METEOROLOGICAL APPLICATIONS OF FRACTAL ANALYSIS
by
Michael E. Rocha
B. A., University of California at Los Angeles
(1980)
Submitted to the Department of
Earth, Atmospheric and Planetary Sciences
in Partial Fulfillment of the Requirements for the
Degree of
MASTER OF SCIENCE
at the
MASSACHUSETTS INSTITOTE OF TECHNOLOGY
@ 1984, Massachusetts Institute of Technology
.1I
A
-- %
Signature of Author
Npartment of Earth, Atmospheric
ang Planetary Sciences, May, 1984
z/
I
/
1
Certified by:
Professor Frederick Sanders
Thesis Supervisor
.1/
/~"' I'
Accepted by:
Professor Theodore R. Madden
Chairman, Departmental Committee
on Graduate Students
FRON V27 1984
MIT LBR
METEOROLOGICAL APPLICATIONS OF FRACTAL ANALYSIS
by
Michael E. Rocha
Submitted to the Department of Earth, Atmospheric, and
Planetary Sciences in May, 1984 in partial
fulfillment of the requirements for the Degree of Doctor of
Science in Atmospheric Sciences
ABSTRACT
Fractal analysis was utilized in a manner similar to Lovejoy
(1982)
to investigate atmospheric scale selectivity.
Midlatitude
cloud and precipitation areas associated with baroclinic, convective
In addition, 500 mb
and baroclinic/convective regimes were examined.
isohypse and isotherm fields were studied on a seasonal basis.
Results from the midlatitude cloud and precipitation data were similar
to Lovejoy's findings for analogous tropical structures:
fractal
analysis indicates no scale selection for atmospheric phenomena with
horizontal length scales between m10 and 103 km.
The upper level
This may be
data, however, indicate a change in regime at =10 4 km.
interpreted as representing the change from synoptic scale forcing
mechanisms to planetary scale dynamics.
Thesis Supervisor:
Title:
Dr. Frederick Sanders
Professor of Meteorology
TABLE OF CON TEN TS
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DATA ANALYSIS.
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Satellite
Radar Maps.
500
SUMMARY
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REFERENCES
CCNCLUSICNS.
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48
BACKGROUND
The
research
mathematical
delineated
concept,
this
that of
coined by its creator,
is
in
paper
the
utilizes
relatively
new
neologism
was
This
'fractal.'
Benoit B. Mandelbrot,
a
about eight years ago and
derived from the Latin adjective fractus, meaning broken (a fairly
apt label, as will be seen).
The determination of an object's fractal character is
a
comparison
its
of
topological
Besicovitch dimension, D.
et cetera.
its
to
or Euclidean,
The topological,
dimension of
commonly
(more
Besicovitch dimension
The Hausdorff
Hausdorff
a line has a Euclidean dimension of
an object is a familiar notion:
one,
DT,
dimension,
based upon
referred to as the fractal dimension) is not quite as straightforward.
In
practice
there
two
are
methods
of
calculating
D.
first
The
involves a comparison of measurements of an object's perimeter made at
The finer the precision of measurement,
different resolutions.
slope
of
a plot of
perimeter
measurement
dimension.
The
resolution
second
perimeter versus area
formula:
versus
resolution.
by the
The rougher
the more sensitive its perimeter value is
outline of the object,
the
represented
The fractal dimension is
the perimeter.
larger
and,
procedure
values
hence,
consists
the
of
greater
a
for a set of shapes.
its
the
to
fractal
of
the
utilizes
the
comparison
It
the
p = AD/2
where P = perimeter
A = area
D = fractal dimension
This
is
analogous
to
previous
the
resolution of measurements
described,
method
fractal
perimeter
versus
area
implies a breakdown in
be interpreted
to
proportional
is
dimension
the
on a single object being replaced by the
In this case
area values for a set of geometrically similar objects.
the
with
the
slope
a plot of
of
fractal
dimension
the geometric similarity assumption,
which can
values.
as representing
A change
the
in
the
transition between
two different
physical processes.
if
said to be a 'fractal'
A shape is
greater than its topological dimension.
its fractal dimension is
It
can readily be seen that
for simple shapes such as circles and squares the fractal dimension is
equal
to
fractals.
the
Euclidean
Hence,
dimension.
these
objects
are
not
But, for any more complicated forms the fractal dimension
tends to be greater than the Euclidean dimension, thus qualifying them
for Mandelbrot's classification as fractals.
