A QJABI-GEOST
lWIC DIAGNOSTIC INVZ8TIOATION
oI A TRQPICAL 87TO9
by
ROBIET WILLIAM BURPES
B.As,
"sMsT. TECh,
arvard Ufveretty
(196s)
LINDGBRA
UBMITTED IN PARTIAL
ULFILLIMNT OF ThE
REIUUMRiNTa FOR TH DEORS& OF
MATU 0r
F CWINCE
at the
MASACRUSEM
INSTITUTS OF TSCHGLWcGY
Aupaut 190
Stnature of Author
Dpartgaet of me
rology# 10 August
Certified by
besls Advisor
Aoopted by_
by -~- -'
Aooeptd
---
I-
Graduate Students
GradSat. Studants
A QUASlGXiESTROPHIC DIAGNOSTIC INVESTIGATION
OF A T'OPICAL STORM
by
ROBERT WILLIAM BURPE
Submitted to the Department of Meteorology on 10 August
19"i in partial fulfillment of the roquirement for the
degree of Master of Belooe
Large-scale diagnostic vertical velocities as determined by
a quei-goostropho, te-alevel model were ouputed for trpical
stor Debbie of September 19S. An isagoei-lotoh analysis of
the horslontal wind field we used in order to calculate a height
field suitable for input into the geotrophic model. Results
tadioate that it is possible to desarbe acourately tropical
motiems using the data network in the viotiity of the Oult of
Hdeoao and the Caribbean Sea. Computatios ware made with and
wtthut latest heating. Te ealculated vertisa velocities
quattively agree with the eloudifes sheon is Tire asphs
1wspered aIaal
itstswly ms etSeth
et
I me
rates.
Imtappes that am iateretSe
ioumiAlusesl
between eelee-
mtions occurs in the reios of u' ma largem
soal, upwrd vertical velocitties thus produc g an area of embasned upwrd notion in this region.
Thesis Supervisor:
Title:
Frederlck Sandoer
Associate Proftesor
ACKW~OLEDMEN
I wwzld like to thank Professor Jule Charny and Prwtessor
Frederick Sanders for their seollent advice and continue
throghbout this investigation.
interest
I am also very grateful to Professor
kdward Lorenz and Dr. Terry Williams for the many discussion e vhioh
I had with them concerning this paper.
The constructive criti csn
and guidance provided by each one of them contributed substantia IYl
to the broading of my knowledge.
Gardner Perry's programming efforts in alteritng the 10-level
wre greatly appreciated.
lodl
Special thanks to Miss Wanda Burak, Mi s
Ann Corrigau and Miss Isabel Kole for doing an excellent Job tn preparftg the data and to Mrs. Jane McNabb for typing the manuscrip;.
I
ld also like to thank the National Burricane Research
Laboratory for providing the data for this study.
Numerical compu-
tations wre performed at the CoWqutation Center,
Iassaobusetto
Institute of Technology.
TABLE OF CONTENTS
Introduation
1
The Diagnostic Model
3
Beight Calculations
Mthod of Analyss
Inoorporatlo
of Latent Heat
12
Tropical Store Debbie
14
Reomts
10
Conclusions
25
Figures
27
aeterences
40
LIST OF
1.
IGURES8
87
Course followed by troptcal store DebbU*
2, Ouface sp for 25 Sepmbewr 198
at 100
W
28
9
3.
Calculated beight field at 860 ab
4.
Calculated beight field at 600
30
3b
b.
Calculated height field at 260
81
3b
0.
8ketoh of T7os nsphs for Septmbwer
7.
Comptd 800 b adiabatio v
8. Compute
800
th
ical notims
32
33
b total vertical notions
34
35
9.
Compted 600
b adiabstic vertical
10.
Computed 000
b total vertical notln
11.
Computed 400 ab total vertical motims
37
12.
Computed 200 ab total vertical motioas
38
18.
Cgmuoted
0 b height
Mmqie
otlsm
3
30
INToDJUCTION
Multilevel dynamic models of the atmosphere have boen used recently
to calculate diagnostic vertical volocitlas in an ettort to doeterino
the relatioBship between vertical notion and cloudiness and precipitation.
Studios by Barr and Lawrence (194) and benders (1905) have shown that
horiAontal motions are equally as important as vertical motions in
Indeed,
acounting for observed cloud systems in middle latitudes.
in order to adequatel
represent cloudiness by vertical notions of
ultilevel models, the availability of
oisture and allomnoe for
hortaeatal changes in static stability must be included.
The main difficulty in associating upward vertical notion with the
cloudiness show in Tiros satellite piotures is the fact that cloud
elemeats move with a spectrum of horisotal velocities.
dhile it
is
true that the larges-cale vertical notion pattern and its movement
in the flow pattern determine the location of the large-scale cloud
mass, horisontal winds frequently adveot saturated parcels of air into
descent regions and unsaturated air into ascent regions.
It
the moist
air in descent regions has not had sufficient time to evaporate and
the unsaturated air in the asoent region has not condensed, a lack of
correlation between the dynamically-calculated vertical velocities and
the cloud pictures will be found.
Agreement between cloudiness and
vertical notion is best in the presence of abundant moisture and in the
early stages of cyclone formation before the horinontal notions have
fully developed.
In lower latitudes *abrehoriaontal velocities are
2.
smaller ad warm ocean surfaces provide a readtly available source
of moisture, it
is antioipated that Tiros pictures aight be more
reprosentative of actual vertical motions.
With the exception of nnsoon ctralations and hurricanes the
horlontal gradients of geopotential height and temperature are amch
smaller I& the tropics than the correpondiug gradients at aiddle
latitudes.
