ON
BAND
FORMATION
IN
A NEW
ENGLAND
WINTER
STORM
by
DAWN
B.S.,
GUSTINE
University
of
WOLFSBERG
California,
Los
Angeles
(1982)
SUBMITTED
EARTH,
TO
THE
DEPARTMENT
ATMOSPHERIC
AND
PLANETARY
IN
PARTIAL
FULFILLMENT
REQUIREMENTS
FOR
MASTER
OF
at
MASSACHUSETTS
Signature
of
SCIENCES
THE
DEGREE
OF
SCIENCE
the
INSTITUTE
May
Massachusetts
THE
OF
OF
OF
TECHNOLOGY
1984
Institute
of
Technology,
AutD
Certif:
T'
Accetedby
Chairman,
,VN
WITH DRAW
FROM
MIT LIBRARIES
Departmental
Co-
1984
ON
BAND
FORMATION
DAWN
in
IN
A NEW ENGLAND
by
GUSTINE
WINTER
STORM
WOLFSBERG
Submitted to the Department of
Earth, Atmospheric and Planetary Sciences
partial
fulfillment
of the requirements for
the degree of Master of Science in
Meteorology
ABSTRACT
This case study addresses mechanisms of band formation
in
a New England winter storm.
The structure
of the bands
and
their
environment
are
documented
with
synoptic
observations,
Doppler
radar
data,
and
analyses
of
instrumented aircraft
flights
through the bands.
The paper
postulates
that processes on three scales are responsible for
the
bands
observed.
The
bands
are
a
manifestation
of
mesoscale symmetric instability.
Potential
energy for the
instability is generated by synoptic scale differential lapse
rate
advection
and
converted
to
kinetic
energy
by
the
symmetric overturnings.
There is a rough equilibrium between
large scale generation and mesoscale depletion of potential
energy.
Frontogenetical forcing results
in
an intermediate
or sub-synoptic
scale region of ascent.
The sub-synoptic
scale vertical
motion brings the atmosphere to saturation
and
enhances the release of potential
energy.
Thesis Supervisor:
Kerry A. Emanuel
Title: Assistant Professor of Meteorology
Index
page
Abstract.............
.......................................... 2
Index.................................................................3
Chapter I Introduction
......................................... 4
Chapter
II Data
Chapter
III Description of
Chapter
IV Role
and Experimental Design
........................... 10
of Symmetric Instability......................43
Chapter V Role of
Chapter VI
Bands...
........................ 7
Frontogenetical Forcing.....................71
Discussion
Acknowledgements..
and Conclusions.........................94
............................................. 97
References..........................................................98
I. INTRODUCTION
snow
of
formation
was
as
observed
of
all
a
an
the
energy
equilibrium
scale
England
field
project
of
a
of
Technology
in
of
to
available
between
An
of
the
release
ascent enhances
The
weather
analysis
radar.
of
Northwest.
Houze
bands
They
(1960)
cells
et al.
observed
noted
that
of
(1976)
in
The
supply
of
represents
by
generation
large
energy by mesoscale
scale
frequently
attracted
one
in
fell
region
of
potential instability.
organization
by Austin
noted
December
England
motions
sub-synoptic
have
cyclones
Linear
attention.
energy
to kinetic
two-dimensional
extra-tropical
11
the
by
argued that the bands
unstable
potential
or
New
Central
instability.
the
intermediate
On
that
storm
winter
(MIT).
the
for
account
conducted
It is
symmetric
processes and conversion
processes.
was
a New
precipitation
manifestation
potential
in
well-organized bands.
from weak,
are
paper
bands
part
to
is
this
Institute
Massachusetts
1982,
of
purpose
The
of
the
in
deal
of
great
structures
precipitation
studies
first
with
presented a comprehensive
cyclonic
bands
a
found
can
in
storms
occur
in
the
Pacific
every
sector
of
the
number
bands
storm.
of
Elliot
bands
tend
in
along
A number
carry
1976),
instability
"conveyor
because
there
very
the
temperature
ducted
of
was
gravity
strong
microbarograph
traces
This paper
and
data
are
description
important
band
during
scale
features
instability.
The
An expression for
for
symmetric
the
atmosphere
the
instability
to
chapter
which
the
theory
time
symmetric
1978).
a
candidate
buoyancy.
Although
have
evidence
was
growth.
the present
not
into six chapters.
in
Chapter
structure.
and
atmosphere may
storm,
frontal
in
allowed
the
form
not found
in
hourly pressure reports.
bands,
in
the
the
or in
described
the
thermal
is
the
(Lindzen and
ruled out in
model
of
have been
band
1982),
pressure fluctuations
is divided
of
of
gravity waves
belt"
large
vector.
that
regions
can be
structure of
waves
small
to
little
a
axes
Bennetts and Hoskins,
these mechanisms
Harrold's
the
shear
(Carbone,
(Emmanuel, 1979;
case.
that
suggested
air
currents
observed
for band formation
include ducted
density
Some of
noted
(1973)
buoyant
(1964)
vertical
of mechanisms
Other possibilities
Tung,
and
the mean
Harrold
circulations
Hovind
California
to lie
suggested.
and
large
IV
its
rate
symmetric
scale
flow
the
a
of change
instability
III
role
analysis
in
a
of
of
of
symmetric
are
reviewed.
of potential
The
is
discussion
and
predictions
derived.
experiment
Chapter
includes
discusses
and
is
II.
The
energy
susceptibility
this
particular
of
6
case
and
the
evaluated.
generation
strength
of
Estimates
rates
are
of
also
frontogenetical forcing.
and
their
Chapter VI
effect
provides
on
updrafts
which it
potential
given.
The
energy
Chapter
strength of
symmetric
would
V
produce
are
depletion
and
discusses
the
the forced updrafts
instability
is
discussion and conclusions.
considered.
II.
Field
radar
data
DATA AND EXPERIMENTAL DESIGN
observations
instrumented
data,
and surface
There
are
wavelength
1982,
to
radar
11
Position Indicator
of
the
Volume
bands.
Velocity
construct
At
from
the
cloud
1830
National
passes
airplane instruments
and
km
physical
long
above
and
radar.
moved
the
was
derived
the
MIT's
air
the
(GMT)
10
December
and
case
Altitude
is
Plan
behavior
Corbin's
technique
of
cm
of
are analyzed for
sections
11.03
period
Constant
the
Center
(1979)
used
quantities
to
derived
Queenair,
for
an instumented
Atmospheric
to
snow
Research,
bands.
temperature, velocity,
parameters
every
second.
At
The
humidity
both
times
at approximately 1 km, 2 km, 3 km, and
surface.
the
GMT,
perpendicular
centered
During
in
Time
(VVP)
recorded
there were four passes
4
hour
1982.
Processing
GMT and
horizontal
flew
24
from
Waldteufel
cross
upper
velocities.
1230
airplane
the
Meridian
additon,
Doppler
These are described below.
(CAPPI) displays
time-height
from Doppler
for
include
measurements,
December
In
study
observations
Greenwich
GMT
case
aircraft
continuous
18
19
the
observations.
Doppler
from
study,
for
Each
on
the
last half
band-normal
pass
bands
of
the
direction.
from the airplane data
was
approximately
observed
1830
GMT flight
Bandpass
are
by
cross
50
the
km
MIT
the band
sections
presented in band-relative
than
rather
with
advected
band
was
The
The National
soundings
at
at
each point
were
12
the
in
3 km
of
the
and
while
completed
4
the
varying
from a running mean were
over
ten
Meteorological Center's
GMT
by
supplemented
are
GMT and
18
wind
measurements
flights,
averaged
band
seconds,
or
about
distance, before analysis.
data
surface
launches
both
are
data
one kilometer air
and
at
were
ms- 1 i, from
4
band,
two standard deviations
The
discarded.
the
axes
x-component
the
subtracting
of
-In
cross
band-relative
two bandpasses
lower
stationary.
than
by more
by
velocity
band
cross
km passes.
The
velocity
band-normal
observed
band.
calculated
was
component
the
the
coordinate
The
coordinates.
physical
at
Chatham,
special
There are
GMT at MIT.
18
MA
upper
standard
(CHH)
and
air
radiosonde
also special
Albany,
NY
(ALB).
Fig.
1
upper air network
passes
the
shows
sites
mentioned
in the Northeast,
and vertical
in
this
and locations
cross section are
indicated.
paper.
of
The
the band
Stations mentioned or
Locations mentioned in the text.
Fig. 1
A vertical
codes.
letter
three
by
used in calculations are shown
of the
locations
The
line.
dashed
a
by
cross section is shown
band passes are indicated.
mmhr~
is
bands
rates
in
this
In
.
section
a
nature
of
the
environment
analysis and airplane
The
850 mb,
air
flow
The
average
GMT.
centered
scale
and
10- 2
backs
with
as
at
height
be
seen
in
Fig.
500mb
than
at
at
can
This phenomenon
is
in
850
The
mb
discussed
in
Bay.
The upper
west-southwesterly.
some
2.
is
temperatures
12 GMT.
at
towards
directed
deep
over
s-1
and
is
shear
vertical
exceeds
shear
general
in
Hudson
on
winds,
heights,
is
England
large
geostrophic
southerly
cyclone
New
over
700
England,
deep
England
New
in
pattern
700 mb,and 500 mb at 00 GMT and
at
about
Environment
weather
show geopotential
2a-f
Figs.
a
by
dominated
Scale
scale
synoptic
a
bandpass analysis.