Generally speaking,
more contorted the outline of an object is,
the larger the value of
its
corresponding
particle represents
this
contortion.
fractal
an extreme case
If
the
The
dimension.
particle's
(in
fact,
Brownian
motion
of
the limiting case)
path were
traced
out
in
the
a
of
two
it
dimensions
the
fill
eventually
would
its
Thus
plane.
entire
trajectory, which has a Euclidean dimension of one (that of a line),
nature lie somewhere between the simplicity of
most objects found in
and
circles
the
and
squares
of
complexity
their associated
one to
This points out a useful aspect of
of
it
makes
dimensional
chacteristics
non-integral
value,
any
of
of
to
being
the
quantify
in
shape
of
terms
to
constrained
a
very
the
the
So
dimension.
Euclidean
the
fractal analysis:
precisely
given
to
opposed
values
integral
general
as
possible
it
range from
fractal dimensions
Hence,
implementation
motion
Brownian
trajectories.
two.
The outlines of
two (that of a plane).
has a fractal dimension of
calculation of D gives one the ability to distinguish between forms
that were heretofore lumped into the same topological category.
This
analytic sensitivity is especially important in the examination of the
complex spectrum of shapes that occurs in the natural world.
The
question
what
Of
analysis.
determination
applications
now
of
an
as
to
the
applicability
importance
is
it
to
arises
object's
have
dimensionality?
a
fractal
of
refined
more
first
Mandelbrot's
of the fractal dimension involved the examination of a
hodgepodge of naturally occurring as well as man-made forms
from soap bubbles
to Koch curves.
ranging
His calculations demonstrated
the
flexibility of this anlysis to topologically categorize virtually any
shape.
This
categorization
is
scientifically
physical process has its own unique value of D.
reasonable since it
a
given
useful
if
This is
intuitively
seems likely that the geometric characteristics of
resulting from different processes would not be the same.
structures
Using this idea Mandelbrot showed the role his new concepts played in
the
of
study
problems
specific
some
different
regions
turbulent
of
terms
of
theoretically
dimension
the
fractal
determine
values
and
as intermittency,
quiescent regions.
in
The
physical
variations
approach would hopefully aid in
whichmost
could be quantitatively categorized
dimension.
the
between
boundaries
These boundaries,
flows.
likely constitute a fractal set,
in
those
notably
most
the
concerning
information
useful
physics,
He suggests that fractal analysis may
involving turbulent processes.
yield
in
(if
next
of
significance
any)
that
would
step
are
the
be
to
fractal
found.
This
the interpretation of phenomena such
turbulent flows
which highly
contain scattered
fractal analysis may be applied to
In addition,
for turbulent
equations
the solutions of the Euler or Navier/Stokes
flows, in that the singularities of these flow fields may be a fractal
set.
Other aspects of turbulent flows such as particle trajectories
and imbedded vortices may also prove to be fractals,
fractal analysis might be useful in
also
is
hypothesizes
a fractal
analysis
their
indicating that
examination.
Mandelbrot
that the flow associated with clear air turbulence
set.
This
to meteorological
made quite recently be S.
leads
us
to
studies.
Lovejoy
the
application
of
fractal
The first attempt at this was
(1982).
. He examined fairly
high
resolution satellite and radar data from the Indian Ocean region and
calculated the fractal dimensions
His
results
indicate
that
D
of cloud and precipitation areas.
for
both
of
these
quantities
is
four-thirds
approximately
that
and
this
does
value
not
vary
significantly over six orders of magnitude in area.
These findings
fact
have
some interesting
out
to be
that D turned
significance
in
itself,
on
speculation
five-thirds
minus
and
Secondly,
have
areas
precipitation
be
does
a
not
that
noted
predicts
law
power
four-thirds for isobars].
cloud
should
[it
this
Lovejoy
some
have
may
four-thirds
although
First,
implications.
fractal
the
physical
offer
any
Kolmogoroff 's
dimension
of
these results imply that tropical
length
horizontal
no preferred
scale, as indicated the scale independence of D.
