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POSTGRADUATE SCHOOL
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INITIAL STAGES AND SOME CHARACTERISTICS
OF WEST AFRICAN LINE SQUALLS
by
JOHN AKINTAYO ADEDOYIN
B.Sc.(HONS) PHYSICS (IBADAN)
\ I
A THESIS IN THE
DEPARTMENT OF PHYSICS
.>
SUBMITTED TO THE FACULTY OF SCIENCE
IN PARTIAL FULFIU1ENT OF THE REQUIREMENTS
FOR TIlE DEGREE OF
DOCTOR OF PHILOSOPHY
OF THE
UNIVERSITY OF IBADAN
MARCH, 1983
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MLAORCK.. ,U
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DEDICATION
To
God for being, as ever, faithful to His promises, my father
and late mother without whose prayers and support, this work
would have been impossible.
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ABSTRACT
A review of West African synoptic weather pattern reveals that
the sub-region experiences a special kind of atmospheric disturbance -
the line squall - whenever the south-westerlies cover, approximately,
the whole of Nigeria. Various methods that have been used to study
the squalls (i.e. observational investigations, satellite investi-
gations and modelling) have not been very successful in isolating
the trigger mechanism of the phenomenon.
It is been proposed that line squalls are initiated through
the amplification (with time) of any wave-like perturbation along
the surface of discontinuity between the south-westerlies and the
north-easterlies. The amplifying perturbations could block the
650 mb. mid-tropospheric jet which further distorts the 'bump' formed
by the undulating perturbation. This distortion forces the south-
westerlies further up and they could condense. The precipitates
fall into the underlying, dry jet and some of them evaporate; the
latent heat of evaporation being supplied by the jet. The iet, now
cooler, sinks. On sinking, the jet could hit the surface of the
earth on which it forms the squall front and crawls; thereby lifting
the south-westerlies ahead of it. The cycle of condensation, evapo-
ration and sinking then continunes.
A gravity-wave model of this mechanism is presented through
the solution of a frequency equation with the aid of a two-layer
atmospheric model. The solution is an eigenvalue problem from which
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many modes of different growth rates and phase velocities could be
obtained. Some of these phase velocities ,,;111 be complex - the real
part representing the pt::1sevelocity Hhile the imaginary part represents
the ampLf.f LcatLon , Waves 'td.ththe largest amplifications (Le. the
largest imaginary part) are those th'lt could possibly block the 650 mb.
mid-tropospheric jet and trigger off line squalls.
Among others, this proposal on the trigger mechanism of line
squalls is able to explain:
(i) the preference of highlands as places of origin of
line squalls,
(ii) the close association between the speeds of propagation
of line squalls and the mid-tropospheric jet and
(iii) the observed overturning of the atmosphere after the
passage of line squalls.
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ACKNO\VLEDGEMENT
I am grateful to my supervisor, Professor F.B.A. Giwa, who
not only laid the foundation of Meteorology in the Department of
Physics, University of Ibadan, Nigeria, but also sugp,ested this
project and painstakingly guided me from the beginning to the end.
I am indebted to the following people for their assistance,
in various ways, during and after the trying days of 1979:
Professor O. Awe, Dr. C.E.F. Oni and Dr. I.A. Babalola.
The University of Ibadan contributed greatly to this project
by granting me graduate assistantship for some part of this course
and later on, a bursary award.
Finally, I most sincerely thank my wife, Bola, for her patience,
loyalty and co-operation throughout the period of this work.
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CERTIFICATION
This is to certify that the work described in
tbis thesis was carried out under
my supervision by
John Akintayo Adedoyin
in the
Department of Physics,
University of Ibadan,
Ibadan,
Nigeria.
F.B.A. Giwa, B.Sc., Ph.D., D.I.C.
Professor in the Department of Physics
and Director of the Computing Centre
University of Ibadan
Ibadan,
Nigeria.
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CONTENTS
Page
Title ... 1
Abstract 3
Acknowledgement 5
Certification 6
List of plates 10
List of figures 11
Notations and symbols 12
CHAPTER 1
I N T ROD U C T ION 14
... CHAPTER 2
A REVIEW OF LINE SQUALL OBSERVATIONS
2-1 West African synoptic weather pattern 20
2-2 Observational investigations of tropical line squall 25
2-3 Satellite investigations of line squalls 41
CHAPTER 3
A REVIEW OF LINE SQUALL MODELS
3-1 Mathematical models 50
3-1-1 Gravity wave models 50
3-1-2 Non-linear steady state model 51
3-2 Numerical models 52
3-2-1 Two-dimensional models 53
3-2-2 Three-dimensional models 58
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Page
3-3 Other theoretical models 60
3-3-1 Plume model 60
3-3-2 Bubble-theory model 61
3-4 Syntheses of line squalls 61
CHAPTER 4
EVOLUTION OF LINE SQUALL
4-1 Necessary atmospheric features in the development of
line squall 69
4-2 Initial stages and trigger mechanism of line squall 73
4-3 Fully-developed line squall 74
CHAPTER 5
FUNDAMENTAL EQUATIONS
5-1 Basic and perturbation equations 84
5-2 Frequency equation 87
5-3 Effect of atmospheric stability on frequency equation 90
CHAPTER 6
SOLUTION TO THE FREQUENCY EQUATION
6-1 Boundary conditions 94
6-2 Mathematical derivations 98
6-3 Special case 102
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CHAPTER 7 Page
SUMMARY AND CONCLUSION 104
LIST OF REFERENCES 115
._. _ ._.-.
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LIST OF PLATES
PLATES TITLE PAGE
l(a) Line squall at ~innaon 12 September 1977. View
to the E at 1555 GMT showing advancing line of
cumulonimbus 29
l(b) Line squall at Minna on 12 September 1977 View
to the NE at 1611 GMT showing roll cloud ••• 30
l(c) Line squall at }unna on 12 September 1977. View
to SSE at 1613 GMT showing front of cumulonimbus
belt and edge of rain area 31
1(d) Line squall at Hinna on 12 September 1977. View
to W at 1648 GMT showing individual cumulonimbus 32
2(a) The 'Fortune' squall of 5 September 1974. Visible
SMS - I image of West Africa at 0300 43
2(b) The 'Fortune' squall of 5 September 1974. Visible
SMS - I image of West Africa at 0930 44
2(c) The 'Fortune' squall of 5 September 1974. Visible
SMS - I image of West Africa at 1100 45
2(d) The 'Fortune' squall ot 5 September 1974. Visible
SMS - I image of West Africa at 1230 46
2(e) The 'Fortune' squall of 5 September 1974. Visible
SMS - I image of West Africa at 1400 47
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LIST OF FIGURES
FIG. TITLE PAGE
1. Three disturbance lines in West Africa 16
2. North-South section of the prevat l.mg winds over
~-1estAfrica 21
3. Monthly frequency and diurnal variation of squalls
over 15 mls at various Nigerian stations 24
4. Number of disturbance lines \olhichgenerated and
decayed in each 50 square over the contient and
ocean during GATE 35
5. Time~height section of wind component over a GATE
ship (2 - 17 September 1974) 37
6. Tephigram showing T9 Td for varous post-squall
soundings 39
7. Two types of arc lines in satellite imagery 48
8. Idealized history of a line squall 63
9. A sketch of vertical profile of wind over West Africa
between April and September 70
10. The process of formation of line squalls 75
11. Schematic croaa-eect.Ion through a class of
squall system 80
12. A section of the interface between the air masses in a
two-layer atmospheric model 95
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NOTATIONS AND SYMBOLS
u x - component of velocity
v y - component of velocity
w vertical velocity
p pressure
.!2E..
w Dt
p density of air
g acceleration due to gravity
v coefficient of viscosity
f Corioli's parameter
s twice the angular velocity of the earth
L latent heat of vaporisation
H scale height
r condensed water vapour per unit mass of air parcel
q specific humidity
C specific heat at constant pressurep
Cv specific heat at constant volume
e potential temperature
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c - C
P v
K
Cp
T temperature
z vertical height
a frequency of waves
wave number along x-direction
B wave number along y-direction
R gas constant
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CHAPTER 1
INTRODUCTION
The weather pattern in West Africa could be classified into two
main seasons: the wet and the dry. One of the distinguishing factors
between the seasons is the amount of rainfall. Almost all the rain-
fall within the sub-region are recorded during the wet season which
is approximately from April to September.
About half of the total annual rainfall recorded in any particular
location within West Africa is due to the precipitation which usually
accompanies isolated thermal convection in the atmosphere (Obasi, 1976).
Such pockets of convective processes are referred to as LOCAL CONVECTIVE
STORMS. As expected, the frequency and intensity of local convective
storms vary from place to place; hence, total annual rainfall is not
uniform in the sub-region.
Apart from local convective storms, there exists a special kind
of storm that travels and thus covers a large area of land whenever it
occurs. These 'travelling' storms are also accompanied by heavy rain-
fall. More than half of the total annual rainfall in West Africa is
attributable to these storms which are kno~m as LINE SQUALLS. The
extent of 'travel' of line squalls (about two to three nights of
'travelling' at an average rate of 15 m/s (Fortune, 1977» puts them
in the class of mesoscale systems.
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r
Line squalls can be defined in various ways depending on the
mode of identification and place of occurrence. However, the definition
given by Zipser (1977) seems all-encompassing and therefore adequate.
According to him, line squalls are:
cumulonimbus clouds, organised in linear fashion,
associated with a pseudo-cold front (squall front)
at the surface. propagating with considerable speed
with respect to the ambient low-level air~ in the
general direction of the squall wind in the cold
air behind the squall front.
Line squalls have been referred to by various names at different
times and places. Hamilton and Archbold (1945) called them DISTURBANCE
LINES (D.L. 's) because the squalls occur along a fictitious bow-shaped
line that has N-W to S-E orientation (Fig. 1). Other names are:
chubascos (in Central America), haboobs (in Sudan) and sumatras (in
Malaysia).
These various names indicate the geographical spread of the
occurrence of line squalls. Thus, they are not entirely a local West
African (or tropical) sub-synoptic phenomenon. Marriot (1892),
Prohaska (1907) and Browning and Ludlam (1962) reported the existence
of this kind of storms in the mid-latitudes. Although mid-latitude
and tropical line squalls propagate at speeds close to the speed of
the mid-tropospheric winds in their respective regions (ioe about 15 m/s
in West Africa), they differ in structure because the anvil in the
latter extends behind rather than in front of the convective elements
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~D-Ll --- - - -D-L 2--- -._ .D._L._3 .-.
o 480 960kms
I I
Fig. 7.' Three dis turbance lines (DL1,DL 2, DL3) in
West Africa (the fict itious bow-shaped lines
propagate westwards across West Africa)
(trom Garnier, 1967-)
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(Fortune, 1977; Mansfie1d~ 1977; Zipser, 1977). This difference in
structure is a direct consequence of the variations in wind profile
between the mid-latitudes and the tropics. In the mid-latitudes, wind
speeds increase almost steadily with height and as such, the high
wind speed aloft makes the updraught air, which originates ahead of
the storms, flow out at upper levels whereas in the tropics, there is
a low-level jet rather than strong upper level winds (Bolton, 1981).
The importance of line squalls lies principally in their contri-
bution to the economic growth of West Africa (vis-a-vis agriculture).
This economic importance is most appreciated if it is realised that
the total annual rainfall in the Sahel region of West Africa is due,
mainly, to these storms. This implies that years of little squall
activities are usually dry in the Sahel. The consequent drought leads
to loss of life (both human and animal) and poor agricultural output.