Because of the anmal a
of these gradients it
is not
intuitively apparent that differeatial vortletty and therml advoctions
on
soberi surfaces play a prominent role in the productlon of rising
motion
In tropical regions the Rossby aumber, *ich is the ratio of
the horisental relative accelerations to the Coriolis accoleration,
approaches one and it
can no longer be regarded as muah less than one
in the scaling of the geostrophic equations.
This means that for the
same velocity and length scales the relative amportance of nonageostrophic
effects I increasing nar the equator.
knohaa
_lor
For these reasons it is not
ihether the quati-geostrophic model will give qualitatively
acceptable results In tropical regions.
It Is the purpose of this
study to determine how well diagnostic results of a ten level quastgeostrophic
model with incorporation of latent heating are correlated
with cloud cover as seen by Tiros wather satellites in tho tropics.
Tropical storm Debbie of deptember 196
was chosen for analysis because
a depression was wanted for the study ~bich had closed Isobars at the
surfae
but vrhih was not so intense as to Invalidate the geostrophic
assumption.
THE DIAW44L.TC MUDEL
The large scale motions of middle latitudes are nearly in
geostrophic and hydrostatic balance and are well represented by quastgeostrophlc models.
A diaganostic, geostrophic equation for calculating
the large-scale vertical motion field and a geostrophil
height tendency
equation can be obtained by combinng hydrodynamic and therodynamic
equations appropriately scaled for middle latitude flow where the
Roesby number is
small
The ftor
of the vorticity equation in
X
,
-
oordinates aonsistent with this type of scaltg to
1)
V2+fo 3VO+
_7
tois the absolute vortictty,
is geopotentlal,
where
a constant value of the Coriolis parameter choose
grid,
and
at the center of the
represents the vertical motion.
40
is
fo
Stallar
scaling on the therodynamie equation resulnts In
00.--
(a tfuctia only of pressure) is
where
diabatlo heating.
a
to the
measure of static stability and
#
(2)
(r &J
By combining these equations so as to eliminate
, a diagnostic
I1-
equation results.
To the eatent that vertical motions are sinusoldal the left
this equation is
negatively correlated with WA itself
side of
The resulting
baroelIte vertical notions can be partitioned in order to determine
ich serve as
the unique contributions of the synoptic mechaniAeni
forcing fanctions for the large scale ascending or descending motions.
The first term on the right side of equation (3) is the variation of
absolute vorticity with height.
An uplard increase of vertiCity
advetion with elevation results in ascent and an upward decrease
the horizontal Laplacian
contributes to descent.
The second term is
of thickness advertion.
Mawia of warm advetio
s
ent and cold advecteon,
heatting effect.
Ascent is
descente
The reman in
tend to produce
ters is
the diabatic
found in regions of maximm heating and
descent near regions of maxtmsm coollag.
A simple parameterisation of latent heating is
to that the diabati
Phillips' (1096)
used in this study
heating term will be included in
equation (3).
scale analysis of t ththermodynamte equation shows
that the maximum amount of latet
geostrophic assumption is
beat release consistent
ith
the
assoc ated with two centimeters of precipitation
over a period of tuwnty-four hours and over synoptic saale distances.
Although precipitation rates associated with troploal storm Debbie cannot
be accurately deterained because of its
track over ocean areas, it
tis
felt that this masiaam rate may well have been exceeded by a factor of
In these
two or three in certain localized areas around the storm.
is
areas the quasi-geostrophic theory Is not strictly valid; but it
hoped that these large rainfall rates are associated with small-scale
features and that the large-scale flow is
well represented by the
geostrophic model.
Appropriate boundary conditions are that W
vanish at the top of
the atmosphere and at the lateral boundaries of the computational grid.
At the lowest level of Input, the 1000 ab surface, it
where
5
is
assumed that
subscripts refer to values at the surface of the earth, 10
is
Cp
subscripts to 1000 ab, and
the drag coefficient.
These terms
horizontal
are respectively the contributions due to local pressure change,
motion over sloping terrain, and frictional divergence athin the surface
friction layer.
The height tendency equation is
=
-f0i
Again to the extent that
differential operator and
The terms in equation (5)
vorticity advection,
lab
2.
r1
'4 ja )
(
has a sitausodal distribution,
itself
the
are negatively correlated.
represent respectively the effects of absolute
differential temperature advection in
and diffterential dabatic heating in the vertical.
the vertical,
AssuIng a
sinusoidal distribution, equation (5)
states that cyclonic vorticity
advection, warn advection above and cold advection below, and an upward
increase of diabatic heating at a given point in
the atmosphere tend
to produce height falls at that point while their opposites tend to
An appropriate boundary condition is
produce height rises.
that
vanish at the lateral boundaries.
The input data are values of ground elevation (taken from Berkofsky
and Bertoni' s (195)
smoothed topography) and of calculated heights at
the mandatory pressure levels up to 50 ab excluding 200 and 100 mb.
The base nap is
a Lambert conformal projection with a noesh length of
165.1 km at 30
and 60
Values o
Of
and the computational grid is
32 by 24 points.
are obtained from a horisontally averaged sounding.
UEIGHT CALCULATIONS
The vertical structure of the low latitude wind field Is not as
coherently organized as the vertical structure of middle latitudes.
Wind speeds are lighter and wind directions are quite variable in
vertical.
At middle and high latitudes the
lnad field is
usually
established indirectly from an analysis of the height field on a
pressure surface
the
constant
These conventional middle latitude techniques do not
accurately describe the synoptic-sccale motions of the tropics since
the horixontal gradients of height are of the same order of magnitude
as the error involved in reporting the observed heights.
field of motion is
The horizontal
most accurately described by the winds themselves
7.
and an isogon-eotach analysis of the wind field represents the flow
better than a direct analysis of the observed heights;.rom this
analysis it
possible to deduce a height field which is
is
consistent
with the observed winds.
