diurnal winds in the himalayan kali gandaki valley. part

2042 VOLUME 130M O N T H L Y W E A T H E R R E V I E W

q 2002 American Meteorological Society

Diurnal Winds in the Himalayan Kali Gandaki Valley. Part III: Remotely PilotedAircraft Soundings

JOSEPH EGGER,* SAPTA BAJRACHAYA,1 RICHARD HEINRICH,* PHILIP KOLB,* STEPHAN LAMMLEIN,#

MARIO MECH,* JOACHIM REUDER,* WOLFGANG SCHAPER,@ PANCHA SHAKYA,1 JAN SCHWEEN,* AND

HILBERT WENDT*

*Meteorologisches Institut, Universitat Munchen, Munich, Germany1Department of Hydrology and Meteorology, Ministry of Science and Technology, Kathmandu, Nepal

#Fachbereich Maschinenbau, Fachhochschule Regensburg, Regensburg, Germany@Astrium, Friedrichshafen, Germany

(Manuscript received 26 July 2001, in final form 18 February 2002)

ABSTRACT

In 1998 a field campaign has been conducted in the north–south-oriented Kali Gandaki valley in Nepal toexplore the structure of its extreme valley wind system. Piloted ballon (pibal) observations were made to mapthe strong upvalley winds as well as the weak nocturnal flows (Part I). The stratification of the valley atmospherewas not explored. In Part II of this multipart paper, numerical simulations are presented that successfully simulatemost of the wind observations. Moreover, the model results suggest that the vigorous upvalley winds can beseen as supercritical flow induced by contractions of the valley. Here, the results of a further campaign arereported where remotely piloted airplanes were used to obtain vertical profiles of temperature and humidity upto heights of ;2000 m above the ground. Such profiles are needed for an understanding of the flow dynamicsin the valley and for a validation of the model results. This technique is novel in some respects and turned outto be highly reliable even under extreme conditions. In addition four automatic stations were installed along thevalley’s axis. Winds were observed via pibal ascents. These data complement the wind data of 1998 so that thediurnal wind system of the Kali Gandaki valley is now documented reasonably well.

It is found that the fully developed upvalley flow is confined to a turbulent layer that tends to be neutrallystratified throughout the domain of observations. The stratification above this layer is stable. A capping inversionis encountered occasionally. This finding excludes explanations of the strong winds in terms of hydraulic theoriesthat rely on the presence of strong inversions. Pairs of simultaneous ascents separated by 5–10 km along thevalley axis reveal a remarkable variability induced by the topography and, perhaps, by an instability of the flow.The analysis of the surface data as well as that of the soundings shows that the flow above the neutral layeraffects the surface pressure distribution and, therefore, the acceleration of the extreme upvalley winds.

1. Introduction

The Kali Gandaki valley in Nepal stands out bothbecause of its extreme geometry and the intensity of thediurnal upvalley winds. The Kali Gandaki River orig-inates near the town Lo Manthang (see Fig. 1) and flowssouthward through the Mustang basin. It cuts throughthe Himalayan barrier between the villages of Marphaand Ghasa forming there one of the deepest valleys onEarth. Farther south, the river rushes down into a gorgeto reach the lower parts of Nepal at an altitude of ;1000m above MSL. The Mustang basin extends from Marphato Lo Manthang. It is confined by the towering mountainchains to the east and west and by the Himalayas in the

Corresponding author address: Joseph Egger, MeteorologischesInstitut der Universitat Munchen, Theresienstr. 37, 80333 Munchen,Germany.E-mail: [email protected]

south. A mountain pass leads to the Tibetan Plateauabout 20 km to the northeast of Lo Manthang.

Before 1998, scattered information was available in-dicating that the diurnal wind system of the valley ex-hibits rather strong upvalley winds (;20 m s21) betweenMarpha and Kagbeni (see Egger et al. 2000, hereafterKG1, for details and relevant literature). This upvalleywind is called Lomar by the locals (‘‘southerly wind’’;we change here the spelling ‘‘Lhomar’’ as used in KG1and Zangl et al. (2001, hereafter KG2) to ‘‘Lomar’’ tobe consistent with the spelling of Lo Manthang wherelo refers also to the south). Nocturnal downvalley windsappeared to be weak.

In fall 1998 the Meteorological Institute of the Uni-versity of Munich and the Department of Hydrologyand Meteorology in Kathmandu conducted a joint fieldcampaign in order to explore the structure of this windsystem in detail (KG1). The following conclusions ofKG1 are based on about 100 double-pilot ascents per-

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AUGUST 2002 2043E G G E R E T A L .

FIG. 1. Map of the Kali Gandaki valley: airplane ascents, dots;permanent surface stations, crosses with circles; villages and sitesmentioned in the text, crosses. Pibal ascents were made in Jomsom,Dhumpha, Chuksang, Tangye, and Lo Manthang. LK is LangpoghyunKola. Height contours, solid (m); contour interval is 500 m. Hori-zontal distances as indicated at the axes (km). The map is based ontopographic data with a resolution of 300 3 300. These data havebeen interpolated to a 1 km 3 1 km grid. See also Fig. 1 of KG1.Dashed, Kali Gandaki and LK.

FIG. 2. Isentropes (contour interval, 1 K; solid) and wind (vectors)in a section along the Kali Gandaki valley as obtained in the referencerun REF of KG2 in the afternoon (t 5 15 h). Shading: light, windspeeds 10–15 m s21; medium, 15–20 m s21; dark, .20 m s21. Seealso Fig. 7a of KG2. The bold letters mark the locations of Tukuche,Marpha, Jomsom, and Kagbeni. Height is above MSL. The narrowestpoint of the valley is located near Marpha.

formed at eight locations covering the distance fromLete to Lo Manthang.

1) The upvalley winds start near the surface beforenoon. The layer of strong winds with typical veloc-ities of ;15 m s21 grows over about 1 h to a depthof ;1000–1500 m.

2) The breakdown of the upvalley wind regime aftersunset begins close to the ground. The upvalley flowceases before midnight.

3) Nocturnal downvalley flows are quite weak.4) Upvalley winds are less powerful both in the so-

called exit region between Chuksang and Lo Man-thang and in the entrance region (Ghasa–Tukuche)than in the core region (Marpha–Kagbeni).