Apart from their ,economic and social implications, hazards in
the field of aviation is another reason why studies on the initiation,
maturity and dissipation of West African line squalls have engaged
the attention of researchers for almost four decades now (Hamilton and
Archbold, 1945; Dhonneur, 1970; LeRoux, 1976; Fortune, 1977; Okulaja,
1978; Bolton, 1981). On the global scale, the contribution of the
phenomenon to the overall transfer of mass, energy and momentum is of
interest to meteorologists. In chap~ers 2 and 3, reviews of some of
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the methods that have been employed in the study of line squalls are
presented. Broadly, these methods could be classified into three:
obse~1ational investigations, satellite investigations and modelling.
Whichever method is used to study line squalls however, complete
theories on them must account for their growth, the fully-developed
stage and dissipation. Furthermore, such theories must account for
their orientation, direction of movement and the almost-uniform speed
of propagation that is common to West African line squalls. It is also
desirable to explain the limitation of line squalls to a perticu1ar
season of the year and the similarities between a local convective
storm and line squall.
Almost all features of the fully-developed (or mature) line
squall have been documented (Hamilton and Archbold, 1945; Mansfield,
1977; Bolton, 1981). Outside West Africa, but within the tropics, the
following works on line squalls are notable: Zipser (1969, 1977),
Belts, Grover and Moncrieff (1976) and Miller and Betts (1977) while
the description of the Wokingham storm by Browning and Ludlam (1962)
gave a good picture of mid-latitude 'travelling' storms. Data from
satellite photographs have also added to our knowledge of the matured
line squall.
In spite of these works, however, there is still some doubt as to
the meteorological factors on which the origin of line squalls are
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dependent. More precisely, the trigger mechanism which at verious
times have been linked with insolation, orography and synoptic conver-
gence, is still not fully understood. This study has, therefore, been
directed towards the ~-::'1.:!s.ttaigaels in the development of line squalls;
paying special attention to the trigger mechanism. In chapter 4, the
mechanism of initiation of line squalls, along with a description of
the atmospheric features during the matured stages, are described.
The process of initiation of line squalls is mathematically
modelled in chapters 5 and 6. A frequency equation is derived in
chapter 5 from the basic set of hydrodynamical equations of motion.
This frequency equation is solved in chapter 6. Since line squall
is not a single mode but rather an amplifying patch of convection that
consists of many modes of different growth rates and phase velocities
(Bolton, 1981), solutions to the frequency equation are expected to
reveal modes with the greatest amplifications. It is proposed that
such modes play vital roles in the initiation of line squalls.
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CHAPTER 2
A REVIEW OF LINE SQUALL OBSERVATIONS
2-1 West African synoptic weather pattern
As a background to understanding the various stages in the
development and study of West African line squalls, a brief review of
West African synoptic weather pattern is essential.
The air masses and the prevailing winds in West Africa are shown
in figure 2. One of these air masses is labelled the MONSOON. According
to Hamilton and Archbold (1945), the monsoon winds originate off the
coast of South Africa and blow south--easterly for some time before
they veer to their direction of south-west in which they arrive at the
coast of West Africa. The long track of these winds over the Atlantic
ocean accounts for their considerable moisture content by the time they
arrive at the southern part of the sub-region. On the other hand,
HARMATTAN WINDS blow across the northern part of West Africa. In
contrast to the monsoon south-westerlies, this air mass is very dry
because of its long track over the Sahara desert.
The north-easterlies and the south-westerlies meet at a zone of
strong convergence called the Inter-Tropical Convergence Zone (ITCZ)
(Fig. 2). Other names for this zone are: Inter-Tropical Front (ITF) and
Inter-Tropical Discontinuity (ITD). The ITCZ moves North or South in
sympathy (but with a phase lag of about one and a half months (Bolton,
1981» with the motion of the sun between the tropics of Cancer and
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21
metres
6000
Westerlies Easterlies
3000
Harmattan
(north ecsterly)
Monsoon
(sout h westerly)
Zone A Zone B Zone C
1Z'N-- - --- - -- - -8°N- - - --- -- - 6N (January)
(July) 2d'N- - - - - - - - - - - - - - - -6°N
Ag·2: North - South secti on of the prevailing
over West Africa
(from Hamilton and Archbold, 7945)
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Capricorn. This zone is farthest North (about 20oN) around July and
at its southermost extent (about 8oN) around January. When the zone
is far North, West Africa is covered by the monsoon south-westerlies;
this is the wet season. Towards the end of the year when the zone is
far South, West Africa experiences the dry season because the sub-region
is almost entirely covered by the dry north-easterly aarmattan winds.
In discussing the characteristic weather pattern of West Africa,
Hamilton and Archbold (1945) divided the sub-region into four weather
zones that run East-West (Fig. 2). Balogun (1974) adopted the suggestions
of Hamilton and Archbold (1945) and Walker (1959) and added a fifth
zone to the four previously mentioned. Zone A of Hamilton and Archbold
is the region north of the rTCZ while Zone B extends from the ITCZ to
about 150 km southwards. Zone C, with a width of about 500 km lies
south of Zone B. Zone D~ which is over the land between July and
September, lies south of Zone C. In other months of the year, Zone D
is over the Atlantic ocean.
Within Zone D, there is an isothermal (or inversion) layer
between 800 mb. and 850 mb. pressure levels. As we shall discuss
later, the existence of this inversion might have something to do
with the observed dissipation of line squalls around the coasts of
West Africa. Zone C is distinguished from other zones because it is
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periodically traversed from East to West by D.L.'s (Hamilton and
Archbold, 1945).
According to Garnier (1967), the movement of the ITez controls
both the number and duration of the weather types experienced in
different parts of West Africa. Broadly, the resulting pattern is
a latitudinal one. For instance, Ibadan (70 26' N 30 54¥ E)
experiences weather conditions in the following way:
Zone A late December to part of January
Zone B February and part, or all of March
Zone e April (or part of March) to about mid or late July
Zone D late July and part, or all of August
Zone e late August to end of October or early November
Zone B briefly in November and early December
Zone A late December to part of January
Since Zone e is, as said earlier, traversed by D.L.'s, a station
like Ibadan experiences intense D.L. activities twice in a year as
the break-down of weather conditions given above shows. This explains
the two peaks observed on the chart of monthly frequency of line squalls
at some Nigerian stations (Fig. 3). The peaks coincide with the periods
when the stations experience Zone e weather conditions. In section 4-1,
it is shown that all the essential atmospheric features that favour
the development of persistent storms are present at such stations during
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24
>.
u
sc
0"
.•e....
>.
x..:-
co
~
A 0 0
c
o
+'"
d 20
\...
d
>
d 10
\c...
::J
a 0 '=O"'-:-4""'-r-;-o....,.T'<"><""T'" 1-:::,.....-norr'="T:7-r=<T ':..,..-,-""":"==-==-r .",.,.....,..,..,~~
2 6 10 1418 22
Moiduguri
Fig. 3: Monthly frequency and diurnal variation of squalls over
15 m/s at various Nigerian stations
(from Hamilton and Archbold)945)
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the periods indicated by the peaks. Hence, it is not a coincidence
that line squalls are, for instance, most-frequently observed in
Ibadan at about April/May and September/October.
2-2 Observational investigations of tropical line squalls
As mentioned earlier, West African line squalls cover an
extensive area of land whenever they occur. As a resu1t~ keeping
track of them is very expensive in terms of human and material resources.
Hence, very few large-scale experiments, aimed at ana1ysing the various
stages in their development, have been conducted up to data. Some of
these experiments are: "OperatLon Niger:~ (1972), 'Operation pre-GATE
ASECNA *1 (1973) and the Atlantic Tropical Experiment under the Global
Atmospheric Research Programme GATE) (1974). Outside West Africa, but
within the tropics, some experiments that are equivalent in scale are:
Line Islands Experiment (1967), Barbados Experiment (1968) and the
Venezuelan International Meteorological and Hydrological Experiment
(VIHHEX) (1972). Before we review these experiments, the worthy
contributions of a few -~~'ldividualwsho have studies various aspects in
the evolution of line squalls would be mentioned.
(a) Hamilton and Archbold (1945)
Hamilton and Archbold studied the meteorology of Nigeria and
adjacent territories. In their report, they mentioned that D.L.'s
*ASECNA - Agence pour 1a Securite de 1a Navigation Aerienne
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could be identified on weather charts as species of fronts which move
in a west-south-westerly (WSW) direction with a speed of 10-15 m/s.
The authors describe how a D.L. manifests itself to a surface observer
thus:
Beforehand we have normal south-westerlies with normal
or almost normal cloud development appropriate to the time
of day. Then in the east we see a dark heavy band of Cb.
approaching. This will be heralded at night by an impressive
display of lightning which can be seen whf.Le the storm is as
much as 100 miles (160 km) away... In the case of an active
D.L. the Cb. will certainly reach to 20,000 feet (600Om) or
more probably to 30,000 feet (9000m) and may even be much
higher •.•
At the forward edge of the cloud base there is a well-
marked roll of low cloud» roughly between the levels 600 and
3000 feet (200 - 1000m) though the top is not always plainly
seen. Just before the arrival of the roll cloud the surface
wind usually falls light or calm. ~~en this cloud passes
overhead, there is a sudden squall from between S.E and N.E.
At 1agos the anemometer commonly records a squall of 30-40 m.p.h.
(12 - 20 m/s); stronger gusts are experienced in the north
but more than 60 m.p.h. (25 m/s) is uncommon though possible.
The squall is the last warning to take cover. Two or
three minutes later, a heavy downpour of rain sets in, reducing
visibility to 1000 - 3000 yards (about 1000 - 3000m) or less
and effectively to nil for pilots inside aircraft. At the same
time the cloud base becomes ragged and very low '00
About 2 to 3 hours after the beginning of the storm
the rain ceases entirely and the surface wind falls calm or
blows lightly once more from S.E. to S.H. After 2nother hour
the medium cloud lifts further and begins to break.
Hamilton and Archbold did not find any general method of fore-
casting the mode, place and time of origin of D.L. 'so We know, how-
ever, from their work that the passage of line squalls, whether over
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land or the oceant overturns~ and therefore stabilises~ the atmosphere.
After the passage, the surface dry-bulb temperature usually approxi-
mate the 700 mb. (or higher) wet-bulb potential temperature before
the squall.
Analyses of the D.L. development by these authors, though commen-
dable, was limited in scope because they relied mostly on ground
meteorological stations and these were sparse at the time of their
study. The situation had not markedly improve at the time Bolton (1981)
carried out his investigations despite a time interval of over thirty
years.
(b) Bolton (1981)
Bolton (1981) studied line squalls that affected Minna
(90 37' N 60 34' E) over a period of about two years using mainly
surface data. Because it is difficult for a single observer to
distinguish between local convective storms and line squalls, Bolton
set out the following criteria for identifying the latter:
(i) maximum squall speed at least 12 mis,
(ii) rainfall at least Smm,
(iii) storm duration at least 3h and
(iv) pressure jump at least 0.5 mb.
Using these conditions, sixty-six line squalls were identified
during the period of study. The speeds of propagation of these storms
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agree well with the results of Hamilton and Archbold. Plates l(a)-
l(d) show the onset of a typical West African line squall as recorded
by Bolton.
The author associated the trigger mechanism of line squalls with
solar heating and orography; specifically mentioning that a combination
of the two might be necessary before these storms develop. However,
as he correctly pointed out, the fact that line squalls are generated
over the sea indicates that these two mechanisms might not be very
essential. There must then be other mode(s) of generation of line
squalls. Large-scale experiments, which we shall now consider, did
not fair better than individual contributions as far as isolating the
mode(s) of generation of line squalls is concerned.
(c) 'Operation Niger' (1972)
This experiment, conducted on the bend of river Niger (O-SoW)
in mid-July 1972, was limited in affair and throughout the period, no
long-lived D.L. was observed. Analysis of the only D.L. that passed
through the experimental array revealed a weakening of the mid-level
jet is the viscinity of the disturbance. From other works
(Zipser, 1977; Bolton, 1981) and our own studies, the weakening of
this jet is likely due to the fact that it descends to the surface
of the earth in the course of initiation of line squarls. More will
be said on this in chapter 4.
