Abstract

A method is presented for analyzing and forecasting the occurrence of lower-atmospheric undular bores over the western and central Gulf of Mexico (GOM) and adjacent land areas using standard operational forecasting and analysis techniques. The method is based on research that has identified a set of sufficient conditions associated with these occurrences realized by observations of the phenomena in recent years and is grounded in the theoretical understanding of undular bores developed during the last century by fluid dynamicists. The approach discusses practical approximations to the theory that allows the operational forecaster to use output from standard numerical and statistical forecast models. In addition to providing an operational method for forecasting the time and location of occurrence, the technique provides a methodology for analyzing and anticipating the strength, forward speed, and horizontal wavelength of the phenomena.

1. Introduction

Undular bores in the lower atmosphere are disturbances produced when a surface boundary such as a cold front, prefrontal trough, or thunderstorm outflow boundary acts as a density current and intrudes upon a less dense, stable layer of air capped by an inversion. The leading edge of the disturbance causes waves to form on the inversion and propagate along it, and its passage is characterized by a sudden jump in pressure and rise in the height of the inversion. The disturbance lasts for several hours or more and produces velocity changes in the layer beneath it. The waves produced are smooth, nonturbid, and amplitude oriented, with the leading wave having the highest amplitude.

Research leading to this understanding of undular bores was first done with water wave experiments by Bélanger (1828) and Froude (1872), followed a century later by descriptions of undular bores in water (Tepper 1950; Benjamin and Lighthill 1954). The concepts developed were extended to analogous phenomena in the atmosphere in the 1980s by researchers such as Clarke et al. (1981), Simpson (1982), Clarke (1984), Crook (1988), Smith and Reeder (1988), Rottman and Simpson (1989), Christie et al. (1978), and Christie (1989). They established that the characteristics of atmospheric undular bores are largely determined by the initial speed of the density current relative to the speed of the waves produced and the resulting changes in the properties of the stable layer. This theory and its practical application, including quantitative measurements of atmospheric undular bores, has been further refined in recent years, notably by Koch et al. (1991), Locatelli et al. (1998), Koch and Clark (1999), Koch et al. (2008a), and Coleman and Knupp (2011). The undular bores examined in this study were consistent with this theory: all were produced by a density current that forced a low-level atmospheric stable layer upward, producing smooth, nonturbid waves contained along the top of the stable layer.

In the cited work from the last two decades, bore strength is traditionally calculated using the ratio of the raised height of the stable layer to its original height before bore passage. Undular waves are not produced at values less than 1; at values between 1 and 2, the density current lifts the stable layer in a nonturbulent motion and produces smooth, undular waves at the interface. As values exceed 2.0, the system evolves into a turbulent bore, with loss of the smooth, undular wave structure. For undular bores, if the lifted layer is sufficiently moist, a series of cloud lines forms in the rising air at the wave crests, with cloud-free zones between the wave crests where subsidence occurs in response to the restoring force of gravity (Clarke et al. 1981). When few clouds occur in the layers above the inversion, the resulting cloud signature is readily detectable in visible satellite imagery. This cloud signature may also be detected in Doppler radar imagery, depending upon the size of the cloud particles (or insects carried aloft), the amplitude of the waves, and their distance from the radar site. Undular bores produce discrete, measureable changes in the lower atmosphere and at the surface, with a notable veering of the wind, rise in barometric pressure, and without decrease in temperature (Locatelli et al. 1998); this allows researchers to identify their passage and, as will be shown, can also be used to forecast their occurrence.

There is practical value in forecasting undular bores, since these phenomena frequently create lower-tropospheric turbulence, adversely affecting aviation interests (Christie and Muirhead 1983). They can also produce mechanical lift in sufficiently unstable air, initiating deep moist convection and the possibility of severe thunderstorms (Clarke 1998). This has led to an increasing number of studies in the last two decades, including some that used existing theory and past observations to successfully anticipate the time and place of bore occurrence. These include Clarke et al. (1981), Mahapatra et al. (1991), Fulton et al. (1990), Locatelli et al. (1998), Koch et al. (2005, 2008b), and Coleman et al. (2010). Their results have established an important basis for the current understanding of undular bores. A listing of important bore characteristics from these studies is referenced in the following sections.

This study was restricted to undular bores over the western Gulf of Mexico (GOM), where large undular bores, 300 km or greater in the along-wavelength dimension (hereafter bore length), occur irregularly and infrequently. However, a review of data from 2009 through 2011 suggests that undular bores of lesser extent and shorter-lived large events may be more common than earlier believed, most likely because they are often not well observed or reported. The effort presented here, based upon recent observations of undular bores in this region and building upon previous studies of undular bores elsewhere, summarizes a set of necessary conditions associated with the development of bores in this region and an operational technique for forecasting them. Particular attention is focused on the seasonal influences that produce these conditions, as they are unique to this region of North America and are also nearly optimal for producing a wave cloud signature visible in satellite images.

2. Theory

a. Density current and bore properties

In order for a frontal boundary to induce an undular bore in a stable layer, the boundary must behave as a density current (see Smith and Reeder 1988; Fulton et al. 1990; Schultz 2004, 2005; Geerts et al. 2006). In general, these studies show that mature frontal boundaries or those undergoing frontogenesis are most likely to behave as density currents while those undergoing frontolysis are less likely to do so. These boundaries typically exhibit a wind speed (U) behind the frontal boundary and in the direction of bore propagation (the “feeder flow”), particularly just above the surface (above frictional effects) that is greater than the forward speed (c) of the density current (Smith and Reeder 1988). This faster speed behind the boundary leads to upward motion directly behind the head of the bore (Fig. 1) and facilitates lifting of the inversion (Young and Johnson 1984). This condition was of notable importance in this study and is examined in the next section.

Fig. 1.

Schematic overview of an advancing gravity current moving into a stable layer (Smith and Reeder 1988).

Fig. 1.