Stimulated by these results, Zangl et al. (2001) per-formed numerical simulations of the Kali Gandaki windsystem using the Pennsylvania State University–Na-tional Center for Atmospheric Research fifth-generationMesoscale Model (MM5). The total flow domain of the

simulations included the Tibetan Plateau as the domi-nant topographic feature of the region. Five nests wereneeded to resolve the core region reasonably well witha grid size of 800 m. There are 38 levels in the verticalwith a maximum resolution of 100 m near the ground.The initial state is in thermal wind balance with themeridional temperature gradient. The level of no windsis chosen such that the atmosphere is almost at rest inthe Kali Gandaki valley. The model calculations werequite successful in that nearly all the observed featuresof the wind field were reproduced in a reference run.Sensitivity experiments were carried out in order to elu-cidate the mechanisms driving the valley wind systemincluding precipitation [see also Barros et al. (2000) fora recent precipitation analysis of the area].

It is a key result of KG2 that a rather stable layerwith strong upvalley winds forms during the day in theentrance and, in particular, in the core region (see Fig.2). This layer of 1000–1500-m thickness is found up-stream of the widening of the valley near Marpha. Theisentropes descend between Marpha and Jomsom. Theflow accelerates in the descending branch to attain amaximum speed of 23 m s21 near Jomsom. The layerof rapid flows stays close to the ground farther to thenorth and ascends toward Tibet with slightly reducedflow speeds. Note, however, that inflow of moderatespeed occurs above this layer up to a height of ;5000m. The stability of the Lomar layer increases from Tuk-uche to Jomsom. The Brunt–Vaisala frequency is N ;1.2 3 1022 s21 in Marpha and 1.6 3 1022 in Jomsomwhere the isentrope u 5 315 K and the ground are



FIG. 3. Monthly mean values of the hourly mean wind speed U (ms21) as observed in Kagbeni in Feb–Mar 1990 at a height of 9 m.

chosen as reference surfaces (u equals potential tem-perature).

The flow pattern in Fig. 2 is reminiscent of the resultsof laboratory and hydraulic model studies of flowsthrough lateral contractions. For example Arakawa[(1969); see also Baines (1995) for an updated outlineof the theory] analyzed single-layer flow through a val-ley of variable width where the Froude number F 5U(g9H)21/2 is the key parameter (U, flow speed; g9, re-duced gravity; H, depth of the flow). There exists a classof solutions where F 5 1 at the narrowest point andwhere F . 1 downstream. This type of flow is similarto that in Fig. 2 where event indications of a hydraulicjump are seen downstream of the maximum contractionnear Marpha. Such models have been invoked by Pettre(1982), Jackson and Steyn (1994a, b), Pan and Smith(1999), and others to explain strong wind storms invalleys and gaps, Structures as displayed in Fig. 2 canalso be found in two-layer models (Armi 1986; Baines1995) and in continuously stratified flows underneath afree surface representing an inversion (Armi and Wil-liams 1993).

In principle, hydraulic theory cannot be applied tothermally driven flows simply because sources and sinksof heat are not included. Here, however, the situationis somewhat different. As explained in more detail inKG2, the diurnal heating of the Mustang basin generatesa pressure difference between, say, Jomsom and the‘‘free’’ atmosphere to the south of Ghasa. The strongwinds can be seen as a response to this pressure gradientin much the same way as gap winds respond to imposedlarge-scale pressure fields. Moreover, Fig. 2 suggeststhat the Lomar is separated by a rather stable top layerfrom the atmosphere above. All this indicates that anapplication of hydraulic theory might help to better un-derstand both the observations and the model results.In turn, information on the thermal structure of the val-ley atmosphere is needed in order to better understandthe vigorous Kali Gandaki valley winds. Such infor-mation was not provided by the campaign of 1998. Stan-dard instrumentation for vertical soundings [see e.g.,Clements et al. (1989) for the design of a valley flowexperiment] like radiosondes is impractical in these re-mote areas where the gas needed to fill the balloons hasto be carried up by porters. Tethersondes cannot be usedfor the same reason. Moreover, the winds are too violent.However, battery-powered model airplanes with remotecontrol (remotely piloted vehicles, RPV) are highly suit-able for this purpose. They are light, their energy de-mand is low, and they can be carried by porters to almostany starting position in the Mustang region. Their max-imum ascent height of ;2000 m is sufficient to penetratethe layer of strong winds. Such planes were used duringthe field campaign in 2001 to be described in this paper(18 February–27 March). Double theodolite pilot bal-loon observations were carried out simultaneously. Inaddition, an array of surface stations was installed. Itwas the main goal of this effort to collect information

on the stratification of the upvalley flow both in thenarrow part of the valley and in the Mustang basin andto verify in this way the model results of KG2 as dis-played in Fig. 2.

The observations described in KG1 were made in thefall. By choosing February and March we wished tolearn more about the seasonal variability of the KaliGandaki valley wind system. The mean wind speeds inFebruary and March 1990 in Kagbeni are shown in Fig.3. These curves are fairly similar to those for Septemberand October (see Fig. 2 of KG1) except that the windsare slightly less vigorous than in the fall. Note that Fig.2 represents equinoctial conditions in the simulations ofKG2 that do not take into account the observed cli-matological mean flow. Therefore, a validation of Fig.2 is possible in March as well as in September. Theclimatological flow has a southerly component at upperlevels in the fall while westerlies prevail in the spring(e.g., Ramage 1971). Throughout the campaign the flowat 500 hPa was generally westerly although many per-turbations moved over the area from the west.

This paper is organized as follows. The equipment isdescribed in section 2. The surface observations arepresented in section 3, the soundings in section 4. Adiscussion is presented in section 5, and concluding re-marks are given in section 6.

2. Instrumentation

a. Airplanes

The idea to use model airplanes as a carrier of in-struments in meteorology is not new. Correspondingexperiments were conducted successfully in the 1970s(Konrad et al. 1970; M. Reinhardt 1997, personal com-munication). More recent activities are described inRenno and Williams (1995), Chilson et al. (1999), andStephens et al. (2000). The new airplanes were designedby W. Schaper (WS) and built by WS and S. Lammleinin cooperation with Modellbau Ulrich and Blue Airlines.They can be flown up to heights of at least 5000 mabove sea level in highly turbulent wind fields. The



FIG. 4. (a) Prototype Kali. (b) The pilot is wearing the specialbinoculars needed to follow the plane at heights of more than 1000 mabove the ground.

prototype Kali is shown in Fig. 4a. The plane has alength of 1.29 m and a wing span of 2.10 m. The totalmass in 3 kg. Flight velocities are in the range of 10–40 m s21. The optimum climb rate for highest altitudeis ;5 m s21. The propeller is driven by an electromotor(Hacker HBR 50S26). Power is provided by 14–16 re-chargeable NiMH cells. To ease control of the planesin turbulent flow, the planes are equipped with gyrosystems for stabilization around the roll axis. Specialbinoculars have been developed by Firma Zeiss so thatthe plane can be followed visually up to heights of;2000 m above the ground (Fig. 4b). To increase colorcontrast, blue blocker sunglasses, also made by Zeiss,were used. The pilot controls the plane by a Robbe radiogear control. Nets are used for landing in difficult terrainand in strong winds. A sounding is normally completedwithin 15 min. The flight path during ascent is chosento maximize the ascent height; that is, the pilots try toexploit slope winds and thermals. Descent is performedso as to equal time spans for ascent and descent. Theplanes were steered by several of the authors (P. Kolb,S. Lammlein, and WS).