This is
atmospheric
dimensions.
somewhat disturbing in
processes
are
Specifically,
that it
associated
generally assumed that
is
with
spatial
characteristic
from research- concerned with the
results
determination of the atmospheric kinetic energy spectrum point to the
existence of different energy regimes for different scales.
alii (1970)
used wind data in their examination of the energy spectrum
and found a k- 3 dependence
(k being the wavenumber)
wavenumbers ranging from approximately six to twenty.
used covariance functions
to analyze Southern
and
dependence
found
wavenumber
variability
s-ales
and
that
this
the
-3
By
thirty-five.
studies,
Gage
found
a
wavelengths <-1000 km.
that
Julianet
kinetic
(1979)
k- 5 /
3
data
extended
variation
Desbois (1972)
Hemispheric wind data
continued
compiling
for systems with
out
from a
to approximately
number
this analysis
for
of
the
troposphere
disturbances
exhibits
wind
to smaller
The combined result of these studies,
energy
of
having
then, is
a
k-5/3
dependence
at small
scales
and
a k-
3
dependence
at
larger
scales
with a transition occurring at a horizontal length scale of about 1000
km.
A -5/3 wavenumber dependence is indicative of a three-dimensional
isotropic
regime
regime.
a -3
while
dependence
indicates
a two-dimensional
Gage's data show that the three-dimensional
well into
the mesoscale,
hypothesizes
that
two-dimensional
and enstrophy
this
flowing
that seems
extension
refects
energy cascade,
reverse
The energy
transfer.
something
downscale,
is
transfer
rather
the
extends
unusual.
existence
He
of
a
with energy flowing upscale
opposite
the
regime
of
three-dimensional
associated with a k- 5 /
and the enstrophy transfer with a k- 3 spectrum.
3
Actually,
spectrum
this idea
of the existence of a two-dimensional inertial subrange was not new;
it
was originally postulated by Kraichnan
(1967).
Steinberg (1972)
had questioned the validity of explaining the -3 wavenumber dependence
in terms of a two-dimensional,
isotropic flow.
wavenumber range 7 < k < 15 the flow is
dominant .energy
conversion
(eddy
He argued that in
not two-dimensional since the
available
to
eddy
kinetic)
also not isotropic at scales this large.
three-dimensional and is
suggested instead that the agreement between Kraichnan's
the
actual
data
may
be
coincidental
the
and
proposed
is
He
theory and
that
the
-3
dependence may be associated with the imaginary part of the wave phase
speed.
This hypothesis is based on purely dimensional grounds,
that the dominant spectral parameter
Desbois
(1972)
namely
has the dimension inverse time.
expressed a similar doubt as to the applicability of
two-dimensional inertial
theory
to waves in
the 7 < k < 15
range,
pointing out
that the flow is
proximity
the
Gage,
to
however,
excitation
found it
assumption to explain
not truly inertial
wavelength
to its close
baroclinic
instability.
to use this two-dimensional
reasonable
the results
of
due
of
spectral
that phenomena such as individual thunderstorms,
and
suggests
wind shear, breaking
et cetera represent the small-scale source of the energy that
waves,
is
studies
flow
transferred upscale.
two-dimensionally
Lilly (1983)
examined this
proposal by analyzing the wakes of moving bodies in stratified flows.
These wakes evolve in much th'e same way as the processes suggested by
Gage as being associated with the mesoscale energy profile and thus
provide a means of testing the reverse
energy cascade
the
initially
laboratory.
Lilly
found
that
hypothesis
in
three-dimensional
isotropic turbulence divides into approximately equal parts of gravity
waves and quasi-two-dimensional
The
stratification.
gravity
turbulence in
waves
the presence of strong
away
propagate
while the two-dimensional turbulence propagates
is
responsible
portion
credence
of
for
the
horizontal
to Gage's
that the -3
the wavenumber
kinetic
hypothesis.
from the
to larger scales and
dependence seen in
energy
Charney
the mesoscale
spectrum,
(1971)
source
thus
lending
showed theoretically
wavenumber dependence for 7 < k < 20 can be explained in
three-dimensional
terms
by
considering
the
conservation
of
pseudo-potential vorticity, analogous to the conservation of vorticity
in two-dimensional systems.
At any rate, the gist of all of these studies is
preferred
length
scales
for
particular
the existence of
atmospheric
phenomena.
it
appears
he examined did not extend
the quantities
as if
although
seem to provide a counterexample,
Lovejoy's initial results
out to
large enough scales to exhibit the dimensional transition found in the
Later work by Schertzer and Lovejoy (1983)
wind variability studies.
expands
upon
the
idea
atmospheric
that
have
processes
no
characteristic length scales, specifically that there is no transition
from three-dimensional
to two-dimensional
from small to large scales.
flow patterns
as one goes
they hypothesize the existence
Instead,
of a single dimensional parameter for all scales, something they refer
This quantity is somewhat analogous
to as the 'elliptical dimension.'
although instead of comparing the
to Mandelbrot's fractal dimension,
perimeter
to the area of an object,
comparison of its
eddies
involves a three-dimensional
with major
axes
in
the
vertical
the horizontal.
dimension ranges from two (for flat objects)
Schertzer
gets its
quantifies the transition from small-scale
elliptical eddies with major axes in
objects).
