Oklahoma), with high vertical fidelity (0.5 Hz); and
NCAR’s 915-MHz, 33-cm MAPR (Cohn et al. 2001), which provided 5-min-averaged observations of radar backscatter, and horizontal and vertical winds at 60-m vertical resolution up to 5 km AGL.
Map of the IHOP_2002 experimental domain showing the lat–lon extent, state boundaries, and key observation sites. The measurement facilities utilized in this study are the WSR-88D network, ARM special sounding sites, NWS ASOS, the S-POL radar at Homestead accompanied by the MAPR profiler, and the Oklahoma Mesonet, color coded in the legend. Blue lines indicate rivers.
Citation: Monthly Weather Review 145, 10; 10.1175/MWR-D-16-0415.1
An example of the RFL marking method for 27 May 2002: (a) a composite radar image, the yellow arrow indicates the RFL of interest; (b) the composite radar image superimposed over a political map of the IHOP_2002 domain, with the transparency increased (the red arch marks the location of the RFL of interest); and (c) the RFL map for 27 May 2002 after all of the RFLs for the night have been analyzed and marked on the map. The color couplets (blue/red; green/black; orange/purple) indicate when fine lines become multiple fine lines and different color couplets are used to assist the eye of the reader when distinguishing overlapping RFLs.
Citation: Monthly Weather Review 145, 10; 10.1175/MWR-D-16-0415.1
Next, we incorporate surface observations to categorize the RFLs as either undular bores, nonundular bores, density currents, heat bursts, retrogressing drylines, cold fronts, or stationary fronts. The categorization system is shown in Table 1. Convectively generated boundaries, aside from heat bursts, are categorized according to their progression in a commonly observed chain of events (e.g., K06) as in Fig. 4. For clarity, we define an undular bore in surface observations as a pronounced and sustained rise in the surface pressure, a rise or no change in the temperature and wind oscillations coincident with pressure oscillations (K91), while a solitary wave has similar characteristics, yet no sustained pressure rise (Christie et al. 1979; Christie 1989). The drawback to our definitions is it opens our analysis up to overestimating the amount of bores that do not evolve into a solitary wave. For example, K08a and K08b recognize a soliton (a group of solitary waves) as an amplitude-ordered bore, an intermediary between our definition of a bore and a solitary wave. Our analysis would count this as an undular bore even though it exhibits the behavior of both an undular bore and a solitary wave. Meanwhile, the use of a finite domain implies that some solitary waves may have been counted as a bore because the evolution to a solitary wave took place outside the domain. We leave the true nature of this evolution to later studies that use numerical simulations to alleviate these observational constraints.
Definitions for characterizing observed boundaries on radar and in surface observations. SFL and MFL stand for single and multiple fine lines, respectively.
Schematic of (a) a density current in supercritical flow that transitions to blocked flow and develops (b) a nonundular bore. The nonundular bore may evolve into (c) an undular bore and eventually (d) one or more solitary waves if the nonlinear components of motion become important. This is similar to the K06 description. White and Helfrich (2012) describe variations of this evolution.
Citation: Monthly Weather Review 145, 10; 10.1175/MWR-D-16-0415.1
MAPR observations (Fig. 2) provide useful information about the vertical structure of events. First, observing the vertical displacements of scattering layers in the signal-to-noise ratio (SNR) reveals whether layers are vertically displaced for at least two hours (indicative of a bore; Carbone et al. 1990; K06). Second, MAPR provides vertical motions. When an event experiences stronger upward and weaker downward motions, we argue this implies semipermanent lifting. In most cases, the Homestead surface observations or Oklahoma Mesonet sites, in close proximity to MAPR, provides the supplementary surface data to categorize the event. For examples of bores in surface data and vertical profilers, see Tepper 1950; K08a,b; Tanamachi et al. 2008; or CK11.
Each event is given a qualitative indication of confidence. An event meeting (i) all requirements (in Table 1) is deemed “well determined,” (ii) less than all the requirements is deemed “adequately determined,” and (iii) radar image inferred requirement only is deemed “poorly determined.” The lowest confidence classification, (iv) “undetermined,” is given for an event that did not fit any categorization. Instead we descriptively name the phenomenon based on its behavior and/or appearance.
4. Overview of observed RFLs
Our results depict an environment that quite often supports a bore or solitary wave (65 of the 152 categorized boundaries, ~43%) initiated by a density current or cold front (Fig. 5, 62 density currents; 3 cold fronts). There is a reasonable amount of confidence in this result since 44 of the 65 were classified as well determined, 8 are weakly determined, and only 13 bores were poorly determined. Moreover, atmospheric bores and solitary waves made up 62 of the 98 convectively generated boundaries (~63%), a category that also included density currents and heat bursts.