The planes carry sensors for an observing systemdesigned and built by Ingenieurburo Wurtenberger con-sisting of a miniaturized datalogger and specially adapt-ed sensors for pressure, temperature, and humidity. Thepressure sensor is based on the Motorola MPX 2100.Its signal is amplified and compensated for changes oftemperature. The resolution is ;1 hPa. Humidity is re-corded via the HIH-3605-B Honeywell sensor with0.6% resolution. The response time is ;5 s. Temperatureis obtained from a LM50 C National semiconductorsensor. The resolution is ;0.2 K and the response timeis #1 s. Ascents for an adiabatic lapse rate must beexpected to be 0.38C warmer than descents because ofthe related hysteresis effect. Three wind generators pro-vided the electric power to recharge the batteries of theplanes and all portable computers. One plane out of 11was lost during the campaign, while another 1 was dam-aged but could be repaired. It is impossible to fly theplanes during the night; therefore, we concentrated onthe daytime flow evolution during this campaign.

b. Permanent stations

Four permanent stations were installed. They con-tained instrumentation for temperature, wet-bulb tem-perature, pressure, and wind speed and direction. Theseinstruments were mounted on a 3-m mast. Power wasprovided by a solar panel. Global radiation was recordedat one station (Jomsom). Observations are made with atime step of Dt 5 120 s.

The surface stations were positioned in Kagbeni onthe roof of a house, in Jomsom close to the airstrip, inMarpha in an open field near the river, and in Tukucheagain on the roof of a house (Fig. 1). It would havebeen preferable to have all stations located close to theaxis of the valley. The stations in Marpha and Jomsomsatisfied this requirement quite well, and that in Kagbenicame close as the house chosen (Hotel Niligiri) is ex-posed to the full force of the wind. On the other hand,the village of Tukuche is sheltered to some extent andso were our instruments. Wind speeds at this locationunderestimate the wind strength found in the riverbed(see KG1). The stations were in operation as follows:kagbeni, 25 February–23 March; Jomsom, 22 February–26 March; Marpha, 23 February–21 March; Tukuche,23 February–21 March.

c. Pilot balloons

Two theodolites were used to track a helium-filledballoon. The system is identical to that described inKG1. Observations were made in Jomsom, Dhumpha,Chuksang, Tangye, and Lo Manthang (Fig. 1).

3. Surface observations

The surface observations are presented first becausethey contain detailed information on the diurnal cycle



FIG. 5. Time series of wind speed y (m s21) (bold) and direction (dots), temperature T (8C) and wet-bulb temperature Tw (8C) (dashed),specific humidity q (g kg21), and surface pressure p (hPa) in Marpha for 19 and 20 Mar.

of flow conditions in the valley. This information helpsin the interpretation of the soundings. The days of 19and 20 March are selected to demonstrate the main char-acteristics of the time series in Kagbeni and Marpha(Figs. 5 and 6). Marpha is located in the narrow part ofthe valley, which opens toward the Mustang basin justnorth of Marpha (see Fig. 1). According to Fig. 2, Mar-pha is situated upstream of the region of flow acceler-ation while Kagbeni is downstream. The wind speedsrecorded in Marpha were moderate, with maximumspeeds of ;7 m s21. There were weak northeasterlieson 19 March until the Lomar set in and continued untilmidnight. The next day was essentially calm until 0900LST when the Lomar set in again. The global radiationdata of Jomsom (not shown) indicate that the afternoonof 19 March was partly cloudy. Visibility was unusuallylow on the morning of 20 March. Clouds covered thesky at ;1300 LST and rain fell for a few hours. Thus,20 March is a day with perturbations in the afternoon.Temperature and the wet-bulb temperature were risingrelatively late on both days because of the shadow castby the Annapurna massif. Moisture decreased in themorning of 19 March as is typical at this station. Itincreased after the onset of the Lomar, which appearedto bring moister air up the valley. However, the differ-

ence between temperature and wet-bulb temperaturewas quite large during the day. Relative humidities were;35%.

The diurnal pressure oscillations are representedclearly in the surface pressure time series. Accordingto Dai and Wang (1999) both the diurnal and the semi-diurnal tidal oscillation reach maximum values of 0.6–0.8 hPa in Nepal, the semidiurnal one being slightlylarger. By and large, our observations (see Table 1) arein agreement with this finding. However, as can seenfrom Table 1, the amplitude of the diurnal (semidiurnal)pressure oscillation increased from 0.6 (0.8) hPa in Tuk-uche to 1.2 (1.1) hPa in Kagbeni. This must be a localeffect that is presumably related to the strong upvalleywinds. In Figs. 5 and 6 there is a pressure maximumabout 2 h after sunrise and another one before midnight.The minimum in the afternoon is more pronounced thanthat early in the day.

The Lomar set in at Kagbeni later on both days andwas more powerful with maximum speeds of 18 m s21.The weak winds in the morning of 20 March were main-ly from the south. There was, however, a distinct onsetof Lomar on the day as well. Temperatures in Kagbeniwere quite similar to those in Marpha. The diurnal andsemidiurnal oscillations of temperature were about the



FIG. 6. The same as Fig. 5 but for Kagbeni.

TABLE 1. Amplitudes of the diurnal (first entry) and semidiurnal(second entry) oscillations of pressure, temperature, and wind ve-locity at the four permanent stations.

StationPressure

(hPa)Temperature

(8C)Wind

(m s21)

TukucheMarphaJomsomKagbeni

0.6/0.80.7/1.01.0/1.01.2/1.1

3.8/1.64.2/1.94.4/1.74.1/1.8

2.2/0.92.6/0.83.8/1.56.0/1.9

TABLE 2. Mean delay (min) of the onset of the valley wind regimeat the station in the left column with respect to the station in the toprow. In parentheses, number of days with negative delays, total num-ber of days, and propagation velocity in m s21.

