accelerations in hurricanes - atlantic oceanographic … in hurricanes by william m. gray colorado...
TRANSCRIPT
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REPORT NO. 54
On the Balance of Forces and Radial
Accelerations in Hurricanes
U. S. DEPARTMENT OF COMMERCELuther H. Hodges, Secretary
WEATHER BUREAUF. W. Reichelderfer, Chief
NATIONAL HURRICANE RESEARCH PROJECT
REPORT NO. 54*
On the Balance of Forces and Radial
Accelerations in Hurricanes
by
William M. Gray
Colorado State University, Fort Collins, Colo.
A report on research conducted under contracts between the U. S. Weather Bureau,Colorado State University, and The University of Chicago.
Washington, D. C.May 1962
NATIONAL HURRICANE RESEARCH PROJECT REPORTS
Reports by Weather Bureau units, contractors, and cooperators working on the hurricane problem are preprintedthis series to facilitate immediate distribution of the information among the workers and other interested units. .this limited reproduction and distribution in this form do not constitute formal scientific publication, reference to apaper in the series should identify it as a preprinted report.No. 1. Objectives and basic design of the NHRP. March 1956.No. 2. Numerical weather prediction of hurricane motion. July 1956.
Supplement: Error analysis of prognostic 500-mb. maps made for numerical weather prediction of hurricanemotion. March 1957.
No. 3. Rainfall associated with hurricanes. July 1956.No k Some problems involved in the Btudy of storm surges. December 195&. ,* «.,So! 5. SurveJ^f Serological factors pertinent to reduction of loss of life and property in hurricane situations.
March 1957-
IV 1 An^e^ftir^^Ude1^8^^ rfr ^Atlantic and Gulf coasts of the United States. May 1957.So! I'. £rt1? Hurricanel^nd toe sea fufface temperature field. Part II. The exchange of energy between the sea
and the atmosphere in relation to hurricane behavior. June 1957. ,.,.„,. ,. _, „«. „„,,, „+< „„No. 9. Seasowl^ariations in the frequency of North Atlantic tropical cyclones related to the general circulation.July 1957.
No. 10. Estimating central pressure of tropical cyclones from aircraft data. August 1957-No. 11. Instrumentation of National Hurricane Research Project aircraft. August 1957.No. 12. Studies of hurricane spiral bands as observed on radar. September 1957.No! 13. Mean soundings for the hurricane eye. September 1957.No. Ik. On the maximum intensity of hurricanes. December 1957.No. 15. The three-dimensional wind structure around a tropical cyclone. January 1950.No! 16. Modification of hurricanes through cloud seeding. May 1958.No. 17. Analysis of tropical storm Frieda 1957. A preliminary report. June 1958.No '18. The use of mean layer winds as a hurricane steering mechanism. June 1950.No 19 Furtbe? examination^ the balance of angular momentum in the mature hurricane. July 1958.Nh 20 Onthe energetics of the mature hurricane and other rotating wind systems. July 1958.So! Si SrmatioToftropical rtorms related to anomalies of the long-period mean circulation. September 1958.No. 22. On production of kinetic energy from condensation heating. October 195».No! 23. Hurricane Audrey storm tide. October 1958. „No 2k Details of circulation in the high energy core of hurricane Carrie. November 195B.No. 25. Distribution of surface friction in hurricanes. November 1958. m,*™™ 1Q«MZ2o! Anote on the origin of hurricane radar spiral bands and the echoes which form them. February 1959-No. 27. Proceedings of the Board of Review and Conference on Research Progress. March 1959-Mo •?8 A model hurricane pi"" for a coastal community. March 1959» ».»—n mwSo! g! ExSange oTh^S! moisture, and momentum between hurricane Ella (1958) and its environment. April 1959.No. 30. Mean soundings for the Gulf of Mexico area. April 1959.No. 31. On the dynamics and energy transformations in steady-state hurricanes. August 1959-No *2 An interim hurricane storm surge forecasting guide. August 1959' . i*,**.^'S°! If. MttetrtS^icarcontiderations pertinent to standard project hurricane, Atlantic and Gulf coasts of the United
States. November 1959. , „ v ,n„No 3U. Filling and intensity changes in hurricanes over land. November 1959. „„„„.. ,QV.So! 35! SlnV^ressure fields inthe stratosphere over the West Indies region in August 1958. December 1959.No. 36. Climatdlogical aspects of intensity of typhoons. February I960. ... ,oft)No 37. Unrest l/the upper stratosphere over the Caribbean Sea during January i960. April i960.No. 38. On quantitative precipitation forecasting. August I960. ^„+om>,OT. io*nNo 39. Surface winds near the center of hurricanes (and other cyclones). September I960. a***** IQ60SS Ub oTiStiSion of tropical depressions and convection in aconditionally unstable atmosphere. October i960.So! 5l On tte SS balance of the troposphere and water body of the Caribbean Sea. December i960.No. U2. Climatology of 2U-hour North Atlantic tropical cyclone HSrs^n STfIt Easi by statistical methods. May l96l.No k*> Prediction of movements and surface pressures of typhoon centers inthe tar tasi oy swiB**^ ™r*~" T» "So! It MarSfSanges in the characteristics of the eye of intense typhoons between the deepening and filling states.No. U5. The occurrence of anomalous winds and their significance. June 1961.No. U6. Some aspects of hurricane Daisy, 1958. July 19°1-No. U7. Concerning the mechanics and thermodynamics of the inflow layer of the mature hurricane. September 1961.No. US. On the structure of hurricane Daisy (1958). October 1961.No. 1»9. Some properties of hurricane wind fields as deduced from trajectories. November 1961. ~.No. 50. Proceedings of the Second Technical Conference on Hurricanes, June 27-30, 196l, Miami Beach, Fla. torch 1962.No. 51. Concerning the general vertically averaged hydrodynamic equations with respect to basic storm equations.
April 1962. . ., ,.£_No. 52. Inventory, use,and availability of NHRP meteorological data gathered by aircraft. April 190Z.No! 53! On the momentum and energy balance of hurricane Helene (1958). April I962.
in
As
CONTENTS
Page
1. INTRODUCTION . 1
2. EQUATIONS OF MOTION AND COMPUTATIONAL PROCEDURE 7
3. RESULTS
Radial Accelerations Relative to a Fixed Storm Center, v ) - . . 16' r gr
Radial Accelerations Relative to the Moving Storm Center,
v ) 17rr gr •L*
Relation of the Fields of v ) to v and the maximum wind . . . 19r gr r
Comparison of the Asymmetries of gdD/dr and (fv~ + v2/r) ' •
Around the Storms . 21
Further Analysis of the v ) Field in Stationary Coordinates . 21r gr
Preliminary Comments on the Magnitude of F 2^
Radius of Trajectory Curvature 25
Condensation Heating Related to Radial Friction and
Deepening Tendency 26
Condensation Heating Related to the Wind and Pressure Field
on August 25 28
h. 3UMMARY OF PRINCIPAL RESULTS . , 67
5. CONCLUSION 67
ACKNOWLEDGMENTS 68
REFERENCES 69
APPENDIX
1. Reliability of the Computations 71
2. Equality of Frictional Acceleration in Fixed and Moving
Storm Center 71
ON THE BALANCE OF FORCES AND RADIAL
ACCELERATIONS IN HURRICANES
William M. GrayColorado State University, Fort Collins, Colo.
1. INTRODUCTION
Hurricanes have often been described as vortices in which the cyclo-strophic wind equation should be valid in the core of maximum winds and thegradient wind equation in the periphery. But the degree to which theseequations really describe the balance of forces has not been determined.This has been due to lack of reliable wind and pressure-hei^it information.With the installation of the National Hurricane Research Project of the UnitedStates Weather Bureau in 1956, and with technical development of the gyro-navigation system and other instruments, these observational inadequacies'have been reduced and to some extent overcome. The Doppler navigation systemhas given for the first time accurate wind data over ocean areas. Within thelimits of accuracy of observations, the question concerning the form of thelaws of motion applicable in hurricanes can now be answered in certain respects for several storms which were thoroughly covered by research missionsof the National Hurricane Research Project.
This paper presents a first attempt to take up some special aspects ofthe balance of forces computed from the fligit data. Computations are madefor Atlantic hurricanes Carrie (1957), Cleo (1958), Daisy (1958), and Helene(1958).
Instrumentation of Aircraft. - During the hurricane seasons of 1957-1958,the National Ifarricane Research Project operated three aircraft in conjunctionwith the U. S. Air Force. There were two B-50fs and one B-lj-7. The instrumentation installed and maintained in these planes has been described byHilleary and Christensen [h]. It will therefore suffice to note that the instrumentation of interest for this study consisted of a Doppler (gyro) Navigation wind measurement, radio and pressure altimeter for determination ofthe altimeter correction ("D"), and a vortex thermometer for temperaturemeasurements. Punch card systems were installed in the back of the aircraft.Instantaneous values of the above parameters were punched at specified timeintervals - one card every two to ten seconds - with the more rapid punchingin the hurricane cores. The punch card decks were taken to National HurricaneResearch Project after each mission and evaluated by machine processing. Notewas taken of the hurricane displacement over the duration of the mission andall quantities were plotted in a latitude-longitude coordinate system fixedwith respect to the storm center.
In this study, use is made of the horizontal winds and the observed "D"values. Temperatures are used only for checking on the computed verticalthicknesses, obtained from the "D" values between isobaric surfaces.
Research Missions in Hurricane Daisy (1958). - Hurricane Daisy began toform on August 2k, just to the northeast of the Bahama Islands (fig. 1). By1200 GMP on August 25, the maximum winds in the core were 70 kt. It was atthis time that National Hurricane Research Project aircraft first enteredDaisy. One Air Force B-50 made radial penetrations into the eye at 330 mb.;additional legs were executed at 1600 and 2800 ft. The second B-50 operatedin the middle troposphere at 570 mb. The B-k'J aircraft carried out a missionin and around the storm at 237 mb.
Two missions were flown into Daisy on the 26th, but without radial penetration. On August 27 and 28, radial penetrations were made at 630 and 620mb. respectively. Daisy reached its peak intensity on the 27th, with maximumwinds above 115 kt. A plot of central pressure with time is given in figure2.