The Large
a.
bands
radar
formed,
they
which
in
2
through
explored
is
bands
the
The
discussion
12
of
presented.
the
of
description
detailed
very
but
exceeded
never
England
New
to
four
of
organized
well
were
the
24 hours,
the next
lifespans
with
decay
the
GMT
2230
By
For
two bands.
The.
Snowfall
weak.
December.
and
each.
hours
10
form
to
continued
six
on
organized into
echoes were
bands
GMT
1930
at
radar
MIT
the
on
observed
storm were
of
echoes
radar
first
The
OF BANDS
DESCRIPTION
III.
layers.
layers
isobars
both
chapter
00
IV.
over
are
GMT
The
New
more
and
Fig.
2
Upper
air
charts
heights
(solid lines)
therms
(dashed
indicated
1 barb
=
lines)
with barbed
10
kts,
are
triangular
on
11
December
drawn every
are
drawn
velocity
barb
every
50
Geopotential
3 dam or 6 dam and iso-
vectors.
=
1982.
kts.
5 0 C.
(1/2
lkt
Winds
barb
A
.5
=
are
5
kts,
ms-1
)
Fig.
2a
Fig .
2b
Fig.
2c
Fiq.
2d
Fig.
2e
Fig.
2f
00
GMT.
of
slope
1
than 1 cms~
is
somewhat
the
than
20% less
and
there
is
the
isobars
more
cold
The
a
model
rough
This
surfaces.
less
of
velocities
850 mb and
are
generally
magnitudes
typically
The
ageostrophy
over
12
GMT.
ageostrophic
warm
at
advection
700 mb
PWM
and
At
advection
has
are
cold
a
at
local
PWM
mb.
700
at
maximum
in
discussed at greater
England.
New
3a-b show surface maps
front
west
Wisconsin
at
from
Its
CHH
at
is
cold
geostrophy
implications
chapters IV and V.
length in
Figs.
of
deviation
The
New
There
ALB.
in
(Fig.
mb
850
also ageostrophic warm advection at CHH and ageostrophic
advection
the
times
speed
of
velocity
scale vertical
wind
at
large
with
at
pronounced
substantial
ageostrophic
but
values.
geostrophic
becomes
England
2d),
of
GMT.
velocities
The
subgeostrophic.
direction
negligible
England at
over New
GMT the wind
At 00
is
over New England.
levels
at all
it
from
vertical
warm
geostrophic
12
at
New
of
west
and
is
temperature
scale
large
yields
calculation
However
horizontal
potential
the
there
2,
The synoptic
the
half
approximately
south
estimated
be
baroclinic instability.
is
the
levels
all
can
motion
upward
scale
Fig.
at
and
GMT
00
at
mb
700
at
mb
850
at
advection
in
seen
be
can
As
England.
area
Tennessee,
in
and
Plains
to
well
Georgia,
and
Alabama,
Great
the
in
are
motion
upward
the
of
bulk
the
presumably
and
storm
the
of
center
The
for 00
GMT and
GMT
advances
00
12 GMT.
eastward
The
at
1
(30
about
8
cross
sections
is
ms
before
does
not affect
the
case
about
18
GMT.
bands
in
New
the
the
warm
inversion
on
a
shows
which
the
surface.
700
mb.
The
strongest
2x10-
2
s~
soundings
V.
There
air
shear
.
are
There
1982.
December
11
sounding,
typical
The
is
is
high
generally
occurs
illustrated
the
00
at
GMT
650
mb
thermal
air
above
Fig.
3c,
for
ALB
sounding
above
layer
950
mb
mb.
600
and
700mb
in
almost
is
below
the
the
stations,
in
structure
length
through
taken
between
saturated
greater
sometimes
unstable
thin
stability
between
atmosphere's
discussed
are
a
is
that
troposphere
upper
to
on
soundings
many
At
V.
Most
The
stable.
features
These
moist adiabatic.
less
middle
the
through
layer
inversions.
strong
generally
is
rate
lapse
show
fronts
of
evident
Soundings
troposphere.
lower
the
into
extends
Northeast.
boundary
the
in
stability
high
show
south
chapter
buoyancy
thermal
the
in
stations
from
soundings
in
in
Forcing of
3.
Fig.
discussed
no
virtually
is
There
front is
warm
just
fronts
the Southeast in
Massachusetts and across
late
quite
until
England
probably
front
cold
The
over
isotherms
potential
warm
the
Note
period.
the Massachusetts
of
slope
there
that
shown)
(not
vertical
with
ascertained
west
to
east
east-west
England
New
is
It
1).
oriented
little
very
kmhr
and
chapters
and
The
and
is
other
IV
and
Fig.
are
3a-b
drawn
Surface
every
omited.
Wind,
indicated
for
4
mb.
on
charts
The
temperature
each station.
11
thousands
in
*F,
December
and
and
1982.
hundreds
present
Isobars
places
are
weather
are
Fig.
3a
673
.+30*
7 2f
1'DECEMBER
SURFACE
65
+125
Fig.
3b
-
1982
12 GMT
23
300 -
500 -
900 -E 6003
N
F
N,
NII
N
000
Fi00
3cSudnNo
a
skew-T,
psuedo
dashed
L
log-p
temperature
N
are
diagram.
shown
in
t0
The
heavy
M
o
1Dcme
sounding
solid
adiabats and moist adiabats
lines.
Winds are
on
ploted
92potdo
temperature and
dased lines.
and
are
the
given
side.
in
light
dew point
Reference
solid
and
are
bands
illustrated
GMT
1130
through the
shown in
December is
four
bands
oriented
toward
are
oriented
about
bands
shear
geostrophic
the
more
width
bands,
of
oriented
some
of
the
at
30
is
about
bands
it
and
80
there
breaks
up.
is
mb
1040 GMT
irregular
into
to
the
two
to
to
original
suggestion
Twenty
or
of
forty
to
The
the
mean
of
The
average
distance
between
mb.
500
and
three
from North).
left
the
wavelength,
At
up
a
to
150
km.
angles
(clockwise
550
the
breaks
various
instances
before
km
is
reflectivity
typically
are
there
850
between
to
December
taken at
4.0 km
in
4a.
Fig.
stage,
first
the
During
to
km
between 2.5
layer
radar
of
display
A CAPPI
December.
11
10
GMT
23
from
lasts
which
stage,
first
band
basis
the
precipitation occurs
The strongest and most extensive
11
Three
factors.
these
the
on
identified
been
have
behavior
band
of
stages
of growth,
bands.
the
of
strength
and
orientations,
wavelengths,
the features'
regions
show preferred
they do
time,
continuity in
do not indicate
these displays
reproductions of
the
Although
displays.
CAPPI
radar
with
of
behavior
and
character
the
section,
this
In
CAPPI Displays
Analysis of Radar
b.
intervals,
four
one
bands
smaller
band
waviness
minutes
or
axis.
In
a
band
in
after
the
Fig.
4a
stage
1, taken
at 1040. GMT
Fig.
Fig.
4c stage
4b
3, taken
stage
2,
at 1822
taken
at
1325
GMT
GMT
Fig. 4 CAPPI displays of radar reflectivity through the layer
between 2.5 and 4.0 km.
Solid lines show contours of 0 dbZ.
regions with dbZ>10 are shaded.
10(D5-
IOdbz
Odbz
00
100
50
DISTANCE
(KM)
Fig. 4d RHI display of radar reflectivity taken at 953 GMT
toward 1400.
sufficiently
it
that
robust
configuration
original
the
that
interesting
is
It
paper.
this
of
scope
the
beyond
is
phenomenon
this
of
Consideration
behavior
themselves.
bands
the
in
instability
an
represents
the
while
this
that
speculated
is
It
decay.
bands
other
550
to
rotates
bands
smaller
the
of
one
break-up,
is
reestablished
no
precipitation
is
a
after
break-up.
Connecticut.
The
the
onset
12
GMT,
the bands
GMT.
England
is
begins
range
of
its
also
at
the
width 20 km,
north at
oriented
splits
the
decays
same
toward
out of New England.
New
as
(Fig. 4c)
area
by
separated
three,
as strong
1737 GMT
toward
band
km,
northern
not
are about
three
the
between
one.
At
single
band
band
The
is
about
two parallel
into
km.
25
Hampshire
New
At
1345
GMT
GMT
and
which rapidly decay.
1345
between
precipitation
little
During stage
but
4b.
a
by
the
from
55*
Fig.
into four bands
split
There
15
signaled
in
GMT
later,
of
with widths
bands
1737
hour
An
45 km wide.
is
toward-
1202
at
shown
as
than
western
in
or
stage
in
weaker
two
stage
of
southeastward
extending
coast,
are
bands
the
along
Island,
Rhode
in
Massachusetts,
of
shore
south
is
there
two,
stage
During
the bands
southern New
form in
England.
The
bands
stage
in
Stage
three
when a band moves into the
radar
those in
MIT
after an
and
stage
CHH.
During
55*-60*.
By
The single
Another
hour.
time.
two.
stage
1940
band
three
GMT,
the
the
band,
forms
band
storm
to
is
moves
5
about
at
is
reflectivity
width
band
the
the
of
top
the
that
that
and
km
taken
Indicator)
Height
shows
RHI
The
GMT.
953
at
1400
toward
(Range
RHI
an
shows
4d
Fig.
is
about 75 km.
of
maximum
The
ms
there
are
Fig.
Because
the
Thus
vertical
bands
generally
probably
are
are
than
less
ms~
1
centered
not
the
resolution,
coarse
the
.05
about
is
at
velocity
actual
As
.
at
velocity
is
times
when
illustrated
over
the
radar.
velocities
vertical
updrafts
maximum
in
bands.
