NCAR/TN-149+STRNCAR TECHNICAL NOTE

December 1979

Isentropic Trajectories forDerivation of ObjectivelyAnalyzed Meteorological Parameters

Philip HaagensonMelvyn A. Shapiro

ATMOSPHERIC QUALITY DIVISION

NATIONAL CENTER FOR ATMOSPHERIC RESEARCHBOULDER, COLORADO

I

I - I - - _~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

I

iii

PREFACE

This technical note presents a numerical technique for calculating

isentropic trajectories, and the objective analysis of trajectory-

derived meteorological parameters. A brief historical summary of the

isentropic trajectory concept and objective analysis method is presented.

The requirement for large numbers of trajectories on a global scale

necessitated development of an objective computer program. A brief case

study illustrating the objective analysis of trajectory-derived meteorological

parameters is included.

v

ACKNOWLEDGEMENTS

We wish to thank Rainer Bleck, Dennis G. Deaven, Jack Fishman and

Louis Gidel for their contributions and helpful discussions.

I

vii

CONTENTS

PREFACE................... iii

ACKNOWLEDGEMENTS . . .... ............. v

LIST OF FIGURES . . .ix

SYMBOLS. xi

I. INTRODUCTION. . . . . . . . . . . . . . . . . . . . . 1

II. OBJECTIVE ANALYSIS ON ISENTROPIC SURFACES . . 3

III. ISENTROPIC TRAJECTORY METHODS ..... ....... 7

A. Earlier Techniques. 7

B. Modifications Leading to Present Trajectory Package 8

IV. TRAJECTORY-DERIVED METEOROLOGICAL FIELDS. . . . . . . . . 16

A. Objective Analysis. ............... . 16

B. A Case Study . 17

V. PROGRAMMING ASPECTS ....... ........ . 23

APPENDICES

A. Transformation Procedure for Isentropic Coordinates . 27

B. Coordinate Conversion for Latitude and Longitude. . . 28

REFERENCES. ......... 30

ix

LIST OF FIGURES

Figure Title Page

1 Observed wind vectors and geopotential height contours 9

2 Trajectory stream pattern on 325 K a surface ... 11

3 M x 10 , observed wind and pressure analysis, 00 GMT 12

4 Same as Fig. 3 for 1200 GMT .............. 13

5 Stream pattern on 325 K e surface. . . . . . . . . . 14

6 Trajectory-derived w field on 325 K 0 surface. .. 18

7 Pa (x 10 ) analysis on 325 K 0 surface, 00 GMT. . . 19

8 Same as Fig. 7 for 1200 GMT. . . . . . . . . . . . . . 2Q

9 Trajectory-derived P0 (x 10 5 ) change field. 21

10 Same as Fig. 9, but contoured using a bilinear trans-form 22

11 NMC 47 x 51 octagonal grid . . . . . . . . . . . . . . 24

12 Program Flow Chart ... ....... ........ 26

xi

SYMBOLS

T temperature (K)

p pressure (mb)

6 -1 -1lR gas constant for dry air (2.8704 x 106 erg.g K )

c specific heat of dry air at constant pressurep (1.005 x 107 erg.g K- 1 )

K R/c (0.286)

e potential temperature (K)

eE equivalent potential temperature (K)

-2g acceleration of gravity (980.6 cmnsec 2)

z geometric height above sea level

2 -2M Montgomery stream function (cm *sec )

f Coriolis parameter

P8 potential vorticity (K sec mb )

w vertical velocity (cm sec )

-1Ce relative vorticity (sec )

V horizontal wind speed (mosec )

u component of the wind (mesec )

v component of the wind (m*sec )v component of the wind (mesec )

1

I. INTRODUCTION

Though isobaric surfaces represent the conventional way of analyzing

atmospheric flow patterns, air flow, when viewed upon these quasi-

horizontal surfaces, does not preserve the quasi-conservative parameters

within the atmosphere which characterize three-dimensional motions. The

quasi-conservative three dimensional air motions take place on surfaces

1000 KO = T (-- ) . Therefore, the most realistic approach for calculating

trajectories is to choose e as the vertical coordinate, i.e., to analyze

air flow on e surfaces (isentropic).

