POSSIBLE MECHANISMS OF EXPLOSIVE MARITIME CYCLOGENESIS
by
PAUL JOSEPH ROEBBER
B.Sc. McGill University
(1981)
Submitted to the Department of Meteorology and Physical Oceanography
in partial fulfillment of the requirements of the
Degree of Master of Science
at the
Massachusetts Institute of Technology
June 1983
Signature of Author
Department of Meteorology and Ph sical Oceanography
Ma,316
1983
Certified br
Thesis Supervisor
Accepted by
Chairman, Departmental Committee on Graduate Students
UndgrMJ
JUN 2WFROM
Mg; ARARIES
POSSIBLE MECHANISMS OF EXPLOSIVE MARITIME CYCLOGENESIS
PAUL JOSEPH ROEBBER
Submitted to the Department of Meteorology and Physical Oceanography
on May 6, 1983 in partial fulfillment of the requirements for the
Degree of Master of Science
ABSTRACT
A statistical analysis of 12 and 24 hour deepening
rates
for all
surface
lows analyzed on at least two successive
NMC 12 hourly hemispheric surface charts was performed
for
one full
year of data. The basis of the analysis was the
Central Limit Theorem of Statistics, which would require an
approximate
Gaussian distribution of
deepening rates,
provided the mechanism(s) of deepening was (were) identical
in all
cases.
Statistically significant deviations from
the normal curve were found in both
the
12 and 24 hour
studies, with the largest deviations occurring along the
tail of
the distribution associated with most
rapid
deepening.
An attempt was made to iteratively fit two
normal curves of different means and standard deviations to
the deepening spectra;
the attempt was successful in both
the 12 and 24 hour cases, suggesting
that most cases of
explosive
cyclogenesis are the result of some physical
mechanism in addition to or other than ordinary baroclinic
instability.
The climatology of explosive cyclones (Sanders and Gyakum,
1980) was updated to include the 1979-82 cold seasons.
In
addition, a climatology of formation positions, maximum
deepening positions,
and dissipation positions
for all
cyclones in a one year data sample was
compiled.
These
studies
suggested
that the preferred regions of explosive
the
baroclinic
zones;
cyclogenesis
are
primarily
climatological and statistical evidence therefore suggests
that the explosive mechanism is a combination of the
baroclinic process and some other mechanism.
A
quasi-geostrophic
investigation
of
a
particular
formulation of the wave-CISK hypothesis was carried out on
a sample of 18 explosively developing cyclones, using a
modified version of the analytic baroclinic model suggested
PAGE 3
by Sanders (1971) and Sanders and Gyakum (1980),
with
the
parameterization
detailed
in
Mak
(1982).
An
extrapolation of
the
instantaneous model
results
(with
friction) was able to account for 67% of
the sample
variance in 12 hour deepening rates, and 14% at 24 hours.
This
was a significant
improvement over the baroclinic
model, which could account for
only 2% of
the
12 hour
variance, and 27% at a range of 24 hours.
It was suggested
that the improvement in the baroclinic model at
longer
range
was
related
to the problem of
extrapolating
instantaneous results rather
than any
improvement
in
forecast skill.
Evidence was
found to suggest that the
postulated explosive forcing of atmospheric bombs, the CISK
mechanism, is
operative only for the period of most rapid
deepening;
subsequent balance is achieved between friction
and the weaker, baroclinic forcing. A logical relationship
between convective heating
intensity and
geographical
location was established, with the axis of maximum heating
intensity located slightly to the warm side of the mean
Gulf Stream position.
CISK
An attempt was made to apply the model to
the operational
forecasting
of explosive cyclogenesis, with generally poor
results.
It is suggested that this failure is
related
to
the
problem of forecasting the occurrence, duration, and
intensity of the operative CISK mechanism, and that the key
to forecasting explosive cyclogenesis is essentially in the
better understanding
of the
initiation and
continued
evolution of
cooperative convection embedded within the
large scale flow.
Thesis Supervisor:
Title:
Dr. Frederick Sanders
Professor of Meteorology
PAGE 4
TABLE OF CONTENTS
Introduction.........
.........................
.. a
Statistical Analysis and Climatology of Cyclones. .10
.1
Statistical Analysis...
.........................
.12
.2 Climatology............. .........................
.24
I Statistics of Deepenin g Cyclones ...............
.24
2 Geographic Distributio n of Cyclones......
.28
Summary................. .........................
.51
Application of an Analyt ic Model of Wave-CISK ....
.52
Objectives.............. .........................
.52
Review of Wave-CISK..... .........................
.53
Model Formulation....... .........................
.56
.4 Model Climatology of Explosive Cyclones........... .93
Operational Forecasting of Explosive Cyclogenesis. 108
Conclusions....................................... 116
ACKNOWLEDGMENTS...................................118
APPENDIX..........................................119
REFERENCES........................................126
PAGE 5
LIST OF FIGURES
2.01 12 hour deepening distribution (warm iseason
)......16
.02 24 hour deepening distribution (warm Iseason )......17
.03 12 hour deepening distribution (cold iseason .
.04 24 hour deepening distribution (cold iseason ......
.
18
19
.05 24 hour deepening distribution / cold seaso n sum.. 20
.06 12 hour deepening distribution (full kyear). .......
21
ear). .......
22
.07 24 hour deepening distribution (full
.08 24 hour deepening distribution / full year
sum.... 23
.09 Table of confidenc e intervals
Table of deepening times (GMT)........ ............
26
.10 Length of deepenin g periods........... ............
27
.11
Formation positions (warm season)..... ............
32
.12
Formation positions (full year)....... ............
33
.13 Formation positions (Bombs)........... ............
34
.14 Maximum deepening positions (warm seas on).........35
.15 Maximum deepening positions (full
y eaT
)...........36
.16 Maximum deepening positions (Bombs)...............37
.17 Maximum deepening positions, Bombs 1976-79. .......
39
18 Maximum deepening positions, Bombs 1979-82. .......
40
19 Maximum deepening positions, Bombs 1976-82. .......
41
.20 Dissipation positions (warm season)........ .......
42
.21 Dissipation positions (full year).......... .......
43
.22 Dissipation positions (Bombs).............. .......
44
PAGE 6
.23
700 mb height contours, May 1980....... ........... 47
.24 700 mb height contours, June 1980...... ........... 48
.25 700 mb height contours, July 1980...... ........... 49
.26 700 mb height contours, August 1980.... ........... 50
1.01 Model heating profiles................. ........... 6 1
.02 Model temperature profile.............. ........... 64
.03 Model stability profile................ ........... 65
.04 Vertical motion profiles (baroclinic mo del).......67
.05 Vertical
profile of friction........... ........... 70
.06 Vertical motion profile (CISK model)... ........... 73
.07 Vertical profile of frictional updraft. ........... 74
.08 Type B CISK (geopotential tendency)
Type B CISK (phase speeds) . ............... ........ 79
.09 Presidents'
Day deepening profile (baroclinic).... 85
.10 Presidents' Dagy deepening profile (CISK)..........86
.11 Presidents' Day deepening profile (CISK cut off)..87
.12 Presidents' Dag surface analysis (model)..........88
.13 Presidents' Day 500 mb analysis (model)...........89
.14 Presidents' Day surface analysis (observed).......90
.15 Presidents' Day 500 mb analysis (observed)........91
.16 Presidents' Day 500 mb vertical motion
(observed).92
.17 Table of explosive cases modeled................. .95
.18 Heating intensity distribution................... 100
.19 Table of explosive cases (model results)......... 103
.20 Table of deepening periods
(modeled/observed).... 107
.21 Cases within satellite coverage 1980-81(Nov-Feb). 111
PAGE 7
.22 Cases within satellite coverage 1982-83(Nov-Feb).112
.23 Table of operational forecasting results.........115
PAGE 8
1.0
Introduction
The
emphasis
of
this
research
on
explosively
developing extratropical cyclones (bombs) has been twofold.
The
first
goal
statistical
has
been
to
evidence
that
such
manifestations
ordinary
of
a
baroclinic
obtain
different
some
storms
physical
instability.
significant
are
in some way
process
Up to this point, there
has been some question as to the actual existence
phenomena
as
distinct
physical
entities,
inadequate data coverage over the oceans.
approach
than
The
of
such
owing
to
statistical
provides a means of addressing this issue despite
the minimal data available.
The details of
this
analysis
are discussed in the first section.
Assuming that such evidence is found, it would further
be
possible
to formulate a definition of a "bomb" that is
consistent with this evidence, and
more
meaningful
importantly,
investigate
than
however,
the
the
it
possible
is
current
would
therefore
definition.
provide
incentive
heat
continuing
deepening
hoped that
release.
effort
via
such
to
Related
to
adequately
method
could
convective
to these questions is the
forecast
computer or graphical methods.
a
Most
physical mechanisms responsible
for explosive cyclogenesis, such as the role of
latent
somewhat
be
realized
explosive
It has been
based
on
PAGE 9
of
parameters
however, it
thickness);
indeed
are
bombs
if
that
seem
would
1000-500 mb
(e.g.,
flow
scale
large
the
a manifestation of some physical process other than
such
instability,
baroclinic
ordinary
or in addition to
doomed to mixed success, since presumably the
efforts
are
mystery
deepening
some
in
related
be
would
process
fundamental way to smaller scale flow parameters.
The
of
effects
bulk
such
investigators
(1981), Sanders and Gyakum
Gyakum
as
(1980), and Bosart (1981) have suggested
critical
which
convection,
cumulus
possibly
of
are
importance to the development of such storms, can
be represented in terms of the large scale flow through the
wave-CISK
hypothesis
discussed
others.
Building
(1981) among
analysis,
statistical
analytic
explosive
model
of
wave-CISK
formulation as a means for
cyclogenesis.
on
second
the
cyclogenesis,
by Lindzen (1974) and Mak
and
the
as
the
results
section
applied
of
the
deals with an
to
cases
of
tests the feasibility of the
forecasting
of
explosive
PAGE 10
2.0
Statistical Analysis and Climatologg of Cyclones
In order
deepening
to
carry
rates,
it
adequate data base.
baroclinic
successive
tracked
February
12
the
1980
(i.e.,
that
NMC
for
was
a
first
statistical
to acquire an
a
were
analyzed
on
at
extent
of
their
not
believed
continuing
we
two
1981)
from
to
beginning
through January 1981.
the
result
M.I.T.
of
missing
archives.
cold
The
The
season
be serious, since the data set still
represents an unbiased sample of one
case,
least
existence,
resultant break in continuity for the 1980-81
is
predominantly
hourly hemispheric surface charts were
and
(February
of
of
thermal lows and tropical storms
skewed one year sample was chosen as a
data
analysis
necessary
All surface lows
nature
excluded)
were
out
full
year;
in
any
do not know a priori which particular season (if
any) will
show
deepening,
since
statistically
the
significant
deviations
in
cold season is more baroclinic, but
not necessarily more or less Gaussian than the warm season.
The
pressure
latitude/longitude
coordinates
and
central
as analyzed on the NMC charts were recorded at 12
hour intervals for as long as each storm was maintained
a
distinct
entity.
In some cases, the surface low moved
off the map or else maps were missing;
a
as
in other instances,
surface system of a tropical nature was transformed into
PAGE 11
a middle latitude baroclinic disturbance.
were
noted
as
they
occurred,
as
they
subsequent treatment of the raw data.
recorded,
it
was
condensed in the following manner:
the
position
of
maximum
the
deepening,
hour
and
were recorded for each system, along
with the date, time, and, magnitude of the maximum
24
the
was
position
Once
affected
data
formation position,
dissipation
These situations
pressure falls.
12
and
The formation position was taken
to be the first analyzed position of the storm, provided it
had not simply moved into view from a previous location off
of the map.
the
The maximum deepening position was taken to be
position
of
the
most rapid deepening;
storm centered over the interval of
thus, the
position
of
maximum
24
hour deepening was the position of the storm at OOZ, if the
most rapid deepening occurred from
position
last
analyzed
and
the
taken
to
be
position of a storm before it lost its
by
another
storm
or
decay, so that the low center was no longer analyzed
on successive maps.
was
12Z,
The position of dissipation was
identity, either through absorption
final
to
at 12Z, if the most rapid deepening occurred from
OOZ to OOZ.
the
12Z
also
recorded.
The duration (in days) of
each
storm
All pressure falls were subsequently
adjusted geostrophically, based on the principle
that
two
storms of identical pressure gradient would not produce the
same
Thus,
maximum
the
geostrophic
wind
at
different
latitudes.
pressure falls can be adjusted according to the
PAGE 12
formula:
dP(adj) = [sin a / sin b3 dP
where "a" is
some
latitude
the
of
reference
storm.
latitude,
and
Traditionally,
"b"
is
a bomb has been
defined as a total 24 hour pressure drop equivalent
mb
at
the
to
24
60N, so that for the above definition, with "a" set
equal to 42.5N, dP(adj)=-19 mb corresponds to one bergeron,
the lower limit of deepening to qualify as a bomb, adjusted
to middle latitudes.
definition
is
used
This altered version of the
in
explosive cyclones occur
either
case,
these
this study, since the majority of
about
the
axis
the
bomb class.