It
is well known from the Relaholtz theorem that a two-dinensional
vector field can be partitioned into non-divergent and irrotational
In particular applying this theorem to the horizontal
components.
wind field
V
where
is
function (
A x 7 1 + V7X
(0)
the unit vertical vector,
7
x
is
velocity potential (
observed wind is
non divergent),
Is
and
itos irrotational).
partitioned between
choice of boundary conditions.
the horisontal stream
and
is
the horizontal
The manner in
X
which the
depends on the
Since the goal of this analysis is
to calculate a height field from the wind field for input into a quasigeostrophic ten-level model which Is non-divergent, the wind field
us
partitioned so as to minsize the kinetic energy of the divergent
part of the wind field.
equation (6)
is
If
the vertical component of the curl of
taken, the result is
4-A
,7
x V
(7)
Thus the horlaonta l.aplcian of the streatuanction equals the vertical
8.
) of the wind field.
component of the relative vorticity (
a Posson equation which can be solved b
equation is
numerloal techniques
computed from the observed wind field and
is
if
This
normal
or its
/
specified on the boundary of the computational grid.
derivative is
Taking the component of the wind field parallel to the boundary
()
= ? 0 +
where
n
and
S
are in the normal and tangential directions: pn
5
positive in the outward direction and
clockwise direction along the boundary.
unknoma quantity
The normal derivative of P
terms of one known quantity
specified in
therefore, is
positive in the counter-
As stated above it
d
is
Vs
,
and one
desired to decompose
the wind so that the kinetic energy associated with the velocity
potential is
The boundary conditions which accomplish this are
minimtIed.
X Y 0zo
(9)
These boundary conditions also associate% the nondivergent, irrotational
component of the vector field with the streamfunctlon.
now be solved for
'psoudoheight"
P/ by
is
may
the Liebmann overrelaxation process and a
field may be calculated by using
(1)
4r5J+
where
Equation (7)
gravity and C is
an arbitrary constant.
Even though the
9Q
is
absolute magnitude of
not precisely determined,
thies heght
because only
into the ten level model
input
for
field is suitable
(3)
needed in equations
of geopotential is
gradient
the horiontal
and (5).
ETW3D OF ANALYSIS
data in the
reporting upper air
stations
of
network
Although the
the data
in other tropical regions,
than
dcase
more
Caribbean area is
middle latitudes.
to that available in
cotlared
sparse
quite
is still
as accurate as possible
.e reported winds were
tL
that
insure
to
In order
and pibal reports
winds from radiosonde
time seotons of horitmonti
of
were plotted for eaoh
I
(,
table 1.
e stations listed in
This mws
28th. These
September 26 to September
Lit
data
12Z
and
done for OOZ
both in
for correcting inconsistencies
ae,s
a
provided
time sections
in
smoothing the wind data
subjectively
or
:
time and in the vertical,
wind field, and for
iale features of the
small-.
elimanate
to
order
stations *bich report
wind:i at those Mexican
12Z
the
interpolating
only at OOZ.
The resulting wind dati
ealyses
was plotted and ison-Isotacb
by the tea-level
the input levels required
time
13Z was chosen for the
to the ; 8th at 12Z.
25th
the
from
model
oft the Tiros satellite
was 11oser to the time
it
because
of analysis
level
model has its lovest
previoyliv .y the ten-level
stated
As
passage.
condition on the
Snace \be botton boundary
ab.
1000
at
input
of
each ol
were carried out at
I
10.
w-equation includes frictional effects and since the geostrophlc
assumption is valid only above the ftrctional boundary layer, the
1000-mb winds do not represent trictionless flow suitable tor the
lowest input level.
A study of the time sections revealed that the
2000-foot winds were relatively free from triotional intluence and
yet resembled closely the 1000-mb flow; therefore, the 2000-toot
field was substituted for the 1000-mb tield.
In regions where the
surface was above this level the flow at upper levels
guide in the 2000-foot analysis.
fitting the analysis to the
Table 1
72201
76225
70256
76394
76844
76679
76092
78016
780883
78076
78118
78367
78384
78397
78601
78"26
78800
78881
78860
78897
7884
78967
78988
80001
80222
was used as a
At all levels an effort was made
for vertical consistency while still
tind
Key West
Chihuahua, Mexico
Guaymas
Monterrey
Merida
Mexio City
Veracruz
Bermada
Grand Bahama
Eleuthera Island, Bahamas
Turks Island
Guantanamo
Grand Capman
tinagton, Jamaea
Swan Island
Puerto Rico
Balboa, Canal Zone
Alntigua
St. Maarten
Guadelupe
Barbados
Trinidad
Curacao
San Andrea
Bogota, Columbia
I.
observed winds.
In the northeast and southwest corners of the grid
where upper air data was lacking the
lov was analysed In the simplest
possible manner so as not to create any large relative vorticities
which night incorrectly intluence ensuing computations at interior
points.
Reconnaissance data provided by the National Hurricane Research
Laboratory was used where applicable in the vioinity of the depression.
A nine point, two pass smoother-unswoother ws
unwanted smal-ocale motions.
elSiate
the smoother-unsmoother
calculation of the
used into order to
Computatlons were made when
se applied to the vorticity field before
field and in a second case to the
itself after relaxation of unsmoothed vorticitles.
W
field
Very little difference
f ields and the former method was
was noted between the resulting
used throughout the computations.
It Should be noted that the resulting height field was not always
consistent in the vertical.
In those regions of weak flow near singular
points in the isogon-isotach analysis where the irrotational component
was nmoe representative of the actual wind field than the non-divergent
component,
displacements of singular points from the analysed location
sometimes were as great as three grid distances.
The pseudoheight field as determined by equation (io) was used as
input for the ten-level model.
Since input heights were limited to
positive values and three decimal digits, the constant in
mwas
equation (10)
hosen in each case so that the lowest height use identically zero.
This permitted input heights in
whole feet from the lowest input level
up to 400 mb and in tens of feet at the remaining levels.
