Calhoun: The NPS Institutional Archive

Theses and Dissertations Thesis Collection

1952

A statistical study of the relationship between local

changes in relative verticity and convective weather phenomena.

Stephens, Russel Thurmond.

Monterey, California. Naval Postgraduate School

http://hdl.handle.net/10945/24851

Library

U. S. Naval Postgrafluato School

Moiiterev. Califoinia

• ^^'"'OfiiV/A93943-5002

A STATISTICAL STUDY OF TH3 EUL^TIOiiSEIP

B3T;yE3i; local ghai^gss hi R^iILaTIVS vorticityMB CGiiV3GTIVE al^TEIR PHiIL^0^i3L;A

R. T. Stephens

A STATISTICAL STUDY OF THZ RELATIONSHIPBET'-vlIBlT LOCAL CHAIIGSS IK HEIATIVS YORTICITY

AiO CONVSCTIVJil .mATiiSR PKSHOl^HaiA

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Hussel Thurmond StephensLieutenant Commander, Un'ited States llavy

Suhmitted in partial fulfillmentof the requirementsfor the degree ofMi'iSTBR OF SCIKUCS

In Aerology-

United States I'laval Postgraduate SchoolItonterey, California

1952

This work is accepted as fulfillingthe thesis requirements for the degree of

MASTER OP SGISKCBm AEROLOGY

from theUnited States ivfaval Postgraduate School

(i)

/ BQ^9

PREFACE

This paper represents a statistical evaluation of the relationship

between relative vorticity and convective weather phenomena. It con-

tains a correlation between vorticity at the 700 millibar surface ajid

convective phenomena, and a correlation between 2^ hour changes in

these two variates. The data for this evaluation were obtained from

700 millibar and surfe.ce maps during the spring and early summer

months of 19^9 through 1951.

This work was conducted at the United States Naval Postgraduate

School, Monterey, California in the spring of 1952 in partial fulfill-

ment of the requirements for the degree of Master of Science in

Aerology,

The author wishes to express gratitude for the assistance of

Associate Professor G, J, Haltiner, Department of Aerology, U. S,

ITaval Postgraduate School, Monterey, California.

(ii)

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TABLE 0? C0KT!E1.'TS

CERTIFICATE 0? APPROVAL i

PHEFACE ii

TABLE OF CQLJTENTS iii

LIST OF ILLQSTRATIOHS iv

TABLE OF SYKBOLS AKD ABBREVIATIONS v

CHAPTER

I. ITJTROKUGTIOK 1

II. THE THEORY OF RELATIVE VORTICITY 3

III. ME3TH0D OF GOLLECTIOH OF DATA 5

IV. THE LINEAR GORRELiiTIOii OF SEVERilL VAHIATES 8

V. lilTERPRSTATIOS AiJD EXTENSIOil OF RESULTS l4

VI. COIICLUSIQK 19

BIBLIOGRAPHY 20

(iii)

LIST 01 ILLUSTHATIOKS

Table 1. Scatter Diagram of Relative Vorticityand Convective Index

Table 2» Scatter Diagrajn of 2^-ho-ar change inRelative Vorticity and Convective Index

Table 3» Scatter Diagram of 2^-ho\rr chajage inRelative Vorticity and 24-houT change inConvective Index during the same period

Table 4, Scatter Diagram of 2^-hour change inRelative Vorticity and the subsequent2^-hour change in Convective Index

Table 5« Skeleton Forecasting Diagram

Table 6, "Best Fit" Forecasting Diagram for700 millibe,r observs-tions of RelativeVorticity, April 1952

Table 7. "Best Fit" Forecasting Diagram forsurface observations of RelativeVorticity, April 1952

Page

10

11

12

13

16

17

18

(iv)

M

TABLE OF SYIffiOLS AilD aBBRUVIaTIQKS

^ Vector vorticity

^xV C\irl of velocity

J Scalar relative vorticity

'V— Wind speed

Kj Curvatirre of e. streamline

^^ Wind shear along the normal to the flow

IM Unit position vector normal to the flow

/>\, Element of length along |f\

^ Scalar absolute vorticity

V^ Two dimensional grad

V V/ind velocity

A Goriolis parameter

X 2^-hour change in convective index

y 24-ho-ur change in relative vorticity

B Clear skies

BC Partly cloudy skies

C Overcast skies

P Precipitation resulting from convection

f^ Thunderstorm

(v)

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(v)

I. IHTRODUCTIOSI

The inspiration for this investigation came from a paper on a,

similar suhject by a former student of the U, S. ilaval Postgraduate

School, Lieutenant 3. J. Writt L6j( , In this paper the frequently

used forecasting rule that v/ind flow from the south causes convergence

and resulting convective phenomena was examined on a statistical basis.