A numerical method for calculating isentropic trajectories, and the

objective analysis of meteorological parameters derived from such

trajectories are discussed in this paper. This isentropic trajectory

technique was originally developed by Danielsen (1961, 1966) and Danielsen

and Bleck (1967), and imposed conservation of total energy for the

parcel to derive trajectory motions. Subsequently, the technique has

evolved from a combination of hand and computer analysis procedures

into an objective computer method. Earlier modifications were due

primarily to the need for accurate analysis for trajectory calculations

whereby time-saving approximations used in other methods could be eliminated.

Bleck and Haagenson (1968) introduced an objective analysis technique

using multiple linear regression, developed by Eddy (1967), which enabled

them to do this. The basic reasons for the evolution, however, were

requirements for bulk production of trajectories, and trajectories on a

global scale. Bulk production (defined here to be trajectories calculated

from every point composing a large array of uniformly spaced grid points)

may be used for generation of vertical velocity fields and potential

2

vorticity change fields as calculated from changes in pressure and

potential vorticity of each air parcel. The global aspects pertain to

recent modifications employing analyzed winds provided by the National

Meteorological Center (NMC) and observed winds from radiosonde data.

The original trajectory package developed by Danielsen and Bleck required

use of the balanced wind (derived from the gradient of the Montgomery

stream function TM = cpT + gz), but could not be used at low latitudes

because of difficulties in solving the balance equation. Even at mid-

latitudes, the balanced wind direction is frequently shown to deviate

considerably from the observed wind direction, and the elipticity

condition required for solution is sometimes not met on the anticyclonic

side of strong jet streams.

The main objectives of this paper are to review and expand upon the

objective analysis method described by Bleck and Haagenson (1968) and to

summarize briefly the trajectory calculation techniques discussed in the

earlier publications of Danielsen and Bleck (1966, 1967). Because of

the limited availability of the 1968 technical note, the objective analysis

method is reviewed below. We then present a detailed description of

recent improvements and the requirements that led to their implementa-

tion. The merit of using the energy constraint is also examined.

Objective analysis applied to "trajectory-derived" meteorological

parameters is discussed and a case study is presented, The final section

describes programming aspects and specific information concerning use of

the computer program.

3

II. OBJECTIVE ANALYSIS ON ISENTROPIC SURFACES

The objective analysis method was designed to generate grid-point

data of 'Y (Montgomery stream function), balanced wind, pressure, and

relative humidity on isentropic surfaces in 12 h. time increments. The

general balance equation in x, y, 0, coordinates can be written as

3e '- + =0 Dv ()f %; - Vu a -v - (2 + u) 1)

where u and v are the x and y components of V, f is the coriolis parameter,

and f = df The basic procedure used to generate the analysis consistsdy'

of two steps:

1. Analyzed grid-point data fields of height, temperature, and

humidity on seven pressure levels (850, 700, 500, 300, 200,

150, and 100 mb), are provided by NMC. These data are co-

ordinate transformed to an isentropic framework by means of

the Duquet (1964) transformation procedure. (The equations

are given in Appendix A.) Due to the large vertical distance

between the pressure surfaces, the conversion process produces

approximate "first guess" grid point values of the desired

parameters on 0 surfaces with a maximum vertical resolution of

2.5 K on the NMC mesh.

2. Radiosonde data, including both mandatory and significant

level data, are processed by means of the same coordinate

transformation and then introduced to improve the analysis,

using a multiple linear regression technique.

Considering an individual 0 surface, each sounding location can now

be assigned a "difference" between the actual YT value inferred from

4

the radiosonde observation and the YM value obtained from the first-

guess field by interpolation. If the resulting field of differences is

sufficiently smooth (i.e., if it contains no features smaller than two

or three times the mean distance between adjacent radiosonde stations),

then the first-guess field can be improved by multiple linear regression.

