THE POSSIBLE ROLE OF SYMMETRIC INSTABILITY
IN THE FORMATION OF PRECIPITATION BANDS
by
MARC ALLEN SELTZER
B.S.,
McGill University
(1982)
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
LocX
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
May 1984
@
Massachusetts Institute of Technology, 1984
Signature of Author
Department of EE rth, Atmospherric and Planetary Sciences
May 1984
Certified by
I
-
Richard E. Passarelli
Thesis Supervisor
/-
Accepted by
Theodore R. Madden
Chairanhepdktmenftal Committee on Graduate Students
M1T L A!rJU
THE POSSIBLE ROLE OF SYMMETRIC INSTABILITY
IN THE FORMATION OF PRECIPITATION BANDS
by
MARC ALLEN SELTZER
Submitted to the Department of
Earth, Atmospheric and Planetary Sciences
in partial fulfillment of the requirements for the degree of
Master of Science in Meteorology
ABSTRACT
Fifteen cases of banded and non-banded precipitation
that are not associated with any surface frontal regions are
An attempt is made to relate the atmospheric
presented.
conditions associated with the banded observations to the
theory of symmetric instability. Linear perturbation theory
and parcel theory are used to assess this instability and to
make predictions of the strength and orientation of the
The theory does explain many of the banded
observed bands.
all of the bands are
features considered in this study:
strong shear and
wind;
aligned parallel to the thermal
when bands
observed
are
near-neutral static stabilities
related to
is
that
wavelength
a
have
occur; multiple bands
isentropic
of
slope
the
and
region
the depth of the unstable
the
explain
fully
not
does
The theory, however,
surfaces.
flow.
mean
the
to
relative
propagation of some bands
Furthermore, symmetric instability is an imbalance that can
The obserarise in a geostrophically balanced atmosphere.
is not in
flow
the
cases,
some
in
that
vations indicate
symmetric
of
indications
the
of
Most
balance.
geostrophic
not seem
do
that
regions
shear
narrow
to
due
instability are
balance.
to be in geostrophic
Thesis Supervisor:
Dr. Richard E. Passarelli
Title:
Professor of Meteorology
1Bd
1.
Introduction.
"It
is
now
well
established
that
the
regions
of
heaviest precipitation in extratropical cyclones are often
organized
on
the
mesoscale
(Parsons and Hobbs, 1983).
in
the
form
mesoscale
bands
forcing.
caused
by
forcing.
symmetric
In
frontogenesis
this
instability in
Others occur far from
While much
known about those bands that occur
frontal
rainbands."
Some of these bands are directly
associated with frontal circulations.
any regions of frontal
of
far
thesis,
alone,
little
from the
the
the formation
is known about
region of
possible
of
these
is
role
of
precipita-
tion bands is examined.
NEWS
The data set used in
this study comes from the M.I.T.
(New England
Storms)
winters,
storms
1981
with
a
to
Winter
1983,
Doppler
observations
radar,
National Weather Service
intense
precipitation
frontal
in nature.
regions
of
very
Project.
data.
bands
These
strong
an
During
were
taken
aircraft,
In
occurred
many
and
of
and
from
storms
that were clearly
near
embedded
neutral
20
standard
these
bands were usually
shear
three
not
in
static
stability.
This paper will attempt to answer three questions:
1.
How do the observed atmospheric
these
storms
compare
conditions during
to those assumed
models of symmetric instability?
in theoretical
Do the observed precipitation bands exhibit those
2.
symmetric
of
theory
the
by
predicted
features
instability?
3.
predict
the
occurrence
doing,
say
something
how
theory,
linear
simple
Using
and
strength
their
about
so
in
and
bands,
these
of
one
can
well
orientation?
It was Hoskins (1974) who perhaps first suggested that
instability.
bands may be caused by symmetric
precipitation
Many of the qualitative properties of symmetric instability
in
a
dry
along
example,
Elliott
classical
of
theory
Solberg
(1936),
among
others.
This
of
instability
(1962)
and
Ooyama
derived
for
a
Eliassen
theory,
for
Harrold,
the predictions
baroclinic
symmetric
(see
and
Browning
well with
fit
by
developed
(1966)
19 76)
precipitation
direction
shear
1964;
Hovind
and
et al.
1969; Houze
vertical
BH).
to as
referred
of banded
observations
the
oriented
the
the
that
noticed
by
summarized
been
hereafter
(1979;
Bennetts and Hoskins
BH
have
atmosphere
inviscid
two
dimensional situation, predicts an instability that produces
roll
coriented
circulations
W hile
geostrophic wind.
in
considered
presence
of
the
In
instability
for
this
the
can
shear
effects
of
moisture
lead
section
dry
the
BH
theory,
original
moisture
convection.
the
along
the
inviscid
to
proposed
the
the
are
not
that
the
observed
criterion
case
of
will
for
banded
symmetric
be .reviewed
following
BH.
The
to
results
these
of
extension
include
(1979
viscosity and moisture following the work of Emanuel
and 1983) will then be discussed.
can
occur
is
instability
Symmetric
instability
that
balanced
atmosphere.
It
a geostrophically
in
arises from an imbalance of
and buoyancy forces.
the pressure
Normal
perturbations.
can be taken
One approach is to examine the
mode
of
solutions
infinitesimal
to
state
basic
a geostrophic
of
Coriolis
gradient,
Two different approaches
to assess this instability.
stability
a type of
the
linearized
equations of motion are sought and
the maximum growth rate
of the perturbation is determined.
The second approach is
from
to consider the finite displacement of an air parcel
its
This
initial equilibrium state.
to
is analogous
the
parcel method for assessing convective instability.
In
the
reviewed
stability analyses
in
this
section,
four further assumptions are made:
1.
The
flow
is inviscid
and
Boussinesq,
and
takes
place on an f plane.
2.
The basic state flow is in the y direction and is
in geostrophic balance.
3.
The basic state and perturbations vary only in the
x and z directions.
4.
The
flow
has
constant
horizontal
and
vertical
shears.
Let V(x,z)
and G(x,z)
denote the basic state velocity
and potential temperature fields.
Geostrophic balance imp-
lies that V is related to e by the thermal wind relation
av
9
az
e
a5X
that
showed
(1966)
Stone
99V varying
from
perturbations
infinitesimal
height,
for
linearly
only
in
baroclinic
this
basic state are unstable if the Richardson number Ri is less
than unity,
where
9 30
(1V)
BH
extended
by
analysis
Stone's
allowing
the
basic
state
velocity V to have constant shear in both the x and z direcThis introduced a minor modification to the stabi-
tions.
lity criterion.
The flow is found to be unstable when
(n/f)-Ri < 1
where n = f + aV/ax is
state
the
flow.
the absolute vorticity of the basic
instability
presence of negative
condition
absolute
is
equivalent
vorticity on
to
isentropic
The orientation of the most unstable perturbation
surfaces.
was
This
(1.3)
shown
to
lie
roughly
halfway
between
the
absolute
vorticity vector and the isentropes.
BH
also showed
that for a saturated atmosphere,
the
effect of latent heat can be included in the stability criterion by replacing
perature.
e by
0e
, the equivalent potential
tem-
In general it is found that the stability of the
flow is decreased when the effects of moisture are included.
5
The inviscid analyses of symmetric instability by Stone
and BH both conclude that the most unstable disturbance has
an
infinite
there
theory,
is no
that the
(1979) showed
to a mesoscale wavelength for
leads
of viscosity
addition
inviscid
Indeed, Emanuel
cutoff.
shortwave
In
wavenumber.
