>. Avurip oo Cug, TR 76-3 ‘ : é Vol. TE Storm Surge Simulation in
Transformed Coordinates |
VOLUME I
Program Documentation
by John J. Wanstrath
TECHNICAL REPORT NO. 76-3 NOVEMBER 1976
y DOCUMENT |
Approved for public release; distribution unlimited. Prepared for
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COASTAL ENGINEERING =) 6 RESEARCH CENTER
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Reprint or republication of any of this material shall give appropriate credit to the U.S. Army Coastal Engineering Research Center.
Limited free distribution within the United States of single copies of this publication has been made by this Center. Additional copies are available from:
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Contents of this report are not to be used for advertising, publication, or promotional purposes. Citation of trade names does not constitute an official endorsement or approval of the use of such commercial products.
The findings in this report are not to be construed as an official Department of the Army position unless so designated by other authorized documents.
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READ INSTRUCTIONS T. REPORT NUMBER 2. GOVT ACCESSION NO|| 3. RECIPIENT'S CATALOG NUMBER TR 76-3
4. TITLE (and Subtitle) 5. TYPE OF REPORT & PERIOD COVERED STORM SURGE SIMULATION IN TRANSFORMED COORDINATES VOLUME I. THEORY AND APPLICATION VOLUME II. PROGRAM DOCUMENTATION
Technical Report
6. PERFORMING ORG. REPORT NUMBER
8. CONTRACT OR GRANT NUMBER(s)
7. AUTHOR(S) John J. Wanstrath, Robert E. Whitaker, Robert O. Reid, and Andrew C. Vastano
DACW72-73-C-0014
10. PROGRAM ELEMENT, PROJECT, TASK AREA & WORK UNIT NUMBERS
9. PERFORMING ORGANIZATION NAME AND ADDRESS
Texas AGM Research Foundation oe.) Box \H
College Station, Texas 77843 11. CONTROLLING OFFICE NAME AND ADDRESS Department of the Army
Coastal Engineering Research Center
Kingman Building, Fort Belvoir, Virginia 22060 14. MONITORING AGENCY NAME & ADDRESS(if different from Controlling Office)
A31231
12. REPORT DATE November 1976 13. NUMBER OF PAGES
WO ko LOS Woil, iti, LYo
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UNCLASSIFIED
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16. DISTRIBUTION STATEMENT (of thie Report)
Approved for public release; distribution unlimited.
17. DISTRIBUTION STATEMENT (of the abstract entered in Block 20, if different from Report)
18. SUPPLEMENTARY NOTES
19. KEY WORDS (Continue on reverse side if necessary and identify by block number)
Computer program Hurricane Gracie
Coordinate transformation Numerical modeling
Hurricane Camille Orthogonal curvilinear coordinates Hurricane Carla Storm surge
20. ABSTRACT (Continue on reverse side if necesaary and identify by block number) A two-dimensional time-dependent numerical storm surge model using orthogonal curvilinear coordinates is presented. The curvilinear coordinate system is based on a conformal mapping of the interior region bounded by the actual coast, the seaward boundary (taken as the 180-meter depth contour) and two parallel lateral boundaries into a rectangle in the image plane. Three regions of the Continental Shelf of the Gulf of Mexico and two regions of the eastern seaboard of the United States are mapped.
(Continued
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Since the transformation is conformal, the associated modifications of the vertically integrated equations of motion and mass continuity are minimized. The coast, seaward boundary, and the lateral boundaries of the computing grid are straight lines in the image plane thus facilitating the application of the boundary conditions. The final coordinates allow for the greatest resolution near the coast in a central area of principal storm surge development and modification.
The model is employed in the simulation of the storm surge induced by Hurricanes Carla (1961) and Camille (1969) which crossed the gulf coast of the United States and Hurricane Gracie (1959) which crossed the east coast. Ana- lytical interpretations of the wind and atmospheric pressure-forcing functions are used in the computations.
2 UNCLA B SECURITY CLASSIFICATION OF THIS PAGE(When Data Entered)
PREFACE
This report is published te vrovide coastal engineers with the results of a study to develop an oper. .onal program for numerical simulation of storm surges on a given segment of the Continental Shelf, using a curvi- linear coordinate system. The report consists of two volumes. Volume I discusses the theory and application of the transformation procedure for generating the curvilinear shelf coordinate system for particular regions, and the theory, numerical algorithm, and application of the storm surge program for simulation of Hurricanes Carla (1961), Camille (1969), and Gracie (1959). Volume II presents the program documentation and the coded programs for carrying out the coordinate transformation (CONFORM), for establishing the spatial lattice (GRID), and for carrying out the storm surge calculations on the shelf (SSURGE). The work was carried out under the wave mechanics program of the U.S. Army Coastal Engineering Research Center (CERC).
This volume was prepared by John J. Wanstrath; Volume I was prepared by John J. Wanstrath, Robert E. Whitaker, Robert 0. Reid, and Andrew C. Vastano, Department of Oceanography, Texas A&M University, College Station, Texas, under CERC Contract No. DACW72-73-C-0014. Most of the computational work in the development and application was carried out at the National Center for Atmospheric Research which is supported by the National Science Foundation.
The authors express their appreciation to Thomas J. Reid for assist- ance in program coding, and to Dr. D. Lee Harris, CERC, for very con- structive comments on the draft of this report.
Dr. D. Lee Harris, Chief, Oceanography Branch, was the CERC technical monitor of the report, under the general supervision of Mr. R.P. Savage, Chief, Research Division.
Comments on this publication are invited.
Approved for publication in accordance with Public Law 166, 79th Congress, approved 31 July 1945, as supplemented by Public Law 172, 88th Congress, approved 7 November 1963.
Colonel, Corps of Engineers Commander and Director
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CONTENTS
CONVERSION FACTORS, U.S. CUSTOMARY TO METRIC (ST)
I INTRODUCTION.
Il COMPUTER PROGRAM DOCUMENTATION FOR PROGRAM-CONFORM.
ONAN HRWNe-
Program Purpose Program Description .
. Type of Computer.
Input Data.
Mathematical peoeeduees a Soy ye eean fami eaeions Flow Chart.
Glossary and Description ioe Terms
Input and Output.
IIl COMPUTER PROGRAM DOCUMENTATION FOR PROGRAM-GRID .
ONIADMHHRWNe
. Program Purpose .
Program Description . Type of Computer. Input Data.
. Mathematical procedures rl Proce Lamitersions
Flow Chart. 6 Description of Teance Input and Output.
IV COMPUTER PROGRAM DOCUMENTATION FOR PROGRAM-SSURGE .
APPENDIX
ONAN HBWNE
Program Purpose . Program Description .
. Type of Computer.
Input Data.
. Mathematical Beawedhontes an ceoenam Lantlzerhions
Flow Chart. 5 6 Glossary and Doseeipelon “of Teens Input and Output.
A FORTRAN LISTING OF PROGRAM CONFORM.
B FORTRAN LISTING OF PROGRAM GRID .
C FORTRAN LISTING OF PROGRAM SSURGE .
CONVERSION FACTORS, U.S. CUSTOMARY TO METRIC (SI) UNITS OF MEASUREMENT
U.S. customary units of measurement used in this report can be converted to metric (SI)
units as follows:
Multiply by To obtain inches 25.4 millimeters 2.54 centimeters square inches 6.452 square centimeters cubic inches 16.39 cubic centimeters feet 30.48 centimeters 0.3048 meters square feet 0.0929 square meters cubic feet 0.0283 cubic meters yards 0.9144 meters square yards 0.836 square meters cubic yards 0.7646 cubic meters miles 1.6093 kilometers square miles 259.0 hectares acres 0.4047 hectares foot-pounds 1.3558 newton meters ounces 28.35 grams pounds 453.6 grams 0.4536 kilograms ton, long 1.0160 metric tons ton, short 0.9072 metric tons degrees (angle) 0.1745 radians Fahrenheit degrees 5/9 Celsius degrees or Kelvins!