However, first it is necessary to convert the difference fields, defined

above with their irregularly spaced data points, into fields of grid-

point values.

Given n'm observations of a variable Dk at n locations (k = l,...,n)

and at m different times (t = 1,...,m), the issue is to find the best

linear prediction of this variable at location (n+l):

n

Dn = k1kn+llt -l k Dkt

If "best" is interpreted in the Gaussian sense, then the following

expression must be minimized:

m7t=l

n

(Dn+lt k1 Dkt)2 = minNn+l-,t k-l k kt

(2)

By setting the derivatives of this expression with respect to the ak

equal to zero, the following n equations are generated:

1 (Dit% Dkt) k D Dt ( i=l.. m)Dkt) k = n+l,t it

k=l,...,n t=l,...,mt=l,.. .,m

If it can be assumed that

m 2A D

t=l

m mm D2 2

-L D2t - -= Z D2t - n+ltt=l t=l

(t=l�---�M)

5

the above equations can be rewritten as

n

k rik k = r,n+l (i=l,...,n) (3)k=l

where the r k are the spatial auto-correlation coefficients defined by

mZ Di

t=l it ktrik = - --

/m 2

Z Dit ktt=l t=l k t

Hence, the weights ak' by which the "predictors" D must be multiplied,k , Dkt must be multipliedk

are given by the solution of a system of linear algebraic equations

whose coefficients are the auto-correlation coefficients between any two

of the predictors.

If the data Dkt are homogeneous, the auto-correlation coefficient

becomes a function of only the distance dik between two predictors i,k.

Computations indicate that the auto-correlation curve most appropriate

for meteorological parameters in general is

r(d) = cos (12 -O.0015d (4)

where d is the distance (km) between two predictors.

To determine the grid-point values of a difference field from data

given at the radiosonde locations, each consecutive grid point is now

chosen as the location (n+l) in Eq. (2), whereas the n predictors are

defined as the radiosonde data surrounding that grid point. Given a

6

spatial auto-correlation function rik = r(dik), the weights k which are

needed in Eq. (2) to compute the grid-point value are found by solving

the linear system, Eq. (3).

After the grid-point corrections are computed in this way, the

"second-guess" fields can be generated by adding the corrections to the

first-guess fields. If the new fields are treated as the first-guess

fields were before (i.e., if an attempt is made to improve them further

by again comparing them to the actual radiosonde data), it is found that

the remaining spatial auto-correlation between the new difference values

is negligible.

Certain deletions and additions have been made in the analysis

since the Bleck and Haagenson (1968) report was published. The most

pertinent change is removal of the relative humidity field, thus eliminating

the ability to compute moist-isentropic trajectories discussed by Danielsen

and Bleck (1967). Two basic reasons for excluding moisture are (1) the

inclusion of moisture greatly increases the number of calculations; and

(2) multiple surfaces of equal equivalent potential temperature, 9E, can

exist (the conservation of equivalent potential temperature is a criterion

for moist-isentropic trajectory calculations).

The additions include objective analysis for the u and v components

of the observed or NMC-analyzed wind and analysis of thermal stability

-e that are needed for generation of potential vorticity fields. Potential

~p

vorticity, P0, in isentropic coordinates is defined as

P ( +f v Dau) (5)P0 = (%0

+ f -y)

7

III. ISENTROPIC TRAJECTORY METHODS

A. Earlier Techniques

Danielsen (1961) solved for isentropic trajectories using a graphical

technique. To construct an isentropic trajectory, he solved for the end

point of the trajectory by simultaneously satisfying the energy equation

(derived from the dot product of the horizontal velocity with the equation

of motion) and a distance-traveled equation. If diabatic and turbulent

mixing processes are neglected, the energy and distance-traveled equations

that must be satisfied are:

^2t2 I dt - 12 2 (| I t dt (M)2 M)1 _ (V2 V (6)

i (V1 + V2) (t2 - t1) _ (length of trajectory) (7)

where subscripts 1, 2 refer respectively to the beginning and end of the

trajectory, and V is the balanced horizontal wind speed.