The
and
The
present
of
calculated
The
no bands
4,
maximum downward
GMT.
GMT.
19
at
1
12
ms-1
.3
is
calculated
velocity
vertical
GMT and
1015
4 GMT,
in
in
5a-b.
Figs.
-. 5
sections
shown
are
velocity
vertical
and
reflectivity
radar
Time-height cross
horizontal divergence.
integrating
by
calculated
are
velocities
Vertical
technique.
VVP
(1979)
Corbin's
and
Waldtuefel
using
cylinder,
radius
km
sections
cross
VVP time-height
velocity and divergence are computed over a
Average wind
30
of
Analysis
c.
maxima
in
convergence
maxima
2 km
and
the
the
VVP
calculations
circulations
is
about
in
(not
shown)
are
reflectivity
suggest
1.5 km.
that
the
are
between
km
higher.
slightly
base
1
of
the
band
'2
I-
09
10
Il
12
Fig.
Fig.
5
VVP
5a
14
13
TIME (GMT)
average
time-height cross
15
ON
16
12/11/82
reflectivity
sections
in
between
17
19
18
dB(Z)
9 GMT and
20
GMT.
3
r2
L-
13
TIME (GMT-)
Fig.
5b
vertical
velocity
from
OJN
12/11/82
integrated
divergence
in
cms
d.
The
band
contoured
moisture
The
cross
1450
distribution
in
along
given in Figs.
Figs.
200Y
and
a
measurement
Rain
in
Figs.
particle
the
subjectively
the
airflow
to
band,
the
is
and
band.
oriented
The y-coordinate
Schematics
observed
is
showing
the
radar
are
the MIT
on
of'reflectivity
of
the
location
band
afternoon flight,
ice
are
the
that
therefore
be
are
the
wider
band
cross
than
is
probe
the
times
measure
airplane
precipitation
more
The
as
is
reflective
five
flight
located between
times
radar.
The
indication
an
section.
than
reflectivities
recorded by a
the
a
The
approximately
taken
which
diameters
with
provides
encountered.
reflectivity
is
array
particles
five
which would
within
(Z) measured by
concentration.
actually
values
high
optical
assume
7a-b
contours
the
to
Reflectivity
pm.
approximately
than
flight,
and
calibrations
greater
band
as
in
perpendicular
550*.
toward
bands
4500
the snow which was
given
plane
toward warmer air.
water
ym and
is
illustrate
one-dimensional
precipitation
rain.
a
presented
6a-b.
counts
300
is
perpendicular
band,
the
Analysis
which
7a-b show contours
probe,
between
the
and
flight paths
of
runs
and increases
oriented
sizes
analysis
sections
x-coordinate
at
a
pass
Band Pass
In
the
of
morning
In
legs.
0 km
and
the
30
km
Fig.
1230
GMT flight
Fig.
1830
6a
6b
GMT flight
Schematics of band passes showing
6
Fig.
superimposed on radar CAPPI display.
flight legs
Fig.
in
km.
7
Band
Abscissa
pass
shows
cross
sections.
distance
across
Ordinate
the
shows
band in
km.
height
4
,25
25
25
,3
2
20
0
15.
20
31
DISTANCE(KM)
Fig.
7a
I _
-20
reflectivity
-10
Fig.
7b
in
dB(Z)
for
40
1230
GMT
0
10
DISTANCE (KM)
as
if Fig.
7a except
for
flight
20
1830
50
GMT flight
30
-2
0
2
--
2
I
cross
7c band relative
GMT flight
band
50
40
30
20
DISTANCE (KM)
10
Fig.
1230
iii
I
I
I
0
velocity,
u,
in
ms
1
for
4
-34
-0
2
--
DT
0
4
Fig.
7d
20
10
0
DISTANCE (KMV)
-10
-20
as
in
Fig.
7c except
for
1830
GMT
flight
30
2-
o
Fig.
1230
40
30
20
10
50
DISTANCE (KM)
7e band relative
GMT flight
streamlines, drawn
every 2x10-5 m2 s-
for
4
3-
w
2-
-20
0
10
DISTANCE (KM)
-10
Fig.
7f
as
in Fig.
7e
except
for
20
1830 GMT
flight
30
4
0
,3
2
2O
H4
0
.2
0
.2
2ii
X'20
0
10
Fig.
20
30
DISTANCE (KM)
7q vertical velocity,
w,
in ms-
40
1
50
1230 GMT
for
flight
4
2
3-
.2
--.2
.6
:1:
0
-
~.4
Ej-
.2
0
-20
10
0
DISTANCE (KM)
-10
Fig.
7h
.4
as
in Fig.
7g except
for
20
1830
GMT flight
30
4
1
IIII
95
95
90
w
85
90
280
0
Fig.
10
20
30
DISTANCE (KM)
relative humidity,
7i
in
percent,
40
for
1230
GMT flight
4
80
90
95
3-
0
I I-
-20
-
I
I-
-N
-10
Fig. 7j
-
0
as
in Fig.
L
20
10
DISTANCE (KM)
7i except
for
1830 GMT
f light
50
on
MIT
(Fig.
radar
updraft
on
visible
be
not
may
regions
downdraft
so
an
in
only
grow
droplets
cloud
and
Precipitation
itself.
band
the
than
wider
probably
is
band
the
with
associated
circulation
the
that
noted
be
should
It
6).
the
by
indicated
those
with
agree
measurements
reflectivity
aircraft
by
indicated
locations
band
The
x-axis .
the
the
radar.
Figs.
7c-d
show
isopleths
the
band.
In
to
relative
cold
velocity
is
band wind
cross
toward
and
band
cross
the
both flights
levels
low
at
air
warm
toward
the
u,
of
at
air
upper
levels.
winds
and
v
CHH, and
from PWM,
as
most
one-seventh
3v/ay
magnitudes
12
GMT
and
(Portsmouth N.
av/ay
are
to
appears
of
less
au/3x and
18
than
and
10-
of
magnitude
Buoy 44005.
s-
upper
estimated
winds
surface
It
of
The
au/ax.
at
av/3y are
across
the
to
justified
be
with
GMT
H.),
The
magnitude
the
are
au/3x
and
over
also
and
3v/3y
is
levels.
the
from
assumed
ocean
BOS,
u
along
assumption
At both times,
is
level
The
(Yarmouth, Nova Scotia).
above.
defined
winds
YQI
band
upper
with
GMT
12
and
GMT
00
respectively
refer
velocities
band
small
at
estimated
are
3v/3y,
divergence,
of
components
horizontal
The
small.
the
along
variations
that
assuming
by
calculated
are
velocities
vertical
and
streamlines
Band-relative
at
of
The
at
PSM
au/ax and
that
the
fields
of
u
of
shown
This
observations.
DT/3x
-u,
=/3z
av/3y
0,
=
Contours
differences,
are
branch
7e-f)
of
an
not
is
hour
strictly
a
and
the
slope
half.
1984)
satisfies
In
streamlines
found
also
(1)
calculated
about
is
defined as
Y
7e-h.
the'
of
1/30.
both
in
slopes
afternoon
circulation.
of
branch
The
downdraft
reflectivity
between
reflectivity
maximum
5
at
the
circulation.
It
to
snow
from
section.
falling
The
is
circulation
the
is
km
almost
in
25
km.
and
km,
sloped
reflectivity of
the
strong
but
smooth
updraft
likely
that the circulation
as
indicated
that
updraft
the
in the
a
1983.
region
is
pattern
streamlines.
In
roll
upward
high
of
also
a
branch of
maximum
above
in
1/30
The
downward
this
(personal
closed
There
the
Such
by
the
morning flight
featureless.
fairly
in
speculated
a
vertical.
occurs
x = -5
a
show
streamlines
upward
of
the
the
flights,
Sanders
streamline
finite
the
study of a banded snow storm on 11-12 February
7f),
mass
with
his
(Fig.
as
valid,
z=0.
V-v,
w,
Figs.
circulation
communication,
<
Y and
in
at
with T=0
w,
of
shown
the
=
d(lnp)/dt
and
continuity.
(Figs.
assumption
set
synchronous
a
relative stream function T, is
The band
3
represent
7c-d
each lasted about
the flights
If
Figs.
in
is
the
due
cross
(Fig. 7e)
is
suggests
a
It
morning experiment is
seems
also
roll
but
observe
the
a
The
flights
These
values
radar
velocities.
morning
base
between
1
updrafts
is
and
km,
km
2
in
morning
the
found
occur
in
the
2
at
km
flight.
afternoon
to
Doppler
from
is
convergence
the
and
respectively.
ms-
calculated
estimated
as
.6
and
those
at 1 km
in
7g-h)
strongest
and
flight
ms- 1
with
The
the
of
well
agree
four kilometers.
(Figs.
.4
are
both flights
in
apparent
is
extend above
strengths
updraft
afternoon
It
downward branch.
to
enough
long
not
wer-e
legs
flight
the circulations
that
the
the
that
at
in
The
altitudes
Doppler
velocity
analysis.
Figs.
air
is
updrafts.
the
lower
afternoon
is
There
left
a
the
morning
flight
recent
descent
as
on
the
lower
The
characterized
The
of
of
developed
by
section
cross
strong
95%)
in
humidity
corresponds
The air in this
(Figs.
sections
are also
the
the
in
In
the
to
the
region
experienced
probably
reflectivities
the cross
than
sections.
cross
region
dry
well.
