I=IIi~
a""s~l~
l"El_~_~
C
-
-Ofi E:Cr -
i.
4T TH
TC
ON TUE NUERICAL PREDICTION OF
HURRICANE TRAJECIDRI
WITH VERTICALLY AVERA(GMD WINDS
by
George Willard King, Captain, UBAF
B.S.
Tufts University
L,
(1955)
SWMITTED IN PARTIAL FULFILLMENT
OF THE REQUIREMNTS I0R THE
DsIEE Or MASTER OF
SCIENCE
at the
MASSACHUBETTS INSTITUTE OF TECHOLOGY
January 1966
Signature of Author
Certified by
Accepted by
......................
Department of Meteorology,
20 March,
......
............
Thesis Supervisor
.........
/..
....
........
1966
.....
..
Chairman, 9'epartment Committee on Graduate Students
'i
/
Table of Contents
Page
Abstract
i.
ii.
List of Figures
Introduction
1
The Choice of a Hurricane Steering Level
2
Some Limitation
of the Baotroropic Prediction Model
4
Preparatior and Analysis of Data
15
Forecast Results
24
Acknowledgments
45
Appendix
46
References
50
List of Figuree
Page 12
1.
Numerical Grid
2.
Shuman's 9 Point Operator
19
3.
Response of 9 Point Operator
19
4.
500 mb Observed Wind Field 10 September 1961 000Z
20
5.
500 ab Nondivergent Wind Field, 10 September 1961 OOO0Z
21
6.
500 ab forecast Wind Field, 11 September 1961 000OZ
22
7.
500 mb Observed Wind Field,
000Z
23
8.
Forecast and Observed Trajectories, Hurricane Carla
25
9.
Forecast and Observed Trajectories,
urricane Flora
26
10.
Forecast vs. Observed Position at 24 Hours - 500 nb
27
11.
Forecast vs.
12.
Forecast vs. Observed Position at 24 Hours - 10 Levels
29
13.
500 mb Stream Function Field, 06 September 1961
30
i4.
10 Level Strea Function Field, 06 September 1961
31
15.
500 nb Stream Function Field, 03 October 1963
32
16.
10 Level Stream Function Field, 03 October 1963
33
11 September 1961
Observed Position at 24 Hours - 4 Levels
ii.
28
ON THE NUERICAL PREDICTION OF HURRICANE TRAJECTORIES
WITH VERTICALLY AVERAGED WINDS
George Willard King, Captain,
USAF
Submitted to the Department of Meteorology on 14 March 1966
in partial fulfillment of the requirement for the degree of
Master of Science
ABSTRACT
Three wind fields are examined with the object of improving
hurricane trajectory prediction with the barotropic prediction
model.
As low latitude wind systems are subject to large variations in vertical structure, it is difficult to represent an
equivalent-barotropic atmosphere over a large area with the winds
at one pressure level.
As mature hurricanes are steered by winds
at all levels in the troposphere, it seems reasonable to establish
a large scale hurricane steering current as the vertical average
of the winds at a number of levels.
The wind fields resulting
from an average of 4 levels after Birchfield and an average of
10 levels after Sanders are used with the barotropic model to
predict hurricane displacements.
The results are compared with
similar forecasts made from the 500-mb winds.
In this experiment the quasi-geostrophic barotropic vorticity
prediction equation is integrated on a numerical grid of 165 km mesh
length.
The vorticity and stream function are computed directly
from the observed winds.
The hurricane circulation is included in
the wind field and hurricane-steering flow interactions are implicit
in the model.
Twelve hurricane cases are predicted, six from hurricane Carla
1961 and six from hurricane Flora 1963.
On the assumption of independence of six cases, the 10 level wind field is shown to be a significantly better hurricane steering field than either the 500-mb
or average of 4 level wind fields.
Thesis Supervisor: Frederick Sanders
Title:
Associate Professor of Meteorology
Introduction
During the past ten years the numerical-dynamical prediction of
extratropical cyclone scale motions has become competitive with forecasts prepared by subjective methods.
The most successful has been the
quasi-geostrophic barotropic prediction model,
with short range improve-
ments added by baroclinic and "primitive equation
"
models.
Several
attempts have been made to use these techniques to predict the movement
of hurricanes and typhoons.
However,
none have yielded results that
approach the skill of experienced hurricaue forecasters.
Three major problems are implicit in the numerical-dynamical prediction of hurricane motions.
The first
equations that describe the atmosphere.
is
lodged in the dynamical
These equations are highly non-
linear and must be simplified before we can solve them.
simplifying assumptions include:
Some of the
the assumption of quasi-geostrophic
motion and the equivalent barotropic and finite level baroclinic atmospheres,
The second problem concerns the numerical techniques used to
solve the simplified equations.
For reasons of economy one must use a
rather coarse finite difference grid to depict and predict the cyclone
scale atmospheric motions.
And the third problem concerns the lack of
observations in regions where hurricanes and typhoons occur.
There are
simply not enough observations to accurately define, within the first
two limitations, either the hurricane or the large scale features of
the atmosphere.
Within the framework of
the barotropic prediction model,
the 500-mb
wind field in current operational use does not accurately represent the
large scale steering-flow for low latitude disturbances.
While the 500-mb
wind field is a good approximation to an equivalent-barotropic atmosphere
at middle and high latitudes, at low latitudes no pressure level approximates an equivalent-barotropic atmosphere over large areas.
(1961)
and Sanders (1961)
Birchfield
have suggested that a vertically averaged wind
field is a better steering-flow for hurricanes than 500 mb or any other
standard pressure level.
