GRAVITY WAVES

AN UNDULAR BORE OVER THE GULF OF MEXICO - MARCH 2008

by Philip Lutzak – September 2008

Figure 1a. Gravity Waves moving southward off the Texas coast and out over the western Gulf of Mexico . 2008-03-15 at 1650 UTC. Courtesy MODIS. Larger image.

Figure 1b. Video of the gravity waves. Courtesy MODIS.

The beautiful gravity waves in Figures 1a and b are similar to the waves fanned out by a boat moving slowly through water, producing extended areas of ripples just ahead of and on either side of the hull. In this case, the "boat" was a large high pressure area and its associated cold front moving eastward to the north of Texas. As the cold front moved towards the northwestern Gulf and slowed down, it pushed out a series of smooth, undulating waves to its south. These waves are called gravity waves because as they push dense air upward into less dense air above them (the crest of the wave), the force of gravity soon pulls the dense air back down (the trough of the wave). If the original push that produced the leading wave is strong enough, and other conditions are right, it can produce a series of these gravity waves behind it. Of course eventually the high pressure and cold front moved on, just as a boat will, and the series of waves settled down and died out.

The type of gravity current that produced these waves is known as an undular bore. The following is a discussion of what causes gravity waves and undular bores, and why the atmospheric conditions were perfect that day to produce the beautiful cloud evidence of the waves shown above.

GRAVITY WAVE & UNDULAR BORE DEVELOPMENT

The undular bore is produced when a gravity current medium, or stable layer of air, is invaded by a gravity current that pushes into it. The gravity current medium is then lifted gently by the gravity current, producing smooth, undular bore waves ahead of the bore itself. Here are the components:

a. Gravity Current Medium In this case the stable layer of denser air that was intruded into and then pushed upward was an inversion layer. Inversions in the atmosphere are an example of a stably stratified fluid, where the fluid density increases with depth, i.e. the cooler air lies at the bottom of the layer, with warm air settled above it. Being quite stable, air parcels within it will not rise unless forced upward mechanically. Because this stable layer of air is a very suitable medium for a gravity current to course through, the stable layer is also called a “wave guide.”

b. Gravity Current Gravity currents, also called density currents, can be produced by a number of different atmospheric phenomena, but they can be defined generally as a fluid of one particular density intruding into a fluid of a different density. In this case the advancing gravity current was caused by a pre-frontal trough preceding a cold front. The cold front was in advance of high pressure moving from the Rockies into the Great Plains states north of Texas.

c. Undular bores According to Rottman and Simpson 1989, "a density current impinging on a low-level stable layer is a common scenario for the initiation of atmospheric bores." Additionally, bores like this are caused by a gravity current that causes a “step-like increase in the thickness of the waveguide layer of stable air” that they push through (Mahapatra, Doviak and Zrinc 1991). We will show how the gravity current forced the stable inversion upward slowly and steadily to produce a series of smooth, non-turbid waves such as those in Figure 3, and that this is classically characteristic of an undular bore.

FIGURE 3. Advancing gravity current becoming an undular bore. Courtesy Simpson 1987-1988.

We first need to look at the synoptic level characteristics over the northwestern Gulf of Mexico and Texas coast on 03-15 (Figure 4 below) to show how the advancing current that triggered the bore and its gravity waves was a pre-frontal trough, and that the stable layer of air that the wave traveled through was an inversion layer, in this case a marine inversion layer. In the section following that, we'll show evidence that the gravity current that produced this series of atmospheric waves was the special case of an undular bore.

SYNOPTIC LEVEL CONDITIONS (a and b):


	FIGURE 4a. Surface analysis for March 15, 2008 at 12Z. Larger image. Courtesy HPC.		FIGURE 4b. Surface analysis for March 15, 2008 at 18Z. Larger image. Courtesy HPC.

a. GRAVITY CURRENT MEDIUM - AN INVERSION LAYER AS A WAVE GUIDE

The pronounced inversion over the Texas coast and out over the northwestern Gulf that night was a special type called a marine inversion, caused by warm air (associated with light southwest winds at the surface) getting cooled down as it lay over the mid-sixty degree Gulf waters. Note that 17-19C (63-67F) water temperatures are typical over the western Gulf in early March. At the same time the air in the layers just above it maintained much warmer temperatures. This warmer air layer above the marine cooled surface layer was caused by fairly strong low to mid level southwest winds blowing very warm dry air as high as 28-30C (83-86F) from the much warmer and more arid Mexican and south Texas mainland.

