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FMU Rep. No. IV-10
(JANUARY 1 9 7 4 )
INDIA METEOROLOGICAL DEPARTMENT
FORECASTING MANUAL
PART IV
COMPREHENSIVE ARTICLES ON SELECTED TOPICS
0 : MOUNTAIN WAVES
BY
R. P. SARKER.
ISSUED BY
THE DEPUTY DIRECTOR GENERAL OF OBSERVATORIES
(FORECASTING)POONA-5
FORECASTING MANUAL REPORTS
No.I-1 Monthly Mean Sea Level Isobaric Charts - R. Ananthakrishnan,
V. Srinivasan and A.R. Ramakrishnan.
No.I-2 Climate of India - Y.P. Rao and K.S. Ramamurti.
Nc.II-1 Methods of Analysis: 1. Map Projections for Heather Charts -
K. Krishna.
No.II-4 Methods of Analysis: 4. Analysis of Hind Field - R.N.Keshava-
murthy.
No.III-1.1 Discussion of Typical Synoptic Weather Situations: Winter -
Western Disturbances and their Associated Features - Y.P.Rao
and V. Srinivasan.Weather
No.III-2.2 Discussion of Typical Synoptic/Situations: Summers Nor'westers
and Andhis and large scale convective activity over Peninsula
and central parts of the country - V. Srinivasan, K.Ramamurthy
and Y.R. Nene.
No.III-3.1 Discussion of Typical Synoptic Weather Situations: Southwest
Monsoon: Active and Weak Monsoon conditions over Gujarat
State - Y.P. Rao, V. Srinivasan, S.Raman and A.R.Ramakrishnan.
No.III-3.2 Discussion of Typical Synoptic Weather Situations: Southwest
Monsoon: Active and Weak Monsoon conditions over Orissa —
Y.P. Rao, V. Srinivasan, A.R.Ramakrishnan and S. Raman.
No.III-3.3 Discussion of Typical Synoptic Weather Situations: Southwest
Monsoons Typical Situations over Northwest India —
M.S.V. Rao, V. Srinivasan and S. Raman.
No.III-3.4 Discussion of Typical Synoptic Weather Situations: Southwest
Monsoons Typical Situations over Madhya Pradesh and Vidarbha -
V. Srinivasan, S.Raman and S. Mukherji.
No.III-3.5 Discussion of Typical Synoptic Weather Situations: Southwest
Monsoons Typical Situations over Uttar Pradesh and Bihar -
V. Srinivasan, S. Raman and S. Mukherji.
No.III-3.6 Discussion of Typical Synoptic Weather Situations: Southwest
Monsoons Typical Situations over West Bengal and Assam and
adjacent States - V. Srinivasan, S. Raman, and S. Mukherji.
No.III-3.7 Discussion of Typical Synoptic Weather Situations: Southwest
Monsoons Typical Situations over Konkan and Coastal Mysore —
V. Srinivasan, S. Raman, S. Mukherji and K. Ramamurthy.
No.III-3.8 Discussion of Typical Synoptic Weather Situations: Southwest
Monsoons Typical Situations over Kerala and Arabian Sea
Islands - V. Srinivasan, S. Mukherji and K. Ramamurthy.
()Contd. on back cover page)
FMU Rep. No.IV - 10
(January 1974)
FORECASTING MANUAL
Part IV - Comprehensive Articles on Selected Topics
10. Mountain Waves
by
R.P. Sarker
Contents
1. Introduction2 . Observational Evidence2.1 Mountain Clouds2.2 Experience of Glider Pi lots2.3 Effects observed from powered a i rc ra f t3 . Flying Aspects of Mountain waves3.1 Vertical currents3.2 Turbulence3.2.1 Turbulence at mountain top level3.2.2 Turbulence above mountain top level3.2.3 Turbulence below mountain top level3.3 Errors in al t imeter readings3.3.1 Graphical Determination of the errors in Altimeter Heights3.4 The effect of the variat ion of horizontal wind speed3.5 The effect on a i rc ra f t icing4. Detecting the Presence of Mountain waves-observational evidence5 . Requirements for Mountain wave formation6. Theoretical works and the i r applications to the prediction of
mountain waves.6.1 Queney's works6.2 Work of Scorer6.2.1 Conditions for occurrence of lee waves6.2.2 Verification of Scorer 's r esu l t s6.3 Works of Palm and Foldvik, Foldvik, Doos, Sarker6.4 Wavelength6.5 The lee wave amplitude
6.5.1 Effect of a succession of ridges
6.5.2 Effect of the profile of air mass stability
6.5.2.1 Inversions
6.5.2.2 Adiabatic lower layer
6.6 Separation of the flow — effect of lee standing eddy
6.6.1 Mountain shape
6.6.2 Stability conditions
6.6.3 Sudden Disturbances
6.7 Short ridges and single peaks
7. Application to Aviation Forecasting
7.1 Observing and Reporting Orographic clouds
7.2 Reports by Pilots
7.3 Application of theoretical and observational results.
Contd.
7.3.1 The likelihood of s ignif icant waves
7 .3 .1 .1 Scale of t e r ra in
7 .3 .1 .2 The presence of Je t streams
7 .3 .1 .3 Irregular Topography7.3.1.4 Changing synoptic conditions7.3.1 .5 Diurnal and Seasonal var iat ions7.3.1.6 Other Effects
7.3.2 Descending current on the lee of a mountain barrier
7.3.3 The level of maximum amplitude
7.3.4 Forecasting Turbulence
7.3.5 Aircraft Icing
8. Some suggested safeguards for flying in Mountain waves
9 . Mountain waves over Western Ghats.
9.1 Mountain waves during Monsoon season
9.2 Some indirect verifications of mountain waves over Western Ghats
9.2.1 Lee waves from cloud observations
9.2.2 Turbulence reports by Aircraft
10. - Mountain waves over Assam-Burma H i l l s .
APPENDIX-A S c a l e for computation of 12
APPENDIX-B Graphical Determinat ion of wave l eng th
REFERENCES
DIAGRAMS.
1. Introduction
In the past, numerous aircraft accidents have occurred over mountains, for
which there was at the time no satisfactory explanation. It has, of course,
long been known that the airflow over mountains or hilly terrain is usually
more disturbed that over flat country. Until recently, however, little was
known concerning the nature and magnitude of disturbances in the airstream caused
by mountain barriers, and the meteorological conditions which have a bearing on
them. In order to make flying over mountains safer, a considerable amount of
research has been conducted in recent years. As a result, much useful informa
tion has been gathered which, if properly utilised, would be instrumental in
reducing the number of air accidents over mountains.
In particular, it is now known that the influence of mountains on the
airflow is much greater than had been suspected. Indeed, there is circumstan
tial evidence that under certain meteorological conditions which are not at all
uncommon, the influence of even small hills extends to surprisingly great
heights.
Perhaps the most important information, from the aeronautical view point,
gained in recent years on airflow over mountains, concerns the standing waves,
which under favourable meteorological conditions, form on the lee-side of
mountain barriers. Many past air accidents can now be explained in terms of
these so-called "mountain waves" which, because of the vertical currents they
set up, the severe turbulence they sometimes generate and their effect on the
accuracy of pressure altimeter readings and air navigation, can constitute a
real danger to the unwary pilot.
During the last few years there has grown a considerable body of experi
mental data on the subject from various sources and notable theoretical
researches into the problem have been carried out, so much so that over at least
an important part of the subject a logical system of ideas has been developed
and can now be presented for the benefit of both forecasters and pilots.
Glider pilots have learnt a great deal of the special air-flow effects which
2
occur in the neighbourhood of mountains. Accordingly much of practical evidence
has come from glider pilots and the potentialities of gliders for research have
been exploited in a number of field investigations. The experiences of the
pilots of powered aircraft, studies of orographic cloud formations as well as
comprehensive programs of observations using the most modern equipment and
techniques have recently rendered a coherent picture on the subject. These
observational data have been supplemented by the recent theoretical studies and
experiments with laboratory models.
In this report an attempt will be made to describe the known properties of
mountain waves with particular reference to their effect on flying conditions.
Methods of detecting and forecasting the occurrence of the mountain waves will
also be discussed together with the measures which should be taken in order
to minimize the inconvenience and hazard of flying over a mountain or hilly
terrain.
In this note it would be desirable to lay more emphasis on the phenomenon
in the Indian regions. But unfortunately, little work has been done on this
problem in India. Most recently only, a few theoretical studies have been
made. No observational data is yet available. We shall, therefore, take
examples from the other parts of the world where data from organised field
studies are available.
Similar notes have been written by Corby (1956) and Alaka (1958).
To start with we shall mention very briefly some of the observational
evidences to make clear the nature of the phenomenon.
2. Observational Evidence
2.1 Mountain clouds
Lenticular clouds are frequently seen over mountainous terrain in all
parts of the world. These clouds remain more or less stationary relative to
the ground and the wind blows through them, so that such clouds are continu
ally reforming at their up-wind edges and dissipating at the down-wind edges.
3
This has been confirmed by time-lapse photography. On many occasions, specially
to the lee of long ridges, a succession of lenticular clouds, parallel to each
other and to the ridge may be seen; the suggestion that they constitute visible
signs of atmospheric lee waves is very strong.
A similar phenomenon can sometimes be observed when an airstream contain
ing a layer of stratocumulus crosses a mountainous area. Organized stationary
clearance of the cloud in the form of holes or zones parallel to the lee of
hills or ridges can be seen. The inference is that the clearances are located
when the airstream executes downward undulations associated with topography.
A number of field studies have been carried out aimed at elucidating the
mechanism causing particular cloud formations which are characteristics of
given hills or mountains. Amongst these are the exhaustive study by Manley
(1945) of the Crossfell Helm Wind and its associated clouds and the study by
Kuettner (1939) of the so-called Moazagoti clouds which form over the Riesenge-
birge in Bohemia. Although on different scales, both these phenomena have much
in common and in particular, rotar clouds are commonly observed at levels of
the hill tops, whilst lenticular clouds may also be seen high above the rotor
clouds.
More recently Ludlam (1952) has discussed the formation of orographic cirrus
over the hills of the United Kingdom, and has put forward evidence which supports
his suggestion that a great deal of cirrus forms in atmospheric waves initiated
by topographical features. His observations show that hills in U.K. can produce
waves having sufficient amplitude at 6 km and above to produce cirrus cloud.
According to Stormen (1948) the astonishing mother-of-pearl clouds sometimes seen
high above the Norwegian mountains at 20-25 km also owe their origin to topography.
This has been subsequently theoretically supported by Palm and Foldvik (1960).
Clearly, in the light of such observational evidence, the view that mountains
have no effect on the airflow above more than three times their heights is no
longer tenable.
4
Most recent ly evidence of clouds of orographic origin has come from
s a t e l l i t e cloud photographs and have been reported by Doos (1962), Fr i tz (1965)
Cohen et al (1966). In India, orographic clouds are seen in the Assam and Burma
Hil ls as evidenced from the s a t e l l i t e photographs (De 1970). In the Western
Ghats near the Matheran area, a case of orographic cloud has been reported by
Sinha (1966) from visual observations.
2.2 Experience of Glider Pi lots
In the early days of g l id ing, the sources of ver t ica l motion used were f i r s t
the up-slope motion to be found on the windward side of h i l l s and ridges and l a t e r
thermal up-currents . During the 1930's the pos s ib i l i t i e s of soaring in atmos
pheric waves began to be explored, special ly in the continent. Numerous ascents
were made on the lee s ide of the Alps in fohn wind conditions and by 1939 several
g l ider ascents to beyond 9 km had been made in the air r i s ing on the up-wind s ide
of wave clouds. The s ignif icant aspect was that many of these ascents were made
well to the lee of the highest ground and could not, therefore, have been simple
cases of f l igh ts in the air stream ascending the windward s lopes. The f i r s t
important ascent by a glider in standing wave in U.K. was made in 1939 by McLean
who reached 3.4 km in the helm wind in Cumberland. On th i s f i r s t occasion,
McLean had d i f f icu l ty in ge t t ing down, and was only able to do so by locating
the down-draught of the wave.
Since the war, wave soaring has become common-place and from the various
reports i t i s found tha t , over the lee ground, waves, i f any, increase in ampli
tude upwards to some level , above which the waves decrease and eventually die
out . They die away downstream, but a succession of several observable waves
has often been noted in U.K. The order of magnitude of the wave-length is
usually in the range of 2-20 km, but wave-lengths upto 70 km or so are sometimes
observed.
In the spectacular Bishop wave which i s charac te r i s t i c of the airstream
over the Sierra Nevada range in California during winter many ascents to
5
well above 12 km have been made by g l i d e r s . In these wave systems ve r t i ca l
currents of 10 m/sec are common, whilst 20 m/sec has been recorded and i t i s
believed that the ve r t i ca l components exceeding 25 m/sec may occur on occasions.
In addi t ion, turbulence of phenomenal in tens i ty occurs in t h i s area. Field
invest igat ions have been carried out at t h i s location as part of the United
States Mountain Wave Project .
2.3 Effects observed from powered a i rc ra f t
There has been a s teadi ly increasing number of reports of mountain airflow
effects from the p i lo t s of powered a i r c ra f t during the las t few years . This
may be because a i rc ra f t are now flying with increasing regular i ty and frequency
over mountain ranges which l i e across a i r rou tes . The reports confirm tha t
areas of l i f t and sink are commonly to be found over and to the lee of mountains.