A pie chart depicting the distribution of characterized RFLs during IHOP_2002. The shade of red represent atmospheric bores, purple are gravity waves, blue shades are density currents, orange are heat bursts, and green shades are frontal surface boundaries. Within the bore, density current, and dryline shades are tints to indicate well-determined (W), adequately determined (A), and poorly determined (P) events. Undetermined cases are broken into a secondary pie chart, where orange is warming events, white is no characterization, gray is a single fine line, and the purple are undular waves (refer to Table 1 for clarification on characterizations).
Citation: Monthly Weather Review 145, 10; 10.1175/MWR-D-16-0415.1
For the 57 well-determined convectively generated boundaries, 44 density currents generated a bore (~77%), and 5 density currents did not (~9%) (Fig. 6). Thus, density currents initiate a borelike response in the environment 90% of the time {
A pie chart depicting the distribution of convectively induced RFLs for well-determined cases only. Color coding is identical to Fig. 5.
Citation: Monthly Weather Review 145, 10; 10.1175/MWR-D-16-0415.1
Our finding that bores are common in the nocturnal environment is qualitatively consistent with the conclusions of WR06. An important distinction between our two systematic studies is that 16 of the 39 bores (not including solitary waves) in the well-determined category are nonundular bores. We attribute the difference to our inclusion of surface thermodynamic data. Based on their definition of bore, these nonundular bores fall outside of the WR06 classification and were potentially grouped into the gust front category. Thus, it is possible that WR06 underestimated the number of bores, which may impact how gust-front driven convection is framed.
The bores in our study were observed on 23 of the 32 (70%) days when convection occurs in the domain (Fig. 6). In total, 14 of the 32 days with bores (~61%) occurred on nights with a synoptic boundary present. Over the entire IHOP_2002 campaign, bores were present on 23 of the 42 (55%) days. During the convectively active period of 15–20 May, 24 bore events occurred, while only 2 density currents did not generate a bore {
Bar graph of characterized phenomena during IHOP_2002 by day; the color coding follows Fig. 5. Undetermined cases are not included.
Citation: Monthly Weather Review 145, 10; 10.1175/MWR-D-16-0415.1
5. Preferred times for blocked flow regime
In this section, we approximate the characteristic flow regime for the entire IHOP_2002 campaign, regardless of whether convection occurred on that day. The purpose is to gauge how often the environment is predisposed to develop a bore if perturbed with a density current. We do so by calculating the nondimensional parameters Fr and H following previous case studies (K91; Koch and Clark 1999; Kingsmill and Crook 2003; K06; K08a,b), but within a bootstrap resampled statistical model. A bootstrap is appropriate for capturing the “true” distribution, since our dataset of observed thermodynamic density current properties is relatively small (26 events). Our statistical approach moves beyond case studies that rely on local measurements at a fixed site or aircraft measurements.
We determine the values for ho, 
Our resampling technique used to construct Fr and H is as follows (Fig. 8):
The three density current properties, do, Cdc, and orientation/direction of movement,8 are resampled independently, constructing an “artificial” density current. The independent resampling removes the dependence on a small sample of density currents. An unavoidable consequence is the increase in the spread of our final distribution.
The three environmental conditions, ho, Δθ, and Uinv, are resampled dependently. A dependent resampling is appropriate, since our pool of soundings is likely representative of the environmental variability during the campaign.
Flowchart illustrating the bootstrap resampling process utilized for estimating the 2D density estimate of flow-regime responses (Fig. 9). (top left) Four different bootstraps of the density current and environmental conditions are performed and (top right) used to calculate the Fr and H values. (bottom left) The 100 000 resampled pairs are passed through a quality check to remove nonrealistic values.
Citation: Monthly Weather Review 145, 10; 10.1175/MWR-D-16-0415.1
Contours of the 95th, 50th, 25th, 20th, 15th, 10th, and 5th percentile of the densest points in the 2D density estimate as a function of time. Color fills are the magnitude of the 2D density estimate normalized by the densest value among all four panels (analogous to a measure for the likelihood of observing a regime relative to the most likely regime at any point during the night), dashed lines represent bore strength: (top left) 2100, (top right) 0000, (bottom left) 0300, and (bottom right) 0600 LST. Modeled after RS89.