This was done by meas\u'ing the time rate of change of relative vorticity

under nearly steady state conditions and correlating the results with

the average prevailing convective phenomena. Surface maps were used in

measuring relative vorticity. A linear correlation coefficient of

0,0513» based on 70 observations, was obtained.

The subject has been of interest to other investigators. Nsunias

sjid Clapp L^J computed inertia trajectories on 10,000-foot maps under

steady state conditions. These trajectories were such that the vertical

component of the absolute vorticity of a moving parcel of air remained

constant. They were determined by initial latitude, curvature, lateral

shear, speed, and direction. These trajectories were not parallel to

the isobars, presuma,bly due to the modifying effects of convergence or

divergence. It was found that if the isobars curved more cyclonically

than the trajectories, convergence was present; if more 8.nti-cyclonically,

divergence was present.

Another paper of a similar nature was published in 1950 £2J .

Relative vorticity was computed at the 700-millibar level by use of the

(1)

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expression for vorticity in natural coordinates and correlated with

the heights of the tops of cumuliform clouds over the ocean along

the 'Vashington-Bermuda airwa^/s. In a ssjnple of ^5^ ohservations, a

linear correlation coefficient of 0,84 was found between relative

vorticity and cloud-top height. A later sample of 210 additional

observations yielded a coefficient of 0,65. These results indicate

a somewhat strong relation between relative vorticity and convective

phenomena.

In view of the above papers, it is believed that convective

phenomena can be predicted through the use of relative vorticity

measurements. The goal of this paper is, therefore, to determine a

simple device which the forecaster may use to improve his forecasts

when convective activity is likely.

(2)

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II. 'TliS TH30RY OF RJIu^SlY'lH YOaTIOITY

Yorticity may "be defined as the curl of the velocity. In

rectangular coordinates.

(1)

In horizontal flow, only the last term appears. However, in

computing the scalar component of vector vorticity, it is more con-

veniently expressed in natural coordinates.

^ ^ ao^ (2)

where

,

A^ s wind speed,

K3 « curvature of the streamlines (positive for cyclonic

curvature, negative for anti-cyclonic curvature)

11^ s unit vector normal to the flow

/y\ = element of distance along irv

^^ a wind shear along the normal, l;egative values are

termed cyclonic shear; positive^ anti-cyclonic shear.

The theorem of absolute vorticity states that the rate of change

of the absolute vorticity of a moving element is proportional to its

horizontal divergence []3j • It may "be written as

4^ = ~ XVV,-V (3)

(3)

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where C component of a.bsolute vorticity and

>^ . V m horizontal divergence,

v/hen absolute vorticity in equation (3) is replaced hy its

equivalent in terms of relative vorticity and the coriolis parameter,

we ohta-ln

It is apparent that a relationship exists between changes in relative

vorticity and horizontal divergence.

Since it has long been established that a direct correlation

exists between the horizontal convergence in the lower atmosphere and

convective activity, it follows that a relationship exists between

changes in relative vorticity auid convection.

The ptirpose of this paper is to determine the nature of this

relationship by extending the above theory to non-steady state con-

ditions and computing local time change in relative vorticity rather

than individual change.

(4)

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III. IvSTHOD 0? GOLLEGTIOM OF Di^TA

Previous investigators have obtained "better results through

the use of upper level rather than surface charts in determining

relative vorticity. Moreover, the theorem of absolute vorticity

as stated hy Holm"boe» Forsythe, and Gustinr3j was derived on the

"basis of frictionless flow. Therefore, the 700 millihar level was

chosen for use in computing relative vorticity. Ohservations were

taken from the O3OO2 charts covering spring and early summer of

19^9 through 1951* This period was selected "because it was "be-

lieved that the changing situation which usus-lly ensues at that

time of the year would yield a, representative spread in values of

"both relative vorticity and convective phenomena, O"bservations of

convective phenomena were taJken from the OO3O i surface charts cover-

ing the same i^eriod.