Summary
bands
both
cases
both
low relative
very
the circulation.
left corners
e.
of
this
descending branch of
of
region
corner
flight,
greater
humidity
(relative
saturated
In
humidity.
relative
show
7i-j
7a-b)
a
in
low.
Band Characteristics
in
a
vertical
large
shear.
scale
The
environment
geostrophic
shear
with
backed
England.
A
stability
low
levels.
small
Very
inferred
from
observed
on
levels,
the
mb
and
these
maps.
features
of
Warm
high
were
upper
at
velocity
was
ageostrophy
was
observed
the
Strong
fronts
New
over
saturation,
vertical
maps.
mb
stability
lower
synoptic-scale
synoptic
850
and
extensive
showed
sounding
at
500
between
height
on
surface maps.
Gross
CAPPI
The
displays.
were
oriented
shear.
about
The bands
moved
The
were
about
5
The
bands
observed.
to
40
minutes
with
particular
slowly.-
of
The
Multiple
were
lasted
1/30
saturated,
a
and
and
a
The
and
between
.05 ms-
mean
observed on
bands
km
to
50 km
3
to
6
hours.
was
the radar
frequently
were
wide,
and
geostrophic
maximum convergence
20
passes.
sloped
formed
There
was
vertical
was
the bands were
The
roll
vertical
there
downdrafts.
the
shape
of
in
20
time-
the bands.
band
showed
of
left
structure and circulation of
airplane
slope
the
in
quasi-linear
reflectivity tops
dependent behavior in
The
to
km.
and
were
bands
15*
at about 1.5 km.
at
the bands were inferred from series
afternoon
circulation
low
relative
velocity
the bands.
was
flight
with
downdraft.
.6
The
an
1
in
in
updraft
updrafts
humidity
ms-
observed
in
the
were
the
bands
ROLE OF SYMMETRIC INSTABILITY
IV.
The Parcel Model of Moist Symmetric Instability
a.
Bennetts
suggested
neutral
large
Hoskins
and
that
some
bands
which
ascent
Symmetric
may
be
form
caused
instability
in
below closely follows Emanuel
and
centrifugal
instability
velocity
and
convectively
of
imbedded
by
have
stable
in regions
symmetric
or
of
instability.
in
sloped,
moist symmetric instability
(1983b,d).
arises
fields
when
are
gravitational
and
layers
(1979)
rolls aligned with the geostrophic
The parcel description of
Symmetric
Emanuel
manifested
is
two-dimensional, mesoscale
density
and
thermal stratification and are
scale
shear.
(1979)
unstable
force.
inertially
perturbations
stable
The
to
the
yet
the
combined
atmosphere
and
in
may
be
symmetrically
unstable.
We
zonal
will
flow
temperature
the
base
potential
consider
is
in
a
base
state
geostrophic
zonal
temperature,
geostrophic
6
v,
are
which a
balance.
is a function only of
state
in
steady,
Then
x and z.
velocity,
the
purely
potential
The equations
vg,
and
virtual
for
where
the
dVg/dt
=
0
f(dvg/dz)
=
(g/Ov
subscript
"g"
in
the
y
a
direction
displacement
vg,
e\7,
equations
tubular
is
and
pressure
to
the
(3)
geostrophically
fluid,
proceed
are
and
extending
in
the
slowly
x-z
so
undisturbed.
infinitely
plane.
that
The
fields
The
of
perturbation
are
where
the
shown
that a
dvp/dt
=
-fur
dup/dt
=
f(vp -
dwp/dt
=
g/vo(6p
subscript
"p"
refers
conserved
(4)
vg)
(5)
-
to
the
Oy)
parcel.
pseudo-angular momentum, M,
M
is
of
displaced
assumed
)36e/ax
state.
parcel
is
0
denotes
hydrostatically balanced base
Suppose
(2)
following
unidirectional.
Eq.
the
= v
(6)
It
can
defined as
+ fx
parcel,
be
(7)
if
(5) can be written
the
vertical
shear
is
-
dup/dt= f(M
If,
(6)
after
and
ating,
(8)
so
are
the
the
meaning
and
6
they
the
displacement,
positive,
displacement
of
Eqs.
increase
lower
values
force
is
M.
In
If
than
the
slope
of
between
parcel's
having
than the
environment.
be
unstable
Symmetric
the
if
a
If
for
environment
(8)
to
is
the
is
right
slope
lifted
its
the
The
the
0 to
is
Figs.
0 is
the
8b,
of
M and
a
the
have
so
the
M is
less
a
with
result
slope
in
value
the
of
M
is
6
has a component in
Slanted displacements
sometimes
less
referred
is
account
for
release
not
saturated,
but
than
restoring
of
higher
so
stable
will
The
6 will
environment
moist
it
slope
M
figures
displacement,
the
of M and
of
of
8a-b.
less
shown
restoring force
slope
is
in
environment.
the displacement.
the
illustration
displacement
slopes
value
of
Eqs.
acceler-
An
both
of
inertially
as
of
A
sides
continue
in
and
Fig.
6.
will
shown
the
8a,
In
lower
instability
convection."
substituted
of
unstable.
direction
stable.
the direction
is
0 than
is
intermediate
parcel
parcel
and
opposite
the
Fig.
a
of M
displacement
left-hand
gravitationally
configurations.
of
and
the
(8)
the
and
upward
represent
slope
(6)
Mg)
than
to as
saturated,
of
that
Oe
latent
a
may
of
6.
"slantwise
should
be
heat.
If
quantity
S
is
M3
Fig.
8a
Stable
configuration.
M
surfaces are steeper than
A
parcel
6
surfaces.
lifted
as
shown
is
subject to gravitational
and centrifugal
restoring
forces.
3-\
Fig.
8b
Unstable
configuration.
0
surfaces are steeper than
M
surfaces.
A
parcel
lifted
as
shown
is
subject to forces in the
direction
of
the
displacement.
Fig. 8
Schematic illustration of symmetric stability and
instability.
6 (dashed lines) and M (solid lines) both
increase upward and to the right.
The directions of the
resulting gravitational (Fg) and centrifugal (Fc) are
indicated.
introduced.
parcel
has
defined
neutral
uniquely
a
and
SPA),
for
.
for
slantwise
and
area
surface
is
is
lifted
that
level.
Fig.
and
S
a
negative
and
positive
temperature,
and
parcel
S
the
dry
lifting condensation
Mg
level
of
in
9 shows
area
area
(SNA
between
shown.
neutral
level
saturated
positive
are
Potential
energy
is
The
(SPA)
of
Slantwise
potential
(8).
the
of
which
The
buoyancy
the
level
(LNB)
convection are also indicated.
b.
displacement
shape
The
its
and
parcel
(LFC)
along
above
negative
and
surfaces
parcel.
it reaches
convection
The
each
Slantwise
the
environmental
free
The
configuration
envrironment.
of
are
moist adiabatically
possible
and
S
buoyancy.
adiabatically until
(LCL)
of
Surfaces
the
path
potential
given
available
integral
to
of
energy,
by Emanuel
Energy
or
(1983b)
a
slantwise
the
forces
in
the
slantwise
eqs.
(6)
positive
is
LN Bg
SPA
f [f(My -
=
Mg)i
+
9
v0
LFC
where
the
of
the
level
the
LNB
of
and
free
LFC
are
the
convection.
integrand is
zero.
If
(_vp
-
level
It
can
eyg)k]-ds
of
be
(9)
neutral
shown
the integrand
is
buoyancy
that
F then
the
and
curl
S
LNB
SPA
LFC
LCL
SNA
Fig. 9 Schematic illustrating a possible configuration of M and S surfaces for a parcel subject to slantwise convection in a satufated environment.
The S surface is a contour of potential temperature below the lifting condensation
level (LCL) and a contour of * above that level. e e* is the equivalent
potential temperature a parceI at a given temperature would have if it were
saturated. e * is conserved only if the air is saturated. The level of free
convection (LCE), the level of neutral buoyancy (LNB), slantwise negative area
(SNA), and slantwise positive area (SPA) are indicated.
VxF
Mg)
=
-j[-fMg
=
-j[-f
VxF
g(p
-(
+g
implies
integrand
eq.
in
LN(B
LFC Mg
SPA
by
point
are
vertical
a
constructing
temperature
read
evaluated in
(TD)
an
along
the
Mg
time
-
yp
rate
as
described
cross
is
the
Emanuel
in
The
values
on
a
change
of
Change of
of SPA
is
SPA
of
(1983a)
and
0,
Mg,
of
plotted
surface,
(13)
section
surfaces.
Time Rate
of
of
eyg)k-ds
standard manner.
c.
The
g
v0v
evaluated
be
can
0
it
If
term
first
result
out so
(9) drops
=
SPA
the
surface,
This
relation.
independent.
path
is
(12)
ax
wind
thermal
Mg
an
along
evaluated
a vg]
g
ev0
3vg +
integral
the
that
the
by
zero
is
(12)
vg
-
az
Eq.