Kagbeni Jomson Marpha

KagbeniJomsom

0—

24 (3/22/6)0

50 (1/21/5)22 (1/23/4)

same at all stations (Table 1). The evaluation of therelated phases shows that temperature maxima occur atabout the same time (1310–1325 LST).

The data for Jomsom (not shown) were similar tothose in Kagbeni except that the Lomar commencesearlier but with reduced intensity. Maximum windspeeds were ;12 m s21 on both days. Those in Tukuchewere ;8 m s21. The increase of the intensity of thevalley winds from Tukuche to Kagbeni is also reflectedin the diurnal and semidiurnal variations of the windspeed in Table 1.

Both the delay of the onset of the upvalley windregime in Kagbeni with respect to Marpha and the re-duction of the diurnal pressure variation in Marpha withrespect to Kagbeni were found almost every day. Table

2 gives the mean delay between all pairs of availablestations. Here, we define the onset time as that momentwhere the upvalley wind reaches half the maximumstrength of that day. This definition avoids ambiguitiesdue to the existence of weak upvalley winds before theonset of the strong winds. It is clear from Table 2 thatthe upvalley wind regime moves from Marpha to Jom-som and farther up to Kagbeni. Typical speeds of prop-agation are 5 m s21. The variability of the delays fromday to day is large. For example, the maximum delaybetween Kagbeni and Marpha is 116 min while lomarwas recorded a few minutes earlier in Kagbeni on 7March. Data from Tukuche are not included because ofthe general weakness of the winds there, which makesit sometimes difficult to determine an onset time.

The winds are driven by pressure gradients. In prin-ciple, pressure gradients can be evaluated by reducing



TABLE 3. Mean ratio of the pressure decrease between the maximumin the morning and the minimum in the afternoon at the station inthe left column and that in the upper row. The second entry in thefirst row is the mean pressure decrease (hPa) at the station indicatedon top.

Kagbeni Jomsom Marpha Tukuche

KagbeniJomsomMarpha

1.0/4.5——

1.22/3.91.0—

1.44/3.41.141.0

1.67/3.11.351.11

FIG. 7. Correlation of the temperature and the wind velocity inMarpha (bold) and Kagbeni (dashed) for positive lags; i.e., windperturbations are shifted by t (h) with respect to temperature. Diurnaland semidiurnal cycles are removed.

all pressure to the height of Jomsom. However, the ba-rometers used at the permanent stations differed in therange of ;1 hPa. This deviation is too large for a pres-sure reduction to be useful. Instead we simply arguethat the nocturnal winds are extremely weak. Therefore,there is no appreciable pressure gradient force in theMustang basin at night. The pressure decrease duringthe day is different at the various stations and the stationwith the largest decrease is the one with the lowestpressure. The mean value of the pressure decrease fromthe morning maximum to the minimum in the afternoonis given in Table 3 for each station (first row). Thedecrease is smallest in Tukuche and largest in Kagbeni.This yields a mean pressure difference between Kagbeniand Jomsom of 0.6 hPa in the afternoon. The meanpressure difference between Marpha and Jomsom is 0.5hPa. The reference simulation in KG2, where the af-ternoon pressure minimum is located slightly north ofJomsom, is broadly consistent with this finding.

As has been mentioned, the daily ranges of temper-ature are about the same in Marpha, Jomsom, and Kag-beni. This implies that the enhanced pressure changesin Kagbeni are due to dynamical processes on top orabove the boundary layer. Pressure difference betweenvarious stations can be explained by variations of thedepth of the upvalley wind layer if there is a pronouncedinversion on top of this layer. As will be shown in thenext section, such an inversion exists only occasionally.One may, however, argue that the observed wideningand intensification of the flow between Marpha and Kag-beni is possible only if air descends from above intothe Lomar layer. The related warming and pressure de-crease appears to be largest in Kagbeni.

The northward delay of the onset of strong winds asquantified in Table 2 is seen in the diurnal and semi-diurnal wind oscillations as well. The wind maximumin Kagbeni as descried by these oscillations lags that inMarpha by 30 min and that in Tukuche by 35 min. Therelated pressure extrema are delayed in an oppositesense. The pressure maximum in the morning occurs inKagbeni at 0817 LST, in Jomsom at 0830, in Marphaat 0850, and in Tukuche as late as 0947. The pressureminimum in the afternoon is recorded in Kagbeni at1525 LST, in Jomsom 1530, in Marpha at 1535, and at1635 in Tukuche. This shows again that the evolutionof pressure gradients between the various stations ishardly linked to the temperature of the Lomar layer.

Differential heating from the ground between, say, Tuk-uche and Kagbeni is presumably not important in gen-erating the strong upvalley winds.

The covariance functions for all time series have beenevaluated as well. Of course, the diurnal and semidiurnalsignals are dominant. However, long timescales prevaileven after these periods are removed. The first zerocrossing of autocorrelations of single station data isfound in the case for lags of 4–6 h. Delays are the mostinteresting features to be extracted from these functions.For example, Fig. 7 shows the correlation of temperatureand wind in Marpha for positive lags. The wind veloc-ities peak at a lag of about 25 min. Thus, the reactionof the winds in the narrow part of the valley to changesof the temperature is quite fast. On the other hand, thewinds have cooling impact on the temperature (notshown). The correlation of temperature and wind inJomsom reaches a maximum after ;60 min, that is,considerably later than in Marpha (Fig. 7). This differ-ence may be due to the fact that Marpha is located ina valley where the flow is more constrained.

4. Vertical structure

As has been mentioned, the airplane soundings wereperformed to explore the stratification of the valley at-mosphere during the day and to validate the model re-sults of KG2 as presented in Fig. 2. The model predictsa deep inflow layer for Marpha and shallow and ratherstably stratified flow in Jomsom.