Figure 5 at right is an IR satellite image from 12Z on the morning of 2008-03-15. The deeper the red and orange the colors are, the warmer the temperatures detected by the satellite. I have circled KBRO and 28N,95W, two locations which lay within the inversion layer and were later transgressed by the undular bore. Notice the medium to dark orange-red colors lying offshore and out into the Gulf - this is the area of cooler temperatures representing the marine inversion that lay out over the warmer (darker red) Gulf. Now note the sounding data from these two locations pictured in Figures 6a and b below. We can see the cool and moist layer at the surface, above which there was a large wedge of warm and quite dry air up to about 900-925mb - the inversion layer. This feature was clearly evident at all other locations over the Texas coast and out into the Gulf (more of these skewTs are shown later in this report). This stable layer, denser at the bottom than on top, is a very eligible candidate for a developing gravity current to move into and lift upward - our wave guide.

Note also the wedge of lighter orange colors along the immediate northeastern Texas coast. This is the first evidence of the forward moving arc of rising and cooling air condensing into clouds that was associated with the leading edge of the bore-induced waves.

FIGURE 5. IR satellite image from 2008-03-15 at 12Z. The two locations noted correspond to the soundings in Figures 5a and b below at the same time. Courtesy RAP/UCAR.

NOTE: Sounding data for KBRO are from actual atmospheric sounding data; all other soundings used in this report are from the ARL re-analysis using interpolation by the EDAS or NAM-12 models.


FIGURE 6a. SkewT analysis for March 15, 2008 at 12Z for KBRO. Click on location name for full-sized image. Courtesy Plymouth State.	FIGURE 6b. SkewT analysis for March 15, 2008 at 12Z for 28N95W. Courtesy ARL. Larger image.

b. GRAVITY CURRENT SOURCE - A COLD FRONT & PREFRONTAL TROUGH

Zeroing in on the synoptic charts from Figure 4, the following surface charts in Figure 7 show a "sideswipe" push into the western Gulf by a cold front. Although originally pushing directly towards the northern Gulf, the cold front slowed down as the high continued eastward rather than south or southeastward. As the morning wore on, the front slowed down over the Gulf considerably but continued more rapidly eastward over the southern states. Note at 18Z how the HPC has drawn the surface front as stopped along the Texas coast but they have drawn in a pre-frontal trough to show the continued forward push of a surface or low-level wind shift ahead of the cold front. This pre-frontal trough continued southward over the Gulf without any significant temperature changes, but it had a strong northwesterly push of drier, more dense air accompanying it, and this is what produced the bore that forced the inversion upward.

FIGURE 7a, b, c, and d. Surface analysis for March 15, 2008 at 12Z, 15Z, 18Z and 21Z. Click on time to see full-sized image. Courtesy HPC.

Figure 8 below shows the wind vectors at 925mb (roughly 2,500 feet above the surface), which show wind direction, but also strength (by the length and size of the arrows). In the 12Z chart, note the strong southwesterly flow over the northwestern Gulf. Then note the low over Arkansas ahead of the Rockies/Plains states high pressure, and how it brings a strong north to northwesterly wind flow across the Texas coast up against the southwesterly flow in the Gulf. By 18Z, the Arkansas low and the high behind it to the north have started moving eastward away from the area, and the stronger north to northwesterly flow in the northwestern Gulf has weakened considerably by 18Z.

FIGURE 8a, b, and c. 925mb vector wind analysis for March 15, 2008 at 12Z, 15Z, and 18Z. Click on time to see full-sized image. Courtesy ARL.

Figure 9 below shows the actual wind speeds at 925mb. Note how the strong northerly winds at the Texas coast at 09 and 15Z weaken considerably and stop moving southward at all by 18Z as the cold front loses its southward momentum. The initial area of 40-50 knot northwesterly winds at 09Z in northeast Texas represent the high winds of the pre-frontal trough that initiated the bore. These winds, averaging 45 knots or 23.1 m/s, represent the actual initial propagation speed of the gravity current that initiated the bore. We will use this bore propagation speed in later calculations. The elongated and slowly weakening band of winds at 35-45 knots at 15Z that weakens further by 18Z reveals the speed of the bore as it moved out into the Gulf; note how well this band of winds corresponds to the gravity waves pictured in Figure 1.

FIGURE 9a, b, and c. 925mb vector wind analysis for March 15, 2008 at 09Z, 15Z and 18Z. The warmer colors (yellow/orange) denote higher wind speeds. Click on time to see full-sized image. Courtesy ARL.

Further up in the atmosphere, at 700mb or roughly 10,000 feet (Figure 10 below), the vector wind shows the cold front and pre-frontal trough in the upper levels were barely present any more at 12Z and not present at all by 18Z as the initially strong push towards the south-southeast had died out.