' To obtain Celsius (C) temperature readings from Fahrenheit (F) readings, use formula: C = (5/9) (F — 32). To obtain Kelvin (K) readings, use forumla: K = (5/9) (F — 32) + 273.15.
STORM SURGE SIMULATION IN TRANSFORMED COORDINATES
Volume II. Program Documentation
by John J. Wanstrath
I. INTRODUCTION
Presented in this volume are the program documentation and list- ings of the coded programs for carrying out a simulation of a hurricane-induced storm surge on the Continental Shelf in curvilinear coordinates for a given reach of shelf. There are three separate programs detailed here for this purpose.
The first program, CONFORM, is employed for evaluation of the transformation coefficients which map the given reach of the Conti- nental Shelf into a rectangle in the image plane, the shelf region being delineated by a smoothed version of the coastline, the shelf break (taken as the 180-meter depth contour in the examples), and bounded laterally by two parallel straight lines. The theory and several example applications of the transformation procedure are given in Section II of Volume I. The example input for CONFORM con- tained in the listings here are for the mapping of the region from a section across Laguna Madre about 90 kilometers south of Brownsville to Marsh Island. Particular care must be taken to follow the pro- cedure exactly for the example if CONFORM is to be verified by obtain- ing the given transformation coefficients. This procedure is given explicitly in the CONFORM documentation.
The second program, GRID, develops the detailed computing grid information, based on the mapping coefficients evaluated by CONFORM plus coordinate stretching information supplied by the user (see Section III of Volume I). Part of the output of GRID is a listing of the grid positions which are required by the user in order to read from appropriate charts the detailed bathymetry field, which is necessary input for the final program SSURGE. The example data supplied here for GRID are for the Laguna Madre to Marsh Island region.
Program SSURGE (Shelf Surge) carries out the numerical integra- tion of the storm surge equations in the transformed coordinate system supplied by CONFORM and GRID, using a parametric representation of a hurricane wind field and pressure field. The theory is given in Sec- tion III of Volume I. The particular example data given here are for Hurricane Carla and the Laguna Madre-Marsh Island grid system.
The appendixes to this volume contain detailed FORTRAN listings of the three programs in this application. The data to be supplied by the user for other applications are discussed in the documentation of each of these programs.
II. COMPUTER PROGRAM DOCUMENTATION FOR PROGRAM-CONFORM
1. Program Purpose.
The purpose of the program is to determine the transformation coefficients which will conformally map the interior region bounded by the actual coastline, a seaward boundary curve, and two parallel lateral boundaries into a rectangle in the image plane.
2. Program Description.
The program is written in FORTRAN IV language. This program and the program GRID provide all the necessary computing grid data for input to program SSURGE. The program GRID takes, as input, the transformation coefficients and determines the computing grid infor- mation (such as, scale factors, grid point locations, and, at each grid point, the orientation of the &-axis to the x-axis).
The program CONFORM is composed of:
MAIN Defines constants. Reads and writes the coordinates delineating the given coastline and seaward boundary curve. Calls Subroutine COEFFS.
SUBROUTINE COEFFS Determines the transformation coefficients. At the completion of each iteration, the coefficients, the variance between the transform-generated curves and that Speci- fied, and other pertinent information are written. COEFFS interfaces all other program subroutines and functions.
FUNCTION XTRAN Is the transformation function x(&,n) FUNCTION YTRAN ' Is the transformation function y(é,n) SUBROUTINE SLFAC Determines the scale factor and derivatives,
ox/9—& , dy/d& , for a given value of €& and m6
SUBROUTINE CURLYB Determines the necessary parameters to fit a spline under tension through the given coast- line coordinates. The spline is fitted with Y2 as a function of X2
FUNCTION CURVYB Interpolates the given coastline, returning a value of y at a specified value for x.
FUNCTION CURDYB Differentiates the given coastline, returning a value of dy/dx at a specified value for xe
SUBROUTINE CUR2YB Determines the necessary parameters to fit a spline under tension with X2 as a function of coastline arclength.
FUNCTION CUR4YB Interpolates the given coastline returning a value of x at a specified value for arclength.
SUBROUTINE CURSYB Determines the necessary parameters to fit a spline under tension with Y2 as a function of coastline arclength.
FUNCTION CURSYB Interpolates the given coastline returning a value of y at a specified value for arclength.
There are identical subroutines and functions as delineated above for the seaward boundary curve specified by coordinates X2P and Y2P . These subroutines and functions are recognized by the same names as their counterparts with a terminal letter A. For example, SUBROUTINE CUR1YA determines the necessary parameters to fit a spline under tension through the given seaward boundary curve.
3. Type of Computer,
The program CONFORM can be run on any computer with minimum core requirements of approximately 24K (based on the present sample program). However, significantly more computer memory would be required if one desires a large number of coefficients and/or numerous integration points, The program has been executed successfully on IBM 360, CDC/ 6600 and 7600, and GE/635, The present sample program requires no auxiliary storage devices, peripherial devices, or magnetic tape input or output. No site-orientated computer plot. routines are involved in the program, Approximately 20 minutes of machine time on a CDD/7600 is required for the sample program (total number of coefficients,
2 x NMAX = 220; number of integration points, 0 < —€ <r , = 110; and number of iterations = 80).
4, Input Data,
Input data, other than constants defined in MAIN, are read in MAIN on IBM cards prepared according to the following list:
Gl) Gases i IWANT, MQ, MOP, NMAX, Continuation flag, number of shore- JMAS1, IL, VARWT line and seaward boundary coordinates
number of mapping coefficients, maximum number of iterations, number of integration points, and the convergence criterion in format 615, F5.3.
(2) Card Group 2
NZ NZ The x,y coordinates (units in x,y space) of the given coastline in the region 0 <x<A are read with one pair per card in format 3X, B76 Sho 764 (laine i'50)) .
(3) Card Group 3
X2P,Y2P The x,y coordinates (units in x,y space) of the given seaward boundary curve in the region 0 <x < are read with one pair per card in format F7.2, 3X, F7.2 (limit 150).
Optional Card 4, Card Group 3, and Card 6
If IWANT = 1, indicating the program is being re- started, the following cards must be supplied:
(4) Card 4
B, BZRO whe values of s)8ayand s Bossi units of length of x,y space from the last iteration of the prev- ious run in format 2E14.7.
(S) Card Group 5
COB, COC The NMAX cards containing the dimensionless Fourier-type transformation coefficients from the last iteration of the previous run in sequential order with one pair per card (format 2E14.7). If more coefficients are desired in the present run than the previous one, blank cards should be supplied for the difference.
(6) Card 6
WSX, WSY The value of the weighting factors for the sea-
WCX , WCY ward boundary curve (x and y component) and coastline (x and y component) from the last iteration of the previous run (4E14.7).
5. Mathematical Procedures and Program Limitations.
The conformal mapping relations, solutions for the transformation coefficients, and the iterative procedure for determining the coeffi- cients are presented in Volume I of this report. The mapping equa- tions are sufficiently general to treat the situation where either or
both given curves are multivalued in y for a specified x value. The only program limitation relates to the computer memory capacity.
No program error messages or consistency checks are incorporated in this routine.
Ge RlownGhantr
MAIN - CONFORM
Read constants; MQ, MQP, NMAX, IL, JMAX1, VARWT, IWANT
Read and write coordinates delineating the coastline and seaward boundary curve; X2 WA goal XA 5 AAP
a
inaktevauleezen Ay bee ROMs m NS X WCX, WCY , COB(N) and COC(N)
Read transformation coefficients and weighting factors
CALL SUBROUTINE COEFFS
SUBROUTINE COEFFS
Compute coastline arclength
Arclength computed directly from given coordinates, O25 2)
Yes Consequently, assumed that used has sufficient data points to accurately define the curve.