42.5N.
rapid
In
deepeners
The statistical analysis of deepening
rates will make it possible to formulate a
consistent
of
are arbitrary definitions, presumably
formulated so as to include only the most
in
Bergeron
new
definition
with the analysis, and therefore less arbitrary
than the current definition.
2.1
Statistical Analysis
The basis
Theorem
of
of
this
Statistics,
analysis
which
is
states
the
Central
that under certain
conditions probability distributions will tend to
the normal curve.
Limit
approach
If we let Xn, n= 1,2,...,N be a set of N
PAGE 13
stochastic variables (e.g.,
the
random
outcome
denote the
number
of
of
the stochastic variable
process
number
independent
X1
is
1), and we let j
realizations
of
that
process, then we can define a variable Yj such that:
=
Yj
X1,j +
X2,j +
...
+
XN,j
Thus, the summation is over processes, and
variables
restrict
Yj
sums
such
density
limiting
infinity
the
Xn, j
the
probability
the
are
of
that
the
additive effects.
all
have
an
distribution
of
the
is the normal distribution.
Yj
as
identical
N
approaches
This is the simplest
Limit
Theorem.
Liapounov Central Limit Theorem requires only that the
random variables Xn,j be independent, and
identically
distributed
not
necessarily
for the distribution of the Yj to
be normally distributed.
would
If we
with mean u and standard deviation s,
(and most restrictive) form of the Central
The
stochastic
In terms of
this
analysis,
one
expect the distribution of the deepening rates to be
normal provided the N underlying processes (the
nature
which
baroclinic
have
processes,
vorticity
not
correspond
advection,
(realizations).
processes
been
be
to
etc.)
specified,
warm
are
but
advection,
the
same
for
of
differential
in
This does not require that each
all cases
of
these
identically distributed, but only that these
processes and no others are operative in all
realizations.
PAGE 14
The
restriction
correct,
of
although
interact.
independence
some
storms
should be approximately
in
proximity
may
We can say little about the interactions between
cyclones, and the effect of cyclones
flow
close
and
subsequent
systems,
on
the
upper
level
but it seems reasonable to
assume that such effects would be statistically negligible.
Analyses of 12 and 24 hour deepening distributions for
warm
season,
cold season, and full year data samples show
statistically significant deviations from the normal curve,
the
largest
deepening
deviations
curve
associated
(fig.
with
2.01-08).
The
the tail of the
analysis
was
accomplished using a chi-square goodness of fit test at the
5% level of significance;
of
erroneously
thus there is a
rejecting
the
null
5%
probability
hypothesis
that the
distributions are Gaussian and concluding the distributions
are not normal.
An attempt was made to iteratively fit two
normal curves with different means and variances to each of
the
cold
season
and
full
year
24
hour
distributions;
it was discovered that the sum of
normal
provided
there
curves
were
no
longer
observed distributions.
a
deepening
the
two
good fit to the data, that is,
significant
deviations
from
the
The statistics of the distribution
associated with most rapid deepening with mean -22.3 mb and
standard
deviation
6.9
mb
indicate
definition of a bomb is quite adequate;
that
68%
the
of
Bergeron
explosive
PAGE 15
cases
(as
defined by the deepening distribution with mean
-22.3 mb) are included by the definition, while only 2%
of
non-explosive cases are included, at least in this sample.
These results provide strong evidence
cyclogenesis
is
analysis,
based
on
the
pressure analyses over data
demonstrate
conclusion.
information
this
an
on
sparse
feature
the
oceans,
further
physical
explosive
investigation,
that
such
oft-times underestimated NMC
one
this was attempted, and will be
chapter.
The fact
process
deepening
must
was
able
emphasizes
Unfortunately, such a procedure
responsible for the
such
explosive
a process distinct in some meaningful way
from ordinary baroclinic instability.
an
that
can
or
give
to
this
no
processes
mechanism.
For
turn to other methods;
dealt
with
in
the
next
PAGE
16
75
WARM SEASON
-12 hour
65EP
55-
(MAY
80-AUG 80)
APadjusted
=
3.54
X
z 0.66
a-
=
3.12
y
= 3.76
N
a
365
Cn)
w
(n
45-
0
( 35-
z
25-
15-
5
15
10
5
0
-5
APadj
-10"
-15
-20.
-25
(mb)
FIGURE 2. 01
Deepening distributions with computed
sample statistics,
Figures 2.01-08. AP and 0 are the sample mean and standard
deviation, N is the total number of cases, '(I and 12 are
the coefficients of
skewness and kurtosis, respectively,
measures of the relative centeredness and peakness
of the
distributions.
The
dashed lines represent the two normal
distributions of different means and variances;
the
solid
line in those figures represents the sum of the two distributions.
-30
PAGE 17
15
10
5
0
-5
-10
APadj (mb )
FIQURE 2.02
-15
-20
-25
-30
PAGE 18
65
55
U)
w
45
U.
0
35
z
25
15
15
10
5
-5
0
APodj
-10
(mb)
FIGURE 2. 03
-15
-20
-25
-30
<n
LU 4
5
-
0356
z
2515525
P0
15
1O
5
0
-5
-10
-IS
APadj( mb )
FIGURE
2. 04
-20
-25
-30
-35
-40
-45
-50
APodj ( mb )
RICURE 2. O~
TI
0
P1
p3
0
PAGE 21
APadj (mb)
FIGURE 2.06
w
-
4 5
o35 -
BOMB
(BERGERON)
z
25-
15-
525
20
15
10
5
0
-5
APodj
-10
( mb)
-15
-20
-25
-30
-35
-40
-45
-50
FIGURE 2. 07
fi
lu
Cl)
u45u-
0 356
z
25-
520
15
10
5
0
-10
-5
APodj
-15
-20
-25
-30
-35
-40
-45
-50
(mb )
FIGURE 2. 08
-u
(0
PACE 24
2.2
Climatology
Statistics of Deepening Cyclones -
2.2.1
A wealth of climatological data can
the
one
year
be
gleaned
from
cyclone sample, and in the process, one can
learn some additional information about explosive cyclones.
Consulting
the data in figures 2.06-07, it is evident that
the average pressure fall is
greater
in
only
one
to
two
millibars
the 24 hour sample than in the 12 hour sample.
This suggests that, on average, most of
the
deepening
is
accomplished in a time period less than 24 hours.
The main
increase is in the dispersion (broadening) of the
24
distribution,
continue
confirming
that
deepening throughout the period.
implications
some
cases
do
hour
This result has important
for any attempt to model the evolution of the
cyclogenesis.
Since it has been shown that the bimodal distributions
of
the
24 hour cold season and full year sample deepening
rates can be approximated by the sum of two normal
it
is
possible
to say something about the true means and
variances of the distributions.
table
2.09.
4.5
The results are
shown
in
One can say with 95% confidence that the mean
deepening of the 24 hour cold
between
curves,
and
5.5
mb,
season
baroclinic
lows
is
and that the mean deepening of
PAGE 25
explosive cyclones in that period is between 21.0 and
mb.
The
23.5
results for the full year data are similar.
mean deepening of the baroclinic laws is
between
The
3.4
and
4.1 mb, and the mean deepening of the explosive cyclones is
between 21.2-and 23.4 mb.
The question
bias in the sample was examined.
of
possible
time
The results of table 2.09
seem to indicate no discernible bias in the 12 and 24
deepening
cases
samples.
according
No
to
attempt
ocean
hour
was made to stratify the
basin,
where
the
potential for bias might have been expected;
in the study of Sanders and Gyakum (1980),
greatest
this was done
and no bias
was
resolution,
the
found in their sample of bombs.
Within
question
of
the
the
limits
length
of
a
of
12
the
hour
deepening period of all
storms and of bombs can be examined.
The deepening
period
is the time interval over which successive 12 hour analyses
indicate a continued fall in storm central
results are plotted in figure 2.10.
pressure.
The
By 36 hours, more than
75'4 of all the lows in the sample had ceased deepening,
compared
to less than half of the explosive cyclones.
average deepening period of all lows was
that
24
hours,
of explosive cyclones was about 45 hours.
bear in mind when
interpreting
these
results
as
The
while
One should
that,
for
example, a 45 hour deepening period is really 39h+/-6h, and
that this analysis can reveal nothing about new sources
of
PAGE 26
energy
introduced
source
has
interpreted
been
into
a system after the initial energy
dissipated.
Thus,
a
system
that
is
as deepening 30h+/-6h may really have deepened
for 18 hours, remained steady, or filled for 12 hours,
and
deepened again for the last 6 hours when a new pulse of
--124h
124h
----124h
124h
Cold
Cold
Full
Full
95% CONFIDENCE INTERVALS
DISTRIBUTION
N
6
------------------------------------------------------Season(baroclinic)!634l
4.5<Ckp
5.5 1 6.6<6<7.4
Season(bombs)
11191 21.1C1p1C23.5 I 6.1<C7.9
------------------------------------------------------------------------------------------------------------Year(baroclinic)
99751
3.4< p< 4.11 6.0<6<6.5 I
Year(bombs)
11551 21.2CiW,<23.4 I 6.2CdC7.8 i
Confidence intervals for the true mean, p, and
standard deviation, do of particular deepening distributions of
sample size N.
DISTRIBUTION
TIME DISTRIBUTIONS
-i
N
11 12Z
112h
(12Z-OOZ,OOZ-12Z) 1
124h
(12Z-12Z,00Z-OOZ) :
"Bombs(12Z-12Z,OOZ-OOZ) I
1157
1178
115
H1
!H
H
'FIGURE 2.09
619
579
58
%
OOZ
.
53.5 1
49.2 1
50.4 1
538
599
57
46.5 1
50.8 I
49.6
1
PAGE 27
DEEPENING
PERIODS
1001
.J
0
H80
Lt.
0
z
W
0
Cr 60
a-
T (hours)
all
Cumulative distributions of deepening periods for
The statistics of the deepening perstorms and for bombs.
deepening
T is the mean
iod distributions are also given;
e is the standard deviation, and 11 and 12 are the
period,
coefficients of skewness and
kurtosis, respectively.
FIGURE 2. 10
PAGE 28
energy was introduced.
resolution,
This is an artifact of the 12
hour
and there is little that can be done about it,
except to urge caution when interpreting the results.
This
point is stressed, as the length of the deepening period is
crucial to the model analysis of the next chapter.
2.2.2
Geographic Distribution of Cyclones
A
series
of
climatological
maps
data
were
drawn
concerning
plotted
utilizing
positions
maximum deepening, and dissipation.
was
up,
-
of
the
formation,
The raw frequency data
on a 5 by 5 degree latitude/longitude tessera
grid, and smoothed by taking 4 times the central value plus
the 4 adjacent values and dividing by 8.
The
warm
season
and
formation positions (fig.
full
2.23-26,
distributions
of
2.11-13) both show marked maxima
east of the Rocky Mountains.
(fig.
year
Mean 700 mb
height
patterns
obtained from Monthly Weather Review) for
May-August 1980 show that in the first 2 months of the warm
season
there
was
substantial
flow
across
the mountain
barrier, but, by July, the flow had all but ceased over the
southern
portion
of
the
mountains.
August 1980 saw the
return of the flow southward, though it could not penetrate
as far south as in the first two months of the warm season.
Thus, the warm season maximum can probably be attributed to
PAGE 29
the
same
cause as that of the cold season, the process of
lee cyclogenesis, though it is not clear why
is
comparable
this
maximum
to the cold season value unless some of the
warm season events represent thermal lows as
well.
Minor
maxima are also located over Japan, and to a lesser extent,
over the southeastern United
(45N,45W),
the
and
Alaskan
the
Atlantic
Ocean
the lee side of the Canadian Rockies along
These - appear
border.
cyclogenesis
States,
and
to
reflect
lee
the baroclinic zones associated with the
warm ocean currents off the coasts of Japan and the eastern
United
States.
actually reflect
currents.
The
The
a
areas off of Japan and the U.S.
combination
of
orography
and
may
ocean
maximum out over the Atlantic is likely an
area of redevelopment, with old lows
cutting
off
in
the
cold air and new lows forming alongside the baroclinic zone
associated with the Gulf Stream.
In the full year data, these maxima
are
intensified,
increases
occurring along the eastern
seaboard-of the United States.