12.
NCORPORATION OF LATENT HEAT
Latent heat plays a very iaportant role not only in influencing
the long-term motions of the general circulation but also in the
short-term synopticscale notions.
for a developing cyclone is
Although the main source of energy
the eddy available potential energy re-
sultinag from horizontal temperature gradients, recent studies have
shown that diabatic heating resulting from condensation is
tant process in
the intensiication and strengthening of large-scale
middle-latitude disturbaneoos and in
in
an impor-
the lower troposphere.
accelerating the movement of cyclones
In tropical latitudes small-scale convection
processes are the principle mechanism through uhich latent heat is
released.
Convective cells act as a heat source for organised tropical
depressions thus providing for the maintenance of these depressions
against frictional dissipation at the surface.
Many attempts have been made to parameterise the influence of heating
due to microscale convective activities on the large-scale tropical
motions.
The present method closely parallels those proposals of Ooyama
(1983) and Charney and Eliassen (1984)*
boundary layer exists which is
It
is
assumed that a frictional
full of saturated water vapor.
Area
of convergence in the boundary layer give rise to ascending ourrents
which advect water vapor upward into the column of air above.
The
water vapor transferred through the top of the friction layer is
assumed
to condense Iamediately and simultaneously warn the air column over the
area of convergence*
The heating is
parameteriaed in
the following way.
13.
the frictional vertical velocity
LetW()
H
represents the diabatic heating, L
(12)
is the latent heat of
is the satUrtioa specific humidity and I
atiou,
onde
(11)
= heating per unit mass
9
where
inhere
d
HI=
and
represents the heat released per unit
.
the surface pressure.
is
ass to the column of air above by an uprd flux of saturated air from
It
the friction layer.
G- is now averaged from the bottom to the
top layer
e
4
N
and
R
aB
aI aAO
Then
(1=
w 'A
L
L
-
(14)
(15)
-q
let
q(6
therefore
where
1
0
dJ
>0
and
q
3
for
Withthhe latrodUotion of the diabstic heating term
4< 0
the ther.m~
for
so
equatien becomes
14.
at
vV
-3FPS
It was suggested by Charney that
be deterined by use
of the formula
where S
subscripts refer to the surface and
ficients
This is done rather than assume some average value for ]V1
and have
W~L
lowest layer.
C,
is the drag coef-
directly proportional to the vorticity at the
In this way the present work deviates slightly from
the method used by Qoyama and Nitta.
This approach is a simpler
method than that used by Charney and kaasseen since the horisontal
transport of water vapor above the friotmon layer was not considered.
TROPICAL STORM DSBBIE
Debbie ws first observed by reconnaissance aircraft on Beptember
24, 196,
to the west of Swas Island as part of an easterly wave
was slowly moving through the western Caribbean.
ichh
At that time a
troploal depression ~&th lowest surface pressure of 1003 ab, maximum
winds of 20 knots, and no well-defined circulation was found.
Debbie
progressed on a northwestward course crossing the northeastern tip of
the
ucatan Peninsula on the 26th and emerging into the Gulf of Mexico
15.
somewhat weaker than before as shown by an increase in minimum suTrface
During the 26th and 27th the storn's course
pressure to 1007 ab.
Early on
became more northerly and forward speed increased slightly.
the 27th maximnm low-level winds increased to 30 knots but no organitation was apparent and highest winds and shower activity were for the
most part confined to the east and northeast of the stomr,
the previous days.
similar to
On the 28th Debbie's course became northeastward,
her forward speed increased, and her surface pressure decreased to
1004 ab indicating slight intensifioation of the depression.
activity
uas beginnin
Some
to be noticed in the northwest quadrant with
lorw level winds of 25 knots while winds to the northeast reached 45 knots.
From the 25th to the 28th Debbie'* center was never well defined
and there was no strong coupling between activities in the lower and
upper troposphere.
The msin feature of the depression
as a region
of strong southerly and southeasterly flow (20 to 40 knots) to the east
of the storm center.
It
was in this region that reconnaissance air-
craft reported oloudiness and moderate to heavy rain and Tiros pictures
Indicated solid overcast.
The dominant feature of the upper troposphere on the 25th of
September was a very sharp trough over the eastern United States.
Debbie
was located several hundred ailes west and south of the trough line
under a region of relatively weak flow aloft.
As this trough moved
further eastward height rises began to occur over the southeastern
United 8tates and eastern half of the Coulf of Mexico.
Consequently
weak antloycloase circulation appeared at 250 mb on the 26th to the
16.
north of Debbie.
By the 27th a new upper level trough had moved into
the eastern United States; at higher latitudes
it
was located over
the eastern Great Lakes region and at lower latitudes over Texas.
The anticyclonie region aloft tntensitied, moved eastward, and was
located just east of Debbie as the tropical storm moved into the
Gulf.
A day later that part of the trough which had been over Texas
on the 27th phased in
with a now higher latitude trough as it
into the central United States.
Debble,
moved
which was very close to
Louisiana and Just east of the upper level trough axis on the 26th,
was becoming more directly
soon lost its
tropical
ntfluenced by the middle latitude flow and
characteristics as it
moved over Thand.
RBSULTS
Vertical velooittes and height tendencies were computed for each
day from September 25th to September 28th at 12 Z. Since the important
features of the oloulations are essentially the same for each day, only
results for September 25th wll be discussed in detail.
On this day
Tiros X passed almost directly over the disturbed area of cloudiness
assoclatod with Debbie thus providing exoellent satellite ooverage. Also,
Debbie's center was located further south than the other days which were
analysed; it
is
felt,
therefore,
that the results of this day provide a
better means for determining the degree to which the geostrophic model is
capable of representing tropical notions.
Because the center of the
tropical storm was not well defined, reconnaissance aircraft could not
17.
accurately position the storas
location.
approximate course followed by Debbie.
