188 MONTHLY WEATHER REVIEW MAY 1959
A GRAPHICAL METHOD FOR COMPUTING HORIZONTAL TRAJECTORIES
IN THE ATMOSPHERE*
HUGO V. GOODYEAR
Hydrologic Services Division, U S . Weather Bureau, Washington, D.C.
[Manuscript received November 19, 1958; revised April 30, 19591
ABSTRACT
Graphical solutions of the horizontal equations of motion :are derrlo1)ed for computation of dynamic tra-
jectories. Expedient techniques for using these solutions are discnssc~d. An approximation for the frictional effect
is suggested, and graphical application is outlined.
1. INTRODUCTION
The purpose of this paper is (1) to study the quantita-
tive relationships beheen the cllanges of the pressure field
encountered by a parcel and the c l ~n g e s i n its velocity,
and (2) to apply the relationships to the construction of
horizontal trajectories.
Trajectories of air parcels have been computed by vari-
ous investigators using t.he actual winds, geostrophic
winds, gradient winds, constant-absolute-vorticity con-
siderations, and combinations of the above. One of the
first contributions to a dynamic nppro:tch was that of
Machta [ 11. Recently additional work on "dynamic"
computat,ion of trajectories has been published by Txlm-
hash'i et al. [2], Francescllini and Freeman [3] , Wobus
[4], the Air Weat.her Service [5], and Hubert [6]. The
methods to be presented in this paper give ex:tc-tly the
same resu1t.s obtained by Wobus' method, and by the Air-
Weather Service method for the frictionless case. Fr:111-
ceschini and Freeman's method differs slightly. Hubert's
method also differs, his approach being bas,ed entirely 011
hourly forecasts of geopotential or stream function fields
by numerical methods. The llandling of the frictional or
damping terms herein differs considerably from that of
Wobus. The other investigators have applied the methods
only in t,he higher tro'posphere and the stratosphere, w11iIe
in t,his paper the methods are applied near the ground.
2. THEORY
The problem is to find the trajectory of a parcel of air.
in the frictionless atmosphere when its initial velocity is
known (or can be closely approximated) and the contour
or pressure field is kno~vn or can be approximated c l ~l r i l l ~
a later interval of time. The sinlplified equations of
motion in the atmosphere without, friction are
*Thin paper was written as part of the work under P.L. 71, 81st Congress
under funds transferred from the Corps of Engineers.
d u =fv- fv,= fv' (1)
and
dv= - fu+ fu,= - fu' dt
where u and u are the east-west and north-south compo-
nents of the wind, f is the Coriolis parameter, the subscript
g denotes talle geostrophic wind, and u' and v' denote the
geostrophic deviations; i.e., u'=u-u, and V ' E V -V ~.
Haurwitz [7] shows these equations have the simple
geometric interpretation t'hat t'lle wind is accelerated along
a line a t right angles to the geostrophic deviation vector
V' and in a right-hand or clockwise sense, as illustrated in
figure 1. Xote that - must always he directed to the
right of V' in the Northern Hemisphere.
For computing the change in the wind during a certain
interval of time, one nlay use the average geostrophic wind
on the parcel during that interval, t,hus assuming that it. is
constant' in magnitude and direction. Then, it follows
from equations (1) and (2 ) that
dV
dt
dV'
" - d t - fu'
during the interval. Now, from figure 1
u{= I" COS e' (5)
v'= V' sin e' (6)
where V'=IV'l=~'u'2+c'2. Use of (5 ) and (6) to trans-
form equat,ions (3) and (4) into a p i r for V' and e' gives
/-
I* v
at -=O
MAY 1950 MONTHLY WEATHER 189
x
- d v
Y
%Vd+
0
FIGURE 2,”Geometric relationships among the initial wind Vt, the
geostrophic wind V,, the initial geostrophic deviation Vi‘, the
final geostrophic deviation VI‘, and the final wind VI.
FIQURE 1.-Geometric relationship between the acceleration vector
d V /d t and the geostrophic ‘deviation vector V’.
and de‘ -=-
dt
These equations mean t,hat the ageostrophic win,d vector
B constant in magnitude ana! rotates anticycbonicnlly with
a period of om-half a pendulum day.