The maxima forced by vortex
with
the
stretching
greatest
on
the
lee
side
increased proportionately;
are
most
reflecting
There
strongly
the
of
it is the
activated
enhanced
mountain
in
coastal
are
areas
that
winter
the
baroclinicity
barriers
of
these
months,
areas.
is a noticeable maximum in the eastern Pacific Ocean
(42N,1 55W) not apparent
in
the
warm
season
climatologg
PAGE 30
which
becomes
established
in the winter months.
probably associated with the breakdown of the
Pacific
high aloft (figure 2.24):
the wintertime regime.
shows
maxima
The
associated
areas discussed previously, with
year
Pacific
maximum
does
season
bomb
distribution
all the major cyclogenetic
the
induced by orographic effects alone.
the
warm
and the establishment of
full
with
This is
not
exception
of
those
It is noteworthy that
appear
on
the
bomb
distribution, suggesting that this area is not a birthplace
of explosive cyclones.
The positions of maximum deepening (fig.
2.14-16) are
well correlated with the positions of initial cyclogenesis,
located slightly upstream.
The Pacific Ocean region
where
the maximum in formation position was found also displays a
maximum in deepening;
initial
formation
this maximum is not downwind of
position, and the magnitude is greater,
suggesting that other cyclones moving through
are
intensifying.
This
area
this
quite
baroclinic,
owing
to
moving through the Bering Straits
ocean
water.
region
has no warm ocean currents
akin to the Kuroshio or the Gulf Stream, but it
be
the
outbreaks
and
can
still
of Arctic air
across
the
warmer
The climatology suggests that storms that do
form in the area are redevelopments of existing storms that
subsequently
move
into
the
Gulf
of
Storms moving through the area deepen in
Alaska
and decay.
response
to
the
PAGE 31
enhanced baroclinicity.
The maximum deepening positions of the
year
sample
in
and
the
baroclinicity
cyclones.
Gulf
These areas are general baroclinic zones, however,
so the climatology does not suggest that any process
than
the
are well correlated with the positions of the
warm ocean currents, primarily the Kuroshio
Stream.
bombs
is
other
necessarily operative in explosive
I
Geographic distribution of formation position, maximum
deepening
position, and
dissipation
position, Figures
2. 11-22. Frequencies are smoothed from the 5 degree
latitude by 5 degree longitude tessera grid of raw data, according to the formula of 4 times
the central
frequency
plus each of the 4 adjacent frequencies, all divided by 8.
FIGURE 2. 11
IT
FIGURE 2. 12
FIGURE 2. 13
FIGURE 2. 14
PAGE 26
w
Li
120.
1W1
I
-u
IT'
FIGURE
2.16
'-4
PAGE 38
The maximum deepening positions of
bombs
(12Z-12Z)
for
the
1976-82 are plotted (fig.
periods
all
1976-79,
2.17-19).
the
24
1979-82, and
There are some
but non-negligible differences in the samples.
hour
small
The maximum
off the eastern coast of the United States strengthened and
moved
closer
to land in the later three year period.
Pacific Ocean maximum around 145W was also greater
latter
period,
somewhat less.
between
mean
in
The
the
but the maximum off the Japanese coast was
There
flow
appears
to
be
little
correlation
fields at upper levels (e.g.
700 mb),
and the day to day c ccurrence of of explosive
cyclogenesis
in
judging
these
areas.
A
better
method
relationship between the upper level
and
of
intense
the
surface
would be t o perform an analysis as in Sanders and
features
Gyakum (1980), where
with
individually
study, the
authors
each
explosive
event
was
respect to the flow at 500 mb.
found
the
relationship
examined
In that
between
the
explosive cyclones a nd the upper trough to be qualitatively
typical of deepening baroclinic
storms
intensifying
lows,
with
such
a
studu
of
the
within or just poleward of the region
of maximum 500 mb wi nd and baroclinicity.
that
most
It is not likely
could explain the small variations in
distributions of explosive cyclones found in the two 3 year
samples, however, and was not attempted here.
FIGURE 2. 17
PAGE 40
oot
+~
ol, 0t
+.
w
w
0
LL
(to
VTO~
40
41
FIGURE 2. 19
FIGURE 2.20
4)
P1
4~I
K'
2"
0
FIGURE 2.21
4:4
C~)
FIGURE 2.22
UE
r
4.4
PAGE 45
The 6 year pattern of bomb distribution clearly
the
eastern
Pacific
maximum.
shows
Apparently, the blasts of
cold, Arctic air were sufficient to compensate for the lack
of exceptionally warm ocean currents, since this feature is
almost of the same magnitude as the maxima associated
the Kuroshio current and the Gulf Stream.
with
Another possible
explanation is an increase in available data, but it
unlikely
that
this
location
seems
is frequented by ships more
than some other areas of the Pacific
Ocean
which
do
not
show any such maximum.
The distribution of dissipation positions are shown in
figures
2.20-22.
diffuse, though
suggestion
of
The
maxima
warm
are
season
pattern
discernible.
is somewhat
There
is
the
a preference for the Gulf of Alaska region,
as well as the area between Newfoundland and Greenland, and
to the south of Iceland.
in the midwest of the U.S.
Soviet
Union.
These
There are slightly greater maxima
and the easternmost tip of
surface
features
appear
reflected in the 700 mb pattern for June, which
to
show
the
be
mean
low centers in the Alaskan coastal region and the Kamchatka
peninsula.
the
The full year distribution greatly
Gulf of Alaska as a region of cyclolysis, unparalleled
anywhere else in
about
accentuates
the
the
southern
Northern
tip
Hemisphere.
The
regions
of Greenland again show relative
maxima, and there is a maximum
over
the
Aleutian
chain,
PAGE 46
with
no
evidence
of
a
maximum over Kamchatka.
maximum over Manchuria suggests this
final
resting
continent.
place
of
lows
area
migrating
located
cyclones,
and
cyclones.
It
the
in
the
Gulf
of
Greenland/Iceland
likely
the Asian
area
for
for
Pacific
Atlantic
is clear that the observed mean low centers
rather
than
regions
Cyclones move into the regions
fixed
over
Alaska
in these areas are indications of the final
lows
the
The dissipation positions of bombs appear to be
primarily
of
as
A minor
of
active
resting
place
cyclogenesis.
and remain in a more or less
position until another storm moves into the area and
absorbs them, or else simply decay until they can no longer
be identified as cyclones.
PAGE 47
4
MAY 1980
Mean 700 mb height contours (dam) for May 1980.
Mean 700 mb height contours for
selected months,
published by Monthly Weather Review, Figures 2.23-26.
FIGURE 2. 23
as
PAGE 48
Mean 700 mb height contours (dam) for June 1980.
FIGURE 2. 24
PAGE 49
Mean 700 mb height contours (dam) for July 1980.
FIGURE 2. 25
PAGE 50
Mean 700 mb height contours (dam) for August 1980.
FIGURE 2.26
PAGE 51
2.3
Summary
This
section
cyclones,
and,
has
dealt
with
particularly,
the
climatology
of
of explosive cyclones.
The
question of explosive cyclogenesis as a phenomenon distinct
from
ordinary
baroclinicity
has
been
examined,
and
statistical
evidence
hypothesis.
The climatology of cyclones indicated that the
was
found
that
supported
preferred regions of explosive cyclogenesis
baroclinic -zones,
are
this
primarily
areas which support the development and
continued existence of ordinary low pressure systems;
finding
is
consistent
with
that
of
Sanders and Gyakum
(1980), that explosive cyclones exhibit a
the
relationship
storms.
mechanism
The
operative
evidence
thus
suggests
that
mechanism.
explosive
(1921),
who
studied
mechanism.
baroclinic
particular
cyclogenesis in detail, and suggested
that the bulk effects of cumulus convection in
the
other
This is entirely consistent with the results of
Gyakum (1921) and Bosart
of
the
in cases of explosive cyclogenesis is
some combination of the baroclinic process with some
with
to
upper level flow that is qualitatively similar to less
intense
cases
this
process
was
the
combination
explosive forcing
An investigation of this process was conducted;
the results will be detailed in the next section.
PAGE 52
3.0
Application of an Analytic Model of Wave-CISK
3.1
Objectives
The results of the
statistical
evidence
previous
that
section
there
process operative in most cases of
that
is
in
some
baroclinicity.
cannot
other
tell
methods
strong
a distinct physical
explosive
cyclogenesis
fundamental way different from ordinary
Unfortunately,
us
is
provide
a
statistical
analysis
anything more about the mechanism itself;
must
be
utilized
to
continue
the
evidence
that
investigation.
Gyakum (1981) has provided substantial
in
the
GE
II
effects
of
convective
baroclinic
Presidents'
mechanism
case, the storm was driven by the combined
process.
Day
condensational
heating
and
the
Bosart (1981), in his analysis of the
storm
of
1979,
suggested
that
was operative in that case as well.
this
Rather than
perform additional case studies, where the data sets
would
likely be less complete, it might be helpful to analyze the
convective condensational heating process in
explosive
cases.
This
approach
sample
of
may make it possible to
ascertain whether or not this combination of
and
a
baroclinicity
convective latent heat release is a likely explanation
for the incidence of explosive cyclogenesis.
PAGE 53
A
sample
cyclogenesis
of
21
was
cases
of
analyzed,
east
within
analytic, quasi-geostrophic model
wave-CISK
(conditional
suggested
incorporate
diabatic
the
by
the
of
Mak
to
the
(1982)
effects
explosive
context
modified
instability
parameterization
coast
of
an
include
a
second kind)
in
order
of convection.
to
It was
hoped that the evolution of these storms could be accounted
for
within
the
limits
of
this
simple
model, and that
a
simple
forecast
scheme
for
subsequently,
explosive
cyclogenesis could be developed.
3.2
Review of wave-CISK
In
wave-CISK,
interacts
with
cumulus
convection
cooperatively
the large scale pattern of convergence and
divergence to
amplify
conditionally
unstable
the
synoptic
disturbance.
In
a
environment, forced ascent through
low-level convergence associated with the propagating large
scale
disturbance produces convective towers, which act to
produce a solenoidal field of vertical motion
release
of latent heat.
is
an
important
wave-CISK;
the
CISK
pumping
the
boundary
in
the
Thus, the large scale disturbance
is amplified, and a positive feedback loop is
There
through
distinction
mechanism
layer,
established.
between
operates
whereas
CISK
through
the
process provides moisture convergence through the
and
Ekman
wave-CISK
inviscid
PAGE 54
increased convergence)
modifications (i.e.
itself;
wave
are made possible through the inclusion of viscous effects.
The
advantage
theoretical
is
mechanisms
wave-CISK
in
of
the
both
flow
scale
large
velocity at some low level.
to
the
vertical
the
formulation,
a
such
Without
of
parameters
specifically
field,
and
relate the
to
ability
magnitude of the convective heating to
CISK
the
one would have to model the convective elements themselves,
and
calculate
the
heating
at
directly,
substantial
increases in difficulty and cost.
In CISK parameterizations, the latent heating is taken
to
be
distributed
specified profile.
the
in
the
Unfortunately, it has been
scale
instability
a heating maximum at low levels
a profile would
entrainment
version
or
presumably
shown
that
shallow
the
of
specifying
results
as
the
convection.
Arakawa-Schubert
vertical
to
either
severe
(1974)
cumulus
this avoided the problem
heating
mechanism
Such
Stark (1976) used a
profiles
a by-product of the calculations.
suggested that the CISK
associated
(Davies, 1979).
correspond
parameterization in a CISK model;
of
some
CISK mechanism is sensitive to the vertical profile of
heating, with maximum large
with
to
according
vertical
was
not
since
it
His results
effective
in
amplifying a disturbance, but the study has b.een criticized
PAGE 55
(Lindzen and Stevens, 1978) as an inappropriate usage of
sophisticated
cloud model.
heating is implicitly
that
is,
the
time
In the usual formulations, the
assumed
to
occur
that
is
infinitesimal.
of
such
a
instantaneously,
it takes the entire cloud ensemble to
process the low level moisture supply
heat
a
noticeable
on
and
the
convert
large
it
to
scale
is
There is little data on the true time scale
process,
but
estimates
have ranged up to 12
hours.
Probably the most serious defect of wave-CISK
lack
of a short wave cutoff.
is
the
The fastest growing scale is
the smallest one, that of an individual convective element.
Even
when a closed parameterization scheme is used (Stark,
1976), this feature prevails.
with
this
problem
has
been
The usual method for dealing
to
introduce an additional
frequency dependent heating parameter to suppress the short
wave modes.
This prescribes a scale, and thus represents a
quick-fix rather than a solution to
(1979)
included
mode
problem.
Davies
a time lag in the heating response to the
forcing, and found that for a lag of
unstable
the
12
is of finite wavenumber.
hours,
the
The time scale of
12 hours seems long, and Davies himself noted that the
effect
was
lag
not a viable means of scale selection based on
this consideration.
some
most
Thus, it is clear that there are still
significant theoretical problems in current wave-CISK
PACE 56
formulations that remain to be solved.