Figure 1 represents the
The surface
ap for September
28th at 12 Z is shown in figure 2; the low near the western coast of
Mexico
is
associated with hurricane Hasel.
Since the hurricane was
loeated almost on the western boundary of the computational grid, the
results for this region are not necessarily realistic because the
height tendenoies and vertical velocities were set identically equal
to sere on the boundary.
Analyses of the "psedoheight" field derived from equation (10)
have shoun that these height fields are quite representative of the
observed Sands.
fro
As a check of the method a wind field ws calculated
the "pseudoheight" field.
It
as found that these
inds wore
systematically weaker than the observed winds but retained at least
80h of the speed and normally differed in direction from the observed
winds by less than 18 ,.
xamples of the "pseudoheight" field at 880
ab, 800 Nb, and 250 ab are given
n figures 3 - 8.
The close agreement
between the derived height field and the observed winds in
middle, and upper troposphere Is revealed in these
in
the introduction it
Is
tigures.
the lower,
As stated
espected that Tiros pictures will be quite
representative of the actual vertical notions in the tropics; but because
of the relatively small geopotential height and temperature gradients it
is
not )mnown It
the meteorological network of upper air stations is
sut-
ficiently dense to adequately represent thoe gradients or ift the diagnostic
vertical motions computed from the geostrophic model wil1 compare favorably
with atampheric motions.
Uesults for each of the days in
the study
18.
upUard notion and maximm
indicate that the location of the maximau
domward motion are well correlated with the cloudy and clear regions
respectively in
the Tiros pictures.
September 35th is
showr
in
figure 6g
A sketch of the TLros neph for
it
should be noted that the
satellite picture mew taken several hours after the meteorological
observation time.
The adiabatic vertical notion at 800 ab is
figure 7 and at 600 ab in figure 9.
upward notion and cloudiness is
per second:
a value which is
The correspondence between maimum
clearly evident.
maximum upard vertical motion is
The magnitude of the
on the order of a half a centimeter
charactoerstt
of the adiabatic calculations.
Simple calculations made on a pseud adiabato chart assu
rate of condensation equals the rate of precipttation,
of air 500 jb in the vertical is
showa in
nag 1) that the
2) that a column
being lifted, and 3) that no divergence
occurs in the lifted column show that vertical motion of this sise Is
more than an order of magnitude smaller than that required to produce
rainall
rates in
excess of one inch per day.
With the introduction of latent hoseating parameteriaed as discussed
prevlously, a diabatic heating term is
heat source may be varied in
added to the W1 -equation.
the vertical by changing the coefficient
associated with the diabatic term.
heating has been distributed in
For the purposes of this study the
the following nsaner.
900 mb
.275
800
100
600
S00
.237
.225
.160
.075
mb
Mb
ab
ab
This
19.
At all other levels only adiabatic motions have been considered.
Figures 8 and 10 show the vettical notion at 800 ab and 600 ab when
the diabatic term is
The magnitude of the ascent regions has
included.
been increased by a factor of two or three with maxitum ascent in exooes
In this case the maxtimm rainfall rate calculated by
of 1-1/2 cm/sec.
the method mentioned above is on the order of a halt inch per day: a
ach maller than typical observed rates.
magwitude which is still
is assumed that the upward motion is moist adiabatic, the
If it
thermodynatc equation for these motions is
4 ~
dL
dQ
(19)
di1
where .rt
is the saturation
s
moist adiabat, and
is taken along a
xating ratio,
r. includes diabatic and adiabatic efftcts.
This
equation offers a means of checking the validity of the 10-level model
calculations with the diabatic term included.
When the heating that is
specified by equation (12) is substituted into equation (19),
that the resulting
4 '5
it
is found
are smaller by a factor of three or four than
those derived by the 10-level model.
This implies that the heating
speoifled in the model should be larger in order to account for the larger
10-level
)
;
but, if
the heating is tncreased then the 10-level
will be correspondingly larger.
.' 5
Because the heating model used here is a
very slaple M jgg representation of the actual latent heating in the
atmosphere, it
is unlikely that a repeated iteration process involving
equation (19) and the 10-level
4J $S wuld converge.
Nevertheless, even
this simple type of diabatic heating has produced a qualitative improvement
20.
of the W
field.
In order to improve the consistency of the results
a parameterization of the heating is needed which also considers motions
above the friction layer.
It should be noted that the structure of the
tropical atmosphere is such thmt the process of condensation is closely
assooiated with cuulus-scale motions.
A parameterization of latent
hoeating in a large-sale model for use in the tropics is, therefore, a
means of representing the dynamics of the small-scale motions in terms
of the large-scale motions.
When the present model for latent heating
is useod
the large-srale upward motion calculated by the geostrophic
model is
well correlated with the cloudy region in the Tiros picture
but not quantitatively consistent with the upward notion
thich seems
necessary in order to produce the observed rainfall; it appears, therefore, that these large-scale motions are acting so as to produce an area
where small-scale updrafts account for a major portion of the precipitation.
The cuaulus- and cyclone-scales are thus cooperating to produce
a region of enhanced upward notion.
A similar conclusion concerning the
interaction between large- and small-scale motions was reached by Charney
and Sliassen (1983) in their discussion of the tropical hurricane.
The vertical notion field in the upper troposphere at 400 mb and
200 mb is
shown In figures 11 and 12; no latent heating was specified
above 800 mb so that these
W
4
represent only adiabatic forcing.
In order to determine more precisely the correlation between the
and the cloud cover, tables were made in which the 10-
computed
WJ '
level 6W
at each grid point was categorised according to the cloud cover
al.
in the Tiros nephanalysis which corresionded to it.