In this paper we will compare vertically aver-
aged wind fields proposed by Birchfield and Sanders with the 500-mb wind
field used with the barotrople prediction model to predict hurricane displacements.
The Choice of a Hurricane Steering Level
The vertical structure of low latitude winds is not coherently organized like the vertical structure at higher latitudes.
Speeds are often
light and directions may change abruptly through a shallow layer in the
troposphere.
At any one time there may be a considerable range in the
equivalent-barotropic levels computed from a number of low latitude upper
wind observations.
sphere,
As a mature hurricane extends to the top of the tropo-
its path is influenced by all tropospheric winds.
In view of the
erratic character of the vertical wind structure, it seems reasonable to
cast aside the equivalent-barotropic model and develop a hurricane steering field as a density weighted ;ertical average of several tropospheric
levels.
In Birchfield's research he chose to derive a vertically averaged
flow from four levels:
1000 mb, 700 mb, 500 mb, and 200 rb.
chosen as the bottom and 200 mb as the top boundary with
boundaries.
1000 mb was
w = 0
at the
In choosing his weighting factors:
3
(where
a.
are the weighting factors) he represented
at each
level by a cubic polynomial and developed a set of four linear equations
for the four weighting factors.
His weighting factors were:
s
14
V:4'+'V
In Sanders'
/35V44'I7
()
research he chose to represent the vertically averaged
flow by the 10 standard pressure levels between 1000 mb and 100 ab.
1000 mb was chosen as the bottom and 100 mb as the top boundary with c
at these boundaries.
=
In choosing his weighting factors Sanders approximated
the average wind field by the trapeziodal rule:
His weighting factors were:
V:
/,l/O}
i./3iI
/,d
/6~5
(2
(2)
t.
3V
0
-.
3.
3.
Sanders' wind field has two advantages over Birchfield's winds.
Because of the erratic vertical profile of the winds, the greater vertical resolution of Sanders' winds yields a better estimate of the
average wind.
Second,
there is considerable evidence that the circula-
tion of mature hurricanes extends to 100 nrb.
The layer between 100 mb
and 200 mb represents more than 10 per cent of the hurricane steering
field.
Therefore,
one would expect the 10 level averaged wind field to
be a better estimate of the hurricane steering field.
Some Liminations of the Barotropic Prediction Model
A comparison of the physical and dynamical characteristics of
hurricanes and the barotropic prediction model is of interest for two
reasons:
(a) does the hurricane violate the physical assumptions of
the model and cause prediction errors and
(b)
can one use or modify
the model to obtain useful information about the hrricane.
One of the basic assumptions of the barotropic model is
the motions are quasi-geostrophic; that is,
gradient and Coriolis forces are in
that
that the horizontal pressure
approximate balance.
and others have expressed this kind of motion in
Phillips (1963)
terms of characteristic
parameters of the motion and a non-dimensional Rosasby Number:
C-(where C is
and 2n_. SIje
the characteristic velocity,
is
(3)
L is
the characteristic length,
the Coriolis parameter at latitude&
6
).
For quasi-
geostrophic motion
R0 z
I
and is about 0.15 for cyclone scale atmo-
spheric motions at middle latitudes.
Consider the Roesby Number for a moderately intense hurricane
(based on detailed hurricane wind profiles reported by Hawkins (1962)).
Let us assume a hurricane whose maximum winds are 40 m sec "1 at 500 ab,
a distance of 60 km from the hurricane eye to the region of maximum wind,
and -& at 25 0 N.
The Rosaby Number for this circulation in 19 or two
orders of magnitude larger than typical values for cyclone scale motions
at middle latitudes.
Non-adiabatic heating through the first law of thermodynamics may
also violate the quesi-geostrophic assumption.
Phillips has estimated
-1
a critical precipitation rate of 2 cm day"1 for this heating.
Precipita-
tion rates in excess of this rate are frequently observed in the forward
or rear right quadrants of hurricanes.
The characteristic length of the motion is also important in the
finite difference schemes used to solve the barotropic prediction equation.
Serious truncation error will result to components of motion whose
wavelengths are less than twice the distance between adjacent grid points.
In operational use of the barotropic prediction equation, the grid distance is 300 km (about 400 km at hurricane latitudes).
hurricane the component of motion of
In our model
the maximum winds has a wavelength
of about 120 km.
To mitigate the non-geostrophic character of hurricane motions,
a number of investigators have partitioned the motion of the atmosphere
between a large scale steering-flow and the small scale hurricane circulation.
Using the quasi-geostrophic barotropic vorticity equation:
V
(where
is
the horizontal wind vector,
component of relative vorticity,
the flow is
vorticity),
(where
V
, then
is
/lA
the vertical
is the earth's
-VxY
;
the hurricane wind vector and "
is
and
is
7.:- Vx7
the steering-flow
Equation 4 becomes:
wind vector).
In
2 __P
and
/
partitioned between the hurricane and steering-
), : Xy
flow as:
-V7
early experiments with steering-flow models,
the hurricane cir-
culation was not allowed to influence the evolution of the steering-flow.
was defined as an axially symmetric vortex
The hurricane circulation
field:
T
()- f
A e)d
-o)
and was subtracted from the observed flow field.
(6)
Equation 5 was then
partitioned as
/Pt-,
and
7
VV(8)
)
(7)
where equation 7 is suitable for numerical integration.
In these experi-
ments Kasahara (1959) found a right bias of the predicted vs. observed
hurricane displacement.
