A case study of a mesoscale trapped gravity wave accompanied by severe weather

Met Apps 1, 2lJ-231 (1994)

A case study of a mesoscale trapped gravity wave accompanied by severe weather Erik A Rasmussen, Niels Bohr Institute for Astronomy, Physics and Geophysics, Geophysical Department, University of Copenhagen, Denmark

The unexpected development of a mesoscale disturbance accompanied by severe weather in the form of strong convection, dense snow and strong gusty winds in a synoptic situation where most of Denmark would normally be sheltered by the Norwegian mountains is discussed. Using all the meteorological data available, including a variety of non-synoptic data, supported by a linear model, it is concluded that the phenomenon which caused the outbreak of severe weather was a low-level ‘trapped‘ or ‘ducted’ gravity wave.

I . Introduction

In this paper we discuss an outbreak of severe weather in the form of unusual dense snowfall and large wind velocities associated with several lines of forced convection over Denmark on 3 March 1980. The unexpected appearance of severe weather in a synoptic situation which would normally be characterised by little cloud development over most of the country caused a tragic accident in which two people were killed due to the crash of a small aircraft. Poor visibility was prob- ably the main cause of the aircraft accident but strong wind gusts may have contributed.

Mesoscale features in the form of rainbands and squall lines occur frequently in most regions of the earth. They ‘affect society in major ways because of their wide geographical range and frequency of occurrence and because they typically contain high rainfall rates’ (Hane, 1986).

Rainbands in extratropical regions occur primarily in association with well-organised extratropical cyclones and ‘the preponderance of research on rainbands has been carried out in the extratropical cyclone setting’ (Hane, 1986). However, rain(snow)bands can occasionally be found in other synoptic-scale environ- ments, as discussed below.

Rainbands in extratropical cyclones have recently been explained as the result of conditional symmetric instability (CSI) (see for example Bennets & Hoskins, 1979; Emanuel, 1983). In the present case CSI is not a likely explanation for the observed series of cloud/snow- bands, which form well away from the nearest frontal zone. Instead, we will demonstrate that the convective cloud bands leading to the severe weather outbreak were most likely caused by a ‘trapped’ or ‘ducted’ gravity wave train.

Gravity waves are easily set up in a stable stratified atmosphere, but since in most cases they can propagate freely in the vertical their residence time in the troposphere is generally short. O n the other hand, in certain situations gravity waves can propagate horizontally even over large distances. If, for example, a stable lower troposphere is bounded upwards by a region which effectively reflects the vertically propagating waves, a duct is created in which the waves can propagate horizontally without losing their energy. Ducted or trapped gravity waves have been extensively discussed by Gossard & Hooke (1975) and by Lindzen & Tung (1 976).

Gravity waves ‘are virtually ubiquitous’ and may be considered as ‘the simplest, most fundamental motions that exist on the mesoscale’ (Hooke, 1986). They are known by forecasters as a cause of clear air turbulence and of some downslope windstorms such as chinooks and maybe foehns, but apart from that their effect on the weather is generally considered to be small.

Atmospheric gravity waves with short periods, typically of around 10 minutes, pressure amplitudes around 100 pb and with phase speeds of 30 m s-I, have been investigated by many researchers (see for example Gedzelman & Rilling, 1978). These waves generally move with a velocity characteristic of the upper troposphere and exhibit features which indicate that they originate from shearing instability at upper levels. These high frequency waves may, as shown by Gedzel- man & Rilling, be used as predictors of rain but have no direct impact on the weather.

Provided the right synoptic setting is present, gravity waves may also give rise to important mesoscale weather phenomena, and they are increasingly, especially among American meteorologists, recognised for their ability to modulate cloud and precipitation fields,

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to produce and interact with deep convection, and to transport momentum and energy both horizontally and vertically (see Bosart & Cussen, 1973; Eom, 1975; Uccellini, 1975; Miller & Sanders, 1980; Bosart & Sanders, 1986; Uccellini & Koch, 1987; Koch & Dorian, 1988a, b; Koch et ul., 1988, 1993).

The 3 March case over Denmark was characterised by a weak surface pressure signal but a strongly fluctuating wind field. This is in accordance with Eckart (1960), who pointed out that oscillating, and occasionally, fairly strong winds, are characteristic of gravity waves.

In a European connection, observations recently made during the FRONTS 84 field experiment in south- western France revealed the existence of a ducted gravity wave with a 90-minute period (Ralph e t ul., 1993). The perturbations in the wind field associated with this wave were of the same order of magnitude as the corresponding ones discussed in this paper, and both waves were associated with the occurrence of deep convection.

As with many other studies of gravity waves our study has been hampered by scarcity of data. Hourly data, for example, are virtually useless for tracking the waves or determining their structure. In the present study a combination of all available data, including ‘standard data’ from the synoptic surface and upper-air network as well as some special high-resolution data, have been used together with a theoretical model to explain the observed phenomenon.

Only through the use of a theoretical model can the very diverse data be put into a common context. The simplest possible model suitable for our purpose has been used; this consists of an atmosphere extending upward to infinity and bounded below by a rigid, level surface. We consider the atmosphere to be made up of two layers separated by a discontinuity in density at a height, H, above the ground. In each of the two layers we assume that plane-wave solutions apply. Plane- wave theory from linear models is only an approximation to reality, but, as noted by Gossard & Hooke (1975), ‘these simple models provide a remarkably good prediction of observed wave phenomena over limited regions of the atmosphere’, and also provide a conceptual framework which may be used by opera- tional meteorologists to interpret high-resolution satellite images.