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Plate l(a): Line squall at Minna on 12 September 1977. View to
the E at 1555 GMT showing advancing line of cumulonimbus
(from Bolton, 1981).
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Plate l(b): Line squall at Minna on 12 September 1977. View to the
NE at 1611 GMT showing roll cloud (from Bolton, 1981).
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Plate l(c): Line squall at Minna on 12 September 1977. View to SSE
at 1613 GMT showing front of cumulonimbus belt and edge
of rain area. (from Bolton, 1981)
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Plate l(d): Line squall at Minna on 12 September 1977. View to W
at 1648 GMT showing individual cumulonimbus
(from Bolton, 1981)
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(d) 'Operation pre-GATE ASECNA' (1973)
This operation, centred in Bamako (Mali), was an intense analysis
of ten days of observation from the sparse network of meteorological
stations in West Africa. Five major disturbances (three of them line
squalls) intercepted the array during the experiment. Some of the
findings of the experiment are:
(i) line squalls are formed often to the east of latitude
SOE and they cover about 1200 km. with an average
speed of close to 15 mIs,
(ii) precipitation distribution is highly irregular,
(iii) there is a noticeable jump in the surface pressure
upon the arrival of line squalls and
(iv) the atmosphere is homogenised after the passage of
line squalls from a pre-squall stage of sharp contrast
between low and mid-level air masses.
(e) GATE (1974)
This is, perhaps, the most intensive investigation conducted up
to date on West Africa disturbances. The experiment made use of
radar, radiosonde ascents, aircraft traverses and satellite data to
study the storms over the eastern Atlantic (i.e. off the coast of
Senegal). Data collected during the experiment are still being
ana1ysed by various scientists. We shall only mention the works of
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Aspliden, Tourre and Sabine (1976) and Mansfield (1977) because this
study has been able to justify some of their observations.
(i) Aspliden et.al. (1976)
These authors identified a total of 176 active D.L.'s during
the three phases of GATE (i.e. 28 June - 16 July, 28 July - 15 August
and 30 August - 19 September). The criteria for identification are:
the D.L. should consist of large cumulonimbus clouds
which tops to about 15 km (this implies that only the
brightest cloud images on the geo-stationary Synchronous
Meteorological £atellite - I (SMS - I) were considered;
although not all cloud clusters qualify to be D.L.'s)
such bright clouds should attain a £ize of at least 20 x 20
during their life time
the D.L. should be active for a minimum of 6h.
Very few (about 8%) of all the line squalls observed during GATE,
as recorded by Aspliden et.al., formed over the ocean. Majority were
generated and decayed over land. Figure 4 shows the number of dis-
turbances which were generated as well as these that decayed in each
5° square over the continent and ocean during GATE.
Although these authors did not explain the process of formation
of line squalls, we know from their work (Fig. 4) that the preferential
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35
o = num ber of cases
generated
o = number of cases
25 N decayed
( 1/1 0/1 2/1
20N
0/1 1/0 2/7I' 7/15 13/lD 15/12 7/3 16/715N 1\I \
\
1/3 / 2/9', 5/~ 13/7 2~20 '1/lD 24/7
10N / , \2/18I
0/2 1/3:<, 4/1 J 0/4 1"/8" 1/3 0/4 1/2
5N "
~
// ~ ~ ~/ D
,/ -
0/1 0/1 "
a (
30W 25W 20W l5W lOW 5W 0 5E 10E
Ag.4: Number of disturbance lines which
generated and decayed in each
5° square over the continent and
ocean during GATE
(from Aspliden et af-,1976)
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places of initiation of line squalls are: the Jos plateau of Nigeria/
Adamawa highlands of Cameroun,Mossi: plateau of Upper Volta and the
Fouta Djalon highlands of Guinea. Further, their report shows that,
over the land, the preferred time of generation of line squalls is
the period of maximum insolation (1400 - 1700 LST).
Aspliden et.al (1976) noticed a peak in the frequency of generation
of line squalls between 0o and 50W (Fig. 4) and postulated that there
must be other forcing agents, apart from highlands, because they
thought the area is devoid of high grounds. Further investigations
in this study reveal that the Mossi plateau of Upper Volta is within
the square 0-5W, 10-15N (Fig. 4) and this probably explains the observed
peak in the frequency of generation of line squalls.
(ii) Mansfield (1977)
This author studied four occurrences of line squalls during phase
phase III of GATE. As his time-height section of W-E wind component
during the period shows (Fig. 5), the speed of the 650 mb. mid-tropospheric
jet is usually a maximum just before the onset of line squalls. He
then proposed that the attainment of the maximum speed within the
troposphere of this jet, prior to the occurrence of line squalls,
might have something to do with the initiation processes of these
storms. From this study, such a link is still obscure.
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37
15 u- component m 5-1
.,,
, I, lO", I
\ . , 'II • 1/\, I,\
..:E:t: \ .....I
I,t
\ I
I I
•.. I II I
..c
.O-l 5
. I
(1)
:r:
a
3 ---- -5 7 9 11
15 V - component m 5-1
..:E:t: 1"0
.•...
..c
Ol
ell
:r: 5 0
9 11
Fig. 5: Time- height section of wind component over a
GATE ship 2 -17 September 1974.
(from Mansfield, 1977)
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Before GATE, some experiments were conducted outside West Africa
but within the tropics.
(f) Line Islands Experiment (1967)
This experiment was conducted early in 1967 near Palmyra,
Fanning and Christmas Islands (150oW - l70oW). Zipser (1969) analysed
a particular squall that crossed the region on 1 April 1967. This
squall moved at about the maximum tropospheric wind speed. Other
features of the storm were essentially the same as those obtained in
experiments conducted in West Africa. A separate analysis by Zipser
(1977), based on this experiment and the Barbados experiment of 1968,
show that there are two scales of downdraughts associated with line
squalls. These downdraughts were also observed by Bolton (1981).
(g) Barbados Experiment (1968)
Zipser (1977) intensively studied a line squall that passed
Barbados on 18 August 1968 using a combination of aircraft and sur-
face - based instruments. Vertical soundings in post-squall areas
of this experiment, and others, show that there is a maximum separa-
tion between temperaturep T, and dew point, Td, at about 900 mb. When
the soundings approached the base of the anvil, which is slightly
above 600 mb., this separation narrows and thereby create an 'onion'
shape for characteristic soundings in post-squall areas (Fig. 6).
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39
300
.._-.- ..-- ..-..~
- 5-0
70
1000
/
() 5 10 15 20 25 30
Fig. 6: Tephigram showing TJ Td for various pos t -
squall soundings
(a) GATE Queen air sounding, mostly in rain, 5 Sept. 1974·
(b)(c)(d) Various GATE oceanic soundings, 12 Sept. 1974
(e) (f) Venezuelan soundings, 28 Aug. 1969.
(g) Barbados sounding (no Td available), 18 Aug. 1968
(from Zipser; 7977)
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(h) VIMHEX (1972)
Betts, Grover and Moncrieff (1976) identified seven line squalls
over Venezuela between June and September 1972. From their report,
we now know that there is a single reversal of the shear at 700 mh. in
pre-squall soundings made in Venezuela. This is a Venezuela feat
which most other line squalls do not possess. Apart from the reversal,
there exists some inflow, the origin of which is not clear, behind
the squall. It might be necessary to study the role(s) of the reversal
and the inflow in the propagation and/or initiation of line squalls.
Unfortunately, the reports from observational investigations of
West African (and other tropical) 'travelling' storms did not synthesise
the various analyses in order to present a clear picture of the evolu-
tion of line squalls. In West Africa, this may be due to the enormity
of the problems involved in keeping track of a sub-synoptic phenomenon
in an area of scarce aircraft flights, sparse network of meteorological
stations, large expanse of semi-arid land that is not habitable and
a people to whom nature is so kind that th~y care less about weather.
Satellite investigations of tropical storms provide better opportu-
nities for synthesising the various events that occur during the
evolution of line squalls.
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2-3 Satellite investigations of line squalls
Investigation of West African severe ~;Ga~l::.seyrstems by
satellite is relatively nsw when compared with other forms of studies.
Although satellite investigation of atmospheric disturbances is
aesthetically appea1ing~ it is not as popular as would be expected
because access to the satellite data is very limited. Furthermore,
among those who have access to satellite records, the criteria for
identifying different atmospheric phenomena on satellite photographs
vary. However, there have been few cases wherein satellite images
have been most wonderfully utilised to study specific atmospheric
disturbances; the work of Fortune (1977) is a case in point.
Fortune (1977) ana1ysed 48 hours of time-lapse satellite imagery
of a family of line squalls in Hest Africa. These images were received
from SMS - I which provided visible and infra-red images of West Africa
every 30 min. lIe (Fortune) examined particular images, taken during
phase III of GATE, on the University of Wisconsin (U.S.A.)
Man-Computer Interactive Data-Access System (HcIDAS) which displays
single images or time-lapse sequences on a television terminal.
Sequential thirty-~inute SMS - I images have the advantage of display-
ing instantaneous atmospheric features over a large area (e.g. West
Africa), Thus, in terms of cost-benefit ratio, satellite investigations
seems to be the least eJrpensive method of studying tropical disturbances.
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Line squalls can be identified in both visible and infra-red
images as distinct cloud masses characterised by high brightness and,
generally, a convex leading edge (Fortune, 1977). On satellite,
images, the roll cloud (Plate l(b» can sometimes be seen, detached
from the main cloud mass, as a low-level feature on the visible images
only. Behind the main cloud mass, points of brilliance could be seen
while the rear of line squalls are usually indistinct and fibrous.
Plates 2(a) - 2(e) show different positions (with time) of the line
squall of 5 September 1977 which Fortune analysed.
Latrasse (1972) classified African squalls into two types
according to the pattern of the satellite pictures. Type I images
have a single well-defined arc-shaped front (Fig. 7) that faces the
direction of propagation. At the rear of this arc is a compact and
very bright cloud mass. The rear edge of the cloud mass is not as sharp
as the rear of the arc. Type II images (Fig. 7) appear like indented
arcs. This type of squalls usually arise from the merging of Type I
cloud masses. Type II squalls often split into sister squalls during the
course of propagation. Apart from these two types , other forms of gust
front exist. Some of these have been reported by Gurka (1976); although
his observations were over the temperate region.
It is possible for SMS - I images to miss the process of initiation
of line squalls because of the relatively large time interval between
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/
fate 2(a): The 'Fortune' squall of 5 September 1974. Visible
SMS-I image of West Africa at 0800
(Arrows indicate position of squall line)
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Plate 2(b): The 'Fortune' squall of 5 September 1974. Visible
SMS-I image of West Africa at 0930
(Arrows indicate position of squall line)
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plate 2(c): The 'Fortune v squall of 5 September 1974. Visible
SMS-I image of West Africa at 1100
(Arrows indicate position of squall line)
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Plate 2(d): The 'Fortune' squall of 5 September 1974. Visible
SMS-I image of West Africa at 1230
(Arrows indicate position of squall line)
)
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Plate 2(e): TIle 'Fortune I squall of 5 September 1974. Visible
SMS-I image of West Africa at 1400
J (Arrows indicate position of squall line).
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f..8
"Type I"
Fig. 7: Two types of arc lines in satellite imagery
(frem La trasse ,1.972)
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successive pictures. It is therefore, not surprising that satellite
imagery has not been very successful in identifying the formative
causes of the disturbances. We do not have empirical evidence for the
time lapse between initiation and full development of line squalls but
from the results of numerical experiments available so far, we suppose
that this time would be less than thirty minutes - the time interval
between successive SMS - I images.
Satellite investigations cannot provide all the information
necessary in the study of the origin, maturity and decay of line squalls.