Mission into Hurricane Cleo (1958). - Cleo formed in an easterly wavewest of the Canary Islands on August 9. It moved steadily westward for sixdays to approximately 15°N., U8°W., and then turned northward moving steadilyin this direction until it took on extratropical characteristics southeast ofNewfoundland on the 19th. It reached maximum intensity on the iHh, just before turning northward. The maximum winds were approximately 120 kt. andminimum pressure 9^5 mb. A three-plane NHRP mission was flown into the stormon August 18 when Cleo was moving in an approximately steady state toward thenorth-northeast at 1*4- kt. Maximum winds were approximately 95 kt. and minimum pressure 970 mb. The storm eye was nearly 25 miles in diameter and theeye wall clouds quite extensive. One mission was accomplished at 820 mb., another at 580 mb., and a third at 255 mb. See the storm track in figure 3«
Missions into Hurricane Carrie (1957). - Carrie formed from an easterlywave near 20° N.., 35°W., on September 5. It moved steadily west and northwest. By September 15 it was located at 30°N., 58°W., moving to the northwest at 12 kt. in approximate steady state. Its central pressure was 965 mb.and maximum winds 85 kt. A mid-tropospheric probe (605 mb.) was made at thistime. Another mid-tropospheric.flight into Carrie (690 mb.) was made onSeptember 17 when the storm had changed direction and was moving toward theeast-northeast at 13 kt. Minimum pressure and maximum winds were nearly thesame as on the 15th. See the storm track in figure h.
Missions in Hurricane Helene (1958). - This storm started to develop onSeptember 22 at 22UN., 65°W., or 200 n. mi. north-northeast of San Juan, P.R.It moved almost directly northwestward for the next four days, while slowlyintensifying (see storm track fig. 5)» A flight at 635 mb. was made onSeptember 2h when the central pressure was approximately 995 rob» and maximumwinds were* 60 kt. Another flight was made on September 25 at 810 mb. when thestorm's central pressure was approximately 980 mb. and maximum winds approximately 90 kt. Other flights into Helene were made by NHRP aircraft on September 26. Only the results of the two flights on the 2Uth and 25th are presentedin this paper.
Each day's data was collected in six to ten hours between the approximatehours of 1^00 and 2U00 GMP. For computational purposes, an essentially steadystate must be assumed for this period of flight, even though some changescould be noted when aircraft repeatedly passed the same point with respect tothe storm center.
O
Table 1 contains the storms, levels, dates, maximum winds, etc., forwhich computations were performed. The ten levels with the asterisk are thosewhere at least six radial legs were flown (except Cleo 820 mb. which had fiveradial legs). It is at these levels that field analysis of computed valuescould be made. A typical sample of a flight path along one of these levels isshown in figure 7*
Table 1. - levels to which radial acceleration computations were made.
Storm Date
Pressure
Level
(mb.)
Approx.Max. Wind at
Flight Level(kt.)
Approx.
Central
Pressure
(mb.)
Central
Pressure
Change
Storm
Direc
tion
Storm
Speed
(kt.)
Daisy 8-25-58 950 65 990 deepening 320 7
*Daisy 3-25-58 830 60 990. deepening . 520 7
*Daisy 8-25-58 570 60 990 deepening 320 7
Daisy 8-25-58 237 ^5 990 deepening 320 7
*Daisy 8-27-58 630 120 9U0 deepeningthen stead}
030r
8
*Daisy 8-23-58 620 105 9U5 filling
rapidly010 18
*Carrie 9-15-57 605 85 965 steady 310 12
*Carrie 9-17-57 690 95 970 steady 080 13
*Cleo 8-18-58 820 90 970 steady 015 Ik
*Cleo 8-18-58 580 85 970 steady 015 15
Cleo 8-18-58 255 50 970 steady 015 Ik
*Helene 9-2U-58 635 60 995 deepening 320 10
*Helene 9-25-58 810 80 980 deepening 310 6
Helene 9-26-58 265 75 950 deepening 300 11
* Used in 10-level composite.
** Composite l600 ft. and 2800 ft.
29/1
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Figure
2.-Centralpressure
vs.
date/time
for
hurricane
Daisy.
10
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Figure
7.-Example
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ON
2. EQUATIONS OF M)TION AND COMPUTATIONAL PROCEDURE
Aircraft observations of necessity are taken in a mixed space-timecoordinate system. But, as already stated, usually the complexity of thereference frame can be reduced by the assumption, usually valid in the firstapproximation, that the storm remains in essentially steady state for theduration of a flight mission. Granted this premise, all observations can beassigned a location in a coordinate system fixed with respect to the movingcenter. Given a plot of the data in this form, an investigation of the forcesand accelerations present may be provided from a variety of viewpoints. Threeof the most common approaches are the following:
(1) Computation of the actual forces with respect to the instantaneouscenter position along the radial and tangential axis of a polarcoordinate system.
(2) A similar computation in a moving reference frame where at firstthe translatory storm velocity is subtracted from all winds, sothat the wind field studied is that relative to the moving center.
(3) Computations in a natural coordinate system in which the forcesacting on individual air particles are determined.
The last system, a Lagrangian one, is in some respects the most attractive. But computations depend on knowledge of the radius of trajectory curvature, a quantity which is not measured but must be obtained by indirectmeans based on the field distribution of data. In such a procedure the shortcomings of the steady state assumption and of the frequent wide gaps betweenflight legs are essential.
Now in hurricanes, much interest reasonably centers on the radial accelerations since they are related to the mass flow through the system and havethe energy releases which produce and maintain these cyclones. Given theobjective of investigating these accelerations, the first two coordinatesystems are not subject to the difficulties encountered in the natural frame,so all requisite data are furnished by single radial flight legs of perhaps15 minutes duration and the whole job can be done by machine process if desired. In contrast, calculations along the tangential direction not only giveless promise of insight into the hurricane but they are also difficult toaccomplish since airplanes cannot fly circles around hurricanes and since gradients of wind and pressure usually are very weak along this axis compared tothe radial axis.
Since the course to be pursued will be to determine whatever is possibleabout radial accelerations, most calculations will be made with respect to thedisturbance's center position since it is felt that the greatest importanceresides in the radial coordinates. Later, however, the transformation of moving coordinates will be made as there is the advantage that local time changesare eliminated, and advective changes are then referred to the storm's centerrather than to a point fixed on earth. Finally, radii of trajectory curvaturewill, be determined from the polar coordinate grid at certain special points.This calculation will be made and the results compared with approximations ofthe radius of trajectory curvature from kinematic methods.
8
The polar coordinate system will be applied to constant pressure surfaceswhich then are treated as quasi-horizontal surfaces, where it may be notedthat even in the areas with strongest pressure gradient the slope of a constant pressure surface will not exceed the order of 1:100 (pressure gradientof 2 mb./mile). The radius r will be considered positive outward and thepolar angle 0 positive counterclockwise. Given the horizontal velocity components v and v along r and 0, the altimeter correction D, the acceleration
of gravity g, and the time t, the cylindrical equations of motion are
v2v =JL + f v - g dD/dr + F (l)r r 0 ' r
• - VQVr ^ dDv© " " r r " g rd9 ' "0- fv - g-^+F^ (2)
Here the dot denotes substantial time differentiation and Fr and Fg arethe components of the frictional force along r and Q, respectively. vr ispositive when directed outward along r. Only the first'of these two equationswill be treated in view of the preceding statement of objective.
Given v =0, F = 0, a type of gradient wind balance along r may be de-r r
fined to be given by2
-Z_ + fv
-i-BT-*--* (3)
The extent to which this equation is applicable will be determined.
Defining v ) =vr -Fr (k)gr
equation (l) may be written
2
J) = !i. +fv.g|2 (5)v
r rgr
'0 &dr
The first calculation will be concerned with the question to what extent
v ) may be neglected compared to the centrifugal, Coriolis, and pressuregr
gradient terms; thus to what extent the pressure gradient balances the centrifugal and Coriolis terms. Later, the imbalances between these forces (if present) will be apportioned to the amounts of advective and/or frictional acceleration occurring.
These calculations were performed for flight legs which were radial orwhich departed little from the radial direction. This restriction would notbe required for storms with tangential symmetry. Such symmetry, however, wasnot observed.
Computational Procedure - The computational procedure begins with the
plotting of v and of "D" values vs. radius on graph paper and the drawingof smooth curves of best fit through the data points (figs. 8 and 9)« Scattering of individual values was not great and representative curves were obtained in all cases for v ; however, there remained some leeway in drawing
the "D" profiles, especially close to the center.
Next, the curves of v and "D" were divided into finite radial intervalsstarting at 5 n.mi. from the center, going outward. Integrating from r^ to
f2r2 wliere r2 > rl> g J d D/dr dr = g(D - D ) where the total derivativerl
has been substituted for the partial derivative. This is permissible sinceonly a one-dimensional equation with r as independent variable is beingsolved.
Integration of the centrifugal acceleration is given by
2 v^/rdr =v In iJT"[ >v^ere v is the mean value of v. between
ri2
In r and In r . This quantity is obtained by plotting vQ vs. In r and taking mean values between chosen intervals of In r. Figure 10 illustrates the
2method for graphical determination of v between the 10 and 15 n.mi. radii.
rr2The Coriolis acceleration is obtained by solving J fVQdr =^g (r2 "
rl
r ), where v~ is the mean of v between r and r and f is considered con-19 y & j.
stant. Graphical determination of v~ is illustrated in figure 8.
The value of f at the location of the storm center was used for each computation period. The greatest latitudinal extent of the computations was only2.50°. This is equivalent to a maximum variation of f at 25° of only 9 per*cent. Since the magnitude of the Coriolis acceleration was usually much lessthan that of the centrifugal term - especially at high wind speeds near the
center where fv was less than 5 percent of vQ /r -the error in assuming fconstant over the whole storm area is negligible. Therefore it was permissible to use a single value for each flight.
10
We may define
g(V D^ =vQ2 in ^+f̂ (p2 -ri) (6)where the subscript "gr" denotes the D-difference which the airplane shouldhave observed with the wind field present (i.e. - in the case of v ) = 0),
r gr 'This result may be compared with the observed or actual D-value gradient denoted by the subscript "act". We shall define
g(Da-nl)' =e(°2 -Dl)gr -g(d2 -Vact (7)
where the prime denotes the "unbalanced" portion of the D-difference.Then
r2
ri
V ) dr = + g (D„- D,)r'gr B v 2 1'
or (8)
— (yyV ) = g ±r'gr r2 - rx
The computations corresponding to the radial traverse along the easternleg of Daisy on August 25 at 830 mb. are shown in Table 2. Increments of r
- r-were taken as 5 or 10 n.mi. from 5 to 20 n. mi. radius, and as 10 n. mi.farther out. The largest radius to which computations could be carried was60 to 80 n.mi. in most cases. These distances are quite sufficient for adescription of the acceleration field in the outer storm areas. Inside ofthe 10 n.mi. radius computation could not usuallybe made due to uncertaintiesof center location. Results in the interval from 10 to 15 miles are, attimes, questionable for the same reason.