Captain D. Mason (1954) of Bri t i sh European Airways Corporation gave a detai led
description of an incident over the mountains of northern Spain on December 18,
a1952 when he was flying viking a i rc ra f t from Madrid to London. He approached
the mountains north of Madrid at 3.4 km and the a i rc ra f t subsequently f e l l to
2.7 km and then rose to 4 .3 km. This was repeated three times in what were un
doubtedly powerful standing waves. The height changes took place in sp i t e of
his using maximum power in the areas of sink and almost closing the t h r o t t l e s in
the areas of r i s ing a i r . I t can be inferred from th i s that the ve r t i ca l com
ponents exceeded 2.7 m/sec.
3 . Flying Aspects of Mountain Waves
3.1 Vertical currents
One of the important aspects of mountain waves from the view point of av ia
tion i s the ve r t i ca l currents associated with these waves. There may be
updrafts and downdrafts associated with the waves. From the view point of
powered a i rc raf t the low-level downdrafts are most important, since they con
s t i t u t e one of the principal threa ts to a i r c ra f t safety. Reports from p i lo t s of
powered a i r c r a f t , during the l a s t few years , confirm that ve r t i ca l currents of
6
the order of 5-10 m/sec associated with standing lee-waves are quite common in
various parts of the world. The danger to aircraft from downdrafts of the
above magnitude can be easily appreciated. An aircraft flying more or less
parallel to a ridge might remain in a downdraft continuously until the
whole length of the ridge is traversed. In such circumstances catastrophic
loss of height might occur.
When waves are present, areas of descending currents generally occur
immediately downwind from a mountain ridge. An aircraft flying upwind towards
a mountain ridge, if caught in a strong downdraft near the ridge, might not
be able to regain enough altitude in time to clear the mountain.
The danger to aircraft is enhanced by the fact that flying through waves
is often remarkably smooth even when the rate of lift and sink may be consi
derable. At night when no warning wave clouds can be seen, or when the sky
is completely overcast, indication of loss of height is given to the pilot
only by the altimeter or the rate of climb indicator, and flying by automatic
pilot can result in disaster to an unwary pilot.
3.2 Turbulence
Pilots have often commented on the extremely smooth flying conditions in
mountain waves, specially in the higher levels. Yet mountain waves also
generate turbulence which can be more violent than any encountered in most
thunderstorms. Of importance to aviation is the fact that the smooth and
turbulent areas are often in close proximity.
Turbulence in mountain waves may occur at, above and below mountain
top level.
3.2.1 Turbulence at mountain top level
The most common and most important seat of severe turbulence in
mountain waves is the area of rotor clouds. The clouds form in standing
eddies under the wave crests at an altitude which is comparable with the
height of the mountain which produces the wave. Measurements made in standing
7
eddies downwind from the Montagne de Lure in France (height 1400 m above
surrounding terrain) have revealed that strong variations in the wind speed ran-
ging from 10 to 25 m/sec occur inside these eddies and that the vertical speeds
can vary from + 8 m/sec to -5 m/sec in 2 or 3 seconds. This is equivalent to a
5
vertical acceleration of 2 to 4 g (Berenger and Gerbier 1946).
Rotor turbulence is much more intense in waves generated by the larger
mountains. Violent sharp-edged gusts exceeding 12 m/sec have been measured in
some Sierra waves, and experienced pilots have reported complete loss of control
of their aircraft for short periods while flying in the rotor areas. According
to Kuettner and Jenkins (1953) high speed aircraft flying with the wind through
such well—developed rotor areas will undergo a breaking effect of such magni
tude as to endanger the structure of the aircraft.
The danger of rotor turbulence to aviation is accentuated by the fact that
the downdraft in the lee of the rotor and the updraft on the other side of it
can drag an aircraft into the rotor cloud. In a dramatic account of an upwind
flight in a mountain wave, Kuettner and Jenkins describe how their aircraft was
caught in the descending current downwind from the rotor and actually "fell"
into the rotor cloud from above. The buffetting which the aircraft was subjected
to inside the cloud was worse than any the authors had experienced in thunder
storms.
The most dangerous situation occurs when lack of moisture prevents cloud
formation or when the sky is completely covered by a thick layer of low cloud.
In such cases no prior visual warning is given of the existence of the turbulent
area.
3.2.2 Turbulence above mountain top level
Although exceptionally smooth flying conditions prevail as a rule in moun
tain waves, this is not invariably the case. Much stronger turbulence has been
experienced in mountain waves over the United States. A report by Harrison(l956)
shows that during the first nine days of 1956, a series of waves occurred down
wind from the Continental Divide in which severe clear air turbulence was
8
reportedby civil and military planes. Injuries to occupants of the aircraft were
reported on four of these days. The transition from smooth to rough flying
conditions in waves is often rapid. The smooth laminar flow suddenly breaks
clown into a chaotic pattern of turbulence extending throughout the vertical
extent of the wave. Such cases are accompanied by a change in the appearance
of the usually smooth lenticular clouds which now acquire a rough, turbulent
appearance.
3.2.3 Turbulence below mountain top level
Apart from turbulence within the wave system, there may often be turbu
lence at low levels within the friction layer. Typical mountainous terrain is
quite irregular and there is evidence that while the main airflow aloft may be
smooth and wave like, the surface irregularities are filled in by turbulent
eddies. The intensity of this type of turbulence is determined by the same
factors which govern turbulence in the frictional layer elsewhere.
3.3 Errors in altimeter readings
Altimeter readings near mountain tops are often subject to errors large
enough to constitute a source of danger to aircraft. Some aircraft accidents
near mountain peaks may indeed be ascribed to an underestimation of the mag
nitude of the altimeter error possible under certain circumstances. Special
studies in the United States to determine the effect on pressure of strong
wind flow over mountain barriers indicate that the effect is in the form of a
pressure reduction which is proportional to the square of the wind speed. For
a wind speed of 45 m/sec the altimeter reading was 100 m too high for unsatu
rated air while the discrepancy was doubled when the air was saturated. Pilots
sometimes report even higher discrepancy. In one case a pilot in the Owens
valley in California reported an altimeter reading nearly 1000 m higher than
the actual altitude.
It is probable that a substantial part of the error in estimating heights
near mountain tops in standing waves results from the waves themselves. When
9
waves are present, there is generally a zone of descending currents immediately
downstream from the summit and a zone of ascending current farther downstream on
the otherside of the wave. A pilot flying up-wind in the direction of the moun
tain may take his reading (in the region) where the air is ascending and set his
automatic pilot to conserve his cruising altitude. Almost immediately afterwards,
the aircraft reaches the descending zone where the unwary pilot may lose altitude
at the rate of 500 m/min. or even more. Thus a few minutes after the altimeter
reading is taken, the aircraft may be at an altitude more than 1000 m lower than
that indicated by the instrument.
Although research, in its present stage, gives no clear clue with regard to
the source of errors, we cannot ignore the possibility that large pressure varia
tions may exist near mountain peaks during high wind. And since these pressure
variations seem always to be in the direction of indicated altimeter heights
which are too high, they constitute an aspect of mountain flying which cannot be
overlooked by pilots.
3.3.1 Graphical Determination of the Errors in Altimeter Heights
In the region of any real atmospheric vortex the meteorological elements
obey very complicated laws. However, under some simplifying assumptions for
mountain top vortices, the following approximate formula may be shown to be true
where po is the pressure in the centre 0 of the vortex, and pA , p and
V are respectively the air pressure, air density and wind velocity at some
point A situated in the region of the vortex. Since the pressure diminishes
towards the centre of the vortex, the difference po - pA characterizes the
drop in atmospheric pressure between the point A and 0. Formula (3.1) shows
that this pressure drop is equal to the product of the density and the square
of the velocity.
It is well—known that very high velocities on the leeside are associated
with the phenomenon of air masses crossing a mountain ridge. In a curved
10
stream, vortices of various intensities may arise and the wind velocities may
reach 100 m/sec and more. In such a case the pressure drop will be
The error in the altimeter readings will be 1000 m. The following table of
possible altimeter errors h for various wind velocities, V, in a mountain
region was drawn up on the basis of such considerations.
Table I
Vm/Sec 10 20 30 40 50 60 70 80 90 100
h(metres) 10 40 90 160 250 360 490 640 810 1000
This table was calculated on the basis of formula (3.1). This formula is only
approximate because of the series of approximations adopted for its derivation.
The value of the atmospheric air density, appearing in the right hand side,
depends on the three spatial coordinates as well as on time. Therefore, the
data of Table I are approximate and, of course, can be recommended to a pilot
only after verification under field conditions.
However from Table I it is possible to draw one useful conclusion, appa
rently in accordance with reality, namely that the greater the velocity in a
mountain region, the larger may be the altimeter error. The altimeter error
in metres is approximately equal to the square of the wind velocity divided
by 10.
In calculating Table I the density was considered constant everywhere.
In reality, the density depends essentially on height. In Table 2 values of
the standard density in the meteorological system for several heights are
given.
Table 2
h Km 1 2 3 4 5 6 7 8 9 1 0
p Kgm/m3
1.15 1.033 0.927 0.829 0.739 0.658 0.583 0.516 0.454 0.399
11
I t follows from Table 2 that for an increase in the height from 1 to 10 km the
density decreases by almost th ree t imes. But since the error in the al t imeter
readings h depends on the density, the variat ion of the density with height
has to be allowed for in calculat ions of h. Table 3 gives values of h
(in metres) for various heights ,
TABLE - 3
p Kgm/m3
Vm/Secp Kgm/m
3
10 20 30 40 50 60 70 80 90 100
1.15 (h=1 km) 11.5 46 .0 103.0 184.0 287.5 414.0 563.5 736.0 931.5 1150.0
1.03 (h=2 km) 10.3 41 .2 92.7 164.8 257.5 370.8 504.7 659.2 834.3 1030.0
0.93 (h=3 km) 9 . 3 37.2 83.7 148.8 232.5 334.8 455.7 595.2 753.3 930.0
0.83 (h=4 km) 8 . 3 33.2 74.7 132.8 207.5 298.8 406.7 531.2 672.3 830.0
0.74 (h=5 km) 7 . 4 29.6 66.6 118.4 185.0 266.4 362.6 473.6 599.4 740.0
0.66 (h=6 km) 6 . 6 26.4 59.4 105.6 165.0 237.6 323.4 422.4 534.6 660.0
0.58 (h=7 km) 5 . 8 23.2 52.2 92 .8 145.0 208.8 284.2 371.2 469.8 580.0
0.52 (h=8 km) 5 . 2 20 .8 4 6 . 8 83.2 130.0 187.2 254.8 332.8 421.2 520.0
0.45 (h=9 km) 4 . 5 18.0 40.5 72.0 112.5 162.5 220.5 288.0 364.5 450.0
0.40 (h=10 km) 4 . 0 16.0 36.0 64.0 100.0 144.0 196.0 256.0 324.0 400.0
Fig.1 is a nomogram for determining altimeter errors from a known value of the
wind velocity, allowing for density variation with height. The horizontal axis
gives the wind velocity and the vertical axis gives the altimeter error. From the
given density for each height, a curve - a parabola - is constructed by means of
formula 3.1 (Table 3). Thus the family of parabolas is obtained. This nomogram
is quite simple and can be easily used.
Suppose, for example, that the airplane flies at a height of 4 km with a wind
velocity of 50 m/sec. In order to find the possible altimeter setting for this,
it is necessary to find on the horizontal scale the point V = 50 m/sec. At this
point a perpendicular is constructed to intersect the curve corresponding the
12
h = 4 km. Then, drawing a para l le l from th i s point to the horizontal ax i s , we
find that in th is case the al t imeter error may reach 200 m. I t i s seen from the
nomogram that the alt imeter errors should markedly decrease with height, and
for a given wind veloci ty and a given height the error cannot exceed a certain
value. Thus, for a wind velocity V = 60 m/sec the error in the al t imeter
readings at a height of 10 km cannot exceed 150 m whereas at a height of 2 km
t h i s error can be more than 350 m.
I t should be borne in mind that an airplane may lose height when flying in
mountain waves. If the p i lo t takes the alt imeter reading in a region of a
mountain wave cres t and then immediately enters a strong descending current on
the lee of the wave c r e s t , the airplane s t a r t s to lose height rapidly , as if
diving into the trough of the wave. If, for example, the velocity of the des
cending current i s 12.5 m/ sec . then during one minute the height los t by the
plane should be 750 m. Since such ve loc i t ies (sometimes greater ones) are
encountered not infrequently in a descending current on the lee of a c r e s t ,
the importance of allowing for th i s phenomenon in f l ights over a mountainous
t e r r a in becomes obvious.
3.4 The effect of the variat ion of horizontal wind speed
Mountain waves in the usual sense, i . e . a l ternat ing zones of ascending
currents extending over the barr ier as well as downstream, are primarily caused
by the appearance of an orographic ver t ica l velocity component at the moment of
encounter of the undisturbed flow with the ba r r i e r . But the nature of atmos
pheric a i r i s such that an air particle i s more easily displaced horizontally
than v e r t i c a l l y . I t i s l ikely that simultaneously with orographic ve r t i ca l
veloci ty component an orographic horizontal velocity component appears at the
moment of encounter of the undisturbed flow with the ba r r i e r . Musaelyan(1960)
by solving equations of hydrothermodynamics found tha t disturbances generated
by mountain bar r ie rs in the horizontal velocity component f ie ld extend in
wave form on both sides of the barr ier (hor izonta l ly) , as well as downstream
13
(Fig.2). There also exist undisturbed surfaces along which the orographic
horizontal velocity component vanishes.