An outline of this paper is as follows. A brief descrip- tion of the synoptic situation on 3 March 1980 is presented in section 2 followed by a discussion of some special observations in section 3. In section 4 the basic theory explaining the structure of the observed phenomenon will be outlined. Section 5 contains a discussion and a comparison of the theoretically derived results with observed data, and finally some concluding remarks are presented in section 6. A map showing

Figure 1. Geographical map showing locations. 1: Aalborg, 2: Schleswig, 3: Aarslev, 4: Sproge 5: Copenhagen.

the geographical locations mentioned in the text is shown in Figure 1.

2. The synoptic situation

O n 3 March 1980 a major low was centered over eastern Russia (near 55 N 30 E) and a deep northerly flow of polar air prevailed over southern Scandinavia. In such situations, Denmark would normally be sheltered by the mountains in southern Norway, which locally reach a height of almost 2500 m, and the lee effect sometimes may be felt as far south as Berlin. Figure 2(u) shows the surface map and Figure 2(b) the 700 hPa upper-air map at 3 March, 0000 UTC. The 300 hPa map (not shown) shows that the northerly flow north and north-west of Denmark extended through the entire troposphere. A jet streak at the 300 hPa level from the southern part of the Norwegian Sea to North Greenland was moving south on 3 March.

Figure 3 shows the radiosonde ascent from 3 March 1980, 0600UTC from Aalborg in northern Jutland. The ascent reveals a low-level stable layer, 1200 m deep, capped by a weak inversion and a deep neutral layer with a dry adiabatic lapse rate. The formation of the dry layer from the surface up to around 700 hPa (including the neutral layer) can be ascribed to vertical stretching due to descent in the lee of the Norwegian mountains about 200 km upstream. Gravity waves cannot propagate through a neutral (dry adiabatic) layer (see section 4), which therefore acts as a reflecting region forming a low level duct for the ‘trapped’ waves. In our case (as pointed out by a anonymous reviewer) the depth of the neutral layer might not be sufficient to ensure a ‘strong trapping’ of the fairly long waves observed, so the waves are likely to be ‘leaky’. O n the

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10’ 20’ E N

60

Figure 2. Synoptic charts for 3 March 1980, 0000 UTC. (a) Isobars at 5 hPa intervals. (b) 700 hPa contour heights at 4 dm intervals.

other hand, the signals of the waves are strong at the surface and coherent waves are observed at two locations. A possible ‘leaking effect’, which may be felt over long distances, therefore does seem to be critical for the analysis carried out in this study.

Figure 3. Radiosonde ascent, Aalborg, 3 March 1980, 06011 UTC (skew T-In p diagram). Inversion between lower stable layer and upper neutral layer arrowed.

The gravity waves were observed in a region including most of Jutland, Funen and Zealand. In the following analysis we assume that the temperature profile from the Aalborg radiosonde ascent from 0600 UTC is representative for this region during the time the gravity waves were observed. This assumption is justified because the large-scale flow changes only very little during this time. The radiosonde ascent from Kastrup, Copenhagen, from 1200 UTC (Figure 4(b)) shows some of the same features as seen on the Aalborg ascent, i.e. a low stable layer and a neutral layer aloft separated by a stable region. The stable layer observed at Kastrup, however, extends over a deeper layer than the inversion at the Aalborg ascent. Further south, in a region not influenced by the Norwegian mountains, the 12OOUTC ascent from Schleswig (Figure 4(a)) shows an approximately moist-adiabatic stratification up to 600 hPa. As is typical for such situations the weather forecasts for Denmark only warned of isolated showers, especially in the south-western part of the country. No major changes were expected in the general weather situation during the day. A satellite image from 0742 UTC (not shown) shows only a few scat- tered clouds over western Denmark.

The surface pressure analysed at 1 hPa intervals at 0900 UTC (Figure 5) shows a weak trough over Jut- land. As the trough moved south the winds ahead of the trough backed temporarily from a north-westerly to a westerly direction. The trough, though weak, was important because the westerly surface winds advected moist low-level air over land. An unusual feature was the packet of waves with a wavelength around 100 km and an amplitude around 0.5 hPa embedded within the ‘large-scale’ trough.

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Figure 5. Surface map, 3 March 1980, 0900 U T C . Isobars at 1 hPa intervals. Broken lines indicate positions of small-scale ridges and troughs associated with the gravity wave packet.

It should be noted that:

Figure 4. Radiosonde ascents, 3 March 1980, 1200 U T C (skew T-ln p diagram). (a) Schleswig. (b) Copenhagen.

At 1005 UTC a NIMBUS 7 satellite image (Figure 6) shows a line (A-A) of convective clouds over the central and southern part of Jutland, including two major storms over the southern part of the country. The cloud band over Jutland was preceded by another north-south orientated cloud band (1-1) over Funen stretching northwards. One more-weakly developed cloud band can be seen still farther east. The best- developed cloud line, A-A, can be recognised farther east on a satellite image a few hours later. The cloud bands all extend far into the lee region south and south-east of the Norwegian mountains.

0 The cloud bands, forced by lifting due to the low- level gravity waves, are situated approximately along the wave crests in the surface pressure field. The surface wind speeds have their maximum along the pressure wave crest lines (see section 4), where also the vertical displacements of the air parcels have their maximum. The phase velocity of the waves and the direction of the horizontal wind perturbation are both per- pendicular to the orientation of the pressure wave crests (cloud lines).