Stored in ARCYB
COASTLINE HAS FOLDBACK, TYB=1
No
_ CALL CUR1YB Arclength computed from a spline fitted through Y2 as a function of X2 with 9 additional points between the given coordinates to further resolve the curve. Stored in ARCYB.
CALL CUR2YB Fit a spline through X2 as a function of ARCYB
CALL CUR3YB Fit a spline through Y2 as a function of ARCBY
Compute seaward boundary arclength; the same
procedure as for the coastline is followed.
Arclength computed directly from X2P, WAP.
Stored in ARCYA
CALL CURLYA Arclength computed from a spline. Stored in ARCYA
CALL CUR2YA
CALL CUR3YA
Write coastline and seaward boundary coordinates and arclengths.
Compute CS and SN _ where
CS = cos (nké) , : for n=1,2 --+ NMAX SN = sin (nk ‘ Zl ee ema ES(iet) Gite) where dai sAi oo Wb/A.)\ MWh ChyGine or i=1,2 °° IL/2+1, IL odd.
Determine the x and y coordinates of the given coastline and seaward boundary curve from the arc- length of the transform-generated curves.
ATXB , ATBY and ATXA , ATYA
Compute B and _ BZRO
Compute COB and COC for n=1,2 -- NMAX
Write transformation coeffi- cients, iteration number, NMAX and IL.
Compute the error function and variance between the trans- form-generated curves and that specified. Write the values. The x and y coordinates of the given curves are deter- mined from the arclength of the new transform-generated curves for the next iteration. Write the values.
COMPUTED VARIANCE
ICOUNT = ICOUNT + 1
7. Glossary and Description of Terms.
Arrays
X2 Dimensioned MQ. Contains values of the x coordinate in units of x,y space of the given coastline (max, 150).
WZ Dimensioned MQ. Contains values of the y coordinate in units of x,y space of the given coastline (max. 150).
X2P Dimensioned MQP. Contains values of the x coordinate in units of x,y space of the given seaward boundary curve (max, 150).
WiZB. Dimensioned MQP. Contains values of the y coordinate in units of x,y space of the given seaward boundary curve (max. 150).
COB Dimensioned NMAX. Contains values of the dimensionless transformation coefficient BY (max. 200).
COC Dimensioned NMAX. Contains values of the dimensionless transformation coefficient ce (max. 200).
DUMB Dimensioned NMAX. Contains the iterative value for BO (max, 200).
DUMC Dimensioned NMAX. Contains the iterative value for Co (max. 200).
ARCYB Dimensioned MQ. Contains values of the arclength for the given coastline coordinates X2 , Y2 (max.150).
ARCYA Dimensioned MQP. Contains values of the arclength for the given seaward boundary coordinates X2P , Y2P (max.150).
SN Dimensioned NMAX x IL1 where IL1 is IL/2 for even IL or IL1 is IL/2 + 1 for odd IL. Contains values of sin (nké).
CS Dimensioned NMAX x IL1. Contains values of cos (nké).
ATXB Dimensioned IL. Contains values of the x coordinate of
the given coastline as determined from the arclength of the transform-generated curve (max. 400).
ATYB Dimensioned IL. Contains values of the y coordinate of the given coastline as determined from the arclength of the transform-generated curve (max. 400).
ATXA
ATYA
Dimensioned IL. Contains values of the x coordinate of the given seaward boundary curve as determined from the arclength of the transform-generated curve (max, 400).
Dimensioned IL. Contains values of the y coordinate of the given seaward boundary curve as determined from the arclength of the transform-generated curve (max.400).
Dimensioned IL. Contains values of & (max.400).
? Dimensioned IL. Temporary storage.
Constants
MQ
MQP
NMAX
IL
IWANT
B
BZRO
WSX, WSY
WCX, WCY
Number of coordinates delineating the given coastline (max.150).
Number of cgordinates delineating the given seaward boundary curve (max.150).
Number of transformation coefficients, BO or Cc. (max. 200). Number of equally spaced integration points for 0 < € < xX with IL > NMAX (max. 400).
If IWANT = 1, the program is to be re-started requiring input from the previous run. If IWANT # 1, it is the initial mapping of the region.
Desired variance (in units of x,y space squared) between the transform-generated curves and that specified. .
h = X2(MQ) = X2P(MQP) k = a/d
-B
B
BS
The x and y component of the weighting factor for the seaward boundary curve.
The x and y component of the weighting factor for the coastline.
8. Input and Output.
The input data required by CONFORM to determine the transforma- tion coefficients which conformally map the Laguna Madre, Mexico, to Marsh Island, Louisiana, region into a rectangle are presented here as an example, The first card image gives the continuation code, number of coastline and seaward boundary coordinates, number of coef- ficients desired, maximum number of iterations, number of integration points, and the convergence criterion, The next 47 paired numbers are the coastline coordinates and the last 40 card images give the seaward boundary coordinates,
Program CONFORM provides detailed and voluminous output concern- ing primarily the rate of convergence. These output statements are not necessary for program completion and can be easily deleted with little alteration to the sequence of instructions. Optional output statements are indicated in the program listing (App. A) by an arrow (<) on the right-hand side of the page.
The results from CONFORM required by Program GRID are 8, B_ , and the coefficients, B, and C, . These are given in the next section as input to program GRID.
Reference to Table 4 in Volume I of this report shows the values for the number of coefficients, maximum number of iterations, and number of integration points are only indicative of the final steps of this particular application. Explicitly, in order to obtain the given coefficients to conformally map the Laguna Madre to Marsh Island region into a rectangle in the image plane, the following steps must be followed:
(1) Set NMAX to 40 and IL to 80 for the first 20 iterations, (2) For the next 10 iterations, NMAX is 60 and IL is 120,
(3) Set NMAX to 80 and IL to 160 for iterations 31 through 50.
(4) Take NMAX as 90 and IL as 180 for the next 10 iterations,
(5) For iterations 61 through 70, NMAX is 100 and IL is 200. (6) Over the next 10 iterations, NMAX is 110 and IL is 220. (7) Starting with the 8lst iteration, WCX amd WSX are set
to 0 and NMAX is 110 and IL is 220 through the 100th interation.
(8) From iteration 101 through 110, NMAX is 130 and IL is 260,
(9) Over the last 10 iterations, NMAX is 150 and IL is 300.
Note that steps (7) through (9) utilize the alternate solution to the mapping equations (9) and (10) in Section II of Volume I of this report, The instructions which must be altered or removed are indicated in the program listing by parenthesis with the proper instruction enclosed.
The following input is required by CONFORM to conformally map the Laguna Madre, Mexico to Marsh Island, Louisiana region into a rectangle in the image plane. Note the fourth and fifth parameters on the first card image are only indicative of the final results. See Input and Output text for explanation of procedure used to obtain the mapping coefficients given as input to Program GRID.