Thus it seems that the mountain waves are not only waves in the vertical
velocity component field, but also superimposed on them and inseparably linked
to them are wave disturbances of the horizontal velocity component. Measurements
of fluctuations in the horizontal wind speed between crest and trough were made
during wave conditions in the lee of the Montagne de Lure by tracking a constant
pressure balloon with radar. The results showed a variation in wind speed from
16 m/sec in the trough of the wave to 26 m/sec in the crest (Berenger and
Gerbier 1956). This variation was associated with a wave amplitude of 1350 m.
There is no doubt stronger fluctuations would accompany more intense waves.
Under wave conditions, an aircraft flying parallel to a large ridge lying
across the wind could be subjected to a horizontal wind materially different from
that prevailing a few kilometres away, and could easily get off the course. If
the wind measured in a wave is used as an average over long distances, the ensuing-
error could amount to many kilometres. Thus, the importance of accurate naviga
tion over mountains, particularly in cloud at night and when ground clearance is
not large, cannot be overestimated.
3.5 The effect on aircraft icing
The vertical displacement of the air in mountain wave is accompanied by a
fluctuation of the temperature between crest and trough. These fluctuations are
reflected in a corrugation of the 0°C isotherm. Measurements by Berenger and
Gerbier (1956) have revealed that the temperature changes are nearly adiabatic,
and since the waves have their largest amplitudes in layers of atmosphere having
great static stability, the level of the 0ºC isotherm in these layers can be sub
stantially lower in the wave crests than would be indicated from a radiosonde
ascents made in the same air mass, but in a locality which is undisturbed by
waves. Awareness of this possibility may help in avoiding unexpected ice accre
tion on the aircraft.
14
Apart from the possible lowering of the 0ºC isotherm, icing conditions are
sometimes aggravated by the aerodynamic peculiarities of airflow over mountains.
Icing depends to a great extent on the concentration of supercooled liquid
water in the potential icing cloud and it is an observed fact that clouds formec
by ascent over mountains have a much greater liquid content than clouds formed
in the free air.
Thus if conditions are favourable for icing generally, a greater liability
or intensity would be expected over mountains.
4. Detecting the Presence of Mountain Waves - Observational Evidence
The effect of mountain waves on the performance and safety of aircraft makes
it important that pilots should be aware of their presence.
The earliest waves to detect are those which are accompanied by the typical
mountain wave clouds described before. Under favourable conditions, a pilot
is able to see these clouds from a long distance so that he can take the neces
sary precautions to minimise the discomfort or danger of flying through them.
Alternatively, the presence or risk of mountain waves can be communicated
to the pilot by the briefing meteorologist on the basis of his analysis of
reports from other pilots of of reports of orographic clouds received at the
forecasting centre. In this connection observers should be encouraged to
note carefully the presence of cap clouds, rotor clouds or lenticular clouds
in the "Supplementary Information" section of the hourly weather reports.
Identification of the clouds will, of course, be difficult on dark nights.
Mountain waves, however, may occur with cloudless skies or with comple
tely overcast skies.
5. Requirements for Mountain Wave Formation
Although the method described in the preceding section for detecting the
presence of mountain waves is useful, it would be necessary for meteorologists
to determine the meteorological conditions which are favourable for the for
mation of such waves, so that they would be in a position to advise pilots
15
when these conditions are satisfied and waves are likely to occur.
From simple physical reasoning it becomes clear that both static stability
and geostrophic forces are important as restoring forces for the formation of
mountain waves. However, the degree of importance of these factors depends on
the dimensions of the barriers causing the waves. If we consider a very small
hill, a few metres high and not more than 100 m wide, both stability and geostro
phic forces axe negligible, because on such a scale the atmosphere behaves as
though the lapse rate were adiabatic and the time taken by the air in crossing
the hill is too short to allow the geostrophic forces to come into play. When
the hill is a few kilometres in width, the geostrophic forces remain negligible
whilst stability becomes significant. On an even larger scale as in the case
of the Alps or the Rockies or the Himalayas, for instance, both stability and
geostrophic forces are important.
The requirement of a stable stratification of air mass for wave formation
has been studied by Larson (1954) and Georgii (1956) and others. Besides
stability, another important factor which determines whether the deformation
of anairstream by a mountain ridge is likely to lead to occurrence of standing
waves is the vertical variation of wind. This has been studied in detail by
Forchtgott (1949). Also observations indicate that for standing waves to
occur, the wind speed at some level above the surface must exceed a certain
minimum, which, however, seems to be slightly different in different locali
ties. This has been studied by Pilsbury (1955), Larson (1954), Jenkins and
Kuettner (1953), Manley (1945) and Colson (1954). However, without going in
detail to these studies, we shall briefly summarise below the results of these
observational studies on the meteorological requirements for the formation of
waves:
(i) Marked stability in the lower layer with comparatively low stability
aloft. The stable layer need not necessarily extend to the ground.
(ii) Wind speed at the level of the summit exceeding a minimum which varies
from about 8 to 13 m/sec depending on the ridge generating the waves, and
16
either increasing or at least remaining constant with height upto the
tropopause.
(iii) Wind direction within 30º of the direction normal to the ridge and not
changing substantially with height.
A recent study by Gerbier and Berenger (1961) shows that if the direction
of wind suddenly changes by 180º, say from westerly to easterly, at a particular
level, then there will be rotor and so turbulence at that level (Fig. 2(a).
6. Theoretical works and their applications to the prediction
of Mountain Waves
Although field observations and measurements constitute a very important
source of information on waves, they are not without their limitations. Syste
matic exploration of wave conditions is often very difficult and expensive.
Moreover, field work cannot normally be planned to coincide with the most impor
tant wave occurrences. Finally field observations and measurements are possible
only at a finite number of points — on the ground or where a cloud is visible
from the ground, or where a plane, balloon or some other instrument happens to
be situated. To determine the flow at all points recourse must be had to hydro
dynamic theory.
In the recent years, theoretical studies on mountain waves made quite a
good progress. However, most of these studies, because of the great mathemati
cal difficulties, have almost invariably made use of the perturbation method
in which the motion is considered as a disturbance superimposed on a given
basic current which is assumed to be steady, laminar, isentropic and inviscid.
The hydrodynamic equations of motion are combined with the adiabatic equation,
equation of continuity and the equation of state. The perturbation quantities
are assumed to be small in comparison with the values of the basic Current.
This justifies linearising the equations by neglecting the squares and products
of the perturbation quantities. The resulting partial second order linear
differential equation can then be solved either analytically or numerically.
17
The limitations of this method are inherent in the assumption underlying
it. The most serious of these are the assumptions of steady, laminar flow and
of small displacement. The restriction to steady laminar flow ignores the fact
that flow over mountains is commonly both unsteady and non-laminar. The assump
tion of small perturbations restricts the validity of the results to mountains
whose height is small in comparison with their width.
In the following paragraphs we shall give a very brief account of some of
the important theoretical investigations of mountain waves.
6.1 Queney's works
Queney(1947, 48) applied hydrodynamic equations to the flow of a stably
stratified current crossing a mountain. He considered a uniform airstream with
constant static stability and made use of a smooth bell shaped profile for the
mountain. The profile is given by
where is the height of the mountain at z = 0 and b is the maximum height
of the mountain and a is the half-width.
He found that the disturbance pattern varied widely according to the width
of the mountain range. Specifically,
i) If 'a' the half-width of the mountain is of the order of 1 km. there is a
system of short stationary lee waves, or gravity waves,
ii) If 'a' is of the order of 100 km there is a complex system of gravity-
inertia waves. The wave length is of the order of a few hundred kilometres
and the wave amplitude increases upwards. Furthermore, the projection of
the ground streamline on a horizontal plane shows a marked horizontal oscil
lation.
No wave train is present when the width of the mountain is between the
above two critical values; instead there is only one wave crest and/or trough
on each streamline.
18
6.2 Work of Scorer (1949, 53, 54)
The absence, in Queney's r e s u l t s , of s ta t ionary wave t ra ins in the lee of
mountains for values of ' a ' between 1 and 100 km is at variance with observations
Scorer recognised that th i s unrea l i s t i c resu l t i s due to the assumption of a
uniform airstream. He therefore provided for var iat ion of lapse r a t e and wind
speed along the v e r t i c a l , but confined himself to f r i c t ion le s s , steady laminar
and adiabatic flow over mountain ridges small enough to allow geostrophic forces
to be neglected.
Both s t a b i l i t y and wind figure in a fundamental manner in Scorer 's solu
t i o n . The relevant parameter i s defined by the expression
where g = acceleration due to gravity
U = horizontal wind component normal to the mountain ridge
z = ver t i ca l co-ordinate posi t ive upwards
θ = potent ial temperature
T = absolute temperature
Y* = adiabatic lapse r a t e
Y = actual lapse r a t e
6.2.1 Conditions for occurrence of lee waves
Scorer found that standing lee waves that are usually observed in prac t ice ,
would be possible only if 12 i s less in some fa i r ly deep upper layer than in
a layer below. This requirement led Scorer to introduce a two layer model
with a value of 12 constant in each layer, but less in the upper than in the
lower layer . There i s a considerable l a t i t ude in the choice of wind and tem-
perature which would sat is fy th i s model, since 12 depends on both the s t a b i
l i t y and the wind. Fig.3 gives an example computed by Scorer using the charac
t e r i s t i c s of wind and temperature shown on the l e f t of the diagram. The lee
19
waves in this figure are in good accord with observations. In particular, the
waves have a maximum amplitude at some middle level and die away higher up
and the Wavelength is of the order of a few kilometers.
The two layer model requires, for wave formation, that the decrease in 12
from the lower layer to the upper layer should attain a certain minimum magni
tude depending on the depth of the lower layer. The more shallow the latter,
the greater the decrease in 12 must be. In fact this is given by the relation.
where and are the values of 12 in the lower and the upper layer respec
tively and h is the depth of the lower layer.
Subsequently some models have also been examined by several authors e.g.
Doos (1961,62), Palm and Foldvik (1960,61), Foldvik (1962) and Sarker (1965)
where the continuous variation of 12 with height has been considered. In these
studies also it is seen that in order for lee waves to occur 12 must assume
lower values through some fairly deep upper layer than in some layer below; but
i t is not possible to specify quant i ta t ively the decrease in 12 which would be
necessary. The most that can be stated is that greater the decrease in 12 with
height i s , the greater i s the likely-hood of wave formation.
6.2.2 Verification of Scorer 's r esu l t s
The second term in the expression for 12 in equation (6.2) concerns the r a t e
of change of wind shear with height. This term i s zero if the wind speed i s con
stant or changes uniformly with height. I t assumes importance only when the
wind shear changes rapidly with height and th i s i s rare ly the case except over
shallow layers or in the upper troposphere near the core of j e t streams. Accor-
dingly i t i s convenient to use the f i r s t term only in computing 12 . Thus for
prac t ica l purpose we may wri te
An idea of magnitude of var iat ion of 12 with height observed in wave conditions
20
may be obtained from the study, mentioned by Corby (1957), of 37 reports of marked
waves made by p i lo t s of Br i t i sh European Airways. On the average, the minimum of
12 a lof t was found to be one ninth of the maximum below. This r a t i o i s conf i r
med by observations made at St.Auban-Sur-Durance on 25 January 1956 when vigorous
waves occurred on the lee of the Lure Mountain r idge. Conputation of
in th i s case shows that the average value between 1 and 5 km i s about 9 times
tha t between 5 and 10 kms. While the value of one ninth for the decrease in
12 with height should not be used as a quant i ta t ive l imit in forecasting waves,
i t does give an idea of the magnitude of the decrease which is observed during
v/ave s i t u a t i o n s .
6.3 Works of Palm and Foldvik (1960), Foldvik (1962),Doos (1961,62), Sarker (1965)
After Scorer ' s work (1949) with two layer model there were several studies
dividing the atmosphere in two or th ree layers or by numerical computations;
e.g. Palm (1958), Sawyer (1960). The study was then extended by continuous repre -
sentation of the parameter 12 . I t was seen by Palm and Foldvik (1960), Doos
(1961,62) that 12 generally decreases exponentially with increasing height and so
they represented 12 by the form
where fo and λ a re constants . Sarker (1965) found that such a representa-
t ion of 12 i s very sa t i s fac tory in the Western Ghats region during the winter
months, December - March, when the airstream has s tab le s t r a t i f i c a t i o n .
With such a representat ion the ve r t i ca l velocity associated with a wave
of length for a symmetrical mountain p rof i l e given by (6.1)
i s given by
21
where k is given by , m being the roots of
and J i s the Bessel function of the f i r s t kind. The lee wave amplitude i s
given by
In the above, suffixes 0 and z denote values at the surface and at height z
respect ive ly .
The corresponding expression for leewave streamline displacement for Scorer ' s
two layer model i s
Here, H i s the height of the lower layer and the origin of z co-ordinate i s
chosen the re . are the stream functions at the lower and the upper
layers given by
and are the constant values of 2 in the lower and upper layer
respect ive ly .
Without going to de ta i l s of these s tud ies , we shal l l a t e r on brief ly s t a t e
the r e su l t s of our study on Western Ghats.