Observations have shown that gravity waves sometimes remain coupled with the storms they trigger, whereas at other times they pass right through the convection because of the high phase speed of the waves (Koch et al., 1988). In the present case the convective clouds seem coupled to the wave.

Cloud line A-A, along which the strongest convection is observed at 1005 UTC, displays a slightly arc- shaped appearance. The formation of curved wave- fronts is typical for many gravity waves and has been associated with the development of strong thunder- storms along the wave crests, which locally change the wave propagation behaviour (Koch et ul., 1988).

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Figure 6. NIMBUS satellite image, 3 March 1980, 1005 UTC, showing cloud band A-A over Jutland (marked with black arrowheads). Another cloud band I-I (marked with white arrowheads) is seen over Funen stretching northwards. Photograph courtesy of Satellite Receiving Station, University of Dundee.

Only a low-resolution image ((not shown) is available, from a TIROS-N passage at 1.227. The configuration of the cloud system at this time is very much like that from 1409 UTC, shown in Figure 7.

Cloud line A-A is clearly recognised in Figure 7 at a position about 150 km farther east than in the previous satellite image (Figure 6). At 1409 UTC, cloud line A- A has started to merge with the preceding cloud lines in the wave packet, forming a fairly large cloud complex which may be characterised as a small mesoscale convective system (MCS). Very dense snow was reported over southern Zealand during the afternoon.

Part of a new cloud band, B-13, which formed behind the major cloud band A-A, can be seen in Figure 7.

The broken line of clouds seen west of cloud line B- B may indicate another cloud line about to form in the next wave crest.

The synoptic surface maps from 3 March, 1200 UTC and 1500 UTC are shown in Figures 8 and 9. Also the 850 hPa map from 1200 UTC is shown in Figure 10. The trough situated over northern Jutland at 0900 UTC can still be seen on the 1200 UTC surface analysis, but is not discernible at the 850 hPa level (or at higher levels). The 1200 UTC surface analysis, like the one at 0900 UTC, reveals the existence of a wave packet of small-amplitude waves with a wavelength of around 100 km. The amplification of the wave crest over the south-eastern part of Denmark (arrowed) in the region where the cloud lines merge may be a

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Figure 7. TIROS-N A V H R R satellite image, channel 2, 3 March 1980, 1409 UTC. Cloud band A-A (black arrowheads) over Zealand is merging with preceding bands, forming a small mesoscale convective system capped by dense cirrus. Parts of cloud band B-B (white arrowheads) are separated from band A-A by a region of less-developed clouds. Photograph courtesy of Satellite Receiving Station, University of Dundei .

hydrostatic effect from cold downdrafts originating from cells of strong convection.

3. Special observations

In this section we consider data from sources not used for routine meteorological work. Data from Aarslev, a station in the network of climatological stations, will be applied, and also data from Sprogs where a meteorological mast is operated by the National Laborat- ory, Riser (for the location of the stations see Figure 1 ) .

3. I The Aarslev data

The Aarslev data set includes measurements of the total incoming solar radiation, the surface temperature and the surface wind field.

The maximum lift of air particles due to the passing gravity waves will occur at the ridges in the surface pressure field where the surface wind also has its maximum (see section 4). However, the formation and the decay of the clouds are associated with the total history

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7' 10" 13' E Figure 8. Surface map, 3 March 1980, 1200 U T C . Isobars at 1 hPa intervals. Broken lines indicate positions of small-scale ridges and troughs associated with the gravity waves.

N

65'

5 5 O

Figure 10. 850 hPa map, 3 March 1980, 1200 UTC. Contour heights at 4 dm intervals.

The radiation measured by two different instruments increased rapidly during the morning from sunrise until around 0900 UTC, when some irregular oscillations were observed caused by cloud band 1-1 leading the first major cloud band (Figure Il(a)). From 1100 to 1200 UTC the radiation temporarily decreased due to increasing cloudiness associated with the passage of cloud band A-A. The second minimum around 1330 GMT was caused by cloud band B-B (Figure 7). Finally, a third minimum in the incoming radiation was observed around 1440 UTC. The pronounced very regular oscillations in the incoming radiation reflect the change between an almost clear sky and almost or total overcast conditions. The variations in the amount of clouds are also reflected in the surface temperatures, which oscillate in phase with the incoming radiation.

The passage of the wave train was reflected in both the strength and the direction of the surface wind field

7' 10' 130 E measured at a height of 10 metres (Figure l l(b)). The wind speeds at Aarslev were measured as ten-minute mean values, whereas the wind directions were instan- taneously measured every ten minutes. As discussed in the following, the seemingly erratic wind field is the result of a superposition of the slowly changing mean wind field with a passing gravity wave train.

Provided the mean (synoptic scale) wind is approximately constant it is possible, following Gossard & Munk (1954) and Gossard & Hooke (1975), to resolve the wind field into a mean vector and an orbital wind

22 I

Figure 9. Surface map, 3 March 1980, Il iOO UTC. Isobars at 1 hPa intervals.

of the wave, and the locations of the cloud lines (the minimum of incoming solar radiation) do not neces- sarily coincide with the locations of maximum surface wind and pressure.