Model formulation
3.3
his earlier theoretical
from
follows
which
(1982),
Mak
findings (Mak, 1981), will be used.
suggested
Mak
short
the
that
problem
cutoff
wave
In
1973).
reaches
a
Some investigators
value.
critical
heating
heating
intensity
is
larger than that found in the atmosphere, one can
ignore the result;
a
CISK
B
the
when
scale
have argued that since the critical
somewhat
is
B CISK, the growth rate becomes
Type
infinite for a particular length
intensity
work,
earlier
In the
intimately related to the phenomenon known as Type
(Bates,
by
suggested
the wave-CISK formulation
In this work,
that argument seems unsatisfactory from
theoretical point of view, and Mak demonstrates that the
arises
phenomenon
same
the
problem,
cutoff
shortwave
from
namely,
as
mathematics
the
relation
the
of the
heating to the divergent (secondary) component of the flow.
If
the forcing of the heating is related to the rotational
(primary) component of the flow alone at some
problems
velocity field still appears
term
of
in
non-zero
the
divergence
the vorticity equation, and is expressed in terms
of the vertical
While
both
The divergent component of the
eliminated.
are
level,
this
derivative
avoids
the
of
the
necessity
vertical
of
p-velocity.
introducing
a new
PAGE 57
parameter and allows the flow to determine the scale of the
cutoff,
it
between
the
formulation
is
not clear how the atmosphere distinguishes
two
vertical
introduces
velocity
an
intrinsic
parameterization (on
the
Coriolis
approximately
parameter,
components.
order
of
time
the
2-4
lag
in
the
inverse
of
the
hours
for middle
latitudes), since the rotational component of the
strongly
influenced
by
This
flow
the rotation of the earth.
the internally determined time scale
of
the
is
Thus,
large
scale
(rotational) flow dictates the time scale of the conversion
of the low level moisture to heating that is noticeable
on
the large scale.
The problem
can
quasi-geostrophic
be
omega
easily
formulated
equation
generalized to include diabatic effects
in
through
the
p-coordinates,
(Haltiner,
1971).
Thus,
CDP?
LF(dynamic/adiabatic)
JLF(diabatic)J
The vertical velocity can then be written as the sum of two
components,
above:
directly
related
to
the
two
forcing terms
PAGE 58
W = Wa + Wd
where
is the vertical p-velocity induced by the
Wa
dynamic/adiabatic forcing
Wd
is the vertical p-velocity induced by the
diabatic forcing
problem
The parameterization
vertical
dynamic
is
(convective latent heat)
in
relating
at some low level;
p-velocity
q
the
to
this was
done in the following way:
4 = -Cp
E h(p) Wa*
is the heat capacity at constant pressure
where Cp
E
is a heating intensity parameter (K/mb)
h(p)
is the vertical heating profile
Wa*
is the vertical p-velocity at p=PL induced by dynamic and adiabatic effects
A continuous, moist (but
not
necessarily
layer from p=PO to p=PL is postulated;
saturated)
the moisture supply
to the condensational process is assumed to be proportional
to
-Wa*,
moisture
certain
the
and to the specific humidity of the layer.
flux
total
heating
is
entirely
precipitated
out,
This
giving
a
heating related to the vertical integral of
profile,
h(p).
The
heating
intensity
PAGE 59
parameter,
E,
the layer.
One can therefore relate the heating
to
the
crudely
ratio
mixing
represents the moisture content of
intensity
the layer, as will be discussed
of
later.
One can, of course, relate the moisture flux to the
entire
vertical
p-velocity
(rotational plus divergent) at
some low level (p=PL), but the resultant flow would exhibit
the
unwanted
feature
of Type B CISK.
An example of this
will be provided later, within the context of the model.
For the vertical profile of heating, two idealized but
reasonable
profiles
have
been
chosen.
<-H(p)
p / PO I
They are of the
form:
hI(p) =
>
h2(p) = { K H(p) / p
where H(p)
= 0 for p > PL,
p < PU
= 1 for PL > p > PU
K
= (PU -
PL) / 2 PO ln(PU/PL)
These are essentially top-hat profiles, one with a low
level
maximum
at
p=PL,
the
other
maximum at p=PU.
These profiles
3.01,
850 mb and PU= 400 mb.
with
PL=
are
with
an upper level
pictured
level
with the cloud top.
figure
One can associate
the PL level with the lifting condensation level,
PU
in
and
the
Although 850 mb is probably
PAGE 60
850 mb may represent the level of
large scale instability.
field is stronger than
profile
represents
adiabatic
at
effects
more
slightly
of
that
between the
profiles
the
with
the
two.
two
effective (15%) in producing deepening
this
level maximum;
than the high
with
performed
were
the
not
the results indicated that the low level maximum
profiles;
was
calculations
sample
(and
of
air
heating
subsidence),
hI(p) profile is probably the more realistic
Some
the
alone
compensating
the
while
Since
moist.
diabatic
through
warming
the vertical motion
levels,
lower
relatively
still
is
itself
since
convergence,
moisture
optimum
this level yields the maximum
showed
calculations
sample
atmosphere,
maritime
slightly too high for the LCL in the
two
with
Davies,
although
profiles
maxima
at
to
be
result
is
consistent
he found the discrepancy
greater,
with
heating
low levels to be about twice as
effective as those at high levels.
PAGE 61
250
HEATING PAOFILES
300350Pu
400 --
450500550600-
h (p)
a 650-
H(p) P
P
700h 2 (p) =KH(p)
750-
P
800850----
PL
9009501000
.2
.4
,
.6
FIGURE 3. 01
.8
1.0
PAGE 62
(unconditional)
This parameterization allows negative
this is done primarily for convenience, as it is
heating;
clearly unphysical.
been
In the
atmosphere,
it
has
total convective heating can be
the
that
suggested
tropical
looked upon as the sum of mean and perturbation quantitiesi
the
mean
is taken to be of the same magnitude as
heating
the perturbation, so
never
any
negative
It is not clear that this interpretation remains
heating.
in
valid
is
there
that
middle
the
in
activity
convective
there,
however;
latitudes,
a sense propagates along with the
and
large scale pattern of convergence and divergence,
The
single mode.
appears
is
akin to a nondispersive wave packet than a
more
therefore
the
main
heating
unconditional
of
effect
to be an enhancement of the effective amplitude of
thus,
the CISK heating, and
of
consequent
growth
rates
(Davies, 1979).
The
CISK
parameterization
was
into
inserted
an
analytic, baroclinic model after Sanders (1971) and Sanders
and Gyakum (1980).
relationships
In
between
this
the
way,
model
one
can
examine
the
parameters and the CISK
mechanism, and hopefully learn something about the process,
accept
the black box approach of a numerical
rather
than
model.
The main disadvantage of such an approach
the
is
that
results, strictly speaking, are diagnostic rather than
prognostic;
the instantaneous results must be extrapolated
PAGE 63
in
some
meaningful
way if one wishes to make a forecast.
Instead of starting ,with an
approximate
exact
system,
and
obtaining
results, as with a numerical model, one begins
with an approximate system, and obtains exact results.
The following thermal structure was assumed:
A
T(xip) = Tm(p)
-
( ay + T Cos kx Cos ly
)
6
Tm(p) = Tm(PO){ p/PO )
The term
variation
in
of
the
representing a
with
parantheses
temperature
constant
intensity
a,
represents
and
field,
meridional
the
the
the
horizontal
first
temperature
second
term
gradient
representing
a
two
A
dimensional
harmonic
horizontal
variation
with
amplitude
T.
variation is taken as constant in the vertical,
representing fairly well the small change in
intensity
horizontal
the
temperature
troposphere.
wavelength
The
contrasts
through
of
maritime
The term Tm(p) represents the average over
in
a
X and Y on a constant pressure surface, and
essentially provides a label
the
isotherms
at
Tm(p)
reveals
that
the
temperature drops off monotonically with height,
that
is,
level.
there
The
is
no
expression
isothermal
appears to represent
the
for
for
region above the tropopause.
atmosphere
quite
well
in
each
It
the
PAGE 64
troposphere, the region of primary interest for this study,
as shown in figure 3.02 for the Presidents' Day case.
P(mb)
200
210
220
230
240
250
T (*K) J
FIGURE 3.02
260
270
280
290
PAGE 65
In
factor,
this
at
study,
the
expression
The
the
is derived as in Sanders (1971);
value of temperature adopted is
Tm(p).
for
the
stability
the constant
vertical
average
adoption of the constant value of temperature
in the expression for the stability factor appears to
little
difference,
except
in
the
troposphere, where it is too large.
small,
however,
Presidents'
of
as
shown
in
upper
make
reaches of the
The discrepancies
are
figure 3.03, again for the
Day case.
P(mb)
250
300 -m(Tm)
400-
500-
PRESIDENTS' DAY CASE
2 -19- 79 ,122
600-
STABILITY PROFILE
700-
800-
-900-
1000,
5
15
25
35
45
a- (P) (10
-3
55
65
J Kg~ mbz )
FIGURE 3. 03
75
85
95
PAGE 66
The geopotential field is determined at
the
atmosphere,
as
in
Sanders
hydrostatics, the geopotential
From
the
relations
is
relative
(1971);
known
at
base
thus,
of
from
levels.
all
for temperature and geopotential, one
can easily obtain relations for the
geostrophic
the
vorticity
geostrophic
wind
and
at any point in the three
dimensional model atmosphere.
The
identical
procedure
for
solving
to that in Sanders;
the
omega
equation
is
the boundary conditions are
that W go to zero at the top and bottom of the troposphere,
p=PO
and
p=PT.
the appendix.
The forms of the solutions are listed
in
The forcing of the vertical motion terms are
identified as in Sanders (1971),
to wit:
W11;
the advection of the component of thermal vorticity
due to the gradient of the perturbed temperature
field by the component of the thermal wind due to
the meridional temperature gradient.
W12;
the advection of planetary vorticity by the component of the thermal wind due to the gradient of
the perturbed temperature field.
W2 ; the advection of surface relative vorticity by the
component of the thermal wind due to the meridional
temperature gradient plus the advection of the
meridional temperature gradient by the surface wind.
W3 ; the advection of the perturbation temperature by
the surface wind.
Profiles for the maximum values of these parameters
in figure 3.04 for the Presidents' Day storm.
appear
PAGE 67
P(m b)
wA
(10-
3 m
mbs-
)
FIGURE 3.04
PAGE 68
Although
in
a
wave-CISK
formulation
it
is
not
necessary to include viscous effects, common sense suggests
that
friction
evolution
of
likely
an
plays
intense
intensifies rapidly.
an
important
storm,
role
particularly
The inclusion of such
in
the
one
effects
that
would
also make it possible to examine, within the limitations of
the model,
attained
the
supposition
prior
to
occlusion
derivation begins by
pressure
gradient
that
frictional
in
postulating
explosive
a
balance
balance
is
cases.
The
between
the
force, the Coriolis force, and friction
in the planetary boundary layer:
f
K X ( V -
Vg
Applying the operator K.(VX
)
( d'/dz ) /p
)
to
the
= F
above
yields,
in
pressure coordinates:
K-(V X F )
-g K(
neglecting the variation of the
V
X dr/dp
Coriolis
)
parameter.
The
form of the stress is specified according to:
=ot
( p/PO
where v is the unit vector
in
evaluating
IV!
the
parameter
7O
P Cd IV! v
the
direction
in
the
of
equation
V.
In
for the
PAGE 69
stress,
it
seems
reasonable
geostrophic surface velocity;
i
= Vg(max)
to
choose
the
maximum
thus:
( b $ ) / 8 fO
For n sufficiently large, T(p) goes to zero above the
The
expression
established
available
form
data.
for
the
in
fairly
The
formulation,
designed
while
capturing
still
stress through the PBL.
profile
of
the
surface
good
vertical
to
the
maximize
stress,
TO,
agreement
profile
analytic
exponential
is a well
with
is
an
ad-hoc
decrease
of the
vertical
force through the PBL for the
Presidents' Day case, with the value of n equal to 9.
profile
of
the
frictional
the
convenience
Figure 3.05 displays the
frictional
PBL.
force
The
is quite close to that
produced by the formulation used in Gyakum (1983b).
PAGE 70
7
1
E
0.
7
9
F (10-5 m- 2
FIGURE 3.05
To
determine
frictional
the
vertical
convergence
the
motion
following
field
induced
by
equation
must
of
vertical
be
solved:
Assuming
the
horizontal
distribution
the
velocity field is the same as that of the Forcing:
A
W4(p) = W4(p) Cos k(x+A) Cos 19
In the same manner as before, one can then solve for W4(p),
PAGE 71
subject
to
the
boundary
conditions
velocity vanishes at the surface
that
and
at
the
the
vertical
tropopause.
The expression for W4(p) is detailed in the appendix.
This is the complete set of equations
the
for
diagnosing
vertical motion field for the case without the effects
of latent
heating.