Citation: Monthly Weather Review 145, 10; 10.1175/MWR-D-16-0415.1
The most likely response for every resampled time lies in the partially blocked flow (Fig. 9). Our application implies that convective outflows in the nocturnal environment are often predisposed to producing bores. This result echoes our observational findings in section 4 (Fig. 5). From 0000 to 0600 LST, the distributions transition to a strong, unimodal distribution, centered at Fr of 1 and H of 1–1.5 within the partially blocked regime. The distribution contracts with time around the densest points, where the densest points fall in the partially blocking regime. As the distribution contracts with time, the bore strength settles near 2 (dashed lines, Fig. 9). According to RS89, this bore strength should produce laminar bores (i.e., nonturbulent). Unfortunately, the IHOP_2002 dataset lacked an extensive vertical profiling network well suited to examine the evolving structure of bores. Investigation of this issue is beyond the scope of the present study.
To determine what aspect of the environment may be controlling the changes in the distribution, we investigate changes to the inversion properties and wind speed in the observed environment. The average strength and the height of the inversion increase twofold from 21 to 6 LST (100–200 m; 2.65–5.50 K), while the average strength of the horizontal wind in the inversion increased by only 7% from 2100 to 0600 LST (5.06–5.41 m s−1). The inversion height is in the denominator of Fr [(1)] and H [(2)] so as the inversion depth increases, both the Fr and H diminish; specifically, H should approach or pass unity. The net effect of the deepening inversion is to move the peak of the 2D density estimate toward a partially blocked flow regime (Fig. 9). Therefore, we argue that the likelihood of generating bores increases through the night due to the strengthening and deepening inversion, where the mean flow associated with the NLLJ in the inversion layer generally play a lesser role. One caveat is that we did not account for the strengthening inversion diminishing the height and changing the shape of the density current (Liu and Moncrieff 2000) or environmental shear amplifying the leading edge of the density current (Liu and Moncrieff 1996). In that case, it is not clear if the density current depth, and thus H, increases or decreases.
We compare our bootstrap resampled distributions of hydraulic theory to the observed times of when density currents and bores appear within the radar reflectivity mosaic. For our analysis, we use the initial and final observed times of the RFLs and calculate their duration. We also assume that all of the events begin as a density current, implying that the first observation of a RFL is, in general, associated with a density current. We assume that the final observed time of a RFL is associated characterized state in section 4. Additionally, we bootstrap resample the mean initial and final times, along with the duration. We plot these alongside the observed times to assist our interpretation of statistical separation.
In Fig. 10a, the observations (black dots) of density currents that do not generate a bore overlap in time with density currents that did generate a bore (Fig. 10a). Physically, this implies that we did not observe a time threshold for bore formation. However, 74% of the density current bootstrapped means fall outside of the 1% whisker for undular bores (Fig. 10a), implying that density currents were more likely to generate a bore as the night progressed. Earlier findings and bootstraps of the mean final observed time, where 99% of the bootstrapped means for undular bores fall outside of the 99% whisker for density currents, are consistent with this result (Fig. 10b). Also, there appears to be no statistical separation between bores and solitary waves. However, we do notice that solitary waves were observed (black dots) exclusively later in the night (0700 + UTC) relative to observed bores (0300 + UTC). This result may be due to the development of more favorable environmental conditions or it may simply reflect the evolution of the bore system as described in Christie (1989).
Boxplots of the resampled means for the (a) initial observance, (b) final observance (UTC and LST), and (c) duration (h) of RFLs during IHOP_2002 according to their final observed state: solitary waves, undular bores, nonundular bores, and density currents. Black dots represent observed data; black dotted line represents the value associated with the first percentile whisker for undular bores.
Citation: Monthly Weather Review 145, 10; 10.1175/MWR-D-16-0415.1
We can say that bores tended to last 2 h longer than density currents on average (Fig. 10c) with a high amount of certainty (74% of the undular bore resampled means do not overlap with density currents). We hypothesize that the bores’ lifetime is at least partially explainable due to wave trapping. We explore wave trapping in the next section.
6. Vertical shear and wave trapping
In the atmosphere, bores degrade into a packet of waves that either are trapped within a wave duct or vertically propagate away. For this reason, we treat these bores as a packet of infinite wave modes. We assume that we can discuss the longevity of the bores observed if we determine what wave modes, given a vertical profile of the nocturnal environment, are sufficiently trapped. One favorable condition for trapping is a heavily sheared layer containing a critical level (LT76), where the wind vector at some height is equal to the bore motion vector. Previous studies point to the vertical shear above the NLLJ for trapping of waves (e.g., Crook 1986; K91; K08b). Another condition is the vertical advection of vorticity by the vertical wave motion [i.e., the curvature term in Eq. (5); Crook 1988].