22
6.4 Wave Length
The horizontal scale of the disturbance imposed on an airstream immediately
above a mountain ridge is determined almost entirely by the scale of the ridge
and the question of wavelength in this vicinity does not arise. In contrast, the
wavelength of any lee wave above level ground downstream from the ridge is
directly dependent on the characteristics of the airstream and is therefore
amenable to calculation, in theory at least. However, the amount of calcula
tion would be quite prohibitive for routine application. However, a fair esti
mate of the lee wavelength, which is quite adequate for many purposes, can be
made from Scorer's 12 parameter defined in simple form by
Theory indicates that wavelength will be somewhere between the maximum and
minimum values of through the troposphere. This
means that light winds and strong stability are associated with short wave
lengths while strong winds and small stability are associated with long wave
lengths. Another important conclusion follows from
that the value of is more dependent on the flow velocity than on the
stability. This means that the wavelength of mountain waves is determined to
a larger extent by variations in the flow velocity than by variation of lapse
rate. Again, although the static stability in shallow layers varies widely
from one air mass to another, the mean stability through the whole troposphere
does not. Thus we should expect the lee wavelength to be approximately pro
portional to the mean tropospheric wind speed. If we assume a mean stability
corresponding to about half the adiabatic lapse rate we obtain for the lee
wavelength λ = 1/2 U, where λ is in km and U in m/sec.
In support of this rough theoretical estimate, Corby (1957) found in
the study of waves from routine radiosonde soundings a correlation coefficient
of 0.91 between the observed wavelength and the mean tropospheric wind speed.
His regression relation is
λ (km) = 0.585 U (m/sec) - 2.8
23
This relation is not very far from the approximate theoretical estimate
λ = 1/2 U. Larson (1954) also found in his study of wave clouds that simple
estimates of wavelength obtained by this approach showed good agreement with
observations. As the wavelength is rarely of vital importance in aviation fore
casting such estimates should suffice when they are required.
From Sarker's (1965) theoretical investigation of mountain waves on the
Western Ghats, there appears to be a suggestion that wavelength increases with
mean tropospheric wind speed. However, as the cases studied were very few, no
relationship of the type given above was possible.
The above wavelength considerations apply to lee wave of the most common
type i.e. those which have their greatest amplitudes in the lower or the middle
troposphere and decrease above. Occasionally lee waves occur with their greatest
amplitude in the upper troposphere or lower stratosphere. Once again the general
wavelength equation cannot be solved rapidly, but by treating the stratosphere
as of infinite stability one obtains an approximate value of the wavelength from
the relations:
where h is the height of the tropopause and is the mean value of 12 for
the troposphere. The above equation appears to be consistent with observations
in that with typical values they suggest much larger wavelengths for this type
of wave, but the relation has not been verified in detail. Although the point
may only rarely be of importance in aviation forecasting, it is appropriate to
mention here that the crest of the first lee wave downstream of a mountain
ridge is commonly observed to be less than one wavelength from the mountain
crest. Indeed for a symmetrical ridge theory indicates that the spacing should
be 3/4 λ .
24
6.5 The lee-wave amplitude
The amplitude of any lee wave is naturally of the greatest importance for
aviation but is unfortunately almost intractable from the forecasting point of
view. This is because, as is evident from equations (6.8) and (6.9) of section
6.3, the amplitude depends in a complex way on both topography as well as on
the properties of the airstream. These aspects were the subject of a theore
tical study of Corby and Wallington (1956). However, although, it is not yet
possible to predict the lee wave amplitude quantitatively, a knowledge of the
relevant factors governing amplitude is a valuable background information for
a forecaster.
For symmetrical mountain ridges the lee wave amplitude depends on the
height of the ridge above the surrounding countryside and also on the horizontal
scale of the ridge. The dependence on height, viz. proportionality is to be
expected. That is, other things being equal, the higher the mountain, the
greater the amplitude.
The dependence on horizontal scale may be regarded as a resonance effect.
The power spectrum of the mountain cross section necessarily has one or more
concentrations in the vicinity of particular wavelengths (of the usual idea-
lised symmetrical cross-section often adopted in theoretical work there is
of course only one such concentration). If the topography posseses one of th
these maxima near the natural wavelength of the airstream, then the lee wave
amplitude will be much greater than otherwise. Put in another way, we may
Say that if the horizontal scale of the mountain roughly coincides with the
lee wave length, the amplitude will be much larger than for both broader and
narrower mountains. The critical manner in which the amplitude varies with
mountain width is shown by the dashed curve in Fig.4 for an airstream with a
wavelength of 2 km. The curve shows a pronounced maximum when the half
width of the mountain is 1 km and falls off sharply for broader
and narrower ridges. This effect is akin to resonance. The natural wave-
lenath depends on the characteristics of the airstream, but the lee wave
25
amplitude is likely to be small unless the width of the mountain matches the wave
length of the airstream.
If the height and width are increased in the same proportion, the variation
of the amplitude is that given by the full curve in Fig.4. Thus an airstream
which is favourable for the formation of vigorous waves to the lee of a small
mountain ridge will not necessarily produce bigger waves over larger mountains.
Also the natural wavelength of an airstream increases with the wind speed.
Therefore, larger mountains would require stronger winds for larger amplitude
wave than small mountains. This conclusion is in line with that drawn by
Forchtgott (1949) from his observations in Czechoslovakia.
However, one should be cautious in making use of the above results. For,
most mountainous terrains contain irregularities on a wide range of scales and in
particular, broad mountains often have superposed short wavelength features.
Furthermore, broad mountains may have steep lee slopes and it is of course the
character of the lee slope which is of most importance.
The dependence of lee wave amplitude on the airstream characteristics is
quite complicated. However, this effect can be stated in plain language as
follows. The amplitude of lee waves is subject to wide variations even amongst
airstreams having similar profiles of wind and stability because of the critical
way in which the amplitude depends on the airstream characteristics. Other
things being equal, the largest amplitude lee waves occur when the airstream
satisfies the condition for waves by only a small margin, and in this region
large changes in amplitude may result from small changes in the airstream.
Apart from the question of this sensitive region, it may be said that larger
amplitude waves are theoretically more likely in airstreams containing a
shallow layer of great stability than in conditions of lesser stability through
a deep layer.
A theoretical result which can be exploited in forecasting is concerned
with the amplitude variation of lee waves with height. Generally speaking, the
maximum amplitude is attained in or near the layer of maximum stability. If
26
the stability is concentrated in a shallow layer, as for example, at an inver
sion, the amplitude has a sharper maximum near this level and falls off
rapidly both above and below. If the stability maximum is diffuse, the varia
tion of lee wave amplitude with height follows the same pattern but is more
gentle. This result is well supported by observations and enables forecas
ters to provide useful advice as to the choice of flight level.
6.5.1 Effect of a succession of ridges
A succession of ridges in the direction of the wind may either intensify
or weaken the waves generated, depending on the position of the ridges with
respect to the wavelengths. The influence of the phase of the windward
undulations on the effect which a mountain ridge exerts on an airstream has
been discussed by several investigators. Georgii (1956) has noted that the
best gliding localities for wave soaring have in common a main ridge and a
"counter—ridge" downstream from it at a distance which corresponds nearly to
an exact multiple of wave length. In this manner the wave generated by the
first ridge is reinforced by the counter ridge. This effect may be repeated
several times if there is a succession of ridges so spaced that they are in
harmony with the natural wavelength of the airstream.
6.5.2 Effect of the profile of air mass stability
6.5.2.1 Inversions
It was once thought that the existence of an inversion was essential for
wave formation. This impression might have come from the fact that the
inversions help make the waves visible, since they are usually accompanied by
a concentration of moisture in the layer just beneath them where clouds are
likely to form in the wave crests. However, it is now established from
theory as well as from observations that existence of an inversion is not
essential for the formation of the waves, although, it does play an impor
tant role in determining the structure and intensity of waves. Corby and
Wallington (1956) have shown that greater amplitudes are possible, if there is
27
considerable static stability through a shallow lower layer than with smaller
stability through a deep layer. The presence of an inversion in the lower layer
would, therefore, be a factor contributing to greater amplitude waves.
The existence of an inversion allows useful conclusions to be drawn with
regard to variation of wave amplitudes with height. In practice the level of
maximum amplitudes is near the level of maximum 12. The presence of an inver
sion, by concentrating the stability in a shallow layer, brackets the level of
maximum amplitudes within narrow limits and allows its determination with
accuracy. This fact is of considerable importance for aviation. Theory also
indicates that the sharper the inversion, the more pronounced is the maximum wave
amplitude. This means that the intensity of waves will decrease rapidly above
an inversion. This inference is borne out by observation.
6.5.2.2 Adiabatic lower layers
Waves are often observed when the lapse rate near the ground is steep or
even adiabatic. Stability conditions in the lowest layer do affect the structure
of the wave and likelihood of their occurrence. According to Corby and Wallington
(1956) the main effect of an adiabatically mixed layer near the ground is to
reduce the amplitude and increase the wavelength. It, of course, happens parti
cularly on sunny afternoon, that the adiabatically mixed layer is so deep that
waves cannot exist. In such cases, the stabilisation of the lower layer in the
evening may make conditions favourable for wave formation. This, perhaps,
explains why cirrus clouds are reported more in the mornings and in the evenings
than during daytime and accountS for the fact that after a sunny day glider pilots
soaring over hills often find increasing lift which carries them to heights they
have been unable to attain all the day.
6.6 Separation of the flow-effect of Lee Standing Eddy
However disturbed the flow over mountains may be close to the ground, it is
often remarkably smooth higher up. Timelapse pictures made in connection with
the Sierra Wave Project have shown that the smaller corrugations in the rugged
28
terrain are filled by eddies, while the smooth flow aloft follows the broad
outline of the mountain range. Quite frequently separation occurs, that is,
the lowest smooth streamline leaves the surface completely and reversed flow
sets in.
The importance of separation lies in the fact that it modifies the effec
tive shape of the mountain. If a mountain is to have its maximum effects, the
low level flow must follow the contours. The presence of a large eddy filling
the space in the lee smooths out the streamline immediately above the mountain
and thus reduces its potential effect on the air flow. When waves are set up,
their phase is such that they generally induce strong flow at the surface down
the lee slope. This fact is confirmed by the curling of the cap cloud down the
lee slope. Conversely, it is sometimes evidenced by the sudden appearance of
waves on the onset of katabatic winds that the descent of air down the lee
slope is favourable for wave formation. It is, therefore, important to know
the factors which have a bearing on separation, particularly that which takes
place near the mountain crest and results in the formation of lee standing
eddy. Scorer (1955) has shown that the following factors play important role
in separation.
6.6.1 Mountain shape
Separation occurs very readily at a sharp edge in the profile of the
ridge. It also occurs more readily when the lee-slope has a precipitous drop
than when it has a smooth and gentle gradient.
6.6.2 Stability Conditions
When the surface is heated anabatic flow is encouraged. Separation is
therefore likely when the lee slope is facing the Sun. On the other hand,
separation is inhibited by differential radiational cooling towards the dusk,
which constrains the wind, to flow down the slope, specially the lee slope.
29
6.6.3 Sudden Disturbances
A downdraft impinging on the lee slope of a mountain range will cause the
air to flow down the slope and may thus prevent separation. Intermittent bursts
of rain with their accompanying downdrafts may thus force the air to flow down
the lee slope and may be instrumental in triggering the occurrence of the waves,
provided the other conditions are favourable.
6.7 Short Ridges and Single Peaks
The results we have described so far apply to two dimensional flow over
long ridges. Flow over short ridges and single peaks presents a problem in
three dimension, for much of the air stream flows round the sides of these obsta
cles and only the layer in the upper levels goes over the top. Mathematically
the problem is much more difficult than that for the airflow over long ridges.
We must, therefore, rely mainly on observations in order to get the information
concerning the disturbance in the air stream caused by short ridges and single
peaks.
As in the case of stably stratified flow over long ridges, there must be
different types of streaming over short ridges and single peaks. However, no
systematic classification of this type has yet been made. Forchtgott (1951) has
described a type of flow the characteristics of which agree well with some of the
observed occurrences and configuration of clouds in the vicinity of isolated
mountain peaks. Figure 5 illustrates this type of flow. Since most of the air-
stream in the lower layers flows round the sides, there are great horizontal
deflections in the streamlines near the foot of the obstacle, while in the higher
levels the deflections are mainly in the vertical direction. The lower level
streamlines diverge on the windward sides of the hill and return to their original
position in the lee of the obstacle. The lower diagram shows three zones where
whole of the streamlines converge, namely to the and to the left of the
front side and directly to the lee of the obstacle. These are regions where the
vertical component of the wind is upwards. The most marked low level convergence
30
of the streamlines occurs directly behind the hill causing the air to flow up the
Lee slope. This is sometimes attested by cloud patches ascending the lee slope of
isolated hills while the windward slope remains clear. The clouds which are
sometimes seen streaming away downwind from mountain peaks can be explained on
the basis of convergence caused by the air encircling the peak. This phenomenon
is known as "Smoking Mountain".
Although the picture presented by Forchtgott appears to fit in well with
observations such as those mentioned above, there are undoubtedly frequent
occasions when the windflow in the vicinity of mountain peaks is materially dif
ferent. For instance, orographic clouds sometimes have the form of a symmetrical
collar surrounding the mountain or of a cap covering the peak.
The airflow around the sides of a short ridge or an isolated peak reduces
the effect of the mountain on the vertical deformation of the air stream and
thus makes the development of the waves less likely. The occurrence of the
waves is, however, possible. But such waves would in general have a relatively
small amplitude and would die away downstream much more rapidly than the waves
generated by an extended ridge. Scorer and Wilkinson (1956) have shown that
the lee wave pattern produced by an isolated hill is very similar to that pro
duced by a ship, the waves being confined within a wedge-shaped region.