Along the flight leg illustrated in Table 2, the gradient along r of Dx gr
exceeded that of D along r, so that v ) was positive and directed out-acx r gr
ward. Profiles of both D and D ^ are shown in figure 9 for the wholegr act
traverse from west to east side of Daisy on August 25. Note that while theslope of the D profile was greater than that of D ^ in the east, the re-
gr act 'verse was true in the west. The accumulated differences are substantial -
about one-third of the total D-drop. Up to.this point, the procedure wasnearly rigorous. Finally a more qualitative step was taken.
Computed values of v ) for each radial interval did not always yieldr gr "
smooth radial curves. This was caused partly by the integration intervals -i.e., only 5 to 10 n. mi. - but primarily it was due to the limit of accuracyto which the gradient of "D" could be measured. Whereas the winds are integrated over the radial interval of r.. to r , the D2 - D, values are obtained
Table
2.-Computationsofv)
alongeasternradial
legofDaisyAugust
25,
830
mb.
rgr
——
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.i
D-values
are
giveninfeet
asintheoriginal
data.
Accelerations
areexpressedinkt./hr.
They
have
notbeen
convertedtoc.g.s.
unitsbecause
itwas
felt
that
1kti/hr.
impartsafeelingforthephysical
magnitudes
involved
more
readilyto
most
readersthat
l.kx10"
cm.sec-."2,
its
c.g.s.
equivalent.
12
by reading off values at individual radii r^ and r. The original readings(by eye) of radio and pressure altimeters from films of the photo-panel1 cannot be considered to be more accurate than ± 10 feet, so that the observed"D" differences over small radial intervals cannot be regarded as exact, evenwhen read from curves of best fit. In order to overcome this difficulty itwas often necessary to increase the radial interval of integration to doubleor triple that chosen originally, especially at the inner radii, where triplingincreased the radial interval from 5 to 15 n. mi. This step, although possiblyeliminating microscale features, does not obscure hurricane-scale features.
Values of v ) were plotted on maps at the mid-point of each radial in-r gr
terval r - r , and isolines were drawn for selected values. A grid was then
placed over the analyzed v~") field in order to obtain averages around the
storm at different radii. This averaging yields mean radial profiles of v~) .r gr
It should be noted that interpolation by qualitative chart analysis was left
to the very end and that all results not depending on this step are free fromthe subjectivity of line drawing.
Henceforth in this paper the symbol v~) , i.e., the mean value of v )* r gr ' r gr
between the radial intervals rp - r_, will be denoted just by the symbol
v ) where the bar has been deleted. Only when averaging is performed aroundr gr
the storm at selected radii will the bar be included to denote circular aver
aging.
Nearly all computations performed and conclusions drawn in this paper arebased on data obtained from the ten levels listed in table 3- These levelswere the ten best covered by NHRP aircraft during the 1956 through 1959 seasons—where reliable wind and D-values were obtained.
The altimeters were not included in the automatic data punching system.
13
Table 3» - Data on which computations based.
Storm and date
Level
(mb.)
Max. WindDifference
right-lef^bquad, (kt.)
Speed of
storm x
2 (kt.)
Column kRight quad,
(kt.)Left quad,(kt.)
Column 5(kt.)
Daisy Aug 25 830 65 35 30 Ik . «* 2
Daisy Aug 25 570 60 U5 15 Ik «* 1
Daisy Aug 27 630 120 85 35 16 ^2
Daisy Aug 28 620 105 55 50 36 ~1.5
Oleo Aug 18 820 90 50 ko 28 «*1.5
Cleo Aug. 18 570 85 U5 ko 28 <**1*5
Carrie Aug 15 605 85 50 35 2k »1.5
Carrie Aug.17 690 95 65 30 26 ^1.25
Helene Sept 2k 635 60 55 5 20 & .25
Helene Sept 25 810 80 55 25 12 -A2
11*
60 40 20
s r (n. mi.)Center 20 40
r (n. mi.) i
60 80
Figure 8. - Profile of the tangential wind (v ) on a west-east traverse
through the center of Daisy. August 25, 83O mb.
40
«•— r(n.mi.)
Figure 9.* - West-east "D" value profile through the center of Daisy that goeswith the v profile in figure 8 . Curve (a), Actual "D" profile. Curve
(b), Computed "D" profile (i.e. D ).
l'/t 10
DAISY
25 AUG.
830 mb
East leg
15 20 30 40 50 60 70 80 90
in r(n.mi.) —*•.
900
800
700 a
600 Jo
500 *
400 e
300 >°
200
100
15
2 2 2Figure 10. - Comparison of v (m. /sec. ) profiles along the east and west
legs of the traverse through Daisy an the 25th at 830 mb. Values of v
are plotted against In r.6
llllllllirillll places of computorion
25 AUG.
B-47 FLIGHT TRUCK
237 mb
160
I
120I
80 40I I
r (n. mi.)
40 80I I
r ( n. mi.) -
120 160I I
Figure 11. - Track of B-U7 on the 25th at 237 mb. Cross-hatching on the tracklines indicate places where the v ) computation was performed.
16
3. RESULTS
.^.i ^eleratior- - ~t«™ to afired storm center, (v^).Kigores 12, 16, S»,28,3a,56,UO,^8 portray the calculated values
of v ) along tne radial legs at the levels with the asterisk in table 1.It?levels have large v^ values, which shows that the r^& S™ -tL equation as here definitive to the fixed storm center (equation 3)Z nl express the halance of forces ^s^^js^^S^^^<* cases, the areas of strongest winds were places of large pos^ v)and the areas of weakest winds places of large negative r,) . Areas p^ eT) were always found in the right grants* where the winds were ^eT \f and negative v ) values were found in the left quadrants whererr•- r—i*~——vrmr£storm,"these would he the proper places for the respective v^ acceler"^ asymmetric wind field related, at least in part, to the storms movement. JZ, this asymmetry in the wind field can seldom he accounted forby subtracting out the speed of the storm (table 3).
The order of magnitude of Vgr ~ ^ *^ f«f' ^fj^etions of the hurricane with range from zero to approxi-«7 ^ ^ **v^ «n q-horm levels was between 100 and 200 kt./ nr.
«vpraee absolute maximum for all storm xevej.*Tese LUudes are alarge fraction of the radial pressure gradient or ra-dial Coriolis and centrifugal accelerations.
is 1Q 25 27 51, 35, 59, *3, *7, 51 contain area distributions, Figures 15, 19, 25, ^«, ?x> ?J> <&> <" '
*4 T^ /(fv +v 2/r) for the ten levels with the asterisk inof the ratio ^W(^V£* would be s0 large afraction of the cen-tablel. It was unexpected that vr; wo
trifugal plus Coriolis accelerations. In many places, «^^*"western quadrants (Daisy August f^f^Carrie,September 17), this ^.^SSl^rtfugpl and Coriolis accelera-gradient force greatly exceeded the radial cenxruugtion.
. , .. r,o +rt on0 sector - when the direc-*Ihe right front ^drant is the g^**0^ f^fline on a560« com-^^SSSclockwise ™" s^ors'have corresponding relations.
17
Figures 52-55 portray mean radial profiles of v ). /(f v + v /r) for_ _ —-p *" 6* ^ 9
all ten levels. There v ) /(f v + v /r) was averaged around the storm at
30° intervals without respect to sign. There are rather large variations be
tween the individual curves, but the storm levels varied in height, and thefour storms differed in intensity, maturity, deepening tendency, etc. Thelarge magnitudes of the ratios are, however, evident at all levels except inDaisy, August 27, 630 mb.
Figure 56 contains a radial composite of v ) /(f v + y /rj for allten flights. There v ) averages between 50 and 60 percent of the radial
r gr ^centrifugal and Coriolis accelerations. A similar composite of the integrated.
radial profile of v ) /(gdD/dr /shows that v ) averages kO to 50 percent ofr gr V / r gr
the radial pressure gradient force. The difference between these two compo
sites is due to the fact that the radial pressure force averaged 10 to 15 per
cent more than the radial Coriolis and centrifugal accelerations (to be dis
cussed later). Figure 6l is a 10-level composite of the absolute values of
v ) , gdD/dr, and (f v + v /r) plotted against radius. An area compositer gr 9 9'
v ) , both with and without respect to sign is shown in figures 57 and 58.r gr'
The area composite with respect to sign clearly shows the asymmetry of v )r gr
between right and left quadrants of the storm. All area compositing was done
with respect to the direction of motion of the storms.
Radial, accelerations - relative to the moving storm center (v ) ).
The above results have shown the radial accelerations relative to the in
stantaneous fixed point of the storm's center. Accelerations relative to themoving storm center will now be considered.
By assuming that the storm moves at constant speed and direction, one canobtain these radial accelerations within the limits of accuracy of the data bysubtracting the tangential component of the storm motion (c ) from total tan-
3gential wind component in equation 5« Other terms which arise from thischange of reference to the moving storm center are less than 5 kt./hr. and canbe disregarded. This is over an order of magnitude lower than most of the computed v ) values. The acceleration of the storm displacement also enters into
rr'gr
5Rigorous verification of this has been accomplished from the absolute equa*-
tions of motion in both vector and Cartesian form.
18
the calculations, but at the times of the calculations the storms were movingat nearly constant direction. Changes in the rate of displacement were alsosmall - less than 1 or 2 kt./hr. Thus the acceleration of the hurricane'smovement was neglected.
With these omissions, the equation for the radial acceleration relativeto the moving storm center - fcere denoted by v ) - is
rr gr
v ) = -g %r + f(v~ " O +rr'gr & dr v 9 9'
K -cj2
or 2 2
rr'gr V s dr 9 r / *• 9 r r /
(9)
where c is the tangential component of the storm's motion. It is definedpositive in the direction of the storm motion, c is thus positive in the
right quadrant and negative in the left quadrant. Substituting from equation5 we obtain
or
where
2.S 2v c crt "\v ) + (a. +_SLa.-s_)=V) ,rr'gr V 9 r r J r gr
v ) - v ) - M (10)rr gr r gr
'The term fcQ was always less than 5 kt./hr., hence had little effect onthe calculations. The absolute value of 2vQc /r was usually k to 6timesgreater than c /r. It is then the 2vQc /r term that largely determines thevalue of M. Thus approximately
2vrtc„v ) =V) --£-£rr gr r gr r
In general v ) <C v ) . In the right quadrants, where*v) is positive,rr gr r gr r gr #
v and c are also positive so that the corrective term subtracts from v ) .9 9 r gr
In the left quadrants, with vr) rnegative, vQ positive, and cQ negative,v ^ is also reduced. Since vA is greater in the rigit than in the leftr'gr w
19
quadrants, the effect of the correction is strongest in the right quadrants.In the front and rear of the hurricane the correction, of course, is not present as cQ vanishes.