In a later paper Kasahara and Platzman (1963) examined the other
interactions suggested by equation 5.
They were particularly concerned
with the advection of steering-flow vorticity by the hurricane circulation,
the last term in
equation 5.
In a simplified treatment they again assumed
the hurricane circulation to be represented by a simple axially symmetric
mathematical function.
They allowed the axially symmetric function to
advect steering-flow vorticity and influence the evolution of the steeringflow.
These simple vortices were found to experience a small acceleration
in the direction of increasing absolute vorticity of the steering-flow.
The acceleration may be to the left of the steering flow or northward
depending on the relative magnitudes of the gradients of the relative
and earth's vorticities.
Their findings have been offered to explain
the right bias of previous steering-flow experiments.
In the barotropic model in operational use at NMC (Vanderman (1962)
see also Morakawa (1960,
tions 5 through 8.
1962)),
the 500 mb flow is partitioned as in equa-
To account for the vortex acceleration,
an axially
symmetric vortex velocity field is added to the steering flow velocity
field to advect the vorticity of the steering-flow.
In this model equation
7 becomes:
(where
C
is the vortex velocity field).
7.
The field of r
is empirically
determined from the eye radius, maximum wind speed and outside mean radius
of the real hurricane.
Each of the above assumptions of axial symmetry leads to a possible
source of error.
axially symmetric.
In the real atmosphere the hurricane circulation is not
An unreal steering-flow may result (Jones (1963))
when a circular flow pattern is removed from the observed wind field.
In
the second assumption, the advecting field on the right side of equation 9
does not physically correspond to the advecting field of the observed
winds.
One may a gue that the real flow is subject to serious truncation
error and that the pattern of
error.
F
can be chosen to minimize truncation
There is no specific evidence to show which advection pattern
yields the smaller error.
In another approach to the hurricane prediction problem,
Birchfield
(1960, 1961) used a 150 km grid to depict the hurricane circulation and
its steering environment.
This approach reduced but did not eliminate
the serious truncation error of the hurricane circulation with a 300-400
km grid network.
As the real hurricane circulation evolves in a non-
geostrophic manner, the assumption that the goneral character of the circulation does not change during the forecast period is implicit in all
versions of the barotropic prediction model.
In addition, hurricane and
steering-flow interactions are implicit in Birchfieids approach and do
not have to bc provided in some artificial manner as iii the steeiing-flow
models.
For our project we have chosen the fine grid suggested by Birchfield.
The objective of our experiment did not include the effects of hurricane
and steering-flow interactions.
them.
We would prefer not to be concerned with
As Birchtield's results were comparable with the best steering-flow
models, the fine grid seemed to be the best choice for our work.
The Numerical-Dynamical Model
At middle and high latitudes the wind field is usually established
indirectly from pressure-height analyses using simplified equations of
At tropical latitudes synoptic scale wind fields cannot be ac-
notion.
curately established with conventional middle latitude techniques.
Uncertainties in local pressure-height observations are of the same order
of magnitude a
the height gradients required to define the wind field.
While actual wind observations serve only as an aid in middle latitude
analyses,
low latitude wind fields are most accurately established directly
from the observed winds.
The Relmholts theorem:
(where
P
k
is
the unit vertical vector,
tion, and
/f
is the horizontal velocity potential) allows decomposition
is
the horizontal stream func-
of the horizontal wind vector into its nondivergent and irrotational components.
The stream function field is
diction.
Thus, It
required in numerical weather pre-
seems reasonable to compute
observed wind field.
directly from the
--- -------- ~
11111 b---
--
41
-~sla~rerrr~
--r
-
'^*"~
'~1~8~8WI I
A number of investigators have proposed numerical techniques
for computing the stress function field directly from the observed
Taking the vertical component of the curl of equation 10:
winds.
/171
yields the vertical component of the relative vorticity of the wind
field.
Equation 11 is a Poisson equation which may be solved by numer(a)
ical techniques if
(b)
P
r
is
computed from the observed winds and
specified on the boundary of the
or its normal derivative is
region of concern.
If we take the component of (10)
parallel to the
boundary:
(12)
(where
n
and
are in the normal and tangent directions,
a
along the outward normal,
tion along the boundary),
a known and
/
a
and
/
a
positive
positive in the counterclockwise direcj
is
specified in
terms of
Vs
an unknown quantity.
A two dimensional vector field that is
both nondivergent and ir-
rotational may be represented by either the stream function or velocity
potential.
In the numerical solution of equation 11, the choice of
boundary conditions determines the way that the nondivergent irrotational
field is partitioned between the stream function and velocity potential.
The vertically averaged wind field is
highly nondivergent and should be
accurately approximated by the stream function.
10.
Therefore,
we have
chosen boundary conditions:
_(13)
that minimise the kinetic enorgy of the velocity potential and associate
the nondivergent irrotational component of the wind field with the stream
A proof of this procedure is included in the appendix.
function.
In our numerical scheme
V
is prescribed at each of 1715 points
on a 49x35 point finite difference grid (see figure 1).
vorticity at each interior point is
approximated by:
' sV
(where
1
and
J
The relative
are the row and column indices, U
itudinal and meridional wind components and
A
is
'2
and
(14)
v
are the long-
the grid distance).
Equation 11 is solved by Liebmann relaxation of the form:
(where
/
is
the iteration index and
o
11.
is the overrelaxation coefficient)
.
..
k
"
'S . o
.
.
. .
.
.
0
..
,
. .
.
.
gas..
. . .
. .
*
..
..
0
-. .• . . . . .
. ..