He showed that the unstable
the most unstable disturbance.
circulation has the following properties:
the
take
plane)
the x-z
(in
Streamlines
1.
form of
closed vortices elongated along isentropic surfaces.
The horizontal wavelength of the disturbance
2.
not depend on
viscosity except in
does
limit of weak
the
diffusion.
such that the
The wavelength of the disturbance is
3.
sloping
do
vortices
The wavelength
direction.
is
horizontal
the
in
overlap
not
determined by the depth
of the unstable layer and the slope of the isentropic
surfaces.
Emanuel
(1983) looked
at the
of
stability
the
He assumed that
basic state to finite parcel displacements.
With
V varies linearly and only in height.
this choice
can easily be shown that the quantity M which
fx+V is conserved.
thus
A parcel displaced
conserve
position,
will
values.
Accelerations
for
both
geostrophic
it
is equal to
from its equilibrium
its
a displaced
original
parcel
M and
(or
0
more
precisely a tube extending infinitely in the y direction) in
the x and z directions are
du_V j( -
dw
dt
(MF-,(i
9(15
99
P
4-)
9
where the subscript g refers to the geostrophically balanced
state
basic
the
to
refers
p
subscript
the
and
parcel's
Displaced parcels can accelerate both vertically and
value.
horizontally due to the surplus (or deficit) of the parcel's
M and 0 with respect to the environment.
in
parcels displaced
stability characteristics
the
Fig. 1 illustrates
the x-z plane.
A possible basic state
M is chosen
configuration of M and 0 surfaces is indicated.
to
o is chosen
to
independently
to
right.
the
to
and
upwards
increase
for
increase upward (a stable stratification).
inertial
and
gravitational
from
the
basic
determine
the
displaced
parcel
displacements used
vertical
to
displacements
instability.
used
be
can
1.5
and
1.4
Eqs.
of
stability,
symmetric
a
and
gravitational
assess
to
used
of
horizontal
with
inertial
assess
the
assessment
The
state,
stability
instability
is
made by imagining the displacements of parcels slantwise in
the x-z plane and evaluating the accelerations of the parcel
using both Eqs. 1.4 and 1.5.
For the basic state shown in Fig. 1, it is found that
parcels displaced vertically or horizontally are stable (see
for
example
acceleration
points
is
A
back
and
B).
toward
the
The
parcel's
parcel's
slantwise displacements, however, it
is
resultant
origin.
found
that
For
if the
IB
M2
\\
o
\A
X
i
x
Fig. 1. Schematic vertical cross-section illustrating the
use of the parcel technique in assessing symmetric instability. A possible basic state configuration of M and 0 surfaces is indicated.
has
displacement
less
but
surfaces
acceleration will
(see
point
reveals
C).
be
The
in
0 surfaces,
will
be
It should
to a good approx-
that for a saturated atmosphere,
imation, the stability can be
replacing 0 by
assessed
in the same way by
0e.
This parcel technique can also be applied
Emanuel
sounding.
linearly in
analysis
possible only
when M surfaces are more shallow than 0 surfaces.
be noted
resultant
a stability
such
instability
the
M
the displacement
the direction of
result of
symmetric
that
of
that
than
that of
than
greater
is
that
a slope
to a single
(1983) has shown that for 0 and V varying
the stabili-ty of a slantwise displaced
x and z,
parcel can be evaluated by adding the value
to
the reversibly
temperature calculated
lifted parcel
the usual way on a thermodynamic diagram.
Parcels
temperature.
can
then
Tv is the virtual
lifted
be
in
from
different
heights in a manner analogous to that of assessing ordinary
(buoyant) convection.
unstable parcels
varying
both
the
for
The level and orientation of the most
finite
initial
lifting
height
and
can
be
determined
orientation
of
by
the
parcel (or tube).
The nature of the circulations predicted by the linear
theory
of
symmetric
instability
suggest
that
symmetric
Figure
explain banded precipitation.
instability
might
(reproduced
from Emanuel,
1979)
shows
the
2
function
stream
associated with the onset of symmetric instability.
Recall
that the closed vortices in the x-z plane extend infinitely
In the presence of moisture, precipita-
in the y direction.
tion
is likely
form
the
in
ascending
branches
a radar,
altitude with
take
would
the
form
the
linear theory can explain the occurrence
of
a
In this
band oriented in the y direction (along the shear).
respect,
of
The pattern of precipitation, when viewed at
circulation.
constant
to
of preci-
pitation bands.
Further support
comparison
between
of this
the
explanation must come from a
observations
of
atmospheric
condi-
tions in the vicinity of banded precipitation and the linear
theory.
For example, there are several observational checks
that can be made:
1.
The
sional
and
atmospheric
with
most of
conditions
the
kinematic properties
should
variations
occurring
in
in
be
two dimen-
thermodynamic
the
cross-band
direction and in the vertical.
2.
the
There should be a region in the atmosphere where
Richardson
number
instability
criterion
(e.g.
Eq. 1.3) is met.
3.
The bands should be aligned along the thermal wind
and should be strongest in the region of instability.
10
{0
.3
0
0 --.
s
*6
-*.~
1.20
0.
.6
0*
12
3 -6
.
/0
9
-1.2/
~
.30
0
0
3
_)o
00
Fi
-S.
-2.
S
m0.5trei
d o50-05i
u
r
at
--
0
c0
0 12
0-.5T20
and3
01
3
t
8
,
r
s
ctiv 211
0
C
Fig. 2. Streamfunction associated with the onset of symmetric instability (reproduced from Emanuel, 1979). Plots
(a) and (b) are solutions for free-slip boundaries with
diffusion coefficient T=10O
and l.6x10-3 , respectively.
Plots 4(c) and (d) are solutions for no-slip boundaries with
T=10- and 10respectively. The abscissa spans one full
wavelength. The dashed line indicates the relative orientation of the potential isentropes.
4.
If
bands
the
to be moving
are observed
in
the
cross-band direction, they should be moving approxithe
mately with
that
that
results
the
the
flow
mean
of
in
this direction.
Note
theory
imply
linear
the present
circulation
not
does
pattern
propagate
relative to the basic-state flow in x.
5.
depth
Spacing
of
between bands
the unstable
be related
should
layer
and
the
slope
of
to
the
isen-
tropic surfaces.
6.
The
include
circulation
a region
pattern
of ascending
along isentropic surfaces.
near
the
motion
bands
should
sloping nearly
2. Observations
National
Weather
Massachusetts,
launched
all
were
properties
Portland,
at nominal
from
taken
Service
times
and
kinematic
data.
Standard
thermodynamic
of
measurements
The
sounding
New
Albany,
and
Maine
special
as well as
Chatham,
from
soundings
balloon
York,
launch times
Rawinsondes were also launched from M.I.T.
were used.
addition, aircraft measurements made by
the
NOAA
In
P-3 were
taken in a sounding mode at locations chosen to best supplement the other soundings.
cant
levels
of
the
NWS
Data from mandatory and signifiwere used.
soundings
All sounding
data were interpolated to a 250-meter vertical grid.
The cases presented in this study were chosen to meet
three requirements.
1.
The low pressure
center,
face analyses, was
tion.
as determined
the
not near
from sur-
region in ques-
This requirement excluded regions that had
a generally more
complicated
mesoscale structure
involving frontal zones and strongly curved flow.
2.
Precipitation was observed on the M.I.T. radar.
3.
Surface fronts were not present in the region.
Table 1 lists the fifteen cases that will be considered
in
this study.
A
measure
of
bandedness
determined for each case.
group
of
four scientists
of
the
precipitation
was
This was done subjectively by a
from radar
reflectivity patterns
13
Table 1.
2
B.O.