It
horizontal and vertical dimensions.
name from the fact that it
elliptical
it
to
large-scale
The elliptical
to three (for spherical
and Lovejoy show empirically and theoretically
that the atmosphere has an elliptical dimension value of 23/9 (-2.56),
indicative of the fact that it
consists
of flow patterns
that are
somewhere between flat and spherical.
As of now it
appears as if
as to scale selection (if
no definitive conclusions can be drawn
any) of atmospheric motions or the dynamical
reason behind any such scaling.
Energy spectrum studies are limited
by the paucity of wind data and the inherent errors associated with
12
interpolation
or
covariance
calculations.
Fractal analysis
offers
another method of determining the scale of atmospheric processes.
In the present work,
an analysis similar to Lovejoy's
made of midlatitude cloud and precipitation areas.
to extend
fractal analysis to larger scales,
temperature fields are examined.
(1982)
is
Also, in an effort
hemispheric
height and
DATA ANALYSIS
outlines
cloud
were examined:
parameters
Four meteorological
obtained from enhanced infrared satellite imagery, precipitation areas
from radar maps,
and
satellite
analyzed--three
500 mb height contours and 500 mb isotherms.
radar
from
baroclinic/convective,
of
data
of
each
nine
the
individual
following
convective,
(2)
categories:
(3)
and
episodes
baroclinic.
The
were
(1 )
The
baroclinic/convective cases involve synoptic-scale cyclones containing
what appear
to be a
significant number
of
convective
cells.
The
baroclinic episodes consist of synoptic-scale cyclones associated with
a minimal amount of convection
large-scale disturbances,
cells.
in
mind
and the convective
cases involve no
only relatively large numbers of convective
The baroclinic/convective situations were chosen with the idea
that these
events
had a reasonable
chance
of exhibiting a
spatial dependence in the fractal dimension--a phenomenon not observed
in
the
tropics.
Individual
convective
cells
tend to be
small in
comparison with rather expansive cirrus cloud shields associated with
mature
baroclinic cyclones.
Assuming that a given physical process
results in a type of cloud mass with a particular fractal dimension
value,
the small-scale cloud elements should have a fractal dimension
representative of convective processes while the larger cloud s hields
should be associated with a dimension characterisitic of baroclinic
phenomena.
To contrast the hybrid baroclinic/convective cases, purely
convective
was
situations were also examined.
and purely baroclinic
hypothesized
that these
each yield
would
episodes
a
It
different
value of the fractal dimension.
Satellite Imagery
Cloud data
satellite photos.
DB5
infrared
from GOES-East enhanced
were acquired
The enhancement begins with the -32*C
isotherm--id
<-32 0 C appears as a
est, any cloud top with a blackbody
temperature
flat gray on the satellite picture.
A different shade of gray (the
scale actually ranges
from white
In
to black)
this
study
only
=10 0 C
used for every
is
the
-32
0
was
C contour
increment
below -32*C.
utilized.
This choice was made somewhat arbitrarily, with convenience
being a not unimportant factor.
There is
in examining this particular isotherm,
to
correlate
Oliver,
fairly well
1977).
corresponding
Hence,
to
the
some physical justification
however,
with precipitation
these
physical
cloud
areas
process
of
as it
has been found
regions
may
be
(Scofield
thought
precipitation
and
and
of
as
their
associated fractal dimension value may then be directly compared with
that obtained from radar map data.
cloud
regions
patterns,
an
implications.
reflect
It
may also be argued that these
mid-tropospheric
interpretation
that
may
latent
lead
heat
to
and
more
advection
far-ranging
With the adoption of this viewpoint, the exact value of
temperature used as a contour outline becomes relatively unimportant.
In
a
previous
investigation
of
this
sort
(Lovejoy,
1982),
the
-10*C
used
was
in
examination
the
was found that varying
It
masses.