Schematic overview of an advancing gravity current moving into a stable layer (Smith and Reeder 1988).

Figure 2 is a schematic diagram of an undular bore adapted from Hartung et al. (2010). Research by Koch et al. (2005, 2008a) has shown that as the density current intrudes into the stable layer, a portion of the stable layer begins to envelope the head of the density current, eventually completely enclosing it, and the enclosed circulation becomes a solitary wave at the leading edge of the undular bore. Additional waves form in the same manner behind the first, and the entire wave set (the soliton) continues propagating along the inversion ahead of the density current. At this juncture the undular bore and its wave set becomes a separate, distinct entity from the density current that propagated it. As these waves move forward along the inversion, they mix warmer air downward, causing a slight increase in the surface temperature if this air is mixed all the way to the surface; this result is generally not observed until a bore is well developed. The leading wave of the developing bore lifts the inversion and the relative strength of the lifting, the bore strength, determines whether smooth undular waves will develop. Per Rottman and Simpson (1989) and Koch et al. (1991), the bore strength (f) can be approximated as the lifted height of the stable layer (h1), divided by its height before lifting (h0):

 
formula

At values between 1 and 2, the waves will exhibit an undular form. Per Clarke et al. (1981) and experiments by Rottman and Simpson (1989), the horizontal wavelength of these undular waves can be approximated by

 
formula
Fig. 2.

Schematic overview of the generation of an atmospheric bore of depth h1 by an advancing density current intruding into a stable layer of depth h0. [Adapted from Hartung et al. (2010).]

Fig. 2.

Schematic overview of the generation of an atmospheric bore of depth h1 by an advancing density current intruding into a stable layer of depth h0. [Adapted from Hartung et al. (2010).]

Per Locatelli et al. (1998) and corroborated by all of the studies cited in this work, it is expected that the lifting caused by an undular bore will last for at least 2–3 h.

b. Density current speed

Per Hartung et al. (2010), undular bores are primarily propagative and not advective; that is, the bore and gravity waves propagating ahead of the density current lose connection to the advection of mass from the feeder flow associated with the density current. The density current slows down as it experiences a decrease in the current depth (Koch et al. 2005) while the developing bore continues to propagate forward unabated. This effectively allows the bore to outrun the density current. Figure 3, a representative diagram of these speeds over time (Fulton et al. 1990), shows the bore speed remaining constant or increasing slightly while the density current decelerated noticeably. Given optimum conditions for containment of the wave energy within the stable layer, discussed in the next section, the undular bore and its wave packet may then continue to propagate unimpeded along the inversion for many hours (Locatelli et al. 1998). Thus, for forecasting purposes, the operational forecaster is most interested in the density current speed at the time when the bore first develops. The technique for measuring it will be discussed in section 3.

Fig. 3.

Density current vs undular bore speed comparison for an undular bore (Fulton et al. 1990).

Fig. 3.

Density current vs undular bore speed comparison for an undular bore (Fulton et al. 1990).

c. Bore speed

Significant advances in developing practical calculations for undular bore wave speeds were made by Rottman and Simpson (1989). They showed that the Froude number (Fr), a measure of the relative strength of a bore, is represented by the ratio of the initial density current velocity to the induced gravitational wave velocity in a shallow layer and is represented by

 
formula

where Cdc is the speed of the advancing density current and Cgw is the speed of an internal gravity wave produced in the shallow stable layer upon which the density current intrudes. The value of Cgw is calculated by

 
formula

where g is the gravitational acceleration, Θυ is the average virtual potential temperature through the inversion layer, and ∆Θυ is the change from the bottom to the top of the inversion layer. Froude number values between 1 and 2 will produce undular waves (Rottman and Simpson 1989). Thus, for undular bores, given the speed of the initiating density current or the gravity waves produced, an approximate range of the other can be determined.

Based on the equation for the Froude number and the relationship of the stable layer heights before and after bore passage, a solution for bore speed was developed by Koch et al. (1991) and has been used to analyze bores in recent studies such as those by Locatelli et al. (1998), Koch et al. (2008a), Coleman et al. (2009), and Goler (2009):

 
formula

Or, substituting from Eq. (4),

 
formula

In attempting to forecast Cbore using (6), complexities arise as most skew-T diagrams do not provide virtual potential temperature, necessitating conversion to Θυ, which in turn requires approximations for mixing ratios, compounding calculation errors. For the operational forecaster, these calculations make forecasting a discrete bore speed prohibitively complex. However, a method of calculating a forecast range can be developed by exploiting the relation between the Froude number and the traditional method of calculating bore strength. Since Fr has the distinct characteristics of yielding values between 1 and 2 for undular bores, we can substitute (3) into (5):

 
formula

Substituting into (7) the accepted range Fr = 1.1 and Fr = 1.9 and f = h1/h0 from (1) produces the following inequality:

 
formula

The value of the inequality lies in combining the two most important relationships established in the theory, that of the Froude number’s relation of the speed of the density current to that of the waves produced and the bore strength equation’s relation of pre- and postbore inversion heights. This inequality produced verifiably reliable results in this study. Its accuracy for operational forecasters is discussed in the following sections and a practical application is illustrated in section 6.