Another effect is that the amplitude falls off much more rapidly with height.
As a consequence forecasters and pilots can afford to pay less attention to
isolated mountains in comparison with long ridges of similar height.
7. Application to Aviation Forecasting
The important effects of mountain waves on the performance and safety
of aircraft make it necessary for the pilot to know beforehand whether he is
likely to encounter waves of any consequence during his flight. Information
to this effect by the briefing forecaster would therefore be most valuable.
31
7.1 Observing and Reporting Orographic clouds
In attempting to supply this information the forecaster would be greatly
helped by careful reports of clouds which are typical of mountain waves and it
would be advantageous for meteorological service to encourage and train their
observers to detect and report such clouds. Provision for reporting orographic
clouds in the International Cloud Code will also be useful.
7.2 Reports by Pilots
In-flight and post-flight reports by aircraft pilots would also constitute
a helpful source of information to the briefing forecaster. The pilots should
therefore be encouraged to spot and report waves and orographic clouds during
their flight. Even mild waves should be reported, since these waves may inten
sify by some development in the synopric situation or as a result of the usual
diurnal variation in the lower layer of the airmass.
7.3 Application of Theoretical and Observational Results
Apart from the reports from observers and pilots, the forecaster must
ultimately rely to a great extent on his own analysis of the situation and on
the application of the observational and theoretical results,
7.3.1 The likelihood of significant waves
The first thing to determine is whether standing waves are possible in the
airstream under consideration. This point can be decided by considering the
profile of 12 which should decrease with height if waves are to be possible.
Theoretically, this condition is achieved either by a decrease of stability or
an increase of wind with height. Analysis of numerous cases of waves have
revealed that there is both a substantial increase of wind and a decrease of
stability with height whenever waves are observed. In practice, most situations
favourable for wave development can be identified by an inspection of a repre
sentative radiosonde ascent in the undisturbed current, without the necessity
for computing the profile of 12. Examples of such ascents made at Stormway
32
and Liverpool on 11th March 1953 are given in Fig. 6(a) and 6(b), in which the
marked stability between 900 and 800 mb and the substantial increase of wind
with height are seen clearly. The computed profiles of 12 are shown for com-
parison in figures 7(a) and 7(b) which show pronounced maxima of 1 around
800-900 mb with much smaller values higher up.
Confirmation that the airstream sampled by the above two soundings was
favourable for waves was provided by an aircraft flying in the same airmass at
a mean height of about 1700 m on the morning of the same day, which passed
through about six smooth waves giving up and down currents of the order of
3.5 m/sec.
In marked contrast, another sounding is given in Figure 8(a), where the
wind speed decreases with height between 800 and 650 mb and which features no
2deep stable layer in the lower troposphere. The corresponding 1 profile given
in figure 8(b) does not show its decrease with height. No wave was observed
in the airstream sampled by this sounding.
Between the two extremes illustrated above, there will be cases when it
will not be possible to decide from an inspection of the upper air data whether
the 12 profile will decrease with height or not. In such cases, it will be
necessary to calculate 12 for various levels and thus find out how this parameter
varies with height. An easy method for performing the computation is given in
2t h e appendix A. A g r a p h i c a l method for determining wave lengths from 1
p r o f i l e i s given in appendix B.
If by applying the 12 criterion the airmass is found to be capable of
containing the standing waves, the next step would be to determine whether
waves are likely to occur on the lee of the mountains along the air route. The
Synoptic situation must be studied to find out whether the wind is expected to
blow from a direction approximately normal to the mountain ridges along the
route with speed exceeding the minimum speed appropriate to the ridges. In
addition, factors such as the probability of separation of flow should also
be taken into consideration.
33
It should be noted that, from the view point of aviation, waves are impor
tant only in as much as they affect the performance and safety of aircraft. It
is, therefore, necessary to try to determine whether any waves which are expected
to occur would be vigorous enough to be of importance to aircraft. In doing so,
the following factors should be taken into consideration.
7.3.1.1 Scale of terrain
Haves of large amplitudes require that the natural wavelength of the air-
stream should be in harmony with the scale of the terrain. If the forecaster,
therefore, wishes to assess the magnitude of the wave effects associated with
the particular ridge, he should consider its size in relation to the wavelength
appropriate to the airstream. Unfortunately it will not be practicable to do so
quantitatively on daily basis. However, both theory and observations indicate
that the larger the mountain the stronger are the waves necessary to produce
maximum effect. To determine the wind speed associated with the optimum wave
conditions over a particular locality requires careful observations over a fairly
long period. The critical minimum value which the wind speed at mountain top
level must attain before waves of any importance are observed, has already been
determined for a number of localities and appear to have a fairly narrow range
between 8 to 13 m/sec.
It is, of course, more important for a forecaster to recognise the situation
giving rise to powerful waves than those in which ordinary waves will occur.
The largest waves will be generated by the largest mountains when there are
strong winds. It is useful to remember that for a given wave, the vertical cur
rents will be larger the higher the speed blowing through them.
7.3.1.2 The presence of jet stream
The presence of a jet stream with its high wind speed and strong vertical
wind shear is an important factor in the occurrence of powerful waves particu
larly in the lee of large mountains. The presence of a jet stream is, of course,
not essential for the formation of waves by small individual ridges. But,
34
sometimes a series of such ridges is so arranged that the overall scale of the
terrain as a whole is equivalent to a very large ridge. In such cases, the
presence of a jet stream may, under favourable conditions, be conducive to the
formation of a powerful wave system of long wavelength in the upper layer above
the shorter waves generated by the individual ridges. Observational data have
shown that the presence of a jet stream over such large mountain systems as the
Rocky Mountains is a rather favourable condition for the development of power
ful mountain waves.
7.3.1.3 Irregular Topography
Mountainous terrain is generally composed of a series of individual ridges
or hills. Disturbances generated by each of these individual features will be
superposed on one another and may give rise to a complicated pattern in which
there is no regular sequence of lift and sink. Sometimes the disturbance
immediately above a ridge may be in phase with the lee waves from other hills
upstream and may result in a large single wave in the vicinity. In general,
the result of the superposition of the successive waves is not forseeable, but
long experience in a given locality may enable the forecaster to associate
peculiar wave features with certain conditions of wind speed and airmass cha
racteristics.
7.3.1.4 Changing synoptic conditions
In attempting to forecast wave effects, the forecaster should of course
Consider not only the state of the atmosphere as indicated by the available
upper air ascents, but he must also attempt to predict those changes in the
air stream which may have an effect on the likelihood and intensity of waves.
For example, given an airmass which satisfies the stability requirements
but which does not contain waves because the wind blows more or less paral
lel to the mountain ridge, a predictable change of wind direction in the
right sense should be taken into consideration when formulating a forecast
concerning the wave effects. The forecaster should in many cases be able
35
to predict variations in wave condition from the respective variations in stabi
lity and wind conditions of the airstream.
7.3.1.5 Diurnal and seasonal variations
The diurnal changes which occur specially in the lower layers and their
effect on the likelihood and intensity of waves must also be taken into consi
deration when forecasting wave conditions. The radiational cooling which sets
in at dusk on clear days may be instrumental in the occurrence of waves during
the evening when this may not have been possible during the day due to the mixing
of the lower layers of the atmosphere. This happens either because nocturnal
cooling produces the requisite lower stable layer or, in the case of the larger
hills, by inducing katabatic winds, thereby impeding any separation which may
have prevailed during the hours of strong insolation.
The diurnal effect is an important one. It is probable that waves develop
ing during the evening as a result of the stabilising of the lowest layers gene
rally have the greatest amplitude near the ground and weaken rapidly upwards.
They are, therefore, of importance only to low flying aircraft.
Forecasters should also be aware of this fact that there is a seasonal
variation in the frequency of wave effects. The greater tendency for low
level stability in winter airmasses and the greater frequency of situations with
a marked increase of wind with height would indicate more frequent wave effects
in the winter half of the year. Observations have shown that this is in fact
the case over the British Isles and in the United States. A winter maximum
does not however seem to be universal. Observations by Larson (1954) have
shown that the maximum frequency of waves generated by Ovik mountains in Sweden
occurred in March and April with a secondary maximum during September and
October. This may be due to the orientation of the mountain ridge in question
with respect to the direction of the prevailing wind in different seasons.
The theoretical study of Sarker (1965, 66, 67) has shown that the frequency of
wave in the lee of Western Ghats is more during December to February and the
36
wavelengths and ampli tudes a r e more during t h i s per iod than t h o s e during t h e
southwest monsoon p e r i o d . The s tudy of De (1970) from s a t e l l i t e p i c t u r e s r e v e a l s
t h a t t h e mountain wave in t h e Assam-Burma H i l l s i s a l so more frequent during the
win te r months — December to February .
Knowledge of t h e seasonal v a r i a t i o n s in wave e f f ec t s for t h e d i f f e r e n t
l o c a l i t i e s would be useful in guiding t h e forecaster with regard to t h e impor
t a n c e which he should accord with t h e problem of waves in his da i l y r o u t i n e
f o r e c a s t s .
7 . 3 . 1 . 6 Other Effects
Short per iod changes in wave condi t ions a r e brought about when convec t ive
a c t i v i t y in t h e lower l aye r s prevents s t r e a m l i n e flow and impedes wave forma
t i o n . Again assuming t h a t o ther cond i t ions for waves a re s a t i s f i e d , a shower
or a bu r s t of i n s t a b i l i t y r a i n wi th i t s accompanying s t rong downcurrents may
force t h e a i r s t r e a m to flow down t h e l e e s lope and induce waves which, though
s h o r t l i v e d , may be q u i t e i n t e n s e . I t would, however, be d i f f i c u l t t o t ime
such an occur rence , but t h e p i l o t should be warned of t h i s p o s s i b i l i t y .
7 . 3 . 2 Descending c u r r e n t on t h e l e e of a mountain b a r r i e r
Every p i l o t should be a b l e to r ecogn i se t h e phenomenon of a descending
c u r r e n t on t h e l e e of a mountain b a r r i e r ( F i g . 9 ) , in which t h e a i r c r a f t gains
and loses h e i g h t . This i s caused by a descending c u r r e n t whose fo rce i s
l a rge r than t h e l i f t of t h e a i r p l a n e , which i s determined by i t s v e l o c i t y .
In such c i rcumstances t h e a i r p l a n e often c r a s h e s .
Let us cons ider an a i r p l a n e ga in ing he ight on a s t r a i g h t course and then
en te r ing a descending c u r r e n t on t h e l e e of a mountain b a r r i e r . Let us
denote t h e v e c t o r of t h e a i r p l a n e ' s v e l o c i t y imparted to i t by t h e engines
by and t h e v e l o c i t y vec to r of t h e descending c u r r e n t by We
d i v i d e t h e v e c t o r i n t o two mutual ly perpendicu la r components, one of
which i s d i r e c t e d along t h e axis of t h e fuse lage (oppos i t e t o t h e d i r e c t i o n
of ) . We denote t h i s by and c a l l i t t h e t a n g e n t i a l component.
37
The second component (perpendicular to ) we denote by and call it
the normal component of the descending current's velocity vector
Obviously, the existence of the component reduces the airplane's velocity
and it will be moving with a velocity
For different angles α the magnitudes of the vector and
will be different. Thus when the angle α decreases, approaching zero, the
tangential velocity component of the descending current will increase
tending to and tends to zero. On the other hand, if the angle α
is increased, the normal component will increase and the tangential component
will decrease. At the limit, when will tend to zero. Both
limiting cases are dangerous. For very small values of α the airplane may be
carried by the current into the mountain, and for very large values of α
a strong descending current may throw it off. Particularly dangerous is a strong
descending current for an airplane whose course is parallel to a mountain ridge.
It should be borne in mind that when the dominating wind direction is
normal to the ridge, the descending current on the lee side is particularly
strong. The velocity of a descending current may reach 5 m/sec and even more.
A pilot caught unawares by a foreceful descending current is in a particu
larly dangerous situation, specially if the airplane is at that moment in proxi
mity of the mountain. Any delay in righting the machine is almost certain to
lead to a crash. The situation is worse if there is danger of icing.
At the peaks themselves the velocity of a descending current may attain
such high values that an airplane in such a current may disintegrate.
The success of the flight in a strong descending current on the lee of a
ridge is wholly dependent on the pilot's correct selection in each specific
case of the best angle and flight course.
It follows from (7.1) that as long as the velocity of the descending
current is lower than the velocity imparted to the airplane by its engines the
38
airplane wil l move in the same di rec t ion, but with a lower veloci ty than
However, the a i rc ra f t veloci ty in a very strong descending current may be
insuff ic ient to overcome the velocity of the descending current i . e . i t may be
that This i s one of the three dangerous cases in which the
a i rc ra f t i s bound to meet with accident and even catastrophe.
7 .3 .3 The level of maximum amplitude
After having determined whether waves of important amplitude are l ike ly
to occur, an attempt should be made to provide the p i lo t with information con
cerning the level at which the waves wil l have the maximum amplitude. Such
information would enable the p i lo t to decide whether to climb or descend,
should be encounter troublesome waves. When the level of the maximum ampli
tude i s near the level of mountain top, a knowledge of t h i s would be useful
in deciding on a safe f l ight l eve l .