Erik A Rasmussen

. . I...... . :

Figure 11. Measurements from Aarslev, E'unen, 3 March 1980. (a) Radiation and temperature measurements. Curves I and I I show the short-wave radiation measured (in relative units and by two different instruments) at the surface. Curve I I I shows surface temperature in "C. (b) Wind measurements. Curve I shows the wind velocity Ifj 10-minute mean values in m 5-l) and curve I I the instantaneous wind direction (dd) measured once every 10 minutes, both at 10 metres height. A, B, C and D mark the passages of four wave crests with a period of 90 minutes. The secondary maxima B'$ and C" indicate waves with half the period of the major oscillations.

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vector associated with the passage of the wave. How- ever, in our case the mean wind is far from constant over the relatively long time required for the waves to pass the observation point. Also, the wind velocity, and especially the wind strength, is affected by the surface and is not fully representative of the velocity fluctuations associated with the wave. For this reason we have chosen a slightly different way to that suggested by Gossard and his collaborators to illustrate the effect of the passing wave train on the surface wind field.

The surface wind direction (Figure l l(b)) veers fairly regularly from west to north from noon (1200 UTC) to around 1600 UTC, except for a temporarily but pronounced backing around 1500 UTC. The most prominent feature in the graph showing the surface wind strength is the maximum around 1330 UTC.

The first wave crest (A) passed Aarslev at 1200 UTC, when a pronounced minimum in the incoming solar radiation was observed. The second wave (B) passed at 1330 UTC, and the third (C) at 1500 UTC, corresponding to a period of 90 minutes. The passages at 1330 and 1500 UTC are clearly indicated by changes in the wind field through a peak in the wind strength at 1330 UTC, and a pronounced backing at 1500 UTC (Figure II(b)). A minimum in the incoming radiation is observed almost simultaneously with the passage of the wave crest B, and slightly before the passage of crest C. Finally, the backing wind at 1630 UTC indicates the passage of a fourth wave (D), again corresponding to a period of 90 minutes.

The variations in the surface wind field at Aarslev may be explained by assuming that the field is a superposition of a slowly varying (synoptic) mean field and a field caused by the gravity wave. The satellite images show that the wave fronts are oriented approximately north-south, which means that the component of the surface wind associated with the wave oscillates between east and west. In Figure 12 there is a sketch of how the wind varies as the waves pass and as the mean field slowly changes. The sketches show how the resulting surface wind can be regarded as the sum of the two contributions from the mean field and the wave, and features such as the peak in the wind strength at 1330UTC and the strong backing of the wind around 1500 and 1630 UTC are easily explained through this assumption.

The minor maximum (B") in the wind strength around 1420UTC and another more pronounced (C':) at 1550 UTC indicate the existence of waves with half the period of the ones considered so far. The shorter waves only partly modify the wind and radiation field at Aarslev, but at the time the wave train passes Sprogs wave B'$ has increased significantly (see next section).

In conclusion, all significant variations in the rather complex wind field observed at Aarslev between 1200

B- T

-0

M 'I b Figure 12. Schematic diagram illustrating the changes in the horizontal wind field during the passage of wave crests B, C and D indicated in Figure 11. (a) Crest B (1330 UTC): Both the mean wind @I) and the orbital wind component (0) are westerly resulting in a peak in the total wind speed (T) during the passage of the wave crest. (b) Crest C (1500 UTC): The mean wind has veered but the speed is approximately the same. During the passage of the wave crest the wind temporarily backs and increases a little. (c) Crest D (1630 UTC): The mean wind has veered a little further and increased a little. The wind backs temporarily as crest D passes.

and 1700 UTC on 3 March 1980 can be explained as a superposition of a slowly changing mean wind field and a passing gravity wave.

3.2. The Sprogra data

Sprogar is a small island situated in the middle of the Belt between Funen and Zealand (see Figure 1). An instrumented tower on Sprogs run by the National Research Laboratory (Ris0) measured wind and temperatures at different levels at lo-minute intervals during the passage of the wave packet. Due to the position of the tower on a small island the contaminating effects on the data from the surface were minimised.

Wind directions and wind speeds at 69.7 metres height are shown in Figure 13(a). The wind directions are 'instantaneous' (not averaged) values measured every 10 minutes and as such are rather 'noisy'. Also, the maximum gusts for every 10-minute interval were recorded and shown in Figure 13(a). Temperatures (10-minute mean values at 69.7 metres height) are shown in Figure 13(b).

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I

Figure 13. Measurements from Sprog0 3 March 1980. (a) Wind measurements. Curve I shows the mean wind speed over 10-minute intervals at 69.7 metres height, and curve I I the maximum gust values, both in m s-'. Curve I I I shows the instantaneous wind direction in degrees measured every 10 minutes. (b) Temperature (10-minute mean values at 69.7 metres height) and surface pressure.

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Sproger is situated not far away from the region where the waves were merging to form a small mesoscale convective system. Nevertheless, the wave packet first seen in the Aarslev data is clearly discernible in the Sproga data.

Figure 13(b) shows a temperature oscillation with an amplitude around 1.5 "C caused by variations in the

(a) 02

+k .4- - A3. - - - - - *.

+0 .-., -.. +9

incoming radiation due to varying amounts of cloud and, as in the Aarslev data set, 180" out of phase with

x + 10

the wind speed.