LMSW
FFS
DDS
Nd 2
Nw
12/05/81
45
4.75
13.5
37
1.4
.8
12/06/81
0
2.75
13.3
5
1.6
1.2
8.3
12/16/81
45
1.75
18.6
39
2.0
1.1
7.1
12/11/82
70
4.00
11.2
55
6.9
4.0
5.2
12/12/82
50
3.75
17.1
51
2.3
1.7
14.6
12/20/82
0
2.00
14.5
-2
0.7
-3.8
5.1
01/11/83
0
3.00
19.4
14
2.5
2.5
4.3
01/24/83
-
4.25
8.2
129
1.5
0.6
5.0
02/03/83
-
3.25
7.9
90
2.4
2.1
5.8
02/12/83
80
5.50
13.4
66
1.8
1.4
9.5
11/28/83
135
2.75
21.2
117
3.0
4.3
5.0
12/03/83
90
5.75
11.5
115
.8
.4
6.0
12/04/83
-
5.25
22.4
63
2.2
1.8
5.7
12/06/83
-
2.00
9.6
120
3.8
01/11/84
45
3.75
19.5
34
3.4
Bandedness
Case
LMSW - level of maximum wind shear magnitude
B.O. - band orientation [deg]
-
[deg]
wind shear direction
(dry)
frequency
: Brunt Vaiscala
: Brunt-Vaiscala frequency (moist)
Note:
FFS, DDS, Nd 2,
3.5
[km I
[m/s/km]
FFS
DDS
Nd 2
Nw2
wind shear magnitude
-0.4
[10 -4sec[10 -4sec-
Nw 2 , computed at LMWS
2]
850-500mb
FFS
8.0
5.8
8.5
such as
The criteria used to define
that shown in Fig. 3.
bandedness were:
The elongation of the reflectivity distribution in
1.
the horizontal
The strength of the reflectivity factor anomaly,
2.
and
The band's temporal and spatial coherence.
3.
with
scale was used,
A four-point
0 signifying non-banded
and 3 meaning strongly banded.
The synoptic situation in a representative case
2.1
The
looking
at
one
surface analysis
center
pressure
representative
for
Figure
5
field
temperature
the
can be summarized
the
for
of
4 shows
The
1981.
and
Boston
geopotential
500-mb
the
Fig.
case.
east
common
some
the
lowany
far from the New England area.
are
same
gradients
temperature
horizontal
to
is well
surface fronts
have
6 December
OOGMT on
obvious
shows
banded
These features
synoptic-scale features.
by
as
classified
cases
time.
height
Note
corresponding
the
and
strong
to
large
geostrophic wind shears in the New England area.
The corresponding radar reflectivity pattern was shown
This
previously in Fig. 3.
represents
the average
reflectivity factor in a layer from
2.5 to 4.0 km altitude.
band of very
constant-altitude presentation
The
dominant
heavy precipitation
feature
is a large
oriented north-south.
This
NC
it
C0
0
rio
I-0
0
O
it
1-h
0-.t
0
Ore
Mto
i0
0rt
00
M*
o CD
00rt
oaD
00
00
t-h I-'
,- t.Ct
0
c- ri. 0
t-h
F-
a art
M0
ACDnIt
.....
::::...:::..#S..
. ...
.. ... ...
/.
' .. 4
~
'-e
.......... ...
-
~
Y
'
S-
'
O"
a...
~
-
#z-~
*A
4
-O mv
-...........
..-........
...........4
..................
: :,.......
: : : : :.
'4..-
~
/
' /
....................
.....
::: .....
..
.....
.............
*,
A/.'
-
'.'..'.
...
/'~/-..o
.y
.......
..
~~
..........
c~:..................."..
....
-...
.
/..-.
A
- Y
.
~
.
r-,
.......
.
. .
.
A.
.
.
.
.
.
.
.
.
.
.
.........................
........
.......-....
............-....................
/...................
.
........ ....... .
.
.......................................~
rA% :::A............................
. .....
%
A ..-....................
t~
$V
4',s/
/~
/a.........
,..
o-w,~.-...
-/
"
V /'/
'41b..................
~ ~ ...
.. .4........
X , ,, ., ,...... .............................
/
4.-A
0 .... .
..
~7
... ...
:. .:
...........................
.. . .. .
......
.............
.A...../
........ .......................
... . . . .
*tw~...........
........
Fig. 4. Surface analysis for OOGMT on 6 December 1981,
adapted from National Meteorological Center (NMC) analysis.
17
Fig. 5. 500 mb geopotential height (dam) and temperature
(C) anlayses for OOGMT on 6 December 1981, adapted from NMC
analyses.
area
England
and
shows that the band
in
rates
of
more
5 with
and
temperature,
a
from
winds
of
2
Figure
3
sounding
shear
strong
noted
is
is
as
includes
at
a
Again the
location 60 km from the band are shown in Fig. 6.
region
than
region of very
the
is embedded within
New
Analyses of potential temperature, equivalent
strong shear.
potential
hours
Figure
A comparison of
per hour.
inches
snowfall
produced
the
14
for almost
persisted
band
particular
layer
a deep
of
the layer of banded
positive static
stability that
precipitation.
There is no obvious frontal zone and there
are
large
no
of
neutral
to
seem
precipitation,
rule
These
occurrence
the
out
elevated
stability.
stable.
is gravitationally
atmosphere
would
regions
convection
or
simple
Most
of
the
observations
of
frontal
convective
adjustment through upright convection.
While many of the band
the
one
just
described,
synoptic scale features.
all of
the
candidate
band
for
most
of
them
exhibit
similar
The presence of strong shear in
cases makes
the
cases are not as striking as
symmetric
mechanism
instability
responsible
for
the
a prime
banded
precipitation.
2.2
Mesoscale features of the cases studied.
In this section, the atmospheric conditions that exis-
ted in each case are presented.
the common features are deduced.
The cases are compared and
19
R.H. ()
9.-
8.
THETA
-
69
THETA EQUIV.
71
7.
74
78
-6.
F-
-
7
81
84
/
5.
8
4.-
00 00,9
0 -89-
3.-
88Il
86
86-
.-
-81
.1e
I.
I7782
0
270
'
275
280
285
290 295 300
305
TEMPERATURE (*K)
310
315
320
Fig. 6. Analyses of potential temperature (K), equivalent
potential temperature (K), and winds from Portland, Me, at
OOGMT on 6 December 1981.
As
tion
implied in
is
often
the previous
embedded
in
section,
regions
banded precipitaof
strong
shear.
Table 1 lists for each case the shear direction, the shear
magnitude,
and
the
Brunt-Vaisalla
moist adiabatic ascent.
frequency
of
the
850-500
mb
wind
calculations
and
come
from the
Also shown are the
shear,
bandedness, and the band orientation.
these
dry
All of these quantities are compu-
ted at the level of maximum wind-shear.
magnitude
for
the
estimated
The data used to make
sounding
that was
taken
closest to the band.
On comparing
the
cases,
a strong
correlation
between
the bandedness and the maximum wind shear becomes apparent.
The
six
cases
bandedness
and
20
have
m/s
non-banded
classified
a
level
a maximum wind-shear
per km.
or
with
The
weakly
or
magnitude
remaining
banded,
2
cases,
with
the
level
3
between
13
classified
as
exception
of
4
December 1983 have a maximum wind shear magnitude between 7
and
12 m/s per km.
A much weaker correlation
when the level of bandedness
is
is evident
compared with the magnitude
of the 850-500 mb wind shear.
Fig.
versus
7 is a scatter diagram of
the
1000-500
mb
geostrophic
the
band orientation
direction
and
the shear direction at the level of maximum wind shear.