FIGURE 10a, b and c. 700mb vector wind analysis for March 15, 2008 at 12Z, 15Z, and 18Z. Click on time to see full-sized image. Courtesy ARL.

c. UNDULAR BORE CHARACTERISTICS

Figure 11 (below right) shows a schematic diagram of an internal bore caused by an advancing gravity current. The following definitions will be critical for measuring our gravity current and internal bore, and will be referred to frequently:

After Koch 2004, the 3 most important characteristics of an internal bore are:

h0 = depth of the stable layer that the gravity current intrudes into, in this case the depth of the low-level inversion before the passage of the bore. This is also called the waveguide depth.

h1 = maximum height of the stable layer that gets lifted by the gravity current of depth d0. This is the height to which the top of the inversion layer is pushed by the bore intrusion.

d0 = depth of the advancing gravity current/undular bore. A good proxy is the maximum height of the wind shift after passage of the bore.

Figure 11. Schematic overview of an internal bore. Courtesy Koch 2004.

SATELLITE AND RADAR EVIDENCE OF THE BORE-INDUCED WAVES

Below in Figures 12 and 13 are infrared satellite and velocity radar images that show evidence of the bore and the waves it produced as it moved outward across the Gulf. I used the infrared satellite images for continuity, since the bore/waves began before visible images were available. Note the presence of the inversion (medium oranges) over the entire northwestern Gulf in Figures 12a and 12b. The radar images in Figures 13a-c are the first images at each location that day that showed the appearance of the gravity waves.


FIGURE 12a. Infrared satellite image at 2008-03-15 1045Z. Courtesy RAP/UCAR.		FIGURE 12b. Infrared satellite image at 2008-03-15 1215Z. Courtesy RAP/UCAR.		FIGURE 12c. Infrared satellite image at 2008-03-15 1525Z. Courtesy RAP/UCAR.


FIGURE 13a. Velocity radar returns from KHGX/KGLS, Galveston, TX at 2008-03-15 1018Z. Courtesy NOAA NCDC archives and Mark Thornton.		FIGURE 13b. Velocity radar returns from KCRP, Corpus Christi, TX, at 2008-03-15 1215Z. Courtesy NOAA NCDC archives and Mark Thornton.		FIGURE 13c. Velocity radar returns from KBRO, Brownsville, TX at 2008-03-15 1547Z. Courtesy NOAA NCDC archives and Mark Thornton.

From these images it appears that the gravity waves first passed through Galveston at about 10Z.

MEASURED CHARACTERISTICS of the UNDULAR BORE

KGLS - Galveston, Texas, was not far south of where the advancing pre-frontal trough first developed. Although atmospheric conditions were not yet right to produce the recognizable cloud patterns that were visible later in the day over the Gulf, the bore first began to produce undulating waves around the latitude of the Galveston area. The radar image in Figure 13a above (and another 1126 KHGX radar image) clearly show the atmosphere there was perturbed in a gravity wave pattern. Thus KGLS is a good candidate to use for studying the bore behavior.

We will use sounding data from Figures 14-16 below to determine the original height of the inversion, how much the inversion layer got lifted, and the height of the bore. We need to establish the following:

h0 = the top of the inversion before lifting, which is the original layer or wave guide that was lifted by the bore.

h1 = the highest level that the top of the inversion got lifted to, a key value for determining wavelength both horizontal and vertical.

d0 = the top of the level where the wind shifted into the northwest, which will tell us the height of the bore produced by the gravity current.

NOTE: The data in all of the following skewT analyses are interpolated values from EDAS, the NAM(Eta) Data Assimilation System and not from actual soundings. It is unfortunately not possible to get actual weather balloon soundings from over the Gulf.

Stable layer height before passage - h0: For this measurement, we only need to determine the original height of the marine inversion. Logically, the original inversion height had to be before 10Z, when the first waves propagated by the bore were passing through Galveston. Thus a realistic time to take the inversion height before the inversion got lifted by the bore is 09Z. Figures 14a and b show the temperature peaked at 28.5C where the 925mb height was 688 meters.

h0 = 688 meters.

Stable layer height after passage - h1: “The passage of the bore results in a sustained elevation of the stable layer” (Koch 2004). So h1 corresponds to the highest level that the inversion layer was lifted to. Looking at the 2 charts after 09Z, in Figures 15 and 16, the maximum height of the inversion was at 900mb, or 946 meters at 12Z.

h1 = 946 meters.

Depth of the gravity current - d0: Since there was no wind shift in the first two soundings, we look to the next available sounding at 15Z. From Figures 16a and b we can see the near-surface winds were southwest to west, but higher up there is a veering to the north-northwest before winds return to the west-southwest and west again at higher levels. This is the nose of the wind shift at the head of the gravity current/bore that has pushed into the inversion. From the numbers in figure 16b we can see that the top (maximum height) of this wind shift is near 925mb, or 726 meters, where the winds veered furthest to the north-northwest at 336 degrees. Therefore the estimated wind shift height after passage of the bore should be at around 726 meters.

d0 = 726 meters.