d. Bore longevity

As an undular bore develops, continued wave propagation along the inversion requires that most of the wave energy does not escape the stable layer but is instead contained along the inversion. Studies by Crook (1986, 1988) found three important mechanisms for trapping the wave energy; the first two involve midtropospheric conditions and thus are not applicable to the distinctly low-level phenomena in this study. The third mechanism describes an opposing jet in the lower troposphere that produces strong curvature in the wave-normal wind. This potential for wave trapping, measured by the Scorer parameter, relies on the difference between its two terms: the first a measure of atmospheric stability and the second the amount of shear in the local environment. Both are taken over the height of the layer:

 
formula

where N = N(z), the Brunt–Väisälä frequency, and U = U(z), the horizontal wind. A positive value at low levels decreasing rapidly with height is most favorable for trapping wave energy and can be produced when a low-level jet crosses the top of a stable layer, causing a rapid decrease in the second (shear) term. This condition has been documented in studies by Fulton et al. (1990), Koch et al. (1991), Koch and Clark (1999), and Coleman and Knupp (2011), and was present in most of the undular bores referenced in this study, all of which were generated by thunderstorm or mesoscale convective system (MCS) outflows associated with a low-level jet (see Table 5). While all of the bores examined in this study were generated by larger-scale frontal boundaries, their environments also exhibited a strong curvature in the wind profile. Calculations by Koch et al. (1991) and Karyampudi et al. (1995) showed a strong correlation of the height at which the Scorer parameter reverses from positive to negative to the height where there is strongest curvature of the wind profile on a plot of the ambient wind in the direction of bore motion (U) against height. This relationship can be exploited in the forecast scheme by using the curvature of the wind profile as a proxy for a favorable Scorer parameter: the forecast skew-T diagrams can be used to plot U against height and analyzed to determine if there is strong enough curvature in the wind profile to support wave ducting at the top of the stable layer. Karyampudi et al. (1995) also defined a bore critical level as the level where U is equal to the bore speed; this critical value should be equivalent to or close to the maximum curvature on the wind profile. For forecasting purposes we can define the critical level as that where forecast U most closely approaches the forecast mean bore speed using the inequality from (8). A calculation of this value is shown in the case study in section 6. The reader is referred to Scorer (1949) and Koch and Clark (1999) for detailed explanations of the Scorer parameter and other examinations of its application.

3. Necessary conditions

Tables 1 and 2 contain data for the 14 undular bores analyzed in this study. Tables 3 and 4 contain forecast versus observed characteristics for the nine bores that were forecast. All data were obtained from an analysis of Geostationary Operational Environmental Satellite-14 (known as GOES-East) imagery and National Oceanic and Atmospheric Administration (NOAA) surface and upper-air analysis data, including rawinsonde and model simulated atmospheric profiles.

Table 1.

Observed bore characteristics: 1) from 925-mb chart, (2) estimated from satellite image, (3) from skew-T analysis, or calculated.

Observed bore characteristics: 1) from 925-mb chart, (2) estimated from satellite image, (3) from skew-T analysis, or calculated.
Observed bore characteristics: 1) from 925-mb chart, (2) estimated from satellite image, (3) from skew-T analysis, or calculated.
Table 2.

Observed bore density currents and speeds: 1) wind speed immediately behind density current in direction of bore during bore formation from ARL skew-T analysis and 2) forward speed of frontal boundary during bore formation from HPC surface analysis.

Observed bore density currents and speeds: 1) wind speed immediately behind density current in direction of bore during bore formation from ARL skew-T analysis and 2) forward speed of frontal boundary during bore formation from HPC surface analysis.
Observed bore density currents and speeds: 1) wind speed immediately behind density current in direction of bore during bore formation from ARL skew-T analysis and 2) forward speed of frontal boundary during bore formation from HPC surface analysis.
Table 3.

Bore forecast characteristics and observed values.

Bore forecast characteristics and observed values.
Bore forecast characteristics and observed values.
Table 4.

Forecast and observed values for density current and bore speeds.

Forecast and observed values for density current and bore speeds.
Forecast and observed values for density current and bore speeds.

From the data examined, the following conditions are determined to be essential for the development of undular bores in the western GOM:

  1. a dual-layer prebore sounding, that is, a saturated or nearly saturated boundary layer beneath a deep layer of warmer, drier air, the interface between the two marked by a substantial temperature inversion; for this region it is typically a tropical maritime layer overlaid by an arid continental air mass;

  2. a dry or relatively dry, elevated mixed layer above the inversion;

  3. the vertical wind profile of the ambient wind in the direction of bore motion shows strong curvature within a few hundred meters of the height of the inversion at the time the bore is expected to occur;

  4. an approaching cold front, prefrontal trough, or other surface boundary that is anticipated to behave as a density current;

  5. winds with a southerly component within the inversion layer that veer to a westerly or northerly component within the intruding density current; this kinematic profile is consistent with the airmass profiles described above; and

  6. a resulting “frontal” lifting of the inversion by a factor between 1 and 2 that lasts for several hours or more.

Atmospheric conditions over the western GOM meet the first five criteria on many occasions in the spring season and less commonly in the fall. It is the last condition, the lifting of the inversion by a factor of between 1 and 2, which seldom occurs.

The first condition, a moist maritime boundary layer surmounted by a deep layer of potentially warmer continental air, is typically present in spring (and less frequently in fall) over the western GOM. [A relatively dry surface boundary layer (SBL) is not conducive to bore formation since it indicates a low-level flow from the north or west; such a low-level wind profile is not present in this region with approaching frontal boundaries.] Although primarily a maritime boundary layer, it is also present well inland overnight, as moist maritime air is advected inland by the prevailing southeasterly flow in conjunction with nocturnal cooling, producing a resultant “marine–nocturnal” SBL that may extend 100 mi or more inland into southern Texas and northeastern Mexico (Fig. 4). The resulting atmospheric profile described above is completed by the prevailing seasonal upper-level pattern that advects a warm, dry southwesterly flow from the Mexican plateau atop the boundary layer. Due to the partially nocturnal character of the inversion inland, undular bore development and longevity over land areas is less common after midday. All 14 observed bores exhibited this atmospheric profile at the time a bore developed. On occasions where considerable moisture was advected into the overlying elevated mixed layer, typically when winds in this layer were more southerly, the layer became unstable, reducing the likelihood for wave trapping. Under these conditions the inversion typically eroded, showers developed, and no undular bore was observed.

Fig. 4.

Mean (1968–96) 925-mb RH March–May from NOAA/ESRL NCEP– National Center for Atmospheric Research (NCAR) reanalysis.

Fig. 4.