Danielsen and Bleck (1966) introduced a computerized version following

Danielsen's earlier concepts. Important assumptions or stipulations

were:

1. The balanced equation was used for calculation of the wind.

2. YM varied linearly in time between initial and final configuration.

3. The trajectory curve is tangent to the balanced velocity vector at

the initial and final points.

8

B. Modifications Leading to Present Trajectory Package

An added option in the current routine allows for trajectory computations

in tropical and subtropical regions through use of NMC-analyzed winds

coupled with the observed rawinsonde winds?. The NMC wind fields are

derived from a combination of radiosonde observations, pressure-gradient

analyses for the geostrophic wind (outside of .the tropic zone), inferred

winds from cloud motions indicated by satellites, and forecast winds

routinely generated by operational models.

The balanced wind is more appropriate than the observed wind over

water (in extratropical regions) because the NMC analysis contains

satellite data pertaining to vertical temperature structure where actual

wind observations are extremely sparse. Conversely, over continents,

where rawinsonde wind data is available, observed winds should be used

because strong cross contour flow of the observed wind relative to the

TM gradient is frequently seen in analysis as illustrated in Fig. 1. If

the balanced wind is used for trajectory calculations over land areas,

the balanced-wind speed component can be combined with the observed wind

direction (this capability is included in the program).

When calculating a trajectory for an air parcel that has been given

precise initialization data (wind, temperature, pressure), as measured,

for example, by an aircraft or determined from hand analysis, an option

is available that allows for assigning initial values of wind and YM to

the parcel as. opposed to objectively interpolating for the values.

Trajectory-derived vertical-velocity, w, and potential vorticity

change fields require.bulk production of trajectories. In the computer

program, bulk production involves trajectories calculated from every NMC

9

Fig. 1 Observed wind vectors and geopotential height contours (m)

measured by a radar altimeter on board the NCAR Sabreliner

research aircraft flying at constant-pressure altitude

(285 mb) on 1 March 1979 along the California coast.

10

grid point contained in a rectangular area not exceeding 576 grid points.

Figure 2 shows a flow pattern for 9 January 1975 generated by 300 12 h,

trajectories on a 325 K 0 :surface. YM, oQbserved wind, and p analysis

corresponding to the data set used for Fig, 1 is given in Figs. 3 and 4.

For our purposes., changes in potential vorticity, (P0)2 - (Pe)l'

along air parcel trajectories are being investigated for evidence of

diabatic and frictional processes in the atmosphere that relate to non-

conservative properties of P0. Vertical velocity fields are calculated

by considering the 12 hr pressure change (P2 - pl)) for the parcels.

(Examples of contoured w and P0 change fields will be given in Section

IV.)

This technique implicity makes some assumptions about the charac-

teristics of the air motion used to compute the trajectories. For example,

the horizontal spacing of the radiosonde stations limits the scale and

magnitude of the absolute vorticity that can be resolved. Also, diabatic

and frictional effects are not calculated along an air parcel trajectory,

and e of the parcel is assumed to be conserved. The kinematically

determined change in Pe of the parcel along the trajectory is, however,

a measure of the importance of these neglected processes.

Another option in the routine is to remove the energy constraint

(Eq. 6) and solve only for the distance traveled (Eq. 7). If we compare

Fig. 2 (trajectories calculated without the energy constraint), with

Fig. 5 (trajectories generated with. the energy constraint), it is apparent

that unreliable trajectories often occur when a simultaneous solution is

sought. .This problem occurs most frequently where the total energy

gradient is weak, i.e., on the anticyclonic side of the jet. However,

11

Fig. 2 Trajectory stream pattern on 325 K 0 surface, 9 January

1975, generated by 300 12 h kinematic trajectories.