The surface pressure measured at Sproga is also shown on Figure 13(b). The pressure falls from around 1100 UTC until around 1400 UTC, and starts rising again around 1600UTC due to the passage of the minor trough. The weak pressure oscillations due to the gravity waves are masked by pressure falldrises caused by the passing trough, but can be seen fairly well at the time the two last major wave crests in the wave train, B'$ and C, pass Sproga. The pressure disturbances are the right order of magnitude according to the theory outlined in the following section (i.e. around 0.5 hPa) and occur simultaneously with the maxima in the surface wind.

The first two maxima (marked A,B) on the graph showing the surface wind strength on Figure 13(a) correspond to the similar marked features in the Aarslev data set. The period for these waves at both stations is 90 minutes and both waves required 35 minutes to travel from Aarslev to Sprognr. The third wave in the Riser data set, B'>, follows relatively quickly after the second, B. This wave corresponds to the wave in the Aarslev data with half the period of the waves A and B. Wave B'$, which was only weakly developed at the time the wave packet passed Aarslev, is much more prominent at Sproga where it caused the strongest winds (gusts up to 20 m s-') observed at Sproga during the passage of the entire wave train.

The method of resolving the wind field into a mean wind and an orbital wind vector due to the wave, as suggested by Gossard and his collaborators, was attempted for the Sproger data set. In Figure 14(a) the wind velocities have been plotted for every 10 minutes for the first period (i.e. from 1235 until 1405 UTC). To avoid spurious noise affecting the wind direction the 10-minute values have been smoothed with a run- ning three-point filter. To get the strongest possible signal the gust values were used for the wind strength, but the mean values would give a qualitatively similar result.

During the first half of the period from 1235 to 1315 UTC the direction of the mean wind was relatively constant. The individual wind vectors were aligned along the direction 265 degrees, defining a mean wind of 9 m s-' with that direction and an orbital wind vector oriented in the same direction, i.e.

N

Figure 14. (a) Mean winds and orbital vectors at Sproge from 1235 to 1405 UTC (see text for details). Horizontal winds have been entered as dots or crosses for every I0 minutes beginning at 1235 UTC and ending at 140fi UTC. The numbers 1 (1235 UTC) to 5 (1315 UTC), marked by dots, correspond to the part of the oscillation between crest A and the following trough. The numbers 6 (1325 UTC) to 10 (1405 UTC), marked by crosses, correspond to the part of the oscillation between the trough and the next crest (B) (see Figure 13(a) ). (b) Schematic diagram showing the influence of a changing mean wind on the direction of the orbital wind. As the mean wind (of constant strength) veers through the positions I , 2 and 3 over a time corresponding to a period of the orbital wind the orbital plane seems to change from an east-west orientation to a north-west-south-east. The mean wind and the orbital winds are indicated by vectors. The total wind is indicated by circles (vectors not drawn).

approximately east-west, with an amplitude equal to 2.5 m s-'. This orientation of the orbital wind vector corresponds to a north-south orientation of the wave (and the cloud bands).

During the second half of the period from 1325 to 1405 UTC the orbital wind vector seems to change. However, the change in orientation of the orbital wind vector is only apparent as it is caused by a change in the direction of the mean wind as sketched in Figure 14(b). This figure shows how the observed change in the mean wind will cause a change in the orientation of the line defined by the end of the wind vectors, even if the orbital motion is constant. Considering this, the orientation of the wave during the second half of the period is therefore the same, north-south, as during the first half period.

Mainly because of variations in the mean winds, the variation in the surface wind field is different as the wave train passes first Aarslev and then Sproger. It is possible, however, at both locations to explain the

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complex variation of the surface wind field within the context of the superposition of a slowly varying mean wind and a passing gravity wave train. The Aarslev and Sprogs data sets are mutually consistent and together they strongly support the basic hypothesis put forward in this paper that the unusual weather development on 3 March 1980 was caused by the development of a ducted gravity wave.

4. Theory

In this section the basic theory explaining the observed phenomenon will be developed following the analysis of Gossard & Hooke (1975), which is denoted by ‘GH’ in the following.

Apart from the satellite images, no ‘direct’ measurements showing the three-dimensional structure of the waves are available. Through the model, the measurements of surface data nevertheless give valuable information about the wave structure aloft.

The equations of motion relevant for this study are somewhat different from the balanced equations presented in most text books. For this reason and in order to provide the ‘non-specialist’ with the necessary back- ground, a brief discussion of the basic equations and of the approximations applied is presented in this section. For a more comprehensive discussion the reader is referred to GH.

4. I . Wave and dispersion equations

Assuming two-dimensional propagation in the xz- plane and neglecting the effect of the earth’s rotation, the linearised perturbation equations of motion for an inviscid, incompressible atmosphere with a basic (zero order) flow u, without shear may be written (GH, p. 75):

a u a w ax aZ -+-=o

where po = po(z) is the density of the basic state and the other symbols have their usual meaning.

Assume solutions in the form of plane waves, so:

u, w, p 0~ exp i (kx + n z - m )

where k and n are the wave numbers in the x and z directions, and the wave frequency o = 2/T, where T

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is the wave period. Then we obtain the following set of equations:

where N is the Brunt-Vaisala frequency and o is the ‘intrinsic frequency’ (i.e. the frequency noted by an observer drifting with the zero order flow) given by:

o - u o k =

From equations (1) to (3) we find the dispersion relation, equation (4), relating the horizontal wave number, k, to the vertical wave number, n, and the intrinsic frequency, o:

n2 = k 2 ( N 2 / 0 2 -1) = ( N 2 / C w 2 - k 2 ) ( 4 4

which for ‘large’ values of N2 and small values of k (large wavelengths) may be approximated by:

n2 = N’IC,‘ (4b)

where C, is the intrinsic phase velocity (the phase velocity noted by an observer drifting with the basic flow), defined by o = k(C-u,) = kc,.