°
o
. . . . .
o
o
S\
.
..
..
"
0
Fi .I.,
"#€
4"
G i
ADO.
SFig
.
.
5
,,?
...
,
-.
Go,
.
.Numericl
Grid
.
....
...
I~-L~LPI-~L-L_ -
-
-
I
-~-~le
sllPaPrWlO~?~a3**s*
~E~
---
-~, "~~*~~Z1IC~r
~
with inward differencing on the boundary of the form:
to a tolerance of
,0
2,
A(,,
YttV1
II
0,0
2
sec-1 for all interior
points.
As there are many discussions of the barotropic prediction model
in recent literature (see Thompson (1961)) we will only briefly review
the dynamical and numerical prediction equations used in this report.
The quasi-geostrophic barotropic vorticity prediction equation 4, with
the aid of equations 10 and 11 may be rewritten as:
V'tY K
(where J
)
is
vYtY)
the Jacobian operator).
equation and may be solved for
region the right side of equation 16 is
?V
t
or its
Equation 16 is
if
Y/
normal derivative is
13.
(16)
(a)
also a Polsson
on the interior of the
specified and (b) on the boundary
specified.
The fields of
~
-
i~ic~C~g3
13~ J~ WII~
and
\7
,
I
-
--_-~ Isr~"--i-~sul
-nenr
r~n~ssaar~--
have already been determined from the observed wind field
through equations 14 and 15; we will assume that
-0oa
on the
boundary.
In the numerical solution of equation 16, V
mRust be specified
on the boundary to compute the right side of the equation.
V
was
calculated at all interior points but not the boundary of the original
grid through equation 14.
Therefore, the prediction grid must include
only the 1551 interior points of the original grid with new boundaries
adjacent and interior to the old boundaries.
Equation 16 is solved by Liebmann relaxation in the form:
to a tolerance of
The new
S6
,-
.-
1 m see
1
at AS
every point.
field is then computed by centered time differencing as:
S14.
(18)
T
-I
-
(where
new
v1t
~~~.
-- I-
t is the time index and At
I
e
r* u.~y~iF~mr~Clb~LI
'--a~slsar~se~iL~-4L131~ar~
I
the time increment and the
is
field by:
A
q
+.
In the solution of equation
increment that is
its
Iq
important to choose a time
less than the time required for a parcel in the field
to move between adjacent grid points;
If the forecast interval is
that is:
4-l
C
A/U
1
larger than this critical value, the numerical
With a wind speed of 40 a sec-l
and grid
we can use a forecast interval of one hour.
The one
solution will become unstable.
distance of 165 k,
4
hour interval was used in this experiment.
Preparation and Analysis of Data
Hurricanes CarlSa
1961,
and Flora, 1963, were chosen for the study
because of the relatively large number of rswindsonde observations over
a broad area surrounding the tracks of these storms.
As three of the four
Mexican stations reported only at 0000Z, most cases were based on this time.
To give greater statistical
independence to the sample,
The study includes six cases from
were separated by at least 24 hours.
Carli,
6-11 September 1961,
individual cases
and six cases from Flora,
3-8 October 1963.
Vertically averaged winds and pressure-heights were computed from
individual rawinsonde observations acording to equations 1 and 2.
observations beginning above 1000 mb, the first
15.
For
standard level (1000 ft
above the ground) of the rawin observation was substituted for the 1000-ab
wind.
For observations terminating below 100 ab (or below 200 mb) the 10
level (or 4 level) averaging was not attempted.
For each case,
a 500 mb height contour analysis was first
prepared
using available aircraft reconnaissance as well as rawinsonde observations.
Height contour charts for the 4 and 10 level average flow fields were then
prepared with attention over space data areas to continuity among the three
flow fields and with charts for the previous day.
Charts for 1000 mb,
700 mb, 500 ab, and 200 mb provided by the National Hurricane Center were
also useful in establishing the 4 and 10 level patterns over the central
Atlantic Ocean.
Analysis of the wind fields included isogons and isotacha
observed winds.
from the
Again, the 600 mb charts were prepared first using recon-
naissance reports with the rawinsonde observations.
Goostrophic winds
computed from the height contours were used as an aid in the wind field
analysis particularly over Canada and the north Atlantic Ocean.
The 4 and
10 level wind fields were also prepared with attention to continuity.
Wind VectOr values were then read at each of the 1715 grid points and
transferred to punched data cards.
The geographical area of concern and the finite difference grid
are shown in
figure 1.
The map is
a Lambert conformal conic projection,
1:13,000,000 scale, with standard parallels at 300 and 60
finite difference interval of
computer output format.
North.
The
i" = 165 km, was chosen to fit standard
Stream function and vorticity fields were
16.
printed at initial, 12, 24, and 36 hour intervals in all cases.
The direc-
tion and speed of the nondivergent wind field was also recovered in selected
cases.
The numerical program outlined in
the previous section was coded for
use with the IBM 7094 electronic digital computer.
During the checkout of
the program, we varied the overrelaxation coefficients to achieve optimum
overrelaxation for this scheme.
tion 17,
\ = .30.
For equation IS, A
= .46 and for equa-
The average number of iterations required to establish
the T field was about 85 and to establish the I
/Ibt field wa about 8.
Average running time for a 36 hour forecast including data and program input and printed results was less than 2 minutes!
During program checkout, some amplification of high frequency components of the flow was observed.
Shaman (1957) and others have observed
this problem and have designed smoothing operators to filter
components.
high frequency
In this experiment we have used the 9 point smoothing operator
(see figure 2) suggested by Shuman to filter the two grid distance component.