To
solve
for
the
heating
induced
vertical motion, one must find the solution of:
Using the parameterization for
q
discussed
earlier,
equation can be written:
where bi = b, for i = 1,2,4
= ba for i = 3
A
where ri = (-R E bi / fO 10 PO) Wa*i for h = hl(p)
=
(ri PO PT / p
)
for h = h2(p)
the
PAGE 72
Thus,
it is necessary to solve Euler equations of the form:
a
P
A
A
h
a
Wdi + B Wdi = ri p
=
for PU
<
p < PL
for p > PL, p
0
Once again, the boundary conditions are that
motion
vanish
at
p=PO
solutions can be found in
and
p=PT.
the
The
< PU
the
vertical
details
appendix.
The
of the
convective
condensational heating induced vertical motion field of the
Presidents' Day
case
is
shown
in
fig.
3.06-07.
The
magnitude of the maximum values are increased from three to
four times the dry model values, and the
maximum
at
low
levels,
(these profiles are for
PL=850 mb, PU=400 mb).
profiles
show
a
where the maximum heating occurs
the
hi(p)
heating
distribution,
PAGE 73
P(mb)
-120
-80
-40
A
0
3m
FIGURE 3.06
40
~
80
120
PAGE 74
P(m'b)
I
IIII
PRESIDENTS' DAY CASE
300
2 -19 -79 , 122
FRICTIONALLY INDUCED VERTICAL
VELOCITY, QG/CISK
400
500-
600-
A
700 -
W4 (CISK)
A
300-
W4 (QG)
)00-
0
0
10
20
20
40
30
60 A
w4(
4
40
80
mbs')
FIGURE 3. 07
50
100
60 W4 (0G)
120 W4 (CISK)
PAGE 75
Once the vertical
field
of
motion
geopotential
vorticity equation.
tendency
The
is
field
can
solution
determined,
the
be derived from the
for
the
geopotential
tendency at the surface low center can be written:
X(center) = X(dyn/adiabatic) + X(diabatic) + X(friction)
The details of the functions are
The
effect
of
heating
is
shown
in
the
appendix.
to increase the instantaneous
deepening rate through the addition of two new
terms,
one
augmenting the effect of the dynamic/adiabatic forcing, the
other opposing frictional dissipation through increased low
level
convergence.
The
scenario
is
as
follows:
vertical motion induced by low level temperature
and
vorticity
advection
aloft
past
conditionally
the level of free convection.
heating induced vertical motion field in
vertical
advection
results in the release of
latent heat through the lifting of
air
the
unstable
This results in a
addition
to
the
motion induced by dynamic/adiabatic effects;
the
increased low level convergence and upper level
associated
with the vertical motion field leads to greater
intensification.
convergence
divergence
and
The
additional
associated
effect
lifting
of
of
frictional
conditionally
unstable air (and resulting convection/latent heat release)
helps
to
oppose
the
damping
effects
of friction.
heating contribution to intensification is through Wil,
The
the
PAGE 76
advection
of the component of thermal vorticity due to the
gradient of the perturbation temperature field.
One can derive the relations for the phase
the
surface
of
features, and their relationship to the upper
level flow (e.g.,
Sanders
speeds
(1971).
500 mb),
in a manner identical to that of
Thus,
Cx(PO) = Cx,Cos kU
+ Cx.
Cy(PO) = Cy(dyn/adiab) + Cy(diab)
where Cx;= Cx, (dyn/adiab) + Cx; (diab)
The specific forms of the terms are shown in the
Two
appendix.
new terms appear in the relation for Cx, the effect of
X(diabatic) in Cx, and the effect
ahead
of
of
latent
heat
the storm associated with W2 in Cxa.
the dry model, Cx, was usually quite small,
X(diabatic)
is
to
greatly
increase
the
effect
on
Cx,,
lows.
The
of
is then to decrease the eastward propagation speed
There is one additional
in
for
expression
Cy,
release associated with W3.
quite
of
retarding
the end result
of the disturbance.
the
effect
on the other hand, is an almost negligible
increase in eastward propagation speed;
heating
Whereas in
the
(enhancing) effect of the term for warm (cold)
release
substantial,
and
related
The
to
increase
heating
term
the latent heat
in
Cy
can
be
there are no retarding effects to
PAGE 77
compensate.
component
The result is an
of
phase
increase
velocity
with
in
the
heating.
northward
The overall
effect of heating is usually to increase the phase speed of
the
disturbance,
direction.
Since
developing
system
most
the
particularly
instantaneous
is
generally
observed "left-movers,"
500
the
mb
northward
flow
west-southwest
surface center, this effect is a
the
in
possible
in
a
over the
explanation
of
cyclones that move to the left
of the instantaneous upper level flow over the
center,
contradiction
An analysis
of the usual baroclinic result.
of 67 explosive
through
cyclones
in
the
a
slight
bias
twice as many of the
right.
September
1981
April 1982 showed 51% of those storms moved to the
left of the instantaneous 500 mb
showed
period
in
flow.
The
left
movers
towards less rapid deepening, with
strongest
deepeners
moving
to
the
A possible explanation of this observation is that
in the weaker deepeners, the
CISK
overcome
tendency
the
right
moving
mechanism
of
is
the
able
to
baroclinic
process.
Type
B
can show that in the case where the heating
is
It is now possible to provide an
CISK.
One
example
parameterized in terms of the total vertical
of
motion
field
(dynamic/adiabatic plus diabatic):
W(PL) = Wa* / ( I - E B
)
where B is some constant that depends on all the parameters
PAGE 78
of the model.
infinity.
Thus, for certain values of E, W(PL) goes to
Some tests were run, using the
example,
a
scheme;
as
plot of phase speeds and geopotential tendency
as a function of heating intensity for the Presidents'
case
are
an
presented in figure 3.08.
Day
As can easily be seen
from the plot, both the geopotential tendency and the phase
speed become unbounded for certain values of E, in contrast
to the parameterization relating the heating
dynamically
and
adiabatically
to
only
the
induced vertical velocity.
It is clear that such a scheme is highly unstable
relative
to small changes in heating intensity, so for the remainder
of this work, the Mak scheme is used.
PAGE 79
50
Il
TYPE
80-
B CISK
40
F a< W (PL)
F.( Wdyn(PL)
60-
30
40
20
20
10
G
0'
'
-0
E
0
-10
-20
-20
.40-
-30
-60-
-80-
-100
-
3
I
I
I
-
5
4
E (10-2 kmb'
FIGURE 3.08
6
7
8
-40
-50
PAQE 80
It is now possible to determine
fields,
the
vertical
motion
phase speeds, and instantaneous deepening rates of
a disturbance at any instant
thickness
field
is
for
which
available.
It
the
1000-500
is not clear how one
determines whether a CISK mechanism is operating,
in
terms
for
the
of
however;
this model, one still must determine a value
heating
fundamentally
intensity.
related
to
Since
the
this
mixing
parameter
which
corresponds
Thus,
it
would
be
necessary
structure
was
to
at
know
that
only
E
the
level
to
This still does not answer the question
determining
the
what
pressure
the
existence
of
a
CISK
consultation of satellite photos would provide
on
of
to the saturation mixing ratio at p=PL.
estimate qs and E.
of
is
ratio at p=PL (see
appendix), one approach would be to choose the value
temperature
mb
existence
of
process;
information
convection (though not whether that
convection was cooperating with the large scale).
Once the instantaneous fields
strictly
speaking,
it
is
then,
is
to
been
determined,
not possible to say what those
fields will look like at any
step,
have
time
determine
extrapolating the instantaneous
a
thereafter.
logical
results
The
procedure
in
time.
next
for
Since
there is no additional information concerning the continued
evolution
assumption
of
the
(for
large
short
scale
forcing,
a
reasonable
periods) might be that the initial
PAGE 81
value
holds
continued
for
the
entire
period.
In
cases,
development will result in an underestimation of
the overall forcing, while in others, the
will
some
initial
forcing
represent the height of the disturbance organization,
resulting in an overestimate.
Still, one might expect that
such
reasonably accurate for time
an
assumption
remains
periods less than some critical time scale for the synoptic
scale, say 12 hours.
The simplest approach is a linear extrapolation.
assumption
is
that
the
forcing
The
and the friction remain
constant over the interval:
DP(t) = 4
( XO + XFO) t / 300 ;
where XO
XF(t) = XFO
= X(dyn/adiab) + X(diab) = const
XFO = XF(dyn/adiabatic)
This will hereafter be referred to as
friction
extrapolation (LCF).
the
linear/constant
It is possible to develop a
scheme that accounts for the increase of friction with time
in an intensifying system.
One can write:
XF(t) = C
(t)
M
In order for the above relationship to be
the
original
form
of
the
stress,
consistent
with
1V(t)H=4V(t0).
In
PAGE 82
reality, in an intensifying system, one
magnitude
would
expect
the
of the wind speed to increase with time, so this
representation is still
frictional
drag.
an
underestimate
of
the
actual
The frictional dissipation is linear in
the sense that a doubling of vorticity with time results in
a
doubling
of
the rate of frictional filling.
Using the
relation:
d C/dt=
. 7X
fO
yields a differential equation for
(t):
d C (t)/dt
+
(b,
It is possible to solve
vorticity
as
a
C / fO) 4 (t) = -b, XO / fO
this
function
equation
of time;
known, it is possible to solve for
thus,
for
relative
once the vorticity is
the
geopotential,
and
the pressure so that:
DP(t) = 4 d C I + XO/XFO 3C 1-exp{-XFO t/
XF(t) = XFO exp{-XFO t/
It is possible to solve
drops
the
t} -
for
k} 3 / 300
XO I 1-expf-XFO t/ fl 3
the
maximum
total
pressure
and the time for the pressure to drop to 90% of that
maximum:
PAGE 83
DPmax
= 4
T(90%)
= 2.3 & / XFO
1 + XO/XFO J / 300
This will be referred
to
(LLF)
this
scheme.
In
as
the
linear/linear
scheme,
the
friction
pressure
change
asymptotically approaches a constant value which is reached
when
the
friction
balances the forcing.
In the constant
friction scheme, the cyclones deepen forever,
since
there
can never be a balance between friction and forcing.
As an
example of an application of these schemes to an
cyclone,
consider
fig.
Presidents' Day case.
explosive
3.09-11, pressure traces for the
The dry model could not
deepening
explosive
of
the
first
6
handle
hours
the
at
all.
Subsequently, it improved, since the real storm had stopped
deepening
while the model continued to develop the system.
The model with the CISK mechanism included performed
well
for
forcing
the
first 6 hours;
resulted
disturbance
had
in
model
ceased
however, the continued CISK
deepening
to develop.
after
the
after
that
baroclinic forcing;
suggesting
that
in
time
would
be
a
that
any
result of the
the forecast trace is quite realistic,
this
case,
the
CISK
mechanism was
operative for the period of explosive development only,
first 6 hours.
real
In the final figure,
the CISK mechanism was cut off after 6 hours, so
development
quite
the
PAGE 84
Figures 3.12-16
observed
and
model
Presidents' Day case.
flow
display
the
surface
initial
and
500
state
of
the
flow
of
the
represent
the
mb
The model appears to
reasonably well, except for the excessive wind speeds
and gross overestimate of the vorticity field
Since
this
results
in
fields
had
500
been
more
the
actual
wind
model
are
probably
and
exactly reproduced, the
vertical motion field and consequent pressure falls in
baroclinic
mb.
greater temperature and vorticity
advection than would be expected if
vorticity
at
exaggerated.
shows even more clearly the failure of
the
dry
the
This result
model
account for the cyclogenesis in the Presidents' Day case.
to
L-LF
-
-
81
C-
-12
OBS
-16---
-- ft --
1,
-
-
-
-
-
-
.0,
.0*
-20-
-24~
PRESIDENTS
DAY CASE
2-19- 79 , '22
PRESSURE CHANGE (mb)
OG MODEL
-28-
-32-
03
0
6
9
12
15
T (hours)
FIGURE 3. 09
1
18
21
24
0
3
6
9
12
T (hours)
FIGURE 3. 10
15
18
21
24
..%-t-6
E
CL
T (hours)
FIGURE 3. 11
0
L/10
L/ 5
3L/10
2L/ 5
L/2
3L/ 5
7L/
4L/5
9L
-U
Model surface analysis, Presidents' Day case (2-19-79,
Dashed lines are
lines are surface isobars.
Solid
12Z).
dekameters.
in
thickness,
mb
isopleths of 1000-500
FIGURE 3. 12
-3L/ 10 |
- /110
I
0
L/5
3L/10
2L/5
L/2
3L/5
4 L/5
(2-19-79,
Model 500 mb analysis, Presidents' Day case
lines are isopleths of 500 mb height, in deSolid
12Z).
vorticiDashed lines are isopleths of absolute
kameters.
ty, in 10''s~
FIGURE 3. 13
PAGE 90
FIGURE 3. 14
PAGE 91
*IIV
~'
p
~
~C~4 ~'
-
>~
500MB ANALYSIS
HEIGH'S/VORTICITY
FIGURE 3. 15
I~I
PAGE 92
FIGURE 3. 16
PAGE 93
3.4
Model Climatology of explosive cyclones
The initial test of the model on the
Presidents'
Day
case suggests that it would be fruitful to examine a larger
sample
of
explosive
performance
of
the
cyclones,
CISK
to
test
the
general
model in those cases.