Thus, during the time interval in which the geostrophic
wind is considered a constant (qual to its average), the
change in the wind vector can be represented by the rota-
tion of the ageostrophic mind vector, keeping constant
length, cloclm4se at a rate equal to the Coriolis parameter.
Now, letting Vi=init,iul wind, and Vf=final wind after
time interval T
Vf=Vi + frV’ X k (9)
where tshe k is the vertical unit vector.
3. GRAPHICAL SOLUTION
THE ACCELERATION TERM
dB‘
dt
evaluate the a.weleration t.elrm.
In figure 2, Vi, V,, and Vi’ represent the init,ial wind, the
geostrophic wind. and the init,ial geostrophic deviation.
Since the vector \I’ is constant in magnitude, and since
we may devise a simple graphical solut,ion t o _= -f ,
Then V’ rotates anticyclonically at the rate f. At the end
of a time increment T , a simple rotation with an angle
8 =f T gives the final geostrophic deviation Vf’. Then a
vector from 0 to the end point P of Vf’ gives the final
wind Vf.
I n t,he application of t,he relations shown in figure 2 to
find the trajectory of a parcel of air during the next hour,
t,he following steps are taken :
(I) Beginning at point 0 lay off on the weather map
a dist,ance OS in nautical miles equal to the observed wind
Vi in knots.
(2) Measure the average geostrophic wind V, along
the path OS.
(3 ) Lay off OR in nautical miles equal to V, in knots.
(4) With pivot of compass on terminal point R strike
an arc SP of angular magnitude f. clockwise from ter-
minal point S.
(5) The distance OP in nautical miles equals the final
velocity Vf in knots at the end of one hour.
(6) The trajectory during this first hour is along a vector
which is the mean between Vi and VI. A simple means
of obtaining this average will be treated later.
The striking of the arc in step (4) explains use of the
term “arc-strike” as a name for the method.
NOMOGRAM TO AID IN DRAWING TRAJECTORY
Trajectories could be computed as described above by
aid of commsses and a straipht-edpe. but since this in-
190 M O N T H L Y W E A T H E MAY 1969
volves measuring distances and angles either on the ana-
lyzed chart or on a working chart it is a rather slow process.
The work can be speeded up by the use of nomograms.
In figure 2 it is obvious t,hat because the angle of strike
is fixed for any given latitude and time increment and
because IVi'l=IV;I point P will lie along a line a through
point S whose angle fl with line m is dependent only on the
angle of strike, 0. Specifically, since triangle RSP is
isosceles
To find the position of P along line a, draw a line n
through P parallel to m and construct a perpendicular
line k from R to line n. Draw line SN. Line SN helps
to locate point P because it can be demonstrated that its
angle y is also dependent only on the angle of strike;
specifically an examination of figure 2 shows
-y=arctan (sin e) (1 1)
It ispossible then to find point P by knowing merely
the positions of the end-points of Vi and V,, and the lati-
tude and tim.e interval.
The next item to consider is the average velocit'y during
the time interval involved in drawing the trajectory. The
velocity with which the parcel is carried forward should he
an integrated average V, from the initial velocit'y Vi to
the final velocity V,.
If VI is not too different from Vi,
1 Va % 3 (Vi + Vr> (12)
If V and V, are very different, this approximation
becomes less accurate and then it is necessary to approach
this from a more rigorous standpoint. Wobus [4] does
this averaging by means of a nomogram called a Traject'ory
Computer. Experience dictates that the shorter the tin:e
interval used the more accurate the trajectory willbe
defined-this is pa.rticula.rly so where the contour or
pressure fields are changing rather rapidly in time or space.
We must look for a system to construct our nomograxs
rapidly for any time interval and latitude desired.
In figure 3 is reconstructed the portion of figure 2 neces-
sary to aid in finding the average velocity V,. S and R,
which are the terminal points of Vi and V,, define V' a t
time t=O. At some later time t , V' has rotat'ed through
angle 0 and point P is then t'he terminal point of V.
It may be seen from figure 3 (see also equations (5 ) and
(6), noting O=a-O') thst
TL*=v;- vi COS e (13)
v*=V: sin e (14)
where U* and v* are components of the velocity vector
V* from S to P.