4.2. The boundary value problem

The model used describes an atmosphere extending upward to infinity and bounded below by a rigid, level surface. The real atmosphere is approximated for this purpose by considering it to consist of two layers in each of which the Brunt-Vaisala frequency and the other parameters are constant. The two layers are divided by a discontinuity in density in the form of a small inversion at height H. The value of such ‘discon- tinuous models’ is their applicability in cases when a discrete, thin, elevated inversion is present in the atmosphere. We further assume that the vertical shear is negligible.

In our model the upper neutral layer is ‘infinitely’ deep, whereas the radiosonde ascent from Aalborg shows that it extends only to around 700 hPa. As pointed out in section 2 the limited depth of the neutral layer may result in a less-effective trapping of the waves. This limitation, however, seems not to be critical for the present analysis, where we only consider the propagation and effect of the waves over a relatively short distance.

The upper winds in the region where the gravity waves are observed may be evaluated from the radiosonde ascents from Aalborg at 0600 UTC, 3 March, and Copenhagen at 1200 UTC. The data are slightly con- tradictory. The ascent from Aalborg at 0600 UTC


Depending on whether M / 0 2 is greater or less than one the vertical wave number n is real or imaginary (equation (4a)). When n is imaginary the waves will be evanescent (i.e. dying off exponentially with height).

shows a “ W l y wind constantly increasing with height up to around 400 hPa (Figure 3). On the other hand, the Copenhagen ascent at 12OOUTC shows a constant wind up to 500 hPa (Figure 4(b)), 1.e. . no vertical shear. Since most of the gravity wave activity was observed close to Copenhagen, at around o r after 1200 UTC, the vertical shear has been neglected in the model.

A more complicated model could of course have been used involving more layers so that a more realistic profile for the temperature could be used. However, the fact that the model results are in good correspondence with the observations indicates that the basic features of the phenomenon are described by the simple model applied in this study.

Apart from giving information about the wave structure aloft, the model also shows whether various types of data obtained from different sources are mutually consistent.

Assuming that N is constant in each of the layers, plane wave solutions of the form:

w exp[z(kx +nz - ot)]

where n given by equation (4), applies for both layers. As usual it is understood that only the real part of the complex expressions is to be taken.

To proceed, boundary conditions must be imposed at the surface, the internal interface and infinity.

The model has been sketched in Figure 15. The lower layer is denoted by index ‘I’ and the upper one by ‘2’. The lower boundary condition at the earth’s surface is given by wl(0) = 0. By requiring that the energy in the wave system goes to zero for infinite heights we find the upper boundary condition w2(z) -+ 0 for 2 + CQ.

2

z =H

z = o Figure 15. Sketch of the model used. Curve I shows a dry- adiubat, and curve I I (schematic) the radiosonde ascent from Aalborg, 3 March 1980, 0600 UTC.

For the lower layer:

w,(z) = [ A exp(in,z) + B exp(-in,z)l exp[i(kx-t)l

where A and B are real.

Using the boundary condition w,(O) = O and assuming that n, is real we find:

w,(z) = -i2B sin(n,z) exp[i(kx-t)l ( 5)

which corresponds to a horizontally trapped, propagating wave with no vertical tilt (the factor ‘-i’ appearing in (equation 5) simply corresponds to a phase shift -n/2 along the x-axis). That n, is real follows from the data obtained from the Aalborg radiosonde ascent from 0600 UTC, 3 March 1980.

In the neutral upper layer N,2 = O and n2 is imaginary (equation (4a)), say n2 = iy2, From the upper boundary condition it follows that the solution in the upper layer must be of the form:

~ ~ ( 2 ) = Dexp [ -y , ( z -H)] exp[z(kx-ot)]

From the kinematic boundary condition w, = w2 at z = H we find that:

D = - i2B sin (n , )

so that wl(z) may be written as:

sin( n,z) sin( nlH)

wI(z) = D - exp [i( kx-ot)]

Using:

exp [i(kx-ot)] (7) poon1 c o s ( w ) p , ( z ) = iD -- k2 sin(nlH)

(see GH eqn. (29.13)) we find that w, = -@,. This shows there is a phase difference of rc/2 between the vertical velocity and the pressure perturbation (and the horizontal velocity).

The dynamic boundary condition, which asserts that the pressure is continuous across the interface, gives, applied at z = H (see GH p. 132):

1 a w 1 aw

2

which may be written:

Erik A Rasmussen

Using the dispersion relation (equation (4b)), this may be written as:

Provided the constant D is known the vertical velocity in the lower layer can be found from equation (6). D (the value of the vertical velocity at the height of the

In deriving equation (9) we have used y2 = k, which for wavelengths around 100 km is very small compared to the other terms in equation (8) and nI2 = (N, lCJ. The parameters Ap/p = AT/T and N, can be found from the Aarhus radiosonde ascent. ATIT -- 2/273, H = 1200 m and N,' = 3 x s - ~ . Computing N 1 2 , the temperature in the lower layer has been corrected for the small temperature rise from the time of the ascent to the time when the phenomena developed.