If
as:
Z
is a two dimensional field, we may express the smoothed field
;
7 +s -(I
z
t
+
to- Z7 )
(20)
(where the subscripts refer to the grid points in figure 2 and
smoothing constant).
the operator will filter
With a value of
k
k
is a
equal to or greater than 0.5,
the two grid distance wave.
However,
the operator
is not highly selective and reduces the amplitude of all components of
17.
finite length.
component,
of
k
After the smoother has filtered the two grid distance
a second pass may be made over the field with a second value
chosen to restore the longer wave components.
smoother-unasoother used the values of
had response characteristics shown in
kI
1
0.5
Our combination
and
k 2 a -0.6
figure 3.
The smoother-unsmoother was applied to the initial
field and fields at 12 and 24 hours.
and
stream function
One might argue that the smoother
would remove much of the character of the hurricane circulation.
4 through 7 show the input wind field, the initial
Figures
and 24 hour forecast
nondivergent wind fields and the observed wind field at 24 hours at
500 ab in the vicinity of hurricane Carla, 10 September 1961,
There is
00002.
good correspondence between the input and nondivergent wind
fields with maximum input speed of 32 m seC" 1 and nondivergent speed
of 26 a sec-
1
,
Correspondence between the 24 hour forecast winds and
observed
winds is
-1-
also quite good with forecast maximum speed of 30 m
"
ec-1 and observed speed of 42 a sec
.
Notice, however,
that the small
scale detail of the maximum wind is lost.
In our scheme,
the center of the hurricane was located and tracked
as a minimum value of the stream function field.
Because of the inherent
limitations of finite difference techniques, the minimum value of strean
function was generally displaced a
usmal distance from the observed hur-
ricane position (see figures 4 and 5).
There was also a small analysis
uncertainty in the location of both initial and forecast positions.
Forecast trajectories were computed from the position of minimum stream
tunction in the initial field to the minimum in the forecast field.
18.
o
7
Fig. 2.
9 Point Operator
e
5
6
o
o
8
0
O
4
o
e
C
I
2
3
20
Fig. 3
Response of 9 Point
Operoaor
1
E
0z
2
0,O
0.2
0.4
0.6
Response
0.8
1.0
li~ii~-rPli~iPTePL~ sil II 1
180
180
1111
- 1
1
- -
- 1
-
-
-
Observed trajectories were adjusted to emanate from the position of
minimum stream function in the initial field.
Forecast Results
Forecast trajectories are compared with one another and with
the observed trajectories in
one notices:
figures 8 and 9.
From these trajectories
(a) a left deflection of the forecast vs. observed dis-
placement in most cases,
observed displacement,
(b) predicted displacement greater than
and (c) vertically averaged winds yielding better
forecasts than 500 ab winds.
Sumaries of forecast vs. observed position at 24 hours are shown
in
figures 10 through 12.
Directions on the polar diagrams are oriented
with respect to the 24 hour observed displacement vector.
observed positions are at the center of the diagrams.
The 24 hour
These diagrams
also show the tendency for forecast positions to lie to the left of
observed positions.
The initial stream function fields of two cases with serious left
bias, 06 September 1961 and 03 October 1963, are shown in
through 16.
figures 13
Stream function fields are presented for 500-mb and 10 level
averaged winds.
Values of stream function are scaled to yield a speed
-1
2
-1
of 10 a sec
with a gradient of 10 a2 sec
across 165 km (one grid
interval).
Analyses are at intervals of 30 a
2
sec
-1
All four analyses show an easterly current in
motion in the vicinity of the hurricanes.
24.
.
the large scale
Notice in particular the
-C-
II~
--
0
0
500mb
/
,
, /
12
12
,
o
\
10 levels
,
/r 12
0
I
oo
0o
0
0I0
4 levels
O
o
"-k
\
09
O
g OO
o
O09
\
\
Observed0
12 R
Observed
OOE
Forecost
Forecast
o 121
Hurricone
Corlo
September
1961
Fig.
O'
O0
;O8
\
\
0
\
\
O
7
7
8.
O0
0
00
S06
Observed
Direction of Movement
i800
500mb -
24 Hours
x - Carlo
@ - Flora
Fig. 10.
I
Observed Direction of Movement
1800
0
0 7
©08
4 Levels x -
24 Hours
Carlo
©-- Flora
Fig, II
28
Observed
Direction
of
Movement
1800
10
Levels x - Carlo
o - Flora
Fig. 12.
24 Hours
20
/
/
/ /
/
/
10
(
I
C::3
20 4,,,..~
Fig.
13. Stream Function Field
-
06 Sept
1961 - 00f
500mb
-90
20
I
10
I
+58
/
/
/
I
I
II
I
I
I20
2O
Fig. 14. Streom Function Field - 06 Sept
1961 -00-
10 Levels
Fig. 15 Stream Function Field - 03 Oct 1963 -OO
-500mb
/
g..
Sreom
Function Field03
Oc
Fig. 16. Stream Function Field - 03 Oct
1963-00
0
Levels40
1963 - OOZ - 10 Levels
r
east-northeast flow from the central Atlantic coast to the Gulf states
on 6 September.
Both 500-b analyses show a stronger and somewhat more
northerly gradient than the 10 level analyses.
hurricanes were moving from the southeast.
At analysis time both
These analyses suggest that
the 500-mb winds are not as accurate as the 10 level winds in representing
the hurricane steering-flow.
As previously noted, our model provides for quasi-geostrophic
interactions between the hurricane circulation and the large scale flow.