-5*C
isotherm
of
tropical
cloud
from
threshold temperature
this
to -15*C did not appreciably alter the results.
areal
The
investigation
of
coverage
the
satellite
the
basically
included
portion of the western North Atlantic:
degrees
65
and
north latitude
object's
two
quantities
of
a
25 to 55
The
west longitude.
eight kilometers.
interest
fractal dimension are its
and
States
United
from approximately
95 degrees
to
resolution of the enhanced imagery is
The
eastern
this
in
used
pictures
in
the
determination
of
an
Hence,
area and its perimeter.
of these parameters were made for each of the relevant
measurements
cloud masses found in the nine cases that were examined.
were made by hand,
Measurements
as digitized satellite imagery was unavailable.
A
polar planimeter was used to determine areal values and a map measurer
was
employed
to
measure
perimeters.
Actually,
even
though
the
crudeness of the measuring devices limited the effective resolution of
the data,
dimension,
this would
a
fact
understood when it
not affect
which
is
seems
the
counter-intuitive.
realized that D is
log-log plot of perimeter versus area.
which would result in
determination
a change
in
of
the
This
basically
fractal
can
be
the slope of a
Thus, a change in resolution,
the perimeter
values,
would not
alter the slope of a log perimeter versus log area graph and,
would not affect D for a given set of data.
Rather,
hence,
poor resolution
associated with crude measurements would show up as an increase in
the
standard deviation of the data from a straight line fit.
The analysis of Case I will serve as a general example
the satellite imagery.
addition to the -32
In
isotherm was included as well.
of
the cloud outline transcribed
Figure 1 shows
techniques used.
higher cloud
It
tops may be
of
the determination
dimension,
values
contour,
from
the -52
0
C
was felt that this representation
interest,
especially
check for the results obtained from the -32*C
in
0C
the
in
convective
Measurements of this contour can be used as a consistency
situations.
aid
of
if any.
for
the
vertical
variation
or they may
of
the fractal
Table I lists the corresponding perimeter and area
each
representation
of
outlines,
of
of
the
these
cloud
data
is
mass
shown
outlines.
figure
in
The
2.
A
graphical
line
was
best-fit. to the set of points and the fractal dimension was determined
from the slope of this line;
here,
the fractal dimension is
equal to
The standard deviation and correlation coefficient
twice the slope.
were also computed.
The
perimeter
versus
baroclinic/convective
episodes
episodes
(cases IV,
V and VI)
baroclinic episodes (cases VII,
individual basis
points and it
as
each case
seemed as if
area
graphs
for
the
remaining
(cases II and III) and the convective
are shown in
figures 3 through 7;
the
VIII and IX) were not analyzed on an
had a rather
sparse number
of data
any line derived from such a limited number
of values would be subject to sampling error.
An examination of figures 2 through 7 reveals
each episode
constitute an extremely
linear
that the data of
set over approximately
FIGURE 1
Cloud mass outlines.
outer contours: -324C
inner contours: -52C
YZ
k4
TABLE I
0000 GMT 3 April 1982
Satellite Imagery
Cloud Mass #
Perimeter
Area
6570
696000
580
9000
360
3600
90
600
80
500
80
400
80
600
50
300
230
2800
570
8300
910
9700
840
22100
170
2200
10440
1343200
310
5200
60
600
90
1100
100
500
TABLE I (continued)
Cloud Mass #
Perimeter
440 km
Area
10000
160
2000
50
500
170
600
150
2300
120
900
50
200
60
300
LOARITHMIC 3 x 5 C*ES
KEUFFEL & ESSER
CO. MADE i U.SA.
*
46 752
2
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four
spatial orders
of magnitude.
The correlation coefficients
consistently high and the standard deviations are rather low.
hypothesized
variation
that
of
the
the
fractal
between
discontinuity
baroclinic/convective
the
size,
It
was
exhibit
possibly
convective
small-scale
large-scale baroclinic regime.
so.
with
dimension
would
cases
are
regime
even
and
a
a
the
This definitely does not appear to be
In addition to the high degree of linearity,
through VI exhibit very consistent values
the data of case I
of the fractal
dimension,
all within a few percent of the four-thirds value obtained previously
in
the study of tropical cloud masses.
The
measurements
meteorological
10.
from
the
three
cases
regime were compiled and appear in
representing
figures
each
8 through
The fractal dimension value obtained from the three baroclinic
occurrences
is
four-thirds;
it
in
noteworthy
is
that
it
deviates
closer to three-halves.
coefficient of this data set is
noticeably
from
Although the correlation
very high, the somewhat limited amount
of measurements makes one hesitant about drawing any conclusions as to
the significance
fractal
dimension
of
this
deviation.
associated
with
Actually,
the
baroclinic
the
fact
episodes
that
the
differs
appreciably from that found in the baroclinic/convective cases may not
be as important as the result that the convective episodes
same fractal dimension as the baroclinic/convective cases.
have the
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Radar Maps
Data
hourly
for
National
delineate
the
of
part
this
Meteorological
precipitation
areas
study
were
derived
from
radar
maps.