Mean (1968–96) 925-mb RH March–May from NOAA/ESRL NCEP– National Center for Atmospheric Research (NCAR) reanalysis.

There was a strong positive correlation of the length of the surface boundary to bore length. Bores 300–800 km in length were observed when extensive Canadian high pressure was moving southeastward over the Rocky Mountains or northern central plains, accompanied by an expansive cold front stretching 800–1600 km from the southern or central states southward into Texas. Shorter bores were produced by less expansive cold fronts extending southward from low pressure over the southern plains.

All of the cases studied exhibited a veering of the surface wind direction within the stable layer to that observed in the density current as the bore passed over a location. This wind shift was observed to be as little as 45°, typically from southeast to southwest along drylines, to as much as 270°, from southeast to northeast along frontal boundaries. The amount of veering in the previously studied cases cited in Table 5 showed less variability due to a more consistent and smaller scale density current source (mesoscale thunderstorm outflow boundaries).

Table 5.

Observed undular bore characteristics from seven significant published studies of undular bores in the United States. Note that none of these undular bores occurred in the western GOM region.

Observed undular bore characteristics from seven significant published studies of undular bores in the United States. Note that none of these undular bores occurred in the western GOM region.
Observed undular bore characteristics from seven significant published studies of undular bores in the United States. Note that none of these undular bores occurred in the western GOM region.

Preliminary analysis of the cloud structures showed that undular bore clouds over the western GOM are distinctly low-level phenomena. The National Aeronautics and Space Administration Langley Research Center Cloud and Radiation Group (NASA LARC) cloud data analysis for the 14 bores in Table 1 showed average cloud bases were 400 m and cloud tops 1000 m, yielding an average cloud thickness of 600 m. Maximum cloud heights were 2000–2500 m. Broadband albedo averaged 60%–70% and τ (visible optical thickness) averaged 2–40. This supports the visible satellite evidence that undular bore clouds in the western GOM are markedly low level and shallow, but because of their origin, saturated or nearly saturated air lifted by mechanical forcing over a short vertical distance tends to be optically thick with high albedo.

4. Forecasting method

a. Density current and bore speeds

From the theory discussed, the initial speed of the density current as it intercepts the stable layer is most relevant for operational forecasting of undular bores. When determining the best proxy for this speed, there were two possibilities of interest: the speed of the boundary itself and the above ground level (AGL) winds behind the boundary. Since there is a proportional relationship between these speeds (Smith and Reeder 1988; Young and Johnson 1984), both were examined for this study. In examining the behind-boundary wind speed, selecting the most representative pressure level was limited to the mandatory levels typically available to operational forecasters. Considering an average density current depth of near 1 km and an average standard height of 750–800 m at 925 mb and 1500 m at 850 mb, the 850-mb level was too high to obtain representative wind speeds for those accompanying the bore. Thus, the 925-mb level was used for density current speed as it is much closer to the average prebore inversion pressure level of 903 mb (Table 1), where the interaction and subsequent lifting first occurs. Comparing the two possibilities for density current speed for forecasting the bore speed range, discussed fully in section 7, resulted in higher accuracy when using the behind-boundary 925-mb wind rather than the boundary speed. Thus, the 925-mb wind speed was found to be the best proxy and should be used in the inequality from (8) to determine the forecast bore speed range.

When examining whether the boundary will behave as a density current, rather than comparing the behind-boundary wind speed to the boundary speed itself, it was sufficient to establish that the forecast 925-mb wind speed was no less than the lower end of the range used in this study: 10 m s−1 (Tables 3 and 4). No undular bore was produced when the forecast 925-mb wind speed was below this value, and the lowest observed 925-mb wind speed when an undular bore was produced was 13 m s−1 (Table 1). Similarly, if the use of the inequality in (8) yields a forecast bore speed below or above the range for observed bores in Table 1 (9.8–18 m s−1), the probability of a bore should be suspect. Forecast surface maps such as those from the Hydrometeorological Prediction Center (HPC) can be used to obtain an isochrone analysis of the expected frontal boundary speed, and the forecast 925-mb wind speed is available from National Centers for Environmental Prediction (NCEP) model forecasts.

b. Atmospheric profile

Forecast skew-T diagrams are essential in determining the predicted state of the atmospheric profile at locations where a bore is expected. A first examination should establish that a saturated or nearly saturated surface layer is capped by an inversion that is beneath a warm, dry, elevated mixed layer (Figs. 5 and 6). The saturated stable layer ensures that lifting will produce clouds along the leading edge of the bore, although the entire boundary layer is often not saturated; in many cases the top few hundred meters will have forecast RH values well below 50%. This is mostly due to the inherently large errors in interpolated soundings at small height intervals, but in the less common cases where the air near the top of the boundary layer is relatively dry, bore development should still be expected, although the cloud signature may be weak or nonexistent.

Fig. 5.

Forecast skew-T diagram, issued at 1500 UTC 7 May for KCRP on 1200 UTC 8 May from NOAA Rapid Update Cycle (RUC) model.

Fig. 5.

Forecast skew-T diagram, issued at 1500 UTC 7 May for KCRP on 1200 UTC 8 May from NOAA Rapid Update Cycle (RUC) model.

Fig. 6.

Observed temperature and dewpoint soundings for Corpus Christi at 1200 UTC 8 May 2010 from University of Wyoming. An undular bore moved through the area later that morning and lifted the inversion to 850 mb.

Fig. 6.

Observed temperature and dewpoint soundings for Corpus Christi at 1200 UTC 8 May 2010 from University of Wyoming. An undular bore moved through the area later that morning and lifted the inversion to 850 mb.