" Values of M were computed and inserted in equation 10. The resultingv ) patterns differed from those of v ) as follows.rr gr r gr
(1) The values of v ) were generally smaller than those of v ) ,rr gr r gr
being approximately 60 to 80 percent of the v ) values.r gr
(2) "Depositions of many of the v ) centers were shifted, from those
of the v ) centers,r gr
(3) Sometimes M exceeded v ) in the right quadrants so that v- ) be-N r gr rr. gr
came negative. In consequence, the area with negative v ) isrr gr
very large compared with that of positive v ) on most of thelevels analyzed.
Ten-level area composites of v ) both with and without respect torr gr ~~~~~
sign are shown in figures 59 and 60. Again compositing has been done withrespect to the direction of motion of each storm. The composite withoutrespect to sign resembles the v ) pattern in the essential features: Maxi-
r gr
mum accelerations to the right and left of tjae direction of motion, respectively (fig. 60). This is an important result in that one might have expected thehurricanes to approach circular symmetry once the effect of center propagationwas removed.
Figure 62 shows a radial composite plot of v ) and v ) without res-rr gr r gr
pect to sign. Although some subjectivity is inherent in the averaging it isfelt that when averaging is extended over ten storm levels, the importantbroad-scale features will stand out. It follows that neither the asymmetrynor the radial acceleration field itself is removed by defining the computation with respect to the moving center. Since most flight levels were in themiddle troposphere, where the net mass inflow or outflow are small, one mighthave thought that v ) should vanish here, especially when integrated around
** rr grthe storm. Such, however, is not the case.
We shall now return to the v ) fields and study their characteristicsr gr
in relation to other storm features.
Relation of the fields of v ) to v and the maximum wind.r gr r
Analysis of the fields of the actual radial component of motion at allmiddle and lower tropospheric levels showed that in general there was a linedividing inward directed and outward directed flow around the hurricane; further that this line usually closely bisected the areas of largest inward directed v ) in the left quadrants and of largest outward directed vr)gr in
r gr &
e
20
the right quadrants (figs. 12, Ik; 16, 18; 20, 22; 2k, 26; 28, 30; 32, 3k; 36,38; kO, k2; kk, k6; kQ, 50).
A coordinate system fixed to the moving storm center (i.e., one givingrelative winds) would of course greatly reduce these winds and would notalways show a similar radial pattern because the storm motion would have beensubtracted from all the winds.
In the right and left front quadrants of the storm, v itself was pre
dominantly directed outward. Inward directed motion occurred mainly in the
rear quadrants. Thus in the right front quadrant v and v ) both pointedr r gr
outward; in the left rear quadrant they both pointed inward. In the right
rear quadrant, where the winds had inward radial components, this component
decreased and reversed to the downstream side in agreement with the v )r gr
field. The reverse was true for the left quadrant. This observation agrees
with findings of a study by Ausman [l] who, working with surface ship data of
many hurricanes, has shown that v is directed inward in the rear quadrants
and outward in the front quadrants at radii from 2° to 6° latitude from the
center. From the present findings this appears to be valid over the whole
hurricane area, not only at these outer radii, but also near the center.
It is an interesting fact that v is zero approximately at the location
of the maximum winds; there its sign changes from negative to positive looking
downstream, v ) is also positive and has maximum values very near thisr gr
change from negative to positive v .r
Qualitatively, this coincidence provides insight on the production of themaximum winds. In the region of weak winds in the left quadrants, air isaccelerated inward and acquires a large crossing angle toward lower pressure.After some time the wind speed increases sufficiently so that gradient' balancein the radial direction is established (v ) = 0). At this point, however,
r gr
the mass is still crossing toward lower pressure and gaining kinetic energy.
It then overshoots the radius of gradient equilibrium and continues toward the
center with v ) directed outward. The speed increases until the air movesr'gr
parallel with the contours of the isobaric surfaces. There the place of
strongest winds and also the maximum v ) is reached. As the air then turnsr gr
outward toward higher contours, both wind speed and v ) again decrease.r gr
21
Comparison of the asymmetries of g dD/dr and (f v + v /r) around the storms.9 9
The question may be asked whether the v ) field arises from asymmetriesr gr
of the pressure gradient force, or of the ( f v + v /r) distribution aroundww • •
the center. In order to answer this question, the fields of D . and D were7 act gr
drawn on maps. At selected radii, the averages of D . and of D were deter-' act gr
mined. From a grid (fig. 6) spaced evenly at 30° intervals, the mean deviation
from the average around the center was computed at each of the chosen radii.
4
Figures 63-72 portray the results. Curves (a) show the deviation of
D (i*e., the wind field), and curves (b) the deviation of D ^ (i.e., thegr act '
pressure field). Though the curves on the different days vary considerably,
certain features nevertheless stand out. At all levels, except Helene Septem
ber 2k, 635 mb., the mean deviation of D . never exceeded that of D . Hence,' ' act gr '
asymmetry of the wind field in nine of the ten cases was then greater than that
of the pressure field. This feature was especially prominent on three of theDaisy flights and in Cleo at 820 mb. This result does not obviously followfrom the common knowledge that winds are stronger to the right than to the leftof the direction of motion. One might have thought that the D field would
grvary concomitantly with D . A radial composite for all ten levels is seen in
ace
figure 73. The deviation of the pressure field varies little with radius,
while the D deviations increase greatly toward the storm center. In the coregr
of maximum winds the deviation of D „ exceeds that of D by a factor of three.gr act
Symmetry in the pressure field is produced as the heat released in ascendingtowers is swept around the center by the strong winds; and the v ) field is
related more closely to variations of the tangential wind around the storm thanto variations of the pressure gradient.
Further analysis of v ) field in stationary coordinates.
As the actual radial wind component is rarely observed to exceed 25 kt.,and as individual air parcels often blow through areas of v ) of 50 to 100
kt./hr. for 30 to 60 minutes, it follows qualitatively that v^ / vr)Kr> atleast over large portions of a hurricane. This interesting observation will
now be analyzed by an attempt to compute v directly. In Eulerian coordinates
22
dv dv dv dv
v = -r-^ + v -^ + v ^-1 + w <5-r- (12)r dt 9 rd9 r dr dz v '
Only the term (v ) cv /dr can be evaluated directly from the flight data; the
range of magnitude was 1 to 10 kt./ hr. after integration over 10 n.mi. radial
legs. Most values were less than 5 kt./hr. however. The vertical advectionterm cannot be computed at all, but from general knowledge of vertical motionand vertical wind shear fields the order of magnitude should not exceed thatof radial advection, again after integration over 10 n.mi. radial distances.The (w) dv /dz term was thus neglected. For computing the tangential advec
tion reliable v fields are available through analysis; the weak link isw
dv /rd9 which was obtained by tabulating the analysis of v prescribed in
this paper on the grid of figure 6. In spite of the weakness of the computa
tion, systematic features of the tangential advection were obtained and reliable features should be revealed by the ten cases. The calculated magnitudeof this term had the order of 10 kt./hr. with range from 0 to 50 kt./hr.
The quantity dv /dt was determined by moving the v field across the
fixed grid of figure 6 with the direction and speed of the storm motion. For
the fast moving storms changes were computed from one-half hour before to one-
half hour after the time when the v field center position was connected withr
the fixed grid center. For more slowly moving systems the time interval chosen
for d/dt was from one hour before to one hour after the tine when the centerposition corresponded with the fixed grid center.
Initial expectation was that v) =dv^dt +(vQ) dv^/rdO .+ (vj dv^/drwithin computational limits. This proved not to be correct by wide margins.The Daisy flight of August 27 at 630 mb. provides a typical illustration.Figure Ik contains the field of v , to be compared with figure 20. Only 1*0-60
percent of the v ) field are accounted for by v . Broadly speaking, the* r gr rorientation of the patterns and the center locations are in fair agreement.
Figure 75 contains the difference field. From the earlier definition
F = v - v ) (13)r r r gr
it must now be postulated that figure 75 represents the Fr field after subtraction of figure 20 from figure Ik in the sense of equation 13. Admittedly
this postulated force field is exceedingly strong (10~ to 3 x 10° dynes/unitmass). But in a strongly turbulent system such as the hurricane, the existence
of such strong frictional forces should not be rejected a priori. Further
comments on the magnitude of F will be offered shortly.
23
Figure 76 contains radial'profiles of v ) , F , and v averaged aroundr gr' r r
the storm at 10 n.mi. intervals for Daisy, August 27, 630 mb. We see that Er
and v are nearly equal and average over half the v ) values. That
IF +v f>| v) Iat all radii is due to the lack of exact superpositioningof v ) and v centers,
r gr r
Similar properties were obtained for the other hurricanes so that it was
considered permissible to form the average of all ten cases, figures 6l, 77,
78. Here also. F and v are nearly equal. F approaches 85 percent of v )r r r r gr
at radii from 30 to 60 n.mi., and 70 percent over the whole radial distance.Since v ) itself was a large fraction of g dD/dr, it follows that friction-
r gr
al and pressure gradient forces have roughly the same magnitude, at leastover substantial portions of a hurricane.
In one respect, Daisy, August 27, 630 mb. was not typical, namely that it
was the only level studied with net positive v ) values after integrationr gr
around the center. The balancing frictional force had to be directed inward.
Daisy on the 27th had reached maximum intensity, its winds were slightlystronger than the pressure gradient and v ) was directed outward. The
r gr
storm was just beginning to fill at this time. In eight of the other ninecases, the pressure gradient force exceeded the centrifugal plus Coriolis
acceleration -in the integral around the hurricanes. The net vr)Kr fieldconsequently was directed inward and F outward.
Figures 79 and 80 portray 10-level area composites with respect to storm
direction of F with and without respect to sign. Figure 79 illustrates ther
predominance of outward directed F especially well. Figures 8l and 82 por
tray 10-level area composites of v both with and without respect to sign. It
is clearly evident that the values of v are not of great enough magnitude to
account for the v ) accelerations even when individual storm level super-r gr
positioning of these two fields is eliminated by the 10-level average.
The above calculations have determined the F accelerations relative to
an instantaneous fixed point of the storm center. The frictional accelerationwith respect to the moving storm center (denoted F ) is identical with that
with respect to the instantaneous fixed center when the storm propagation isk
assumed constant.