These
maps
Center
and
contour
discrete
VIP
(vertically
integrated precipitation) levels corresponding to particular ranges of
intensity.
precipitation
isopleths
were
this
the VIP
investigation
which
examined,
and
precipitation
In
represent
precipitation-free
analysis of the satellite imagery, it
the
level
between
boundary
Analogous
regions.
one
to
the
may be worthwhile to go back and
investigate the fractal properties of higher VIP level contours.
The analysis used on these maps was the same as that used on the
Although the coverage of the radar maps included
satellite photos.
the entire United States,
only the area corresponding to that of the
DB5 satellite pictures was utilized in order that these two data sets
might be directly compared.
Figures 11
through 13 show the perimeter versus area graphs for
the compilations of the three individual episodes comprising each of
the three different types of regimes studied.
low
number
examination
of
of
statistically
measurements
each episode
meaningful.
associated
Due to the relatively
with
individually
a
The
results
of
those
gleaned
from
these
different
masses.
First, the overall fractal dimension of
regions
is
somewhat
higher
than
that
case,
was not considered
somewhat
than
given
of
the
the
to be
compilations
satellite
an
are
cloud
the radar map VIP
cloud
masses.
Its
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value is also more consistent from one type of episode to another--the
radar data do not exhibit a higher value of the fractal dimension in
important similarity in
in
'kink'
exhibit
the two data sets is
the radar
the graphs;
any
An
regime as was the case with the satellite data.
the baroclinic
spatial
the lack of any sort of a
echo measurements
dependence
of
fractal
the
do not appear to
dimension
over
approximately three orders of magnitude.
500 mb Charts
The maps used were NMC's 0000 GMT 500 mb analyses.
were of interest in
large scale,
making
These charts
that they display meteorological quantities on a
it
possible
to extend
the
fractal
analysis
to
(lines
of
contours encompassing larger areas.
Two
constant
types
of
height),
isopleths
in
were
analyzed;
isohypses
six dekameter
intervals
and isotherms,
Since the flow at 500 mb is
intervals.
geostrophic balance,
the wind field.
analyzed.
5*C
to a good approximation
in
the shape of the isohypses corresponds well with
Hence their fractal properties can, in a crude sense,
be related to the advective patterns of the flow field,
fractal
in
properties
of
the
cloud
masses
that
had
as can the
previously
been
The fractal characteristics of the isotherm configurations
can also be linked with those of
radar echoes,
the cloud masses,
as well as
the
since all of these quantities are directly related to
the spatial distribution of latent heat release.
April, July
The 500 mb isohypses for four time periods--January,
and October 1983 (representing each season of the year) were examined
using the previously
delineated
to
be
day-to-day
advantageous
variations
of
for
comparing
patterns
flow
For the January
relatively small.
per
This temporal spacing was
month, at weekly intervals, were utilized.
felt
Five maps
analysis procedure.
individual
at
this
charts,
level
as
episodes,
tend
height contours
from the lowest on a given map out to 552 dkm were measured.
to
be
ranging
Although
the 552 dkm contour correlates approximately with the core of the jet
stream at this time of year,
its choice as an upper-bound
more pragmatic considerations
than physical ones (the planimeter used
to determine areas is
By
July
the
circumpolar
allowing measurements
although for
limited as to the size of what it
this
vortex
has
significantly
of contours with values
study
as
involved
can measure).
contracted,
high as
thus
582 dkm,
the meaningful aspect of a contour is
its
areal extent, not its actual height.
Figures 14 through 17 show the perimeter versus area graphs for
each month's isohypses.
isopleths is
The calculated
fractal
that
of
these
quite a bit smaller than the values associated with the
cloud mass and precipitation area contours.
fact
dimension
the
height
contours
are
perimeter enclosing a given area is
This simply reflects the
somewhat
smoother--id
est,
This implies
that the
smaller.
the
shape of the height lines is more circular and/or less convoluted than
the cloud mass and precipitation
outlines.