(10)
Ovg)
-
so
constant
6,p are
Mp and
-
_-fp(M
-j
6
and
tephigram
dew
TD
and
d
d
--SPA
dT
LNB
g
dT
I
fM
LFC
g
d
[
-SPA=
0
dT
+
dzLNB
-
v0
as
eq.
the
lifted
along
the
ambient
are
zero
-
d
-0
dT
-
(15)
)keds
va
dzLFC
rate
an
of
Mg
change
for
surface
because
the
equal
Ova)
z=LFC
environment.
are
-
dT
a
specific
and
the
The
last
subscript
two
environmental
at
the
LFC
parcel
and
"a"
terms
and
LNB.
of
parcel
Eq.
therefore
LN B
d
-SPA
=
surfaces
d
g
M(
dTv0
Mg
d
(-6
M
Vp
g dT
LFC
time
temperature
potential
is
is
to
(15)
(14)
LNB
z=LNB
is
refers
evo
(6p
d/dT
it
9
-vapkvds
Ova)
dT
where
(14)
vkes
=-(Gp-
g
LEO
are
vertical
in
the
for
system
gives
along
an
a
Mg
parcel
the
lifted
time
surface
in
coordinate transformation is
of
(16)
dr
semi-geostrophic
al.
(1975).
The
in
this
vertically
rate
^
eva)k-ds
--
dr
system described in Hoskins et.
change
d
-- 6vp
change
physical
for
a
coordinate
time
rate of
coordinate
parcel
coordinates.
lifted
The
=
X
x
vg/f
(17a)
ug/f
(17b)
+
y -
Y =
(17c)
Z = z
=
$
$
+
(ug
2
vg 2)
+
(17d)
(17e)
T = t
The
geostrophic
semi-geostrophic coordinates
d/dT
velocities
gradient
of
friction
is
above
The
of
are
+
=
d/dt
+(vp
v-
neglected
the surface
here
radiational heating and latent
d
dT
vp
=s
the
@D is
the
change
in
va)'
X -
and
and
is
VX
because
vertical
horizontal
the
coordinates.
the
(18)
waa/3Z
Effects
circulation
base
case.
potential
parcel
-
horizonatal
the
in this
of
is
VX
semi-geostrophic
in
operator
a/3t
environment
the
rate
and
small
is
=
are
wa
and
va
e
system.
time
The
geopotential.
and
system
coordinate
geopotential in physical coordinates
semi-geostrophic
where
new
coordinate
physical
the
are
letters
the
are
letters
capital
temperature
is
changed
by
heat release.
Q
Lv d
Cp
Cp dT
(
(19)
where
Q is
constant
heat
Z.
in
varies
which
and
vaporization
of
the
of
linearization
Rv
is
the
average values
this
gas
of
approximation,
for
constant
(19)
dT
Eq.
(21)
between
is
valid
the
LFC
dT
(20)
and
6 are
temperature.
With
vapor
water
2
the
LNB.
and
T
-(21)
parel
Qa
=
~-
Cp
is
rate
The
temperature
d
-6va
gives
Lv qs
1 +
Cp RvT8
provided
environmental potential
a
saturated,
Q/Cp
6 p-
and
ratio
is
d
--
is
and potential
temperature
eq.
mixing
e
Lqs
RvT6 dT
dq
dT
where
latent
Lv
equation
Clausius-Clapeyron
is
air,
parcel
the
If
is
mass,
saturation
the
is
qs
the
Cp
unit
dry
of
capacity
heat
pressure
per
heating
radiational
the
saturated,
of
change
it
as
of
is
the
is
Lv d
-qs
Cp dT
(22)
Radiational
heating
environmental
The
small
radiational
saturation
mainly by
is
heating,
mixing
ratio
is
saturated,
d
-6
In
this
be
neglected because
d
the
and
(23),
LNB
LB
eq.
g
f[-(vga
where Oe(g/Ov0)
advection and
the
(16)
-
be
neglected.
the environment
environment.
(22)
. -wa-
is
If
changed
the
can be written
3Lv
-qs
(23)
aZ Cp
of
heating
stable boundary
and friction can
layer ensures
that
the surface do not reach
and eqs.
(18),
can be written
3
g
vgp)*VXOv + wa- -- O0e)dZ
sOv(g/Ov0)
correlation
+ qs(LV/Cp).
(24)
term
integrand,
rate
of
w and 36e/3z.
be
lapse rate
It will be
is important in this
advection,
can
The
SPA are geostrophic
former process
lapse
the
can
so
v
Geost rophic
in
surface
With this assumption
processes which can change
that
Qa,
v
LFC
shown
at the
recently in contact with
mid-troposphere.
d
eq.
case, surface processes
air parcels
(21),
the
va
dT
the
of
upward motion of
atmosphere
except
written
which is
given
by
case.
the
first
-
(Z
Vgp)-V6a = g
f6 0
(vga -
VO)]
x
Z,)[k(V6
(25)
where
(Yga
k
Kgp)
~
g
X
...
V6
fv1dZ
(25)
Zp
is
positive,
SPA
encounters
colder
be
must
the
of
Some
the
the
from
as
discussed
in
is
state
base
is
symmetric
in
chapter
to
III.
geostrophic
M
balance
that
and
is
chapter
III,
inaccurate
clearly
it
is
found
that
in
particular
this
there
is
some
in
a
local
that
deviations
this state are associated with the displacement.
picture
is
case
also assumed
is
It
this
height
with
the
assess
In
varies
theory
the
instability.
unidirectional.
direction
shear
geostrophic
layers
flow
the
if
only
conserved
to
of
derivation
attempting
when
susceptibility
atmosphere's
the
in
Theory
the
of
Limitations
mind
in
kept
temperatures.
environmental
assumptions
parcel
lifted
the
because
is
(25)
eq.
If
Hemisphere.
increased
d.
counterclockwise
turns
it
if
zero
Northern
the
in
height
than
greater
is
with
turn
not
does
shear
geostrophic
the
if
eq.
of
side
hand
right
The
level.
parcel
the
zero
and
height
Zp)
-
is
is
with
f?6dZ
Zp
=
(Z
where
and
relation
wind
thermal
the
through
Such a
case.
In
region
of
above New
ageostrophy
is
intermediate
The
England.
the
between
scale
synoptic
of
the
scale
ageostrophy
and
the
band
scale.
In
analysis
this
separately.
Susceptibility
instability
is
geostrophic
the
of
New
mean
good
of
by
wind
is
It
in
is
small,
large
effect on
the
evaluated
assumed
parcel
scale
the
that
is
best
small
The
effect
of
ageostrophic
winds.
section
in
terms
velocity
the
which
gradient
it
effects
by
fvg
The
of
produces.
the
component
70*,
the
mb and
direction
850
angle
of
that
eq.
(8)
in
The
terms
shear
is
a
along
that
is
its
is
It
Mg.
a
there
effect is
is
displaced
is
not
with
discussed
sub-synoptic
of
instability
associated
Its
mb over
the
balancing
and
of
three-dimensionality
6e
ageostrophy
forcing.
the
only
of
perturbation
on frontogenetical
of
the
of SPA.
distribution
represented
pressure
and
considered
pressure
significant
500
since
is
the
using
that
rate of change
time
of
dM/dt
is
with
between
symmetric
unidirectional,
direction
shear
considered
mesoscale
Mg
the
flow
are
assuming
assumed
approximation.
the
to
calculating
geostrophic
England.
variation
evaluated
flow
geostrophic
the
scales
three
in
a
the
the
considered
scale
vertical
Evaluation of Susceptibility to Symmetric Instability
e.
1
Fig.
to
perpendicular
the
10a-b
show
Clouds
and
12
GMT.
Regions
of
possible
are
The
at
WAL
the very
both
in
the
warm
conditional
at
00
might
GMT.
develop
are
indicated
with
VA)
in
sections.
in
inversion
this
ee
the
than
It
region.
caps
km
3
is
clearly
any
M surfaces
are
surfaces,
are
sections.
the cross
about
chapter
lines.
stations, which
at surface
to
the
00
at
M
and
scalloped
where
i.e.
from the surface
at
CAR
(Caribou,
associated
There
II.
thermal instability below
The
6e
of
layer which slopes
discussed
fronts
of
sections
is
vg
shear.
mean
the
along
velocity
velocity
The
vector.
axis
an
onto
stations
the
of
projection
the
is
cross
stable
The distance
the East Coast.
fx
is
which
section
cross
the
shear.
vertical
mean
indicated below
is
cross
the
the
reported
Island,
(Wallops
ME)
a
shallower
weather
included on Fig. 1,
Note
constructing
instability,
or
to
parallel
shaded.
by
shear
mean
the
of geostrophic
component
Figs.
GMT
evaluated
+
vg
between
distance
physical
=
Mg
calculating
in
to
from stations along
constructed
used
atmosphere
of
location
the
shows
state
to
perpendicular
section
base
the
is
instability
symmetric
cross
of
susceptibility
The
is
a
with
pocket
the inversion at PWM
vertical
motions
which
Fig.
10
Vertical
scalloped
with
are
instability
the
surface
stations
snow,
#
= skies
=
shaded.
light
10
large
is
V
overcast,
&
=
=
skies
surfaces
Clouds
are
indicated
are
Weather
dots.
the
00
broken)
6e
Mg
stations
showers,
shear.
potential
of
below
indicated
rain,
air
the
and
5K
ms- 1 .
regions
Upper
with
axis
ordinate
and
lines
across
every
drawn
drawn every
are
(dashed lines)
sections
lines)
(solid
surfaces
cross
axis.
=
haze,
symmetric
indicated on
observed
( XX
at
light
=
=
fog,
Fig. 10a
00
GMT
5
3
L
2C 00
.
-10
/e
0
WAL
0e
MIV ACY NEL LGA ISP GON PVD
Se
0
0
0
0
0
0
CHH
PSM PWM NHZ
. ~_ -'
V
AUG BGR HUL
)*~
CAR
0
Fig. 10b6-o
12
05
GMT b
5(9297
Lii1
4
/
deR
283
0W AL
40
MIV
A CY NEL LGA ISP GON PVD 7C
.