All
band
cases
judged
non-zero
direction
is averaged
measuring
the
over
geostrophic
are
presented.
a 1 km
shear
shear
layer.
direction,
The
The
and
latter
errors
the
in
mean
140
130120 1101009080 -A
z 700
0 60-
Cr
O 50-
A
A
40-
e
S
30 -A
maximum
A
20-
wind shear
1000-500mb geostrophic
shear direction
a
100
Mean shear direction at level of
-
0
-10
0
10
20
30
80
40 50 60 70
BAND ORIENTATION
90
100 110
120
130
140
Fig. 7. Scatter diagram of the band orientation (deg) versus the 1000-500 mb geostrophic shear direction (deg) and
the shear direction at the level of maximum wind shear.
shear
direction
tively.
are
±10
degrees
and
±5
degrees
respec-
The error in measuring the band orientation is ±10
degrees.
While
both measures
the shear
of
direction are
close
to the band orientation, the geostrophic shear direction is
The RMS scatter is 9.2 degrees for the
in better agreement.
versus
direction
shear
geostrophic
degrees
13.4
shear direction at the level of maximum shear.
for
the
In Table lA,
the total wind shear at the level of maximum shear and its
Most
component in the band-parallel direction are compared.
of the wind shear is
In
of
nine
the
the direction parallel
band
eleven
within
shear,
wind
maximum
in
to the band-parallel
equal
result
of
7 that
Fig.
the
cases,
component.
bands
magnitude
the
error
the
are
to the band.
in
of
measurement,
This underlines
aligned
parallel
the
is
the
to
the wind shear direction.
Vertical
profiles
analysed for each case.
were
stable
layer.
for
dry
of
the
static
stability were also
It was found that all of the cases
adiabatic
ascent
above
the
boundary
In the region of maximum wind shear, three of the
band cases show an atmosphere that is neutral or potentially
unstable for moist
adiabatic ascent.
other regions of the
show
indications
negative
of
moist static
There are,
however,
atmosphere in many of the cases that
potential
stability
instability
(a8e/az<O).
A
is a sufficient condition
Table lA
Case
Bandedness
B.O.
LMSW
FFS
BP FFS
12/05/81
45
4.75
13.5
13.3
12/06/81
0
2.75
13.3
13.2
12/16/81
45
1.75
18.6
18.5
12/11/82
70
4.00
11.2
10.7
12/12/82
50
3.75
17.1
17.1
12/20/82
0
2.00
14.5
14.5
01/11/83
0
3.00
19.4
17.8
01/24/83
-
4.25
8.2
02/03/83
-
3.25
7.9
02/12/83
80
5.50
13.4
13.0
11/28/83
135
2.75
21.2
18.5
.12/03/83
90
5.75
11.5
11.3
12/04/83
-
5.25
22.4
12/06/83
-
2.00
9.6
01/11/84
45
3.75
19.5
19.3
LMSW -
level of maximum wind shear magnitude
B.O. -
band orientation
[m/s/km]
FFS - wind shear magnitude
BP FFS - band-parallel wind shear magnitude
Note:
[kn]
[deg]
FFS, BP FFS, computed at LMWS
[m/s/km]
for
saturated.
only
overturning
convective
For
the
cases
if
atmosphere
the
considered
in
this
study,
is
the
regions of potential instability that were judged saturated
are weakly unstable and can be considered,
of measurement, to be neutral.
having
substantial
potential
Regions that are assessed as
instability
In
associated with humidity gradients.
atmosphere
is not
instability
static
may
stability
uniformly
never
be
with
the
either dry or moist
within the error
this
saturated
released.
and
is
no
in
all
instance,
the
When
bandedness,
conditions
are
cases
the
potential
comparing
the
correlation
for
apparent.
This
lack of
correlation, however, does not preclude the possibility that
potential
instability may play a role
in
the formation of
the precipitation bands considered in this paper.
Some
evidence
assumption
The
that
bands
geostrophic
the
are
has already
bands
observed
shear
are
to
vector
been given
to
justify the
two-dimensional
be
aligned
which,
in
in
close
a
nature.
to
the
geostrophically
balanced atmosphere,
is the direction perpendicular to the
average
gradient.
temperature
properties
can
therefore
The
band's
be approximated
thermodynamic
as having most of
their variations in the cross-band direction.
Doppler
radar
and
synoptic
evidence
that
most
of
direction
with
the
variations
scale
flow
in
wind
is
in
this
predominantly in the cross-band direction.
In addition,
analyses
the
provide
band-parallel
flow
occurring
As
appear
stable
to be
or
motivation
associated
near-neutral
for
with
fact
strongly
atmosphere
considering
primary mechanism.
is the
it
earlier,
remarked
sheared
which
symmetric
that
the bands
flow
in a
the
provides
as
instability
a
The two-dimensional nature of the bands
also justifies the application of simple models of symmetric
instability and as will be shown
the
observations
can
be
used
in the following section,
to
support
assumptions made in the theoretical models.
many
of
the
3.
Comparisons between theory and observations.
observations
The
the
in
presented
previous
section
support the use of simple linear theory and parcel theory in
Most
assessing the symmetric instability of the atmosphere.
are observed
properties
The bands can,
strong
very
in
to be
the cross-band direction.
to a good approximation,
two-dimensional.
with
thermodynamic and kinematic
in
of the horizontal variations
be considered
The observed bands were always
shear of
vertical
the
to be
associated
band-parallel
wind
component.
The assessment of symmetric instability
3.1
In Section 1, two techniques were discussed for asses-
the
unity,
infinitesimal
is
heat
the
atmosphere.
One
If
this parameter is less
symmetrically
from
perturbations
latent
of
effect
Where
atmosphere
balanced basic state.
of
the symmetric stability parameter
from observations.
(n/f)-Ri
than
to calculate
is
technique
stability
symmetric
the
sing
can
be
geostrophically-
the
the atmosphere
included
to
unstable
is
saturated,
the
by
replacing
the
potential temperature with the equivalent potential temperature in the expression for Ri.
second
The
derived
by
technique
Emanuel
(1983).
is based
on
the
The environment
parcel method
surrounding
given parcel is assumed to be in geostrophic balance.
parcel's
stability
can
be
assessed
by imagining
a
The
its disp-
lacement from an equilibrium state and by evaluating the net
acceleration of the parcel resulting from the imbalance in
pressure gradient, Coriolis and buoyancy forces between the
parcel and its environment.
The
ments
the
of
stability
in
can be assessed
two ways
of the observational base.
displace-
to parcel
atmosphere
depending on the quality
Given adequate sounding coverage
across a band, a vertical cross-section of 0 and M can be
Then,
constructed.
as
shown
in
2, parcels
section
unstable to symmetric overturning where M surfaces
are
are more
shallow than 0 surfaces.
The
parcel
stability
can
assessed from a single sounding.
also
be
approximately
This method is similar to
the analysis of upright convection on a tephigram.
atmospheric
stability
sounding
is
of
temperature
and
assessed by adiabatically
winds,
Given an
the parcel
lifting a parcel and
modifying the lifted parcel temperature by adding the value
I
Tr
2- 9
where v
is
f
32-
Nqa
V-V
the environmental velocity and v
is
the initial
velocity of the parcel.
Stability
analyses
done for all cases.
which
vertical
available
in
based
a single
sounding were
Multiple soundings across a band from
cross-sections
only
on
three
of
could
the
be
band
constructed
cases.
were
Stability
analyses from a single-sounding require the knowledge of the
For
absolute vorticity, n=f+av/ax as a function of height.
av/3x
The
fields
neglecting
av/ax
to
the
is
vorticity
stability.
the
from
LFM
500mb
was
found
that
analysis
would
number
ten
about
only
relative
It
cases.