- interface H) may be estimated from equation (7) using observed values of p (the pressure perturbation), o (or C,) and k. Insertingp = 0.5 x 102 Pa, C,, = 10 m SC',

and L = 77 x 103 m (k = 2n/L), we find that D = 0.43 m s-I.

The amplitude of the wave-induced vertical displacement from the mean state within the duct is given by:

/ / 4

q(z) = j w ( z ) n sin O, t dt = w(z ) /w ,

giving a maximum displacement at z = H equal to 528 metres. Since the flow will be nearly hydrostatic we .~

Equation (9) may be solved numerically, giving the solution C,, = 10 m spl. Higher-order modes are possible as solutions to equation (9). The next mode corresponds to a slower-moving wave, C,, = 2 m sp', but no sign of this wave has been found in the data.

For small values of n l H , equation (8) may be further simplified to:

may compute the surface pressure perturbation corresponding to the maximum lift by:

PI(Z = 0) qmax g Ap

which upon inserting the relevant values gives p,(z=O) = 0.5 hPa in accordance with the observations.

which for L, >> H gives:

A0

POI

This is an equation similar to equation (16b) in Gos- sard & Munk (1954). Equation (10) is also well known from oceanography, where the phase velocity for long waves in a system composed of a shallow layer over- lying an infinitely deep fluid is given by equation (lo), the so-called 'shallow-water approximation' (see for example Kundu, 1990).

Using the simplified equation (10) we find that C,, = 9 m s-', which is a value close to that obtained for the fundamental mode from equation (9).

The group velocity C, = do/dk corresponding to equation (10) is the same as the phase velocity, i.e. the waves are non-dispersive. The non-dispersive character of the waves is evident in the observations when the data from Aarslev and Sprogs are compared.

Using the simplified dispersion relation (equation 4(b)), the vertical wavelength is found as LZ,, = 10 km. Because of the large vertical wavelength compared with the depth of the lower layer the horizontal velocity is practically independent of depth in this layer.

5. Discussion and comparison with data

The ground-relative phase speed of the waves, C, may be estimated from the data and compared with the intrinsic phase speed calculated from equations (9) or (10) using the relation C-u, = C,. In doing this, only the (constant) wind component of the basic flow (zero-order wind speed) in the direction of the wave propagation, uo, should be considered.

Considering the slope of the waves (relative to the north-south direction), a mean phase speed of 15 m s-l between Aarslev and Sprogs may be computed from the observed passages of the waves at the two locations. The same value for the ground-relative phase speed is obtained by considering the movement of the main cloud band on the NIMBUS satellite images from 1005UTC (Figure 6) and 1409UTC (Figure 7), assuming that the cloud band moves along with the wave.

As the lower stable layer where the propagating wave forms is only around 1200 metres deep, an appropriate mean wind may be defined as a mean between the surface and the 850 hPa level (excluding the value next to the surface, which will be strongly influenced by friction). The mean wind may be estimated from the radiosonde ascents nearest (in time and location) to the region of interest (i.e. the ascents from Aalborg at 0600 UTC and from Copenhagen at 1200 UTC). Both ascents show a north-westerly wind around 330" and 10 m SC'. This corresponds to a zero-order flow, uo, in the direction of the wave propagation of around

228


It is of interest to compare the present work with a study by Matsumoto & Ninomiya (1965) in which they consider a system of line echoes associated with the occurrence of heavy snow falls in the vicinity of a cold vortex centre. The phenomenon studied by Matsumoto & Ninomiya takes place well within a cold air mass and shows many similarities to the case considered here. The observed phase velocity was of the

'order of 100 km h-'. The fluctuations of temperature were hardly noticeable, the amplitude of the pressure changes was, like here, small, around 0.7 hPa, and the wavelength of the order of 300 km. Using a simple model in the form of a wave at a density interface (inversion) they arrived at an expression for the phase velocity similar to equation (lo), corresponding to a phase velocity around 90 km h-'.

5 m s-', which gives a good correspondence between the observed phase speed of around 15 m s-' and the intrinsic phase velocity of 10 m SC'.

To check whether the observations and the model- derived parameters are mutually consistent we can, using the so-called impedance relation (GH, p. 94), compute the amplitude of the pressure perturbation at the surface. The impedance relation may be derived in scalar form from equation (I) as:

p = POC,,U

and, inserting po = 1.15 kg m-3, C, = 10 m s- ' , and u = 5 m s-l we find p = 0.5 hPa. The pressure signa- ture of the wave at the surface is therefore small and can hardly be detected by standard barographs. The waves in the pressure field are nevertheless revealed by the 1 hPa surface map analysis shown in Figures 5, 8 and 9.

Since w, cc -;PI and p l u, the maximum vertical displacement of an air particle due to the passage of the wave will occur simultaneously with the maximum in the wind velocity induced by the wave. The maximum vertical displacement will in turn be correlated with the maximum cloud cover. A relation between maximum surface wind speed and the maximum cloud cover is indirectly verified by the oscillations in the surface temperatures at Sprogs where the minimum temperatures occur in phase with the maximum wind speeds (Figure 13).

The gravity waves did not seem to decouple from the convection they triggered. Instead, the cloud bands propagated with the waves for long distances and the waves remained remarkable coherent even in the pres- ence of the strong convection as shown by the Aarslev and Sprogs data sets. Ultimately, over Zealand the waves began to merge and a region of intense and pro- longed precipitation in form of snow was formed.