>
HH
PSM PWM NHZ
AUG
e
BGR HUL
0
e
CAR
At
km
from 2.5
The
ME).
to
3.5
unstable
(Providence, RI)
layer
is
km
at
that
compute
to
are
Cross
constructed
not
sections
at
00
constructed
on
skew T,
unstable
difference
also
of
GMT
in
M,
12
GMT.
=
-10
log p charts
parcels
between
were
parcel
and
the
of
thin
of
of
the
of
actual
potential
cross
only
computed
within
velocity
at
thin
balance.
Energy
(not
shown)
Soundings
to
smaller
geostrophic
Potential
in
unstable
somewhat
were
m/s.
the
layers
the
TD
using
constructed
extent
shown
lifted
(Islip,
thickness
also
geostrophic
and
the
3 km.
of wind
6,
Mg
The
on
Slantwise
and
along
from 4 km at ISP
winds
changes
of
GMT,
12
subgeostrophy
appear
to be
Evaluation
At
sections was
geostrophic
likely
deep.
The
the
were
Large
(Houlton,
were
M.
to
not
the
levels.
f.
most
there
km to
cross
due
did
because
mandatory
plotted
10
that
sections
are
Fig.
these
at HUL
ME).
shown)
winds
However,
instability
.5
(not
in
winds.
layers
from
to 5 km
extends
sections
actual
km
(Augusta,
AUG
region indicated on
than
2
about
layer varies
Cross
the
at PVD
slopes
instability
possible
of
region
unstable region
potentially
NY)
the
GMT,
00
The
were
of
6
and
11a-b.
achieve
the
environmental
TD
were
soundings
Figs.
also
If
the
greatest
potential
N
N
N600
a%
-
Go
22
IN
2
N
IN
N
Fig
Fig
11aN
11b12GM
00G00M
10002
263
Fig.
11
Reference
Soundings
dry
Reference
The heavy
along
adiabats
pseudo-moist
solid
temperature
and
line
dew
ms-1
on skew-T,
log-P
diagrams.
M =-10
are
shown in
light
dashed
lines.
adiabats
are
shown in
light
solid
lines.
and heavy dashed line are sounding
point
temperature,
respectively.
temperature,
for
each
these
there
sounding.
layer,
the
approximately
The
buoyancies
considerations.
is
was
vertical
can
With
be
H-2
conversion
of
4
degrees
velocities
the
potential
buoyancy
resulting
estimated
km,
of
from
depth
energy
of
to
from
dimensional
the
unstable
kinetic
energy
roughly
KE - SPA
The
fraction
motion is
the
the slope
slantwise
potential
from
of
of
energy
-
1
z
.4 ms
1
to
.5
.6 ms-
1
as
(26)
vertical
1/50.
ms-
(27)
could
yield
the
convert
the
observed.
Generation of SPA
overturnings
energy
kinetic
energy.
friction,
and
vertical
-L
energy
potential
symmetric
through
or about
-V2-272 m s~
50
Depletion and
to
expressed
the Mg surface
available
precipitation,
released
kinetic
1
g.
The
288K
-/2eKE
50
vertical velocities
4K s 272 m 2 s-2
2 0 0 0 m9.8ms~
e0
of
W -
The
A H 1 -A6
motions,
irreversibly
Irreversibilities
disipation.
so
an
come
Buoyancy
appropriate
is
time
H~
and
2
dB
-tdz
dt
is
SPA
case,
geostrophic
lapse
rate
geostrophic
shear
backed
12a-b
CHH
show
at
height
mb
and
GMT
and
12
between
850
mb
00
300
estimated
mb
at
by
thickness
of
geostrophic
the
and
GMT.
in
height
that
Note
500
at
mb
left-hand
The
constant
according
850
between
coordinates
00
is
roughly
from
layers.
some
with
shear
the
shear
GMT
and
side
the
because
case
of
pressure
at
backs
with
between
850
eq.
(25)
is
maps
into
The
eq.
(17).
mb
and
500
to
Figs.
height
to
proportional
the
discussed
are
contribution
this
with
sub-synoptic
the
motion
in
GMT.
is
of
scale
synoptic
The
advection
layer
which
and
a
coordinates
the
semi-geostrophic
wind,
12
motion
is
transforming
semi-geostrophic
thermal
of
variation
upward
by
There
section.
next
the
In
upward
forced
frontogenetically
scale,
above.
effects
The
(28)
discussed
advection.
small.
is
s-(8
-. 07 m 2-3
2km
affected
rate
lapse
velocity
vertical
be
can
2
processes
the
by
generated
geostrophic
in
.5ms9.8ms- 2
8K-4KO
2km
288K
B
T
- TH -
SPA
this
to
taken
is
temperature,
before.
be 4 K as
f
the
between
difference
the
AO,
potential
environmental
and
parcel
where
B=g/60A,
as
defined
is
The buoyancy, B
energy.
kinetic
length for vertical
a mixing
as
seen
be
can
layer,
unstable
updraft,
observed
an average
is
the
depth of
the
km,
~ .5 ms~
W
T=tW/H.
is
scale
V6.
pressure-weighted
mb
in
The
shear
,g500-
Vg85o
850 -. V gSFG
V0300
- lO5oo
5 MS- 1
Fig.
Fig.
12
rotation
of
12a
00
GMT
the geostrophic
shear
at CHH with height
300 -V, 500
50o-Xeso
,gSFC
5 MS- 1
Fig.
12b
12
GMT
the
local
thickness
The
between
semi-geostrophic
for
direction
of
the
figures.
The
effect
of
SPA
change
value
of
the
where
v
is
500
A
mb.
temperature
evaluated
at
thickness
and 500
of
at
of
the
stations
be
can
temperature
The
at
850
typical
value
of
500
increases
25
va
is
in
100
are
in
the
in
regions
the
on
GMT.
mb
850
at
is
the
12
and
GMT
both
rate
typical
A
ms~
at
evaluated
semi-geostrophic
direction
layer
the
of
K
2.5
is
mb,
and
VXe a,
the
in
gradient
mb
semi-geostrophic
dimensionally.
(vp-va)
of
maps
advection
rate
estimated
vector difference
evaluated
most
in
wind
lapse
the
00
mb
thermal
mean
and
500
positions
reference.
indicated
of
and
mb
850
mb
500
at
temperature
potential
thermal wind is
mean
semi-geostrophic
show
13a-b
Figs.
positive.
the
along
gradient
temperature
VG.
where
regions
in
(25)
to eq.
contribution
a positive
is
There
thickness,
of
contours
along
flows
layer,
the
through
km.
between
of
H
e
(vp-v~a
su 4
km,
the
mb
850
mb.
-SPA
- --2vg850
-
vg500VX6- H
(29)
60
dt
9. 8ms-
2
288K
e
.09
m2s-3
(25ms-
)(2.5x10-5 Km-1)(4x10
3
m)
Fig.
13
Semi-geostrophic
Thickness
of
lines)
drawn
mb
is
(dashed
the
every
lines)
semi-geostrophic
reference.
layer
is
6
between
m2 s2
drawn
500
mb
Potential
every
coordinates
on
maps
at
5K.
850
December
and
850
mb
temperature
Station
mb
are
1982.
11
(solid
at
500
locations
in
given
for
Fig.
Fig.
13a
13b
for Symmetric Instability
h. Time and Length Scales
symmetric
for
the
are
calculated
a
and
geostrophic
zonal,
growth
in
are
flow,
state
base
hydrostatic
and
case
instability
symmetric
of
mode
unstable
most
the
for
rate
scales
linear
and
the' wavelength
respectively
a,
and
L
these
actually observed.
time scales
length and
scales
this
for
parameters
with atmospheric
compared with the
section
this
In
modes.
unstable
most
time
length and
instability and derived
for
problem
stability
linear
the
solved
(1983b,c)
Emanuel
depletion
(28).
SPA given in eq.
of
the
as
of magnitude
order
same
the
of
is
value
This
approximately
(30)
L - VzH/f
f
2
2
/N
the
circulation,
the
Typical
H
(Vz
is
Vz
where
2
2
s
10-2
values
length
Vz
km,
s-1.
of
With
scale
and
2
(31)
1)
-
and
parameters
s~
0 2
these
the
is
N
these
shear,
vertical
average
,
values,
growth
rate
Brunt-Vaisala
during
and
eqs.
of
the
H is
s
N
(30)
200
case
the
of
depth
frequency.
are
study
F
((1/6e)@6e/3z)
and
km
yield
(31)
and
10~
s
a
,
respectively.
The
length
larger
than
the
growth
rate
calculated
hours,
which
observed.
during
is
The
the
band
widths
later
were
trend
that
above.
The
than at
00
smaller
in many areas
this
vertical
at
a
greater
the
be
related
12
GMT.
of
The
about
time
3
scales
to grow smaller and weaker
can
period.
to
generally
depth
magnitude
observed.
that
case
the
of
scale
the
was
order
time
of
shear
However
an
wavelengths
observed
periods
obvious
or
is
suggests
probably
bands
GMT.
scale
of
the
is
It
the
stronger
unstable
not
parameters
at
12
GMT
layer
was
V.
which
Situations
likely
usually
that
baroclinic
associated
waves
Heckley
1979;
and West,
Sanders
related.
were
11-12
February
has
considered
and
state
It is
1982).
well
e.g.