Richardson
the
of
error
an
reason,
in
estimated
was
all
for
on
effect
small
vorticity
relative
vorticity
lead
a
have
to
appears
relative vorticity
study, the
the cases considered in this
percent.
neglected
For
this
in
the
single-sounding stability analyses.
3.2
An example of the assessment techniques as applied to
a single sounding.
The
sounding
in
used
this
example
taken
was
from
Chatham, Massachusetts at 12 GMT, on 5 December, 1981.
At
(judged level.3) was observed in
the
this
time,
vicinity
a snow band
Chatham.
of
approximately
three
As
hours
the
later,
band
the
moved
band
Chatham
over
orientation
was
045 degrees.
Fig.
8a-d
shows
vertical
profiles
of
Ri~1,
O,
Oe,
and the magnitude of the component of wind shear parallel to
the band.
Ri
is
calculated from the observed band-parallel
wind shear, which is considered to represent the geostrophic
basic state, and dry adiabatic ascent is assumed.
Note that
the layer between 4.0 and 6.0 km is nominally symmetrically
unstable.
The potential
temperature and equivalent poten-
29
10
-9I
8
76-
5I
I.--.
4-
3-
.I
2
~
-
I 01
-3
-2
-I
0
(Ri)~
I
2
3
4
Fig. 8 a-d. Plots of (a) inverse Richardson number, (b)
potential temperature (K), (c) equivalent potential temperature (K),, and (d) the magnitude of the band-parallel wind
shear (m/s/km) as a function of height (km).
10
9-
8-
7-
6-
5-
4--
3-
2-
0
280
Fig.
8b
290
300
310
320
THETA (K)
330
'340
350
31
9
8-
7-
6-
-
94
270
280
290
300
310
320
THETA EQUIV. (*K)
Fig.
8c
330
340
350
10
9876-
04-
-4v
0
-5
-10
0
-- (ms-' km-')
a
Fig.
8d
5
10
t5
on the other hand,
tial temperature profiles,
is
atmosphere
for
stable
and
dry
both
show that the
adiabatic
moist
The profile of the shear magnitude indicates that
ascent.
between 4.0 and 6.0 km
the subcritical Richardson numbers
result from the very strong shear in this layer.
for the
the modified parcel temperature
Plotted is
9.
Fig.
in
is illustrated
case
this
for
method
parcel
The
slantwise adiabatic ascent of an air parcel originating at
The orientation of the parcel is
4.0 km height.
be
that
a parcel
the
along
direction
an
to
refers
of
(045*/2250).
direction
band-parallel
the
along
long
infinitely
)
band.
the
assumed to
(Recall
oriented
tube
shown
Also
the
are
environmental and dewpoint temperatures, and the 283 0 K moist
adiabat that passes through the parcel's starting position.
is
instability
gravitational
Conventional
by
evaluated
lifting the parcel vertically along its appropriate moist or
dry
adiabat
and
parcel
along
temperature
of
slantwise
analysis
This
near
displacements,
and
figure)
the
in
comparison.
parcels originating
vertical
line
the
temperature
parcel
the
Symmetric instability is evaluated by lifting
environment.
the
to
temperature
the parcel
comparing
4.0
they
km
are
are
(labeled
ascent
making
shows
completely
unstable
to
the
that
same
while
stable
to
slantwise
displacements up to a height of 6.25 km.
Fig.
difference
10
shows
between
a
contours
of
parcel
and
the
its
maximum
temperature
environment
during
10
9
I.-
2LEGEND
--
0t
-70
-
TEMPERATURE
.- PARCEL TEMPERATURE
PSEUDOADIABAT
DEWPOINT
-60
-50
-40
-30
-20
TEMPERATURE
-10
0
10
Fig. 9. Plots of the modified parcel temperature (C) for
the slantwise ascent of an air parcel originating at 4.0 km
height, the environment and dewpoint temperature (C), and
the 283K moist adiabat that passes through the parcel's
starting position. The orientation of the parcel is assumed
to be along the band-parallel direction (045*/225*).
7.50 -~4
5.50--
4.50 -00
2 4.00
0-0
.0
-
or 3.50-
-0.4
2.50
31~
01
2.00
1.50
-_-2-j___-_-
--
0.00
0.
0.D
45.
90.
135. 180. 225.
ORIENT ANGLE
270.
315.
350.
Contours of the maximum temperature difference (C)
Fig. 10.
between a parcel and its environment during lifting to 9.5
(km) and parcel
km as a function of starting height
orientation (deg).
lifting to 9.5 km for different starting heights and parcel
orientations.
There is a well-defined maximum of 2.5*C at a
height of 4.0 km for an orientation angle of 030*/210*.
analysis
further
indicates
that
originating
parcels
all
The
between 3.25 and 5.5 km with an orientation angle between 30
The depth
and 60 degrees are unstable to slantwise lifting.
and
of
height
the
layer
of
maximum
instability
symmetric
agree with the Richardson number analysis.
3.3
Results of single-sounding analyses.
The parcel method can be used to predict the strength
In the above exam-
and orientation of banded precipitation.
ple,
parcel
that
theory predicts
at 030*/210*.
be oriented
the band will
The strength of the band should be related to
the predicted parcel temperature surplus.
The actual corre-
lation between this temperature surplus and bandedness will
be examined after assessing the symmetric stability for all
band cases.
Table 2 lists for each band case the minimum value of
the Richardson number for dry and moist adiabatic ascent and
descent,
number.
and
the
level
of
the
dry
Richardson
Notice that in three of the cases, the Richardson
number for moist ascent is equal
corresponding
presence
minimum
of
value
for
humidity
dry
to or greater
ascent.
gradients
that
This
result
than
is due
in
a
to
it's
the
static
stability for moist ascent that is greater than that for dry
Table 2.
Case
Banded- B.O.
ness
LMWS
LMI
Ri
dry
Ri
moist
12/05/81
:5
4.75
5.00
.76
.46
12/06/81
0
2.50
2.50
.93
.93
12/16/81
:5
1.75
1.75
.58
.32
12/11/82
O
4.00
none
.931
1.53'
12/12/82
>0
3.75
4.00
.43
-0.11
12/20/82
0
2.00
2.00
.32
-1.79
01/11/83
0
3.00
3.00
.79
0.78
01/24/83
-
4.25
none
.101
1.901
02/03/83
-
5.50
none
.491
8.451
02/12/83
80
5.50
5.25
.68
0.34
11/28/83
13 5
2.75
2.75
.86
1.23
12/03/83
90
5.75
5.75
.62
0.29
12/04/83
-
5.25
5.25
.55
0.44
12/06/83
-
2.00
none
.131
1.24'
01/11/84
45
3.75
3.75
.92
0.94
B.O. : band orientation [deg]
LMWS : level of maximum wind shear
: level of maximum instability
LMI
[kIn]
[kin]
i -
value calculated at
LMWS
Except
ascent.
for
instability to
regions of potential
upright convection, the level of minimum Richardson number
is
the
same
for
both
to Table 1,
comparison
dry
and
saturated
many of
are included in Table 2.
conditions.
For
listed
there
the parameters
The parameters listed are computed
at the level of minimum Richardson number calculated for dry
ascent.
This level will subsequently be referred to as the
level of maximum instability
The boundary layer up
(LMI).
to 1.5 km is excluded from all calculations in an attempt to
exclude regions where surface viscous forces are important.
In these regions the inviscid theory may not be valid.