By far the largest number of ascents were made atairport in Jomsom (15–17, 19–21, 23–25 March), thatis, in the core region of the Kali Gandaki wind system.A selection of these results will be presented first. Par-allel ascents upstream of Jomsom were made in Dhum-pha (21 March) and Marpha (16 and 17 March). Theonly soundings in the entrance region took place nearTukuche (23–25 March). Further parallel ascents in Ek-lobati (see Fig. 1; 19 and 20 March) provide informationon the stratification downstream of Jomsom. In addition,



FIG. 8. Potential temperature u (K) as a function of height duringascent (solid) and descent (broken) at 0620 LST 25 Mar in Jomsom.Also given is the specific humidity q (g kg21) during ascent anddescent (triangles). The vertical coordinate z is defined such that z5 0 at Jomsom airport at 2751 m above MSL.

the flow in the exit region was investigated in Tangyeon 27 and 28 February, in Lo Manthang (3, 4, and 6March), and in Chuksang (10 and 11 March). It wasintended to obtain a reasonably complete dataset of tem-perature and moisture profiles throughout the Mustangbasin during the day by undertaking this rather strenuouspart of the campaign.

a. Core region

1) EARLY MORNING

Only one ascent was made early in the morning inJomsom (Fig. 8; 25 March). Flight operations at theairport usually began before 0700 LST and were ter-

minated before noon. Model plane ascents were not per-mitted during that time. Visual control is impossiblebefore sunrise, leaving little time for early aircraft ob-servations. A ridge was moving toward the Mustag areaon that day but gradients at 500 hPa were weak. Lightnortheasterly winds prevailed throughout the first 2 km(not shown). The morning atmosphere was stably strat-ified at least up to the maximum ascent height of 1450m above the ground (N ; 1.3 3 1022 s21). The air wasslightly cooler during descent. The loop during descentclose to the ground reflects, of course, a flight maneuvrebefore landing. The temperature profile in Fig. 8 de-viates substantially from those found quite often earlyin the morning in valleys (Brehm and Freytag 1982;McKee and O’Neal 1989; Whiteman 1990) where aninversion extends from the ground to a height of a fewhundred meters. This difference is presumable due tothe absence of nocturnal downvalley flow in Jomsom.

The moisture decrease linearly with height. Ascentand descent values are almost identical. The absence ofa clearly visible hysteresis effect in the moisture profilesuggests that the difference of the potential temperaturebetween ascent and descent are real because the re-sponse time of the temperature sensors is shorter thanthat of the humidity sensors.

2) UPVALLEY FLOW

A fairly complete set of ascents has been obtained on19 March and will be presented in detail below. A warmridge was centered over Tibet on that day.

The upvalley wind regime was just beginning to es-tablish itself at 1100 LST (Fig. 9a). Rather weak north-easterlies are found up to a height of 2000 m above ashallow layer of upvalley flow. Convection appears toerode the stable layer established during the night sothat an almost neutral layer of ;300 m depth is seenabove a shallow superadiabatic layer. Higher up, thestratification is stable (N ; 1022 s21). Such profiles arecommon in valleys late in the morning (e.g., Whiteman1990). Ascent and descent temperatures differ by 1–2K close to the ground presumably because of additionalheating before the start. The moisture is well mixed upto a height of ;700 m and is constant above z 5 1400m. The u profile in Eklobati is similar in shape to thatin Jomsom but the air was warmer by about 1 K andalso drier. One hour later (1200 LST; Fig. 9b), the upval-ley wind was quite strong in the lowest 250 m of thevalley atmosphere. A layer of weaker upvalley windsextended at least up to a height of 1300 m. The velocityspike at z 5 1300 m is presumably real because trackingproblems tend to occur only during the first minutes ofan ascent when the balloon is close to the observers.This profile of velocities is reminiscent of those reportedin KG1 (Figs. 10 and 11 of KG1) for the early stagesof the Lomar. In general, ascent temperatures are higherthan those found in descent. The systematic temperaturedifferences of more than 2 K as recorded up to heights



→

FIG. 9. Potential temperature u (K) and specific humidity q (g kg21) as a function of height during ascent and descent on 19 Mar inJomsom and Eklobati. Also given is the wind speed (bold) and the wind direction (dots) as obtained at the base in Jomsom. As can be seenfrom Fig. 1 upvalley winds occur for 1808 # dir # 2408: (a) 1100, (b) 1200, (c) 1300, and (d) 1400 LST. Wind velocities too noisy to beshown. Jomsom (Eklobati) is at 2751 m (2810 m) above MSL. Jomsom airport: z 5 0 m.

of 1200 m are so large when compared to our estimateof hysteresis effects that we speculate that these struc-tures are linked to deep eddies. Rapid and strong var-iations of the vertical motion of the RPV have also beenreported by the pilots. The mountains along the valleyaxis are quite rugged and have heights of 1000–2000m with respect to the valley bottom. It is conceivablethat vortices are generated by the interaction of theupvalley flows with these obstacles. Moisture appearsto be affected by the eddies as well. Humidity is higherduring descent in contrast to what one would expect fora hysteresis effect. All in all, the onset of the Lomarled to a decrease of the temperature in the lowest 250m, as one would expect for thermally driven flows.However, the temperature rose in the layer above thestrong winds. The pronounced increase of moisturewithin 1 h up to a height of 700 m must be due toadvection.

Given a flow speed of 10 m s21 it takes just 10 minto advect air from Jomsom to Eklobati. Indeed, tem-peratures in Eklobati decreased in the lowest 500 m sothat Eklobati was no longer warmer than Jomsom. Thereis some similarity is both profiles, but many details donot coincide. Advective moistening is seen in Eklobatias well.

A well-mixed layer of 1000-m depth was capped bya pronounced inversion at 1300 LST in Jomsom (Fig.9c). A stable layer was found higher up. The strongwinds were confined to the neutral layer. The moisturewas well mixed in the neutral layer and the moisturecontent was strongly reduced above the inversion. Ascompared to Fig. 9b, the potential temperature is nowlower below the inversion and about the same above.The close proximity of ascent and decent in Fig. 9c issurprising given the large flow speeds in the Lomarlayer. Moreover, the noisiness of the wind speed profileindicates that the flow was highly turbulent. The u pro-file in Eklobati at 1300 LST is quite similar to that inJomsom but there is warming above the lomar layer.Warming from above is common in valleys during theday (e.g., Brehm and Freytag 1982) and is thought tobe due to the downward branch of the cross-valley cir-culation with slope winds forming the upward branch.Given the large width of the Mustang basin it is doubtfulif this explanation is sufficient in this case. It is likelythat upvalley flow descending above the Lomar layercontributes to the warming (see also Fig. 2).

The neutral layer in Eklobati was not as deep as inJomsom. Therefore, we have here an example of adownward sloping inversion where hydraulic modellingmay be appropriate. The potential temperature ‘‘jumps’’

by about 3 K on top of the neutral layer so that the flowis supercritical with F ; 1.5 in Jomsom. The relatedpressure difference between Jomsom and Eklobati is 5–7 Pa.