See Appendix for clarification of this point.
2k
Preliminary comments on the magnitude of F .! — — J.
From the discussions of Osborne Reynolds [13] it appears possible to represent Fr by the Reynolds form of eddy stresses. In cylindrical coordinates,F then has four components,
d
"rd9"
where the wavy bar denotes area average and the prime represents deviations
from this mean. In this form the term d/dz v^*w^ would be the verticalgradient of the horizontally integrated product of the radial and vertical
wind eddieso
Evaluations of v^ and v' values and their tangential and horizontal gradients were made on a number of mid-tropospheric levels here studied from con
tinuous pitot tube recordings mounted on the aircraft. These pitots measuresmall changes in atmospheric pressure as the aircraft travels along the radialand tangential legs. These minute pressure changes are converted into theirequivalent eddy wind variations. From such information, the Reynolds stressescan be computed.
d v'v* drv'2 v'2From the preliminary computations —^rr-- . —J-L, and -^- turn out to be
rd9 rdr r
quite small and cannot account for F . An order of magnitude estimate of the/**-->-/ r
d/dz v' w' term, using determined values of v' from the pitot tubes and assum-5 r
ing Malkus' [10] and Riehl's calculations of vertical velocities in hurricane
cumulonimbus clouds indicates that this term may have the required magnitude.
Vertical gradients of v' w' were determined with a two-layer model, i.e., esti
mates were made for the mid-tropospheric levels and zero values were assumed
for the surface and the tropopau.se. This is a crude approximation, but it
encourages one to seek further experimental data. With improved sampling and
measuring techniques a reliable estimate of the importance of the d/dz v'w'
term can be achieved. New instrumentation and special flight planning with
NHRP aircraft is being contemplated for the I961 and' I962 seasons with this
purpose in mind.
5Calculations performed in University of Chicago meteorology laboratory classgiven by Prof. H. Riehl (i960).
25
Radius of trajectory curvature.
The gradient wind equation in natural coordinates on a constant pressuresurface is
v20 = -g oD/dn + fV +~. (x4)
where V is the wind speedn is distance measured perpendicular to V, andR is the radius of trajectory curvature.
As mentioned in the introduction, this equation is difficult to solve since Ris not observed directly; moreover, dD/dn also is not always obtainable. However, at places where v = 0, n is along r, dD/dn = dD/dr, and V = v . At
p 0these locations v /R + fvQ = g dD/dr, and
2
VR= dD ft (15)
The radius of trajectory curvature, as given by equation (15) can now be computed directly along the lines of v =0 and may be compared with radii deter
mined from kinematic methods. It is apparent that in view of the possibleimportance of frictional forces, deduced earlier, this comparison should notsucceed. R was evaluated with equation (15) and kinematically with Blaton'sformula for many places of the ten storm levels where v = 0 occurred along or
.close to a flight leg. As expected, R as given by equation (15) was consistently too small to the left and too large to the right of the direction ofmotion. If equation (15) is amended to include radial friction we have
2
^ +%=gd7-Fr <16>R
Addition of equation (16) with the earlier obtained equation, i. e. —
v fV = -g ^— + F ,r r 9 & dr r'
2 2v v
Q O »yields -2- .- -i- + v = 0 ,
R r r '
or R =—g (17)
I?--vr r
This formulation should yield correct values. Computed radii of trajectorycurvature with equation (15) often are as low as 7-8 miles to the left and aslarge as 35 miles to the right of the direction of motion at radii of 20 milesfrom the center. Comparable values obtained with equation (17) are 12 and27 miles respectively. These values obtained with equation (17) agree much
26
with the kinematically determined radii from Blaton's formula. Especially inthe left quadrants of the storm is this discrepancy between equations (15) and(17) large.
Another approach, showing the non-validity of the gradient wind assumptionin natural coordinates, is to assume dD/dn = dD/dr in the areas where v ) is
r gr pnegative and compute the values of R necessary for balance from equation V/R +fV = g dD/dr, or
R--^- (18)eo7" w
In practically all places where this calculation was performed values of Rnecessary for gradient wind balance in natural coordinates were smaller thanthe values of R determined kinematically by Blaton's formula or with equation
(17).
This method has the advantage that it can be applied at places wherev f 0. The assumption dD/dn = dD/dr where dD/dn always ^ dD/dr is entirely
permissible at places where v ) is negative. This assumption decreasesrather than increases the difference between the R obtained from equations(15) and (17). Thus substitution of dD/dr for dD/dn in equation (lU) increasesR and causes computed values of R under the gradient wind assumption in naturalcoordinates to compare more favorably with kinematic and equation (17) determinations. Even with this substitution the computed differences in R in thefrictional and non-frictional cases are appreciable.
It follows that the gradient or cyclostrophic assumption, usually made inhurricanes, can be in error by as much as 50 to 100 percent. One cannot hopeto obtain the proper constraints in hurricane growth and maintenance with thecyclostrophic approach.
Condensation heating related to radial friction and deepening tendency.
If one averages the D . values around the storm at constant radius at
the grid points shown in figure 6 and plots these Dact values with respect toradius at 10 n.mi. intervals he obtains a radial profile of the actual "D"values as discussed at the beginning of the report. A similar profile of theD values can also be made. If the two profiles are plotted together begin-gr
ning at a uniform value at an outer reference radius, say 80 or 100 n.mi., theaccumulative inward imbalance between the D and D fields is portrayed
(fig. 83). The 80 to 100 n.mi. reference radius is a reasonable startingpoint, since the major condensation warming occurred inside of this limit inall cases.
This integrated height difference between the D . and D profiles can
be converted into a virtual temperature excess or deficit with the hydrostaticequation from the tropopause region downward, holding a high pressure surface,say 100 mb., constant. A positive integrated value of (D - Dact) means
27
that the atmosphere is warmer than would be computed with the assumptionv ) =0. A negative value of (D - D .)has the opposite meaning,r gr gr act
Heating of the air beyond that demanded for radial balance of forces willbe denoted as "excess heating." Part of this heating, if present, is used tomaintain the storm against frictional dissipation; another part is availableto intensify the storm. There is thus a unique balance between frictionaldissipation, intensification, and excess or deficit heating when values integrated around a hurricane are considered. Thus
[EXCESS HEATUG] - [FRICTIONAL DISSIPATION] = (STORM INTENSIFICATION]
If the integrated value of (D - D .) from the outer to the inner s^orm
radius is negative, then not enough condensation heat is relased to maintainthe pressure gradient at the value needed to go with the wind pattern even inthe frictionless case, and the storm must decay.
Of the ten levels analyzed, eight had large positive integrated values of(D - D ,) ; i.e., strong excess heating. These hurricanes were deepening orN gr act7 ' 'moving in steady state. The other- two cases (Daisy August 27, 630 mb.;August 28, 620 mb.) had slightly negative and slightly positive (D - D .)
values, respectively, when averaged around the storm. The condensation heating, although large, that was occurring in Daisy on the 27th and 28th had nolarge residual to balance the dissipation of kinetic energy by friction. Itwas on the 27th that Daisy reached maximum intensity and began to fill. Onthe 28th Daisy was filling rapidly. Retarding frictional effects thus over-weighed the generating effects of condensation heating on these two days.
Figures 8k and 85 show the predominance of negative v ) and vrr) -- overpositive values of these quantities. Negative values of vr) and v^) correspond to "excess heating" and are proportional to positive integrated valuesof (D - D ) from outside to inside of the storm, as shown by equations(8)
gr actand (9) when integration is performed in a negative sense along r.
These composite radial profiles, which include Daisy levels on August 27and 28, bring out clearly the considerable excess of the observed D-value dropfrom outside to inside of the hurricanes above that required to balance centrifugal plus Coriolis accelerations when composites of the ten levels aremade. In these diagrams Iv ) |and jv ) |have been averaged around thestorm without respect to sign (top curve) and divided into positive and negative parts. The major contribution to |vy) Jand jv^)^/ ls clearly seen tobe made up of negative values of these quantities; i.e., positive (D - T>act)or excess of pressure gradient over centrifugal and Coriolis acceleration. Atsome levels, notably the two Carrie flights (figs. 36 and ko) negative vr)or v ) values prevailed over nearly all of the flight level.
rr gr
28
These large residual negative v ) accelerations are needed to maintainr gr
the storm against frictional dissipation and to deepen it* It is no coincidence that the two levels which showed little negative residual of v ) were
r gr
the two where filling was beginning or taking place. Also it is no coincidencethat steady state storms must have negative integrated values of v ) to balance frictional dissipation. r &
Condensation heating related to the wind and pressure field on August 25.
Daisy on August 25 was in the deepening stage (fig. 1, 2). Radial legswere flown along four distinct levels on this day (950, 830, 570, 237 mb.).An excellent and unique opportunity thus exists to obtain vertical gradientsof the computed v ) fields and perhaps obtain some insight into the deepen-
r gr
ing process that was occurring on this day. At first, vertical thicknessvalues were determined from the actual slopes of the isobaric surfaces andthen converted to mean virtual temperature with the hydrostatic equation.Taking the mean virtual temperature in each layer at r = 100 n. mi. as reference temperature, the inward warming was computed over intervals.of 20 n.mi.and plotted accumulatively from 100 n.mi. to the center (fig. 87). It is seenthat nearly all warming took place inside the kO n.mi. radius. The computedvirtual mean temperature in the eye between 570 and 237 mb. was ^.5°C. warmerthan the temperature in the same layer at the 100 n.mi. radius.
This computed cross-section agrees quite well with that obtained fromthe aircraft temperatures (fig. 88). Figures 87 and 88 corroborate a previous evaluation by Malkus [9], who, using the same data, had found that a goodapproximation to the D-field in the low troposphere could be obtained by assuming an undisturbed top in the high troposphere and integrating downwardhydrostatically with the observed (virtual) temperatures.
We shall now approach the interesting question as to what fraction of thistotal temperature field is due to "excess" heating as defined above. For thispurpose the D -D . profiles were plotted together on one diagram. If we set(D -D .)!=0 af the storm periphery and then plot D -D as a function of ra-dfus at each level, the unbalanced portion of the orop in height of the pressure surfaces will become readily apparent.
At the lower three pressure levels, the radial slope of the actual D-pro-file - after integrating around the hurricane at each level - exceeded that ofthe D profile. At 237 mb., (fvQ +vg2/r) exceeded gdD/dr. But (Dgr -Da<yt)integrated over the entire storm volume was negative. The four radial legsused for computation at 237 mb. are shown in figure 11. It was not possibleto obtain values of v ) within the 20 n. mi. radius, and radial symmetry
r gr
over a limited distance had to be assumed for computations along legs D and B.Nevertheless, the cross-hatched part of the curve in figure 86 is believed tobe reasonably accurate.