It is immediately evident
the
LMI
shear.
is
within
This
instability
that in all but
250 meters of
implies
result
from
that
the
one case,
the level of maximum wind
the
indications
strong shear.
of
symmetric
Any change
in
static stability with altitude has only a small effect on
the
level
of
the
critical Richardson
correlation between bandedness
that was observed
same
as
the
number.
The positive
and maximum shear magnitude
in Section 2 is therefore essentially the
positive
correlation
between
bandedness
and
minimum Richardson number that is evident in Table 2.
Table
3
instability
by
cases studied.
10
is applied.
summarizes
the
single
the
assessment
sounding
parcel
of
method
symmetric
for
all
The same analysis technique used with Fig.
Parcels are lifted
slantwise from starting
levels between 1.5 and 7 km to a height of 9.5 km for all
39
Table 3.
Case
Banded-
B.O.
LMI
MTS
LMTS
P.O.
LUP
ness
12/05/81
45
4.
30
3.25-5.50
12/06/81
0
2.
0
2.00-2.50
12/16/81
45
1.
40
1.50-1.75
12/11/82
70
4.
*
**
12/12/82
50
3.
60
12/20/82
0
2.
01/11/83
0
3.
01/24/83
-
4.
140
7.00
02/03/83
-
3.
130
5.50
02/12/83
80
5.
45
2.50-5
11/28/83
135
2.
120
1.75-3
12/03/83
90
5.
110
4.50-6
12/04/83
-
5.
*
4.50-6
12/06/83
-
2.
01/11/84
B.O.
LMI
MTS
LMTS
P.O
LUP
*
**
-
:
:
:
:
:
:
45
3.
band orientation [deg]
level of maximum instability [kIn]
maximum parcel temperature surplus
level of MTS [km]
predicted orientation angle [deg]
layer of unstable parcels [kin]
no preferred orientation
no unstable parcels
3.00-5
1.50-2
-10
20
*
50
[deg C]
1.50-2
**
1.50-3.50
possible
band orientations.
From the
resulting plots
following quantities are read and listed in Table 3:
and magnitude of maximum parcel-temperature
surplus,
the
level
predicIn
tion of band orientation, and layer of unstable parcels.
of
eight
the
band
eleven
cases,
LMI
the
determined
from
layer of unstable
Richardson number analyses is within the
In two band cases the LMI is within 500 meters of
parcels.
this layer.
One band case has no LMI.
a preference for
maximum parcel
cases to have a larger
the stronger band
temperature
There appears to be
and
surplus
a deeper
layer
of
instability.
Fig. 11 shows the actual band orientation, the predicted orientation and the 1000-500 mb geostrophic shear direcThe predicted orientation angle is calculated to the
tion.
nearest 10 degrees.
In only six of the eleven band cases,
as close or closer to the
the predicted orientation angle is
band
orientation
direction.
predict
Using
than
the
1000-500
the parcel
the band orientation
method
is
mb
geostrophic
stablity
shear
analysis
slightly worse than
to
simply
using the 1000-500 mb geostrophic shear direction.
Other comparisons can be made with the linear theory.
Recall that linear theory predicts
that the spacing between
bands should be related to the depth of the unstable region
and to the slope of the isentropes.
An order of magnitude
for this predicted wavelength can be obtained from a simple
scaling
analysis.
Let
H be
the
depth
of
the
unstable
I
I
I
I
*
Band orientation observed
A
1000-500mb geostrophic
shear direction
X
Bond orientation predicted by
parcel theory
I
I
I
I
I
1/11/84 02
12/3/83 0
--
11/28/83
182
2/12/83
122 -
1/11/83
12E -
-
12/20/62t2212/12/82
I8 -
12/11/82
122
-
12/16/81
02 -
12/6/81
02-
12/5/81 12
I
2
0
I
20
I
-
40
60
I
I
I
80
100
120
I
140
160
180
DIRECTION
Fig. 11. Comparison of the actual band orientation, the
predicted orientation, and the 1000-500 mb geostrophic shear
direction for each band case.
region.
The slope a of the isentropes is given by
30
et=
"
(3-il
.
ax
Thus the predicted wavelength L can be approximated by
L ~ -U.
(3 -2)
IotI
If
the
thermal
is
wind' relation
used
to
express
the
horizontal temperature gradient in terms of the geostrophic
wind shear, L can be written in finite difference form as
L ~
(3-3)
,
-6
A H
where
4& and
AV are
evaluated
across
the
level of maximum
instability.
Multiple
cases.
The
bands
observed
were
wavelength
reflectivity factor maps.
predicted
by
Eq. 3.3
observed
in
was
four
of
measured
the
from
band
radar
Table 4 shows that the wavelength
agrees well with
the
observed wave-
The present linear theory also implies
that the bands
length in three of the four cases.
should
not
propagate
unstable region.
the
band
Figs.
along
with
cross-band wind
speed
(bandedness).
relative
The
to
the
12 a-e show the
vertical
for
level
the
mean
flow
in
the
observed speed of
profiles
five, bands
of
the
judged
observed
level
3
3 band cases were chosen for this
comparison because they were the easiest to follow in time.
Table 4.
Case
Observed
Wavelength
(km)
Predicted
Wavelength
(km)
12/12/82
45
56
01/11/83
55
71
11/28/83
01/11/84
115
65
120
164
9k
BAND
6
SPEED
t
/,
-/
REGION OF MAXIMUM
INSTABILITY
5
V///;
j
S
T
C4
3'
2
5 DECEMBER
1981
OL
-2 0
S
-15
-10
I
-5
U (ms-')
Fig. 12 a-e.
The observed cross-band wind speed (m/s) as a
function of height (km) for (a) 12GMT, 5 December 1981, (b)
OOGMT, 16 December 1981, (c) 12GMT, 11 January 1983, (d)
12GMT, 12 February 1983, (e) 18GMT, 28 November 1983.
10
9
8
BAND SPEED
7
6
5
LA4
3
16 DECEMBER
-10
U (ms~1 )
Fig.
12b
1981
10
9
8
7
6
BAND SPEED
5k
REGION OF MAXIM
1
11 JANUARY
-20
-15
-10
10
-5
U (ms~')
Fig. 12c
1983
15
20
.1
81BAND
SPEED
.47
71-
REGION OF MAXIMUM
IT Y
I STABIL
3k-
12 FEBRUARY 1983
01
-2 5
I
-20
I
-15
I
-10
U(ms~')
Fig.
12d
I
-5
I
48
BAND SPEED
4
I
'3
REGION OF MAXIMUM
INS TAB ILIT Y
j&
w
m
26
NOVEMBER
Otl
-20
1983
i
-15
-10
U(ms~')
Fig.
12e
The
region of maximum instability as assessed from the dry
Richardson
number
condition
shown
hatched.
The
taken from the sounding nearest
is
cross-band wind profile
is
to the band, but not in the band, in order to minimize the
influence
the
of
that
Note
itself.
band
u
positive
is
defined as toward the warm air.
The
five
bands
to
corresponding
are
studied
cross-band
winds
in
the
at
speeds
mid-troposphere
None of the bands,
5 km height).
(between 3 and
moving
all
however,
are moving with the cross-band wind in the region of maximum
The disagreement is not serious. The altitude
instability.
where
the
two
speeds match
unstable region.
is
within
a kilometer
of
the
It is also evident from the figures that
there is substantial shear of the cross-band wind within the
unstable layer.
this layer.
While such shear magnitudes are much less than
band-parallel
the
they
km),
Shears of 2 to 5 m/s per km are observed in
are
shears
not
in
these
regions
The
negligible.
(13-21
present
m/s
per
linear
theoretical model does not include a basic-state cross-band
wind that varies with height.
The observations suggest that
this may be an important addition to the theory.