Thus, it
might be possible to evaluate the wave-CISK hypothesis as a
generally
credible
explanation of explosive cyclogenesis.
With this in mind, explosive cyclones in
January
1978
to
March
the
period
from
1981 which occurred in the region
covered by the GOES-East satellite were
modeled.
In
the
M.I.T.
archives for this period, in addition to the usual
OOZ and
12Z
thickness
hemispheric
and
upper
air
surface
pressure/
1000-500
charts, 3 hourly North American
surface analyses and 6 hourly hemispheric surface
were
available
on
microfilm.
Since
analyses
the upper air (and
consequently, the thickness) charts are available
12
mb
only
at
hour intervals, the instantaneous model results must be
extrapolated from OOZ
charts
make
it
and
possible
12Z.
to
The
additional
surface
extrapolate the results for
periods less than 12 hours, since it is
then
verify
observations.
the
deepening
rates
with
possible
to
In
particular, the desire is to model the period of most rapid
deepening,
so only those cases whose explosive phases were
initiated at 00Z or 12Z were modeled, 21 cases in all.
first
The
6 hours of the rapid deepening phase was modeled, on
PACE 94
the basis that the underlying
forcing
would
introduce
assumptions
the
least
concerning
the
serious error in the
calculations for such a time period, while still preserving
the
important
and
measurable
characteristics
of
the
explosive phase.
Satellite
convection,
pictures
using
were
examined
evidence
of
the necessarily qualitative criteria of
observable small scale structure
tops
for
on infrared.
on
visible
and/or
high
The value of E was chosen such that the
observed 6 hour deepening was approximately
accounted
for
by the diagnostic model extrapolation, provided evidence of
convection was found on
evidence
used.
E
the
satellite
pictures.
If
was found, the model with no diabatic heating was
If no satellite images were available, the value
was
chosen
to
fit
the
observations.
Although
satellite evidence was clear cut in only about 1/3
cases
(the
majority
being
calculations were performed
convection
as
The table (fig.
inferences
cases.
analyzed
no
long
as
the
than
of
the
ambiguous),
the
the
observed
convection
The results of these 18
rather
on
based
the
assumption
of
the evidence did not preclude it.
3.17) lists
concerning
somewhat
of
for
cases
the full 21;
excluded did not exhibit any
and
the
18 of the 21 total
were
statistically
the 3 cases that were
particular
but rather a steady, strong deepening.
data
explosive
phase,
PAGE 95
MODELED EXPLOSIVE CASES
DATE
f TIME i LAT 1 LONG 1
imm/dd/ygl (GMT)! (N) 1 (W)
- - -- - --------------1/03/78
00
44
58.5
03
47.5
57
06
47.5
55
2/22/78
PRESS(mb) 1 SATELLITE 1
NA I NH 1
NOTES
1
986
984
972
988
1001
997
988
1000
Not Avail.
12
15
18
34
3435.5
71
68
66
00
03
06
42.5
42.5
43
52.5
51
50.5
994
980
970
992
3/04/78
12
15
18
42
43
43
67.5
65
63
984
978
970
983
Conv. east 1
center.
i
11/29/78
00
03
06
46
46
47
56.5
55
56
999
995
988
999
989
Not Avail.
No evid.
earlier.
00
03
06
41
41
41.5
67
66
65
995
991
985
996
Not Avail. I
00
03
06
36.5
38
40
78
76
75
992
988
983
992
12
15
163
42
41.5
41
66
64.5
61. 5
1001
994
9e6
998
00
03
06
35.5
33
34
69
69
67
993
990
984
992
12
15
18
41
41
41.5
54
54
50.5
990
990
972
990
00
03
06
34
35
35.5
73
72.5
70
1006
996
996
1006
00
03
06
45
45
46
65
63.5
61
983
978
973
980
2/28/78
12/10/76
12/25/78
1/18/79
2/01/79
2/10/79
12/17/79
1 1/24/80
FIGURE 3.17
Barotropic
4
instabiliti
Structure.
Not Avail.
Eye struc.
earlier.
984
-40C IR
center.
983
-45C IR
near center
987
-SOC IR
center.
980
960
996
972
-50C IR.
Structure
earlier.
-75C IR.
-60C IR.
Ci shield.
-45C IR.
-60C early.
Noth. vis.
I
i
PAGE 96
DATE
1 TIME 1 LAT 1 LONG : PRESS(mb) 1 SATELLITE
t
NH 1
NOTES
(N) 1 (W) 1 NA i
imm/dd/yyl (GMT)!
12
15
18
40.5
42
45.5
61.5
58.5
56
1001
994
984
1001
00
03
06
32
34.5
33.5
77
76.5
75
1009
1000
998
1008
00
03
06
35
34.5
35
74.5
73.5
73
997
986
986
995
11/22/80
00
03
06
36
39
40.5
72
68.5
68
1011
1007
1004
1011
Not Avail.
1/17/81
12
15
18
41
42
42
66.5
66
64.5
1002
996
988
1002
-80C IR
point. Stril
988
ations.
12
15
18
44
42
36.5
65
65
63
998
997
980
996
Not Avail.
2/29/80
3/02/80
1 3/03/60
3/03/81
Cell. struc'
-45C IR.
984
Vis.
NE.
evid.
-50C
IR.
998
986
982
1
PAGE 97
The question as to whether
the
moisture
convergence
necessary
to supply the model heating intensity, E, can be
supported
by
the
relationship
observations
between
was
considered.
A
the heating intensity and the mixing
ratio can be established using the assumption that the rate
of
precipitation
is
proportional
to
the
moisture
convergence into the column plus the
surface
The
can be related to the
precipitation
rate,
in
turn,
evaporation.
integrated heating by condensation so that:
q(PL) >
(E
Cp
( P' - PU )I / 2 Lc PO
(1
+ Wd*/Wa*>
where Wd* is the diabatically induced
p-velocity at p=PL.
The details of the derivation are supplied in the appendix.
Thus,
we
can
evaluate
the
right hand side of the above
equation, and compare this "implied" mixing
saturation
ratio
to
the
value as computed From the observed temperature
at p=PL, in the model, 850 mb.
In 5 of the 1
cases,
the
required magnitude of E implied a mixing ratio greater than
the saturation value in the actual atmosphere, as
by
NMC.
g/kg;
analyzed
The discrepancies were small, on the order of 1-2
since the actual
temperature
at
850
mb
was
not
directly observed (the value used was interpolated from the
analysis),
suggests
this finding may not
that
be
too
serious,
but
it
caution be used in interpreting the results
PACE 98
of this analysis.
support
the
In any case, the computed mixing
idealization
of
ratios
a nearly saturated boundary
layer to the pressure level PL.
An attempt was also made to estimate the stability
of
the atmosphere in the vicinity of the low center for the 18
cases.
Interpolated temperatures were
surface, 850 mb, and 500 mb
on a pseudo-adiabatic
that
in
evidence
atmosphere
4
out
of
of
convection
was
from
the
level NMC analyses, and plotted
diagram.
the
obtained
This
analysis
indicated
8 cases that showed fairly clear
on
absolutely
the
satellite
stable.
photos,
the
Even after the layer
from the surface to 850 mb was lifted (150 mb), 3 of the
cases
remained
absolutely
stable.
8
These inconsistencies
graphically demonstrate the rather substantial
problem
of
data availability in the regions of these storms.
The values of E were plotted on a map according to the
latitude/
longitude coordinates of the storm center, based
on the logical supposition that this parameter
some
recognizeable
would
have
geographic distribution, at least in a
climatological sense.
Figure
3.18
reveals
a
reasonable
distribution of heating intensity, with the axis of maximum
heating intensity located slightly to the warm side of
mean
position
of
the
maximum
gradient
of
temperature, associated with the Gulf Stream.
the
sea surface
This
is
an
PAGE 99
area
one
might expect to be conditionally unstable with a
warm, moist boundary layer;
furthermore, this differs from
the distribution of maximum bomb occurrence detailed in the
previous
(1980),
section
which
and
discussed
revealed
no
by
Sanders
preference
for
and
maximum
surface temperatures, but was associated instead
region
of maximum SST gradient.
the
would
tend
to
baroclinic instability without paying too great a
price in terms of warmth
should
with
sea
This result is consistent
with a CISK mechanism, since such a region
maximize
Oyakum
be
noted,
and
however,
moisture
that
21
availability.
data
points do not
constitute a very complete data set, and provide
of leeway in the analysis.
It
a
degree
PAGE 100
FIGURE 3. 18
PAGE 101
The results for the 18
table (fig.
3.20).
cases
are
presented
in
the
In particular, the poor performance of
the 6 hour baroclinic forecast should be noted, the minimal
reduction
of variance suggesting a no skill forecast.
baroclinic
averaged
mechanism
6
mb
was
clearly
fast.
displacement
speeds,
This is a persistent and
model.
well
correlated
averaging
it
puzzling
with
the
about 7 m/s too
feature
of
the
Krishnamurti (1968) found that the amplitude of the
vertical motion predicted by quasi-geostrophic
too
as
less than the deepening actually observed.
The phase speeds were also not
observed
insufficient,
The
large;
theory
was
the phase speed problem is most likely related
to this feature.
The phase speeds in the
somewhat
correlated
better
CISK
model
are
with the observations, though
the net effect of the mechanism is to increase the velocity
of the systems;
thus, the variance of the speeds is better
explained by the CISK mechanism, but not
The
dramatic
improvement
their
magnitude.
in the reduction of variance of
the northward component of the phase velocity is due
to
large
While
negative
correlation
with the observations.
this suggests a strong statistical link
model
and
the
between
the
a
CISK
observations, and thus makes a forecast of
them feasible, it renders the underlying physical mechanism
somewhat
questionable.
northward
disturbance
phase
The result implies that when large
speeds
exhibits
a
are
predicted,
the
actual
small northward displacement, and
PAGE 102
vice versa.
The statistical
regression
equation
of
the
form:
A
Cy = m Cy + b
where Cy is the model result, and m and
parameters,
b
are
can be used to predict the actual phase speed,
but one can only be confident of the physical
the
regression
results for m-11 and b-40.
validity
of
In that case, there would be
a one-to-one correspondence between the model and the
real
atmosphere.
In the 12 hour forecast, the
baroclinic
model
again
underpredicts the observed deepening rates and overpredicts
the phase speeds.
deepening
The CISK model does much better, but the
rates are too large, suggesting that the forcing
on average has already
applied
weakened.
The
CISK
model,
when
for only the period of most rapid deepening, shows
an increased reduction of variance over the
extrapolations,
standard
CISK
and the mean and standard deviation of the
resultant deepening distribution conform
that of the observations;
more
closely
to
this result lends support to the
supposition that the CISK mechanism is operative
the period of most rapid deepening.
only
for
PAGE 103
EXPLOSIVE CASES (model results)
T=6h
1 Ext I
DP I o I R '1
1 C 1 o 1 R 1
-------------------------------------------1 LCF a -7.31 4.61-.06H1
Cx 119.81 8.11 .261
LLF 1 -7.11 4.51-.0711 Cy 111.41 5.01-.21!
*
I£11~C
123.51 7.49 .081
------------------------------------------------CISK
1 LCF 1 -14.21 5.31 .9311 Cx 115.9110.91 .241
LLF 1 -14.01 5.31 .9411 Cy 121.7110.21-.471
11 C
128.7110.61 .261
------------------------------------------------OBS
1
1 -13.31 5.4!
11 Cx 111.71 5.61
1
1
1
111 Cy 1 6.7111.51
i
1 C
116. 4 1 83. 41
I
------------------------------------------------Model
-----GG
T=12h
1 Ext
DP 1a
R
C I
1 R 1
-------------------------------------------f LCF 1 -14.61 9.31-.1511 Cx 119.81 8.11 .071
LLF
-13.61 8.71-.1511 Cy 111.41 5.01-.241
I C 123.51 7.41 .051
--------- ---------------------------------------i CISK
1 LCF 1 -28.4110.71 .6711 Cx 115.9110.91-.05!
I LLF
-27.0110.11 .6711 Cy 121.7110.21-.191
a1
11 C
128.7110.61 .041
------------------------------------------------'CISK(cut)l LCF 1 -24.2111.8 .81H: Cx 116.8110.51-.051
LLF 1 -22.4111.3 .8211 Cy 118.7110.31-.351
C
127.1110.21 .041
------------------------------------------------LFM
1 -9.31 5.11 .32e' Cx ' 9.31 3.71 .211
Cy 1 7.6110.31 .87
1 C
115.21 4.51 .171
Model
-----GG
1
OBS
1
1 -19.11 9.21
1
1
1
1
1
FIGURE 3. 19
it Cx 113.01 4.91
1# Cy 1 8.01 7.51
11 C 116.81 5.41
1
I
PAGE 104
I Ext 1
Model
GG
CISK
DP
T=24h
o a R
!