Our examination begins by measuring the difference between the shear vectors within the NLLJ (derived from radiosonde data taken at the five ARM sites at 2100, 0000, 0300, and 0600 LST) and the direction of bore motion. We investigate the multiple layers of vertical shear through the lower troposphere, but present just two of the depths that have the smallest difference. These two shear vectors extend from the height of the wind maximum in the NLLJ upward to 1.5 or 2.5 km AGL. We look beyond a simple measure of the direction of the NLLJ because theory (Shapiro et al. 2016) and our observations have peaks in U and V at different heights.
The angular difference between the movement of bores and the two aforementioned shear vectors are shown in Fig. 11. The direction of the bore movement is slightly rotated counterclockwise to the bulk shear vector extending to 1.5 km and slightly clockwise to the bulk shear vector extending to 2.5 km. The alignment between the bore motion and the vectors increases with time. It is possible that the bores are aligning with a bulk shear vector whose depth lies between the NLLJ wind maximum and 1.5–2.5 km. This is plausible because there is great variation in the direction and speed of the wind above the wind maximum of the NLLJ (Shapiro et al. 2016). These results suggest that the NLLJ plays a role in wave trapping that strengthens through the night.
The angular difference between an observed bore direction and the direction of the environmental bulk shear vector contained between the height at the max NNLJ wind and (a) 1.5 and (b) 2.5 km. The values are calculated from both the initial and final observed bore directions. In this plot, the 0° direction implies that the wind shear vector in that layer is aligned parallel and with the direction of movement of the bore, while a counterclockwise CCW (clockwise CW) implies the shear vector is 90° rotated to the left (right) of the bore motion. Contours are in percent of total for bulk shear; 45 cases for each.
Citation: Monthly Weather Review 145, 10; 10.1175/MWR-D-16-0415.1
We investigate how the NLLJ is assisting in wave trapping by analyzing possible wave ducts ahead of observed bores with linear wave theory (described in section 2). We utilized data from the same five ARM sounding sites to calculate the environmental Scorer parameter [(5)]. Our analysis looked for a layer characterized by positive Scorer parameter values adjacent to the surface topped by a layer characterized by negative Scorer parameter values, and critical levels embedded within a sheared layer (Richardson layers <¼; LT76), both of which are favorable conditions for wave trapping. We also derive a vertical profile of the normalized vertical motion by solving for (4) and compare it to observations of bores that passed over MAPR.
We utilize vertical profiles of temperature, moisture, and winds, taken from soundings located within ~100 km of 13 bore events. It is conceded that we are unaware of how well the environmental conditions in the analyzed soundings represented the true prebore environment. Repeating this analysis on data taken from Plains Elevated Convection at Night (PECAN) will be very beneficial because pre- and postbore soundings were launched within close proximity to bore passage. For brevity, we present only one of the 13 case studies, but this case study exhibits commonalities between nearly all analyzed environments.
The case study involves a 4 June 2002 bore (also analyzed by K08a). We will use the 0600 LST sounding at Vici, Oklahoma, for our analysis. In Fig. 12a, the profile of the Scorer parameter is displayed as a function of direction to illustrate how trapping varies with bore orientation. Wave trapping appears unfavorable for a bore traveling from 30° to 150°, where the Scorer parameter exhibits a layer of negative values adjacent to the ground topped by a layer of positive values (Fig. 12b). Critical levels may be present for a wave coming from 150° to 290°. While analysis of the Richardson number and the bore-relative flow appears to suggest a critical layer could be present (not shown), no wave traveled in this direction around 0600 LST.9 For a wave traveling from 290° to 30°, it appears that there is a positive Scorer layer below a large negative layer (Fig. 12a), a favorable pattern for wave trapping.
Analysis of a prebore environment from Vici, OK, at 0600 LST for a bore traveling from 334° at 10.8 m s−1: (a) Scorer parameter as a function of direction and (b) curvature of the wind as a function of direction.