12

7 . 1' ...' :: '' ' ,!*,o

- *...**^"*^7^ '^-7

Fig. 3 T x 10 (solid lines), observed wind and pressure (dashed

lines) analysis on 325 K 0 surface, 9 January 1975, 00

:,--31

Fig.o, 39 I~~

~M x10( o ldlie),osrv e win an rsur dse

~~~~~~~~lns nlss n35K~sr f ae 9 aur 17,0

GMT.

13

Same as Fig. 3 for 9 January 1975, 1200 GMT.

f

I5

Fig. 4

14

Fig. 5 Stream pattern on 325 K 6 surface, 9 January 1975, generated

by 300 12 h trajectories. The trajectories were calculated

with the energy constraint.

15

since the trajectory curve is tangent to the velocity vector at the

initial and final points, the shape of the cuvye usually indicates its

reliability. The trajectory stream field shown in Fig. 5 is typical

when' either' the' balanced wind or the observed wirnd is used' in conjunction

with the energy constraint. Peterson and Uccellini (1979) discussed

this disconcerting aspect of the energy constraint and showed examples

of "energy-consistent" trajectories: originating from the same point but

following significantly different paths.

16

IV. TRAJECTORY-DERIVED METEOROLOGICAL FIELDS

A. Objective Analysis

Trajectory-derived meteorological parameters are contoured from a

network of data points assigned to the. temporal midpoint location of

each trajectory. Consequently, the data array locations are skewed

because of parcel accelerations and trajectory curvature.

Our first attempt to contour the irregularly spaced arrays was

through implementation of a bilinear transformation for a four-point

quadrilateral. Because such a transformation involves highly distorted

grid spacing which frequently results in computer-contoured lines that

cross, we now use the multiple linear regression method discussed in

Section II. The objective analysis technique basically follows that

described in Section II. However, minor variations make a brief des-

cription appropriate.

The first guess to the uniform-grid field is the trajectory-calculated

mid-point values (P0 change or w for example) assigned to each grid-

point location denoting the beginning of a trajectory. Then each skewed

midpoint location can be assigned a "difference" between the actual

value inferred from the trajectory and the value obtained from the

first-field by interpolation. The irregularly spaced difference field

is then converted into fields of grid-point difference values and applied

as corrections to the first guess, using the regression analysis. The

auto-correlation curve, r(d), used for computing the weights which

determine the correction at each grid point, is identical to that given

in Eq. 4, except that the correlation is zero at 750 km instead of at

1000 km.

17

B. A Case Study

The w field for 9 January 1975, 0600 GMT, objectively derived from

the trajectories,.shown in Fig. 2 and cQntoured following .the regression

analysis, is given in Fig. 6. The ascending- motion.along .the California

coast (3-4 cm sec ) can be related to the YM and p field of Fig. 3.

The analysis indicates. that an ups;tream trajectory off the California

coast would initially move toward lower pressure. During the second

6-hour period (downstream, Fig. 4), no significant pressure change for

the same air parcels is apparent.

Figures 7 and 8 show the Pe analysis for 9 January 1975, 00 and

1200 GMT. Figure 9 shows the corresponding P0 change field, again

contoured following a regression analysis of irregularly spaced locations.

The P0 change field given in Fig. 10 applies to the same data set but

was contoured using the bilinear transformation for a four-point quadrilateral.

Comparison of Fig. 9 with Fig. 10 indicates that the multiple linear