FIGURE 14a. SkewT graphic analysis for location KGLS on March 15, 2008 at 09Z. Courtesy ARL.		FIGURES 15a. SkewT graphic analysis for location KGLS on March 15, 2008 at 12Z. Courtesy ARL.		FIGURE 16a. SkewT graphic analysis for location KGLS on March 15, 2008 at 15Z. Courtesy ARL.

edas.mar08.001

Lat/Lon/Elev:  29.30  -94.80      16m

YR: 2008   MON: 03   DAY: 15   HOUR: 09

PRESS HGT(MSL) TEMP DEW PT  WND DIR  WND SPD

HPA M C C DEG M/S

 1002.     0.   17.3   17.3   201.9     9.8

 1000.     9.   17.4   17.4   201.4    10.7

  975.   225.   17.0   16.9   206.6    18.0

  950.   452.   28.0    2.1   219.2    29.4

  925.   688.   28.5   -2.1   223.5    27.3

  900.   930.   27.7   -4.9   230.3    23.7

  875.  1178.   26.0   -7.9   239.4    21.9

  850.  1431.   24.0  -11.7   249.1    20.9

  825.  1690.   21.9  -15.8   260.1    20.7

  800.  1955.   19.7  -16.6   269.0    21.4

  775.  2226.   17.3  -16.9   275.6    22.0

  750.  2504.   14.7  -17.0   281.8    22.4

  725.  2788.   12.0  -16.9   286.2    22.8

  700.  3080.    9.2  -17.3   289.2    23.5

  650.  3686.    3.5  -18.4   291.6    26.6

  600.  4328.   -2.2  -20.3   291.6    30.2

  550.  5011.   -7.7  -27.7   290.7    31.2

  500.  5744.  -12.6  -36.6   285.1    30.6

  450.  6539.  -18.0  -43.4   282.5    31.1

  400.  7408.  -24.0  -48.8   285.8    29.9

  350.  8368.  -31.0  -51.6   291.8    32.2

  300.  9441.  -38.1  -56.1   293.2    33.4

  250. 10685.  -42.2  -71.8   293.2    40.3

  200. 12164.  -51.3  -81.1   290.0    40.7

  150. 13977.  -65.0  -85.8   281.0    41.8

  100. 16390.  -72.2 -101.8   268.9    23.2

   50. 20490.  -68.5 -273.1   235.0     8.4

edas.mar08.001

Lat/Lon/Elev:  29.30  -94.80      16m

YR: 2008   MON: 03   DAY: 15   HOUR: 12

PRESS HGT(MSL) TEMP DEW PT  WND DIR  WND SPD

HPA M C C DEG M/S

 1003.     0.   17.2   17.2   239.8     5.7

 1000.    29.   18.0   16.6   231.0     4.7

  975.   246.   17.5   16.9   240.4    10.4

  950.   471.   25.2   12.8   244.4    23.5

  925.   706.   25.6    9.3   251.7    26.6

  900.   946.   25.8   -1.4   257.6    26.8

  875.  1193.   25.0   -8.5   260.9    26.9

  850.  1445.   23.2  -11.5   264.4    26.5

  825.  1703.   21.0  -12.2   267.4    26.4

  800.  1967.   18.6  -11.8   269.8    26.1

  775.  2238.   16.1  -10.1   272.5    25.6

  750.  2514.   13.4  -10.1   274.8    24.6

  725.  2797.   10.8  -10.9   276.2    23.2

  700.  3088.    8.1  -13.7   276.0    21.9

  650.  3692.    2.2  -16.9   271.9    20.6

  600.  4331.   -2.8  -17.5   272.1    24.7

  550.  5013.   -8.4  -23.4   276.7    29.7

  500.  5746.  -12.0  -34.9   284.0    32.0

  450.  6544.  -17.0  -42.5   292.8    31.3

  400.  7416.  -23.5  -46.8   292.9    29.2

  350.  8376.  -30.8  -50.6   289.1    29.9

  300.  9452.  -37.3  -60.8   283.3    30.7

  250. 10699.  -42.3  -70.0   279.8    32.8

  200. 12174.  -52.3  -82.0   278.8    42.8

  150. 13977.  -65.5  -87.1   279.2    41.4

  100. 16385.  -73.0 -102.4   285.2    23.3

   50. 20486.  -67.5 -273.1   258.1     9.0

edas.mar08.001

Lat/Lon/Elev:  29.30  -94.80      16m

YR: 2008   MON: 03   DAY: 15   HOUR: 15

PRESS HGT(MSL) TEMP DEW PT  WND DIR  WND SPD

HPA M C C DEG M/S

 1007.     0.E  18.1   17.3   270.5     2.9

 1000.    52.   18.7   16.8   256.3     4.2

  975.   269.   17.9   15.0   275.7     8.9

  950.   493.   23.5    5.1   312.3    12.0

  925.   726.   24.2   -4.5   336.0    11.5

  900.   965.   23.6   -7.7   326.7    10.7

  875.  1209.   22.4   -8.3   312.0    12.7

  850.  1459.   20.3   -8.0   305.5    15.0

  825.  1715.   18.4   -7.3   299.6    16.8

  800.  1977.   16.7   -6.7   291.5    16.8

  775.  2246.   15.0   -7.4   275.5    16.7

  750.  2522.   13.1  -10.1   259.4    19.2

  725.  2805.   10.8  -14.9   252.2    22.1

  700.  3096.    8.7  -16.6   253.4    24.2

  650.  3703.    3.8  -18.4   266.6    26.2

  600.  4345.   -1.9  -23.9   276.0    27.6

  550.  5030.   -6.1  -31.8   280.2    27.5

  500.  5769.  -10.6  -36.1   284.0    25.4

  450.  6567.  -17.6  -38.9   282.8    24.2

  400.  7436.  -24.7  -44.9   280.5    26.6

  350.  8393.  -30.8  -56.0   276.9    29.9

  300.  9472.  -36.2  -65.7   269.0    33.2

  250. 10723.  -42.2  -72.9   269.6    37.3

  200. 12193.  -53.3  -78.3   268.3    36.0

  150. 13995.  -64.5  -89.6   288.0    34.3