Per the discussion of bore longevity in section 2d, to determine if the atmospheric profile is conducive for wave formation along the stable layer, a plot of the ambient wind in the direction of expected bore motion (U) against height should be calculated. The graph should show a sharp decrease in U with height at the approximate forecast level of the inversion. If no strong curvature is expected, bore development should be considered less likely. See the case study in section 6.1

c. Moisture and clouds

The elevated mixed layer should be relatively dry. If this layer is forecast to become gradually saturated, the forecast models typically show it becoming increasingly unstable and the capping inversion weakened or broken by the approaching boundary. Such atmospheric profiles indicate that wave trapping is much less likely and the chances increase that an undular bore will not occur or cannot be sustained. This is the primary reason that considerable midlevel clouds were not present in any of the bores in this study. Significant moisture in the upper troposphere did not affect the stability of the overlying layer, and when it was observed, the thin high-level clouds produced did not obscure the bore’s low-level, optically thick roll clouds in satellite images.

d. Inversion lifting

Once it has been established that the necessary conditions are likely, it is critical that the rise in the inversion height will be sufficient to yield a bore strength of 1–2 during the forecast period and remain at that level for at least 2–3 h. The forecaster should examine forecast soundings for multiple locations in advance of the boundary from 1 to 3 h before the expected arrival of the density current to up to 1–6 h after. For accuracy, it is essential to use the raw data values when making calculations. Using forecast inversion heights over a period of several hours, it is necessary to ascertain the maximum forecasted height of the inversion before lifting (h0), and the maximum height forecast after lifting (h1). If h1/h0 is not between 1 and 2, then an undular bore should not be expected, although because forecast soundings have a wide margin of error, the forecaster should allow for a deviation of approximately 0.3 below and above the 1.0–2.0 range. The bore onset can be forecast as the time the inversion is first lifted. However, because the arrival time of a density current is very difficult to forecast accurately, some lead and lag time should be added. Thus, for skew-Ts showing an inversion being lifted at 1200 UTC, a forecast time of 0900–1500 UTC is appropriate. This should be repeated for multiple locations along the expected horizontal length of the bore. When all six necessary conditions are met, each condition should then be weighed as more or less equal and in turn the total possibility of an undular bore can be rated as low, moderate, or high.

5. Verification method

The forecaster should first confirm the undular bore cloud signature through satellite images. Whether it can be verified or not, the forecaster should then examine the surface and upper-air data in the area where the bore was expected. The pressure jump at the leading edge of the bore, the accompanying wind shift, and temperature variations should be verified with data from surface observations. For temperatures, as explained in section 2, undular bores are normally accompanied by a slight increase in surface temperature or no decrease. Any sudden decrease in temperature should be suspect, as this is more likely an indication of the density current passage. When examining atmospheric profile data taken from reanalysis and model soundings, such as the lifted height of the inversion and change in wind profile, it is important to remember that these are approximations of temperature and wind speed changes and do not represent the bore. They must be considered along with all other surface and upper-air data before making a determination that an undular bore has occurred.

Occasionally infrared satellite imagery may also reveal the bore’s wave clouds. Figure 7a shows a GOES 1-km infrared image with the distinct bore wave clouds clearly visible. The bore that produced these waves continued southeastward into the western GOM, producing more undular waves at several points along its path, including a vivid set of waves well east of the Mexican coast between 1600 and 1800 UTC (Fig. 7b).

Fig. 7.

(a) NOAA GOES-East 1-km infrared satellite image at 0801 UTC 11 Apr 2011, showing a set of undular bore waves moving through southeastern TX (SE TX). (b) NOAA GOES-East 1-km visible satellite image at 1655 UTC 11 Apr 2011, showing undular waves moving offshore from northeastern Mexico. These waves were produced by the same undular bore that generated the gravity waves in (a).

Fig. 7.

(a) NOAA GOES-East 1-km infrared satellite image at 0801 UTC 11 Apr 2011, showing a set of undular bore waves moving through southeastern TX (SE TX). (b) NOAA GOES-East 1-km visible satellite image at 1655 UTC 11 Apr 2011, showing undular waves moving offshore from northeastern Mexico. These waves were produced by the same undular bore that generated the gravity waves in (a).

Corroboration may also be established through radar imagery if the bore amplitude is sufficiently deep. Figures 8a and 8b show, respectively, the NOAA 1-km visible satellite image of the western GOM at 1545 UTC and the Doppler radial velocity radar from Brownsville, Texas, at 1537 UTC. Even though the satellite image shows the cloud representation terminating just east of the coast, the radial velocity observed by radar clearly shows the undular waves extending over the Brownsville area as well. Although further radar image analysis was beyond the scope of this study, methods for detailed analysis of bore speed, wavelength, and wave period using radar reflectivity and radial velocity are valuable tools for analysis and are well represented by studies such as those by Mahapatra et al. (1991) and Coleman and Knupp (2011).

Fig. 8.

(a) NOAA GOES-East 1-km visible satellite image at 1545 UTC 8 May 2010 showing the large undular bore over the western GOM. The circled area corresponds to the coverage of the Brownsville radar image in (b). (b) NOAA GOES-East with high-resolution KBRO Doppler radial velocity radar centered at Brownsville, TX. The area within the circle corresponds to the circled area in (a). The reversing wind directions are clearly visible, with lighter colors in the wave bands indicating motion toward the radar and the darker areas motion away from the radar.

Fig. 8.

(a) NOAA GOES-East 1-km visible satellite image at 1545 UTC 8 May 2010 showing the large undular bore over the western GOM. The circled area corresponds to the coverage of the Brownsville radar image in (b). (b) NOAA GOES-East with high-resolution KBRO Doppler radial velocity radar centered at Brownsville, TX. The area within the circle corresponds to the circled area in (a). The reversing wind directions are clearly visible, with lighter colors in the wave bands indicating motion toward the radar and the darker areas motion away from the radar.

While detailed data such as continuous 500-m tower readings (Mahapatra et al. 1991) and profiler observations (Knupp 2006) are typically not available to the operational forecaster in most of this region, verification of the bore passage at the surface can usually be established through National Weather Service (NWS) surface observing stations, particularly through Automated Surface Observing System (ASOS) 5-min data (1-min data lack pressure and temperature readings). For surface observations over the open water, data from the National Data Buoy Center (NDBC) are particularly valuable, as many stations provide observations at 15-min intervals. These readings are taken over short enough time intervals to offer empirical corroboration of an undular bore passage, as will be shown in the next section.