Vertical changes in the profile of (D - D ) and hence also v ) whengr ac*u r gx
integrated around the storm, were quite small from 950 to 570 mb. The greatest vertical change in the profiles occurred between 570 and 237 mb.(fig.86).
29
The portion of the inward temperature increase arising from the (D - D Jgr actf
profiles is shown in figures 89 and 90 in profile form. As noted earlier, computations as illustrated in figures 86 to 90 have a less secure basis thanthose performed along individual radial legs because they are dependent onfield analysis by qualitative methods and subsequent integration around thecenter. If, however, it will be granted that no serious analysis error hasbeen made, a quite interesting result has been obtained. Figures 89 and 90may be interpreted as showing the heating function which gives rise to stormdevelopment and to circulation maintenance against internal and surface friction.
The "free" surface pressure fall at the ground, estimated from figures89 and 90, is about k to 5 nib. - v/ith the assumption of an undisturbed top at50,000 feet. Some of this pressure fall must, by previous reasoning, be used,,up in internal and surface frictional dissipation. The amount of heating that,is left over after this dissipation is used to intensify the storm. In thecase of Daisy on the 25th, internal and surface frictional dissipation wassmaller than the excess heating. It also follows that since development ofthe wind field lagged the pressure field, the drop in central pressure was theprimary feature and occurred before intensification of the hurricane winds.
It would appear that from the local thermodynamic viewpoint, the actionof the cumulus clouds in the initial disturbance must have been such as to produce a net area warming of the troposphere above 500 mb., with magnitudes of1°C. above that ordinarily obtained from convection in disturbances.
If Daisy, as observed onAugust 25, is considered typical of developinghurricanes, then initial warming in the middle and especially in the uppertroposphere may be postulated as a definite initial condition of hurricanedevelopment. In this connection, it is of interest to record that a reconnaissance plane of the U. S. Navy which scouted the cyclogenetic area about 12hours before the research missions reached Daisy (2200-2if00 GMT, August 2^),observed a nearly solid precipitation area on radar with extent of about
k 210 n. mi. and convective echo tops above 40,000 feet in the precise area whereDaisy was forming.
30
DAISY
25 AUG
830 mb
ft
Figure 12,- v ) in )st./hr,r gr
1/
V.
Figure 13,
DAISY
25 AUG
830 mb
Isotochs(kn)
i •
DAISY
25 AUG
830 mb
vr(kn)
Figure 1^. - Radial wind component.
-0.8
4.0 .« -93_o
° \
•}s> -.4.0 -4.0 -}x> ^a -*^ \ 0
DAISY
25 AUG
830 mb
SOn. mi. 40•-p.8.aA 0.10.4 0.4 OJ 0.2 JXI 0.8 0.2 .0.4 0.4
-0.4
-0.4
-J18-9*
-<
-f.1 — •20
-08
-0.6 40
•5.5 60 dial.
-03
-0.J
Figure 15.
31
32
SO a. at. 40 20
n A>a —» i
-II
,50 S2V _C }:2J 20
\ ~\49
\ -28 '-25 -'
--•» _^,
Figure l6. - v ) in kt./nr.r gr
Figure IT-
_^_30_ daisy* 25 AUG
570 mb
"^ Isotochs (kn)
0 0
j ? i ? DAISY\ jJ 25 AUG\y^\ ' 570 mb
?7 / f f I? \ i
Figure 18. - Radial wind component.
DAISY
-q.s
0.1
V
25 AUG
0.2570 mb
40 b. mi.0.3 %
-20*0.1
—8
fv + —1v+ r
0.2 0.2
V -WW "V -06-o.. • *'*.,« * -V
-0.3"l* -q.6-0.fi* -0.8*-09
0 ^ -0.9-q,6
-°,5-q.6
-q.7
601).mi. 40 20 ' 02
-q.9
-q.$
-q.6
-0 3
q.2
0.3
Figure 19.
33
3h
Figure 20. - v ) in kt./br.r gr
DAISY
27 AUG
630 mb
Isotochs(kn)
\ * *
? .2 I * t 0
-.«* -9 -§ '
DAISY
27 AUG
630 mb
vr(kn)
6 -5 0.
Figure 22. - Radial wind component.
-0.2
—40a. mi.
-$2
-0.2
—20-0,2
DAISY
0.2 27 AUG
,*/-„, 630 mb
IV
f* +T
*V*«i «<? a?»-q\i °--2-«3 '̂q-SO-S V "«•• 9 "92 -0-2 -0.2
? O.I*3
0 V • q-5o V q.4
-04
0.5
0.7
Figure 23.
35
*
DAISY
28 AUG
620 mb!0 vr) :
45 0
Figure 2k. - v ) in kt./hr.r gr '
Figure 25.
DAISY
50 28 AUG
\ \ 620 mb
Isotachs (kn)\
T
DAISY
28 AUG
620mb
vr(kn)
-10
Figure 26. - Radial wind component.
V -02
1.4 -12 IT^Im
0,3
0.4,
0.2
. «i * 0.4-0..5 '.-HftS 'A 0..3-0.6 *0,2 .oj > q.4
DAISY
28 AUG
620 mb
%
% + T
-0.4 -0.10.5
/
Figure 27,
37
Fig
ure
28.
-v
)in
kt.
/hr.
rg
r
CL
EO
82
0m
b
Iso
tach
s(k
n)
Co
Fig
ure
29
.
CLEO
820mb
vr(kn)
Figure
30.
-Radialwind
component.
-,f|
-I..0
-L
0
-q.e
-1.3
-1.2
•60
n.m
i.
-1.8
—2
0
CL
EO
82
0m
b
-2
0.1
q.z
0.1
0.2
-i.9
q-6
•-2.
.9q-
*-I
2"I-
2"^
"2.0j0
.*
V^
*°S0
"•mi-
-q.3 -q
.i -V9-
'
-L
2 0.3
Fig
ure
31
*
q.6
0.3
.VO
Fig
ure
32.
-v
)in
kt.
/hr.
CL
EO
58
0m
bC
LE
O
58
0m
b
Iso
tach
s(k
n)
§•
Fig
ure
33
.
CL
EO
58
0m
b
Figure
3h.-Radialwindcomponent,
-0
.9
-V0
0.1
-0
.8
-0.9
-q.e
-0
.9
-0.8
-0
.9
—6
0
—4
0
0—
20
-0
.9
"l-2
,,-0
.9-1
.7•
0.7
•-I.
0Q
"o9°
-"6-1
.4-1.
3"0
.8-J.
2-1
.2-0
.9-1.1
-^6.5
*'\Y
•'
IlE
ye.^
0.1'q
.20
-40
.2
0.3
0.2 q
-0.2
-0
.2
-0.5
Figure
35*
20
CL
EO
58
0m
b
-2
Vfl
fv+
—V
r
40
60
n.m
i.
-0
.3
k2
CARRIE15 Sept605 mb
/60n.mi. 40
•25 ' Mobv \y\ ?5
Figure 36. - v ) in kt./hr.r gr
Figure 37.
CARRIE
15 Sept
605 mb
Isotachs (kn)
N ^\0
. c / a CARRIE8 6 ' 4 xi2 .9 - '/ • ? 3 15 SEPT
•60 /
/ , " yr(kn)
12 »
605 mb
Figure 38. - Radial wind component.
-0.4' -0.4
CARRIE
15 SEPT
605 mb
-0.2
-0.3 —40 a. mi. ' TV0+~T~
-0.6
-0.3— 20 0.2
'-0.2
-0.5 q.i'-L9 <*
0.1
-q.e2.9
°-2.8-t.3-0.5 -0.2 „,. % • • • 0.1 0.4-0.4 ^1-0.3 # .
• -O.l"0.-1 • . -0.3' ft °.s *-q.2-0.3-0.3 '-q."
-0.9 * -0.2
-1.3-0.9 * — 20
-0.2
-0.3
-0.1
-0.8
— 40
Figure 39.
h3
kk
Figure UO. - v ) in kt./hr.r'gr
CARRIE
17 Sept
690 mb
Isotachs (kn)
50 ?
Figure Ul.
CARRIE
17 SEPT
690 mb
vr (kn)
Figure k2, - Radial wind component.
-0.2
* -0.2
N
\-0.7
-0.7
-0;8
-IA
-6.1
-7.9
0-9 „-2*7-1.0 •
• -^-qs i
-02
-qz
-03
-0.3
-03
-0.2
20
-031
Figure ^3.
^5
U6
A HELENE
25 SEPT
is \ 810 mb
« \ *'V
Figure 44. - v ) in kt./hr.r gr
Figure 45.
HELENE
^^50 25 SEPT810 mb
> Isotachs (kn)
HELENE
25 SEPT810 mb
Vr (kn)
-10
-15
Figure 46. - Radial wind component.
60n.mi 40
-0.9
-p.e
-0.61
. p.5HELENE
.0.3 25 SEPT
p810 mb
0
p
-0.1
-0.2
-0.30.1
0.3
-0.1
:P-3-02
1-15-.5.0 P-2P-I o0.1
-I0.5„-3.5' -.2-2.0
-0.5 * •*«
-0.5 -P-3
"P-a -0.6
p.2
-0.9-
0.5
Figure 47.
hi
48
Figure 48. - v ) in kt./hr.r gr
Figure 49.
HELENE
24 SEPT
635 mb
Isotachs (kn)
60 a. mi.
HELENE24 SEPT635 mb
Vr (kn)
Figure 50. - Radial wind component.
-0.2
-p.3
-0.5 -0.2
-p.2"P.2 -p.4 -o.|
-•°-3-0.9 -J,-J-9 0.3
60n. mi. 40 20 a ? '1 I I . • 9
-,2.0
-J.4* J.0-J.O -pi
-0.2
-J-2
-J .5
-1.0
-0.2
'p.7
-0.2
-0.1
0
Figure 51•
HELENE24 SEPT
635 mb
n»+%-v0
0.3
0.5
J-0
P.6
0.5
P.6
P.6
h9
50
i.o
0.8
0.6
0.4
0.2
DAISY
Figure 52.
J L
10 20 30 40
r {n. mi') >
figure 54.
28th(620mb)
580 mb
J L
50 60 70
1.0
0.8
0.6
0.4
0.2
1.0
0.8
0.6
0.4
0.2
,+"N
15 SEPT605 mb
CARRIE
J l I L
10 20 30 40
r (n.mi.)—>
Figure 53*
HELENE
24 SEPT635 mb
50
j l i L
10 20 30 40
r(n.mi.) >
Figure 55*
50 60
v)Figures 52-55: —£J5E— plotted against radius. Values have been averaged
around the storm without respect of the sign of v ; .