  100. 16409.  -74.8 -103.7   282.5    20.1

   50. 20445.  -69.1 -273.1   261.2     7.5

FIGURE 14b. SkewT numerical analysis for location KGLS on March 15, 2008 at 09Z. Top of the inversion height, where the temperature peaked, is marked in red. Courtesy ARL.

FIGURES 15b. SkewT numerical analysis for location KGLS on March 15, 2008 at 12Z. Top of the inversion height is marked in red. Courtesy ARL.

FIGURE 16b. SkewT numerical analysis for location KGLS on March 15, 2008 at 15Z. Height of the wind shift after the passage of the bore marked in red. Courtesy ARL.

Going forward we have:

h0 = 688 meters

h1 = 946 meters

d0 = 726 meters

From Koch 2004, the depth of the gravity current, d0, should be between h1 and h0. (See also Figure 11 above.)

Since 688 < 726 < 946, this verifies.

From Schaub 2005: "Furthermore, the layer immediately above the inversion must be conditionally unstable in order to prevent the wave from propagating its energy out of the stable duct layer. Most studies of gravity wave events have found that these criteria were met."

There are two important points to add about the previous data. First, all of the skewT diagrams show a conditionally unstable layer directly above the inversion, i.e. the air is cooling at a rate near or slightly less than the dry adiabatic rate and greater than the moist adiabatic rate from the inversion up to about 600mb, and then shows cooling close to the moist adiabatic rate from 500mb up to 300mb. Given sufficient moisture at those levels, any upward forcing of air parcels just above the inversion would cause them to continue to rise and cause clouds and/or precipitation. We will show later how this did indeed happen. Secondly, please note in all of the previous skewT diagram sequences how surface temperatures and dewpoints remained nearly the same or actually rose slightly after the bore passage. This a characteristic of an undular bore and we will return to this point later.

UNDULAR BORE STRENGTH:

Per Coleman, Knupp, Herzmann 2007 et al, bore strength can be approximated by h1/h0, or the height of the inversion layer after passage divided by its height before the passage. A value between 1 and 2 indicates a weak, and thus non-turbid undular bore. As values get higher than 2.0, the system will evolve into a turbulent bore, with more rapid wavelets forming along the wave tops and loss of the smoother, more orderly undular bore wave structure. Also, from Locatelli 1998:

“The most basic structural classification of bores is the distinction between undular and turbulent bores. Rottmann and Simpson (1989) have shown from tank experiments that the ratio of the post bore height to the pre-bore height (termed the bore strength) determines whether a bore is undular or turbulent. Strong bores are turbulent, which means that the disturbance is characterized by a single, turbulent transition in height and velocity. Weak bores are undular, which means that the transition in height and velocity occurs over a series of several laminar, wavelike undulations.”

For the bore at location KGLS,

h0 = 688 meters

h1 = 946 meters

d0 = 726 meters

bore strength = h1/h0 = 946/688 = 1.38 This value falls within the expected range of a non-turbulent, undular bore.

For the same measurements at a critical location further out in the Gulf, where the undular bore and the gravity waves were at their peak, please look at the calculations for 28N,95W measurements in the 2008-03-15 undular bore.

These calculations reveal the following bore characteristics, which also point to a non-turbulent, undular bore:

h0 = 475 meters

h1 = 742 meters

d0 = 508 meters

bore strength = h1/h0 = 742/475 = 1.56

Now we'll look at the Froude Number, and more closely at why a number between 1 and 2 is indicative of an undular bore.

THE FROUDE NUMBER...

The Froude number is one of the most important calculations used when examining wave generation and turbidity due to a forced current within a fluid. Named after William Froude, the Froude number was based on his analysis of speed to length ratios of induced waves in water. Although his formulas dealt with the ratio of inertial and gravity forces when waves travel through water, there have since been many different derivations developed for applications of his principles to other fluids such as our atmosphere.