regression technique retains (i.e., does not smooth) the maximum and

minimum values.

18

Fig. 6 Trajectory-derived w field on 325 K 6 surface, 9 January

1975, 0600 GMT. Ascending motion is contoured with solid

lines,

19

Fig. 7 Pe (x 10P ) analysis on 325 K 8 surface, 9 January 1975,

00 GMT.

20

Same as Fig. 7 for 9 January 1975, 1200 GMT.Fig. 8

21

Fig. 9 Trajectory-derived P0 (x 10 ) change field corresponding

to same data set as Fig. 6. An increase in Pe of the

parcels is contoured with solid lines. Values were

contoured following a regression analysis.

46

.1./

·1~

- I · -I I -I --

& i A- -%.a

22

Fig. 10 Same as Fig. 9, but contoured using a bilinear transfor-

mation for a four-point quadrilateral.

23

V. PROGRAMMING ASPECTS

The three-dimensional analysis required to generate multiple Q

surfaces for multiple time periods uses significant amounts of computer

time and space. Because of computer space constraints, the analysis and

trajectory package consists of three concatenate programs.

Program I reads radiosonde data and "standard" pressure level grid-

point data from two NMC tapes, and compiles an output tape containing

data that have been coordinate-transformed into an is-entropic framework.

The radiosonde data tapes provide global coverage, while the most com-

monly used grid-point data tapes apply to the NMC octagonal grid area

shown in Fig. 11. Two other grid data tapes available contain data for

an equatorial projection circling the globe between 480N to 480S latitudes

and a south polar projection similar to the octagonal grid. The program

space allows for computations on a rectangular grid--not to exceed 576

grid points (the most commonly used is 24 x 24). It can be placed

anywhere on the globe when used in conjunction with the appropriate

tapes. (The equations for conversion of latitude and longtitude to grid

point coordinates are given in Appendix B.)

Program II uses data from the output tape of Program I and utilizes

the multiple linear regression technique to compile a tape containing

grid-point values of d6/dp, p, YM' balanced wind, and the u, v components

of the observed wind.

Program III, using the output data tape from Program II, constructs

backward or forward trajectories- and can employ all the option modes

discussed in Section III b. PF is analyzed in Program III instead of II

because more computer space is available.

24

5 10 15 20 25 30 35 40 45

NMC 47 x 51 octagonal grid. The pole point is I, J = 24, 26.

J

50

45

40

35

30

170W-25

20

15

I0

5

I

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I

Fig. 11

25

Program III A uses data tapes compiled by-Pr~ogr~am IIll a~nd contours

trajectory-derived w 'and V chang ye.elds ..Q!Iowing the' technique-describ~ed

in Sect ion I V. 'The'. owI .'hg't. for .'ll the',~ rgias, is. g ivanm -ini FEig. l12,

26

PROGRAM I

Input desired gridlocation and timeperiod.

Input radiosonde andNMC grid data tapes.

Call Rgrid (decodeNMC grid data).

Call Duquet. -

Call Stkadp. *

Call Wzzadp (decoderadiosonde wind data).

Call Sort (test data forcode errors).

Call Duquet. _-

Compile output tapeof values returnedfrom Duquet.

End.

SUBROUTINE STKADP

Call Manadp (decodemandatory levelradiosonde data).

Call Sigadp (decodesignificant levelradiosonde data).

SUBROUTINE DUQUET

Compute p, z, dO/dpand TM at 0 levels.

Call Winder (computeu and v at 0 levels).

PROGRAM II

Input NMC grid data(first guess 0 gridfields).

Input coordinate trans-formed radiosonde data.

Compute differenceUse function Bint tointerpolate values fromgrid.

Call Calcwt. _

Use weights returnedfrom Calcwt to com-pute corrections tofirst guess grid values.

Smooth final fields withfive point filter.