In computing vorticity "by use of equation (2), contour lines

were assumed to "be streamlines. Curvature of the streamlines was

determined "by comparison with a series of arcs drawn on opaque paper.

Reported wind speeds were recorded when a.vailable, otherwise v/ind

speed was estimated with a geostropic wind scale. The units of

curvature were nautical miles ~^, and the units of v/ind speed were

knots, ./ind shear was measured along the normal to the flow in knots

per nautical mile. Hence the units of relative velocity were hours "''.

(5)

It might "be interesting to note at this point that wind shear "became

a contributing fs,ctor to relative vorticity only v;hen the jet stream

was in or near the area under consideration,

Convective phenomena were converted into numerical equivalents

"by use of a "convective index". This index consisted of weights

assigned to cloud type, cloud cover, and present wea.ther phenomena

as designated "by the U. S. v/eather Bureau l5J • Lower cloud types

1 thrau.gh 5 were assigned a value of one (1); lower cloud types 8 and

9 and middle cloud type 7 were assigned a value of two (2). The weight

assigned to cloud cover corresponded to the code numher representing

tenths of cover C^J • ..'eights assigned to present weather phenomena

were as follows: intermittent precipitation 3. steady precipitation 5»

shower activity 7t s.nd thunderstorm activity 9« To determine the con-

vective index from a station model plot, the numerical value equivalent

of present weather phenomenon was added to the product of the numerical

values of cloud type and cloud cover.

Measurements of relative vorticity and convective index were made

over the area "bounded "by latitudes ^0 and i^5 degrees north and longitudes

95 s,nd 100 degrees west. This area was selected "because it was deemed

advisable to avoid one in which topography might contribute to convective

activity. Values of the two variates throughout the area were averaged

and recorded.

(6)

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Relative vorticity measurements were recorded for all days for

which the 0300^ 700 millibar chart was available. However, con-

vective index measurements were recorded only v;hen the area was

sufficiently covered by reports of such quantity and distribution

to provide a good meas\ire of prevailing weather, llo special

treatment was afforded the presence of fronts or unusual phenomena.

(7)

IV. TH'i; LI1.^£ GOPJiSIulTIOri' OF SEVSEUL VaRIATSS

A total of 286 simultaneous observations of relative vorticity

and convective index were recorded. Relative vorticity ranged in

value from -0.3200 to -h 0.3500 and convective index from to 25.

A linear correlation coefficient of + O.605 was determined for these

two variates. Although this coefficient is slightly less than those

determined hy Grutcher, et.a,, ^2] , it appears somewhat significant.

Twenty four hour changes in relative vorticity were correlated

with the convective index computed at the end of the period. In a

sample of 266 ohservations in which the change in relative vorticity

ranged from -0.3700 to + 0,3200, and convective index ranged from

to 25, a correlation coefficient of -t-0.^5 was obtained. This

result is considered insignificant for forecasting parposes.

Twenty four hour changes in relative vorticity were then

correlated with concurrent twenty four hour changes in convective

index. In a sample of 26^ observations in which change in relative

vorticity ranged from -0.3700 to -h 0.3200, and change in convective

index ranged from -17 to -t- 18, a correlation coefficient of •+• 0.78

was obtained. This value appears significant, 8.nd its merits will

be discussed later.

Finally, twenty four hour changes in relative vorticity were

correlated with subsequent twenty four hour changes in convective

index. In a sample of 251 observations, a correlation coefficient

(8)

of -0,37 was obtained, A larger coefficient from this computation

would bave provided a valuable forecasting tool. Possibly a shorter

time change interval, such as twelve hours, would have given a larger

coefficient; however, the necessary charts were not available to

attempt a correlation on that basis.

In the a.bove computations, relative vorticity data were grouped

in class intervals of 5 hours"-^, and convective index data were

grouped in class intervals of 2 units. Scatter diagrams are shown

on the following pages.

(9)

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TABLE 1. SCATTER DIAGRAM OF EBLATIVE VORTICITY AIID COifTEGTI^/E INUEX.