No attempt was made
to adjust the cloud position to the meteorological observation time of
12Z and the outer two rows of grid points were not included because
of the dependence of the
The sum of the results at 800 ab, 700 ab, 600 ab, and 500 ab are
tions.
shoiw
at these points on the boundary condl-
W '4
table 2 for the adiabatically
in
table 3
and in
Wh
These
computed with the addition of the diabatic term.
W 4
for the
omputed
tables show that where there was cloud in ti
was computed almost
ward motion is
subsidence fro
ll
of the te.oo
upward motion
noephanalysis,
The disappointing fact is
that up-
calculated in two regions where one might have expected
the nephanalysas.
Tie first
in the western
of these is
Caribbean south of the region of heavy cloud where surface observations
Indicate the presence of clouds
while the noph shows a region which is
regions is
4
n agreement with the negative
mostly clear.
to the northwest of the frontal oloudiness
4
The second of these
in the western
Atlantic where there is a direct transition from heavy cloudiness to
clear sky In the nephanalysts.
in
The maximum coputed ascent is
the frontal acne and the maximum computed
oscent is
located
In the clear
region but the southeastward displacement of the front in
the four hours
between the asp time and the satellite observation time resulted
'
tlre
being computed in
n nega-
a region corresponding to a clear area in
the noph.
It
is
very little
apparent from an inspection of the two tables that there
difference between the adiabatically doeteormned
W 4
a
and
22.
TABLE 2.
Adiabatic vertical velocities in
units of 10 - 4 pb/se
Cloud amounts are
0 - 20% overage
Open (0)
Mostly opme
20 -
0% coverage
Mostly covered (CIO) 50 -
0% coverage
(NP)
Covered (C)
80
-
100% coverage
Numbers are the totals for calculations at 800 ab, 700 ab, 800 ab
and 5O0 ub were the
)
are categorissd according to the neph
olassifitation at the same point.
0
loP
0
C
49
10
0
0
TABLE 3.
0
WP
MC
C
4/21
81
11
0
0
30
104
0
6
30
a6
43
119
18
80
37
220
Vertical velocIties vwth diabatio term.
Units are the same as above
3
91
0
3
o6eoW5
s
26
32
102
-S(L)
258
92
48
230
those determined with the diabatic ter.
The
aain difference is that
with the addition of the diabatic term the magnitude of the maxima
updrafts has been iaereased.
Since & good correlation between the 10-level tl
np
e
to 12,
'A and the Tiros
sts even though the cloud position In the noph we not atjuated
the region of maximm updraft to the east of Debbie and the
saturated air associated with the asmset mst have moved slowly and at
appros mately the easm
In addition the borinsotal velocities
apoe.
relative to the stor's motion mt
have beena emal
enough so that the
region of havy eloud was tound near the dynamically torcd updraft
rather than being advcoted into regions of weak updraft or subsidence
as frequently happens in middle latitudes.
motions art
Thus, althengh borizontal
equally as iaportant as vertical notion
in
accuntin
for
observed cloud systems in middle latitudes, im the tropics the large-
scale vertical motions seer
to determne the location of the cloud
cover and the horisontal notions are normally of secondary
From studies of middle latStude observatioas it
mportance.
is know that ioo-
therms and height catoars are nearly parallel toin the middle and upper
trposphere.
The thickness adveotion is not small, bower, because
the rapid upward tanrese of wind speed cmpemsates for the small angle
the
between th
others
and height contours.
In a similar manner the
magnitude of the difftterential vortiity advetion
level of the Jet stream core.
reases up to the
Thus in the middle and upper troposphere
the vortiity advection term in the 4) -equation is generally comparable
to the temperature adveotion ters.
In the lower portions of the atmosphere
24.
to the
the wind field has a more pronounced component of flow normal
temperature field.
Temperature advoctlons, therefore, remain large
of the
while vorticity advections are more weakly developed beeuse
smaller horixontal wind speeds.
Thus temperature adveotions play the
major role in t be production of vertical motio
sphere at middle latitudes.
of the thermally Induced
In this lover latitude study the magnitude
0'A
is greater than that due to the vorti-
city term in the lower levels of the atmosphere.
magnitudes of the forcing
equal.
It
in the lower tropo-
At highr levels the
unoctions in the 4) -equation are nearly
is
seems strange that the influence of the vortiloty toer
not smaller in the middle troposphre where the horiotal
vlocitties
to
tend to reach a minimum; this to probably a pbenomenon peculiar
tropical storm Debbie and unrepresentative of average tropical
motions.
to be a
Also to be noted is the fact that at all levels there seems
tendency for the vertical motions due to temperature adveottane
to have
a sign opposite from those resulting from vorticity adveotions.
of the
The height tendencies offer a further check oa the acuracy
calculations.
At all levels the tendencies seem to be consistent with
the observed atmospheric motions.
In particular the height tendencies
be compared
In tens of teet per twelve hours calculated at 950 nb can
as
with the throe hour pressure tendencies in tenths of a millibar
are of
given by the regular synoptic reports since these two quantities
approrimately the same magnitude.
Although the number of reported ten-
of the isallobars
dencaes is not sufficient to permit a thorough analysis
tendencies
the surface pressure tendencies and the calculated height
25.
This is
(see figure 13) are consistent both in sign and saga tude.
quite remarkable since the
4
agaitude of the calculated
to much
smaller than the actual vertical motions.
Petterwsen'sI formula for th
speed of a low cater uwas applied
to Debbie's position on the 23th of September.
A speed of 10 k/hour
in a west north West direotion ms derived; this compares favorably
with an actual notion of 8 km/hour toward the northwest and nd icates
that the formula it applicable to surface lows in the tropics when an
accurate surftse pressure analysis is avalable.