From equation (8) and the relation B =r -O ' , it follows
that O=ft; thus equations (13) and (14) can be integrated
y I
FIGITRE 3,--Geo1netric determination of terminal point T of tra-
jectory from vector relatio8nship.s shown in figure 2.
for the time interval 0 to 7 , giving
Since Vr-Vz=V*, equations (15) and (16) give the
coordinates of t'he parcel relative to point S after a time
interval 7. If SA is the line through S and the point
(xI, yr), making the angle (Y with line m, then from figure
3 and (15) and (16):
Sow since the average velocity v, for the particular
arc-strike, 0, has the components X r /T and '?Jr/T, it follows
from (17) that the end point of the vector V, lies along a
line which makes angle a with line "1, and a is dependent
only on the angle of strike; specifically, from (17):
Similarly, by integration of equations (5 ) and (6)) it
can be shown that X f /7 and Yr/T lieon a line through
point R that makes angle 012 with line m. Thus in figure
M-it-h equations (IO), (11), and (18) it is possibleto
ronstruct quickly a nomogram to compute trajectories at
any latitude and for any time period. Example: to con-
struct a computing nomogram for the 30°N. latitude and
for intervals of 2 hours. The angle of strike e=ft=4Q sin C$
=30° where L?- 24 hr. 3600 2 the angular speed of the earth's ro-
s , rc.=e/2.
-
~
t,ation. From equation (IO), p=-- =75O. 1800-e - 180°-300
2 2
From equation (l l ), y=arctan sin 30°=26.6'
MAY 1959 MONTHLY WEATHER REVIEW 191
R
FIGURE 4.--Non1ographic determination of terminal point T of trajectory. The lines emanating from point S at angles a, j% and Y
are fixed for constant latitude and time interval.
From equation (18), cr=arctan l--cos e the origin S a s i n f i g u r e 4. For convenience thetrajectory
+sin 0 is best drawn on transparent paper. Lay this over the
1 -cos 30° = arctan =NO. map and mark reference points so thvtt the same relative
--sin 30' position of paper and chart may be easily found. Mark 6 the point where the trajectory is to be started. (The
On a piece of graph paper of suitable size draw three initial wind here should be known or closely approxi-
lines with angles of 75", 26.6O, and 80° all emanating from mated.) From this point lay off a line twice the vector
7r
192 MONTHLY MAY 1959
I
I
I
I
I I I
1005
I
I
I
I
I
I
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IO08 I
FIGURE 5.-Example of construction of trajectory for model pres-
sure field with observed wind VC, mean geostrophic wind V,, angle
of arc-strike 0 , and final wind V,. All wind vectors are drawn
to lengths corresponding to 3-hr. displacements so that the %hr.
trajectory is given by the mean wind vector V..
.FIGUBE 6 .4 v e r l a y for rapidly constructing trajectories at latitude
35" for a polar etereograpMc map base of scale 1 : 10'.
Vi, since the time interval is two hours.
As a first approximation assume this is the actual path
taken by the parcel and m0asure the average geostrophic
wind along this path. References [3], [4], and [5] have
some good discussions on several methods of ascertaining
the average geostrophic wind and the relative merits of
t,hese methods.
From the initial point of Vi lay off twice V,. Nowlay
the transparent work-sheet on the computer so that the
terminal end, S, of Vi falls on the origin and the terminal
end, R, of V, falls along the abscissa. This is shown in
figure 4. From R follow the ordinates upward to line 7,
then along t,he abscissa to line ,B and mark point P. A line
from 0 to P is the final velocity Vf at the end of 2 hours.
With a straight-edge along R and P mark point A on line
a and mark a point T halfway from A to S.
The line OT now represents t.he average velocity V,
during the first 2 hours, and since it is aut.omatically
doubled it is also the trajectory of the parcel for the first
2 hours. Vf now becomes the initial velocity for the next
2-hour period and the process is repeated.
I f the pressure or height gradients are not nearly uni-
form along and near the path along which the geostrophic
average was measured, second and perhaps (rarely) third
approximations should be made.