Cc
Cc INPUT OATA
Cc 1 47 40 150 1500¢001 360200 110.00
(2 c
Cc Cc
Cc COASTLINE COORDINATES Cc SEAWARC ECUNDARY COORDINATES
Cc Cc
Cc Cc 000e00 C35e00 000-200 0CzZe00 005e60 040200 007¢50 OCE&e0C 010.50 044.80 016200 010200 018.200 C5100 021-250 014-50 028200 CSSe00 028200 0190200 040.200 O0€Ee80 036200 022200 050e50 C730e00 045-200 028-00 056-50 C7Ee50 054.00 022200 063-200 C8z2e00 060-¢00 036.00 070200 CS3e00 066200 040200 074.200 102.00 077.200 046200 076250 1C€e50 085e00 0£1200 079200 116200 089-00 055e00 082250 127200 095.200 0€4-00 9088.00 127.280 104.006 076200 095200 14€.00 110e00 084-00 106.200 154-00 11200 C8900 121200 1€2200 113-50 0S&.00 136-200 167200 114.50 1C1e00 153200 168.30 117.250 106.00 158e00 16€200 121-00 1120e00 161-50 167230 127.00 116200 166.200 166280 139-00 121-00 171200 169 00 153.00 122.50 183.200 168.00 165600 122200 197.200 16€.00 179200 119200 214.00 1€£206 193000 11560C0 230-00 16€-00 213-00 104-00 242.200 167290 229200 0922006 250200 1€€.270 244-200 080e00 253240 167220 257200 071200 257200 169050 267200 O0€€e00 265-00 169200 276200 062200 280200 166200 289200 O£€e00 291-00 1€2200 298200 049-200 296200 159e30 307200 042.200 298250 156290 318200 03£-00 303200 159-00 333200 027200 314200 193-00 347200 020¢00 323200 148.00 360200 014200 328200 142200 331-00 1372e20 335¢50 129220 342250 121200 349250 11€e00 352250 113200
III. COMPUTER PROGRAM DOCUMENTATION FOR PROGRAM-GRID 1. Program Purpose.
The purpose of this program is to determine the grid point array in the stretched curvilinear shelf coordinate system and appropriate scale factors needed for program SSURGE. The detailed grid is needed in order for the user to read off depths from an appropriate bathymetric chart of the shelf region at grid locations.
2. Program Description.
The program is written in FORTRAN IV language. This program interfaces between Programs CONFORM and SSURGE. It is assumed that the conformal mapping of the storm surge region has been completed to the user's satisfaction. The program GRID takes, in part, as input, the transformation coefficients and determines computing grid information of scale factors, grid point locations, and, at each grid point, the orientation of the €&-axis to the x-axis
The program GRID is composed of
MAIN Defines constants. Reads transformation coeffi- cients outputed from CONFORM. Reads water depths along a line near center of grid from the seaward boundary to the coast. Computes grid point locations, scale factors (uv, v , and FP), and, at each grid point, cos 6 and sine 6 Writes computing grid information.
SUBROUTINE XUT Writes information transferred into XUT.
SUBROUTINE SHCOR Determines and writes grid point coordinates in X,y space and the distance in nautical miles between points.
SUBROUTINE TRAN Computes the x and y coordinates of the trans- form-generated coastline and seaward boundary curve.
SUBROUTINE TRAN1 Computes x(&,n) and y(é,n)
SUBROUTINE TRAN2 Computes 9x/d— , dy/d— and 6 = tan-1 (22/28)
SUBROUTINE CURV9 Contains the expansion curve Y = Z + B(x") , where A,B, and C are constants. The term Y is either Sp (units, nautical miles) or T (units, minutes). The term X is cither S* (units, nautical miles) or T* (mits, minutes). This subroutine computes Y and dY/dX given the coefficients and X .
20
SUBROUTINE CURV1 - Determines the necessary parameters to compute an interpolatory spline under tension through a sequence of functional values contained in arrays X2 and Y2.
FUNCTION CURV2 - Interpolates the given curve, Y2 asa function of X2 , returning a value for Vi Salven gear
SUBROUTINE CURV3 - Determines the necessary parameters to compute an interpolatory spline under tension through a sequence of functional values contained in arrays X2P and Y2P.
FUNCTION CURV4 - Interpolates the given curve, Y2P asa function of X2P , returning a value for Va SAN CT Xane
Type of Computer.
The program GRID may be run on any computer with minimum core requirements of approximately 26K words (based on the present sample program appropriate to the Hurricane Carla surge simulation grid). GRID requires no auxiliary storage devices, peripheral devices, or magnetic tape input or output. No site-oriented computer plot routines are involved in the program. Approximately 25 minutes of machine time on a GE/635 is required for the sample program to determine the computing grid information. This time is based on the following pertinent program parameters:
a) 150 transformation coefficients, BO or C, >
b) 121 evenly spaced values of & for determining the transform-generated coastline arclength as a function OH EG 3
c) 51 evenly spaced values of n for determining the arc- length along a particular isoline of &€ as a function Oi 8
d) the computing grid of 47 € (or S*) lines and 15 n (or T*) lines;
e) for determining the scale factor F , the area in x,y space of each quadrangle is approximated by using 4 evenly spaced increments between €&€ isolines and 2 evenly spaced increments between mn isolines.
2\
For production runs, smaller sampling intervals might be required in) D, © eiatel Csjxscaailily, © -
4. Input Data.
Input data, other than constants defined in MAIN, are read in MAIN and are on IBM cards prepared according to the following list:
(1) Card 1
NMAX, NUMXI, Number of mapping coefficients, number of &
NUMETA, DELSS, lines + 2, number of n lines, AS* in nautical
SOPRA PDE ils miles; first value of S* in nautical miles,
ND, NS AT* in minutes, number of depths, number of points in format 314, 3F5.1, 214.
(2) Card 2
BETA BZRO The value of 8 and By from the conformal mapping solution in format 2E14.7.
(3) Card Group 3
COB, COC - The NMAX cards containing the Fourier-type transformation coefficients, BL and C , in format 2E14.7. ”
(4) Card Group 4
SY Temporary storage for the ND values of the water depth (fathoms) along a line from the seaward boundary to the coast. This information is needed to evaluate the traveltime coordinate T.
5. Mathematical Procedures and Program Limitations.
4
Information concerning the expanding grid procedure and the relations transforming &,n to S*,T* space is presented in Volume I of this report. The user is required to know the coeffi- cients of the expansion function
S, = A + B(S*)”
for each region of the curve where S. is arclength (nautical miles) along the transform-generated coastline. For the sample program, there are five regions of the expansion curve. Selecting AS* = 6 nautical miles, the number of AS* intervals of each region and the value of dSp/3S* at the end points of each region, we can determine the coefficients of each region from three simultaneous equations derived from the constraints:
22
For region I, 176 nautical miles < S* < 236 nautical miles (10 intervals of AS* )
OSD 2 6205 Ae 8
do*
dSp 35*
S p
Los) che | Sy
S ale Se Pp
For region IT, 236 nautical miles < (CMaintervallswore AS)
OSD 3 '1..8 at S*
dS*
eS aye Se
S aoe
For region III, 260 nautical miles < (7 intervals of AS* ) ;
3S*
dSp 3S*
S wy
Nos ge Ss
1.0 ae §=6O
302 nautical miles at S*
= 176 nautical miles
236 nautical miles
236 nautical miles
S* < 260 nautical miles
236 nautical miles
260 nautical miles
260 nautical miles
S* < 302 nautical miles
= 260 nautical miles
= 302 nautical miles
= 302 nautical miles
For region IV, 302 nautical miles < S* < 356 nautical miles (9 intervals of AS* )
SP 1.0 at S* SSP 1.0 aye }8=SR S = 302 nautical miles at S*
23
= 302 nautical miles
= 356 nautical miles
= 302 nautical miles
For region V, 356 nautical miles < S* < 452 nautical miles (16 intervals of AS* )
SSP 1,0 at S* = 356 nautical miles
asp
aoe = Zoll at S* = 452 nautical miles
S
p 356 nautical miles at S* = 356 nautical miles
A similar procedure is followed for the expansion function T=A+t BT*)°
where T is the long wave traveltime (minutes) along the particular isoline of &. The program assumes that there are, at most, two regions of the expansion curve with the second region being AT = AT*. The expansion coefficients are determined by the program. If the user desires only one region (i.e., T = T* and AT* = total long wave traveltime-number of n grid intervals), the program computes AT*.
24
6. Flow Chart.
MAIN - GRID
Define constants NMAX, NUMXI, NUMETA, DELSS, DELTT, SSTRT ND, NS, XLAMDA, SLMAP, and XIDUM
Read conformal mapping transformation coefficients, BETA, BZRO, COB(N), COC(N)
Determine and write transform-generated coastline
and seaward boundary curve coordinates for N4 evenly
spaced ivalues ods: a (Om< ew) ae Store in ATXB , ATYB, and ATXA, ATYA.