(see
fronts
Hoskins,
1983.
in
is
Vertical
becoming
instability
communication,
"megalopolitan
the
The
motions
clear
a
in
found
1984)
snowstorm"
substantial
symmetrically
neutral
modifications
Frontogenetical Forcing of Upward Motion
rate of
frontogenesis
is
given by
that
frequently
are
symmetrically stable case.
a.
known
Hoskins
(personal communication,
Emanuel
frontogenesis
found
It
state.
(personal
active
both
and
symmetric
and
frontogenesis
troughs.
produce
can
base
the
of
picture
deep
with
shears
vertical
accompany frontal circulations and complicate
and ageostrophy
the
Large
also
are
instability
symmetric
favor
to be frontogenetically active.
are
FORCING
OF FRONTOGENETICAL
ROLE
to
that
of
1984)
base
the
---
VV2-7
6 =
where
v2
describes
side
hand
increase
in
advection
of
Williams,
axis
of dilatation,
DEFI
=
#d,
u
ax
given by
-
)2-
ay
where
#d
is
dilatation
deformation
isotherms
=
tan~1[
measured
axis
gives
is
purely
form
an
angle
frontolytical.
(3v +
au
3v
ax
ay
the
ax
(33)
(34)
then
x-axis.
the
from
direction
less
is
5
]
stretching
of
)2
ay
the
and
au
clockwise
dilatation, the deformation
is
+
+
1966;
Stone,
deformation
the
are
av
#d
of
magnitude
The
can
gradients
g.
e.
(see
fields
the
differential
to
due
right
gives
which
temperature
Large
the
term on
first
gradient
isotherms.
1968).
dt
frontogenesis
confluent
deformation
by
formed
be
The
wind.
temperature
the
V d6(32)
-
3z
horizontal
the
is
Vw---
+
-
dt
along
which
the
If
the
deformation.
450
with
frontogenetical.
The
the
axis
Otherwise
of
it
Circulations
b. Frontal
to
be
(1947)
Sutcliffe's
his
is
It
time.
ageostrophic
of
streamlines
forcing
can
be
of
thermally
circulation.
deformation
in
given
by
terms
of
the
distributions.
direct.
If
Negative
Shapiro
and
showed
a
is
equation
the
(1982)
that
that
of
the
two-dimensional,
frontal
transverse
the
expressed
temperature
is
of
1982)
advection
Sawyer-Eliassen
the
as
known
now
equation,
a
developed
Eliassen
problem.
finite
essential nonlinearities
the
geostrophic wind captures
a
in
Hoskins,
and
along
quasi-geostrophic
collapse
f.rontal
(Heckley
believed
inclusion
Eliassen's
a
describe
cannot
equations
assuming
not
but
the
that
showed
(1967)
Williams
it.
front
the
across
balance
geostrophic
by
work
Sawyer's
expanded
(1959)
in
of frontal circulations.
quasi-geostrophic analysis
Eliassen
to
Sawyer
fronts
near
important
is
velocity is
the thermal wind.
is
effect
this
that
showed
(1956)
According
vertical
development equation,
by vorticity advection by
generated
theory.
quasi-geostrophic
from
expected
motions
with upward
zones
frontal
of
association
The
Q is
related
stretching
for
The
Q,
which
positive
the
elliptic
equation,
term
wind
geostrophic
a
the
circulation.
single
Q forces
of
the
circulation
thermally
forcing
and
indirect
term
deformation
to
leads
the
to
and
Q>O
direct
a
relation
between
intimate
The
circulations
Hoskins
al.
et.
described
and
ageostrophy.
velocity
vertical
the
exceeds
shear
that
gradient.
restore
is
which
Vertical
needed
geostrophic
derived an equation for w
w
23 2 w
2
+ f2
( Q2)
=,(
60
2
N V
2
=
et.
al.
occur
balance
pressure
attempt
(1978)
the forcing function
terms of
in
atmosphere's
Hoskins
balance.
the
balance
the
reflect
motions
wind
thermal
to
others.
can
Ageostrophy
satisfy
to
generated
three
between
relationship
the
(1978)
among
(1982),
Hoskins
and
Heckley
by
to
expanded
been
has
equation
Sawyer-Eliassen
dimensions
the
behind
assumptions
some
frontal
governing
equation
the
the
changes
instability
symmetric
is
equation
equations.
the
of
derivation
the
Sawyer-Eliassen
violates
but
that
fact
the
the
of
character
fundamental
and
by
Thus
stability.
symmetric
frontogenesis
illustrated
on
condition
ellipticity
The
is
instability
symmetric
to
while
to Q<O.
leads
when
deformation
shearing
circulation,
2V-Q
Q.
(35)
where
g
3
av
3x
-g
,
ay
V6)
(36)
is
N2
=
is
g/60 (dO/dz)
Upward motions
the
the
Frontal
surface maps
south
of
the
vertical
layers
Massachusetts
slope
Q is
are favored where
c.
On
Brunt-Vaisala
cross
towards
across
sections
the
north
frequency
squared.
convergent.
Structure
(Figs 3a-b),
and
and
coordinates)
phyicial
(in
divergence
horizontal
the
V2
warm fronts
Georgia
(Figs.
and
and
are
Alabama.
10a-b),
become
more
located
the
On
frontal
diffuse
with
height.
At
surface
and
00
front.
velocity
sounding
Note
GMT,
as
that
stable.
is
the
manifested
is
in
00
GMT sounding of
CHH's
shown
a
On
CHH
7*C
in
as a
7*C
Fig.
inversion
atmosphere
the PWM
the
cold
14a.
and
above
sounding
450
just
in
wind
shift
inversion
the
of
the
layer
is
on
800
at
only
the
mb.
slightly
frontal
between
the
humidity
front appears
(not shown),
inversion
north
temperature,
The
a
the
air,
layer
825
is
mb
and
12
GMT
740 mb.
At
12
GMT,
sounding
for
850
There
mb.
the
CHH
is
(Fig.
no
front
is
14b)
wind
much
shows
shift.
weaker.
only
The
a
2*C
The
inversion
atmosphere
is
stable
at
Fig.
as
in
14
Soundings
Fig.
3c.
for
CHH
on
11
December
1982.
Diagrams
are
600DN
U)
U)N
700-*
800%
-
+
900-
\
1000 263
273
TEMPERATURE (K)
Fig.
14a
00
GMT
283
300
600LU
ind
(n
L 700a.
800
900
1000
263
273
TEMPERATURE
Fig.
14b
1
(K)
12 GMT
283
TEMPERATURE (K)
Fig.
14c
18 GMT
had
Neither
and
20 C
a wind
shift.
10 C
had
shown)
the
and
decays
troposphere,
between
period
of
region
the
the
Geostrophic
are
are
winds
The
velocity
is
shifted
Addition
and
field
Multiplication,
with
determined
mb
almost
and
division,
are
and
calculations
lower
the
to
front
to
middle
increases.
Upward Motion
divergence,
the
Q are
above.
formulae
the
from
and
geopotential
temperature
potential
derivatives are approximated
calculated
and
latitude
subtraction
point-by-point
resultant fields
The
the
GMT,
circulations,
and
components
differences,
20
of
Forcing of
according
contoured subjectively.
by centered finite
band
18
and
horizontal
deformation,
subjectively
calculated
analyses.
of
GMT
00
stability
thermal
the
d. Frontogenetical
Fields
525
below
frontal
500 mb.
neutral above
In
mb and
400
little
shows
14c)
stable
is
atmosphere
The
structure.
below
stable
mb.
800
400 mb.
(Fig.
sounding
GMT
18
CHH's
Both were
above
almost moist adiabatic
at
respectively,
inversions,
(not
PWM
and
MIT
for
soundings
GMT
12
The
mb.
650
below
graphically.
subtracted
performed
are
trigonometry
at
then contoured.
station
are
from
Each
itself.
graphically.
accomplished
locations.
The
Fig.
15
Frontogenetical
(solid
lines)
isotherms
(dashed
deformation
potential
shown with arrows.
at
00
units
of
forcing
in
lines)
in
K.
GMT.
Geostrophic
4.5x10-6
Dilatation
s-
and
axes
are
pig.
15a
Fig. 15b
geostrophic deformatio
temperature,
Potential
and dilatation axes
show
15a-b
Figs.
mb
700
and
New
England,
at
The
region.
deformation is
The
Seaboard.
is about 4x10
There
5
00
geostrophic
deformation.
geostrophic
deformation
the
4x10-
about
5
convergence,
e.
Figs.
700
mb.
upward
(not
This
s-1.
which is
Eastern
this
both
levels
The
actual
cold
mb
700
well
shown)
over
is
to
similar
or
500
the
little
very
is
mb.
The
and
to the west,
The
front.
deformation
magnitude
New
England
due
to
is
of
also
ageostrophic
Frontogenetical Circulations
16a-d
show V-Q for
There
is
where
there
mb.
discussed below.
Strength of the
motion
GMT
time is
that
advancing
deformation
actual
is
mb,
850
at
500
at
shown)
12
At
geostrophic deformation at
the
(not
GMT
at
with
at
deformation
deformation
no
deformation
associated
the
.
s~
almost
is
and
and MIT but
frontogenetical throughout
geostrophic
maximum
and
England
New
central
mainly
covers
mb
700
high
high deformation
of
region
The
frontolytical at CHH and ALB.
of
Pennsylvania,
frontogenetical at PWM
is
deformation
The
Ohio.
there
swath
a
is
mb,
central
across
deformation
850
At
mb.
850
at
GMT
00
for
dilatation axes
deformation magnitude, and
geostrophic
isotherms,
potential
00 GMT and
geostrophic
Q
is
12 GMT at 850 mb and
frontogenetical
convergent.