Bore strength and wave characteristics can be calculated using the equations from section 2. Measurements of bore speed, extent, and shape can be obtained through estimates of size and forward movement from visible or infrared satellite images.

The forecaster usually needs to examine a number of sites to find reporting stations with unambiguous indications of a bore passage, especially if the actual location or timing of the bore passage is different from what was forecast. Consequently, it is sometimes necessary to use a different location for verification than that used for the forecast. This will be seen in the case study in section 6. Finally, actual radiosonde data should show a similar profile but with more detail than that depicted in the forecast model sounding (cf. Figs. 5 and 6).

6. Case study: Bore 4 (Table 3)—Forecast developed 7 May 2010 for 8 May 2010

a. Forecast

The surface forecast for 1200 UTC on 8 May indicated a 1030-mb high over the western Great Plains pushing an extensive cold front from the north-northwest (approximately 340°) into the northwestern GOM (Fig. 9a). From the isochrone analysis the cold front was forecast to move at approximately 8 m s−1, with 925-mb winds behind the front (Fig. 9b) from the north to northeast at 15 m s−1. The large area of wind shift at nearly the same angle of incidence from southern Texas to Louisiana indicated that, if an undular bore was produced, it would be exceptionally extensive in length. A representative forecast of those issued for the west-northwestern Gulf coast is shown in Fig. 10a: the 7 May 1500 forecast skew-T plot for conditions on 8 May at 0900 at Corpus Christi, Texas (KCRP), showed a fully saturated surface boundary layer up to 891 m (918 mb) surmounted by a dry, elevated mixed layer. The forecast skew-T profile for 1200 UTC (Fig. 10b) projected the inversion height at 1085 m (898 mb). A lift to this level would result in a bore strength reading of 1085/891 = 1.22 if a bore were to occur. With a forecast lifting of the inversion between 0900 and 1200 UTC, it was reasonable to expect bore passage between 0900 and 1500 UTC. From (8), using the forecast 925-mb wind for the density current speed, Cdc = 15 m s−1, and f = 1.22, the forecast bore speed range was 9.2 m s−1 < Cbore < 15.8 m s−1, yielding a mean of 12.5 m s−1. Figures 11a and 11b show forecast soundings from farther northeastward at Galveston, Texas (KGLS), where a forecast inversion at 1060 m at 0900 UTC and subsequent lifting to 1300 m at 1200 UTC yielded the same f value of 1.22. These soundings were used to calculate the wave-ducting capability within the time frame where the bore was expected to occur per the theory explained in section 2d. Using the forecast north-northwesterly density current direction as a proxy for the expected bore direction, the 340° component of the ambient wind in the direction of the expected bore motion (U) for the two forecast soundings was calculated and plotted against height in Figs. 11c and 11d. The critical bore speed, using the forecast mean of 12.5 m s−1, was plotted with a vertical dashed line. Maximum curvature in both diagrams (horizontal dashed line) was at 620 m, with the area of strongest decrease from 600 to 1000 m. This is quite close to the forecast inversion height of 898 m at KCRP and 1060 m at KGLS. Thus, the atmospheric profile exhibited a clear tendency to support wave ducting along the inversion.

Fig. 9.

(a) NCEP HPC surface conditions forecast issued at 0730 UTC 7 May 2010 for 1200 UTC on 8 May 2010. (b) The NCEP 925-mb wind forecast for the same time.

Fig. 9.

(a) NCEP HPC surface conditions forecast issued at 0730 UTC 7 May 2010 for 1200 UTC on 8 May 2010. (b) The NCEP 925-mb wind forecast for the same time.

Fig. 10.

(a) NOAA/ESRL Forecast skew-T diagram for KCRP for 0900 UTC 8 May 2010. (b) As in (a), but for 1200 UTC.

Fig. 10.

(a) NOAA/ESRL Forecast skew-T diagram for KCRP for 0900 UTC 8 May 2010. (b) As in (a), but for 1200 UTC.

Fig. 11.

NASA LARC RUC forecast soundings for KGLS, for 8 May 2010 at (a) 0900 and (b) 1200 UTC. (c),(d) As in (a),(b), but for the corresponding calculated wind profiles for a north-northwesterly (340°) wind direction. The vertical dotted line on the wind profiles represents the mean forecast speed of the undular bore (critical speed) in m s−1, and the horizontal dotted line the height in km where the forecast wind speed was closest to the critical speed in the lower troposphere. Note that the strong curvature in the wind profile topped by a significant decrease in bore-relative wind corresponds closely to the level where the inversion was expected to occur.

Fig. 11.

NASA LARC RUC forecast soundings for KGLS, for 8 May 2010 at (a) 0900 and (b) 1200 UTC. (c),(d) As in (a),(b), but for the corresponding calculated wind profiles for a north-northwesterly (340°) wind direction. The vertical dotted line on the wind profiles represents the mean forecast speed of the undular bore (critical speed) in m s−1, and the horizontal dotted line the height in km where the forecast wind speed was closest to the critical speed in the lower troposphere. Note that the strong curvature in the wind profile topped by a significant decrease in bore-relative wind corresponds closely to the level where the inversion was expected to occur.