Call Baleq (computesbalanced 4 M.

Compile output tape ofTM' balanced "M, p, u,v and d 0 /dp fields.

SUBROUTINE CALCWT

Find eight nearestradiosonde locations toeach grid point.

Calculate weights Oakusing the auto-correla-tion function andappropriate distance d.

- II~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

PROGRAM III

Input 0 level, geographi-cal coord. and timelength for desiredtrajectories.

Input analysis neededfor calculatingtrajectories.

Call Gridi (computeslat. and long. for allNMC grid points ifbulk production optionis employed).

Call Vortic (computesPo fields).

Compile output tapeof values returned fromVortic.

Call Trajec.

End.

SUBROUTINE TRAJEC

Calculate first guess(kinematic) trajectory.

Call Traj (option toemploy energyconstraint).

Compute APO and Apfor each 12-hr trajec-tory. Use function Bintto interpolate valuesfrom analysis.

Compile output tapeof values needed togenerate w and APofields.

PROGRAM III A

Input values for de-riving w and APO fields.

Calculate w and timemidpoint of eachtrajectory.

Call Conver (computefirst guess grid field).

Call Object.

End.

SUBROUTINE OBJECT

Calculate differencesweights and correctionsfor deriving final wand APo fields.

Program Flow Chart.Fig. 12

27

APPENDIX A

TRANSFORMATION PROCEDURE. FORISENTROPIC COORDINATES

If T and P are known at two adjacent.-levels. i, i + 1 that bracket a

particular potential temperature level j, .then,

9 (T - B Pi K )

Tr =· - I

e - B 1000-

where

Ti +1 i- T.T -T

K pKi+ i

From T. and 0. we can derive P. defined asJ J J

1000Pi Q 1

J- KT.

Wind is assumed to vary linearly with height z where zj at any level

j can be derived by integrating the hydrostatic equation from some lower

reference level is such that,

c (T I -IT.i) Tz. z.; + j- i n + T. -3 i g n, - 1

where

.-.. K)

28

APPENDIX B

COORDINATE CONVERSION FOR 'LATITUDE AND LONGITUDE

1. Octagonal grid

The equations for conversion of latiLtude (4) and East Longitude

(X) to NMC grid point coordinates I, J are,

I = 1 - R cos (10Q - X)

J = J + R sin (100 - )p

where

R 31.4 2 tan ( 9 0 - +(J ) 2R = 31.42 tan (. 2 -= ) + (J - )2 J (I P P

and,

I = coordinate of

= coordinate ofJ = coordinate ofp

north pole in I-direction (Figure 11)

north pole in J-direction (Figure 11)

If I and J are known, ( and X are given by

C = 90° - 2 tan' R/31.42)

100 - tan- , ( ( J - .3 / (I - I) )p -p <

29

2. Tropical Grid

The tropical grid is on a Mercator projection. Given ( and X, we

have

I = + 15

J = yJ + 12

where

yJ = 11.459 In [tan(- + ] ) ]4 2

If I and J, are known, ( and X are given by

X = 5(I-1)

( = 2 arctan eY - 90

where

J - 1211.459

30

REFERENCES

Bleck, R., and P. L. Haagens.on, 1968: Objective analysis on isentropic

surfaces. NCAR.Tech. Note NCAR-TN-39, Boulder, CO., 27 pp.

Danielaen, E. F., 1961: Trajectories;: isoQbaaic, isentropic and actual.

J. eteor., 18, 470-486.

, 1966: Research. in four-dimensional diagnosis of cyclonic storm

cloud systems. Rep. No. 66-30, Air Force Cambridge Res. Lab., Dept.

66-30, Bedford, IA, 53 pp. [NTIS AD 632 668].

___._ , and R. Bleck, 1967: Moist isentropic flow and trajectories in

a developing wave cyclone. Rep. No. 67-0617, Air Force Cambridge

Res. Lab., Dept. 67-0617, Bedford, MA, 1-34 [NTIS AD 670 847].

Duquet, R. T., 1964: Data Processing for Isentropic Analysis. Technical

Report No. 1, Contract No. (30-1)-3317, Air Force Cambridge Research

Laboratories.

Eddy, Amos, 1967: Statistical objective analysis of scalar data fields.

J. Appl. Meteorol. 6(4), 597-609.

Peterson, R. A., and L. W. Uccellini, 1979: The computation of isentropic

atmospheric trajectories using a "discrete model" formulation. Mon.

Wea. Rev., 107, 566-574.