(10)

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(13)

V. IliTilRPRSTATIOU MTD BXTi2ISI01I OP RESULTS

Of the correlation coefficients determined, that of -*• 0,78

"between the twenty four hour change in relative vorticity and the

concurrent twenty four hour change in convective index is the most

significant. A regression equation for these two variates is

X s .60 y -y- .68 (5)

where x s 2^ hour change in convective index

y = 2^ hour change in relative vorticity.

In order to forecast the change in convective index from

equation (5)f a prognosis of the 700 millibar level is necessary.

This value would then he applied to the current convective index to

provide a forecast convective index. However, in most instances, such

a prognostic convective index could he translated into a variety of

convective phenomena. In view of this fact, a forecasting diagram

such as shown in table (5) might prove useful.

This diagram was devised with the U. S. llavy verification system

in mind [^l] . It also requires the use of a prognostic 700 millibar

chart. Forecast weather, provided it is of convective origin, is

determined by entering the diagram adjacent to present weather and

moving right to the change in relative vorticity as determined from

current and prognostic 700 millibax charts. Such a diagram could be

prepared for a given station by making a, "best fit" to observed cha-nges

in relative vorticity and the weather. The two diagrams shown in

tables (6) and (7) have been prepared in that ma,nner.

(1^)

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The diagram shovm in table (6) is the ""best fit" for changes

in relative vorticity at the 700 millihar level. Meas\irements were

made at several stations taken at r9.ndom in the United States diiring

the month of April 1952. It provided 82 percent verification for

12^ cases, llote that this is a rather small sample taken from

several reporting stations, and it is believed that a single station

over a long period of time would provide a higher percentage of

verifications.

The diagram shown in table (7) is the "best fit" for changes in

relative vorticity at the surface. These measurements were made at

the same stations during the sane period as in table (6), The re-

sulting diagram provided 70 percent verification. At the surface

the numerical value of rela.tive vorticity is usually much smaller

than at the 700 millibar level, and the resulting change in relative

vorticity is correspondingly smaller.

(15)

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YI. CONCLUSION

This study indicates that a relationship exists Tsetv/een

concurrent changes in relative vorticity and convective phenomena.

With the aid of fairly accurate prognostic charts, this result

should he of considerable assistance to the forecaster,

No attempt has been made to correlate the moisture content

of the air with convective phenomena, and no doubt such a correlation

would be high. Therefore, it would appear desirahle to modify the

above procedure by introducing a measure of moisture content.

Probably a study of changes in these variates over a shorter

interval of time, say 12 hours, would give better results. This

was impractical due to the unavailability of the necessary charts.

All the above measurements were made during the spring a.nd

early stimmer. This seemed advisable because of the changing

situation in middle latitudes at that time of the year. However,

in order to be of greater value in forecasting, such & study should

be carried out for the entire year, particularly the prepars.tion of

forecasting diagrams similar to those presented here.

(19)

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BIBLIO&RiiPHY

1. Aerographers Manual, Foiirth Edition. HavAer 5O-IIOR-3,United States ¥avy, April 19^5

•

2, Grutcher, H. L. , J. G, Hunter, and R. A. Sanders. Forecastingthe heights of cuinulus cloud-tops on the iashington-Beriiiuda

airways route, American Meteorological Society,Bulletin 31:1-7, January, 1950.

3» Holmhoe, J., G, E. Forsythe, and j. Gustin. Dynejnic meterology,ITew York, John Wiley amd Sons, 19^5*

4. TIamias, Jerome and P. F, Glapp. Hormel fields of convergenceand divergence at the 10,000-foot level. Journal ofMeteorology. Yol. 3, 1: 14-22, March, 19^6.

5. U. S. Department of Gommerce, leather Bureau. Station modeland explanation of weather code figures and syiahols.

Washington, U. S. Government Printing Office, 19^8,

6. .v'ritt, 3. J. A statistical correlation of the time ra,te ofchange of relative vorticity with convective weatherphenomena. Unpublished.

(20)

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U. S. Naval Postgiiuliuitt; 5ciiuoJ

Monterey, Calit'oniia

OCT 2 BINDC1IYSE 2 53 lUTERHR

THesis

S693 StephensA statistical study of

the relationship between

local changes in relative

vorticity . .

.

OCI 2 BINDERY

18059S693

StephensA statistical study of the

relationship betvreen localchanges in relative vorticityand convective weather phenomen*

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U. S. Naval Postgraduate SchoolMonterey, California