They are supposed to complement observations from the ground. Data
obtained from ground stations on in-cloud parameters like vertical
wind speed, moisture content, temperature and pressure would have
been most reliable if they could be supplemented by reports obtained
from direct flights into storm centres. Such direct flights are,
however, hazardous. Consequently, studies on tropical disturbances
have concentrated on modelling. The prohibitive cost of the option
of observational investigations as a method of studying line squalls
also favours the alternative of modelling.
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CHAPTER 3
A REVIEW OF LINE SQUALL MODELS
Models of line squall that start from the complete set of
hydrodynamical equations fall into two main categories: mathematical
and numerical. There are other models that do not start from these
equations but from some assumptions on the nature of motions of severe
storms. Into this class falls models that have attempted to formulate
the trigger mechanism of line squalls. Firstly, mathematical models
shall be considered because of their relevance to this study.
3-1 Mathematical models
These models can be classified into two main divisions: the
simple linear gravity wave models which are mainly appropriate to
initial development of the storms and the non-linear steady state
models that are applicable to the mature stage.
3-1-1 Gravity wave models
Gravity wave models omit non-linear effects in order that the
equations of motion could be tractable. This linearity permits
separation of variables; the variables being small deviations from
steady state values. In the stable gravity wave model,the static
stability of the atmosphere is taken to be greater than zero. Such
positive values for static stability are normally assumed while
simulating convective processes in dry air. Results show that in dry
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air, wave-like perturbations propagate without any amplification. As
will be shown later, release of latent heat in moist thermal convection
modifies the value of static stability tllereby making it negative and
in such environments, wave-like perturbations amplify without any
propagation. Since this study is interested in the initial stages in
the development of line squalls, gravity wave models will be developed
later on.
3-1-2 Non-linear steady state model
An example of non-linear steady state models is the two-dimensional
convection in shear by Moncrieff and Green (1972). They did not
investigate the internal structure of convective storms; rather, their
Cartesian set of axes moved with the speed of the storm such that at
any instant, the motion is stationary relative to these axes. The
flow was assumed to be steady and adiabatic and the entropy of the
system ~rritten as a function of the streamfunction and height. From
this expression, a conservative quantity was derived which allowed the
flow field to be discussed without any integration of advection deri-
vatives.
In discussing the flow field, Moncrieff and Green derived a non-
dimensional number R which lacks any sirect physical interpretation
but is numerically related to the Richardson number Ri• This relation-
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ship is
where H = upper boundary of model (or cloud tops) and
Z* = steering level i.e. the height at which there is
no motion re~_ative to the storm.
With the aid of this reLatLon , the authors tried to predict the
speed of propagation of some 'travelling' storms. Results show that
the predictions are reliable if Ri is of the order of unity. The
reliability however waves for storms that have considerable motion
along the third dimension. This, as the authors pointed out, might
be due to the two-dimensional nature of the model.
3-2 Numerical models
The basic equations used in numerical modelling (as a matter of
fact, in mathematical modelling) are: the first law of thermodynamics,
the equation of state for ideal gases and the equations for the
conservation of mass, momentum and water substances. These equations
have been applied in simulating convective storms in two-dimensions
(Ogura, 1963; Takeda, 1971; Hane, 1973; Sch1esinger~ 1973; Mitchell
and Hovermale, 1977; Moncrieff, 1978) and three dimensions (Ogura and
Charney, 1962; Pastushkov, 1973; Steiner, 1973; Miller and Pearce,1974;
Moncrieff and Miller, 1976; Schlesinger 1978,1980).
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3-2-1 Two-dimensional models
These are simplifications of the dynamical description of the
flow within severe storms. Results from these simulations are
qualitatively representative of the structure and motion within storms.
QuantitativelYt however~ some results (e.g. Hanet 1973) do not agree
entirely with observations because the atmospheric phenomena being
simulated have considerable motions along the third dimension. The
numerical model of deep, moist convection in two dimensions by
Schlesinger (1973) is fairly representative of other two-dimensional
models.
The reliability of numerical simulations of atmospheric distur-
bances as far as predicting and understanding atmospheric phenomena
is concerned depends largely on the amount of physics specified in
the equations governing atmospheric motions and the boundary conditions
applied in solving the equations. Since nature is complex, we cannot
but have some idea1isation and assumptions in setting out the equations.
Some basic assumptions made in two-dimensional numerical models are:
(i) all y-derivatives are zero (i.e. parameters are invariant
along the horizontal direction perpendicular to the
direction of motion of storms)
(ii) the Corio1is force is neglected (most numerical experiments
make this assumption because the square of the time-scale
of the disturbances simulated is usually less than f-2),
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(iii) motions are ane1astic i.e. local time changes of air
density are neglected in the continuity equation (Ogura
and Charney (1962) have shown that this assumption
filters out waves of little meteorological significance
e.g. acoustic waves. Further, Ogura and Phillips (1962)
showed that this assumption along with that of Batchelor
(1953) that the distribution of pressure and density are
always close to the distribution of pressure and density
in an adiabatically-stratified atmosphere are necessary in
deriving the ane1astic equations),
(iv) non-inclusion of the solid phase of water,
(v) no friction on the earth's surface (i.e. no-drag condition
along the surface),
(vi) supersaturated water vapour condenses and water drops in
an unsaturated region evaporate instantenous1y until
saturation is rea1ised in the region and
(vii) both ascent and descent are moist-adiabatic in saturated
air.
Justification of some of these assumptions have been examined in
detail by Schlesinger (1972).
Generally, in two-dimensional numerical models, the equation
of continuity is written as:
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The equation, written in this form, is accurate to about 1 per cent
(Moncrieff and Green, 1972). This equation inp1ies the existence of
a streamfunction, ~, such that
u =
and w =
The streamfunction, the vorticity equation and the conservation
equations form a comparatively simple modelling system. Using this
system, a Poisson-type of equation is solved for ~. This equation
is then integrated rather than the primitive equations.
In the two-dimensional numerical model of Takeda (1971), this
integration was carried out to study the effect of vertical profile
of ambient wind on a precipitating convective cloud. Some earlier
attempts (Amason, Greenfield and Newburg, 1968; Liu and Orvi11e,l969)
had included ambient wind field, vertical shear and precipitation in
their models. Takeda improved on these works by taking cloud physical
process into account more realistically; although he did not include
the ice phase. He grouped precipitating convective clouds into three
types according to the way the clouds develop:
(L) convective clouds formed on both sides of an initial
cloud due to diverging downdraughts in an atmosphere
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of sufficient conditional instability and in the presence
of weak shear,
(ii) short-lived clouds that form when the direction of the
vertical wind shear is constant with height and the shear
is strong and
(iii) 'long-lasting' clouds that are formed when the vertical
wind shear changes direction at a certain level.
Takeda's work indicate that there is a critical range of height
(about 2.5 km) within which the shear must change direction for the
formation of 'long-lasting' (or 'travelling') clouds. An important
deduction from this experiment is that two air masses, oppositely
directed and of different stabilities (e.g. one moist and the other
dry) are necessary atmospheric conditions before 'travelling' clouds
could develop. In particular, models of line squalls should be
formulated using two-layer atmospheric models.
Hane's (1973) two-dimensional model was on the structure and
tendency for self-maintenance of line-squall thunderstorm. He showed
that there is a tendency towards convergence upon a common solution
when various initial perturbations are imposed on the undisturbed
atmosphere and that the time-lapse before convergence is a function
of the initial difference between perturbations. Hence, the problem
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r- -
of initial perturbation isp according to Hane, not very important.
Some of the parameters he calculated (e.g. vertical velocity and
cloud size)t like those of most two-dimensional numerical models,
are larger than the observed values. This is probably due to the
non-inclusion of motions along the third dimension. For instance,
the downward motion, as calculated by Hane, is more intense than it would.
have been in nature where some flow would have taken place out of the
plane of his model. He concluded that the capacity of the dry midd1e-
level air to evaporate rain and to be cooled, along with the capacity
of the moist low-level air to condense and to be warmed, ultimately
control the line squall thunderstorm circulations. This study agrees
with his conclusion.
Schlesinger (1973) investigated the influence of ambient conditions
on the behaviour of deep moist convection in the atmosphere. His
results revealed that the intensity of convection increased with low-
level moisture supply and decreased with increasing mid-tropospheric
wind shear. In cumulus (or cumulonimbus) convection, the intensity
of convection usually decreases with increasing wind shear because
the shear inhibits the formation of new bubbles at the top of the
cloud. In the study of line squalls, however, vertical wind shear
is very vital to the development and sustenance of the squalls.
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The inability of two-dimensional models to correctly predict
the speed of propagation of severe storms that have considerable
motion along the third dimension, the fact that in-cloud parameters
simulated from these models are usually larger than observed values
and the reality that line squalls are three-dimensional in nature
are some of the reasons that, at various times, prompted the exten-
sion of numerical experiments into three dimensions. The initial
problem in three-dimensional numerical simulations was how to extend
the vorticity system to three dimensions. This problem was surmounted
by modelling with the primitive equations.
3-2-2 Three-dimensional models
Ogura and Charney (1962) used the primitive equations in their
three-dimensional simulation of thermal convection. The authors
did not explicitly treat the individual clouds and their internal
circulations; rather, they dealt with the general problem of thermal
convection on a large scale. Their model was designed to study the
dynamical behaviour of all forms of thermal convection. From a fre-
quency equation, which was derived from the primitive equations, they
showed that wave-like disturbances generated in dry air propagate
without any noticeable amplification whereas disturbances in moist
air amplify with time. In chapter 5, their work is extended to describe
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: ,.-
waves generated along the surface of discountinuity between dry and
moist air masses.
The numerical simulation of cumulonimbus clouds by Miller and
Pearce (1974) was also in three dimensions. Their formulations were
in pressure coordinates. An advantage of the use of pressure coordinates
is the availability of data at pressure levels. Also, according to
Miller and Pearce, the use of pressure coordinates could be prompted
by the neatness of handling the air density in the mamentum and con-
tinuity equations. Although the continuity equation appears neat in
the pressure coordinates, it should be mentioned that this is the exact
form of the equation. Since Ogura and Charney proved that the eli:lli-
nation of the local time changes of a t.r density in t lie continuity
equation filtexs out acoustic waves, it implies that waves of no mete-
orological significance would be implicit in pressure-coordinate
formulations. Miller (1974) showed how such waves could be suppressed.
The model of Miller and Pearce responded sensitively to cloud
physical processes in spite of the fact that their parameterization
of liquid water was not too refined. It is therefore essential for
numerical experiments to take into account, in a more realistic manner,
as done by Takeda (1971), the size distribution of water drops, rain-
fall intensity, evaporation, congulation and terminal fall velocity
of water drops.
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A major problem in numerical experimentation is the specifica-
tion of boundary conditions (B.C.). Often, the vertical components
of velocity are taken to be zero at the surface and the top of con-
vective elements; the top being usually near the tropopause. The
most satisfactory way of choosing lateral boundaries would have been
the specification of a large distance from the storm centre. where the
disturbance of the velocity field would be zero throughout the period
of convective activity (Miller and Pearce, 1974). Current computer
facilities would not. however, allow this. Thus, various B.C.'s have
been assumed for the lateral boundaries. Details of some forms of
these B.C. 's could be obtained from Takeda (1970,1971), Hane (1973)
and Miller and Pearce (1974).
3-3 Other theoretical models
Apart from mathematical and numerical models of thermal convection,
there are other theoretical models. Two of these are: the plume(or jet)
model and the bubble-theory model.
3-3-1 Plume model
In this model, the cloud is assumed to behave like a steady
turbulent plume which entrains the air of the environment (Ogura,l963).
This model cannot be easily applied to natural convection because there
is, usually, no well-defined source-region in nature. Another hindrance
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is that the theory disregards any development in time of the depth
or width of the cloud. Mixing at the top of the cloud is also neglected
in the plume model.