Citation: Monthly Weather Review 145, 10; 10.1175/MWR-D-16-0415.1
We observe the 4 June bore coming from 330°, consistent with a favorable profile of the Scorer parameter. The bore is aligned with a curvature term that is large and negative at low levels (due to the denominator, the bore-relative winds, being negative), but positive above 1000 m. Comparing the magnitude of the first and second term of the Scorer parameter (Fig. 13a), we see the variations (Fig. 13b) are dominated by the curvature term. If we attribute changes in the curvature (Fig. 12b) to changes in the Scorer parameter, then the top of the wave duct is correlated with the inflection point above the nose of the NLLJ.
Analysis of the same bore from Fig. 12 for the (a) stability and curvature terms of the Scorer parameter in the direction of the bore; (b) Scorer parameter in the direction of the bore and the possible k2 range (indicated by the hatched orange box, defining a range of horizontal wavelengths for trapped modes between 5022 m and 13 606 m); (c) bore-relative winds with positive and negative Scorer parameter layers superimposed for the lowest three layers (orange hatch is positive layers and blue hatch is negative layers); and (d) normalized vertical wind profile for a trapped wave.
Citation: Monthly Weather Review 145, 10; 10.1175/MWR-D-16-0415.1
Next, we examine the difference between the observed and predicted vertical motion profiles and the horizontal wavelengths. Agreement between these values would provide evidence that the curvature is an important factor for developing a wave duct. Comparison between the predicted normalized vertical profile of w (Fig. 13d) and the w observations from MAPR (Fig. 14) shows good agreement. The height of the maximum w from MAPR and the predicted height of the maximum w are both close to 1000 m. Additionally, the observed horizontal wavelength (calculated from the translation speed of the bore and period of the disturbance from MAPR vertical wind) is around 10 km, falling within the range of theorized wavelengths (5036 and 13 805 m, Fig. 14). Based on our analysis of linear wave theory, we expect this wave to be trapped due to curvature.
Multiple Antenna Profiler (MAPR) for one of the two bores on 4 Jun 2002. (top) A time lapse of the signal-to-noise ratio, (middle) the time lapse of the vertical velocity, and (bottom) the time lapse of the horizontal wind vector as a function of height. All panels are from 1000 to 1200 UTC.
Citation: Monthly Weather Review 145, 10; 10.1175/MWR-D-16-0415.1
If we argue that curvature is not the mechanism but a critical level (LT76), we are unable to explain most of the cases (2 of the 13 cases appear to contain a viable critical layer). Instead, 9 out of the 13 appear trapped according to our linear wave analysis where the curvature term dominates the profile of the Scorer parameter. In total, 2 of the 13 cases did not appear to be trapped according to our two methods. This result is consistent with other studies highlighted in K91. The maximum vertical motion is located near the transition from positive to negative Scorer parameter, and observations (Fig. 14) compare well with this finding. Therefore, we conclude that the curvature above the maximum wind in the NLLJ is generally not correlated with a critical layer as in LT76, but often contains the positive curvature necessary for internal wave trapping in the nocturnal environment of the southern Great Plains.
We admit that our theory requires that trapped waves exponentially decay within an infinitely deep second negative Scorer layer, which is not consistent with observed Scorer profiles. Therefore, our adaptation of linear theory cannot alone explain complete trapping. Interestingly, we see vertical motions extending above the analyzed negative Scorer parameter layer (although the vertical motions from the MAPR data are limited to areas that surpass a threshold for the signal-to-noise ratio). We hypothesize that the shallow depth of the second negative Scorer layer insufficiently traps waves, allowing vertical motions to extend upward into the lower troposphere. Because these waves that emerge from bores appear to last quite a long time, we presume that a more complex interaction of wave trapping is taking place, which is unexplained by our current method. Additionally, observations may contain nonlinear motions that are not represented in our linear analysis. We shall explore these limitations in a model framework within future studies.
7. Conclusions and implications for future work
Our analysis that utilizes radar and surface measurements taken during IHOP_2002 show that convective outflows intrude into the southern Great Plains stable nocturnal boundary layer and frequently generate atmospheric bores. This finding is supported by our application of hydraulic theory, since the dominant flow regime is partially blocked. Observations indicate a likely time to observe bores occurs during 2200–2000 LST, coincident with the strongest signal for a partially blocked flow. The height and strength of the inversion are the most influential factors for developing an atmospheric bore; while the curvature of the NLLJ is found to be a critical determinant for maintaining long-lived bores. As the night progresses, the direction of atmospheric bores generally aligns with the direction of the shear vector defined by the maximum wind to 1.5-km and maximum wind to 2.5-km shear vectors. We also find, as did K91, that the transition of the Scorer parameter from positive to negative values occurs above the wind maximum of the NLLJ, near or at the inflection point in the wind profile associated with the curvature. The preferential direction of bore movement indicates that the effectiveness of the waveguide varies with orientation due to variations in the curvature term. Our analysis of 13 cases supports this theory. Based on our analysis, we argue that while a bore may develop in many directions along an outflow, their preferred motion aligns in the direction that contains maintainable wave modes.