One hour later, the air above 1000 m was cooler anddrier in Jomsom and the inversion disappeared (Fig. 9d).There is no obvious reason for this process. It is aston-ishing that the inversion still existed in Eklobati wherea slight cooling aloft was accompanied by a moistening.The q curves in Eklobati differ greatly between ascentand descent. This difference is not due to an observa-tional error. The plane carried two moisture sensors thatgave almost exactly the same result.

The Lomar layer reached a depth of at least 1500 mat 1500 LST in Jomsom (not shown). Ascent and descentdiffer again substantially at that time so that one cannotassign a clear structure to this turbulent boundary layer.Towering cumulonimbus clouds evolved in the after-noon in the south above the mountains bordering thekali Gandaki valley. This feature is typical of radiationdays with clear skies in the morning. Like on manyother days, the clouds moved northward and showerswere recorded in Jomsom. Thus, this last flight of theday documents a late stage of the Lomar where thevalley flow was affected by convective cells. The par-allel ascent in Eklobati was not successful.

So far we have found that capping inversions mayoccur but do not appear necessary for the Lomar todevelop. The Lomar layer is neutrally stratified in con-trast to the model results. Both warming and coolingare seen to occur above the Lomar layer. This suggeststhat conditions above the Lomar layer proper have animpact on the evolution of the upvalley flows in agree-ment with the numerical simulations.

On 21 March piloted-balloon (pibal) observations andrelated airplane ascents were made in Dhumpha at themouth of the Langpoghyun valley (see also Fig. 1). Thisvalley ascends along the slopes of the Annapurna mas-sif. One expects maximum descent near Dhumpha ac-cording to Fig. 2. The sounding at 1100 LST revealsan inversion at z ; 1200 m that descends until noonby 200 m due to an increase of the temperature aloft.The parallel soundings in Jomsom do not reveal inver-sions but the temperature aloft is increasing as well. Theflow is neutrally stratified in the Lomar layer.

In 1998, tree deformations were mapped in the Lang-poghyun valley (see Fig. 18 of KG1). It was found thatstrong upvalley flows must prevail in this valley. Thesewinds form a branch of the Kali Gandaki wind systembut are oriented almost normal to the standard flow di-rection of the lomar. On 21 March, such winds were



FIG. 9. (Continued)



FIG. 10. Track (dots) of a balloon released at 1026 LST 21 Marnear Dhumpha. Bold shows height lines (m); T1T2 shows the baseline.The numbers along the track give the height of the balloon abovethe starting position; y (x) axis pointing toward the north (east).

observed. The track of a balloon released at 1026 LSTat the axis of this side valley is depicted in Fig. 10. Theballoon followed this valley at a velocity of ;10 m s21.It should be said, however, that the flow above the bal-loon in Fig. 10 was southerly. At the moment, the causeof these rapid northeastward accelerations is unknown.

Ascents in Marpha were made in unfavorable con-ditions. The upper-level flow was southwesterly on bothdays and appears to have induced strong downward mo-tion above Marpha. It was frustratingly difficult for thepilots to gain height. The ascents are, therefore, notpresented.

b. Entrance region

The only soundings in the entrance region of the Lo-mar were made near Tukuche. Examples are given inFig. 11. The pre-lomar u profile at 1130 LST in Tukucheis quite similar to that in Jomsom but temperatures areslightly lower in the lowest 750 m than in Jomsom.Note that the stratification is neutral even before theonset of the Lomar. At 1330 LST the potential temper-ature in Tukuche is essentially constant in the lowest500 m. The stratification is weakly stable above. Theair is clearly warmer above the Lomar layer in Jomsomthan in Tukuche. Moisture is well mixed throughout theascent in Tukuche but decreases slightly above the Lo-mar layer in Jomsom. All this indicates that there isdescent between Tukuche and Jomsom. It is difficult tocompare Figs. 2 and 10 because there are no wind ob-servations available in Tukuche so that the depth of theLomar layer cannot be estimated. The Brunt–Vaisalafrequency is N ; 5 3 1023 s21 when evaluated for the

total profile in Tukuche at 1330 LST. This is less thanin Fig. 2. The warming at upper levels in Jomsom withrespect to Tukuche is found also in the numerical sim-ulations.

c. Exit region

Soundings in the exit region were carried out in Chuk-sang, Tangye, and Lo Manthang. Wind intensities inChuksang were lower than in Jomsom in agreement withwhat had been found in 1998 (KG1). The Lomar layerwas neutrally stratified without a capping inversion. Thevillage of Tangye is located to the east of the Kali Gan-daki River. An escarpment overlooking the main valleywas selected for the observations. Until that time, noflow data had been collected in the eastern part of theupper Mustang basin. Observations were made on 27and 28 February under a prevailing southwesterly upper-level flow ahead of a trough at 500 hPa. Upvalley windswere of moderate strength on 27 February. The nextday extremely strong upvalley winds were observed inthe presence of strong convective activity in the south.Figure 12 shows a situation where at least the lowerpart of the Lomar layer is well mixed with maximumvelocities of ;20 m s21. This establishes that vividupvalley winds occur in the eastern part of the Mustangbasin. The balloons could not be tracked for long enoughnor did the planes reach sufficiently large altitudes asneeded for an estimate of the depth of the lomar layer.The dryness of the flow indicates that the air, say, inKagbeni, is not simply advected up to Tangye. Dry airfrom higher levels must be mixed into the Lomar flow.

It was hoped that strong katabatic winds or cold airoutflows from the Tibetan Plateau would be centered inLo Manthang. Conditions late in winter tend to be fa-vorable for such flows. However, the observations weredisappointing in that respect. The morning observationsof 3, 4, and 6 March revealed that the stratification wasquite stable close to the ground (N ; 0.015 s21). On 3March upvalley winds were quite vigorous and the strat-ification was neutral at least up to heights of ;1000 m.No inversion was found. An interesting situation oc-curred on 6 March with a deep layer of stably stratifiednortherlies on a rather cold morning. The 500-hPa mapshows that a small ridge moved over the area. The stablelayer disappeared quite rapidly (ascent at 0945 LST; notshown) but is was not before late in the afternoon thatsoutherlies were observed. The Lomar layer of a depthof ;1250 m was well mixed with maximum flow speedsof 8 m s21. No inversion was detected. The northerlyadverse flow conditions were almost too dominant forthe Lomar to reach Lo Manthang on that day.

All in all, it became clear that the basic characteristicsof the Lomar layer, namely neutral stratification and theabsence of an inversion, are found also in the upper partof the Mustang basin.