3-3-2 Bubble-theory model
This model assumes that convection consists of separated volumes
of bouyant fluid rising from an ambient fluid (Scorer and Ludlam, 1953),
The most primitive form of this model - the classical 'parcel model'
- does not allow heat and mass transfer through the boundary of rising
ftolumns of air (i.e. no entrainment) and it ignores pressure distur-
bance (Ogura, 1963). The model's assumption of uniform properties
on the horizontal plane is definitely an over-simplification in the
light of more-recent numerical experiments.
3-4 Syntheses of line squalls
In the course of synthesising the varous stages in the evolution
of line squalls, the trigger mechanism of line squalls have been
modelled; albeit with some inadequacies. Generally, these models do
not start from the complete set of the hydrodynamical equations of
motion. Examples of these models are those of Tepper (1950) and
LeRoux (1976).
Tepper (1950) proposed a 'pressure jump line' mechanism for line
squalls. He hypothesised that the initial impetus to the formation
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of line squalls is the acceleration of the cold front in a border
between warm and cold air masses wherein an inversion or an isothermal
layer exists in the warm zone. As a consequence of the acceleration,
a pressure jump is formed (Fig. 8) along the surface of the inversion
or isothermal layer. The air above the inversion is vigorously forced
up as the pressure jump passes and, depending on the stability of
this air, condensation' and precipitation of moisture could occur.
The air below the inversion is also forced upwards and the precipita-
tion falling into it may, depending also on the stability of this air
mass, reach the ground or evaporate completely.
After inducing the pressure jump, the cold front decelerates
and the jump moves ahead of it. The deceleration causes the forma-
tion of a rarefaction wave lo1hich,according to Tepper, moves faster
than the jump and eventually overtakes and destroys it. The destruction
of the jump would apparently mark the termination of line squall. The
total life span of squalls initiated by the Pressure Jump Line mechanism
is thus a function of: the intensity of the initial acceleration of the
cold front, the duration of the acceleration and the nature of the
deceleration which follows the acceleration (Tepper, 1950). Such
squalls could also be terminated whenever the inversion along which
the pressure jump is travelling either descends to the ground or
disappears completely.
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-- .--
'/~7717711111 7777777777777777777
(a) (d)
7777777777777777777
(b) (e)
--------------------~
777777777 777777777777777777/
(e) (f)
Fig . s . Idealized history of a line squall
(0) Initial posit ion of the cold front and the inve rs ion in the warm
s ector
(b) The cold front ccceler ctes inducing a pr-essure jump on the inversion
(c) ThE' cold front decetor ctes while t he jump moves out che cd with the
rore toc tion wove following
(d),(e)&(f) Successive positions of pressure jump being ov er tcke n by the
rarefaction wav e
(from Tepper J 1950)
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Tepper's hypothesis was based on the works of Prieswerk (1940)
and Freeman (1948) which dealt with the theory of a gas flow in gas
dynamics. He (Tepper) adapted the theory to describe the flow of
atmospheric disturbances alopg"an inversion in response to gravity.
In the original theory, Freeman discussed the motion of a disturbance
induced by the acceleration of a vertical-plane piston and propagated
along an inversion. The level of the inversion would rise as a bump
in its immediate surrounding and the wave produced would travel as a
gravitational wave with a speed greater than the wind speed. The
velocity of various points along the wave crest increases as a function
of height and the leading edge of the wave is alsornost vertical; thus,
creating a jump.
The data on which Tepper based his theory were read from the
original traces of pressure, wind, temperature and precipitation
taken at 55 automatic recording stations operated by the United States
(of America) Weather Bureau during the cloud seeding experiment of
the Cloud Physics Project (1948). One-minute synoptic maps were
constructed from the data. Some important features of the maps are:
1. Pressure gradients are very intense (the order of
magnitude is 1.15 rob km-1).
2. The leading edge of the pressure gradient undulates
violently and has an irregular motion.
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3. The pressure jump is independent of the other parameters
like wind shift, rain gush, wind speed maximum etc.
4. Other meteorological parameters, apart from pressure
maximum, have an intimate relation one to another
(e.g. these pare.meters are propagated with about the
same speed - 20 to 21 m/s).
5. The average speed of all meteorological parameters
exceeds the average maximum speed of the surface wind
6. The precipitation belt lies behind the pressure jump.
It has been observed (Mansfield, 1977), in agreement with one
of the conclusions of Tepper (number 5 above), that the speed of the
West African line squall and consequently that of all meteorological
parameters associated \"ith it, is the maximum within the troposphere.
There are, however, other features of the t-lestAfrican line squall that
cannot be easily explained by the Pressure Jump Line theory. Some
of these features are:
1. There is no conclusive evidence that Lnve.rs Lons exist
prior to the onset of line squalls. Our experience in
West Africa is that the layer of inversion observed
during the period of frequent line.squall activities
occurs around the coast rather than the regions of initia-
tion of line squalls (Fig. 2).
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2. The speed of propagation of line squalls, according to
the Pressure Jump theory, depends on the variations in
intensity and duration of the initial acceleration of
the cold front and hence, this speed must vary from one
( occurrence of line squall to the other. This is contrary
to the almost-uniform speed of propagation of West African
Line squalls which have been observed to be, approximately,
the speed of mid-tropospheric winds.
3. The observed overturning of the atmosphere after the
passage of line squalls.
4. The preference of highlands as places of origin of line
squalls.
Another area where the Pressure Jump Line theory seems inappro-
priate as far as West African Line squalls are concerned is the iden-
tification procedure. Tepper identified line squalls through the
showery precipitation that accompanies lines of storm which appear
in the warm sector of cyclones, almost parallel to the cold front,
along which there is intense convective activity. While this procedure
was good at the time of his study because of the little amount of work
done then on the dynamics and kinematics of line squalls, it is now
known that precipitation that accompanies line squall is very erratic.
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With the present-day knowledge (obtained from numerical experiments
and observations) precipitation, being irregular, might not have any
meaningful relationship with other meteorological parameters during
line squall activities. Irregularities in the amount of rainfall
accompanying line squalls make Tepper's identification procedure for
these storms inadequate as far as West Africa is concerned.
Some other attempts were made, after the Pressure Jump Line
theory, at explaining the trigger mechanism of line squalls. In one
of such attempts, LeRoux (1976) explained that the key element is
a surface of discontinuity, between two air masses, upon which a
descending cold-core wind jet impinges; although he did not advance
a viable mechanism for the descent of the jet. A series of events
would then follow the impingement and these will lead to the formation
of line squalls. Danielsen (1974) had earlier ana1ysed cases in which
a descending jet core triggers dust storms.
LeRoux observed no relationship between the intensity and speed
of propagation of line squalls and thereby concluded that line squalls
are energised solely by the easterly jet. In effect, the line squalls
proposed by his mechanism will not be self-perpetuating (Fortune, 1977).
This is contrary to the observations of Hamilton and Archbold (1945),
Browning and Ludlam (1962), Fortune (1977), Mansfield (1977), Zipser (1977)
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and Bolton (1981). It is now known that fully-developed tropical line
squall propagates in a way that suggests an in-built regenerative
mechanism that sustains motion.
Our explanation, in the next chapter,of the trigger mechanism
of 'line squalls will be a modified form of the proposal of LeRoux
(1976). In our discussion, we hope to shed some light on those areas
where the Pressure Jump Line mechanism and the explanation of LeRoux on
the trigger mechanism of line squalls seem to be inadequate in formu-
lating the initial stages in the evolution of West African line squalls.
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CHAPTER 4
EVOLUTION OF LINE SQUALL
4-1 Necessary atmospheric features in the development of line squalls
As stated eariier, two air masses of different stabilities are
essential before line squalls could develop. One of these ,air
masses - the lower one - must be convectively unstable (Hamilton and
Archbold, 1945; Browning' and Ludlam, 1962; Hane, 1973). Some other
features that are essential in the organisation of line squalls are:
cold front zones (i.e. borders between cold and warm air masses)
(Prohaska, 1905; Tepper, 1950; Takeda, 1971), a moisture profile that
drops betwean 600 mb. and 500 mb. (Hane, 1973), a mid-level tropo-
spheric jet maximum (Mansfield, 1977) and strong vertical shear
(Browning and Ludlam, 1962).
Strong vertical shear in the ambient winds aids severe storms
because it separates the dowodraught from the updraught and allows
the former to thrust under the bouyant columns (Browning, 1964).
The development of new cells on the downshear side of a storm is also
enhanced by a strong vertical shear (Fortune, 1977). Figure 9 shows
the vertical profile of wind over West Africa between the months of
April and September. From this figure, it is observed that strong
vertical shear is created in the lower troposphere of West Africa
by the 650 mb. mid-tropospheric jet. Thus, throughout the period
referred to as the wet season, strong vertical shear, an essential
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<,
70
200 mb
650 mb
850 mb
1000 mb
- 30 -5 o 1 -+ velocity m/s
Fig.9: A sketch of the vertical profile of wind
over West Africa between April and September
(Deduced from soundings made at the meteorological
station attached to the airport at Ibadan-
7° 26' N 3° 54 E)
"
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atmospheric feature in the development of line squalls, exists in
West Africa.
Another feature that enhances the development of line squalls
is a drop in moisture profile between 600 mb. and 500 mb. (Hane,1973).
This is likely to be so in West Africa because some mixing might
occur, below the 650 mb. jet (Fig. 9), between the dry north-easterlies
and the humid mansoon winds since the surface of discontinuity is
not a sharp demarcation. This mixing tends to make the layer of air
above the 650 mb. jet to be drier than the layer below it and con-
sequently, there is a drop in moisture profile between 700 mb. and
500~.
In West Africa, an example of a convectively-unstable air mass
is the monsoon wind. If a parcel of the monsoon south-westerlies is
sufficiently lifted, condensation occurs and the latent heat released
makes the parcel warmer than its environment. This results in the
bouyancy of such parcels of air. The south-westerlies are therefore
convectively unstable. Convective instability is, as mentioned
earlier, important in the development of line squalls.
The conclusion from the foregoing is that the basic atmospheric
features that aid the development of line squalls are present in
West Africa during the period of frequent squall activities. However,
the existence of these features alone cannot give rise to line squalls.
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Other factors play vital roles. A probable factor is the African
easterly waves.
Burpee (1972) traced the origin of the easterly waves to an
area between Khartoum (3ZoE) in Sudan and Ft. Lamy (now N'Djamena)
(15oE) in Chad. The waves propagate westwards across West Africa
from June to early October (Burpee, 1972) with a period of about
3.5 days and wavelength of about ZOOO km (Carlson, 1969 a,b). These
waves are sometimes associated with the development of line squalls.
The probable link, from our studies, is the evidence of Fortune (1980)
that a necessary condition for the development of line squalls is a
synoptic pattern that generates convergence and the observation of
Reed, Norquist and Recker (1977) that there is a convergence ahead
of the African easterly wave trough. The deduction from these is
that line squalls could be initiated in the viscinity of the easterly
wave trough. This idea is still not a consensus because of some clear
distinctions between the line squall phenomenon and easterly waves.
One major distinction between line squalls and easterly waves
is the speed of propagation of the two phenomena. Line squalls
propagate at about 15-20 mls (Hamilton and Archbold, 1945 ;
Aspliden et.al., 1976; Fortune, 1977; Mansfield, 1977; Zipser,1977;
Bolton, 1981) while easterly waves propagate at about half this value
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(Carlson, 1969 a,b; Burpee, 1972). As a further distinction between
line squalls and African easterLy waves, Carlson (1971) found that
the waves have a warm core at high levels and a cold core at middle
and low levels '''ithno clear pattern of cloudiness or vertical
motion whereas line squalls are cold-core in Low levels (due to the
downdraught) and they possess distinctive cloud and vertical motion
patterns.