Except for the 500-mb case of 3 October, some northward deflection was
provided in the forecast tracks.
enough.
bowever, this deflection was not large
The discrepancy in direction is
attributed to a nongeostrophic
interaction between the hurricane and large scale flow that was not
provided in our model.
Measures of forecast accuracy are shown in tables 1 through 9.
These include:
Spred ,
the predicted displacement;
Sobs I the observed
displacement;
E
, the magnitude of the vector error of the predicted
displacement;
Rv
, the ratio of the magnitude of the vector error to
the observed displacement; and
Ra ,
the ratio of the predicted to ob-
served displacement.
As indicated earlier, there is some uncertainty in the location
of the center of the hurricane circulation in both the observed and
forecast stream function fields.
This uncertainty is most apparent
in the 12 hour forecasts where it is a large fraction of the small displacements.
In the 24 and 36 hour forecasts it becomes a smaller frac-
tion of the displacement error.
than R,
In
most cases,
for 12 hours.
34.
Rv for 24 hours is less
TABLj
1.
Results of 12 Hour Forecasts -
500 mb
(Displacements in Nautical Miles)
Initial Time
Carla 06 Sep 61 0000Z
8pred
ob s
Ev
R
R
120
110
41
.374
1.091
"
07 Sep 61 00002
70
70
83
1.190
1.000
"
08 Sep 61 000OZ
75
75
19
.252
1.000
09 Sep 61 0000Z
120
95
66
.699
1.263
10 Sep 61 00002
135
95
73
.772
1.421
11 Sep 61 00002
115
75
59
.787
1.533
57
.679
1,218
"
Carla Average
Flora 03 Oct 63 000OZ
150
100
109
1.088
1.500
160
110
95
.864
1.455
"
04 Oct 63 00002
"
05 Oct 63 0000E
60
65
14
1.262
1.091
"
06 Oct 63 0000Z
40
60
24
.396
.667
07 Oct 63 12002
25
40
63
1.579
.625
08 Oct 63 1200Z
35
100
130
1.300
.350
Flora Average
73
.915
.948
Average of All Cases
65
.796
1.083
"
35.
TABLE 2.
Results of 12 hour Forecasts - 4 Levels
(Displacements in Nautical Miles)
Initial Time
Carla 06 Sep
pred
obs
E
R
0000Z
100
110
37
.335
R
.909
"
07 Sep
0000Z
80
70
74
1.054
1.143
"
08 Sep
0000Z
105
75
32
.433
1.400
"
09 Sepr
0000Z
95
100
49
.516
1.053
"
10 Sep
0000Z
135
95
79
.831
1.421
"
11 Sep,
0000Z
90
75
29
.381
1.200
50
.592
1.188
Carla Average
Flora 03 Oct 63 00002Z
95
100
28
.284
.950
"
04 Oct 63 0000z
85
110
64
.582
.773
9
05 Oct
63 0000z
55
55
0
.000
1.000
"
06 Oct 63 0000z
40
60
30
.496
.667
55
40
94
2.350
1.375
15
100
97
.967
.150
Flora Average
52
.780
.819
Average of All Cases
51
.613
07
Oct 63 1200
08 Oct 63 1200Z
86.
1.069
TABLE 3.
Results of 12 Hour Forecasts -
10 Levels
(Displacements in Nautical Miles)
Initial Time
pred
Sobs
Carla 06 Sep 61 0000Z
110
110'
07 Sep 61 0000Z
120
08 Sep 61 0000Z
"
.226
1.000
70
1.361
1.714
85
75
.214
1.133
09 Sep 61 0000Z
105
100
.363
1.050
10 Sep 61 0000Z
100
95
.360
1.053
11 Sep 61 0000Z
100
75
.449
1.333
.495
1.214
Carla Average
Flora 03 Oct 63 0000Z
80
100
.319
.800
04 Oct 63 0000Z
115
110
.488
1.045
05 Oct 63 0000Z
80
55
.501
1.455
06 Oct 63 0000Z
65
60
.406
1.083
07 Oct 63 1200Z
30
40
1.750
.750
08 Oct 63 1200Z
35
100
.915
.350
>v2rage
.730
.914
Average of All Cases
.613
1.065
"
Flora
37.
TABLE 4.
Results of 24 Hour Forecasts
5-00 mb
(Displacements in Nautical Miles)
Initial Time
obs
pred
BV
Carls 06 Sep
0000oz
280
185
136
.735
1.405
07 sop
00008
155
165
126
.813
1.000
S 08 Sep
00002
160
26
.164
.938
" 09 Sep
00002
240
185
107
.576
1.297
10 Sep
00002
260
165
119
.722
1.576
ooo0000oz
220
150
105
.702
1.467
103
.619
1.280
"
"
" 11 Sep
Carla Average
Flora 03 Oct 63 0000oz
230
205
175
.853
1.122
04 Oct 63 0000o
260
200
161
.757
1.300
"
05 Oct 63 0000
95
95
87
.700
1.000
"
06 Oct 63 0000o
90
90
0
.000
1.000
"
07 Oct 63 12002oz
25
115
140
1,215
.217
"
08 Oct 63 1200
15
300
314
1.047
.050
Flora Average
141
.762
.782
Average of All Cases
122
.690
1.031
38.