The
forcing
of
regions
of
in units of 3x10-16 s- 3m
Fig. 16 V-Q
A#
1
85
Fiq. 16a
Fig. 16b
86
Fig. 16c
Fig. 16d
9 .Q
roughly to the maxima
convergent Q correspond
in
geostrophic
However, the Q convergence is not constrained
deformation.
to areas where the deformation is frontogenetical.
At 12 GMT
the region of convergent Q is further north than at 00 GMT.
The strength of the vertical velocity resulting from V-Q
can be estimated by assuming both terms on the left-hand side
of Eq.
Then
(33) are of the same magnitude.
2V-Q or W
f2a2 w
3 z2
If H - 2 km and V-Q
H2 (V-Q)
(37)
f2
10-15 m-1 s- 3 as suggested by Fig.
W = (2x10 3 m) 2
l0-
1 5 m-
1s-3
= .4ms- 1
13
(38)
(10-4s-1)2
Figures 17a-d show divergence for 00 GMT and 12 GMT at 850 mb
and 700 mb.
Note that the convergence occurs in regions
where there is frontogenetical forcing.
There is convergence
in New England, in the neighborhood of the warm front, and in
Pennsylvania and Ohio, ahead of the cold front.
convergence
is
at
850 mb.
and 12 GMT are similar.
The maximum
The convergence fields at 00 GMT
Fig 17
V-v in units of 4.5x10~6 s-
Fig.
17a
Fig.
17b
Divergence
Fig. 17c
Fig, 17d
Divergence
(
W
@z
and
at 700 mb,
W
(1
3z
)70 0 Az + w 8 5
agree
values
between
f.
bands
1983b,d;
postulated
that
by
convection
at
are
scale
plays
that
surfaces
of
Mg
Neither
of
these
along
=
about
GMT.
instability,
in
this
-10ms
1
Mg
parcels
surface
instability
it
1979)
is
and
saturation
of
free
sub-synoptic
scale
case.
(Figs.
,
00
mb
650
they
the
The
ascent.
at
adiabatically
5b)
level
the
to
roll
to
brought
lifted
are
the
is
environment
parcels
release
(Fig.
symmetric
Hoskins,
and
Bennetts
unstable
12
analysis.
(40)
velocity
vertical
ms-1
moist
on
synoptic
along
soundings
1
Frontogenetically Forced Circulation
parcels
here
documented
most
the
unstable
the VVP
in
(39)
.1ms
.07ms-1
.05
the
with
well
papers
(Emanuel,
that
2x10- 5 .1500m +
0
calculated
previous
11500m ~ .07ms-
4.5x10-4 s
Effect of the
In
At 850mb,
the divergence.
)8 5 0 Az
can
New England
in
velocity
vertical
the
estimated from
also be
These
of
strength
The
is
GMT
In
the
11a-b)
the
and
850
saturated.
must
be
to
their
lifted
mb
To
dry-
condensation
The
frontogenetical
and
an
component
as
required.
the
In
magnitude
which w enters
the
is
above
d
forced
change
The
.
upward
motion
is
to
the
from
eq.
due
SPA
in
.8
estimated
velocity
vertical
The
-
d
Pm/
regions,
unstable
same
the
of
are
terms
estimated
as
SPA ~ W- g
. 1 ms _1 9.8ms~
288K
6v aZ
dt
J
The
1
.1ms
W
frontogentically
(24)
the
In
are
(24)
SPA in eq.
w(Lv/Cp)( 3qs/3Z).
these
values,
-. 25Kkm- 1 .
36e/3Z
and
of
change
in
terms
The
here.
estimated
a
The
SPA.
decreases
3ee/3Z
of
that
demonstrated
was
could
it
so
x-direction
and
magnitude.
of
order
the
component
a vertical
both
this
so
buoyancy.
negative
little
it
/3Z)
atmospheric
typical
rate,
in
is
rate
time
w(g/60 )(r m/rd)(3e
For
lapse
and
w
of
effect
this
of
atmospheric
section
previous
correlation
negative
an
circulation has
ageostrophic
the parcels
lift
along
very
by
opposed
is
displacement
surface
Mg
lifted
parcel
dry-adiabatic
almost
an
sees
a
cases,
In both
levels.
decrease
SPA
smaller
magnitude
geostrophic
The
in
-1.7
due
lapse rate
frontogenesis
advection.
Note
in
vertical
to
the
Fig.
given
enhance
2a
that
the
the
in eq.
an
is
velocity
generation
advection
may
25x10- 3 Km- 1 )(2x10 3 m)
(41)
m2 s
x 10
than
(-.
of
order
of
due
to
SPA
(29).
geostrophic
isotherms
at
lapse
850
mb
rate
are
00
The
GMT,
frontogenesis
thermal
of
component
the
balance
primarily
the
to
the degree
the
of
an
the
levels,
to which the geostrophic
gradient.
at
lower
the
in
to
isotherms
Frontogenesis
winds
At
England.
increase
the troposphere.
the
at upper
New
temperature
along
wind
gradient.
temperature
winds
the
requires
geostrophic
component
over
east
increasing
the lower portion of
westerly
respect
is
relation
wind
the
to
west
approximately
oriented
By
affects
increasing
levels
frontogenesis
with
increases
wind backs with height.
were
atmosphere
to
parallel
As
discussed
in
chapter
displacement
is
expected
to
be
the
the
either
does
not
explain
was
of
smaller
converted
have
been
sometimes
rate
at
to
kinetic
a
form
shear,
as
or
The
which
rough
Length and time
6e
of
stability
The
between
the
the
steeper
model
current
bands
of
movies
two
the
were
passes
made
at
observed
roll
the
in
behavior
on
observed
of
slope
However
surfaces.
numbers
symmetric
for
the
IV,
band
airplane
different
perturbations is
The
mean
the
the
took
intermediate
Time-dependent
displays.
CAPPI
Mg
the
times.
structures
the
in
found
than
various
layers.
unstable
bands
two-
in
surfaces.
6e
and
Mg
the
streamlines
the
expected
those
than
smaller
of
to
unexplained.
remain
instability.
slopes
according
instability
data,
radar
and
observations
were
scales
the
the
in
formed
bands
almost
of
theory.
by the
few
A
symmetric
The
rolls
sloped
predicted
model of
aircraft
by
indicated
of
unstable
flow.
dimensional
As
conditionally
parcel
the
of
criteria
portions
Substantial
case.
this
of
bands
the
in
features observed
explains many
instability
Symmetric
AND CONCLUSIONS
DISCUSSION
VI.
from
dimensional
series
rolls
to
not known.
potential
energy
was
equilibrium
energy
was
There appears
estimated.
between
generated
its
generation
and
to
and
depletion.
instability
However
the
conditions
for
symmetric
under
the
assumption
of
two-dimensional
were
derived
The
condition
the
of
rotation
counter-clockwise
generation,
energy
potential
for
responsible
this
violates
vector,
shear
flow.
assumption.
could
lifting
conditionally
the
such
instability
a base
in
results
symmetric instability
degree
The
correspond
that
there
the
to
results
are
to
not
symmetric
to
The
frontogenesis,
also
under
which
conditions
has been derived.
to
observed
which
predictions
of
study.
did
strength
by
warm
the
stratification.
or
attributable
between
case
the
relevant
which violates
state
and
width
shear
vertical
as
apparently
ageostrophy,
band
parameters
to
related
be
to
seem
in
decrease
progressive
of
periods
later
the
as
weaker
and
which
highlighted
is
formation
smaller
became
bands
during
diffused
front
The
the
that
fact
band
and
forcing
frontogenetical
also
may
relation
The
energy.
potential
the
generated
where
level
advection
rate
lapse
geostrophic
the
enhanced
have
by
bands
Frontogenesis
released.
be
could
energy
potential
a
to
parcels
unstable
of
formation
to
contributed
have
process
intermediate scale
This
deduced.
frontogenetical forcing was
to
attributed
ascent,
scale
sub-synoptic
of
region
A
the
deviations
of
theory
from
band
symmetric
characteristics
instablity
relevance
have
some
the
idealizations.
suggests
even
when
Future
analytical
work
and
hypothesis
that
states
may
be
former
condition
lapse
rate
created
processes
land
in
There
the
the
and
absence
by
should
and
The
relevant
hence
of
surface
this
effect
because
mechanism,
and
the
base
of
it
the
allows
of
generation
heating
source of
address
ageostrophic
unstable.
particularly
are an unlikely
the
instability
friction.
These
any bands which form over
winter.
has
presence
been
is
In
from a storm.
have
symmetrically
mainly
the
simulations
three-dimensional
advection
instability.
is
computer
no
been
of
It
speculation
bands
affects
seems
likely
precipitation
forming mechanisms
(Bosart,
the
that
in
1984)
amount
in
New
this
of
to
whether
precipitation
case,
England
discussed been inactive.
as
there
had
the
would
band
Acknowledgements
his
assistance
Dr.
Richard
Experiment
Dr.
and
were
of
advice
Passarelli
Frederick
Garner
and
was
and
her
thanks
author
The
a
Sanders
Peter
the
Kole
drafted
NSF/g
8209375-ATM.
the
during
the
directed
source
also
Neilley
invaluable
advisor,
of
made
assistance
figures.
in
of
course
England
ideas
and
taking
work
research.
Winter
comments.
the
for
Storm
encouragement.
Stephen
discussion
insightful
This
this
New
useful
provided
Emanuel,
Kerry
Dr.
data.
was
and
Isabelle
supported
by
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