CONCLUSION8
Although the upper air data network in the Caribbean region is
more dense than in most other tropioal regions, large areas are present
In the Caribbean, the Gulf of Mexco, and the western Atlantic where
no aeteorological observations are taken; as a result it was not knoan
whether this network ms sufficient to pemit an accurate analysis of
the goopotential gradients for use in a quasi-geostrophic model.
realistei
The
results of the model indicate that by using the beritontal
wind field as initial data a "psaudoheight" field may be calculated
which is representative of the existing tropical
otions.
The verti-
oal motions computed adiabatioally from the 10-level model agree qualitatively but not quantitatively with typtoal observed rainfall rates.
petters.en,
verre
Weather Analrss ad Forecasti.
McGraw-Hill Book Company, Inc., New York. 1980. p. 49.
Volume 1.
260.
The addition of a diabatic latent heating term has inreased the
magnitude of the upward vertical motion but not enough to account
for the rainfall rates.
Admittedly the paraeterizati o
of latent
heating is quite simple but the incorporation of even this sinGle
model of the heating has ~proved the results.
Since the large-
scale vrtical notions are smaller by a ftator of three or four than
those assumed to be required to produce the observed rainfall, it
seems apparent that in the region of prominent
updraft around tropu-
cal storm Debbie the oyclone-sale motions are interacting with the
cumulus-scale motions in such a manner that the vertical notion is
enhaned.
The Tros nephs have been used as a means of verifying
the results of the 10-level model; however, since the 10-level
and the cloudiness rvealed by the satellite pictures aree qualitatively, based on the four days considered
n this study the Tiros
pietures appear to be an excellent representation of the large-scale
vertical notion field in the tropics.
.3
..
.
...
.
.
.
.
.
0
Ago
40
o
~o
O
.
*
.
.
..
0
8
0
8
P6I\
-
4L
0o1*
*a 0
0
4
0T N
a.
0 flat\
0
16-
.
r
Figure 1.
\
&27.
.
.
.
.
C?-
.t
.
006I .
Approximate course followed by tropical storm Debbie, x's
mark the 1200 GMT positions from September 25th to 28th.
L
I
to
O
40
.
.
.
..
0
O a
0s0~
C0
.
"
~
.
.*o
42*
*
. .
.
.
..8
to
0
o . o,
o-. . . . . ..
.
.
...
*
. . i.
.
.•.
0
0
So.
0
. - ,\.
.•
.
.
.
.
..
6
0
to
o
20
O
.
.
*
H
-
°
.
,
6..
40
45
402
VWAL
.
.
11o.
.
. / .-,
. ...
0
L0
20
4
S'WAZ E L
+
.I
.
R~
.
.
.
3
I
o.."s"
o&
.
.
.
..A .. .
.
. . .. ....
.
.
.."..OHZo.. . ..•0,.°
.
79
.
rt
844
.
.
.
.
.
.
.
.
.
9501
900
Figure 2.
8
Ps
.".
.o.
08.
O1/
.
so
- '*06;, •
.. . .
.
.]
o
."."
Loi
0~
)0
.~
.
.
Surface pressure map 1200 GMT 25 September 1965.
.
.
.
100
p.
4
5
.
..
.2
aS,0
V
.
/...................,,...
2
15Q
Figure 3.
.
/
-
9S
..
.
.0;'
, .\
oo
5
/
.,
.
. ,..
-
850 mb pseudoheight field 1200 GMT 25 September 1965.
zero.
equals
are in whole
.
25
&46
.S...,
1*. . •.\.3...
250
•o
Heights
equals zero* feet, x indicates grid point at which the height
•
.. .-.
.....
.
.
I
.. . •
p..
.
..
.00
.
0
.o.
.
vp
.
.
3020
.
.
o* the height equals zero.
4q
4 .which
at
grid point
whole feet,
%
200
235
_P xa indicates
0
L
*
* **"
"*
\ '*""
..
. . .
.
•.. .
. .
.
:. . . .
. . 0. a. 0.. 0
f30
5 00
:- a-... . ...
-. o
0
911
. . ..
500
79
5 50
...
vo
r-
10
VO
550\
.J. . ..
5600
3
20
t
2QO
0~~-
402/55
~
1/
. ..
H
. *i*
40
308
64
550
550
uuC
100.
60"0
. .1.1
.
.400,
.. . 0 .
39
.
.
. . . .. .. .
0'.
6?9
.
4
1
0
L 40
*,
.
-
.
04+..
.
... 500'
0i
Heights are in
Figure 40
500 mb pseudoheight field 1200 GMT 25 September 19650
Figure 4.
500 mb pseudoheight field 1200 GMT 25 September 1965. Heights are in
whole feet, x indicates grid point at which the height equals zero.
O53
.
.
*
*
*
*
.
.
.
.
.
.
.
*•
53Z
. . ,oo
.. .--...
o
3&
0
tea4
S .
°
.
.2
1
.o46
•
o\
* *
*
,,,. ,2oo,,B.o,
toO*2
r4l
S***
*
4 44*
e
got
I~oo
*
*
000
2~
0"1*
• em,/,,,,
/
•
•
•
•
me o
mee
•
•
e.,,•
I .....
-"
•
Q
•
•
•
•
6
mTI
•
it
o.
!
•
•
•
't3'
Figure 5.
Heights are in
250 mb pseudoheight field 1200 GMT 25 September 1965.
whole feet, x indicates grid point at which the height equals zero.
•
•
Figure 6.
Sketch of Tiros X nephanalyses for 25 September 1965. The analyses
on the right is for orbit 1222 at 1619 GMT and the analysis on the
left is for orbit 1223 at 1800 GMT.
'.
0
34(3 ., *
.
o
(,
FarJ
..
s
.
..
.
.
.
•
0
.*
0
..
as~
.
.
.
•o
.
.
sty
•/
.•
.
.
o
4
•
..
..
•
o
o4
-0
. ...
• 0
•,
.
..
.o ..
O
0.•r
•
.
.
.
.