If a second approximation is necessary in the example
of the computation of the trajectory for the first 2 hours,
place the work-sheet back on the chart and measure the
geostrophic wind average along V,. Lay off the ne,w aver-
age Vg and re-compute Vf and V,. Do not change Vi,
however.
RAPID APPROXIMATE CONSTRUCTION OF TRAJECTORIES'
A method for t,he rapid construction of frictionless t,ra-
jectories, based upon equation (12) is presented below.
Although the method applies to any latitude or map base,
an example is given for 35' N. latitude and a map base of
1 :lo7, polar stereographic projection. Figure 5 is a model
pressure pattern showing isobars at a 1-mb. interval. A
trajectory is to be constructed mat 3-hour intervals begin-
ning at point 0. I n order to remove partly the effects of
surface friction, the so-cdled "gradient" wind is used 8s
the initial wind a t p i n t 0. A sheet of transparent paper
is placed on top of the map and this wind is projected for
3 hours, or in this case 90 nautical miles, downstrea,m to
form the vector Vi.
Next, the mean geostrophic wind direction for the
%hour period is determined at the midpoint of Vi (point
">. This same direction is laid off from 0 to form the Vi
2
line g. The 3-hour mean geostrophic wind speed is meas-
ured using a transparent overlay illustrated in figure 6.
The upper part of the overlay gives the distance moved by
a parcel moving with the mean geostrophic wind. The
scale on the left is in nautical miles. The straight and
curved lines are spaced at intervals of20 nautical miles.
*he method was developed by Mr. K. Peterson.
MAY 1959 M O N T H L Y W E A T H E R 193
In order to obtain the %hour mean geolstrophic wind
speed, place the scale on the map so that the straight lines
on the overlay are parallel to the line g and so that point
-lies on the base line. The pain-should be approxi- Vi Vi
2 2
mately mid-way between the poililt where an i s o h r touches
the edge of the overlay at zero and the point on the base
line 2 mb. to the right. Holding R pencil over the point
which is 2 mb. to the right on the bsaseline, move the scale
so that this point is coincident with point 0 and the
straight lines on the scale are pa.ralle1 to line g . Place a
dot at the edge of the scale on line g . This determines the
ector Vg, the distance a parcel starting at point 0 will
travel in 3 hours if the wind is geostrophic. I f a 6, 2, or
ls-hour travel distance is desired, use the overlay 'to
measure across 1, 3, or 4 mb., respectively. Such an o'ver-
lay can be prepared for any latitude, map base, map pro-
jection, and isobar interval with the aid of a standard
geost,rophic wind scale and a distance scale. If overlays
are prepared for every 5 O latitude, the error at Mitudes
between overlays is negligible.
Place the bottom point, of t,he overlay at point V, and let
point Vi touch the left edge of the scale. TJsing the arcs
as a guide, go to the right (anticyclonically) and place a
point at the right e'dge of the scale, locat,ing point Vf. The
vect,or V f represents the actual wind after 3 hours. Bisect
the chord from Vi to VI to obtain the point V,. (It can be
shown t,hat V, should be between the midpoint of the arc
and the chord, but t.he error is negligible). The vector V,
is the mean actual wind given by equat,ion (12) for t.he
time period; it represents the trajectory for the first 3
hours.
The angle at the bottom of the orerlay is obtained from
the relation t9=fT; here f~=30"(sin +)T where 4 is the
latitude and T is the arc-strike interval. At +=3 5 O and
with a %hour arc-strike,
e=30° (0.374) (:%) =r,l.gO
The above equat.ion can be used to obtain arc-strike angles
for any latitude and arc-strike interval.
In order to obtain the next 3-hour trajectory, transfer
the transparent paper to the next map, in this case, 3 hours
later. Take the ve.ctor VI and translate it so that the
initial point 0 is coincident wit11 the point V,. This new
vector is used as the actual wind for the next 3 hours. The
remainder of the technique is the same a,s before-draw
t'he mean geostrophic wind direction, use the overlay to
determine the mean geostrophic movement, and strike the
arc to obtain the new 3-hour final wind and mean wind.