In the remaining sections of this chapter, we shall endeavour
to show the important role(s) played by the observed atmospheric
features that have been enumerated above, in the development and
maturity of line squalls.
4-2 Initial stages and trigger mechanism of line squall
\vave-lik.edisturbances are known to be generated along surfaces
of discontinuity between air masses; the discontinuity between
the south-westerlies and the north-easterlies in West Africa is
not an exception. It will be shown in the next chapter that, depending
on the stabilities of the air masses on either side of the disconti-
nuity, such wave-like disturbances could intensify in vertical velocity,
pressure, temperature etc. as well as propagate through the system
in which they are generated. TIlis 'conflict' between the air masses
was long ago recognised as a cause of violent we athe'r (Hubert, 1926).
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In this study, the initiation of line squalls is associated with these
perturbations.
The five sections of figure 10 show the East-West section
~
through the lower troposphere in West Africa. Figure10 a represents
the undisturbed atmospheric structure of the sub-region wherein
the north-easterlies are on top of, but oppositely directed to,
the south-westerlies. The boundary between the air masses is at a
pressure level of about 850 mb. (Fig. 9). Figure lOb shows the
onset of a wave-like disturbance along the surface of discontinuity.
The disturbance could intensify as shown in figure 10c and this
intensification could be large enough to cause a blockage of the
650 mb. mid-tropospheric jet.
On encountering the surface of discontinuity, the jet further
distorts the surface and in the process, forces the monsoon south-
westerlies to ascend more (Fig. 10d). Due to the forced convection,
a rising parcel of the convectively-unstable south-westerlies could
condense. Precipitations from the rising parcel of air could fall into
the jet that now underlies the monsoon winds. Some of these preci-
pitates evaporate in the dry jet; the latent heat of evaporation
being supplied by the jet. This evaporative loss of heat cools the jet
and it sinks.
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75
a
b
c
d
e
Fig,10:The process of formation of line squalls
(from Le Roux I 1976}
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It is possible for the jet, while sinking, to hit the surface
of the earth; especially if the whole process occurs over a high
terrain. This relative ease with which the jet touches the ground
over highlands explains the preference of such elevated areas like
the Jos Plateau of Nigeria as the source region of line squalls.
The active phase in the evolution of line squalls starts as soon
as the jet hits the surfaee of the earth (Fig. IOe). The jet goes
beneath the monsoon winds and lifts them up. As a result of this
lifting, south-westerlies rise and develop into large clouds. Clouds
as large as the cumulonimbus may form because the depth of the
south-westerlies over West Africa (about 150Om) throughout the wet
season is large enough to sustain such clouds. This explains the
reason why line squalls propagate as a series of cumulonimbus cells.
The horizontal extent of propagation depends on the extent of travel
of the jet over the surface as well as the condition within the
atmosphere ahead of the storm.
4-3 Fully-developed line squall
The mechanism of propagation of fully-developed line squall is
self-sustaining. Moist low-level air is lifted by the mid-tropospheric
jet as the latter moves over the surface of the earth. The rising,
moist air condenses and the precipitates fall into the underlying jet.
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Since the jet is dry, some of the rain evaporate in it. The evapo-
rative loss of heat coools the jet and it sinks; thereby continuing
its motion over the surface and consequently, lifting the moist air
ahead. Hane (1973) presented a similar mechanism for propagating
severe storms. The sustenance of the storm, however, requires an
atmosphere ahead of the storm that is convectively-unstable, sheared
in vertical wind profile and lacking other storms (Dhonneur, 1970).
In the case of an environment that had experienced local convective
storms before the arrival of the squall, precipitation is usually
very low by the time the squall arrives. Thus, it is not uncommon
for squalls to dissipate completely in areas where severe convective
storms had previously occurred.
Vertical wind shear, another vital factor in the continuous
sustenance of line squalls, is responsible for the structure of
inflow ahead and outflow to the rear that is usually observed in
West African line squalls. This structure is an overturning of the
atmosphere because high winds aloft are brought to the surface and
the lower winds transported upwards. This overturning stabi1ises
the atmosphere after the passage of squalls (Hamilton ann Archbold,
1945; Zipser, 1977; Bolton, 1981). The high wind is brought to the
surface through the descent of the 650 mb. mid-tropospheric jet.
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Also, sustenance of line squalls depend on the continuous
deecent of the 650 mho jet. As mentioned earlier, this descent is
enhanced by the evaporative loss of heat by the dry jet as a result
of the precipitates falling, and evaporating, in it. Precipitation
is an integral part of convective instability. Hence, convective
instability in the atmosphere ahead of the storm is essential for
the sustenance of line squalls.
The descent of the jet creates a region of strong convergence
which is often called the squall front. According to Mitchell and
Hovermale (1977), this front is the gusty wind surge that leads the
surface outflow from an intense convective storm downdraught.
Strong convergence and the fact that the descending jet is colder
than the monsoon winds aid the ascent of the latter.
The rate at which convective elements develop ahead of
'travelling' storms depends on how fast the squall front travels on
the surface. The squall front is, however, the leading edge of the
mid-tropospheric jet as the jet crawls over the surface. Hence, the
close association between the speeds of propagation of line squalls
and that of the mid-tropospheric jet. As observed (Aspliden et al.,
1976; Mansfield, 1977), line squalls propagate at speeds of about
1 mls over and above the speed of the mid-tropospheric jet. This
slight increase in speed might be due to the faster rate at which the
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jet is sinking as a result of a combination of its own weight and
negative bouyancy produced by evaporation.
On the average, West African line squalls propagate in a
direction WSW. This is so because the monsoon winds that p,ive
rise to the development of cumulonimbus clouds come from a direction
that is more WSW than SW and the north-easterlies are from a
direction that is more ENE than NE. That is, the jet, whose
leading edge is the squall front, comes from a direction that is
approximately ENE.
Immediately fol.l.owfng the squall front in a 'travelling' storm
is the core of precipitation (Fig. 11). The intensity of rainfall
decreases as one moves further away from the front because the
rising columns of monsoon winds would have lost most of their
moisture content and thereafter, merge with the anvil. However,
Zipser (1977) observed heavy rain in the region 30-160 km behind
the squall front (Fig. 11). This rain is from the anvil base.
The anvil is usually thick and extensive. This thickness is
of the order of 8 to 10 km (Zipser, 1977) in the region 30 - 100 km
behind the squall. Whereas the anvil could extend to about 300 km
behind the storm, the highest could tops is about 15 km (Fig. 11).
This large horizontal extent of the squall system with respect to
the vertical extent justifies the use of the hydrostatic relation
in mathematical modelling of line squalls.
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80
Tops 200mb or higher
300 Ambient
Upper trop => 300
air ~
~
400
1.00
500
600
600
700
700
800
800
Ambient ~d la
900
900
Squall front 100km
Fig.11: Schematic cross section through a class of squall system. All flow is
relative to the squall line which is moving from right to left.
Circled numbers are typical values of Etw In ll(
(from Zipser, 1977)
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The extent and persistence of line squall systems suggest
the possibility of organised ascents in the upper half of the tropo-
sphere. These ascents would be very vital to the sustenance of the
storm. But ascents would normally give rise to downdraughts.
t-fansfield(1977), Miller and Betts (1977), Zipser (1977) and
Bolton (1981) have all observed two scales of down draughts in
line squall systems.
According to Zipser (1977) one of these downd raught s could be
located within the lowest 200-400 m behind the squall front. Its
origin may be the ambient cloud-layer air (900-800 mb).lifted into
the cumulonimbus and sinking after becoming negatively bouyant or
from higher up in the cumulonimbus where downdraughts ncrmally
originate from water loading and entrainment of air with low wet-
bulb potential temperature, 6w , (Zipser, 1977).
The other scale of downdraughts has lower ew . Zipser (1977)
postulated that their origin must therefore be from areas of low
ew in the ambient winds. This source must be typically above 750 mb.
These downdraughts approach the line squall from the front and the
rear (Fig. 11). While overtaking from the rear is more easily
understood because of desceuding 650 mb. jet, the entry from the
front is more difficult to visualise. However, the picture becomes
clearer if it is realised that along the squall front, cumulobimbus
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clouds are not continuous in distance ar in time.
Fujita (1963) noticed mesoscale high pressure regions
immediately following line squall passage as well as the occasional
presence of a mesoscale low-pressure area some distance behind line
squalls. Although this observations was over the temperate zone,
Zipser (1977) has also observed mesohighs and mesolows within the
tr6pics. A large depth of cold saturated air combine with large
liquid water content (Sanders and Emanuel, 1977) to cause a rise
in pressure in regions immediately following line squall passage.
Mesolows exist at a distance of about 100 km behind the squall front.
Data compiled by Aspliden et al. (1976)(Fig. 4) show that line
squalls scarcely propagate beyond the coasts of West Africa. Hence,
it would be right to conclude that line squalls decay around the
coastal areas. This decay might be as a result of extensive convective
activities around the coasts such that squalls propagating into these
areas die out. l\nother reason for the decay might be the fact that
the 650 mb. jet now impinges on the ocean surface rather than the
solid earth on which it could spread out and form squall fronts
which force the south-westerlies to rise. The layer of inversion
(Fig. 2) observed around the coasts could also contribute to the
decay of line squalls by inhibiting convective processes. This layer
acts like a 'lid' that prevents upward growth of clouds(Balogun, 1974).
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This 'lid' is never taken off unless the force of convection is
enought to destabilise the inversion layer. In the absence of
precipitation from convective elements, line squalls would dissipate
completely.
Attention can now be focused on a gravity wave model for the
initial stages in the development of line squalls.
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CHAPTER 5
FUNDAMENTAL EQUATIONS
5-1 Basic and perturbation equations
The basic equations in the theory of atmospheric circulations
are given below in notational forms. All symbols have their usual
meaning. However, for the sake of clarity, the meanings of these
notations and symbols have been given.
au + 2
at \7(~)2 + fAu + \p7P+ g = v\72u ·.... 5.1
...Q.. (Lr +
Dt C e) = 0 ·.... 5.2
P
Dp +
Dt p\7u = 0 ·.... 5.3
DDt (r+q) = 0 5.4
Te = (p rPo ·.... 5.5
5.1 is the momentum equation with viscosity term,.
5.2 is the energy equation and 5.3 is the equation for continuity of
mass. 5.4 represents the conservation of water substance and 5.5 is
the definition of potential temperature.
The Corio lis parameter, f, in the momentum equation could be
neglected because the square of the time scale of line squalls is
much less than -2f • All molecular effects (e.g. viscosity) are
also neglected because the scale of motion is much more than molecular
dimenSions.
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If DDtr = Q, 5.1 - 5.5,
after sinp1ification, become:
· . • •. 5.6
• • • •• 5. 7
aw + aw + dW + waa\z01+ 1. ap + = 0at uax Vay P dZ g • ••.•. 5.8
• . • •• 5.9
· . . .. 5.10
19..+Js+~+~+Q= 0
at dX dy dZ · . . .. 5.11
••••• 5.12
The set of hydrodynamica1 equations 5.6 - 5.12 consists of non-
linear equations and, hence, they are difficult to solve. The
perturbation method reduces them to a linear form.