For our purposes here, examining a gravity current that induces waves downstream (in this case at the top of a boundary layer), we can express the Froude number as the ratio of the speed of the upstream gravity current (mean gravity flow velocity) to the speed of the bore-induced waves downstream (phase speed of the internal gravity wave.) This simply means that we are comparing the speed of the initial gravity current that caused the bore to the speed of the bore and gravity waves that got induced within the stable layer. For Froude numbers less than 1, the flow is called subcritical; this condition represents a smooth, non-disturbing flow of the gravity current into the stable layer. For Froude numbers greater than 1, the upstream speed of the gravity current starts to exceed that of the current and waves that it induces downstream; the invading flow now begins to disturb the invaded layer by lifting it more forcefully. As the Froude number initially exceeds 1, specifically between Froude numbers 1 and 2, the gravity current speed becomes fast enough to cause waves, but they are relatively smooth and undular (Figure 1a). As the Froude number exceeds a value of 2, the invaded layer gets lifted forcefully enough to produce very turbulent wave motion (Figure 1b). From this we can see why Froude number 1 is called the critical number - it is the breaking point where the advancement of one fluid flow into another begins the change to turbulent.

... & THE HYDRAULIC JUMP

Hydraulic jumps occur, by definition, when the gravity current flow has a Froude number greater than 1, implying that this current is pushing forward into a fluid at a speed higher than the speed further downstream in the invaded flow. In some cases, such as in Figure 1a below, where the Froude number was 1.35, the speed of the invading bore/gravity current is not much faster than that of the flow it's invading, so that it causes a relatively smooth "jump" of the flow into undular waves. This is the undular hydraulic jump. Quite often, however, the flow of the gravity current is so fast (Froude numbers of 2 and up), that the flow where this jump occurs changes rapidly from smooth to turbulent, as seen in Figure 1b below, where the Froude number was 7.0.

"For a Froude number slightly above unity, the hydraulic jump is characterized by a smooth rise of the free-surface followed by a train of stationary free-surface undulations: i.e., the undular hydraulic jump (Fig. 1a). For larger Froude numbers, the jump is characterized by a marked roller, some highly turbulent motion with macro-scale vortices, significant kinetic energy dissipation and a bubbly two-phase flow region (Fig. 1b)." - From Hydraulic Jumps: Bubbles and Bores - H. Chanson


Figure 1a is the smooth bore: Undular hydraulic jump. Flow conditions : Fr1 = 1.35, d1 = 0.090 m. Flow from left to right. Courtesy H. Chanson.	Figure 1b is the turbulent bore: Hydraulic jump with roller. Flow conditions : Fr1 = 7.0, d1 =0.024 m. Flow from left to right. Courtesy H. Chanson.

I've included the Froude number calculations for the 2008-03-15 undular bore using location 28N,95W. The results give us a Froude number of 1.65, characteristic of a undular bore.

WAVE CHARACTERISTICS:

For this analysis, we will use location KGLS. Here again, for ease of reference, are the bore characteristics determined earlier:

KGLS: h0 = 688 meters, h1 = 946 meters, d0 = 726 meters

WAVE SPEED Since the bore and its waves were moving almost due south from Galveston into the Gulf, and we've already determined that the bore/wind shift came through KGLS at about 10Z and then through 28N,95W at about 18Z, and the distance between the two points is 100 miles, we can determine a reasonable approximation for the speed of the bore and its advancing waves of 100 miles in 3 hours or 33 mph = 28.7 knots = 14 m/s. Please see the CIMSS analysis which calculated an average speed of 25-30 knots, in very good agreement with our number.

Wave speed in the central Gulf = 14 m/s.

WAVELENGTH HORIZONTAL (crest to crest)

Examining the image in Figure 1, it appears that the length from crest to crest is a little less than the length of the mouth of the Corpus Christi Bay, or roughly 8 miles. 8 miles = 12.8 km ~ 13 km.

Horizontal wavelength = 13 km

A formula from Clarke et al 1981 is that for bore strength between 1 and 2, the value for the horizontal wavelength = (10 +/- 4)*h1; Using h1= 946 meters from above, the range of the horizontal wavelength = (10 +/- 4)*946 m = 6*946 - 14*946 m = 5,676 m - 13,244 m ~ 5.7 - 13.2 km.

Thus the expected range for the horizontal wavelength = 5.7 to 13.2 km.

Our estimate of 13km is within the calculated range.

WAVELENGTH VERTICAL (trough to crest): A formula (per internal bore diagram from Koch, see Figure 5) is:

Wavelength vertical = 2*(h1 – d0). Using the previously established values:

Wavelength vertical = 2*(946 - 726) = 2*(220)

Vertical wavelength = 440 meters.