(a)
T1.0
Composite10 levels
0.6
0.6 - \\^(0)0.4
(bP" *-—""- -"•*.
0.2
___ i i i i i i
v )r y
f VQ + —0 r
10 20 30 40
r(n. mi.) >
Figure 56.
0>)
50 60
v)
"3d"
Values are plotted against radius after averaging around the storm
without respect to the sign of v ) .r gr
51
52
Oirtelion of Motion
Figure 57.
Direction of Motion
Figure 58.
Composite of10 Levels
Vr)gr Kn/hr
With respect to sign
•t o
60 "•'
Composite of10 Levels
/hrVr) Kn
Without respect to sign
60 o
33 53/6l
• \ V\J0 146 34 \#22 22 2055
Direction of Motion
Figure 59.
Direction of Motion
Figure 60.
Composite of10 Levels
VrrJ Kn/hrVrrJ/gr
With respect to sign
Composite of10 Levels
Vrr)/gr
Kn/hr
Without respect to sign
6qb.ov-
J \ 3.4 26 3.' 44 V S-\I \ ^0 *4.5 / V
'.* 17 25 141 44 /38 ^ y50 41 /a 22 2' 19
53
12
0-
10
0
80
60
40
20
gaT
29
r
Co
mp
osi
te1
0le
vels
50
40
30
-
20
10
Co
mp
osi
te1
0le
vels
-I1
1I
' 50
60
10
20
30
40
r(n
.mi.
)—
s*
-1—
11
11
L
Fig
ure
6l.
102
03
04
0
r(n
.mi.
)>
Fig
ure
62
.
Radial
profiles
ofvarious
accelerations
atselected
radii.
Values
have
been
averaged
around
stormwithout
respectto
sign,
50
60
t
DA
ISY
25
AU
G.
83
0m
b
£100
-/*
\(a
)
Deviation o
c o ai
Z
(b)
i•
i1
11
10
20
30
40
50
r(
n.
mi.
)>
60
Figure
63.-830mb.,
August
25.
70
15
0-
a>1
00
-
> a»
Q
10
DA
ISY
25
AU
G.
57
0m
b
20
30
40
50
r(n
.mi.
)*
Figure
6k
.-570
mb.,
August
25,
Figures
63-73.
-Curve
(a),
mean
deviationof
theD
profiles
oftheindividual
radial
flight
legs
at0
gr
selected
radiifrom
theirmean
value
(ft.).
Curve
(b),
mean
deviationof
the
DQct
profiles
ofthe
individual
radial
flight
legs
atselected
radiifrom
theirmean
value
(ft.).
VJl
56
(|93i)U0.U0!AaQ U09(AJ
oE
(|93j ) U0JIDIA3Q UD3^
O
o10
oin
o
o ~
o wCM »-
a>
ONvo
Deviation(feet)-_
roCD^oooo
"^
I1I1
5
i\>
o
•K
^^^,
s"wo*^s•o^•"^^***^3c:•y
Is-
2oiO V
*yTQ*OiLOJ-*
o
+
\
RRIEEPT.
mb
o
Deviation(feet)-
3b
SI
11
-A
ii
-\^-V
-s\ ""^>s"o*/
—i
S^O
/VCD
O(/)>
-/^"OXI
/1°"H73*k__
/Jm
LC
Deviation(feet)
*1
ON-4
5roo
wOoo
o
I
•
111
ro
o
•V
i\«
?w3°
-
ex
•
••ro"
O-
•
00orof—
eno
•1
3*0
o
•M
ODo
ro
_nOI*U)
ooooo
1I
1
i11
M
I oI*•
^ ro_
O
^"^"
OJ
O
/•/
^/ o
crIt1
o71•/
"a2o
<J>o
\1*/•
3m 1o
-g
o
iV»
CDo
*
\
ONCo~
Dev
iati
on
(feet)
ro
OJ
•i>
en
o>->
iC
O
0O
00
OO
O0
0
11
"1
11
11
1
N
^O
H*
/0-
ro3
OJ
O-
s%
a
933
»O
O
1O O
l
0 CI)
0 -J
0
mpositelevels
CD id
3
ro
O
ro3
O
—5
^->
IU1
•
IO O
)O e
nO
3O
l
3°
Devia
tio
n(f
eet)
_ro
o»
oo
o
11
X
11
'"c
y
'1
M
-\ \
HELEN24SEPT.635mb
11
\m
-
Devia
tio
n(f
eet)
r1
•—
1"I
- -
¥"
"a
-
r•*
00ro
X0
^m
./
!
LENESEPT.mb
VJl
CO
DAISY
27 AUG
630 mb
Vr kn/hr
Figure 7k. - Field of v (kt./hr.).
DAISY
27 AUG
f 630 mb* Fr (kn/hr)
Figure 75- - Computed frictional force (kt./hr.).
59
6o
140 -
120 -
100 -
c
t1.0
0.8
0.6
0.4
0.2
0.6
0.4
0.2
Composite
10 levels
1.0 20 30 40
r (n.mi.)——>50 60
Composite10 levels
10 20 30 40
r(n.mi.) —>
50 60
Figure 76. - Curve (a), v ) averagedr gr
around the storm without respect tosign. Curve (b), F averaged around
the storm without respect to sign.Curve (c), v averaged around the
storm without respect to sign,curves: units kt./hr.
All
Figure 77. - Curve (a), —=rV
— 9
9 r
averaged around storm without respect to sign. Curve(b]"J
F
—averaged around storm withoutc>D
g"lr"respect to sign.
Figure 78. - C,urve(a)
9
9 r
averaged around storm without respect to sign. Curve (b),
==- averaged around storm with-oD
g "17out respect to sign.
12
I *\
\\. \
10
36
*40
Direction ofMotion Composite Of
10 Levels
3^ Fr or Frr Kn/hr'
'•7 60"-""-/ v ^^- •.- A0
\ \ 'A* *^« fO , \ 33 ^4, 30 . /
/
? 12 ?°
22 ^25-
25_^ 27>•—-'31
io y
'.3 10
/ §/
Figure 79.
Composite ofDirection of Motion 10 Levels
Fr or Frr Kn/hr
(lWithout respect to sign
I" \ ?* 23 .J8 43/50V9 \\I * 34* ' 157 \ 1
39) .44 N38 # J.54 | .41 $*\ I 36 (56
V 4l/ ' PziZ\J}925
38 4334
25"
Figure 80.
41 30 / 23 '9 !7
61
62
-10 -$
Direction of MotionA
Figure 8l.
Direction of MotionA
Composite of10 Levels
Vr Kn / hr
With respect to sign
Composite of10 Levels
Vr Kn / hr
Without respect to sign
16
21
25—.
43A
15 V
10 20 24 . 32 I.41 M• * I V Sit^-^<38
23f
Figure 82.
60"
69 55) 44 3' V i856
28V S4>
400 r-
DAISY25 AUG
830 mb
East
100 80 60 40 20 center 20 40 60 80
< r(n, mi.) r(n. mi.) >
Figure 83. - Curve (a), observed D-profile
(D .). Curve (b), computed D-profile
necessary to balance observed wind
pattern (D )••gr
63
-i400
100
40
30
t £2
0
10
po
s
Co
mp
osi
te1
0le
vels
JL
JL
1020
30
40
r(n
.mi.
)—
>
Fig
ure
8U
.
SO
60
50
40
30
20
10
-
Co
mp
osi
te1
0le
vels
-J1
11
i10
20
30
40
r(n.mi.)—>->
Figure
85.
SO
60
v)•
Ir
gr
"*IVgrId
iVided
in*°*
»*•*Pos
itive
andneg
ative
parts
after
averag
ingaro
undthe
storms
atselected
radii.
Illustrates
thepredominance
ofpositive
(D-D
)1.
domin»n,„,
r»"5
Kgr
act
,e-'
dominanceof
pressure
gradient
over
centrifugalandCoriolis
accelerations,
orexcess
warming.
ON
60
40
20
•— 0
-20
•40
-60
25 AUG
Composite 1600' 8 2800*
20 30 40 50 60 70 80
r. ( n. mi.)—>-
Figure 86. - Daisy accumulated plot of (D . - D )ace gr
from outside reference of 100 n. mi. to inside
of the storm. Values averaged around storm at
selected radii. Negative values of (D -
D ) indicate stronger pressure gradient thangr
corresponding wind field.
65
66
25 AUG.
7i237mb
f4t3°C 2°C l°C ±°C-~
200 -
.o
E 400
25 AUG.
£ 600-/7/ 570 mb V 600
,4°C 0 e
3 C 2 C l°C -it^4 «3t / -2lr h •«*/
•
K \ / s
5 / / s
v /3 I r "* <A •* •0
"/// / /{ /I 1 0°C
'-%/y •* •JF^^0
•-^—r 1, 1 "
800-830 mb
350 mb
800
10001000- n 1 1 1 1 r-10 20 30 40 50 60
r(n.mi.) >
Figure 87. - Vertical cross-section ofthe computed temperature increasefrom the temperature values outsidethe storm area. Values in °C.
EYE
25 AUG.
2°C
1°C
Nx \ 570 to 237 mb
\ \ 830 to 237 mb\—950 to 237 mb
V
40 60
r (n.mi.)-
l-iiy**-**'80 100
Figure 89. - Excess warming (°C.) between particular pressure layerscorresponding to the imbalanceof the pressure over the
— . vo(f vQ +
0-) acceleration.
EYE
10 20 30 40 50 60
r(n.mi.) >
Figure 88. - Vertical cross-section ofthe observed (aircraft) temperatureincrease (°C.) from the temperaturevalues outside of the storm.
25 AUG.
40 60
r ( n. mi.)-
Figure 90. - Comparision of actualwarming (curve a) with the acceleration warming (curve b) between570 and 237 mb. Values in °C.
61
k. SUMMARY OF PRINCIPAL RESULTS
The principal results from all of the levels within the four stormsstudied are:
1. The gradient wind equation along the radial direction, as defined inthis paper for hurricanes, does not express the radial balance of forces to asatisfactory degree of approximation in nearly all portions of the storm. Thequantity v ) is nearly always a large fraction of the Coriolis and centri-
r .&•
fugal acceleration. In some places v ) is even larger than these acceler-r gr
ations.