The basic assumption of the perturbation method is that
atmospheric motions are made up of small perturbations superimposed
on a basic atmospheric state. Fundamentally, the basic motion as
well as the perturbations must separately and jointly satisfy the
hydrodynamical equations. Thus, the atmospheric variables in
5.6 - S.12'could be expressed as linear combinations of their basic
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;>-
and perturbation values as:
u = u+s u'
v = v+s v'
w = w
e = es+ e' • . . •. 5.13
q = qs+ q'
p = ps+ p'
where subscript's' denotes the basic values and the primed
variables are the perturbation quantities. The set of equations
5.6 to 5.12 could be converted into their equivalents in the
pressure co-ordinates. As stated earlier, an advantage of the
pressure co-ordinates is the availability of more data (on
atmospheric parameters) along isobaric surfaces. Further, the
frequency equation being sought appears neater in the pressure co-
ordinates. (The equivalence of the frequency equation in z-coordinates
have been documented by many authors e.g. Giwa (1965)). The set
of values in 5.13 could be incorporated, along with the co-ordinate
conversion, into the hydrodynamical equations to give, after lineari-
sation:
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;:.-
au' au' dUS 1 p'
at + us -a+x wap+- P = 0s ax
av' + u -d-V+' 1 ap'at s dX psay
= 0
__ l_(dw_+ Us aw)
+1:. ap' + R
Ps3 a ax Ps ap p
T' = 0
s . • .. 5.14
au' + av'+
3y aw = 0dX dP
-a+e' u -ae'
ae
-A s
ax's a+x OJdP
- +LQ
c = 0p
T' = Pe,(~)K
Po
5.14 is a set of perturbation equations which could be used to derive
a frequency equations.
5-2 Frequency equation
The perturbations superimposed on the hydrodynamical equations
constitute the disturbances in the atmosphere. Since atmospheric
disturbances are wave-like in nature, these perturbations could
be expressed in the form:
= i(o t + (Xx + By)u' u(p) e 0
= i(o t + (Xx + By)v' v(p)e 0
etc.
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,,t- ::.~
S'Uch that
-a-u='at ia u'0
au'
ax = iau'
and au'ay = iBu'
Le. a
at = ia0
a = ia •.•.•• a· 5.15ax
a
ay = i8
If it is assumed that density does not vary along the horizontal
(i.e. x direction) in the mean atmosphere,
and so, if G' =
=-aG'ax
and 1 ap' = aaGy'p ay
s
SiRce, as mentioned in chapter 4, the horizontal extent of the line
squall phenomenon is much more than its vertical extent, the hydrostatic
relation could be used in this mathematical formulation.
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In effect, the vertical acceleration terms will be neglected in
the set of equations in 5.14. If 5.15 is incorporated into 5.14 and
tracer element, ~, placed on the latent heat term, the following
set of equations is obtained:
ia*
au
u' + wap+s iaG' = 0 ·... 5.16
iav' + i8G' ::: 0 ·... 5.17
aG' +-RT'ap Ps = 0 •••••5.18
iau' + i8v' + aawp = 0 ·... 5.19
iaS' + ass LQwap+cll ::: 0 ·... 5.20
p
P
T' = e' (-p!!·l ..... 5.21
0
Eliminating u', Vi, 6', T' and G' from the set of equations
5.16 - 5.21, the result is:
2'-
2 ~l+ _R_Ts[.~LgP s ~a_L_~]\1w ar 2a a 2 p2 = 0 •••• 5.22p g R6sCs p
ap Cp dZ
R2T ~LPs ar) aTIf r = -~2L(lC --R6- ap ~, ••.•. 5.23g p s
5.22 becomes
2 at (a2~+ + 82
ap2 p2 )w =2 0 .... 5.24s a
* a = a + u a0 s
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5.24 is the frequency equation being sought and its solution will
be discussed in the next chapter.
5-3 Effects of atmospheric stability on the frequency equation
The measure of atmospheric stability in 5.24 is r and as
defined in 5.23
2
r R T J..lLP dr dT= __ s [-.-8.(1 s
2 C )- ~ dP +~P s dZg
If Y represents the environmental lapse rate and rd denotes the
dry adiabatic lapse rate t
J..lLP '\
r = ___ s ~) _ yJ 5.25R9s dP
In a dry atmosphere, the latent heat term in 5.25 will be non-
existent because there would not be any condensation
i.e. u = 0
Thus, 2
R
rd = --TT f rd-y]
g
where the subscript '2' is used to distinguish the atmospheric
stability within such a dry environment. In the context of West
Africa, 5.26 would represent the stability condition within the
north-easterlies.
In a moist environment, J..l= 1 because of the possibility of
condensation during a convective process. This possibility would
then justify the retention of the latent heat term. The term
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-- - 91 -LP
[ 1 __ 8 ~fd Re op
s
in 5.25 would then represent a modification of the dry adiabatic
lapse rate due to precipitation. If a layer of saturated air is
assumed, this term will be the wet adiabatic lapse rate, fw •
LP
i.e. rw = fd[ 1 - R6s aor?l
s
Since the south-westerlies are moist, the stability of this air
mass could be written as:
• . • . • • .. 5.27
The south-westerlies have been shown in section 4-1 to be
convectively unstable and as such, the environmental lapse rate,
y, within this air mass lies between r w and fd•
This inequality implies that fl (5.27) will be negative. If such
a negative value of r is used in the solution of 5.24, the condi-
tion for obtaining a sinusoidal function of time for any perturba-
tion will be
o2 < O.
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This implies that the frequency of the waves generated is purely
imaginary and such waves are known to amplify, without any propaga-
tion, with time.
In practice, an environmental lapse rate that is greater than
the dry adiabatic lapse rate is a sign of instability. Such an
environment cannot be maintained for along period because mechanical
mixing would occur and thereby stabilise the atmosphere. This means
that the environmental lapse rate, even in dry air, is scarcely
larger than rd and if it does, it is for a short period. Therefore,
r2 (5.26), which is a measure of the stability of the north-easterlies,
will be positive. If such a positive value of r is used in the
solution of 5.24, the condition for obtaining a sinusoidal function
of time would be
The implication of this is that the frequency ~f the waves generated
will be real. Such waves propagate~ without any amplification, with
time.
Given a situation, therefore, where there are two air masses,
whose separate stability conditions give complex and real values for
wave frequencies respectively, one on top of the other (e.g. the
north-easterlies on the south-westerlies) the waves generated
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along the surface of discontinuity would both propagate as well
as a amplify with time. The velocities of such waves would be
complex; the real part represents the phase velocity while the
imaginary part is a measure of the amplification of the wave.
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CHAPTER 6
SOLUTION TO THE FREQL~NCY EQUATION
Equation 5.24 could be solved by assuming a single-layer model
for the atmosphere. This kind of solution had been discussed by
Ogura and Charney (1962), Hiller (1974) and Bolton (1981). However,
since line squalls require two layers of air with different static
stabilities for their initiation ann sustenance, it would be more
realistic to solve equation 5.24 with the aid of a two-layer model
of the -atmosphere.
6-1 Boundary conditions
Figure 12 shows a section of the interface between the air
masses in a two-layer atmospheric model. If wand u are the
vertical and horizontal components of velocity respectively, the
velocity component perpendicular to the surface is
wcose - usine •
This value must be.continuous if no air parcels accumulate at the
interface. For small € (fig. 12),
aax£ = tane ~ eine.
Also, coae ~ I
Therefore, wcose - usine a£~Tjl-U- ax
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95
w
"k1----~u
I
I
I
I
I
I
I
I
I
I
€
I
Fig. 12 : .A section of the interface between the
arr masses In a two-layer atmospheric model
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w = DDet: (fig. 12)
i.e. w'" .... 6.1
= iooe: + a.use: (using 5.15)
i.e. w '"ioe: • • • • • • • • • •• 6.2
From 6.1, 3e: ae:w - uax '"at = ioo£
but w - ae:uax is continuous at the interface, hence
e: is continuous.
If e: is continuous. then (from 6.2)
iW(1 is continuous •
w = H W (discussions Hitlr:Gi~~ra1.,981)
P
where H is scale height.
Since w
o is continuous, then
aapW is continuous.
Infinite pressure gradient are never observed at the interface
between air masses. If ever they occur, it is only for a very
short time because air would quickly rush from a region of high
pressure to that of low pressure. So, pressure is necessarily
continuous at the interface.
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i.e. p(z+£) is continuous.
p(z+£) = p(z) + £~Clz
ClsP(z)= ps(z) + p'(z) + £ Clz
Ps (z) is a constant and the underlined term is a product of small
quantities which makes it negligible; therefore,
ClP(z )
p'(z) + £ ~z is continoous •
By definition, w = .QE.Dt
Using 5.15 and the hydrostatic relation,
w = Lop ' - Pgw
i.e. wia = p ' -~ ia . • . • .. 6.3
It has been proved that
ClPs(z)
p'(z) + c dZ is continuous. So. using 6.2,
dP (z )
p' (z) + 1~a- --Cl~:--=- p'(z) - i~ is continuous.
From 6.3, this means
w
a is continuous.
For'the scale of motion consLdered in this study, the surface of
the earth is usually assumed to be flat. Because of the flatness,
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the vertical component of velocity is taken to be zero at the
surface i.e. w = o.
To summarise, the following terms are continuous at the lllter-
face between the air masses:
W
a
and R W a dWa p 2 2 ap-g(a. +S )
At the surface, w = 0 i.e. R W aa p 2 2 -da=Wp 0 and at theg(a. +S )
top, W = 0 (this is a free surface condition).
6-2 Mathematical derivations
To simplify mathematical derivations in our two-layer atmospheric
model, parameters like density, temperature and horizontal velocity
that should normally vary with height are taken to be constant withic
each layer. Subscripts '0', 'B' and 'T' indicate the values of
these parameters at the surface, boundary between the air masses and
the top of the atmosphere respectively while subscripts 'I' and
'2' are used to identify atmospheric parameters within the lower
and top layers respectively.
We recall equation 5.24 i.e.
2 2 2
~+ =gf- (a +S )
ap2 p2 " W = 0 .
s a "-
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The solution to this differential equation is of the form
m m
w AP 1 + BP 2 . . .. 6.4
where A and B are constants and are the roots of
o .•6.5
From 6.5t let
J !- gr(a 2+82)ml = ~ + 2
(J
and
If A =
w = APl/2+A t Bpl/2-A (from 6.4).
At the boundary between the air massest it was shown that
H w
(J P
and w
(J are continuous.
At the surface, w = 0 and at the top w = O.
Applying these conditions to 5.24 with the following substitutions:
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Ql'= 2 Q2)
g(a + ~
and 2 2 two equations are obtained:
g(a +8 )
1.2
I 1 P 21.2 P
-{[ HB-Q2 (A 2+ -2)] -l HB-Q2(- - A )] (-)
T }- B2 2 P 02 ·Az = 0 ••. 6.6B
. •. 6.7
6.6 and 6.7 constitute a set of simultaneous equations in
Al and A2• For non-trivial solutions, the determinant of the
coefficients must vanish.
If
PT 21.2and R2 = (p-) ,the value of this determinant is
B
2
°2
- -2- = 0 . • •• 6.8
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Recalling from chapter 5,
al = a + aU0 l
°2 = a +0 au2•
Substituting for al and a2 in 6.8,
"0 1 + Rl 1 + R2{2aul[2:>"(11 _ R ) + 1]- 2au2[ 2:>"(21 ~.R ) + 1) }+l 2
2 2 1+R 2 2 1+R
{a "i [2>'1(1-R 11) + 1] - a "z
2
[2k2 (l-R} + 1]} = o • • •• 6.9
At a glance, 6.9 looks like an ordinary quadratic equation in
ao but a closer observation reveals that the coefficients are
not constants. This means that no analytic solution exists for
6.9. Thus, the equation reduces to an eigenvalue problem to
determine ao such that there are non-zero solutions.
Some of the eigenvalues obtained in the solution of 6.9 will
be complex and as said earlier, the imaginary part of such complex
values would represent the amplification of any wave-like pertur-
bation. The modes with the largest amplifications (i.e. the
largest imaginary parts) are those that are proposed as being
responsible for the onset of line squalls. These modes are those
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that will likely awplify up to the extent of blocking the 650 mh.
mid-tropospheric jet as explained in the proposal for the initiation
of line squalls (section 4-2). A single-layer model could be
considered as a special case of equation 6.9.