Theory indicates tha t the level of maximum amplitude must be near the
level of maximum 12 . Since, when waves occur, t h i s parameter has a marked
maximum through some low or middle layer, the forecaster can safely predict
that the level of maximum amplitude will be somewhere in th i s layer, which
in general coincides with the layer of greates t s t a b i l i t y . The more pronoun
ced the s t a b i l i t y , the closer to t h i s layer does the maximum amplitude of
waves tend to occur. In the case of a pronounced inversion, the forecaster
may confidently advise the p i lo t tha t the waves will be strongest at the
inversion level and tha t they wil l decrease rapidly above that l eve l .
Sometimes more than one wave system may prevail simultaneously at d i f f e -
rent l eve l s . This happens when the ve r t i ca l p rof i le of 12 shows more than
one maximum. The wave system associated with the different maxima wil l be
different because of the differences in wind speed at the i r respect ive
l e v e l s . In general , the upper wave system wil l have the longer wave-length.
39
7.3 Forecasting Turbulence
Turbulence within a system of standing lee waves is most frequent and most
severe in the standing eddies under the wave crests at mountain top level. This
turbulence is specially violent in waves generated by large mountains. Rotor
turbulence connected with lesser mountain ranges is much less severe. But, in
general, it is almost always present to a greater or lesser extent. The degree
of turbulence is, of course, greater the better developed the waves are, and it
should be possible for the experienced forecaster to give the pilot at least a
tough indication of the expected degree of turbulence in the rotor cloud area in
any given situation. Forecasting the occurrence and degree of turbulence above
the rotor cloud level is much more difficult. The problem is related to that of
forecasting clear air turbulence. During the last few years a number of papers
dealing with this problem have been published, but no definite clue has yet been
found concerning the factors essential to its occurrence. Among the meteorolo
gical variables which have been variously considered as being of importance are
low values of Richardson's number, vertical and horizontal wind shears as well
as combination of variables such as the product of horizontal wind speed and the
vertical gradient of the wind shear. Georgii (1956) suggested that if the tempe
rature distribution in an airstream containing waves becomes unstable, the waves
lose their characteristics and turbulence sets in. This criterian appears to pro
vide a satisfactory explanation for those cases when turbulence is known to occur
simultaneously with a breakdown of the wave system and should prove of some help
to the forecaster. But it is doubtful whether the criterion is valid for all
occurrences of turbulence in standing waves above the rotor cloud level.
Mountain waves are often associated with jet streams. Therefore the statis
tical results on the turbulence associated with these currents may be of some
value to the forecaster. Bannon (1951-52) has produced statistical evidence that
turbulence is concentrated on the low pressure side of the jet stream - some dis
tance away from the axis. In the sample studied, very few cases of turbulence
were reported on the anticyclonic side and none near the jet stream axis.
40
Jones (1954) has obtained essentially the same results. Out of a sample of
147 cases of heavy turbulence, 75% were on the low pressure side of the jet
stream and 10% on the lower half of the anticyclonic side. The average distance
of the turbulent areas from the jet axis towards the cyclonic side was of the
order of 100-150 km. Predicting turbulence below the rotor cloud level can be
approached by adopting the same principles as those generally used in predic
ting turbulence in the friction layer, since the processes which give rise to
turbulence in this layer operate over high ground too. Indeed there is reason
to expect that the larger scale of surface irregularities over mountainous
terrain should intensify the turbulence associated with such meteorological
variables as strong winds and instability near the surface. Corby (1957)
puts forward the reasonable suggestion that unless there are definite reasons
to the contrary, it would be best to predict that turbulence over high ground
would be one degree higher on the descriptive scale used than that expected
over flat country. Thus if in a strong unstable airstream moderate turbulence
is expected generally in the friction layer and in cumulus clouds, the fore
caster may appropriately add "but locally severe over high ground", if the
air route passes over rugged mountainous terrain. Similarly if severe turbu-
lence is predicted generally, the forecaster may add "specially over high
ground" when appropriate.
Situations in which turbulence may be more specifically associated with
high ground are those in which the airmass is potentially unstable. Forced lif
ting of the air by a mountain ridge may release the instability and give rise to
thundery activity with its associated turbulence.
7.3.5 Aircraft Icing
Stationary lee waves can aggrevate aircraft icing both because of the
vertical displacement they cause in the level of the 0ºC isotherm and because
of higher liquid water content in clouds formed in air which is forced to
ascend a mountain ridge. Advice on the added risk and severity of aircraft
41
icing resulting from the above two factors would be very useful in aviation
forecasting.
Once the pattern of vertical motion is known, it is a straight-forward matter
to assess its effect on the level of the 0º isotherm from a representative soun
ding in an undisturbed current. Since however the vertical motion in the mountain
waves varies with height in a manner which cannot be determined in a routine way,
some realistic assumption with regard to this motion must be made. A reasonable
basis for the present purpose is to assume that the air at all levels follows the
shape of the ground. Thus for an airstream crossing a mountain 1000 metres high
and containing stationary waves, it is assumed that the air is lifted at all
levels 1000 metres above its undisturbed level. The effect on the level of 0ºC
isotherm may then be determined from a tephigram by assuming dry adiabatic cooling
until saturation and wet adiabatic cooling above the condensation level. Some
times it Seems likely that when a suitable airstream flows perpendicularly across a
long ridge, the amplitude of the wave may exceed that of the ground by a factor of
perhaps However if the height of the highest ground including that of the
individual peaks is used in applying the procedure some additional tolerance will
automatically be included. A suitable phrase for use in a forecast for a flight
over Assam and adjacent states in the winter might be "level of 0ºC isotherm
12000 ft (3.7 km) lowering to 10500 ft (3.2 km) over the hills". In general, if
the levels at which icing is expected coincide with a layer in which waves are
likely to exist, the forecaster should predict worse icing conditions than he
would otherwise.
Apart from the possible lowering of the level of the 0ºC isotherm, the effect
of mountain on the intensity of icing due to increased liquid water content need
also be considered. Icing is a complicated matter involving both meteorological
and aerodynamic quantities. The most important of the meteorological variables
is the concentration of supercooled liquid water in the potential icing cloud.
This is a difficult quantity to be measured. But such measurements as have
been made from aircraft have rarely values greater than 1 gm/m3, whereas at
42
mountain observatories values as high as 4 gm/m3 have been measured. However,
the impossibility of predicting the liquid water content of clouds and other
variables necessarily limits icing forecast to qualitative and subjective
terms. In view of the greater liquid water content which is likely in clouds
over mountains, if a forecaster expects some degree of icing generally, he
will be well advised to indicate a greater liability or intensity over high
ground, whatever kind of airflow is expected over the hills. If it so
happens that the levels at which he expects icing to be most likely coinciding
with a layer in which waves are likely to occur with large amplitude, he can
predict worst icing conditions with more confidence.
8. Some suggested safeguards for flying in Mountain Waves
The following flight rules are essentially those proposed by the United
States Weather Bureau (1955) when flying into an area of suspected wave con
ditions:
a) If practicable, avoid flight into the wave area. Otherwise observe the
following precautions.
b) Maintain a frequent watch on the altimeter, especially at night or when
flying in cloud, and remember that the altitude indicated by a pressure
altimeter may be upto several hundred metres higher than the actual alti
tude of the aircraft.
c) Approach the mountain range at a 45 degree angle rather than directly,
particularly when flying upwind, so that a quick turn can be made away
from the ridge if it suddenly appears dangerous to continue.
d) In case of sustained loss of height when flying parallel to a ridge,
rising air will most probably be found by changing course so as to fly
a few miles towards or away from the high ground upwind. If, however,
the aircraft is so near the lee slope that the downcurrent is obviously
caused by air flowing down this slope, look for rising air further down
stream.
43
e) It is possible when flying into the wind to utilise updraft areas to gain
height. In particular look for rising currents upwind of the rotor cloud and
also of the lenticular clouds if they happen to be near flight level. Caution
should, however, be exercised in employing this procedure, since it is not
always possible to pinpoint the updraft areas.
f) Avoid flying into the rotor clouds. Also avoid the lenticular clouds when
their edges are torn and irregular.
g) Avoid flying through a powerful wave on instruments
h) Because of strong downdrafts and turbulence, and the hazards of instrument
flight near mountain tops, avoid flying into a cap cloud, even if it means
turning back.
9. Mountain waves over Western Ghats
Sarker (1965, 67) studied the occurrence of mountain waves on the lee of the
Western Ghats theoretically. The average W-E vertical cross section of the ghats
near the Bombay-Poona region (Fig. 10) was represented by
where the elevation of the ground surface at the level z = -h with the
numerical values h = 0.25 km, a = 1.8 km, b = 0.52 km and km.
Expressions for vertical velocity and streamline displacement due to waves were
established for a two—dimensional motion. Computation for six cases has been made
for the winter season when the wind is more or less westerly and the atmosphere is
dry (Sarker 1965). The wind and temperature profiles for two of these cases are
shown in Figs. 11(a) - 12(a) and the corresponding f(z) profiles are shown in
Figs. 11(b) - 12(b). It will be seen from the f(z) profiles at once that lee waves
are possible in all these cases. The wavelengths and maximum vertical velocity
associated with the waves are shown in Table 4. The variation of maximum ampli
tude and maximum vertical velocity for these cases are shown in Figs. 13(a), 13(b),
14(a), 14(b). The streamline displacement for one case is shown in Fig. 15.
44
This s tudy leads to t h e following conclus ions :
i ) The a i r s tream of winter season has t h e favourable s t a b l e s t r a t i f i c a t i o n
for producing mountain waves on t h e l e e of t h e Wes te rn Gha t s , provided
t h e wind i s w e s t e r l y ,
i i ) The Western Ghats being very broad do not g ive apprec i ab le ampl i tude for
s h o r t e r waves. Only t h e waves of lengths 25 km and more a r e impor t an t .
i i i ) For t h e s e waves t h r e e or more c e l l u l a r motions ex i s t below t h e motion
of e x t e r n a l type above.
i v ) Amplitude of waves i n c r e a s e s wi th wave l e n g t h . For waves of l eng th
26 km t h e maximum ampli tude wi th in 8 km i s 120 m whereas t h e ampl i tude
ranges from 1200 to 2200 m for waves of l eng ths 62 .8 to 78.5 km.
v) In t h e range of wavelengths 25-78.5 km t h e maximum v e r t i c a l v e l o c i t y
i n c r e a s e s wi th wavelength. The maximum v e r t i c a l v e l o c i t y for a wave of
length 26.2 km i s 0.6 m / s e c ; whereas for waves of l eng ths 62 ,8 t o 78.5 km.
t h e v e r t i c a l v e l o c i t y v a r i e s from 4 . 8 to 5 .6 m/sec .
v i ) The v e r t i c a l v e l o c i t y has maximum at a height which appears t o i n c r e a s e
with wavelength.
Case No. Date(Time) Wave l eng thL (Km)
Maximum v e r t i c a l
v e l o c i t y (m/sec)
and i t s he ight
1 . 5 March 1962 (00Z) 26.27 . 84 . 42 . 8
0.4 (8 km)
2 . 6 December 1960 (12Z) 25 .110.0
5 .93.9
0.4 (1,2 km)
3 . 21 January 1959 (12Z) 26.2
8.3
4 . 9
.3.1
0.6 (10 km)
TABLE - 4
45
TABLE - 4 (Contd.)
Case No. Date (Time) Have length
L (km)
Maximum vertical velo
city (m/sec) and its
height
4. 4 January 1959 (12 2) 62,8
10.8
5.6
3.5
4.8 (15 km)
5. 14 December 1960 (00Z) 69.7
10.6
5.6
3.6
2.6
1.9
5.2 (14 km)
6. 26 December 1960 (12Z) 78.5
11.2
5.7
3.6
2.5
5.7(15 km)
9.1 Mountain waves during Monsoon Season
Sarker (1967) also investigated mountain waves over Western Ghats during the
southwest monsoon season. During this season the air mass does not have that
much stable stratification as the winter season. It is more or less neutral for
moist adiabatic processes or even sometimes unstable in some layers. For the
theoretical study Sarker (1967) assumes a saturated atmosphere with pseudo-
adiabatic lapse rate. The wind in the season is westerly below and easterly aloft.
Generally, the westerly wind increases from 10 kt at surface to about 30-40 kt
between 1 and 2 km and then gradually decreases and becomes easterly at 6-7 km.
On a strong monsoon day, the westerly may extend upto 10 km as well and also may
be considerably stronger. Moreover, it has a secondary maximum in the layer
5-6 km.
The mountain waves for five strong monsoon cases were investigated. The
f(z) profile for one such case is shown in Fig. 16. It is seen that these pro
files are substantially different from those presented earlier, which are favoura
ble for mountain wave formation. But still they produce mountain waves. The
wavelengths are given in Table 5.
46
TABLE - 5
Case No. Date Wavelength in km
1. July 5, 1961 19.2
2. June 25, 1961 29.2
3. July 6-9, 1963 20.6
4. July 11-12, 1965 31.7
5. July 21, 1959 26.5
I t i s in te res t ing to note that while during winter season 3 to 4 waves
superpose on one another, there appears to exist only one wave during monsoon
season. Also i t i s clear that i t i s possible to have lee waves excited by a
mountain in a s t a t i c a l l y neutral atmosphere, if the wind shear i s favourably
d i s t r ibu ted .
Inc identa l ly , we note that mountain waves of length 60-70 km can occur on
the lee of the Western Ghats during the winter months and three or four waves
may superpose on one another in that season. But, during the monsoon the waves
are of length 20-30 km only and also not more than one wave appears to ex is t .