This work also highlights that in a general sense the theory is consistent with previous case studies and illustrates that the general behavior of bores adheres to current theory. Although the data we analyze are from a single field campaign, our proposed mechanisms for generation and maintenance of bores are governed by the general dynamics and thermodynamics of the summertime Great Plains nocturnal environment: a stable surface boundary layer, the presence of a southerly NLLJ, and the ongoing production of density currents from mesoscale convective systems.
Our study also raises questions regarding the dynamics of atmospheric bores. Past studies (B95; LT76) suggest that most wave ducts are “leaky” and most, but not all wave energy is trapped within the wave duct. Because a convective outflow is often a 3D response, how the environment traps gravity waves generated in a bore is also a critical, unaddressed question. Effective trapping means that bores will be long lived, while an untrapped wave may not be able to propagate far from its parent density current. However, such untrapped waves will vertically propagate out of the wave duct and disturb more vertical layers above the duct.
We return to our original question: What is the relationship of bores to the nocturnal environment over the southern Great Plains? Our results suggest that bores are a relatively common and thus an inherent component of the life cycle of both density currents and organized nocturnal convective systems. From a dynamics standpoint, a convective outflow will experience unfavorable changes to the pressure gradient force driving its propagation unless a bore displaces fluid upward and away from the leading edge of the outflow. A consequence of this semipermanent lift is induced upward motions. Moreover, our study illustrates that the maximum vertical motions extend well above the shallow nocturnal boundary layer and through the lower troposphere. This lifting will act to erode any convective inhibition within the storm inflow and may release convective instability initiating or maintaining deep convection (CK11). Whether bores enhance the precipitation and contribute to the nocturnal maximum in rainfall over this region is a subject for future research. Idealized simulations of convection in stable environments (e.g., Parker 2008; Schumacher 2009; French and Parker 2010) and simulations of specific events (e.g., Blake et al. 2017) suggest that bores can play a role in initiating or maintaining deep convection. The recent Plains Elevated Convection at Night (PECAN) project (Geerts et al. 2017) will certainly prove valuable in exploring this topic.
Acknowledgments
This research was supported by NSF Grant AGS-1237404. We thank NCAR, the DOE-ARM Program, and the Oklahoma Climate Survey, who provided and maintained the data for this study. We are also indebted to Dr. Michael Richman (School of Meteorology, University of Oklahoma) for explaining how to apply the statistical models, to Dr. Steven Koch (NOAA’s National Severe Storms Laboratory) for effectively guiding this research, to Dr. Howie Bluestein (School of Meteorology, University of Oklahoma) for informal discussions about bores, to Larissa Reames (School of Meteorology, University of Oklahoma) for her thoughtful discussions of nocturnal convection, and to Benjamin Blake for his participation in this joint research.
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A parameter r exists, defined as the ratio between the total depth of a two-layer system and the depth of the lower layer. According to RS89 when r ≪ 1, solutions for the two-layer model are similar to the single-layer model, and are analogous when r = 0.
We define ho in the atmosphere as the depth over which 
The depth of the density current is calculated assuming that the change in pressure is entirely due to hydrostatic pressure changes from the colder fluid. For a derivation, see K91.
Our analysis does not account for nonlinear wave modes, as discussed in Christie et al. (1979) and Christie (1989).
Although LT76 implied the longest vertical wavelength should be the dominant mode, they do not discuss what the distribution of energy is for each trapped wave mode. Thus, we assume that the energy distribution will not determine the dominant wave mode. We concede that Scorer (1949) came up with a different constraint, this time on finding the smallest k trappable. However, this is dependent on knowing the shape of the disturbance. During IHOP_2002, we are unaware of the ever-so evolving shape and height of the density current. Thus, we leave this portion of theory out of our analysis.
We track the density current direction to correctly calculate the U parallel to the movement.
Analysis of earlier Vici, OK, sounding (~0000 LST) does exhibit trapped wave modes according to the methods in section 2 for a wave traveling from 250° at 15 m s−1 (the observed speed of the RFL). However, the favorable conditions degrade with time. This possibly explains why no wave modes appeared to be maintained from this direction.