Determine and write coastline arclength ‘Sp (nmi) as a function of &.
Store Sp in X2 Seo = a WA,
CALL CURV1 Fits interpolatory spline through Y2 as a function of X2.
Defines number of expansion regions (IRG)
for Sp(S*) , coefficients A, B, and C and
number of AS* grid intervals of each region (IIC).
25
Determine appropriate values of & for evenly spaced values of S* .
For i = 1,2 -- NUMXI
* S5 = SSSTRT + (i-1)*DELSS
CALL CURV9
Computes Sp for a given S* .
FUNCTION CURV2 Returns a value for & given Sp. Arrays X2 and Y2 cannot be al-
, tered from the cast call of CURV1.
Store THO YC1l
E
Determine and write coordinates and arclength Sn (nmi) along a particular isoline of € (=XIDUM) for NS evenly spaced values of n ,
=) <S i S Bo S 15 Ao n tim WZ.
Store
Store n
CALL CURV 1
26
Read water depth (fathoms) at ND evenly spaced positions from the sea- ward boundary curve to the coastline.
Determine and write the long wave traveltime T (min) and distance S, (nmi) along XIDUM for the ND positions.
i ne 2P S fm | VWZIP 5 n
Store
Store
CALL CURV3 Fits interpolatory spline through Y2P as a function of X2P.
Define number of expansion regions (RG) for T(T*) ,
coefficients A, B, and C, and number of AT* grid
intervals of each region (IIC). If IIC(1) = NUMETA-1, compute AT*
27
Determine appropriate values of jn for evenly spaced values of T*.
For j - 1,2 .. NUMETA
* T, = (j-1) * DELTT
CALL CURV9
Computes T for a given T*
Function CURV4
Returns a value for S, given
T. Arrays X2P and Y2P can-
not be altered from the last call of CURV3.
Function CURV2
Returns a value for nn _ given
Sn AGkays exe andy 2yican
not be altered from the last call of CURVI1.
Store n in ETA.
The values of & and n are now known. Determine the computing grid data.
28
CALL SHRCOR Determines and writes the grid point coordinates in x,y space and the distance (nmi) between
grid points.
Store x coordinate in X and Store y coordinate in Y for i = 2,3 .. NUMXI-1 j = 1,2 .. NUMETA.
Determine and write scale factor u X,y space-nmi).
Store u in SX for i=1,2 .. NUMXI-2.
(units
Determine and write scale factor vw (units of X,y Space-time minutes). Store v in SY for j=1,2 .. NUMETA.
Determine and write dimensionless scale factor F. Compute area in x,y plane of each quadrangle subdivided into IQUAD intervals between E
lines and JQUAD intervals between n lines.
Store area in X for i=1,2 .. NUMXI-1° 1,2 .. NUMETA. =1,2 .. NUMXI-2. 1,2 .. NUMETA.
Store F anh SC Geo he
29
Determine and write cos 6 and sin 6 where @ is the orientation of the & axis to the x axis.
Store 6 in Z for i=1,2 .- NUMXI j=1,2 .. NUMETA.
Store cos 6 in X and Si -@ min SC stone ils .. NUMXI-2 is
1= j= NUMETA.
30
7. Description of Terms.
Arrays
All arrays except COB, COC, XI and ETA are reused throughout the
program.
The user is cautioned to consult each major program divi-
sion for assessment of array contents.
XO Ge Dimensioned NUMXI x NUMETA. X2,Y2 X2P,Y2P
ATXA,ATYA
ATXB,ATYB Dimensioned the larger of N4, NS, ND, or IQUAD.
Ne 135 (G ee
TIC Dimensioned IRG (the number of Sp(S*) expansion regions).
SX Dimensioned NUMXI-2.
SY Dimensioned NUMETA.
COB Dimensioned NMAX. Contains the conformal mapping transformation coefficients, Be
Coc Dimensioned NMAX. Contains the conformal mapping transformation coefficients, ore
XI Dimensioned NUMXI. Contains the values of & for determining the computing grid data.
ETA Dimensioned NUMETA. Contains the values of n for determining the computing grid data.
Constants
NMAX Number of conformal mapping transformation coefficients, Bo @e G@ . n n
NUMXI Number of computing grid € lines. With respect to the computing grid in Program SSURGE, there is an extra & line at each lateral end. This requirement results from the averaging procedure used in determining the grid data.
NUMETA Number of computing grid n lines. This is the same
number as in Program SSURGE.
3!
DELSS
DELTT
SSSTRT
ND
NS
IRG
The value in nautical miles of AS* . This corresponds to DXI in Program SSURGE.
Since in SSURGE the product, wAS* , is always computed in the surge equations, we determine DXI in meters such that u values are dimensionless, i.e.,
DXI = 1852 x AS * i cami | SLMAP —_2™ yy unit X,y unit The value in minutes at AT* . This corresponds to DETA
in Program SSURGE.
Since in SSURGE, the product, vAT* , is always computed in the surge equations, we determine DETA in meters such that v values are dimensionless, i.e.,
DETA = 1852 —4 nm
x AT* min. x SLMAP amt
x,y unit
The first value of S* in nautical miles.
The number of water depths (fathoms) inputed from the sea- ward boundary to the coast for determining distance as a function of long wave traveltime.
The number of points along XIDUM for determining n as a function of arclength Sh 3
Horizontal extent of the mapped region in units of x,y space.
The particular value of €&€ used in determining n(S_)
The chart scale relating distance in nautical miles to distance in x,y units (i.e., nmi is equivalent to 5l units of length in x,y space).
Acceleration due to gravity (feet-s~*).
The number of points used in determining &€ as a function of arclength Sp.
The number of expansion regions of SP(S*) or t(T*)
32
IQUAD The area in x,y space of each grid quadrangle is sub- and divided into IQUAD intervals between €& lines and JQUAD JQUAD intervals between n_ lines.
8. Input and Output.
The first card input to Program GRID gives the number of conformal transformation coefficients NMAX , the number of & lines NUMXI, the number of n Jlines NUMETA, AS* in nautical miles DELSS, the first value of S* in nautical miles SSSTRT, AT* in minutes DELTT, the number of water depths ND , and the number of points US _ used to establish n = n(S_)
The second card gives 8 and B, and the next 150 cards give the mapping coefficients B, and C, . These 151 cards are the punched output from Program CONFORM.
The remaining 31 cards are the depths picked off a bathymetric chart of the northwestern Gulf of Mexico. These depths are on a constant €&-line selected by the user.
Expansion coefficients, provided by the user, appear as statements within the program after format 135.
The reader must refer to Section III of Volume I of this report for an explanation of the parameters associated with the stretched shelf coordinate system for the Hurricane Carla surge simulation.
Output from GRID consists of the transform-generated coastline and seaward boundary coordinates, the transform-generated arclengths along the coastline and seaward boundary, and for each of the five sections, dSp/aS* , S* , Sp , and €&€ are listed. Additionally, the transform-generated arclengths at values of n for evenly spaced increments of T* and at n values for constant increments of T along the chosen —-line, the n values along the specified €-line such that AT* is constant, and the traveltime and depths along the constant é-line are printed.
The following output is required by Program SSURGE for simulating the Hurricane Carla surge. This includes the x,y coordinates of the grid intersections, the scale factors u and vw related to the trans- HORNACNOM Of & to Se ginvcl’ wy) wo We | megaeeemielhy, ie Seale iie- tor F , and the sines and cosines of theta giving the orientation of the €-axis to the x-axis. The program listing indicates when these may be punched or written on tape or disk for convenient input to SSURGE .