While the larger waves of winter season may have some aviat ional importance,
the shorter waves of monsoon may contr ibute to r a in f a l l on the l ee - s ide .
9.2 Some indirect ver i f ica t ions of mountain waves over Western Ghats
The mountain waves studied over the Western Ghats could not be ver i f ied
due to lack of observational data on mountain waves in th is region. S a t e l l i t e
photographs of cloud pictures did not help verify these waves. The waves can
be manifested in clouds if there i s suff icient moisture in the atmosphere.
But during winter season the atmosphere i s very dry in th i s region, so that
nc wave pattern cloud i s v i s i b l e . During the monsoon the sky i s overcast
from which i t i s d i f f i cu l t to discern the wave clouds. Because of these two
factors perhaps we did not get any evidence of wave clouds from s a t e l l i t e p i c
t u r e s .
47
However, perhaps some indirect evidence can be had from the following:
9.2.1 Leewaves from cloud observations
Sinha (1966) observed on 6 March 1965 orographic clouds at Matheran
(Lat. 18º 58'N, Long. 73º 18'E) which is a hill station situated at the Western
Ghats on the top of a narrow long and rather isolated north-south steep ridge.
The ridge is about 750 m high, about 2 km wide and 9 km long at the top. It is
situated about 50 km to the east of Bombay. Two rows of clouds were estimated
to be at a height of 1500 m. Their thickness was estimated to be 200 m approxi
mately. The wavelength inferred from the estimated vertical angle and the height
of the clouds was between 1 to 2 km.
Our theoretical computation for this case was made by fitting the profile
with numerical values a = 0.5 km, b = 0.75 km for Matheran.
The f(z) profile for this is given in Fig. 17. Fitting a two layer model for
f(z) profile, the theoretical computation shows that wavelength L = 2.7 km. The
wave amplitudes are 426 m, 206 m and 70 m. at heights 1 km, 1.5 km and 2 km
respectively. The maximum vertical velocities at these levels are 3.2 m/sec.
1.9 m/sec and 0.7 m/sec. These theoretical values of wavelength and wave ampli
tude at 1.5 km agree fairly well with the estimated values of wavelength and
cloud thickness at 1.5 km.
9.2.2 Turbulence Reports by Aircraft
i) On 9 December 1964 a Boeing 707 in its flight from Madras to Bombay
reported light continuous clear air turbulence at 1759 IST when its position was
15º 40'N, 76º 10'E, well on the lee of the Western Ghats, at a height of
28500 ft (roughly 9 km). The duration of this turbulence was reported to be
6 minutes. The speed of the aircraft was approximately 700 miles/hour, so that
in 6 minutes it goes about 70 miles, i.e. 110 kms. That is, during this period
the aircraft was on the lee of the Western Ghats.
Our theoretical investigation shows that there were mountain waves both in
the morning and in the evening on that day. The wave length for morning was
48
31 km and for evening 19.4 km. Corresponding to the wavelength of 19.4 km the
maximum ver t i ca l velocity at 9 km was about 10 cm/sec. The a i rc raf t would have
crossed five wavelengths during i t s f l ight on the lee s ide and would have
f e l t the up and down undulations about ten times. This might be the reason for
the continuous l ight turbulence, although i t i s not possible to say if turbu-
lence might be due to other causes.
i i ) On 11 December 1964 a Comet 4 in i t s f l ight from Colombo to Karachi repor
ted l ight continuous clear a i r turbulence at 1335 IST at a height of 32000 ft
(roughly 10 km) while i t s position was 14ºN, 75ºE, on the lee side of the
Western Ghats. The duration of turbulence was 2 minutes. The speed of the
a i r c ra f t was roughly 500 miles/hr so that in two minutes i t crossed 17 miles,
i . e . 30 kms approximately. That i s in 2 minutes i t has come to the windward
s ide , outside the lee wave, if any.
Our theore t ica l computation shows that on t h i s day there were mountain
waves both in the morning and the evening. The wavelength for the morning
works out to be 18 km and for the evening i t i s 44 km. The corresponding
maximum ver t i ca l veloci ty at 10 km works out to be 10 cm/sec and 190 cm./sec
respect ive ly . I t . i s quite possible that the turbulence reported by the a i r
craf t might be in association with the waves, although there i s no direct way
of ve r i f i ca t ion .
10. Mountain waves over Assam-Burma Hills
De (1970) invest igated the presence of mountain waves over the Assam
and Burma Hil ls with the help of s a t e l l i t e p ic tu res . He studied in a l l six-
teen cases for which the wavelengths as observed from the s a t e l l i t e photo
graphs vary between 17-34 km. The observed wavelengths have also been com
pared with the wavelengths computed theo re t i ca l ly . Some of these cases are
presented in Table 6.
49
TABLE - 6
S.No. Date Time ofoccurrencein GMT
Location Meanobservedwave lengthin km
Computed
wave
leng th in
km
S.No. Date Time ofoccurrencein GMT L a t i t u d e Longitude
Meanobservedwave lengthin km
Computed
wave
leng th in
km
S.No. Date
h m s
L a t i t u d e Longitude
Meanobservedwave lengthin km
Computed
wave
leng th in
km
1 . 23.11.66 08 0 22 26 .5-27 .5 ºN 98.5-100ºE 23 19.6
2 . 1.12.66 07 02 00 25-26 °N 99-100°E 20 20 .9
3 . 8.1.67 06 29 15 27.5-29ºN 99-101°E 22 2 6 . 3
4 . 9 .1 .67 07 19 40 25-27 °N 99.5-102°E 22 22.0
5 . 13.2 .67 06 11 30 24-26 °N 98-101°E 31 26.1
6 . 10 .2 .67 07 29 29 25 °N 94-95°E 17 17.1
7 . 14.2 .67 07 02 01 25-26ºN 98-100ºE 22 23 .6
8. 5 .3 .67 07 43 43 23.5-24ºN 93.5-95°E 21 20 .1
9 . 9 .2 .68 05 44 38 27.5°N 99.5°E 23 22.2
It is seen from Table 6 that there is good agreement between the observed wave
lengths and the computed wavelengths. In all the cases the conditions for for
mation of mountain waves were found favourable as evidenced by the wind pro
file and thermal stability of the atmosphere. Computation of vertical velo
cities and streamline displacements for these cases given above in the lee of
the Assam hills are in progress. The satellite cloud photographs depicting the
waves for two cases are given in Figs. 18 and 19.
50
A P P E N D I X - A
Scale for Computation of 12
Wallington (1953) constructed a scale for computation of 12 for the Br i t i sh
Meteorological Office tephigram. Following the same method, we have constructed
a similar scale for the tephigram used by the India Meteorological Department.
The approximate expression for the parameter 12 i s given by
I t i s assumed that the second term on the r ight hand side i s negl igible com
pared to the f i r s t term so that
where
where
θ = potential temperature
τ = absolute temperature
= dry adiabatic lapse rate
Z = height
The assumption involved in equation (2) will not hold good for the saturated
atmosphere, so that the scale given should not be applied during the south-
west monsoon season.
In Fig. 20 let the actual lapse rate at a point x on the T-Φ gram
be represented by the straight line xy. Then we have
51
where the subscripts refer to the points considered. From Fig.20 we get
Zy - Zx = ZL - ZX, the latter quantity representing a height interval along an
isotherm.
By integrating the hydrostatic equation along this interval we get
where p, R, g, A and denote respectively the pressure, the appropriate gas
constant, the acceleration due to gravity, the reciprocal of the mechanical
equivalent of heat and the entropy. (A appears if the work done by gas is
expressed in thermal units).
From equations (4) and (5) we have
and
TN-TM and are measured by the lengths NM and LX respectively.
Therefore,
where K is a constant appropriate to the scale of the diagram. Assuming the
angle XMY constant and equal to 45º, we have
He now construct a triangle XPQ similar to XMY such that PX is proportional to
T—2 . This leads to
52
Hence a scale as shown in Fig.21 can be constructed in which angle APB i s
45º and temperature i s indicated by an inverse-square scale of Absolute Tempera-
ture along PA. This scale i s designed, for the Tephigrams used in the India
Meteorological Department and taking 5.35 cm to represent 300ºA, the distances
from P in centimetres to given points on the Celsius temperature scale are as
follows:
Temperature°C
PA
Cm
Temperature°C
PA
cm
30 5.25 -20 7.53
25 5.43 -25 7.83
20 5.61 -30 8.16
15 5.81 -35 8.51
10 6.02 -40 8.88
5 6.24 -45 9.27
00 6.47 -50 9.69
- 5 6.71 -60 10.62
-10 6.97 -70 11.69
-15 7.24 -80 12.94
The gβ scale is l i nea r . When gβ × 105 i s 100 sec-2 the distance along
PE on th is par t icular scale i s 27.3 cm. The units of gβ × 105 sec-2 i s 0.273 cm
a p a r t . The distances along PB for various values of gβ are as follows:
53
gβ × 105 s e c
- 2PB cm gβ × 10
5 sec
- 2 PB
4 1.09 44 12.03
8 2.19 48 13.13
12 3.28 52 14.22
16 4 .38 56 15.32
20 5.47 60 16.41
24 6.56 64 17.50
28 7.66 68 18.60
32 8.75 72 19.69
36 9.85 76 20.79
40 10.94 80 21.88
Scales for other forms and units can be constructed by applying the
appropriate conversion factors.
Using the Scale
For convenient computation of 12, the above scale is constructed on a
Celluloid. To find the values of gβ at a particular level, say X of a given
curve, the scale is laid on the tephigram so that PA is parallel to
the dry adiabatic with the appropriate temperature at X. The lapse rate in the
vicinity of X is produced to intersect PB where the required value of
gβ × 105 sec-2 can be read off directly. The method is illustrated in Fig. 21.
Division by U2 then yields the approximate value of 12.
In practice it is generally sufficient to evaluate the mean value of 12
for 50 mb layers. In doing so, the following should be noted:
54
(a) The mean lapse rate and mean wind speed over these layers should be used
in the computation.
(b) In estimating the mean wind speed over a given layer, the reported speeds
may be used only if the wind direction shows little change with height
upto, say 500 mb. If the variation of direction is more than 30º, it
would be necessary to use components of the wind in the direction of the
wind in the low levels above the friction layer.
(c) The above scale is not suitable for computing 12 through a continuous
cloud layer, so that we cannot use this scale for the southwest
Monsoon season.
55
A P P E N D I X - B
Graphical Determination of Wavelength
In Appendix A we have given a scale to compute 12 from a tephigram
directly. In this appendix we shall show how to calculate wave length graphi-
cally from the distribution of 12.
I t was shown in section 6.3 that when 12 can be represented by an exponen
t i a l function of the form
the wavelength i s given by
m being the roots of
For a s t a t i c a l l y s tab le atmosphere 12 can be generally represented by equation
(1) so that we can wri te
and then wave number i s given by
The walues of 1(z) obtained by the method given in Appendix A are then plot ted
in a simple logarithmic graph paper as given in Fig.22.
The exponential decrease of 1 i s represented by a s t ra igh t l ine due to the
logarithmic scale used. This l ine determines the parameters 1(0) and C. The
intersec t ion of the l ine with the horizontal log scale gives 1(0) . The
parameter C is derived by the formula
where i s the angle of inc l in ina t ion of the s t r a igh t l ine with the
ver t i ca l and P is a constant dependent on the scale of the diagram. In our
case the value of P i s 0.7675 km-1
. I t can be calculated easily for any
other sca le .
56
Having thus determined t h e parameters 1(0) and C, t h e wave number k i s
determined by equation ( 4 ) , I t i s seen t h a t t h e number of zeros in t h i s equa
t i o n depends s o l e l y on t h e r a t i o 1 ( 0 ) / C . If t h i s r a t i o i s l e s s than a c e r t a i n
minimum va lue 1 (0 ) /C = 2.405 no s o l u t i o n e x i s t s . This l i n e i s shown by
L1 = ∞ in F ig . 2 3 . This diagram dep ic t s t h e g r aph i ca l s o l u t i o n of equation
( 4 ) . In t h i s diagram t h e unbroken and broken l i n e s are i s o l i n e s of cons tan t
wave leng ths and t h e coord ina tes a r e 1(0) and C, t h e s c a l e of 1(0) being l o g a
r i t h m i c . If t h e point ( (o), C ) l i e s to t h e l e f t of t h e l i n e L1 = ∞
no wave e x i s t s , and i f i t l i e s to t h e r i g h t of t h e l i n e a t l e a s t one wave
e x i s t s .
The l i n e s L2 = ∞, L3 = ∞, L4 = ∞ correspond to t h e va lues of t h e
r a t i o equal to 5 .520, 8.654, and 11.792 r e s p e c t i v e l y . I f t h e
po in t { (o), C} l i e s in between L1 = ∞ and L2 = ∞ then only one
wave e x i s t s . I f i t l i e s between L2 = ∞ and L3 = ∞ two waves e x i s t
s imu l t aneous ly . I f t h e point l i e s between L3 = ∞ and L4 = ∞ t h r e e
waves e x i s t ; and i f i t l i e s to t h e r i g h t of L4 = ∞ then four or more
waves e x i s t . The wave systems beyond t h e s e a re not shown he re , because in
normal meteoro logica l cond i t ions such as high values of a re
r a r e l y found and i f at a l l , t h e corresponding wavelengths would be q u i t e
Smal l . I t can be mentioned here t h a t s i m i l a r diagrams for two wave systems
were drawn by Palm and Foldvik (1960) .