This procedure can be repeated indefinitely, b'ut since the
trajectories are frictionless, and since their accuracy is de-
pendent on the accuracy with which the geostrophic wind
field, both present and future, can be delineated, they must
be used with care when projected for much more than 12
hours. While the large-scale features might be forecast
with sufficient accuracy, the fme-grain structure might not,
and many result dependent on the latter will suffer.
4. FRICTION EFFECT
EFFECT O F FRICTIONAL ACCELERATION ON TRAJECTORIES
Without friction, once a trajectory is started it may be
carried on step by step, chain fashion, for as long a period
as the contour field is known or can be approximated.
However, experience has shown that this tends to give
unrealistic results as inertial oscillations dominate the
motion.
The 'introduction of some damping effect will give more
reasonable trajectories but will not, of course, eliminate all
the difficulties. The major effect of the viscous or fric-
tional forces is to slow down the parcel speeds, and this
neglect leads to systematic errors. For this reason, con-
siderable effort has been directed to devising a method of
applying frictional corrections. The simplest hypothesis
regarding the frictional forces is that they are directed
against, the wind in direction and are proportional to the
first or some higher power of the speed. It is suggested
that a closer approximation to reality can be obtained if
a correction is made using such a relationship with the
const,ants of proportionality determined statistically from
a comparison between trajectories computed by dynamic
and kinematic methods for a large number of observations..
The suggested procedures are described here ; evaluation
of the empirical constants will be reported a t a later date.
Doubtless, the more modern theories of the frictional
forces c.an be incorporated by more elaborate procedures.
THEORY
I f the acceleration due to friction of an unspecified
nature is defined such that it is proportional to a constant
power of the speed, the equations of motion may be
written.
"_ du fd-kv" dt-
These equations can be solved graphically if n is known.
probably n is a function of the speed itself, but for sim-
plicity a value of 2 was assumed.
The frictional effect, as defined above, was added at the
end of each time interval to the computations of the
trajectory by the methods already described. This in-
volves finding the time-average of the friction term kV2
during the interval of time T in which V f changes to
Ti,, where TjF is the h a 1 speed after the friction effect has
been added.
whereis average frictional acceleration, V, is speed at
time 7 . The average acceleration a during the interval T is
194 MONTHLY WEATHER REVIEW &lAY 1959
FRICTION N O M O G R A M
k = ,003 T = 1 hour
Putting V= V,+Z t and using equation (20) to eliminate a,
we can eva.luate the integral in the middle member of (1 9).
This g.ives
Putting (21) into (19) gives
GRAPHICAL SOLUTION OF EQUATIONS
Equation (2 2 ) c.anbe solved graphically. See figure 7.
I n equation (22) k~ occurs outside the brackets in the
last term. This suggests that once a system of isolines
of friction deceleration values is calculated this same chart
can beused for other values of k and T by the simple
expedient of multiplying the values ofhhe isolines by an
appropriate factor. This expedites the testing of different
values of IC in different situations, either actual or models.
To use the frict.ion nomogram, figure 7 , enter the abscissa
with the value of the initial wind speed Vi and the ordi-
nata with the value of the final wind speed VI obtained
by the arc-strike technique. At t.he point thus obtained
interpolate between curved isolines. This interpolated
value is subtracted from Vf t.o yield VF, the final speed
after reduction due to friction.
APPLICATION TO TRAJECTORIES
As previously mentioned, the friction effect is added
stepwise a t the end of each trajectory computation. This
can be considered only an approximation. Nevertheless,
if the method proposed is to be useful, it must be able to
predict certain singularities in the field of motion. For
example, any mind, initially not in balance, fin a straight
uniform pressure gradient field will eventually approach
a value such that the accelerations on any parcel of air
total zero. At this point there is a balance among the
forces due to the Coriolis effect, the pressure gradient, and
friction.
Furthermore, if the wind is initially considerably out
of balance the time required for the wind to adjust to
within 5 percent of the theoretical balance velocity is of
the order of 10 hours under the condit.ions assumed. As
an actual example, consider a wind with initial compo-
nents u=20 kt. and v=35 kt.; the geostrophic wind is a
constant at vg=40 kt. and u,=O. For k=0.003 mi.-l and
at 30" N. latitude, the wind will be in balance when
9=34 kt. and u=15 kt. It takes 26 hours, computed by
t,he methods suggested here, for t,he wind to adjust to
within * 5 percent of the balance values.