33
(s
Cc
Cc INPUT OATA 150 47 15 €eC17600 beO
Cc
2ellO9S529EE01 $e5078648EE01 — 403493873EE01-608176823EE01 -301662011EE01 105427266EE01 10242251 3EE0 1-101728220E600 — 50 06365C2EEC0-10447071 8E-02 106711 B64EE0 C= 106921 €334FE00 — 305976 884EE 00-6073 30E€28E-01 1041928E€0EE00 103667185EE00 -101794944EE00-200S13£91E-01 4e1190876E-01 1033577S3E-01 —307759354E-01 48326 202E-03 7eS716016E-02 £03212199E-02 -1200800307E-01 106050344E-01 —102900039E-C2=1 oS5SEB75E-01 —5eSE35S7EE-C2 164022560E-01 - 103795311 E-02-€ 03872108E-02 -305084819F-02 926181509E-02 606049823E-03-4 01274 305E-02 —1LeS301L171E-C2 201339S555E-02 Ge 3927 E01E-0 3-20 286E277E-02 -101320465E-02 263924600E-02 -100776871E-03=262€90670E-03 -102749315E-CZ €032075S5E-03 400667667E-03 104115754E-03 —106674068E-03 40€354670E-04 3099555 78E-0 4-206 3€ SEB TE-04 -1288E€218SE-C3 5e682€825E-04 900772255E-04 70€038S04E-04 10559158 3E-0 3-4 09427928E=04 Gel903310E-C4 30343SCS4E-04 =101438246F-0 3-1 60257530E-04 30e7952217E-04 167815754E-04 -607125224E-04 9.487E230E-05 4037 12E674E-04-507567649E-06 -Se8352281E-C4 10470€E47E-04 108351595E-04-3021 298S4E=-05 —205129E44E-04 300412098E=-04 = 100435642E= 05-2 04449992E-04 100222874E-05 20632669 26E-04 —5e8439003E-0€-10570C8E2E-04 —102788127E-CE 105418€1 6E-04 = 206134 267E-0 £=9 03255390E=05 -1204402S52E-CE 100334477E=04 = 500798231 E-6 €=502775377E-05 -203834472E-0F 3024€6€107E-05 20 6851102E-C £-206723854E-06 ~ 40311551 9E-0£-2003€ 34£2E-06 206351927E-O5 1014225£3E-05 — 203699234E-05 263080753E-06 10060475 2E=-C £-2081665S6E-06 -121909960E-0£F £0396716€2E-06 7e96S51199E-06-703427355E-07 —904350325E-0€ 26056501 4E=-06 205921 445E=-06€-208028600E-06 ~ 30163786SE-CE £020276€6E-06 800697 328E-07-3020493£3E-06 ~3elS5S61S70E-0E€ 20976EE27E-06
31
Si
200821 353E-CE€-3010024£2E-07 —20e7802158E-C6 7037£3£ 32E-07 201309951 7E-CE-7e16EE7C2E-07 —601223962E-C7 124500429E-06 — 108451 7125-C7-10e11971C7E-06 ~303654205E-07 1eldDLES6SE-06 Oe S384S1 SE~0E-405424002E-07 —~3001203E€4E-C7 502354470E-07 ~3e6734064E-056—3e5E2570SE-07 —1041408SCE-C7 4024405E5E-07 703883 EB8SE-CE-10e2054ES8E-07 —3022624E€6E-C7? 5e8860751E-08 20 7314736E-C7? 1200232€4E-07 —20802569SE-C7-202327544E-08 2e23410S7E-07 126C852806E-08 —904987154E-Ce& &.2818260E-08 109075643F-0 E-€ 022565 73E-08 —3e8SSIO06E-C2 706237684E-08 102275 724E-CE-407253453E-CB —202807ESSE-OEF 50952221 76-08 =407725860E-05-3eS1 76467E-08 ~1e 305SS312E-CE 42.9413841F-08 1063466543E=-0 €-10e21 40226F-08 — 301 784570E-CE-4.01840C8E-11 2e5290277E-CE 102313703E-08 20 8336831 E-C e-£ 054265 78E-09 le 39OLSES4E-CE 402592CS21E-09 “100413 04SE-CE 206€28°344F-095 2094255S57E-05-329004823E-09 ~204943488E-CS 7205014S6E-09 ~201406560E-05-6 00237420E-09 1e85C877SE-CS Ee3SS4S90E-09 —307924576E-0 $-6092481S54E-09 1eS02798S5SE-CS Ee0SEE2ZCEE-09 —9e6309918E-10-205732490F-09 -105450222E-0S 12536S5158E-09 1e7949E78E-CS 3¢730S565E-10 —204545E€29E-CS=-S eS SO4IELE-10 109894453E-CS $e3920S30E-10 —20e0184501E-09-€ 03272S67F-10 103540433E-0S 70495€038F-10 ~10e18976S8E-0S—-2e61004€ 3E-10 5e6070780E-10 GSe228SSHEE-11 20872681 8E-1C 30045&C69E-10 ~Se8993701E-11-2e76E4ESOE-10 9eS877297E-11 £olB3SE14E-10 -109201876E-10-308510567E-10 6e7366526E-11 364544254F-10 ~8eS2129S1E-11-201782821E-10 —306849378E-12 12826C0446E-10 ~30e18S53177E-12-00 295704 3E-11 -5e8541698E-11 6GelfESB99F=-11 Se4400211F-11-720616809E-12 —7064321740E-11 203844000E-12 5Se4853889E-11 1e7E€02C39E-11 —5e9319842E-1] 1-101 7E3087E-12 3eSO82522F-11 7e7SEESG7T7TE-12 —303595148E-11 304384777E-12 109693600E-11 368907201E-13 ~2006267S8E-11 SeE511549E-12 100770427E-1 1-1 0€112022E-12 —100974893E-11 €e770SE36E-12
34
30 7490653E-12-500353240F-12 -306€CS5S470E-12 801811602E-12 = 405852351E-1 3-£ 0 946SE72E-12 = 704799452E-12 7643025436-12 = 10297923€E-12-SelE12560E-12
90 3319ES4E-14 503514749E-12 -8e512744B8F—1 3-3 0465ES60E-12 —4e6265112F-12 202230€44E-12
2074291 5AF-1 2-10 365E647F=12 —105414227E-12 Ee 782235HE-13
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Le 42H7FF3E-12 60296€02464F-13 = 104235914F-1 2-1 69242733E-13
8e0951983E-12 204162215 =-13 —6074564SEE-12 161457465E-13
206952090E-13-165477038E-13 —10687.7584E-12 460572501E-13 = 909264500F=1 4-3 0736£145E=13 609077629E=14 “40 7001E€51E=13 - 102943830E-12-2040€C812E-13
10 7026387E-14 20511€468E-13
102898776E-15=102325175E-13 -100197686E-13 $e0332572E-14
90003627 4E- 14-6 03540502E-15 = 10253069SE-12-1 00823757E=15
100618124F-12 4643203E8E-14 —101324192E-12-163624101E-14
7e0773E€32E-14 20450EE13E-14 —602031906E-14 30025€047E-15
304 432727E-1 4-1 01037802E-14 -1097S2800E-14 3e7E0CE72E-14 956 636 53e¢ 460 390 32e 300
IV. COMPUTER PROGRAM DOCUMENTATION FOR PROGRAM-SSURGE. . 1. Program Purpose.
The purpose of the program is to numerically simulate the storm surge in orthogonal curvilinear coordinates with the vertically integrated form of the quasi-linear long-wave equations.
2. Program Description.
The program is written in FORTRAN IV language. It is assumed that the conformal mapping of the region under investigation has been completed. The transformation coefficients for three regions of the continental shelf of the Gulf of Mexico and two regions of the eastern seaboard are provided in Appendixes A and E in Volume I of this report.
An interfacing program is required which inputs the coefficients and generates a curvilinear computing grid to the user's satisfac- tion. The output from the program (and, in part, also the input to SSURGE) must be the scale factors, grid point locations, and, at each grid point, the orientation of the €-axis to the x-axis.