The initial formation of gravity waves has been associated with the pre-existence of deep convection. How- ever, the review of 13 case studies by Uccellini & Koch (1987) indicated that convection did not play a significant role in triggering many of the strongest wave events reviewed by them (for a thorough discussion see Koch et al., 1988). For waves with scales around 100 km relevant to this study, the possibility of deep convection having a role in generating the waves is dubious. Gravity wave development has been observed in the vicinity of upper-tropospheric jet streak exit regions, and it has been suggested that the energy source needed to initiate and sustain the gravity waves may be related to geostrophic adjustment processes associated with jet streaks (LJccellini & Koch, 1987). As mentioned in section 2, an exit region of an upper- level jet streak was present in the region at the time when the gravity waves were observed.

6. Summary and concluding remarks

The development of a mesoscale disturbance accompanied by severe weather which unexpectedly formed over Denmark on 3 March 1980 has been discussed. The development occurred in a synoptic situation with a north-westerly to northerly flow, when most of Denmark would normally be sheltered by the Norwegian mountains.

The main characteristics of the disturbance, which could be followed for several hours, were a number of well-defined bands of convective clouds which moved fairly rapidly east, across the prevailing northerly flow, accompanied by intense snowfall. Pronounced oscillations of the su+ace wind velocity were observed at an instrumented tower during the passage of the waves, while the variations of surface pressure were small.

A line-convection system of fairly large horizontal scale, as seen on the satellite images shown on Figures 6 and 7, and the associated disturbance of the surface wind field can only exist if a strong mesoscale forcing mechanism is present. Upper air and surface observations indicate that the phenomenon under discussion was caused by a wave packet of trapped/ducted gravity waves. This hypothesis was tested by means of a theoretical model consisting of two layers separated by a discontinuity at a height H above the ground. Guided by the observations, the lower layer was assumed to be stable whereas the upper layer was assumed neutral. The upper neutral layer will reflect the vertically propagating waves and a low-level duct is formed in which the waves can propagate horizontally without losing their energy. In the model the upper-level neutral layer was 'infinitely' deep. The fact that the upper-level neutral layer in our case only reached to about 700 hPa did not seem to affect its ability to trap the waves for the relatively short distances considered in this work.

Good correspondence was found between the model- derived parameters, such as phase velocity and the

229

Erik A Rasmussen

magnitude of the surface pressure perturbations, and those observed.

That atmospheric gravity waves may cause pronounced oscillatory motions accompanied by only minimal variations of pressure was noted by Eckart (1960), who pointed out that ‘one may therefore make a tentative identification of the gravity waves with the fluctuating component of the wind’. Hines (1972), in his discussion of gravity waves in the lower atmosphere (troposphere), mentions that ‘Eckart’s tentative identification remains to be established as a general truth unless, retrospectively, one is prepared to admit its validity almost as a matter of definition’. Hines further pointed out that ‘If the fluctuating winds are indeed a manifestation of gravity waves, then in typical circumstances so many of these waves are superim- posed that no single one may be recognized’, but that (as was the case over Denmark on 3 March 1980): ‘Exceptions do occur when individual spectral com- ponents rise well above the general noise level’.

More recently, mesoscale gravity waves have been defined by Uccellini & Koch (1 987) to be periodic wave trains or single solitary-like wave disturbances with periods of 1-3 h and wavelengths of 50-500 km. According to Koch et al. (1993) ‘this class of waves has become increasingly recognized for its ability to modulate cloud and precipitation fields, to produce and interact with deep convection’, and ‘the most dra- matic influence of mesoscale gravity waves is their effect upon sensible weather’. Koch et al. also point out that solitary waves are an exceptionally stable, large-amplitude form of gravity wave, and that the effects of mesoscale gravity wave trains (as considered in this paper) upon the weather appear to be somewhat more subdued or indirect, though no less important as far as the weather is concerned.

Concerning the classical problem in distinguishing between the gravity wave effects and the effects due to deep convection, the idea that the system of convection lines was simply ‘drifting convective cells’ can be ruled out. The highly organised waves persisting for several hours have a horizontal scale of =I00 km (i.e. many times larger than their vertical dimension), while for convective systems typical scale ratios are of order unity. The observed phase speeds are much higher than would be expected for simply drifting systems, and, most important, drifting convection cells cannot explain the pronounced oscillations in the near-surface winds observed at Sprogra.

The nature of the phenomenon was not realised at the time it occurred. The perturbations of the surface pressure field were negligible and the associated strong oscillations in the surface wind field were not detected by the routine observation network. It is important that such cases are thoroughly analysed and explained in order to improve the general understanding of fore-

casters, and to provide guidance for similar cases in the future. The case demonstrates that forecasters need a broad meteorological knowledge, including topics for- merly not included in most meteorological training programmes, in order to understand and handle this kind of phenomenon.

The ‘weather active gravity wave packet’ discussed in this paper developed in a general northerly flow. Such flows occur frequently and trapped gravity waves should be looked for in such circumstances. The Sprognr data showed that gusts associated with such waves may exceed 20 m s - ’ . Gusts of this magnitude may constitute a danger for the construction of high structures such as the bridges presently under construction in Denmark, as well as for the traffic on such bridges.

Acknowledgements

The author wants to thank The National Laboratory, Riser and the Danish Meteorological Institute for pro- viding the data for this study.

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A case study of a mesoscale trapped gravity wave accompanied by severe weather