It should be emphasized
in
the
above
analyses
the basic state.
are
that the observed winds used
assumed
to be
representative
of
Recall that it is a cross-band circulation
that is associated with the onset of symmetric instability.
It
is thus possible
that some
of
the
observed
cross-band
wind speed may be due to the band perturbation in which case
the above comparison of wind speeds is not meaningful.
3.4
*The geostrophy of the observed wind shear.
In
all
the previous
of
of
assessments
the
symmetric
stability of the atmosphere it was assumed that the observed
winds
were
of
representative
the
geostrophically-balanced
In particular, in the Richardson number anal-
basic state.
was
assumed
that
the
observed
band-parallel
yses,
it
shear
is equal to the geostrophic shear.
wind
This assumption
can be justified only if the band is oriented in the direction of the geostrophic shear and if the shear parallel to
the band is equal in magnitude to the geostrophic shear.
It was shown in Fig. 7 that the band orientation tends
to be close
these
to
the
two directions
geostrophic
Even
shear direction.
differ by fifteen degrees,
if
the magni-
tude of the band-parallel wind shear will differ by only 4%
from the geostrophic wind shear as a result of choosing the
wrong
Therefore
direction.
the
cross-band
temperature
gradient can be used to estimate the magnitude of the geostrophic shear.
For each band case, two soundings on oppo-
site sides of the band were used to estimate the temperature
gradient between the
two stations.
geostrophic wind shear is,
az
The
magnitude
of
the
from the thermal wind relation,
feU
ax
where Ax is
the distance between the two stations and AO is
the potential temperature difference evaluated at any given
height.
For
five
available,
cases
profiles
vertical
geostrophic
where
shears
wind
the
necessary
soundings
were
were plotted of the estimated
and
also
of
the
band-parallel
shears.
The band-parallel wind shear is determined from the
sounding
nearest to
that
is
used
the
the
in
This
band.
single
the same
is
sounding
Richardson
sounding
number
analysis.
the profiles it was found that in
From comparison of
many
were
of
regions
the
Table
different.
markedly
wind
band-parallel
the
atmosphere,
in
shear
the
two
magnitudes
shear
lists
the
average
region
of
maximum
5
instability and the estimated geostrophic wind shear in this
region.
The agreement is not good.
observed
wind
maximum
instability.
reassessment
substituting
shear
of
is
In all five cases, the
supergeostrophic
stability of
region
of
prompted
a
the
discrepencies
These
the symmetric
in
the atmosphere
the geostrophic wind shear for the observed.
The Richardson number condition of equation 1.3 was used for
this
reassessment.
This
can
be
horizontal temperature gradient as
RL=
written
in
terms
of
the
Table 5.
Case
12/05/81
12/11/82
Average Band
Parallel Wind
Shear (m/s/km)
12.5
9.01
Average Estimated
Geostrophic Wind
Shear (m/s/km)
6.5
4.01
12/20/82
12.4
6.8
01/11/83
11.3
8.7
01/11/84
12.2
5.4
1 Value computed at level of maximum wind shear
All other values are computed in the region of
maximum instability.
Note: There is no region of maximum instability
for the case 12/11/82.
or equivalently as
>f[ oaI1
(3.4-)
The quantity on the right-hand side of 3.4 is
critical value of the temperature
perturbations
the
gradient for instability.
If the observed temperature gradient
critical gradient,
therefore
is
greater than
this
from the geostrophic basic
state are theoretically unstable.
Vertical profiles of the critical temperature gradient
were constructed for each case.
was
calculated
Figures
dry
using
the
The static stability
nearest
sounding
to
the
ae/az
band.
13 a-e show these profiles computed for moist and
adiabatic
cross-band
ascent
and
temperature
descent
gradient
along with
estimated
the
as
observed
described
above.
In each case there are regions that are convectively
unstable.
gradient
In
can
be
these
regions
computed
only
the
for
critical
dry
temperature
conditions.
The
region of maximum instability as previously assessed from a
single sounding is indicated on each figure.
The
regions
previously
assessed
as
unstable
symmetric overturning are now assessed as stable.
to
dry
Indeed,
there are no regions of dry symmetric instability indicated
for
any
of
the
five
cases.
Saturated
conditions,
in
general, weaken the stability but except in the convectively
unstable regions, the atmosphere is found to be stable or at
R.H.(%)
107
5
DECEMBER
1981
9. -
8.--
55
7.--
-6770
6.:
\74.
OF77
'/REGON
5.
MAXIMUM
INSTABILITY
8083
4. -
85-
88
3.-
55-
43
/
..
33
t
57
71
0
-4.
i62
.
-2.
0.
2
4.
6.
8.
A 9(*K/lOOkm)
10.
12.
AX
Fig. 13 a-e. Cross-band temperature gradient (*K/100km) as
a function of height (km) for (a) 12GMT, 5 December 1981,
(b) 12GMT 11 December 1982, (c) 12GMT, 20 December 1982, (d)
12GMT 11 January 1983, (e) OOGMT, 11 January 1984. Observed
temperature gradient (dashed), critical temperature gradient
for dry ascent (dotted), critical temperature gradient for
moist ascent (solid).
R.H, (%)
I
DECEMBER
1982
9.-
6ee<<0;
61-
8.-
67
6771
7377
8286
--
4.-
3.-
98-
< 0
97
.
97-
..
88
2. F
..-
----
19-
- -
98
Ile < 0
N
N
11e
I
-4.
-2.
0.
~...A
13b
88
77
~
2.
4.
6.
8.
A@ (*K/100km)
--Fig.
93-
10.
12.
R.H. (%)
-2
0.
2.
4.
6.
Ax
Fig.
13c
(*K/lOOkm)
8.
10.
R.H.(%)
s6.
FUI 5.
0
w
m
01
I
-2.
o.
-
II*t.~I
2.
6.
4.
(*K/OOkm)
AX
Fig.
13d
8.
10.
R.H. (%s)
I0.
9.
8
7.
6.
-
I
5.
4
AD
Fig.
l3e
(*K/QO0km)
most
neutral
to
moist
symmetric
While
overturning.
the
estimate in measuring the observed temperature gradient can
be
in error by as much as 1*C/lOOkm, it is noteworthy that
the estimate obtained never exceeds the critical value.
may
be
that
the
atmosphere
is
so
efficient
in
It
restoring
itself to a stable geostrophic balance that the theoretical
instability condition can rarely be observed. However, such
a
possibility
appears
entirely
observed persistence of these bands.
inconsistent
with
the
Conclusions and suggestions for future work
This study was undertaken to examine the possible role
formation of precipitation
of symmetric instability in the
Observations from fifteen cases of banded and non-
bands.
banded precipipation were used for this purpose.
research
The
completed
not sufficient to
to date is
conclude whether or not symmetric instability is responsible
for banded precipitation observed in the storms considered
The banded observations do agree with cer-
in this study.
tain aspects of the present linear theory.
The bands are embedded in regions of strong shear and
are aligned close to the direction of the thermal wind.
nine
of
the
eleven
band
within 15 degrees of both
and
the
actual
1000-500 mb
shear direction
direction
quasi-two-dimensional
variations
orientation was
the
at
level
the
of
maximum
Observations provide evidence that the
instability.
are
shear
band
cases,
the
in
with
thermodynamic
In
most
and
occurring in the cross-band direction.
bands
the
horizontal
kinematic
properties
of
Finally, the spacing
between bands does agree with the predictions of the linear
theory.
middle
Narrow regions of strong band-parallel shear in
troposphere,
typically
having
magnitudes
of
ten
the
to
twenty m/s per km are observed to accompany the occurrence
of banded precipitation.