The Program SSURGE is composed of:
MAIN Defines constants and interfaces the subroutines.
SUBROUTINE ZERO Initializes all arrays to zero. SUBROUTINE FIELD Reads data and writes the water depth field
relative to mean sea level, the wind field parameters and the storm positions.
SUBROUTINE WINDF Calculates the wind and atmospheric pressure fields.
SUBROUTINE ELEV Computes the water tevel anomaly, H . SUBROUTINE FLUX Computes transports, Qox and Qn ¢ SUBROUTINE DRAW1 Outputs H and vertically averaged water
velocities, Qou/D and Qx/D ,
at hourly time intervals and saves the water level anomaly at prescribed grid locations for output at program completion.
35
SUBROUTINE METER Calculates and saves the vertically averaged water velocities at prescribed grid locations for output at program completion.
SUBROUTINE HUV Outputs the saved information of the simu- lated hydrographs, simulated current meters, and observed water levels at program completion.
Type of Computer.
The program SSURGE may be run on any computer with minimum core requirements of approximately 30K words of memory (based on the present sample program appropriate to the Hurricane Carla surge simulation). The program has been executed successfully on the IBM 360, CDC/6600 and 7600, and GE/635. The present sample program requires no auxiliary storage devices, peripheral devices or magnetic tape input or output. No site-dependent computer plot routines are involved in the program. Approximately 14.4 minutes of machine time on a GE/635 is required for the sample program to complete 66 hours of surge simulation.
Input Data.
Input data, other than constants defined in MAIN, are read in SUBROUTINE FIELD. These data are on cards prepared according to the following list.
(1) Card 1
NT1 - Number of cards (max.50) containing on each TIM, ROT, RAD, VRMAX and PZRO (format I5).
(2) Card Group 2
NT1 cards with each card containing values of
a) TIM The time in hours at which the hurricane wind and atmospheric pressure parameters are recorded (format F10.1).
b) ROT The angle in degrees between the direction the storm is moving and the region of maximum winds (format F10.1).
c) RAD The distance in nautical miles from the storm
center to the region of maximum winds (format F10.1).
36
(3)
(4)
(5)
(6)
(7)
d) VRMAX Maximum observed windspeed in knots (format
F10.1).
e) PZRO Atmospheric pressure in millibars of the storm
center (format F10.1).
Card 3
NT2 - Number of cards (max.150) containing on each
TIMPOS, XPOS and YPOS (format I5).
Card Group 4
NT2 cards with each card containing values of
a) TIMPOS The time in hours at which the hurricane
position is recorded (format F10.
A)
b) XPOS The x-coordinate in units of x,y space of the hurricane center (format F10.1).
c) YPOS The y-coordinate in units of x,y space of the hurricane center (format F10.1).
Card Group 5
GRID2 The fluid depth in fathoms relative to level along each column, i=1,2..IM, is a nested do-loop for j=1,2..JM (format The program will zero those values for prior to computations. The depth data
mean sea read with 11F7.2). even itj are posi-
tive numbers which the program converts to nega- tive values (in meters) to be consistent with
the coordinate system.
Card Group 6
S The values of the dimensionless scale factor relating
the (x,y) plane to the (&€,n) plane are read along each column, i=1,2..IM, with a nested do-loop for j=1,2..JM
(format 5E14.7).
Card Group 7
DSDXI The values of the dimensionless scale factor, u,
transforming &€ to S* are read with per card for i=1,2..IM (format E14.7).
SiG
one value
(8) Card Group 8
DTDET Values of the dimensionless scale factor wv transforming n to T* are read with one value per card for j-1,2..JM (format 2X,E14.7).
(9) Card Group 9
HOBS1 The values of the observed hourly water level
in feet at grid location (IH1, JH1) are read with 19 values per card in format F4.1.
(10) Card Groups 10 through 14
HOBXk The values of the observed hourly water level in feet at grid location (IHk, JHk) are read sequentially with the same format as above.
(11) Card Group 15
YOSo 2 LAL The paired x,y coordinates (units in x,y space) of the computational grid points are read along each column, 1-1,2..IM, with a nested do-loop for j=1,2..JM (format 10F7.2).
(12) Card Group 16
COSG, SING The paired values of the cosine 6 and sine 6 where 6 is the angle between the € and x axis are read along each column, i=1,2..IM, with a nested do-loop for j=1,2..JM (format 10F8.5).
A computer printout of the sample program and data cards are given later in this section.
5. Mathematical Procedures and Program Limitations.
The storm surge equations, solutions, and algorithms are described in the text of Volume I of this report.
Basically, the model utilizes a centered difference, leapfrog analog of the vertically integrated, quasi-linear form of the long-wave equations. The algorithm treats the time dependency explicitly and employs a computing lattice in which the transports, Qox and Qa >» are computed at the same location
but are staggered in time and space with respect to the water
38
level anomaly. The program assumes that seabed scouring does not occur. No program error messages or consistency checks are
incorporated in this version.
Flow Chart. MAIN - SSURGE
Define constants
CALL ZERO Initializes all arrays to zero
CALL FIELD Reads data and writes depth field, wind field parameters and storm positions
CALL HUV Output saved information of simulated hydrographs, simulated current meters and observed water levels.
39
CALL WINDF computes wind and atmospheric pressure fields WIND(i,j) , P(i,j)
CALL ELEV computes water level anomaly GRID1 (i,j)
CALL DRAW1 _ outputs H, Qcox%/D and Qrx/D at hourly time intervals and saves information for output at program completion
CALL FLUX computes transports, GRIDI. (i,j) and GRID2(i,j)
40
Glossary and Description of Terms.
Arrays
GRID1 Dimensioned IM x JM. Contains H values in meters
at oad a) and Qo in meters squared per second at even itj
GRID2 Dimensioned IM x JM. Contains fluid depth values, Dy » in meters at odd itj and Qps in meters squared per second at even itj
S Dimensioned IM x JM. Contains dimensionless scale factor, F, determined from the conformal mapping of (x,y) space to (&,n) space.
DSDXI Dimensioned IM. Contains the dimensionless scale factor, u, transforming §& to S*
DTDET Dimensioned JM. Contains dimensionless scale factor, v , transforming n to T*
XX Dimensioned IM x JM. Contains the x-coordinate in units of (x,y) space of the grid point locations.
YY Dimensioned IM x JM. Contains the y-coordinate in units of (x,y) space of the grid point locations.
COSG Dimensioned IM x JM. Contains cosine values of 0 relating the orientation of the &-axis to the X-axis at the computational grid points.
SING Dimensioned IM x JM. Contains sine value of 6 relating the orientation of the &-axis to the X-axis at the computational grid points.
WIND Dimensioned IM + 1 x JM. Contains values of the windspeed in meters per second. At a transport computational grid point (i,j), the S*-component | is stored in WIND at (i,j) and the T*-component is stored at (itl,j).
P Dimensioned IM x JM. Contains values of Hg (the hydrostatic elevation in meters of the sea surface corresponding to the atmospheric pressure anomaly) and stored at odd itj.
HOBSk Each array is dimensioned 67. Contains the observed
en.2. 6 hourly water levels in meters at grid locations a (IHk, JHk).
4I
ROT
TIMPOS
XPOS
YPOS
Constants
IM
JM
NMAX
Each array is dimensioned 300. Contains values of the computed water level H in meters at grid locations (IHk, JHk).
Each array is dimensioned 300. Contains values of the vertically averaged water velocity, Qc»/D , in meters per second at grid location (ITk, JTk).
Each array is dimensioned 300. Contains values _ of the vertically averaged water velocity, Qyx/D , in meters per second at grid location (ITk, JTk).
Dimensioned 50. Contains the time in hours at which the wind and atmospheric pressure field parameters (ROT, RAD, VRMAX and PZRO) are recorded.
Dimensioned 50. Contains the angle in degrees between the direction the storm is moving and the region of maximum winds.