In
this case
it's
most
ZrAUARY
JNIJA-tSSOHY
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FIGURE 14
5
6
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5
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in
be
interpreted
as
large-scale
(synoptic
and
could
dimension
fractal
between
difference
the difference
Physically,
likely a combination of these two factors.
representing
the
planetary)
and
small-scale (mesoscale and convective) processes.
An important difference between the isohypse data and the cloud
data
area
mass/precipitation
is
of
former
the
in
presence
the
a
distinct 'kink'
in
the perimeter versus area graphs at large scales.
deviation
in
slope
This
occurs
to a perimeter of m3x104 km)
the fractal dimension
this.
Table II
(corresponding
a significant change in
and represents
value.
km2
7
m2.5-3.0x10
shows
the quantification
of
Slope and fractal dimension values were recalculated separately
encompassing
for contours
km2 .
at
areas greater than and less than
The change in D at large scales is
a shift
=2.5x10 7
most likely associated with
from a spatial regime characterized
by relatively circular
cut-off features to a larger scale regime characterized by undulating
circumpolar flow.
This is a noteworthy result in that it
evidence
of
episodic
basis.
A result that is
isohypse
fractal
dimension
seasonal
dependence.
lower
in
prevalent.
the spatial dependence
the
spring
Actually,
does
the
the fractal dimension
somewhat surprising
not
appear
it
would
Intuitively
and
of
is the first
autumn,
when
October
data did
to
have
a
seem as if
cut-off
is
that the
significant
D should be
features
have a
on an
are
relatively
more
low
fractal dimension value, but the April flow patterns did not.
The results from the isotherm analyses are shown in
and
19.
In
January,
the
perimeter
versus
area
figures 18
plot
has
TABLE II
Alternate Best Fits
Date
Parameter
Perimeter
1/83
Isohypses
<2.5x104
<2.5x10
7
1.20
>2.5x104
>2.5x10
7
1.00
Isotherms
4/83
7/83
Isohypses
Isohypses
Isotherms
10/83
Isohypses
<3.0x10
4
Area
D
<3.0x10 7
1.20
>3.0x10 4
>3.Ox10
7
1.08
<2.5x10 4
<2.5x107
1.16
>2.5x104
>2.5x107
0.88
<2.5x10 4
<2.0O107
1.18
>2.5x10
>2.Ox107
1.13
<3.Ox104
<3.0x10 7
1.34
>3.Ox104
>3.xO107
1.03
<2.5x10 4
<2.5x10 7
1.13
>2.5x104
>2.5x10
7
0.92
E
ES
x 5 C
"O"RITH
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AESER3MIC
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5
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567
2
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6 7 89
9
6.
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9
8-
7
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characteristics
that are quite similar to the corresponding isohypse
graph--specifically,
P values,
with a distinct discontinuity at large scales.
hemispheric
number
of
gradient
temperature
isotherms
correspondingly
indicate
a very linear relationship between log A and log
that
fewer
there
is
much
significantly
may
not
be
smaller
less.
What values
data points.
discontinuity during the summer.
at this.
is
a
large-scale
In July,. the
and,
hence,
the
Thus,
there
are
there
are
fractal
seem
to
dimension
The July isohypse graph also hints
SUMMARY AND CCNCLUSICNS
has been made of the work done by S.
this study an extension
In
His investigation involved the application of fractal
Lovejoy (1982).
examined
the
precipitation
selectivity.
of
properties
fractal
areas
atmospheric
of
the determination
in
analysis
and
could
find
tropical
no
He
selection.
scale
cloud
masses
any
of
evidence
and
scale
The present research dealt with midlatitude cloud masses
and precipitation areas as well as analyses of 500 mb flow patterns
and temperature fields.
Three different types of meteorological conditions were chosen in
the examination of cloud masses and precipitation areas:
baroclinic,
convective, and baroclinic/convective, in order to adequately test for
500 mb data were incorporated into
the presence of scale selection.
the study so that the characteristics of larger scale phenomena could
be analyzed.
Isohypses
examined
isotherms were
and
on a seasonal
basis.
It
cloud
was found
mass
and
that a fractal dimension
precipitation
atmospheric scale selection,
area
that the
midlatitudes
(namely,
yielded
no
midlatitude
sign
of
any
the same result obtained by Lovejoy in
This was surprising in that it
his examination of tropical data.
hypothesized
data
analysis of
presence
of
convective
different
and
dynamic
baroclinic)
was
processes
in
operating
at
supposedly different scales would be associated with structures whose
geometric
properties
varied
accordingly
with
scale.