Substantial shears of five to ten
m/s per km over very deep layers in the atmosphere are also
observed when bands occur.
The
on
a
two stability analyses used
knowledge
geostrophic
of
the
direction
wind shear.
assumed
shear as measured
the symmetric
is
geostrophic shear
direction.
the band-parallel wind
the
insta-
from single soundings was
It was
state.
insofar as choosing the direction is
direction
of
the observed band-
geostrophic basic
the band-parallel
that
magnitude
for each case,
to represent the
shown that,
and
In assessing
bility of the atmosphere
parallel wind
in this study depend
concerned,
a good approximation
to the
A much weaker assumption
shear
is
These
the geostrophic wind shear.
close
in
magnitude
is
to
two assumptions lead to
the conclusion that regions of the atmosphere for each band
case are symmetrically unstable.
Indications of symmetric instability, however, were in
most
cases
due
to the
strong
shear
regions
troposphere that were confined to layers
kilometer in depth.
in the middle
approximately one
It was noted that symmetric instability
is an imbalance that can arise in a geostrophically balanced
atmosphere.
It is unlikely
that such
a narrow region of
strong shear can correspond to a geostrophic base state, for
in
that event,
small-scale
horizontal temperature
gradients
of a magnitude rarely observed in the atmosphere would have
When such gradients are observed,
to exist in this region.
they
are
dependent
usually
associated
circulations
(e.g.
not with a balanced flow.
with
mid-level
ageostrophic,
frontogenesis)
timeand
In section 3, estimates were made of the geostrophic
wind shear magnitude in five band cases to see how well they
the observed band-parallel wind shear.
compare with
found
that
the
agreement
It
soundings.
was
atmosphere
assessed
Richardson
number
shown
used
to
adiabatic
were
the
of
the
regions
of
the
(using
the
unstable
shear
wind
observed
stability
that
of
it
most
infinitesimal
to
When the
ascent,
the
in
parts
symmetric
the
stable
dry conditions.
in many
is
When the geostrophic shear magnitude
assess
symmetrically
that
analysis),
indications
cases,
poor
symmetrically
as
highly ageostrophic.
was
was
It was
is
stability
found
the
for
troposphere
the
for moist
is assessed
where
regions
is
under
perturbations
that
five
the
atmosphere is not convectively unstable are at most neutral
to symmetric overturning.
Unfortunately,
the
measurement
the
of
geostrophic
basic state in these five studies was only an estimate.
poor
spatial and temporal
used
in
this study may
state conditions.
resolution
not be
of
adequate
the
The
sounding data
to determine base
There are several ways to try to overcome
Single and dual Doppler radar can be used to
this problem.
better determine the basic state wind field surrounding the
band.
To
represent
better
thermodynamic
as
well
as
kinematic features, an instrumented airplane can be used to
make passes through the band at various altitudes.
solution
would
be
to
set
up
a dropsonde
network
Another
in
the
Simultaneous soundings could then be
vicinity of the band.
made with much better spatial
and
than
temporal resolution
the standard synoptic scale sounding network.
estimates
The
the observed
of
component
that
were
of
made
geostrophic
this study suggest
in
wind shear
the
not in
either that the basic state in which bands form is
balance,
or that the geostrophic basic state is
not easily observed.
The observations that the ageostrophic
geostrophic
of
comparable
magnitude suggest that using a linearized model
to examine
and
the
geostrophic
components
of
stability characteristics
are
shear
the
of a geostrophic basic state
It cannot be assumed that perturbations
is not appropriate.
from the geostrophic basic
state are small.
It has also been pointed out that the present theory
does not explain the propagation of the band relative to the
mean
The
flow.
which
observations
a
includes
suggest
wind
cross-band
that
a basic
that
speed
state
varies
with
height may be required in the theoretical models.
Linking
tion
can
symmetric
have
major
instability
with
consequences.
prediction of when and where
banded
precipita-
Operationally,
bands will
form and
the
in what
direction they will move is vital to producing a mesoscale
forecast.
tant
Furthermore, such an instability may have impor-
effects
Emanuel
on
the
large
scale
flow.
Stone
(1972)
and
(1984) have both suggested that symmetric instabi-
lity
can
momentum.
observed
in
result
If
appreciable
symmetric
that
bands
is
instability
occur
responsible
in
often
so
of
transports
heat
and
for the
extratropical
cyclones, the resulting heat and momentum transports should
be
more
transports
studied.
closely
may
be
necessary
A
parameterization
in
the
current
for
such
large-scale
numerical models.
Acknowledgements
I am grateful
to my advisor Prof. Richard Passarelli
for his helpful suggestions,
research.
I thank
ideas,
Stephen Garner
and support during this
for
the
discussion we had concerning this thesis.
to
Spiros Geotis
and
Pauline
Austin
for
many hours
of
Special thanks go
their meticulous
review of this thesis and to Isabelle Kole for her expedient
drafting of the figures.
REFERENCES
Conditional symBennetts, D.A., and B.S. Hoskins, 1979:
metric instability - a possible explanation for frontal
rainbands. Quart.J.Roy.Meteor.Soc., 105, 945-962.
Bennetts, D.A., and J.C. Sharp, 1982: The relevance of conditional symmetric instability to the prediction of
Quart.J.Roy.Meteor.Soc.,
mesoscale frontal rainbands.
108, 595-602.
Air motion and
1969:
and T.W. Harrold,
Browning, K.A.,
Quart.J.
precipitation growth in a wave depression.
Roy.Meteor.Soc. 95, 288-309.
On the vertical circulation in frontal
Eliassen, A., 1962:
zones. Geofys.Publ., Bjerknes Memorial Vol, 147-160.
On convective bands
Elliott, R.D., and E.L. Hovind, 1964:
within Pacific Coast storms and their relation to storm
structure, J.Appl.Met., 3, 143-154.
Inertial instability and mesoscale
Emanuel, K.A., 1979:
Part I. Linear theory of inertial
convective systems.
instability. J.Atmos.Sci., 36, 2425-2449.
On assessing local conditional symEmanuel, K.A., 1983:
Mon.
metric instability from atmospheric soundings.
Wea.Rev., 111, (in press).
The Lagrangian parcel dynamics of
Emanuel, K.A., 1984:
J.Atmos.Sci., 40, 2368moist symmetric instablity.
2376.
The role of potential
Hoskins, B.J., 1974:
instability.
symmetric stability and
Meteor.Soc., 100, 480-482
vorticity in
Quart.J.Roy.
Mesoscale rainbands in
et al. 1976:
Houze, R.A.,
tropical cyclones. Mon.Wea.Rev., 104, 868-878.
extra-
On the stability of baroclinic circular
Ooyama, K., 1966:
instability.
for
criteria
A sufficient
vortex:
J.Atmos.Sci., 23, 43-53.
Parsons, D.B.,
and P.V. Hobbs,
1983:
The mesoscale and
and microscale structure and organization of clouds and
precipitation in midlatitude cyclones. XI: Comparisons
between
observational
and
theoretical
aspects
of
rainbands. J.Atmos.Sci., 40, 2377-2397.
Solberg, H., 1936:
Le mouvement d'inertie de l'atmosphere
stable et son role dans la theorie des cyclones. Memoir
presented to the Meteor. Ass. of the U.G.G.I., Lisbon,
Dupont Press, Paris.
Stone, P.H.,
1966:
On
bility. J.Atmos.Sci.,
non-geostrophic
23, 310-400.
baroclinic
sta-
Stone, P.H., 1972: On non-geostrophic baroclinic stability:
Part III.
The momentum and heat transports.
J.Atmos.
Sci., 29, 419-426.