From Schaub 2005: “For a gravity wave to propagate with minimal loss of energy, there are theoretical requirements (as put forth by Lindzen and Tung, 1976) for the stable layer, as well as for the thermodynamic structure above the inversion. For example, the static stability of the stable layer must be large, and the layer must be deep enough to contain at least a quarter of the vertical wavelength.”

The stable layer depth = 688 meters. (Value of h0, or original depth of inversion.)

¼ vertical wavelength = 440/4 = 110 meters.

Since a 688 meter depth clearly can contain 110 meters, this requirement also verifies.

MORE ON WHY THIS WAS AN UNDULAR BORE

In this section I would like to briefly rule out other atmospheric disturbances that may be confused with undular bores.

"Haase and Smith (1989) described what they call a “modified gravity current,” which had the appearance of an undular bore but differed from a true undular bore in that there was positive flow relative to the disturbing motion behind the leading edge and cooling took place due to the advection of the gravity current air. Such a disturbance occurs when the phase speed of the solitary waves is slower than the gravity current intruding on the stable layer. Since the disturbance does not move ahead of the gravity current, such a complex structure is best referred to as a modified gravity current. These modified gravity currents can evolve into a series of solitary waves as the gravity current weakens and the denser fluid separates from the solitary waves. Fulton et al. (1990) and Doviak et al. (1991)" - Locatelli 1998.

Figure 18c at right shows the configuration of such a modified gravity current. For our case, there was no additional positive flow (no more push) behind the original positive flow of the cold front and pre-frontal trough. As a matter of fact, we have previously shown that the initial southward-moving energy from the pre-frontal trough slowly died out as the Plains states high moved eastward and that any additional energy was deflected eastward. We have also shown that the disturbance (the waves) did indeed move out (and stay out) ahead of the gravity current for the duration of this event. In addition, it is clear from the temperature and dewpoint readings in the skewT diagrams that no cooling at all occurred behind the leading edge of the waves and then the bore; indeed, temperatures remained steady or rose slightly (see Table 1 below.)

"Only a few of the atmospheric disturbances that have been labeled as undular bores actually fulfill the requirement that the stable layer remain elevated for an extended period of time following the passage of the bore." (Locatelli 1998)

The skewT diagrams and numeric charts used previously in Figures 14, 15 and 16 have clearly shown the prolonged elevation of the inversion.

"This cloud system was associated with a surface wind shift but not with a surface temperature or dewpoint change and was referred to by NWS bulletins as a ‘‘prefrontal trough.’’ These characteristics, along with decreasing wave strength (as noted by horizontal width and optical reflectivity of the cloud bands) are indicative of an undular bore." (Clarke 1998).

The synoptic evidence previously presented shows the existence of a prefrontal trough, and the skewT data from Figures 14-16 and Table 1 below show the wind shift (bore passage) with a lack of any appreciable surface temperature or dewpoint change.

FIGURE 18. The true undular bore configuration in a, with similar but not true bore configurations in b and c. Courtesy Locatelli 1998.

Station: Galveston/ TX US KGLS 4 29.30 -94.80 16 72242 81
Data for KGLS from 0000Z 15 MAR 08 to 2300Z 15 MAR 08

ID     TIME   T   TD  RH  DIR SPD GST ALT SLP VIS CIL  COV WX  MAX MIN  PR6 PR24 SC      
KGLS   0852  66  64  93 210  15  22  958 015   6   5  OVC H                       
KGLS   0952  67  64  90 220  15  22  958 018   6   5  OVC H                       
KGLS   1052  67  64  90 250  13  20  962 031   6   5  OVC H                       
KGLS   1152  66  63  90 270   7        968 051   8  16  SCT      67  66             
KGLS   1252  66  62  87 220   7        966 043   4      CLR H                       
KGLS   1352  68  63  84 220   9        968 051   8      CLR                         
KGLS   1452  73  63  71 250   9        972 062  10      CLR                         
KGLS   1552  77  63  62 270   9        974 070  10      CLR

TABLE 1. METARS for KGLS from 09 to 16Z March 15-16, 2008. Note that the cloud/wave layer passage and wind shift was not accompanied by a notable temperature or dewpoint change.

THE VISIBLE EVIDENCE - CLOUDS

Although undular bores are fairly common, very distinct cloud evidence of them is rare. This is because not only must there be a fairly widespread, moist and shallow layer (the wave guide) ahead of the bore, but there must be a conditionally unstable layer above it to allow the moist stable air parcels at the top of it to get lifted into clouds. Finally, most of the unstable layer just above the lifted layer must contain relatively dry air, so that when clouds do form they do not continue upward enough to cause higher, spreading clouds that would obscure the undular formation. Thus we would expect an atmospheric profile much like that in Figure 6b (reproduced at right). Note the wedge of very dry air above the top of the moist surface layer that got lifted. In the case studied here the low level saturated layer was deep enough to begin with (roughly 500-700 meters) and got lifted high enough (200-300 meters) to form shallow clouds along the wave crests that were thick enough to be seen from ground and space. The NASA LARC data from the chart in Figure 19 at right show that the cloud tops reached to no more than 1,000 to 1,500 meters.