As an example, t h e l i n e given in Fig.22 g ives t h e values 1(0) = 1.30
km-1 and C= .0921 km-1 , and t h e corresponding wave lengths from F ig .23
a r e 60, 19, 10 and 6.5 km. The wave length computed t h e o r e t i c a l l y by
De (1970) in t h i s case a r e 75 , 1 9 . 6 , 10.6 and 6/8 km. The observed wave
l eng th from s a t e l l i t e p i c t u r e i s 23 km.
57
REFERENCES
1. Alaka, M.A. 1958: Aviation Aspects of Mountain waves, WMO Tech. Note 18,
p. 47.
2 . Bannon, J .K. 1951: Meteorological Aspects of Turbulence Affect ing A i r c r a f ta t High A l t i t u d e s - Prof. Notes 7 , No.104.
3 . Bannon, J .K . 1952: Weather systems Associated with some occasions of
seve re Turbulence a t High A l t i t u d e , Met. Mag. V o l . 8 1 .
4 . Berenger , M. e t . Gerb ie r , N. 1956: Monographie No.4, de 1a Meteorologie
N a t i o n a l e . Roy. Met. Soc. 87, pp .13 -23 .
5. Cohen, A. Doron, E. 1966: Meteorological satellite data of report of
studies. Report of work performed under contract C.W.B. 11055.
6. Colson, De Ver. 1954: Results of double theodolite observations at B1 shop,
Cal., in connection with the "Bishop wave" phenomenon. Bull.Amer.
Met. Soci. 33, pp.107-146.
7. Corby, G.A. 1957: Airflow over mountains: Notes for forecasters and pilots.
Met.Rep.No.18, H.M. Stationery office.
8. Corby G.A. 1957: A preliminary study of atmospheric waves using radiosonde
data. Quart. J.R. Met. Soc. Vol.83, pp.49-60.
9. Corby, G.A. and Wallington, C.E. 1956: Airflow over mountains: the lee-
wave amplitude Q.J.R.M.S. 82, pp.266-274.
10. De, U.S. 1970: Lee wavesas evidenced by satellite cloud pictures. IJMG
Vol.21, No.4, pp.637-642.
11. Doos, BO.R. 1961: Tellus, Vol.13, No.3, pp.305-319.
12. Doos, Bo. R. 1962: Tellus Vol.14, No.3, pp.301-309.
13. Foldvik, A. 1962: QJRMS Vol.88, pp.271-285.
14. Forchtgott, J. 1949: Wave streaming in the lee of mountain ridges Bull.
Met. Czech, Prague, 3, p.49.
15. Forchtgott, J. 1951: The air flow round a conical hill. Gliding, Vol.2,
p. 147.
16. F r i t z , S . 1965: The s i g n i f i c a n c e of mountain l ee waves as seen from s a t e l l i t e p i c t u r e s - J r . Appl. Met .Vol .4 , No .1 , pp .31-37 .
17. Gerb ie r , N. and Berenger, M. 1961: QJRMS, 87 pp .13 -23 .
18 . Kue t tner , J . 1939: B e i t r . Phys. F r e i . Atmos. 2 5 , pp.79-114.
19 . Kue t tne r , J . and J e n k i n s , C F . 1953: F l i g h t a spec t s of t h e mountain wave.
Air Force Cambridge Research Cen t re , Surveys in Geophysics, Tech.
Report No.35.
20 . Jones , D.C.E. 1954: Fur ther I n v e s t i g a t i o n s of High Level c l e a r Air Turbu
l e n c e . Met. Mag. V o l . 8 3 .
58
2 1 . Larson, L. 1954: Observat ions of l e e wave clouds in t h e Jamtland Mountains,Sweden, T e l l u s , 6, pp.124-138.
22 . Ludlam, F.A. 1952: Orographic c i r r u s clouds QJRMS 78 , p .558 .
2 3 . Manley, G. 1945: The Helm wind of C r o s s f e l l , 1937-1939. QJRMS 7 1 , pp .197-
219.
24 . Mason, D. 1954: H i l l s s tanding waves and s a f e t y he igh t s , W e a t h e r , London.p . 4 5 .
2 5 . Musaelyan, Sh.A. 1960: B a r r i e r waves in t h e Atmosphere. T rans l a t ed fromRuss ian . I s r a e l Program for s c i e n t i f i c T r a n s l a t i o n s , Je rusa lem,1964.
2 6 . Palm, E. 1958: Geophy. Publ . Vol .20, No.3 , Os lo .
2 7 . Palm E. and Foldvik , A. 1960: Geo. Publ . Vo l .21 , No.6, Oslo , pp. 30.
2 8 . P i l s b u r y , R.R. 1955: A pre l iminary a n a l y s i s of s tanding wave r e p o r t srece ived a t Nor thhol t during t h e winter of 1953-1954. Met. Mag. 84,pp.313-318.
2 9 . Queney, P. 1947: Theory of p e r t u r b a t i o n s in s t r a t i f i e d c u r r e n t s wi tha p p l i c a t i o n to a i r f low over mountain b a r r i e r . The Unive r s i ty ofChicago P r e s s , Misc. Rep. No.23.
3 0 . Queney, P. 1948: The problem of a i r f low over mountains . A summary oft h e o r e t i c a l s t u d i e s — B u l l . Amer. Met. Soc . 29 , pp .16-26 .
3 1 . Queney, P. et a l . 1960: The a i r f low over mountains, WMO Tech. Note 34,
p . 135.
3 2 . S a r k e r , R.P. 1965: "A T h e o r e t i c a l s tudy of Mountain waves on Western
Ghats" IJ Met. and Geophy. Vol .16, No.4, pp.565-584.
3 3 . S a r k e r , R.P. 1967: "Some Modif icat ion in Dynamical Model of orographicr a i n f a l l " , Monthly Weather Review, U.S. Weather Bureau Vol .95 , No. 10, pp.673-684.
3 4 . Sawyer, J . S . 1960: QJRMS Vol .86 , pp.326-345.
3 5 . Sco re r , R.S. 1949: Theory of waves in t h e l e e of mountains . QJRMS 7 5 ,pp .41-56 .
36. Scorer, R.S. 1953: Theory of airflow over mountains, II: The flow
over a ridge. QJRMS 79, pp.70-83.
37. Scorer, R.S. 1954: Theory of airflow over mountains, III: Airflow
characteristics, QJRMS, 80, pp.417-428.
38. Scorer, R.S. 1955: Theory of airflow over mountains IV: Separation of
airflow from the surface, QJRMS, 81, pp.340-350.
39. Scorer, R.S. 1956: "Airflow over an isolated hill". QJRMS 82,
pp. 75-81.
59
4 0 . S c o r e r , R.S. and Wilkinson, M. 1956: "Waves in t h e l e e of an i s o l a t e dh i l l " . QJRMS 82, pp.419-427.
41. Sinha, M.C. 19662 "Mountain lee waves over western Ghats". IJMG Vol.17,
No. 3, pp.419-420.
4 2 . Stormen, C. 1948: "Mother of Pear l c louds" , Weather, London, 3 , p . 1 3 .
4 3 . Wal l ington , C.E. 1953: OPS-ENG Report , I n t e r n a t i o n a l Air Transpor t
Association.
DIAGRAMS
Fig
. I
No
mo
gra
m
for
de
term
inin
g
err
ors
in
a
ltim
ete
r h
eig
hts
Fig
. 2
Dia
gra
m
of
dis
turb
an
ce
s
ge
ne
rate
d
by
a
mo
uta
in
ba
rrie
r in
th
e
ho
rizo
nta
l v
elo
cit
y
co
mp
on
en
t fi
eld
Fig
. 2
(a)
Sc
he
ma
tic
flo
w
pa
tte
rn
wh
en
the
re
is
a re
ve
rsa
l o
f w
ind
d
ire
cti
on
.
Th
e
roto
rs
or
bil
low
s
are
u
ns
tea
dy
or
mo
vin
g.
Fig
. 3
An
ex
am
ple
o
f a
tra
in o
f le
e
wa
ve
s2
is g
rea
ter
in th
e
low
er
layers
th
an
h
igh
er
up
. E
ac
h s
tre
am
lin
e
may c
on
tain
man
y
wa
ve
cre
sts
. (A
fte
r S
co
rer,
1
94
9)
Th
ick p
ort
ion
s
ind
ica
te
up
wa
rd
mo
tio
ns
Fig
. 4
Va
ria
tio
n of
lee w
ave am
plit
ud
e w
ith
siz
e
of
mo
un
tain
(A
fte
r C
orb
y
an
d
Wa
llin
gto
n)
Fig
. 5
S
ide vie
w &
gro
un
d p
lan
of
the
str
ea
mli
ne
s
over
a c
on
ica
l h
ill.
Th
e lo
wer
dia
gra
m
sh
ow
s
the
flo
w
in t
he
lev
el "a
".
Up
- &
d
ow
n-
co
mp
on
en
ts
are
d
en
ote
d
by +
8
- re
sp
ec
tiv
ely
.
( A
fte
r F
orc
htg
ott
, 1
95
1)
Fig
. 6
(a
) R
ad
ios
on
de
as
ce
nt
at
Sto
rmw
ay
on
II
Ma
rch
6
3,
03
00
h
Fig. 6(b) Radiosonde ascent at Liverpool on II March 6 3 , 1500 h
Fig. 7 ( a ) Fig. 7(b)
Profile of 2:(a) corresponding to fig. 6a, (b) corresponding to fig. 6 b
( A f t e r Corby, 1 9 5 7 )
Fig
. 8
(b)
The
co
rre
sp
on
din
g p
rofile
of
2
(Aft
er
Co
rby
1957)
Fig
. 8
(a)
Ra
dio
so
nd
e
ob
se
rva
tio
n
at
Ald
erg
rov
e o
n 4 A
pri
l 54
at
1400
G
MT
Fig. 9 Flight of on airplane in a region of descending
current on the lee of a mountain barr ier
Fig. 10 Average west to east vertical profile at the
western ghats. The crest is at X = 5 km & X = - 6 0 km
is the coast.
Fig . 11(a) WIND AND TEMPERATURE
PROFILES.Fig. 11(b) f(z) = 2 - PROFILE A
ON 2 1 . 1 . 1 9 5 9 (12 Z)
FIG. 1 2 ( a ) WIND AND TEMPERATURE
PROFILES
FIG. 12(b) f (z ) = 2- PROFILE AT
SANTACRUZ ON 4 .1 .59(12Z)
Fig. 13 Variation of amplitude with height
(A) For short wave lengths
(B) For long wave lengths
Fig. 14 Variation of wave vertical velocity with height
(A) For short wove lengths
(B) For long wave lengths
Fig. 15 Streamline displacement due to wave at levels of 2 , 4 , 6 & 8 km.
on 26.12.60 (12 Z)
(A) For longest wavelength L1 = 7 8 . 5 km
(B) For two wavelenths L1 = 7 8 . 5 km & L2= 11.2 km
Fig. 16 Profi le of f ( z ) = 2 for July 5 , 1961 for saturated a tmosphere .
FIG. 17
FIG. 18 : 9 JAN 1967
OBSERVED WAVELENGTH IS 22 KM & COMPUTED
WAVELENGTH IS ALSO 22 KM ( D E 1970 )
FIG. 19 : 29 JAN. 1 9 6 7
OBSERVED WAVELENGTH IS 22 KM 8 COMPUTED
WAVELENGTH IS 20 .9 KM (DE 1970)
FIG
. 2
0
FIG. 2I
FIG. 22
FIG
. 23(a
)
FIG
. 23
(b
)
FIG
. 23
(c)
No.III-3.9 Discussion of Typical Synoptic Weather Situations: Southwest
Monsoon: Typical Situations over Interior Peninsula and
Coastal Andhra Pradesh - N.M. Philip, V. Srinivasan and
K. Ramamurthy.
No.III—4.1 Discussion of Typical Synoptic Weather Situations:
Weather over the Indian seas during the Post—Monsoon
Season - V. Srinivasan and K. Ramamurthy.
No.IV-13 Rainfall of India — P. Jagannathan.
No.IV-16 Microseisms and Weather - A.N. Tandon and S.N. Bhattacharya,
No.IV—17 Medium Range Forecasting - K.R. Saha and D.A. Mooley.
No.IV-18.1 On the Criteria for declaring the onset of the southwest
monsoon over Kerala — R. Ananthakrishnan, U.R. Acharya and
A.R. Ramakrishnan.
No.IV-18.2 Monsoons of India: Synoptic Features associated with onset of
Southwest Monsoon over Kerala — R. Ananthakrishnan,
V. Srinivasan, A.R. Ramakrishnan and R. Jambunathan.
No.IV-18.3 Some aspects of the "Break" in the Indian Southwest Monsoon
during July and August - K. Ramamurthy.
No.IV-18.4 Northeast Monsoon - V. Srinivasan and K. Ramamurthy.
No.IV-20 Evaporation - N. Ramalingam.
No.V-1 Techniques of High Level Analysis and Prognosis:
1. Organization and Methods of Analysis - P.K. Das,
N.C. Rai Sircar and D.V. Rao.
No.V-2 Techniques of High Level Analysis and Prognosis:
2. Prognostic Techniques and Assessment of Accuracy of
Forecasts - P.K. Das, N.C. Rai Sircar and D.V. Rao.