TABLE 5. Results of 24 Hour Forecasts - 4 Levels
(Displaceaments in Nautical Miles)
prod
88obs
v
0000Z
215
185
69
.374
1.162
07 oep
00002
190
155
125
.808
1.226
08 Sep
00002
255
160
97
.608
1,594
" 09 Sep
00002
220
185
88
.474
1.189
"
10 Sep
00002
270
165
148
.895
1.638
"
11 Sep
00002
OOX)
195
64
.430
1.300
99
.598
1.351
205
103
.500
.878
200
118
.591
.750
Initial Time
Carla 06 Sep
"
Carla Average
Flora 03 Oct 63 0000Z
180
04 Oct 63 00002Z
05 Oct 83 0000Z
115
95
30
.319
1.211
06 Oct 63 00002
105
90
20
.335
1.167
07 Oct 63 0000Z
TO70
115
173
1.540
.609
08 Oct 63 1200Z
85
300
244
.812
.283
Flora Average
115
.658
.816
Average of All Cases
107
.628
1.084
39.
TABLE 6.
Results of 24 Hour Fbreoasts - 10 Levels
(Displacements in Nautical Miles)
Initial Time
Carla 06 Sep 61 00005
aprod
obs
Ev
Rv
270
185
125
.673
1.459
"
07 Sep 61 00002
240
155
150
.970
1.548
"
08 Sep 81 OOOOZ
185
160
39
.244
1.156
"
09 Sep 61 0000
230
185
70
.379
1.243
"
10 Sep 61 00002
230
165
80
.482
1.394
"
11 Sep 81 00002
180
150
38
.252
1.200
84
.500
1.334
Carla Average
Flora 03 Oct 63 00002
170
205
69
.339
.829
04 Oct 63 00002
235
200
78
.382
1.146
"
05 Oct 63 0000
130
95
39
.412
1.368
"
06 Oct 63 0000Z
105
90
15
.167
1.167
"
07 Oct 63 12002
35
115
131
1.136
.304
"
08 Oct 63 12002
130
300
202
.673
.433
Flora Average
89
.518
.875
AvMrage of All Cases
87
.509
1.105
40.
TABLE 7.
Results of 368 Hour Forecasts - 500 mab
(Displacements
in Nautical
8
iloes)
RV
Ra
Initial Tine
pred
Carla 08 Sep 61 00002
400
255
216
.847
1.569
obs
V
t
07 Sep 61 00002Z
280
230
133
.576
1.217
"
08 Sep 81 00002
250
255
12
.047
.980
"
09 Sep 81 0000Z
375
280
156
.557
1.339
"
10 Sep 61 0000Z
385
245
146
.594
1.571
*
11
340
205
141
.689
1.659
134
.552
1.389
Sep 81 0000Z
Carla Average
Flora 03 Oct 83 0000Z
270
315
235
.745
.857
"
04 Oct 63 0000Z
350
240
151
.628
1.458
"
05 Oct 63 0000Z
150
130
145
.968
.867
A
06 Oct
63 0000Z
120
70
54
.769
1.714
"
07 Oct 63 1200Z
85
210
195
.928
.405
t
08 Oct 63 1200Z
85
520
437
.841
.163
Flora Average
203
.813
.911
Average of All Cases
188
.682
1.150
41
TABLE 8.
Results of 36 Hour Forecasts - 4 Levels
(Displacements in Nautical Miles)
Initial Time
Carla 06 Sep 61 O000Z
8
pred
8
obs
£
v
R
v
R
R
375
255
170
.667
1.471
"
07 Sep 61 0000Z
300
230
134
.581
1.304
"
08 SBep 61 0000Z
400
255
145
.569
1.569
09 8ep 61 000Z
360
280
123
.440
1 286
"
10 Sep 61 000
380
245
168
.687
1.551
"
11 Sep 61 00002
330
205
131
.638
1.610
145
.597
1.465
Carla Average
Flora 03 Sep 61 0000Z
200
315
154
.488
.635
"
04 Sep 61 0000Z
200
240
62
.259
.833
"
05 Sep 61 0000Z
135
150
61
.407
.900
"
06 Sep 61 000Z
155
70
88
1.264
2.214
"
07 Sep 61 1200Z
80
210
232
1.105
.405
"
08 Sep 61 1200Z
210
520
348
.668
.404
Flora Average
158
.699
.899
Average of All Cases
151
.648
1.182
42.
TABLE 9.
Results of 36 Hour Forecasts - 10 Levels
(Displacements in Nautical Miles)
Initial Time
pred
obs
v
R
v
R
Carla 06 Sep
000o0
410
255
224
.879
1.608
" 07 Sep
0000z
310
230
180
.784
1.609
08 Sep
00005
315
255
60
.235
1.235
09 Sep
0000z
340
280
86
.308
1.214
10 Sep
00002
335
245
94
.384
1. J67
11 Sep
00002
290
205
88
.427
1.415
122
.503
1.405
"
Carla Average
Flora 03 Oct i 0000o
220
315
124
.394
.698
"
04 Oct 4
00002
300
240
67
. 280
1.250
"
05 Oct I 00002
150
150
56
.373
1.000
"
06 Oct 4
0000Z
120
70
53
.753
1.714
07 Oct 4
1200Z
145
210
192
.916
.690
08 Oct 4
1200Z
255
520
296
.570
.490
Flora Average
131
.548
.974
Average of All Cases
127
.525
1.191
43.
As shown by Rs , the forecast storm displacements franCarla
were greater than the observed displacements.
Further,
there was
a tendency to accelerate the storm with increasing forecast time.
Jones (1963)
reported similar results with his steering flow predic-
tion models for hurricane Carla.