In summary, as the gravity current pushed the moist air in the stable layer in front of it up into the conditionally unstable layer above it, the saturated air parcels at the top of the stable layer did not have to be forced upward very far before clouds developed. But as soon as these clouds reached a few hundred meters in height, the dry layer above it prohibited further upward development.

FIGURE 6b reproduced. SkewT analysis for March 15, 2008 at 12Z for 28N95W. Courtesy ARL. Larger image.

FIGURE 19. Cloud top heights in thousands of meters. Note the cloud tops associated with the bore in the Gulf did not exceed 2km. Courtesy NASA Langley Cloud and Radiation Research

CONCLUSIONS

The analysis shows that this series of waves was indeed produced by an undular bore. The bore occurred under ideal conditions to produce clouds at the crests and clear skies at the troughs so that we had very fine visual evidence of the bore-induced waves. The bore produced a large number of waves due to a slow, steady speed over a long period of time through a very large area. To summarize:

1. This was an undular bore caused by a pre-frontal trough pushing into a low-level marine inversion layer. It moved south to south-southeastward at about 14 m/s with a vertical wavelength of ~ 440 meters and a horizontal wavelength of ~ 13km. At its widest the bore and resultant atmospheric waves covered an arc from southeastern Louisiana to southern Texas, a distance of roughly 500 miles, and traveled a north to south distance of as much as 200 miles.

2. The bore maintained a steady and long duration because it pushed through a very extensive, very stable inversion layer at a steady rate of flow. The gravity current (bore) strength (h1/h0) and the Froude numbers both exhibited values of between 1 and 2 at all points measured, ideal for an undular bore. In addition, the pre-frontal trough/gravity current that produced the undular bore and its waves maintained an ideal speed to keep the bore strength at these levels, never exceeding a speed that would produce numbers high enough to cause turbulent flow. The bore ended as the inversion deteriorated under daytime solar heating of the layer and mixing of the PBL.

3. The production of these gravity waves was aided by a conditionally unstable layer just above the inversion; in addition the stable layer that was lifted contained sufficiently high RH to produce highly visible low-level cloud evidence of the wave structure.

4. All the measurements conducted in this study against benchmarks previously established in the literature bore out.

One additional point of interest: in the satellite loop of the bore in Figure 1b, there are two clearly visible layers of low-clouds and fog along the lower Texas coast and in the middle of the northwestern Gulf . These two layers appear to get "absorbed" into the wave structure as it moves over them. In reality, the stronger northwest to north-northwest winds at the leading edge of the bore (roughly near the rear edge of the cloud waves) caused enough mixing of the clouds with the drier air above to cause their dissipation. This can be seen if you look closely at the satellite loop. The low cloud areas can still be seen under the forward to middle sections of the cloud waves, and don't really dissipate until they near the back edge of the wave clouds where the mixing winds of the bore were located.

APPRECIATION

A special thanks to Professor Lee Grenci of Penn State for pointing me to this event and providing the initial insight to its causes, as well as providing valuable reference data and guidance. And especially thanks to Steven Corfidi of SPC for his insight and for reviewing this study. Thanks to Penn State's professor Steven Seman and Dr. George S. Young for reviewing this work. And finally thanks to Mark Thornton, Penn State Certificate graduate, for providing the radar images and invaluable support.

REFERENCES

1) Rottman, J.W., and J.E. Simpson, 1989 - The formation of internal bores in the atmosphere: A laboratory model. Quart. J. Roy. Meteor. Society.

2) Mahapatra, Doviak and Zrinc 1991 - Multi-sensor Observation of an Atmospheric Undular Bore

3) Koch 2004 - The Structure and Dynamics of Solitons and Bores

4) Schaub 2005 - Gravity Waves: Those Important to Operational Weather Forecasting

5) Coleman, Knupp, Herzmann 2007 - The Spectacular Undular Bore in Iowa on 2 October 2007

6) H. Chanson - Hydraulic Jumps: Bubbles and Bores

7) Smythe Holloway 1988 - Hydraulic Jump and Undular Bore Formation on a Shelf Break

8) Clarke 1998 - An Atmospheric Undular Bore Along the Texas Coast

9) Wakimoto & Kingsmill - Structure of An Atmospheric Undular Bore Generated From Colliding Boundaries During CaPE

10) Locatelli 1998 - Structure and Evolution of an Undular Bore on the High Plains and Its Effects on Migrating Birds - John D. Locatelli, Mark T. Stoelinga, Peter V. Hobbs and Jim Johnson