C 1
o I
R I
I LCF I -29.2120.11-.521
LLF
-25.2117.61-.521
t
Cx 120.11 9.51-.111
Cy 111.41 5.31-.361
C 124.11 8.31-.16
-55.3121.3 .38
-49.2118.71 .351
i
Cx #16.6112.11-.26?
Cy 118.9: 8.2'-.40!
C
t27.1,10.31-.26!
LCF
LLF
Cx 118.3911.01-.21!
CISK(cut)l LCF 1 -36.8115.7 .191
-29.3112.91 .21K Cy 114.81 5.71-.56
LLF
i
-18.5' 5.9?
LFM
C
.011
Cx
Cy
11C
1
I
1 -31.2110.1?
OBS
i
11 Cx
11
I
'
125.01
8.7,-.28?
8.51 3.11
8.8t
.i
112.91
5.6?
4.71
111.21 5.6?
~Cy 1 7. 01 7. 41
C
1114.91,
5.91
.911
.96
.909
PAGE 105
In the 24 hour extrapolations,
improvement
in
the
baroclinic
there
model
measured by the reduction of variance.
large
negative
the observed
forecast
a
dramatic
performance,
This is due to
as
the
correlation between the model forecast and
data.
improvement
The
with
continued
trend
of
dry
model
time is most likely related to
the method of extrapolating
than
is
instantaneous
results
rather
any improved forecast skill, particularly if the main
deepening period of these storms is
less
than
24
hours.
The model improves, since it develops slowly, and therefore
does not come into frictional balance
real
atmosphere.
It
effectiveness of the linear
frictional balance;
is
as
possible
friction
18
to
model
as
the
examine
the
in
producing
the maximum pressure fall and the time
required to attain 90% of that maximum
the
rapidly
were
computed
cases and compared to the observed data.
The mean
maximum computed pressure fall and the average time to
of
that
90%
maximum were greater than that observed, implying
that the friction is really of a higher order than
model
for
(e.g.
quadratic).
Since
based on a constant forcing, the
the
in
the
extrapolations are
possibility
also
exists
that the forcing itself weakened and frictional balance was
attained somewhat earlier than would
otherwise
have
been
the case.
In the case of strong forcing followed by weak forcing
PAGE 106
(CISK
cut-off),
the maximum pressure falls were still too
large, and the time required to attain 90% of that
was
much
longer than the time to observed balance.
not clear from these results that the
weakens
maximum
before
frictional
these results are also
consequent
necessarily
balance i.s attained;
influenced
underestimate
forcing
of
by
the
The pressure changes following
the
It is
however,
linearity
and
frictional dissipation.
the
cut-off
of
the
CISK
mechanism are small, and a quadratic frictional force could
conceivably reduce substantially
the
time
to
frictional
balance.
Returning to the 24 hour data, there appears to
be
a
dramatic drop in CISK model performance, another suggestion
that
this
time
extrapolating
scale
is
instantaneous
inappropriately
results;
the
long
for
success
at a
range of 12 hours gives some hope of applying the scheme to
the
operational
forecasting
of
explosive
cyclogenesis,
however, and the details of one such study are presented in
the next section.
PAGE 107
DEEPENING PERIODS (Hours)
1Model
!!
Statistics
H
N
Git
! 18
CISK
1. 16I
,CISK(cut)H1 15
B-IOBS
H8
16
TM
DPmx(rmb)!
183.3 1 53.3 !!
C
-98.4 1 69.5
183.3 1 53.3 !1 -188.6
i
4
1 67.5 i
150.7 187.4 H1 -47.9 1 20.7 1
------------------------------------23.8
FIGURE
9.3
3.20
H-31.8
s'13.3
H1
PAGE 108
3.5
Operational forecasting of explosive cyclogenesis
The
climatological
parameter,
E,
provides
map
a
of
means
the
for
heating
intensity
a general forecast
scheme, which would proceed as follows:
1. Evaluate the quasi-geostrophic model parameters in the usual way (Sanders, 1971).
2. Consult satellite coverage of the area in
question to look for evidence of convection
in the storm region.
3. If evidence of convection is found, assume it
is interacting cooperatively with the large
scale disturbance, and determine the value of
E from the climatological map, based on the
position of the storm center.
4. If no evidence of convection is found, set the
heating intensity to zero.
There are, of course, some drawbacks to
Convection
CISK.
scheme.
is a necessary but not sufficient condition for
There must be a symbiotic relationship
observed
this
convection
and
between
the forcing, that is,
convection
which would have some meaningful effect on the dynamics
the
developing cyclone.
tell nothing about this.
is
often
inconclusive;
the
of
Perusal of satellite pictures can
Furthermore,
satellite
evidence
convection may be occurring, but
could be obscured by a cirrus shield or
not show up well on the photos.
else
simply
does
PAGE 109
A second drawback is with the values
climatology
of
explosive
cyclones.
E
Perhaps
cyclones with operative CISK mechanisms
intensities
of
at
from
the
there
are
lower
heating
that do not result in explosive cyclones;
contours may be biased towards
the
most
intense
the
events,
rather than cover a range of development.
Lastly,
extrapolating
there
are
diagnostic
the
results.
assume the heating intensity,
forcing
is
constant
in
time.
the
geographic
areas.
thermodynamic
consequently
does
problems
It
and
would likely vary as the storm
different
usual
is
with
necessary
consequently
the
to
CISK
In reality, the intensity
developed
and
moved
over
The scheme does not consider
properties
of
the
system,
and
not automatically cut off the CISK when
the boundary layer can
no
longer
support
the
continued
convection.
As a test of the scheme on an operational
lows
basis,
all
that formed in or entered the coastal waters adjacent
to the eastern U.S.
through
February
in the period from November
12,
at 12Z.
1982
1983 were modeled and a forecast of
bomb (1) or no bomb (0) was made for
beginning
14,
the
24
hour
period
As a control, the forecast was compared
to that of a subjective forecast made prior to the modeling
for
the same period.
Also, the performance of the LFM was
PAGE 110
noted.
In all, there were 19 cases in this period, of which 2
were
explosive
cyclones.
The positions within the model
area are plotted in figure 3.22.
expected
bomb
This
is
were
bombs,
which
as shown in figure 3.21.
8
season,
of
in
7 bombs were observed in this period.
the two previous seasons, there were
explosive
the
cyclogenesis
in
the
6
sector.
and
17
The 1980-81
season appears to have been fairly representatives
1978-79
1/4
frequency for this period, in this area, as
computed from the 1980-81 climatology, in
storms
about
10
Thus,
cases
the
In
of
the test
period represents an anomalously quiet period for explosive
cyclogenesis.
PAGE 111
FIGURE 3.21
PAGE 112
FIGURE 3. 22
PAGE 113
Two
24
hour
model
predictions
were
made,
an
extrapolation of the 12Z diagnostic results, and the sum of
two 12 hour diagnostic extrapolations, beginning at 12Z and
OOZ.
The
indicate
results,
summarized in the table (fig.
generally
overforecast
the
performance.
none.
the
LFM,
which
difference
in
typical
was
minimal.
deepening
sample.
The
the
baroclinic
CISK model.
model
did
variance
better
of
the
scheme
deepening rates were
case,
forecast.,
in this
storms
was
however;
minimal.
less
excessive,
even
thus,
the
The model average
in
the
baroclinic
indicating that the process or processes responsible
for the lack of development were not handled
basic model.
region
well
by
the
In particular., two cases on December 12, 1982
merit further examination.
was
quite
The baroclinic forcing
strong,
with
estimated
temperature gradients ranging from 1.40 to 1.71
and
in
The models were generally unable to
improve on the subjective
utility
but
The bomb/no bomb forecasts
respect, reflecting their tendency to develop
than
fashion,
rates,
were able, at best, to account for 62% of the
the
schemes
The CISK schemes showed a slightly better
reduction of variance in the actual
the
The
occurrence of explosive cyclogenesis, in
marked contrast to
predicted
poor
3.23),
in
the
meridional
X
104K/m
perturbation temperature amplitudes in excess of 10 K.
The wavelengths of the disturbances were somewhat large, on
the
order
of
4500
km,
but
the
baroclinic forcing was
PAGE 114
sufficiently strong to easily produce model bombs, with
hour
deepening
rates
of
-25
atmosphere was apparently stable
and
-48
mb.
to
small
The
24
real
perturbations,
however, as the surface systems did not develop ( -2 mb and
+4 mb).
the
If these two cases are removed
forecast
performance
is
from
improved
(see
3.23), but the schemes still fall short of
forecast.
The
schemes
were
the
sample,
table, fig.
the
subjective
able to forecast the actual
cases of explosive cyclogenesis;
the trouble was rather in
predicting too many bombs.
If one accepts the CISK hypothesis as a
for
explaining
the
viable
occurrence of explosive cyclogenesis,
then it is necessary to understand the mechanism
inhibiting
the
cooperatively
problem,
then,
with
is
means
observed
convection
the
scale
large
from
flow.
that
was
interacting
The
basic
to understand the interactions between
the mesoscale and the synoptic scale.
PAGE 115
OPERATIONAL FORECAST RESULTS
T=12h
I Ext I DP
Model
GG
i
c-11
R
!!
LCF 1
LLF 1
-6.01
-5.41
7.71 .341
6.6: .351
CISK
1 LCF 1
LLF ,
-8.11
-7.41
8.91 .371
7.91 .39
i
LFM
1
-1.51
4.71 .1211
I
0BS
-3.81
6.21
H
T=24h
Model
GG
I Ext
I
DP
1
I LCF 1 -11.3112.01 .3311
I LLF 1 -9.31 9.91 .32|1
CISK
1 LCF I -15.7115.41
1 LLF 1 -13.2112.71
LFM
I
SUBJ
1 BS
11 R(0/1 FCST)11
R H! All #w/o 1211
6 1
-3.31
1
01
1
1
-8.31
.57 1
.79 i
.2411 .56 1
.2511 .56 1
8.31-.0611
1
.78 11
.78 H
£
11.581
9.61
.78 H
.78 H
11
I!
.80 '1
1
11
T=24h (12+12)
Model
GG
CISK
I Ext
1
DP
I
11 R(0/1 FCST)11
R H' All 1w/o 1211
o I
1 LCF 1 -14.0114.31
LLF 1 -12.4112.31
.331! .49 1
.3411 .57 1
.65 11
.65 11
I LCF ' -16.4115.21
LLF 1 14.7113.21
.35,1 .49 1
.3611 .55 1
.55 11
.55 11
a
LFM
I
1
-3.31
8.31-.0611
OBS
I
1
-8.31
9.61
FIGURE 3.23
Il
1
ii
11
PAGE 116
4.0
Conclusions
the
In
deepening
normal
first
section,
rates
curve,
associated
revealed
along
with
statistical
tail
rapid
of
the
in
some
way
baroclinicity.
With
explosive
the
of
different
the
distributions
deepening.
suggests that the operative process in
is
analysis
significant deviations from the
the
most
a
This
evidence
explosive
cyclones
from
that
climatological
of
ordinary
distribution
of
cyclones, which showed a maximum in frequency in
strongly
currents,
baroclinic
the
evidence
areas
adjacent
to
warm
ocean
suggests a process in combination
with baroclinicity.
In the study of explosive cyclones, the CISK mechanism
was able to account for the observed deepening, whereas the
baroclinic model fell well short.
heating
intensity,
geographic
cases.
a
parameter
considerations,
The
was
A
logical
based
pattern
of
upon physical and
established
from
these
axis of maximum heating intensity was located
slightly to the warm side of the mean Gulf Stream position,
a
prime
moist
area
boundary
explanation
for
for conditionally unstable air with a warm,
the
represents
layer.
for
maximum
strong
the
SST
This
relationship
suggested
an
preference of explosive cyclogenesis
gradients,
baroclinicity
since
yet
is
such
still
an
able
area
to
PAGE
117
for
the
support conditionally unstable air.
The CISK mechanism provided
"left
an
explanation
moving" phenomenon, in which some storms move to the
left of the instantaneous 500 mb flow, contrary to
synoptic
systems.
typical
The evidence was unclear as to whether
the explosive forcing weakens before frictional balance can
be
attained,
seems
to
but
the
indicate
parameterization
possibility
the
that
advisability
includes
thermodynamics,
so
that
convection
be
assured
can
of such an occurrence
an
an
of
a
internal check on the
automatic
when
formulating
the
cut
off
physics
of
the
no longer
supports its continuance.