Experiments were made around a hurricane using dif-
ferent values of IC. A value of 0.020 gave trajectories that
were quite unreasonable in that they headed more directly
across the isobars than is observed. With a value o'f
0.0003 the winds acquired speeds that were much higher
than observed and overshot the isobars leading to con-
siderable motion toward higher pressure. A value of
0.003 gave trajectory winds more nearly like those
observed.
RESULT O F FRICTIONAL ACCELERATION
The fact that adding friction always accelerates the
air parcel toward a certain balance value leads to an
interesting corollary. This 'is that if several winds of
different values (both in speed and direction) are all
subjected to the same geostrophic field and this fieldis
changing with time the different winds will approach the
same value because of frictional acceleration. Thris is
not necessarily the value where balance requirements are
satisfied since the geostrophic wind itself may be changing.
The results of a test of this corollary are shown graph-
ically in figure 8. The four initial winds, A, B, C, and D,
are shown at the bottom left of the figure. After 6 hours
in a geostrophic wind field which appears as at upper
cent,er, t.heir values became as at the lower center, and
after the next 6 hours with geost.rophic wind as a t upper
right the final values are as at lover right,. Note that at
the end of each of the two &hour ,intervals the differences
among the several winds are progressively reduced.
Trajectories in several hurricanes were computed by this
mathod, using actual data for pressure fields. The results
are encoura.ging. The useof k= 0.003 mi.-l yielded tra-
jectorie's t,hat in most respects .are similar to 'traject.ories
drawn from actua.1 wind d a h where available. These
MAY 1959 MONTHLY WESTHER REVIEW
SCALE
20 MI. -
GEOSTROPHIC WIND
DURING SECOND 6 HOURS
INITIAL WINDS WINDS AFTER 6 HOURS WINDS AFTER 12 HOURS
C
195
FIGURE 8.-Illustration of tendency of different initial winds subjected to the same changing Pressure field to approach the same
value because of frictional acceleration.
trajectories together with evaluations of k itself will be
discussed at greaker length in a subsequent article.
ACKNOWLEDGMENTS
The author wishes to express his gratitude to Dr.
Charles S. Gilman, at whomse suggestion this study was
made. His many helpful criticisms were of invaluable
aid. Thanks are also due to Messrs. Kmdall R. Peterson,
Sydney Molansky, Robert W. Schoner, Calvin W. Coch-
rane, Earnest A. Rodney, and Vance A. Myers for their
thought-provoking discussions.
REFERENCES
1. L. Machta, The Effect of Varying Pressure Gradient on S i r
Trajectories, MS Thesis, New Pork Univ., 1946 (Unpublished).
2. K. Takahashi et al., “Analysis of Extraordinarily Heavy Rains
a t the End of Bai-u,” Journal of Meteorological Society of
Japalz, Ser. 2, vol. 32, No. 9/10, Sept./Oct. 1954, pp. 281-289.
3. G.A. Franceschini and J. C. Freeman, “Computation of Hori-
zontal Trajectories in the Atmosphere.” Texas A and Ri
Research Foundation, Suientific Report No. 1 on Contract AF
19 (604)-1302, Jan. 1955,18 pp.
4. H. B. Wobus, “Wind and Pressure Gradient Computation with
Application to Trajectory Computations,” Proceedings of the
Workshop on Stratospheric Alzalysis ami? Forecasting, District
of Columbia Branch, American Meteorological Society, 1957,
5. 11.8. Air Force, Air Weather Service, “Constrant-Pressure Tra-
jectories,” AWSM 105-47, (AFR 1!30-16), Sept. 1956, 86 pp.
(see pp. S 3 8 ).
6. W. E. Hubert, “Hurricane Trajectory Forecasts From a Non-
Divergent, Non-Geostrophic, Barotropic Model,” Monthly
Weather Review, vol. 8 5 , No. 3, March 1957, pp. 83-87.
7. B. Haurwitz, Dynamic Meteorology, McGraw-Hill Book Go., Inc., .
New Pork and London, 1st ed., 1941,366 pp. (see pp. 155-156). ’-
pp. 129-143.