It'
can
thus
not scale-selective out to
be concluded that either the atmosphere is
extents
with areal
structures
premise
Mandelbrot's
of
or
a106 km2
flaw in
to objects
rise
give
that different phenomena
a
is
there
and hence different values of
with different spatial characteristics,
his fractal dimension.
terms
In
of the absolute
of
value
the
fractal dimension,
cases were
cloud masses from the convective and baroclinic/convective
m4/3,
associated with a D of
similar
to
clouds from the purely baroclinic episodes
Lovejoy's
the
findings.
The
yielded a data set with a
significantly higher fractal dimension value--close to 3/2.
Assuming
that cloud masses resulting from purely convective processes and cloud
masses
resulting
fractal
dimension
that the
hybrid
purely
from
values
baroclinic
processes
different
have
the fact
(as indicated by these results),
baroclinic/convective
fractal dimension comparable
situations
to the convective
had an associated
cases indicates
that
convection may be the dominant structural determination mechanism out
to scales
well
beyond
the
size of
This result agrees with Gage's (1979)
individual
convective
elements.
finding, based on data compiled
from a number of wind variability studies,
that the three-dimensional
isotropic regime extends well into the mesoscale.
The precipitation area data had an associated fractal dimension
that was
apparently
information,
resolution
but was
of
higher
than
actually
the radar
that derived
comparable
measurements
was
from
when
greater
the
the
fact
than
satellite picture measurements was taken into account.
cloud
mass
that
the
that of
the
Unfortunately,
a physical interpretation of
the significance of
the particular value of the fractal dimension associated with a given
'beyond the scope of this paper.'
structure is
The results obtained from the analysis of the 500 mb isohypse and
from Lovejoy's
isotherm data deviate significantly
findings
in
that
they definitely illustrate some sort of atmospheric scale selectivity
(assuming,
least,
at
interpretation
of
fractal
well-defined change in
corresponding
deviation
to
occurs
that
well-defined. during
basic
dimension
premise
data
is
involved
correct).
in
all
the
with
seasons,
It
summer.
could
it
appears
to
be
represent
possibly
This result is
analysis
may be a viable
scale
selection.
important in
tool in
Currently,
that it
indicates
the investigation
atmospheric
energy
utilize harmonic analyses of wind variability data.
of
km,
This
transition from synoptic scale forcing mechanisms to planetary
dynamics.
4
wavelengths.
planetary
although
the
A fairly
D occurs at a horizontal scale of =3x10
structures
in
the
less
the
scale
that fractal
atmospheric
spectra
studies
Fractal analysis
offers another means of interpreting this as well as other types of
data.
REFERENCES
J. Atmos. Sci.,
Geostrophic turbulence.
Charney, J. G., 1971:
1087-1094.
Sci., 28,
Large-scale kinetic energy spectra from
Desbois, M., 1972:
Eulerian analysis of EOLE wind data.
J. Atmos. Sci., 32,
1838-1847.
Evidence for a k-5 / 3 law inertial range in
Gage, K. S., 1979:
in mesoscale two-dimensional turbulence.
J. Atmos.
Sci.,
36, 1950-1953.
Julian, P.
R., W. M. Washington, L. Hembree and C. Ridley, 1970:
on the spectral distribution of large-scale atmospheric
kinetic energy.
Inertial ranges in two-dimensional
1967:
Kraichnan, R. H.,
turbulence.
J. Atmos Sci., 27, 376-387.
Phys.
Lilly, D. K., 1983:
Fluid,
10,
Stratified turbulence and the mesoscale
variability of the atmosphere.
Lovejoy, S.,
1981:
1417-1423.
J. Atmos. Sci., 40, 749-761.
The statistical characterization of rain areas
in terms of fractals.
Boston Radar Conference preprint.
Mandelbrot, B. B., 1982:
The fractal geometry of nature.
Freeman, 460 pp.
Schertzer, D. and S. Lovejoy, 1983:
atmospheric motions.
write:
The dimension of
Unpublished.
For a copy,
EERM/CRMD, Meteorologie Nationale, 2 Ave. Rapp,
Paris 75007, France.
Scofield, R. A. and V. J. Oliver, 1977:
A scheme for
estimating convective rainfall from satellite imagery.
NOAA Tech. Memo.
NESS 86, Washington, D. C.,
Steinberg, H. L., 1972:
47 pp.
On the power law for the kinetic
energy spectrum of large scale atmospheric flow.
Tellus, 24, 288-292.