In the operational forecasting
that
in
an
anomalously
baroclinic model
evolution
as
was
low
about
the CISK model.
study,
bomb
as
it
was
found
frequency period, the
able
to
handle
cyclone
This was largely due to the
observation of convection associated with storms, since
the
presence
assumed.
of convection, a cooperative interaction was
The basic problem, and the solution, of explosive
cyclogenesis
appears to reside in the better understanding
of the interactions between the synoptic
scales.
in
and
sub-synoptic
PAGE 118
ACKNOWLEDGMENTS
I must thank Professor Frederick Sanders
me
to
the
meteorology
drafting
the
wild
in
world
I
also
thank
Isabelle
quickly, and neatly.
made
life
at
M.I.T.
by
the
for
help
and
not just possible, but probable.
This research was supported by the Office
and
Kole
Lastly, I wish to
express my appreciation of my fellow students, whose
humor
introducing
of atmospheric bombs, and to Synoptic
general.
figures
for
of
Naval
Research,
Naval Environmental Prediction Research Facility,
under contract N 00014-79-C-0384.
PAGE 119
APPENDIX
The following equations
describe
the
model
atmosphere.
The temperature structure is:
A
T(xiytp) = Tm(p)
( ay + T Cos kx Cos 1y )
-
where Tm(p) = Tm(PO) ( p/PO)b
Tm(PO)
=
k = 2 j/
( kl/47 )
Lx ;
1 = 2
5$
T(xty,PO)dxdy
Ly
and the geopotential structure is:
(x, Yp)
=
m(p)
+ 4. Cos k (x+A)
Cos 19
A
-
R
ln(PO/p)
( ay + T Cos kx Cos 1y )
where Im(p) = R I Tm(p)dlnp = {R Tm(PO)/b> C 1 -
(p/PO) 3
+ *m(PO)
From the geopotential, we can derive relationships for the
geostrophic wind and the geostrophic relative vorticity:
A
u(x~y,p) = { I #.Cos k(x+l)
+ R ln(PO/p)
v(x,g~p) =
<
Sin 19
I a -
A
1 T Cos kx Sin lj 3 } / f0
k(x+J.) Cosly
-k 4.Sin
AA
+ R ln(PO/p) k T Sin kx Cos l
I / fO
C(xtyip) = (b,/fO) C R T ln(PO/p) Cos kx Cos l
A
-
ffrCos k(x+.a)
Cos ly 3
PAGE 120
3
a
where b1 = k + 1
The expression for the stability factor, 6, is:
6 =
<K/p
-
dInTm/dPI
( R Tm / p
)
= tCN R Tm I / p
where Tm =
<Tm(PO)/(PO-PT)(1+b)}
£ 1
-
( PT/PO ) 3
The equation to be solved to determine the vertical motion
field is:
(Fz
+ YA:;
an o$ ak
0
Dop .
TRiKb
o k
x
IA
o s ty
A
Sin
- -- p
Ly
.Oa C
Let:
W = WI + W2 + W3
A
A
&
where W1 = W1(p) Sin kx Cos l
;
A
W1I(p) + W12(p)
W1(p)
W2 = W2(p) Sin k(x+A) Cos 1j
A
W3 = W3(p) Sin
2
1y
= {2RTak/fOXTm}
<
The solutions are:
W11(p)
+
W12(p)
(PT/p) I (PO -
= {kTp /b,'Tm}
(a, /b ) F1(pq
1.
ln(POfp)
p)/(PO - PT) 3 ln(PO/PT) ) p
-CF1 (p,q, ) } p
PAGE 121
W2(p)
=
<akk/f0Fm} <
W3(p)
=
<-klTk-/fOTm(bi
A
F1(p, q, ) I p
~
/b, )Sin kA
where a,
=
<{f0o%
b2
=
41z
q
= -C I + E I + 4(b, /a,
qa
= t
<
F1(pq
/
2
/I R Tm I
)
>
3
/
I + 4(b. /a, ) 3
I + I
9-I
F1(pq) = -
I p
2
-
EPO(p/PT)
-EPO(PT/p)
2
9-I
-
3
PT(p/PO)
PT(PO/p)
3I / [PO PT
-
P0 PT 3
+1
The equation to be solved
the
for
frictionally
induced
vertical motion is:
'
Thus,
CorVp
Cos kLX)Cy
we can write:
W4(p) = W4(p) Cos k(x+A) Cos 19
and solving for W4(PO)=W4(PT)=O yields:
-A
W4(p)
=
<C
3/(n-1+qj )(n-q, )3
+ p ( PO -
-
pn
PT
<<t
)J / ( PO
I-N, n-41
91
p9 '( PO PT --
PT
>>
n
ta
PO PT
)
PAGE 122
p
where C 3 = -C g
For
release
Cd l0j bi
including
on
the
the
o n(n-1)
effects
fO 90 PO
I /
of
convective
latent
heat
vertical motion field, the following equation
must be solved:
..DP
np P l
0H
0o~
The solutions are of the form:
Wdn = A2n
(p
= A3n pol+
-
PO )
<a
A4n p +
- PT
= A5n p9 (
p>PL
rn / (2a, -bn*)}
p
PL>p>PU
)
pCPU
where qn*,bn* = q1 ,b, for n=1,2,4
= qa'b,
'rn
for n=3
= -C- R E Wa*n bn*
/
fO
oP0)
In order to solve for the constants A2n,
vertical
motion
proFiles
matched at p=PL and p=PU.
A2n = An
11
i-all(PL
-
((2-qn*)PT
and
their
A3n,
first
A4n,
A5n,
the
derivatives
are
Thus, the constants are:
1+i
PU )
A3n = An {(2-qn*)PO PT (PU -
a-9
-
PL )
(1+qn*)(PL -
-
a-r
PU )}
(1+qn*)(PO PU
-
PT PL
)}
PAGE 123
A4n = An ((2-qn*)(PO PL -
PT PU
A5n = An ((2-qn*)PO (PL
PU
-
)
)
-
(1+qn*)(PL -
(1+qn*)(PL - PU )}
-
i-2.'
An
=
(-a, rn)
PU )I
/ C (2a,-bn*)(1-2qn*)(PO
-
t-ai
PT ) 3
From the vorticity equation, the geopotential tendency can
be found,
if the vertical motion field is known.
In the model,
the geopotential tendency at the surface center (x=Lx/2-A, y= 0 ,
p=PO)
can be written:
X(sfc center) = X(dyn/adiab) + X(diabatic) + X(friction)
=
where X(d/a)
<kT
9
'-9
(PO PT -f0
X(diab)=
<-A21
X(fric)= {Cg
p
<2Ra-CEb,
I Tm}Sin k
%/bl
p
-
1 a, F1(q,,PO) +
(1-24 ) / b1
b,
3}
lVI
4,
PO I Sin kA
n(n-1)3 /
+ CA24 9fO (1-2q 1 ) / b,
(n-1+q, )(n-q )P033G*
PO')
= XF(dyn/adiab) + XF(diab)
where F* = {(1-2, )PT + 4,(PO PT + PO PT )
-
(1-2q, )PTln(PO/PT)/
F*}
'f0
Cd
9 '-9l
PO PT )3
PO PT I / { PO PT
-
PO PT 5
PAGE 124
G*
=
((1-2q, )PO
+ q, PT
PT
-
PO I
/
<
PO -
PT I
+ n
The equations for the eastward and northward components of
the phase velocity can also be found:
Cx(PO) = { X(dyn/adiab) + X(diab) I Cot kA / k L
-<
2 %a/bfbTm}F*
--
A22 9.fO(1-2q
A,
a
)/b, k-NOPO
I
-
P/bi
-
Cy(PO) = -- 2 %b, KT/bQ 'Tm)F**
Sin kA
+-( 2 A23 99f(1-2q4
)/b. Ito PO'
0
where F** = F*, q, replaced by 4
.
In the expression for the friction, the term IVt
We
can
estimate a value of
V:
appears.
based on our equations for the
velocity:
u(PO) = ( 1 Wo
fO) Cos k(x+A) Sin l9
v(PO) = (-k i / fO) Sin k(x+A) Cos l9
-~
~
2
143
Since "V!=(u + v ), it can be shown from the above that:
AA
Vimax = b
If the assumption is made that
into
the
/ B.
8 fO
the
moisture
convergence
column plus the surface evaporation is approximately
PAGE 125
equal to the precipitation rate, then
due
to
condensation
must
rate of precipitation.
=
Thus,
heating
also be approximately equal to the
<(
-
/ g Lc} dP
From the model parameterization for
Cp E Wa*
integrated
Thus,
P =
P = -
the
( PL -
4,
it can be shown that:
/ 2 g Lc PO
PU
)
q
can
(-q W / g) + Evap
the
mixing
ratio
be
related
to
the
intensity, E, such that:
q(PL) > E Cp
( PL - PU ) / 2 Lc PO ( 1 + Wd*/Wa*
)
heating
PAGE 126
REFERENCES
Arakawa, A.., and W. Schubert, 1974: "Interaction of a
Cumulus Ensemble with the Large Scale Environment. Part I." Journal of the Atmospheric
Sciences. Vol. 31, pp. 674-701.
Bates, J. R., 1973: "A Generalization of the CISK Theorg."
Journal of the Atmospheric Sciences. Vol. 30, pp.
1509-1519.
Bosart, L. F., 1981: "The Presidents' Day Snowstorm of 18-19
February 1979: A Subsynoptic-scale Event." Monthly
Weather Review. Vol. 109, pp. 1542-1566.
Chang, C. B., D. J. Perkey and C. W. Kreitzberg, 1982: "A
Numerical Case Study of the Effects of Latent Heating
on a Developing Wave Cyclone." Journal of the Atmospheric Sciences. Vol. 39, pp. 1555-1570.
Chang, C. P., 1971: "On the Stability of Low Latitude QuasiGeostrophic Flow in a Conditionally Unstable Atmosphere." Journal of the Atmospheric Sciences. Vol. 28,
pp. 270-274.
------------ , and R. T. Williams, 1974: "On the Short-wave
Cutoff of CISK." Journal of the Atmospheric Sciences.
Vol. 31, pp. 830-832.
Charney, J. G., and A. Eliassen, 1964: "On the Growth of the
Hurricane Depression." Journal of the Atmospheric
Sciences. Vol. 21, pp. 68-75.
Davies, H. C., and R. A. de Guzman, 1979: "On the Preferred
Mode of Ekman CISK." Tellus. Vol. 31, pp. 406-411.
-------------
, 1979: "Phase-lagged Wave-CISK." Guarterly
Journal of the Royal Meteorological Society. Vol.
105, pp. 325-353.
DeGroot, M. H., 1975: "Probability and Statistics."
Wesley, 607 pp.
Addison-
Gyakum, J. R., 1983b: "On the Evolution of the GE II Storm.
II. Dynamic and Thermodynamic Structure. Monthly
Weather Review. Accepted.
------------- , 1981: "On the Nature of Explosively Developing
Cyclones in the Northern Hemisphere Extratropical Atmosphere." Ph.D. thesis, M.I.T., 225 pp.
PAGE 127
Haltiner, G. J.,
311 pp.
1971:
"Numerical Weather Prediction."
Wiley,
Krishnamurti, T. N., 1968: "A Study of a Developing Wave
Cyclone." Monthly Weather Review. Vol. 96, pp. 208-217.
Kuo, H. L., 1975: "Instability Theory of Large-Scale Disturbances in the Tropics." Journal of Atmospheric Sciences.
Vol. 32, pp. 2229-2245.
---------- , 1965: "On the Formation and Intensification of
Tropical Cyclones through Latent Heat Release by Cumulus Convection." Journal of Atmospheric Sciences. Vol.
22, pp. 40-63.
Lindzen, R. S., 1974: "Wave-CISK in the Tropics." Journal of
Atmospheric Sciences. Vol. 31, pp. 156-179.
-------------- , and D. Stevens, 1978: "Tropical Wave-CISK with
a Moisture Budget and Cumulus Friction." Journal of
Atmospheric Sciences. Vol. 35, pp. 940-961.
Mak, M.,
1981: "An Inquiry on the Nature of CISK. Part I."
Journal of Atmospheric Sciences. Vol. 33, pp. 531-537.
-------
1982: "On Moist Guasi-Geostrophic Baroclinic Instability." Journal of Atmospheric Sciences. Vol. 39,
pp. 2028-2037.
Sanders, F., 1971: "Analytic Solutions of the Nonlinear Omega
and Vorticity Equations for a Structurally Simple
Model of Disturbances in the Baroclinic Westerlies."
Monthly Weather Review. Vol. 99, pp. 393-407.
-----------
,
and J. R. Gyakum, 1980:
atology
108, pp.
of the
'Bomb'."
"Synoptic-Dynamic Clim-
Monthly Weather Review. Vol.
1589-1606.
Stark, T. E., 1976: "Wave-CISK and Cumulus Parameterization."
Journal of Atmospheric Sciences. Vol. 33, pp. 23832391.
Tracton, M. S., 1972: "The Role of Cumulus Convection in the
Development of Extratropical Cyclones." Ph.D. thesis,
M. I. T., 157 pp.