Because all of the factors were considered likely to occur, the final forecast was for a high chance of a large undular bore, greater than 300 km in length, between 0900 and 1500 UTC on 8 May 2010. The characteristic gravity wave clouds were forecast to develop near the coast from Louisiana to southern Texas in a west-southwest to east-northeast orientation.

b. Verification

The 1500 UTC surface analysis for 8 May 2010 (Fig. 12a) confirms the forecasted position of the cold front along the coasts of Texas and Louisiana during the morning. Figure 12b shows the striking image of a 600–700-km band of gravity wave clouds produced by a large undular bore earlier that morning. Although it appears that the weak, western edge of the bore passed through KCRP between 1200 and 1400 UTC, the data did not conclusively confirm this. In contrast, clear signatures were exhibited over land at Slidell, Louisiana (KLIX), and offshore at NDBC buoy 42362. The raw data from the observed 1200 UTC sounding at KLIX (Fig. 13a) indicate a strong inversion at 867 m (h0) with an elevated mixed layer above it. The interpolated skew-T reanalysis data from the Air Resources Laboratory’s (ARL) Eta Data Assimilation System (EDAS) at 1500 UTC (Fig. 13b) show that the inversion was lifted to 1570 m (h1) between 1200 and 1500 UTC. The 1800 UTC data (Fig. 13c) indicate that the inversion was still elevated at 1570 m 3 h later, an intrinsic property of undular bores.

Fig. 12.

(a) NCEP HPC analysis for 1500 UTC 8 May 2010. (b) NOAA GOES-East 1-km visible satellite image with location of NDBC buoy 42362 at 1515 UTC 8 May 2010. The leading edge of the bore is approximately 200 km (125 mi) ahead of the cold front [see (a)].

Fig. 12.

(a) NCEP HPC analysis for 1500 UTC 8 May 2010. (b) NOAA GOES-East 1-km visible satellite image with location of NDBC buoy 42362 at 1515 UTC 8 May 2010. The leading edge of the bore is approximately 200 km (125 mi) ahead of the cold front [see (a)].

Fig. 13.

(a) Sounding data for KLIX, for 1200 UTC 8 May 2010 from University of Wyoming. Underlined data at the 921-mb pressure level show the original inversion height at 867 m. (b) Sounding data for KLIX, for 1500 UTC 8 May 2010 from NOAA/ARL EDAS40. Underlined data at the 850-mb pressure level show the lifted inversion height at 1570 m. (c) Sounding data for KLIX, for 1800 UTC 8 May 2010 from NOAA/ARL EDAS40. Underlined data at the 850-mb pressure level show the inversion is still lifted to 1570 m after 3 h.

Fig. 13.

(a) Sounding data for KLIX, for 1200 UTC 8 May 2010 from University of Wyoming. Underlined data at the 921-mb pressure level show the original inversion height at 867 m. (b) Sounding data for KLIX, for 1500 UTC 8 May 2010 from NOAA/ARL EDAS40. Underlined data at the 850-mb pressure level show the lifted inversion height at 1570 m. (c) Sounding data for KLIX, for 1800 UTC 8 May 2010 from NOAA/ARL EDAS40. Underlined data at the 850-mb pressure level show the inversion is still lifted to 1570 m after 3 h.

The bore strength f was within the expected range for an undular bore:

 
formula

The observed bore speed estimated from isochrone analysis of visible satellite images was 14.3 m s−1, within the forecast range of 9.2–15.8 m s−1.

Using (2) and (3), the calculation for expected wavelength yields

 
formula

The wavelength range estimated from the visible satellite image yielded 12 km for the leading wave and was within the calculated range.

Visible satellite data show that the bore clouds developed at approximately 1200 UTC well offshore of western Louisiana where only weak surface indications occurred. But a strong signature for the surface passage of the bore, including the pressure jump associated with the leading edge, was measured just after 1500 UTC at NDBC buoy 42362, located approximately 152 mi south of KLIX. Figure 14a shows two equally centered GOES-East 1-km visible satellite images of the buoy location at 1515 and 1545 UTC on 8 May 2012. Figure 14b shows the buoy data from 1400 to 2045 UTC, with data from 1515 to 1900 UTC underlined to show the period where the bore influence at the surface is apparent. The data show that between 1515 and 1545 UTC, the pressure jumped 1 mb from 1020.7 mb to 1021.7 mb as the winds veered from 220° to 340° and began to increase from their prebore average of 2–3 to ~6 m s−1. These conditions continued for several hours, a necessary condition for verification of an undular bore passage. During this period the temperature increased slightly from 24.4° to 26.7°C, satisfying the expected condition of a slight temperature increase or no change. The NCEP surface analysis from Fig. 12a shows that the cold front lagged over 100 mi behind the leading edge of the bore at this time and thus over time the bore had moved considerably farther ahead of the initiating density current, as expected from the theory discussed in section 2.

Fig. 14.

(a) NOAA GOES-East 1-km visible satellite images at 1515 and 1545 UTC 8 May 2010. The images are centered at the same geographical point so that the temporal passage of the bore waves across the location of NDBC buoy 42362 can be visually ascertained. The corresponding buoy data are illustrated in (b). (b) NOAA NDBC buoy 42362 historic data from 1400 to 2045 UTC 8 May 2010. The time frame encompassing the pressure jump is underlined. Note the accompanying wind increase and shift into the north and that the pressure remained elevated for over 3 h. No decrease in air temperature was observed, and clearly the cold front was far to the north at this time (see Figs. 12a and 12b). The underlined data at 1515 and 1545 UTC can be compared to the visible satellite data in Fig. 14a.

Fig. 14.

(a) NOAA GOES-East 1-km visible satellite images at 1515 and 1545 UTC 8 May 2010. The images are centered at the same geographical point so that the temporal passage of the bore waves across the location of NDBC buoy 42362 can be visually ascertained. The corresponding buoy data are illustrated in (b). (b) NOAA NDBC buoy 42362 historic data from 1400 to 2045 UTC 8 May 2010. The time frame encompassing the pressure jump is underlined. Note the accompanying wind increase and shift into the north and that the pressure remained elevated for over 3 h. No decrease in air temperature was observed, and clearly the cold front was far to the north at this time (see Figs. 12a and 12b). The underlined data at 1515 and 1545 UTC can be compared to the visible satellite data in Fig. 14a.