6-3 Special .case
In a one-layer model, ul = u2 = u (say) and ~l:::":2 = " (say).
If these are incorporated into 6.9,
•.. 6.10
The conditions for 6.10 to be true are:
(i) = o
a
Le. oa- = +u •
This represents the velocity of the ambient wind
without any perturbation whatsoever.
(ii) .\= O.
This imlies a2 = 4gra2
For any single layer of air, this condition represents
a phase velocity of
a
::::
a u ± 2/gf
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where 21gf 'represents' the perturbation on the
ambient wind u. As considered in section 5-3, if r
is positive, the perturbation propagates, without any
amplification, through the ambient winds. On the other
hand, if r is negative, the perturbation amplifies
with time without any propagation. These conditions
depict wave-like perturbations in dry and moist air
respectively.
(Hi) = o.
Substituting for R1 and
2HRO2 '
(P.-!')2A 1 + 2A -0::: ...... 6.11
Ps 1 - 2A - 2HO"T
Equation 6.11 had been obtained by Mi11ier (1974) while solving
a frequency equation for the speed of propagation of external
gravity waves. These are 'free-surface' waves which occur in one-
layer models through the use of particular boundary conditions.
Simplified cases of these waves in hydrostatic, neutral and non-
hydrostatic, neutral systems have been discussed by }til1er (1974).
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CHAPTER 7
SUffi<UffiYk~D CONCLUSION
In this study, the importance of line squalls and the reasons
for studying the phenomenon have been stressed. In order to under-
stand the various stages in the development of line squalls, the
West African synoptic weather pattern was revfewed in chapter 2. In
the review, it was noted that line squalls are most-frequently
observed wheneve r an area experiences the weather Zone C of Hamilton
and Archbold (1945). Various methods that had been used to study
these squalls (i.e. observational investigations, satellite
investigations and numerical simulations) were mentioned and the
problems that still exist highlighted; the main problem being the
isolation of the formative cause of line squalls.
Although observational investigations of West Africa distur-
bances were not very successful in isolating the trigger mechanism
of line squalls, they offer good descriptions of line squalls such
that the phenomenon can now be easily identified. For instance,
Hamilton and Archbold (1945) and Bolton (1981) described how D.L
manifests itself to a surface observer.
As recorded by observational investigations, a cross-section
through a D.L. along its direction of propagation shows that the
updraught slants over the downdraught in a way that permits the
precipitation from the former to fall into and possibly evaporate in
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, -
the latter. This mechanism permits the persistence of line squalls.
The main shortcoming of observational investigations is that they
lack any synthesis of the various stages in the development of line
squalls.
Satellite investigations of tropical storms provided a better
alternative to observational investigations as far as the synthesis of
various experimental analyses are concerned. This is so because
sequential analysis of successive satellite photographs could show
the various stages in the development of line squalls. On such
satellite photographs, the following criteria could be used to
identify D.L. 's:
1. a D.L. should consist of large cumulonimbus clouds
with tops to about 15 km;
2. such bright cloud clusters should attain a size of at
least 2°x 20 during its life-time and
3. a D.L. should be active for a minimum of 6h.
Line squalls identified this way fall into two categories according
to the pattern of satellite pictures: type I squalls have a well-
defined arc-shaped front that faces the direction of propagation
while type II squall fronts appear like indented arcs. The amount
of information that could be extracted from a single satellite photograph
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is so much that this form of investigation is probably the cheapest
means of studying tropical disturbances.
Limited access to satellite data along with the hazards, and cost,
of observational investigations were mentioned as some of the reasons
which favour the option of modelling in the study of tropical disturb~~ces.
There are two main categories of models: numerical and mathematical.
Mathematical models could be further separated into gravity wave models
and non-linear steady state models. Non-linear steady state models
are applicable to the mature stage in the evolution of line squalls
while the gravity wave models are appropriate in describing the
initial development of the storms. Since this study is primarily
interested in the initial stages in the development of line squalls,
the gravity wave model was developed.
A review of nunerical models showed how the basic hydrodynamical
equations of motion could be utilised in two-dimensional numerical
experiments; the discussion being based principally on the work of
Schlesinger (1973). Some specific findings of two-dimensional numerical
models are:
1. the conclusion drawn from Takeda (1971) that two air
masses of different atmospheric stabilities and
oppositely directed to one another are essential before
'long - lasting' clouds could develop;
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2. the deduction from the work of Hane (1973) that initial
perturbation is not very important in simulating tropical
disturbances because there is always a tendency towards
convergence upon a cocrmon solution irrespective of the
initial perturbations assumed and
3. the observation of Schlesinp,er (1973) that the intensity
of convection increases with moisture supply from the lower
air mass and decreases with mid-tropospheric wind shear.
Three-dimensional numerical models were discussed as extensions
of two-dimensional models; the extension being necessary because values
of in-cloud parameters obtained from two-dimensional models are usually
larger than observed values. Furthermoret such extensions are more
realistic because the line squall phenomenon is three-dimensional in
nature.
Notable among three-dimensional models is the work of Ogura and
Charney (1962) who derived a frequency equation from \o7hichthey showed
that wave-like disturbances generated in a typical pre-storm 'tropical'
environment amplifies with time. This study extended their work to
describe waves generaged along the surface of discontinuity between
dry and moist air masses.
From numerical experimentst basic atmospheric features that are
essential before line squalls could develop were deduced. These
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features include:
1. a two-layer air maSS wherein the lower layer is
convectively unstable;
2. cold front zones (i.e. borders between warm and cold
air masses);
3. a moisture profile that drops between 600 mh. and 500 mb
pressure levels;
4. a mid-tropospheric wind maximum and
5. strong vertical shear.
These features were shown in section 4-1, to be present in West
Africa during the wet season.
Another possible feature that was mentioned is the existence of
an inversion or isothermal layer in the atmosphere. The Pressure
Jump Line theory of Tepper (1950) on the trigger mechanism of line
squalls was based on this feature. TIlis theory links the formation
of line squalls with the development of a pressure junw which is a
consequence of the sudden acceleration of the cold front in a border
between warm and cold air masses wherein an inversion or isothermal
layer exists in the warm zone. Some observed features of the West
African line squalls that cannot be easily explained by this theory
are:
1. the almost-uniform speed of propagation of line squalls;
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2. the observed overturning of the atmosphere after the
passage of squalls and
3. the preference of highlands as source regions of line
squalls.
LeRoux (1976) also made an attemQt at formulating the trigger
mechanism of line squalls. The defect in his explanation is that his
model could not justify the self-sustaining nature of line squalls.
This study's presentation of the ttigger mechanism of line squalls
was a modification of the proposal of LeRoux.
The trigger mechanism of line squalls proposed in chapter 4 is
based on the amplification of any wave-like disturbance along the surface
of discontinuity between south-westerlies and north-easterlies. The
waves generated along the surface of discontinuity could amplify up
to the extent of blocking the 650 mb. jet. The jet, by impinging on
the surface, further distorts the 1bump , that has been formed and
thereby forces the monsoon air to rise more. The rising monsoon air
could precipitate and the precipitates fall into (with some of them
evaporating in) the underlying jet. The jet, now cooler, sinks and
while sinking, could hit the surface of the earth especially if the
whole process occurred over a table-land. On the surface, a squall
front is formed as a result of strong convergence.
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As the squall front moves over the surface, it lifts the con-
vectively-unstable south-westerlies ahead of it and since t~e depth
of this air mass during the wet season is enough to sustain large clouds,
cumulonimbus clouds could form. The extent of line squall activities
over the surface depends on how far the jet could travel on the
surface. This picture of the trigger nechanism of line squalls seems
to be more adequate than previous presentations because it has been
able to account for a number of observed features of the disturbance.
Among others, this proposal has accounted for the following:
1. the limitation of West African line squalls to a
particular season of the year (line squalls are limited
to the wet season because this is the period when the
weather Zone C of Hamilton and Archbold (1945) cover a
substantial area of land and as shown, all essential
atmospheric features that aid the development of line
squalls are present in any area experiencing the weather
Zone C),
2. the preference of highlands as places of origin of line
squalls,
3. the close association between the speeds of propagation
of line squalls and the mid-tropospheric jet,
4. the orientation and direction of movement (WSW) of D.L. 's,
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5. the role of precipitation in the sustenance of line
squalls and
6. the observed overturning of the atmosphere after the
passage of line squalls.
While discussing the evolution of line squalls in chapter 4, the
two scales of downdraughts in a fully-developed line squall system
were mentioned. One of these downdraughts is within the lowest 200-
400 m behind the squall front. The other scale of downdraughts has
lower wet-bulb potential temperature, 8w , and probably originate
from areas of low 8w (i.e. pressure level 750 mh. and above). In
a fully-developed squall system, heavy rainfall ~hat immediately follow
the squall front cause a rise in pressure (mesohigh) while about
100 km behind the front, there exists a mesolow.
To justify the proposed trigger mechanism mathematically,
chapter 5 and 6 were devoted to the development of a gravity wave model
for initiation of line squalls. From the basic hydrodynamical equations
governing atmospheric motions, perturbation equations were derived
(in pressure co-ordinates). One advantage of the use of pressure co-
ordinates is that more data on atmospheric parameters are available
along isobaric surface. Further, the frequency equation, which was
derived from the set of perturbation equations, appears neater in
pressure co-ordinates.
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In the derivation of the frequency equation in chapter 5, a new
definition emerged for the static stability of the atmosphere. The
effects of this stability term on the possible solutions to the
frequency equation were discussed in section 5-3. It was ShO~4U that
in a moist environment~ the condition for wave-like solution to the
frequency equation is that any wave-like perturbation must amplify
with time without any propagation while in a dry environment, any
wave-like perturbation must propagate. without any amplification,
with time. In a situation wherein two air masses, on.e on top- of the
other, have different stabilities, waves generated along the surface
of discontinuity would amplify as well as propagate with time. This
is the situation along the surface of discontinuity between the north-
easterlies e~d the south-westerlies in West Africa.
Such amplifying patch of convection consist of many modes of
different growth rates and phase velocities. All possible phase
velocities will be solutions to equation 6.9 which was obtained by
solving the frequency equation with the aid of a two-layer model of
the atmosphere. Some phase velocities which are possible solutions
to equation 6.9 will be complex; the real part representing the
phase velocity while the imaginary part is a measure of the amplifi-
cation of the wave-like disturbance. \-7aveswith the largest amplifi-
cations (i.e. largest imaginary parts) along the surface of discontinuity
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between the north-easterlies and the south-westerlies are those that
might likely block the 650 mb. mid-tropospheric jet and thereby
initiate line squalls.
This study is by no means exhaustive on West African line squalls.
For instance, the role, if any, of the African easterly waves in the
process of initiation of the disturbance needs further study.
Dhonneur (1970) noted that the two phenomena - easterly waves and line
squalls - could interact in spite of the fact that they are distinct.
A possible interaction is, as mentioned in the thesis, the observed
coincidence between the arrival of easterly wave troughs and the
formation of line squalls. If this coincidence were true, there
should be a periodicity in the occurrence of line squalls since the
easterly wave trough would normally arrive in West Africa at a regular
interval of about 3.5 days. This periodicity would, however, be
subject to the existence of other atmospheric parameters that are
essential to the development of line squalls.
Hamilton and Archbold (1945) and Aspliden et ale (1976) have
observed that line squalls are generated during the period of maximum
isolation. It will be necessary to investigate how this observation
fits into the model proposed in this study on the trigger mechanism of
line squalls.
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- 114 -
The determination of the pgrticular wavelength that gives rise
to maximum amplification in the possible solutions to equation 6.9
is an avenue for further numerical work.
The dissipation of line squalls needs further investigation.
Although few suggestions were made on the dissipative process of line
squalls, there is need to fully justify the reason(s) why tine squalls
dissipate completely around the coasts of West Africa.
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