Therefore, this author feels that
the error is lodged in the physical assumptions of the barotropic
model rather than in the techniques peculiar to this experiment.
for each wind field at 24 hours were compared
The average I,
for significant differences using the Student's t Test.
Assuming that
all 12 cases were independent with 11 degrees of freedom,
ments are significant at the 95 per cent confidence level.
all improveAssuming
6 independent cases with 5 degrees of freedom in the 12 cases, the
following improvements were significant at the levels indicated:
10 levels vs. 4 levels at 80 per cent, 10 levels vs. 500 mb at 90 per
cent, and 4 levels vs. 500 mb at 85 per cent.
the 10 level wind field is
We may conclude that
a better estimate of the hurricane steering
field than either the 4 level average or 500-mb wind field with the
barotropic prediction model.
In view of the improvement in performance of the 10 level wind
field over the 4 level field, an averaging scheme with even greater
vertical resolution should be investigated.
processing equipment,
it
With high speed data
should be possible to devise a vertical aver-
aging scheme that would include all winds reported in
of the upper air observation.
the rawin section
Acknowledgements
This experiment was performed under the supervision of Prof.
Frederick Sanders.
Numerical computations were performed at the
Computation Center,
Massachusetts Institute of Technology.
The
author received financial support from the Alr Force Institute of
Technology and the Environmental Sciences Service Administration.
The author wishes to thank personnel of the National Hurricane
Rsearch Laboratory for their encouragement and assistance in providing part of the data required for the project.
wishes to thank Prof.
Finally, the author
Sanders for his encouragement,
guidance throughout the project.
45.
support and
Appendix.
On the partition of the observed winds between the stream
function and velocity potential fields.
The Relmholtz theorem:
(where
is the horizontal wind vector,
is the stream function and 2
,
is
the unit vertical vector,
is the velocity potential) allows the
decomposition of the wind field into its nondivergent and irrotational components.
Taking the vertical component of the curl of (1) yields:
(2)
is the relative vorticity of the wind field).
(where
Equation 2 is a Poisson equation which may be solved for
iif
(a)
the
right side of the equation is specified within the region of concern and
(b)
'
or its normal derivative is specified on the boundary.
A two dimensional vector field that is both nondivergent and ir-
rotational can be represented by either the stream function or velocity
potential.
In the numerical solution of (2),
the choice of boundary
conditions determines the way that the nondivergent irrotational vector
is partitioned between the stream function and velocity potential.
In
most meteorological problems, the vector field is highly nondivergent
and it is desirable to partition the nondivergent irrotational component
with the stream function.
46.
Taking the component of (1) tangent to the boundary:
5
(where
is tangent to the boundary and positive in the counterclockwise
direction and Y\
is normal to the boundary and positive in the outward
direction) which specifies
?Ila
//c1
an unknown quantity.
S
boundary, then
L.
in terms of
It
'V
a known and
is specified as constant on the
is specified in terms /
.
To show that the above boundary conditions associate the nondivergent
irrotational flow with the stream function, we will require that they minimisz
the kinetic energy of the velocity potential field expressed as:
From (1) we may write:
To minimise (4) we will set the variation of the right side of (4) to zero.
Note that :
is prescribed at every point and its variation is zero.
a.
V
b.
If the boundary is fixed then we may write:
c and :
47.
(where
is
).
F (X
the variation of the function
We may then
write:
A
(5)
X
xvY)(
Equation 5 may be rewritten and transformed in
a form to identify
the terms of (3) as follows:
(a)
Rewriting (5)
in component form:
A
A(
Aedd.
(b)
CL
Integrating by parts:
Syl09LV
7_ ..,gq' $-ZL~r
c
Grouping terms 2, 4, 6,
Slsu: (oF-
VISlr)-1
#1
~P~ta
d (ITh
(e)
-
'V, ag~ ?q~ d q, aJ ds~ )di~;o
d~t a;l
Ld( ,)
(4)
1;
and:
SI Qva
(c)
gq)
A)*d
t4i.
and 8 from the above
,VVv
and transforming terms 1,
~cV-(vd
48.
,
3,
5,
and 7 into a surface integral:
yieldst
4
jj~ul~m
i~ir~~~tA.0y
9 b1k (6)
To miniamze the kinetic energy of the velocity potential field, equation 6
requires that:
a.
.-
(X
-v7'
on the interior of the region of inte-
gration and
b. VS z '/)
Oo
(ith
region of integration.
) on the boundary of the
The first condition is simply our
definition of stream function at interior points and the
second condition is our specified boundary condition.
There-
fore our boundary conditions minimize the kinetic energy of
the velocity potential field.
The author is indebted to Marvin Geller for his assistance in
the derivation of this proof.
49.
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Birchfield, G.E., 1961: Numerical Prediction of Harricane Movement with
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Dunn, G. et al, 1962:
107-119.
The Hurricane Circulation of 1961.
Mon. Wea. Rev.,
DunnU G. et al
128-138.
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1964:
Hawkins, R.F., 1982: Vertical Wind Profiles in Hurricanes.
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Hawkins, H.F., and Rosenthal,
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Jones,
A.W.,
1961:
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8., 1965: On the computation of stream funcMon. Wea. Rev., 93, 245-252.
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50.
Sangater, W.Z., 1960: A method of representing the horizontal pressure
J. Meteor.,
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O.G.,
Shman,
Smoothing and Filtering. Mon. Wea. Rev., 83, 357-361.
II
Thompson, P.D., 1961: Numerical Weather Analysis and Prediction.
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L.W., 19862
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The
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Mn. Wea. Rev.,
51.
90, 19-22.