←

FIG. 11. Potential temperature u (K) and specific humidity q (g kg21) during ascent (bold) and descent (broken) in Jomsom and Tukucheon 24 Mar. Also given are wind speed (bold) and direction (crosses) as observed at the Jomsom baseline: (a) 1130 and (b) 1330 LST. Jomsom(Tukuche start) is at 2751 m (2670 m) above MSL. Jomsom airport: z 5 0 m.

5. Discussion

We have to conclude on the basis of the soundingsthat the strong winds in the Kali Gandaki valley cannotbe explained on the basis of hydraulic flow theory, andwe have, therefore, to look for other ways to understandthe observations. As a first step let us fit equations tothe surface data in order to obtain information on thedynamics of the Lomar. For example, we may select thestations Kagbeni and Marpha to adopt the equation ofmotion along the valley axis to the available surfacedata of wind and pressure. With

n11 n n n n(u 2 u )/Dt 5 a(p 2 p ) 1 bu (5.1)k k m k k

(t 5 nDt; Dt 5 120 s; subscripts k, m for Kagbeni andMarpha; p, pressure; u, along-valley wind component;u . 0 for upvalley flow), we have a simple equation tobe tested that asserts that changes of the along-valleywind speed uk in Kagbeni are caused by the pressuregradient force estimated by using the pressure data fromKagbeni and Marpha. There is also a damping term. Itis understood in (5.1) that all terms are deviations fromthe time mean. Nickus and Vergeiner (1984) appliedthis equation to observations in the Inn valley. Ofcourse, (5.1) is a rather simple approximation to the fullequation of motion:

]u 1 ] ]u ]u 1 ]p25 2 u 2 y 2 w 2

]t 2 ]x ]y ]z r ]xo

1 f y 2 du, (5.2)

where y is the cross-valley coordinate; y, the cross-valley flow; w, the vertical velocity; and f , the Coriolisparameter. When using (5.1), we assume that cross-val-ley advection, vertical advection, and rotation effectsare negligible. This omission is presumably justifiedwith respect to rotation and cross-valley winds. It is lessobvious if vertical advection is unimportant. Unfortu-nately, no data are available to check this assumption.In principle, a term ;[( )2 2 ( )2] could have beenn nu uk m

added on the right of (5.1) in order to include along-valley advection. It would have been inconsistent, how-ever, to include horizontal advection when vertical ad-vection is excluded.

We wish to introduce another more empirical equationin combination with (5.1). As outlined above, there issome evidence that descent and the related warmingaloft cause the pressure to fall deeper in Kagbeni in theafternoon than in Jomsom or Marpha (see Table 2). Thisdescent is presumably linked to the acceleration andwidening of the Lomar jet toward the north. If so, thewind velocity in Kagbeni provides a gross measure ofthis effect. We assume that

n11 n[(p 2 p ) 2 (p 2 p ) ]/Dtm k m k

n n n5 gu 1 d(p 2 p ) (5.3)k m k

captures this situation. Strong upvalley winds wouldlead to an increase of the pressure difference pm 2 pk,provided g . 0. A damping term is added on the right.Note, that (5.3) is counterintuitive. One could argue thatstrong winds advect mass to Kagbeni. This would re-duce the pressure difference and, then, g should be neg-ative. Of course, (5.1) and (5.3) are simply the equationsof a regressive model of first order relating the variablesuk and (pm 2 pk) to their changes in time.

The coefficients a 2 d in (5.1) and (5.3) are deter-mined by a least square fit using all observations. Theresulting coefficients are given in Table 4 for three pairsof stations. With a 5 0.32 3 1024 (m s21 Pa21) weobtain about half the correct value ; 0.6 321 21r Dxo km

1024 for the pair Kagbeni–Marpha where Dxkm is thedistance of the villages. This means that the true ac-celerations due to the pressure gradients are partly can-celed by an effect that is not represented properly in(5.1). Advection of momentum from above is presum-ably a good candidate. The damping term b is negativeand implies a damping time | b | 21 ; 40 min. Nickusand Vergeiner (1984) obtained a damping time of 30min for the Inn valley.

The coefficient g is positive with g ; 6 3 1025 (Pam21). The pressure difference between Marpha and Kag-beni increases by about 2 Pa within 1 h if the anomalouswind speed is 10 m s21. This is a relatively weak effect.The damping is quite strong with a damping time of| d | 21 ; 20 min. The system (5.1) and (5.3) is, ofcourse, stable in the sense that any initial perturbationwould be damped out in an integration given the co-efficients in Table 4. Without damping, however, thesystem is unstable with an e-folding time of (ag)21/2 56.4 h. The related growth is slow but not negligible.

It is surprising that a is smaller for the Kagbeni–Jomsom pair than for Kagbeni–Marpha despite the factthat Jomsom is closer to Kagbeni than Marpha. Thecancellation of the accelerations due to the pressure gra-dient by unrepresented effects must even be larger inJomsom than in Kagbeni. The damping time is longerand g has just half the value of that of the Kagbeni–Marpha pair. On the other hand, the coefficients attainthe largest values for the Jomsom–Marpha pair. The e-folding time (ag)21/2 is just 3.6 h in this case. Thisindicates that the Jomsom–Marpha section is the mostimportant one for the evolution of the upvalley flow.The interaction of the Lomar with the layers above isstrongest there. However, this interaction extends clearly



FIG. 12. Potential temperature and specific humidity during ascent (bold) and descent (broken)at 1200 LST 28 Feb in Tangye (3600 m above MSL). Also given are wind speed and direction;z 5 0 at the observing site.

TABLE 4. Coefficients a, b, g, and d as determined from the station data by a least square fit of (5.1) and (5.3) for the station pair givenin the first row.

a(1024 m s22 Pa21)

b(1024 s21)

g(1024 Pam21)

d(1024 s21)

Kagbeni–MarphaKagbeni–JomsomJomsom–Marpha

0.300.230.72

24.422.526.5

0.620.310.82

27.827.9

214.7

to Kagbeni. This conclusion is supported by the frequentobservations of warming above the Lomar layer.