7. Forecast accuracy

From Table 6 the probability of detection for an undular bore occurrence was 89% for the sample examined here. Date and location of occurrence were correctly forecast for the seven cases where an undular bore occurred, and the general horizontal extent of the bore was forecast successfully (greater or less than 300 km) for six of the seven bores that occurred. The forecast density current speed was within 80% of the observed speed estimated by isochrone analysis for six of the nine forecast cases (67%); boundary speed underestimation by the forecast models made forecasting this parameter much more difficult. A forecast bore strength of 1–2 was observed for six of the seven (86%) bores that occurred. For the single bore where the strength was much higher than the 1–2 range (f = 2.7 for bore 3; Table 3), the satellite images indicated that this bore produced undular waves at some points east of the Texas coast, and thus most likely had a bore strength between 1 and 2 at times, but data supporting this outcome were not found. Finally, the horizontal wavelength was within the forecast range 71% of the time using the Eq. (2).

Table 6.

Forecast accuracy for nine forecast bores from Table 2.

Forecast accuracy for nine forecast bores from Table 2.
Forecast accuracy for nine forecast bores from Table 2.

As expected from the theory, the behind-boundary speed was higher than the boundary speed in the majority of the analyzed cases (9 of 14) in Table 2. When using the inequality from (8) for the forecast bore speed range (Table 4), the observed value from isochrone analysis was within the forecast range 100% of the time (for all seven cases) when using the 925-mb wind speed as a proxy for the forecast density current speed, but only 29% of the time (two of seven cases) when using the forecast boundary speed from HPC surface forecasts. This difference appeared to be primarily due to a slow boundary speed bias in the forecast models (see section 8).

8. Conclusions

It is shown that a standard set of currently available forecast atmospheric parameters can be used for operational analysis and forecasting of undular bore development over the western Gulf of Mexico region with promising accuracy. This approach is feasible because the development of undular bores in this region is fundamentally dependent upon quantifiable characteristics of the seasonal “dual layer” atmospheric profile in this region and the ratio of the advancing density current speed to the velocity of the induced waves. This ratio, the Froude number, also has a direct bearing on the degree that the inversion is lifted by the bore, and taken together these two factors were exploited to produce an inequality that reliably forecast an accurate range for the bore speed over two consecutive spring seasons. In direct correlation, the majority of forecasting successes occurred in scenarios where the forecast bore strength (f) was within the 1–2 range at multiple points along the expected length of the advancing density current. Failures occurred where only one or two such points existed, and this was most common where the forecast density current speed was outside the 13–26 m s−1 range shown in Table 3. Thus, the forecast density current speed was very critical in making a successful forecast.

Bore wave amplitude was not included as a forecast parameter in this study, primarily because the forecast requires detailed measurements of density current depth and its verification requires detailed analysis of Doppler velocity radar data, both of which are beyond the scope of this study. However, its inclusion in a further refined forecast scheme should be quite possible given a practically developed method for analyzing these data.

Forecasting wave shape and size shows considerable promise given the close relationship of the boundary size to the horizontal extent of the bore and the observed ratio of the horizontal extent to the “front to back” extent of the wave set in visible satellite images. It should be possible to codify this relationship to produce a fairly refined forecast of overall bore shape in the future.

There was a considerably slow bias in the forecast models’ estimation of frontal boundary speeds in this region. Why this occurs was beyond the scope of this study, but it appeared that frontal boundaries initially approaching this area in the spring and fall seasons move at a fairly high speed but slow down considerably as they near the coast due to compressional warming of the air as it descends from the higher Texas plateau region, reducing the across-boundary temperature difference, along with an oft-present resistance from strong subtropical high pressure to the east. Undular bore initiation always occurred before this deceleration. This aspect should also not be confused with the expected deceleration of the density current as it initiates an undular bore. Regardless, much more research needs to be done on the speed of density currents that produce undular bores in this region.

Acknowledgments

The author would like to thank Dr. George S. Young of The Pennsylvania State University’s (penn State) Department of Meteorology for reviewing each phase of the manuscript and providing invaluable help and guidance in discussions of the theory and concepts presented in this study. Thanks to Stephen Corfidi of NOAA’s SPC and Mark Thornton for reviewing a number of drafts and providing valuable insights. Special thanks to Professor Lee Grenci of Penn State’s Department of Meteorology for reviewing this manuscript and particularly for bringing the question of undular bore origins and dynamics in the western Gulf of Mexico to my attention. Finally, many thanks to Michael Foner for reviewing this manuscript; this work is dedicated to your memory.

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Footnotes

1

One of the most difficult aspects of forecasting undular bores in this region is the paucity of meteorological observations, as most bores occur over the open water where there are scant surface and upper-air observations. Also, the relatively short duration for undular bores, usually 3–8 h, relative to the standard twice-daily radiosonde ascents, makes it difficult to use radiosonde observations to examine conditions during bore passage. Observed soundings, except for rare exceptions, are limited to 0000 and 1200 UTC daily. Since most bores over the western GOM region form between 0600 and 1200 UTC but have dissipated by 0000 UTC, the initial state of the atmosphere and inversion layer can usually be verified by radiosonde data, but the lifting produced by the bore cannot. Interpolated soundings from model output are typically the only resource for validating changes in the atmosphere produced by a bore in the absence of an observed sounding. Detailed interpolated data from NOAA/Air Resources Laboratory (ARL) and the Rapid Update Cycle (RUC) model are particularly useful approximations and were used extensively in this study. Also, as skew-T diagrams are point specific, it is critical that multiple locations along the expected path be examined. Finally, data from model forecasts for the original and lifted inversion heights are invaluable in ascertaining the forecast state of the atmosphere when a bore is expected, particularly anticipated temperature profiles, but they are spatially and temporally limited in coverage due to their source from 1- or 3-hourly model output with spatial grid limitations that are too large to represent a mesoscale undular bore event. Thus, it is only these data taken together with all recommended forecast parameters that provide a cumulative indication that such an event is likely to occur.