*
*
*
00
0
3
"*
34a
*
*
*
*
0
:.'
06
.
0
*240
*
*
*
6
0*
*
taken from nephanalyses.
cloud systems
regions of major
a
900
a
0
..
&
IK,
0
0
2a3
t
0
0 a 00
-O
so
o0
0
gas961
4.
7o
67
1
~1.
*90
6.7
t-+
006
Fiur
7
80-b
region
Figure 7.
\
I
\
5 epemer195.Untso
20 GT
dibaicvetialmoin
epesn
1-4mbse-1an sadd res
iedsar
al mtin
vrtca
o
0s
takn fromnephanlyses
of maor clod systms
on
1 September 1965.
4
motion
vertical
adiabatic
800-mb
represent
and shaded areas Units
sec -25
mbGMT
are 10 -1200
fields
motion
50
all vertical
.
Figure 8.
800-mb total vertical motion 1200 GMT 25 September 1965.
0
0
a
0
€
1
016
"....
+
o0
* .
&
.
6~
644
'O
-5
999
'
\.
0i
r-10-1
467
4
O
O
.7
ply
)P1
,
.
.
.
.
.
.
*
*0
r Be
*
*
*
*
*
,"
' •
"
i/I
t °
961'
BB
e*
J
Figure 9.
/
'
~
600-mb adiabatic vertical motion 1200 GMT 25 September 1965.
0
*
.
S\
.. /.
/.
.
*
**
*
**•
* *
*
*
ee
4
*•
*o
*
*
***
o0
s...
*
*
*
*O
. ... .F
*
*~I
. ./ "I . . . ./
.
..
.
..
Figure 10.
....... .
.
,
r.
...
O
*
.
e
.
.
.
*
**
*
*****
. .
.
0
0
*'~
*
* *
.
.
.
.
600-mb total vertical motion 1200 GMT 25 September 1965.
*
*
*
*
*
*
*
*
*
*
*
*
Figure 11.
400-mb total vertical motion 1200 GMT 25 September 1965.
.
.5
.
*
S
.
S-
~~ i~~Jj
il~"."l
O
.
.
.
*
*
*
.
3-
2 ;'ao
-g
-5
;-.-
*
/0
*
*
*
0
*S* ********e
*
**
*. *. \ *.
p* ono**
**5****
*
*
467
*
*
6*7
*16*
*
e
•
O
o.
*
060
o,
46'1
.
.
*
.
.
Figure 12.
•
.
.
.
.
.
.
.
.
.
.
r
.
.
.
80*
200-mb total vertical motion 1200 GMT 25 September 1965.
.
+
~\
.
.
.
6
. . * . /. .
638*
-- 0
-. 2.5.,
.o
n.
•r
.
.
.o
..
O
0
..
"l
s
.
0
2.,
5o
248
'"
* * ,*
*
*
e.r
•
cql
,
\
*
*
*
5
T /o
ri
•
160
*5
.
.
...
-2.5 ,
•1.
40Z
.
.
.
!
,
.
.
.
..
09
.
*
.
.5.
"*
*
*
\
**
2 .
**"
r o
Ole"
**
.
*0*•
**
0
.,
o
0
0 .
\
224no
•o.
5
r
2.5
... . . .
J'
.
...
o.
.
..
..
.
.
. . .
.
,
eo
0
.4
2501
/
7
-)
0
...
86
.
0
961
0
2.5--~
of 90 feet per twelve hours.
Units are in tens
0
'70
ofo
.
.
.jO
.
"-.
*
.*
'
"...
.
.
Units are in tens of feet per twelve hours.
Figure 13.
.
0
.
.
.
0
.
eg o
I
OI
r
0
. 1.-.
e
-
s,
/
5
.. .
,.
950-mb height tendencies 1200 GMT 25 September 1965.
\
40°
Barr, 8., and M. B. Lawrence, 1984: A comparison of large-scale
vertical motion and satelllt-observed cloud systems. Unpublished M. 8. thesis, Department of Meteorology, M.I.T.
Berkofsky, Lo, and E. A. ertoani, 1956: Mean topographic charts
36,
o.,
for the entire earth. Bull. Amer Meteo,
380-34.
. G., and A. Eliassen, 1984:
Charney,
hurricane depression. Jo. Atnos.
On the growth of the
ct., 21, 08-75.
Daunrd, M. B., 1964: On the influence of released latent heat
on oyclone development. J A ie d Meteror, 3, 27-3.
Ba*dmsd,
B. F., and 8. Rosenthal, 1985: On the coputation of
93,
on J M . KA.*,
stream functions from the qind field.
240-252.
Numerical studies of organied cirT. N., 19088
culation in subtropical latitudes. Final report to S48A,
Contract CWB 10877, Department of Meteorology, U. C. L. A.
EKrIshaautt,
titta, T., 1904: On the development of the relatively smllscale cyclone due to the release of latent heat by condener. III, 42, 200-268.
sation. .. Meteor. doo. Jananu,
S185:
Some examples of numerical weather prediction,
with the special emphasis on the development and maintenance
of relatively small scale cyolones.
Ser. II, 43, 148-12.
.7
Mjteor. Ao.anAnan,
Goyama, Ko, 1963: A dynamical model for the study of tropical
cyclone development. New York University. Paper prepared
for the 43rd annual meeting of the A.M. . in New York, Jan.
21-24, 1963.
Phillips, N. A., 1938:
1, 123-178.
Sanders, F., 19688:
photographs.
Geostrophio motion.
eviews
ot Grjeghsin.,
cloud
Large-scale vertical motion end satelite
Final report U. B. Weather Bureoa Contract
Cub-10643, Dept. of Meteorology, M.I.T.
8human, F. G., 197: Numerical aethods in weather prediction:
smoothing and filtering. Mo. Wes,_. Rev., 83, 357-361,
II