We wish to address here one further issue that cameup quite often during the presentation of the soundings.There is good evidence that the strong Lomar flows areturbulent. Profiles during ascent and descent differsometimes so strongly that it is quite unlikely that thisdifference is due to the choice of the flight track (seeFigs. 9 and 11). As has been mentioned, this turbulencemay be generated by the orographic obstacles to theflow. There is, however, also the possibility that theconfiguration of Lomar is inherently unstable. We mayperceive the flow in an idealized manner as a two-layersystem where the strong winds are confined to a mostlyneutral layer. The air is stably stratified above this layer

and almost at rest. Sometimes there is an inversion, moreoften there is none. We have here, therefore, a variationof the classic Kelvin–Helmholtz instability problem(e.g., Drazin and Reid 1981). As outlined in the appen-dix, the stability in the upper layer is only weakly damp-ing. The discontinuity of the upvalley wind at the upperboundary of the Lomar layer supports instability at allwavelengths L 5 2p/k (k is the along-valley wave-number) provided an inversion is absent. However,growth rates increase ;k so that short waves are pre-ferred. It is conceivable that such instabilities play arole in generating the deviations in Figs. 9 and 11.

Finally, let us comment on the relationship of thenumerical simulations of KG2 to the observations.Many aspects of the wind profiles obtained in our cam-



paign are compatible with those presented in KG2. Theonset of the upvalley winds close to the surface and thefollowing buildup of the upvalley wind layer as foundin KG2 were observed this time, as in 1998 (KG1).Moreover, the model results agree with the observationsof strong upvalley winds in Tangye (see Fig. 6 of KG1)and Lo Manthang. It is, however, our impression thatthe turbulence parameterization in MM5 underestimatesthe generation of turbulence in strong flows as in Fig.2. It is presumably for that reason that the model predictsa stably stratified Lomar layer in contrast to the obser-vations. Pronounced gravity wave features have beenfound in KG2 that where generated by the ridges nearMarpha (see Fig. 8 of KG2). It was suggested in KG2that these waves are important for the generation of thestrong upvalley winds. This speculation can be ruledout on the basis of our observations. Internal gravitywaves cannot be excited in neutrally stratified flow.

6. Concluding remarks

The following summarizing statements can be madeon the basis of our observations. The period of weakwinds extends normally from late in the evening till0900–1100 LST. Although weak downvalley flow pre-vails during that time, weak upvalley flow occurs quiteoften. The rather limited number of soundings in themorning suggests that the atmosphere is stably stratifiedat sunrise but that there is no inversion. After sunrise aconvective layer of a few hundred meters depth evolveswith a shallow superadiabatic layer at the ground.

The upvalley winds set in close to the ground. A well-mixed upvalley wind layer is established within aboutan hour with a depth of 1000–1500 m. Sometimes aninversion is found on top of this layer, but more oftenthe neutral layer is simply topped by a stable layer. Thefully developed upvalley flow is presumably unstable.These flows are found at all locations from Tukuche upto Lo Manthang.

The transition to the upvalley regime occurs first inMarpha and Tukuche and moves with speeds of ;5 ms21 upward to Kagbeni and farther on to Lo Manthang.

As outlined in KG1 and KG2, the upvalley winds aregenerated primarily by the heating of the Mustang basinbefore noon. Low pressure is established there with re-spect to the atmosphere to the south of Lete at the sameheight as the basin. The flow driven by this pressuregradient has to pass the narrow part of the valley wherethe Kali Gandaki River cuts through the Himalayas. Thedynamically most interesting part of the flow is that nearand to the northeast of Marpha where the highest windspeeds are found. The suggestion of KG2, that somekind of supercriticality is involved in generating thesehigh velocities, is not supported by the soundings. How-ever, the soundings support and even extend the sug-gestion of KG2 that descent above the Lomar is animportant part of the dynamics of the Kali Gandaki windsystem. Although the soundings provided information

on the stratification of these upper layers, the wind ob-servations via pibals are too inaccurate at large heightsto tell us much about the flow conditions up there.Therefore, important aspects of the dynamics of the KaliGandaki wind regimes have to await further clarifica-tion.

It is an important side result of the recent field cam-paign in the Kali Gandaki valley that soundings up toheights of ;2000 m above the ground can be made byuse of RPV even under extreme conditions. The verticalresolution of the resulting profiles is quite good as isthe quality of the observations. This sounding systemis quite mobile and vertical profiles can be obtainedalmost anywhere. It is a drawback of this method thathighly skilled pilots are needed at least under the ex-treme conditions of the Kali Gandaki valley.

Acknowledgments. The campaign could not have beenconducted without the financial support by DeutscheForschungsgemeinschaft. A great number of people sup-ported the project in many ways. We wish to expressour gratitude to Robbe Modellsport for help with respectto the remote control, to Hacker Antriebstechnik for helpwith respect to the plane engines, to Ingenieurburo Wur-tenberger for support with respect to the sensors, toZeiss for its generosity with respect to the developmentof the optical control systems, to Simprop Electronicfor advice on the implementation of the motor, to H.Muller of Modellbau Ulrich and Blue Airlines for helpin building the plane, to K. Budion for advice withrespect to batteries, and to Optik Schadow. We are grate-ful to G. Zangl for comments and to L. Gantner forproviding information on the synoptic situation. Thecomments by the referees helped substantially to im-prove the presentations of the results.

APPENDIX

Instability of Two-Layer Flow

We consider a two-layer atmosphere. The lower layerof depth H represents the lomar layer with constantmean wind U1 and vanishing Brunt–Vaisala frequency

5 0. A second layer with vanishing wind U2 5 02N1

and stable stratification . 0 is assumed to extend to2N 2

infinity above this layer. Both layers are separated by amaterial surface. There is no density jump. We assumeperturbations of the form exp[ik(x 2 ct)] and solve thewell-known wave equation

2 2] w Ni i1 w 5 0 (A.1)i2 2 2[ ]]z (U 2 c) 2 ki

[i 5 1, 2; wi(z) Fourier coefficient of vertical velocity],under the boundary conditions w1 5 0 at z 5 0, p1 5p2 at z 5 H, and w2 5 0 at infinity. Moreover, thekinematic condition

w (c 2 U ) 5 w c1 1 2 (A.2)



relates the vertical velocities at z 5 H. With w1 ;sinh(kz) and w2 ; exp(2nz), one obtains the polynomial

2 4 2 2 3k c 2 4k U cosh (Hk)c1

2 2 2 2 2 21 [6k U cosh (kH ) 1 N sinh (kH )]c1 2

2 3 2 2 4 22 4k U cosh (kH )c 1 k U cosh (kH ) 5 0. (A.3)1 1

There are two real roots. One of them violates the upperboundary condition and must be excluded. The otherone is c ; U1/2 for L # 4 H. The complex pair is c ;U1(1 6 i)/2 except for L . 4 H, where | c | is somewhatsmaller. In essence we recover the result of the Kelvin–Helmholtz problem.

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