2. Internal radial friction plays an important role in balancing the.cyclindrical radial equation of motion. The frictional acceleration is directed toward the storm center when v ) is positive and outward from the storm
. . r grcenter when v J is negative. The average magnitude of the radial frictional
r gr
acceleration for the 10 levels for which area integration could be performed
is 25 to 30 percent of the pressure gradient force and 35 to kO percent of theCoriolis and centrifugal acceleration. These internal radial frictional accelerations may possibly be accounted for by a vertical Reynolds stress term.
3- All deepening or steady state hurricanes must have stronger pressuregradients than centrifugal and Coriolis accelerations in order to account forfrictional dissipations.
5. CONCLUSION
In this study the balance of forces in hurricanes has been examined alongthe radial direction in stationary and moving coordinates; further,the radiusof trajectory curvature has been computed in selected portions of ten flightmissions. The Coriolis acceleration proved negligible compared to the centrifugal acceleration as expected. But gradient or cyclostrophic balance ingeneral is not achieved by a wide margin, both along the radial axis and innatural coordinates. The pressure gradient far exceeds that required to balance the centrifugal acceleration to the left, and falls somewhat short ofthe requirement to the right of the direction of motion. In moving coordinatesthis asymmetry is reduced but by no means eliminated. Hence the maximum windmay be interpreted as produced by "overshooting" toward low pressure far beyond the point of equilibrium. A similar situation appears to exist withrespect to many extratropical wind maxima, especially higi-speed centers alongjet streams.
The field of radial motion shows general outflow ahead of and inflow behind the hurricane. The zero line is to right and left of the direction ofmotion, approximately where the maximum difference between pressure gradientforce and centrifugal acceleration is found. This geometric relation is considered more than a coincidence and supports the preceding deductions about theorigin of the maximum wind.
68
The air is accelerated radially in the sense prescribed by the differencebetween pressure gradient force and centrifugal acceleration. But the magnitude of the actual acceleration generally falls short of the computed acceleration by a wide margin. Hence it becomes necessary to postulate the existenceof large lateral frictional forces which act opposite to the v ) accelera-
-1 0' r &tions. The magnitude is 10 to 10 dynes per unit mass which corresponds toa pressure gradient force of 1 rob./5 m. Such large frictional forces previously have not been suspected to exist. Nevertheless, there appears to be novalid reason a priori why the friction should not rise to become a substantially larger fraction of the pressure gradient force in hurricane-type systemsthan under normal atmospheric circumstances. If so,an important break on hurricane development has been uncovered, and it remains for the future to give aquantitative expression to enter as a constraint in hurricane models.
A few initial steps were taken to relate dynamics and energetics. Fromthe difference between pressure gradient force and centrifugal acceleration,the departure of central pressure from that required to balance the wind fieldcan be computed, also the temperature field which must be assumed with thepressure departure, assuming hydrostatic conditions with the top of the disturbance at 100 mb. Only one case had sufficient fligit levels to perform allof these calculations.
In this instance there was a "forced" surface pressure drop of k - 5 mb.and an "excess" temperature warming of about 1°C, usually in the high tropo-spheric layer above 500 mb. From this it appears that hurricane developmentis related to the capacity of the atmosphere to effect a large heat transportinto the upper atmosphere which, from heat balance considerations, must beeffected by ascent in narrow channels as demonstrated by Riehl and Malkus [17]for the equatorial trough zone. The relation between dynamics and energeticshas merely been touched on in this paper, but one definite suggestion fromthis aspect of the work is that one should search for conditions in tropicaldepressions which favor banding the mass ascent into narrow walls or lines.
,acknowledgments;
The author wishes to express his sincere gratitude to Professor HerbertRiehl under whose direction this paper was written. Professor Riehl proposedthis topic, encouraged and supervised all phases of its carrying out. Theauthor also wishes to thank the National Hurricane Research Project and itspast and present directors Messrs. Robert Simpson, Cecil Gentry, and HarryHawkins for generously furnishing the valuable data needed to make this study-possible. The author expresses his appreciation to the U. S. Weather Bureaufor providing financial assistance and to Mrs. Delia Friedlander for draftingthe illustrations. Thanks are also due to Dr. T. N. Krishnamurti of the University of Chicago and the above mentioned Mr. Robert Simpson for helpful discussions on this subject matter.
69
REFERENCES
1. M. Ausman, "Some Computations of the Inflow Angle in Hurricanes Near theOcean Surface," University of Chicago Project Report, 1959,22 pp. (Unpublished. )
2. D. Brunt, Physical and Dynamical Meteorology, The University Press-, Cambridge, 1939, 428 pp.
3. B. Haurwitz, Dynamic Meteorology, McGraw-Hill. Book Company, Inc. New York,194l. (See pp. 192-195.)
k. D. T. Hilleary, and F. E. Christensen, "Instrumentation of National Hurricane Research Project Aircraft," National Hurricane Research Project ReporfeNo. 11, U. S. Weather Bureau, 1957> 71 PP.
»
5. L. F. Hubert, "Distribution of Surface Friction in Hurricanes," Journal ofMeteorology, vol. 16, No. k, Aug. 1959, pp. 393-404.
6. L. A. Hughes, "On the Low-Level Wind Structure of Tropical Storms," Journalof Meteorology, vol. 9, No. 6, Dec. 1952, pp 422-428.
7. C. L. Jordan, D. A. Hurt, Jr., and C. A. Lowrey, "The Structure of Hurricane Daisy on 27 August 1958," Journal of Meteorology, vol. 17, No. 3,June i960, pp. 337-3^8.
8. N. E. LaSeur, "An Analysis of Some Detailed Data Obtained by Aircraft Reconnaissance of a Hurricane," Scientific Report No. 5, Contract AF-19 (6o4)-753, Florida State University, 1957-
9. J. S. Malkus, "On the Thermal Structure of the Hurricane Core," Proceedings, Technical Conference on Hurricanes, American Meteorological Society1958, Sec. D. 3, PP. 1-2.
10. J. S. ^fellkus, "Recent Developments in Studies of Penetrative Convectionand an Application to Hurricane Cumulonimbus Towers," Technical ReportNo. 6, Woods Hole Oceanographic Institution, 34 pp. (Unpublished manu*script.)
U. J. S. Jfelkus, "Cloud Patterns in Hurricane Daisy, I958," Te^^^cal ReportNo. 8, Woods Hole Oceanographic Institution, 63 pp. (Unpublished manuscript. )
12. J. S. Malkus and H. Riehl, "On the Dynamics and Energy Transformations inSteady-State Hurricanes," Tellus, vol. 12, No. 1, Feb. i960, pp. 1-20.
13. 0. Reynolds, "On the Dynamical Theory of Incompressible Viscous Fluids andthe Determination of Criterion," Philosophical Transactions of the RoyalSociety of London, Ser. (A), vol. 186, 1958 pp. 123-164.
14. H. Riehl, Tropical Meteorology, McGraw-Hill Book Co., Inc., New York, 392pp.
70
15. H. Riehl, "On Production of Kinetic Energy from Condensation Heating," pp.381-399 of The Atmosphere and the Sea in Motion, The Rossby Memorial Volume,Rockefeller Institute Press in association with Oxford University Press,
New York, 1959-
16. H. Riehl and R. C. Gentry, "Analysis of Tropical Storm Frieda," NationalHurricane Research Project Report No. 17, U. S. Weather Bureau 1958, 16 pp.
17. H. Riehl and J. S. Malkus, "On the Heat Balance in the Equatorial TroughZone," Geophysica, vol. 6, No. 3/k, 1958, pp. 503-538.
18. Staff Members, National Hurricane Research Project, "Details of Circulationin the High Energy Core of Hurricane Carrie," National Hurricane ResearchProject Report No. 24, U. S. Weather Bureau,1958, 15 vpl
19. L. Sherman, "On the Wind Asymmetry of Hurricanes," Journal of Meteorology,vol. 13, No. 5, Oct. 1956, pp. 500-503.
71
APPENDIX
1. RELIABILITY OF THE COMPUTATIONS
The reader's first reaction to this study may be tinged with some skepticism as to the accuracy of the data and the reliability of some of thecomputations made from the qualitative analysis. This may be particularlytrue of computations that made use of the radial wind component.
It must be emphasized that the Doppler navigation system, from which allthe winds were obtained, is an extremely accurate nagivation instrument.Winds above 10 to 15 kt. are accurate to within 2 to 3 percent in both direction and speed - except at isolated points in turn and in a few heavy rainareas. These areas were not used in the computations. 33ie tangential windcomponents are j.ust as accurate as the total wind.
All computations where "D" values were used were dependent only on the'gradient of the "D" values and not on the absolute values themselves.
It should be remembered that the radial winds used in this study werethose *of the actual radial wind components and not the relative radial windcomponents. The magnitudes of the actual radial winds are much greater thanthe relative radial winds and lend themselves far better to evaluation.
Computations using the actual radial winds were made only from carefulfield analysis where isolated non-representative values could be disregardedand^a relatively smooth well defined pattern could be obtained. Evaluationof v ) and v ) was in no way dependent upon the radial wind.
r gr rr gr ^ ^
That there may be a few inherent data and computational shortcomings incertain of the individual levels is not denied. The area integrations did re-quirej qualitative analysis. It is felt, however, that in the 10-level composites these possible individual level shortcomings could not and did notdisguise the broadscale features displayed.
2. EQUALITY OF FRICTIONAL ACCELERATION IN FIXED
AND MDVING STORM CENTER (F )
The substantial derivative relative to the moving storm center v is,
with the previous stated assumption of constant storm translation,
ov dv dvv rr rr ^ rr
0r rdO rr dr^ ^3z^
where vQ and v are the relative tangential and radial wind components. The
local derivative d v /dt drops because of the steady state assumption. The
relation between the substantial derivative with respect to the moving centerand that with respect to the fixed center is then with the statement and footnote of page 17
72
Now
and
2v c c0 0 . 0v - ( - -£-=• + -«L- - fe ) = v . (19)
rr v r r 9 ' r
v ) = v - F , or F = - [v ) - v ] (a)r gr r r' r r gr r
vw)«. =^ "Fw* or F^ *"- ^J^ "^ (b)rr gr rr rr' rr rr gr
2
*„)„ -O +(- -^P-+ ^- -f °e) («)rr gr r gr r r y
2v c c
♦ =v +( £-8 +-|_ -fcj (d)rr r r r 9
from previous stated principles and assumptions. Substituting equations (c)and (d) into equation (6)
F^ - - (v ) -vj =F (20)rr r gr r r
The frictional acceleration with respect to the moving storm center isthus shown to be identical with that with respect to the fixed instantaneouscenter.
USCOMM-WB-DC