Dimensioned 50. Contains the distance in nautical miles from the storm center to the region of
- maximum winds.
Dimensioned 50. Contains the maximum observed windspeed in knots.
Dimensioned 50. Contains the atmospheric pressure in millibars of the storm center.
Dimensioned 150. Contains the time in hours at which the storm position is recorded.
Dimensioned 150. Contains the x-coordinate in units of (x,y) space of the storm center.
Dimensioned 150. Contains the y-coordinate in units of (x,y) space of the storm center.
Number of grid points in the S* direction. Number of grid points in the T* direction.
Maximum number of time steps.
42
INC
DXI
DETA
DELT
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ITk, JTk k=1,2..6
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PINF
Comments
Number of time steps between saving of surge results in HYDk, UCMk and VCMk. INC must be an even integer number.
Grid increment in meters in the S* direction.
Grid increment in units of meters in the T* direc- tion. The units of DTDET-DETA (that is, v-AT*) are in meters.
Time increment in seconds. DELT must be less than that required for numerical stability and, also, be an even integer multiple of 60 minutes.
Acceleration due to gravity (=9.8 meters per second squared).
Dimensionless seabed drag coefficient (=0.0025).
Grid point location for saving the computed water level. The sum of the indexes must be odd. Index THk cannot exceed IM. Index JHk cannot exceed JM.
Grid point location for saving the vertically averaged water velocities. The sum of the indexes must be even. Index ITk cannot exceed IM. Index JTk cannot exceed JM-1.
Coriolis parameter (=6.70875 x 10 © second 1) for latitude: 27s 256252! Ne
Wind ingress angle in units of degrees.
Far field atmospheric pressure in millibars (=1016 millibars).
1. A symmetric analytical hurricane wind field representation as given by C. Jelesnianski (1965, A numerical calculation of storm tides induced by a tropical storm impinging on a continental shelf, Mon. Wea. Rev., 94, 379-394) is employed in the surge model).
2. The wind stress coefficients, K, and Ky , are 1.1 x 107° and 2.5 x 107-© and are defined in SUBROUTINE FLUX.
3. Constants YRANGE, THIT, XHIT and YHIT are not used in this program version.
43
8. Input and Output.
Data statements in Program SSURGE establish the number of computa- tional points, number of time steps, output interval, values of the grid and time steps, acceleration due to gravity, bottom stress coeffi- cient, and locations of the simulated hydrographs and current meters, and the Coriolis parameter corresponding to latitude 27° 23.232' N.
Card input provides the number (NT1) of hurricane description sets, followed by NT1 cards giving the time (in hours after start of com- putations) and the three required storm parameters. These are suc- ceeded by one card giving the number (NT2) of hurricane positions to be used, followed by NT2 cards providing the time and storm center positions in the original arbitrary Cartesian grid (see Volume I of this report). NT1l need not be equal to NT2, nor must the observed storm parameters and positions coincide in time.
For the Carla surge computations, the hurricane characteristics are stipulated at 6-hour intervals for the first 18 hours and at 3-hour intervals for the remainder of the 66-hour prototype time simulation. Note the radius to maximum winds are in nauttcal miles, the maximum winds are in knots, and the central pressures are in mtllibars. Due to the erratic movement of Hurricane Carla, the coordinates of the center of the storm are specified at hourly intervals, except for a single 6-hour interval spanning the end of the simulation. These coordinates are specified in x,y space (100 units = 219 kilometers).
The depth field, in fathoms, is introduced followed by the array of scale factors, S . The array of scale factors, DSDXI (an alias for u ) is read along the specified row and the scale factor array DTDET (an alias for wv) is specified along the chosen column.
The following six arrays are the observed hourly water levels (in feet) from Padre Island (HOBS1), Aransas Pass (HOBS2), Port O'Connor (HOBS 3), Pleasure Pier, Galveston (HOBS4), Mud Bayou (HOBSS), and Sabine Pass (HOBS6). These data are not necessary for any phase of the surge calculations and may be omitted.
The last two arrays input to SSURGE give the coordinates (XX,YY) of the grid points in units of x,y space, and the sines and cosines of theta at the computational points.
All input not in the MKS system of units is converted internally to the MKS system.
The storm parameters, hurricane center coordinates, and depth field are printed out immediately following input. At hourly time intervals, the water level anomalies and the depth averaged velocity component fields are printed. The six simulated hydrographs and current meters are printed out with their positions and the corresponding observed water levels at the completion of the surge simulation.
44
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APPENDIX A
FORTRAN Listing of Program CONFORM
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CT1)MkS*O = €T)N*S*U = YWNS =VANS
( aAouay ) YVeS/EXIAD =XIM ( 8AOWdY ) 4VGS/VXS3 =XSM uVa3/9A43 =ADM
YVGS/VA43 =ASM
O° v7 4335 =4VER
GA 3354GX55S+VASI40 X49 =43 ADM*xLA*GABWNS =GAS4S XIJMeLd*DXSWNS =EX33 ASM* 14% VABWNS =VWASS XSMeLIeVXAWIS =VWX 43 GAWNS*S*U = FAAWNS =BASWNS OXWNS*S°O = GX3SWIS =8XaWNS ¥AWNS*S°O = VARNIS =VABWNS VXWAS&eS°O = WX3WNS =v X3WNS H)v4a9S =(L1)A
vovaoS =C(11)N
c*x( CCT Me (T)GALVISESY )*xS°O exe( (CCTPVLECTIZ+( 136K 4VISBV 2*S°9 cxx( CET )SH=CT)VALYV) SEV bkS°O CxeC COTY (CL IZ4¢CT DVXLVISGV eS °0
GAWNS + DAIZWNS =GASWAS BXWNS + GX3N1VS =X SINS VAWNS + VARANNS =VASWNS
VXWAS + VXSWNS =VX3WNS
ceux (CC LI) M@CLIDGALVISaV) =GAWIS
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dic
85
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GGONX*ENNS =8NTV
: VOINX*VANS =VN1V
CO °2/ IX TS0x( CITA + CLIDA } + BWAS =AwWAS O°c/I XTS0%( €11IN + (4109 ) + VWOS =VWWAS T+il =I
eZ2 Gi OD ( TI 23° LI )SI
cxx (CLT )Me( LI) GALVY SGV + GANVA =GAUVA Cex EC LIDL=C LI YGXLVY SGV + 3XYVA =GXHVA CexX(CLIIS=(LIDVALV JSG + VAYWA =VAUVA c*xx¢€ CLI Do@C LI) VKLVY }SEY + VXHWA =VXUVA
( 41 *© VWWOIS * GNIV )JAASYUND =CLI)GALY
€ Ml * WWOIS * GNIV JdAHYND =C4L1)EXLV
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( 41 © IWNOIS © UNIV FVAHHND =(LI)VXLV SNW =CLIDA
VNIv =( LI )x
ei
Vl*t=il e242 Od
0°0 =BAXNVA
O°0 =GXYVA
O° OG =VANVA
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0°Q =Gwns
0° 0 =vwns
eid
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; (SdX3/° l™=Sdx3)*S° = SHNIS €S73G)dX3 = SdX3Z
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T4715 = IddItsS
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(t}exe(ec)ex = TxX130
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(c@N) cxeCINN)ICX = TWN13d
(TAN) CX#(N) CX = NWBC
CEVCAXED + (ZIZAXED + (TIZAKTD = TddIs EXTVAG/CIXTVSAG/ TX Wad = £9
ex WaG/TIX 139d 721X150 co
TX V304 21X 1307 ( IX 130+21X 150 }= = 13
€ 1) oxX=(t)ex = CIXIG
(c)exel(E)ex = 2x1ad
a ai) GS SWEIPIN Shi!
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I=tdN = AVEI UG eS i ap OG
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eSVIG = t9VId
ecxG = 1XG
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(SHNIS=( (SdxX 3/°T+Sdx3)*S* )kS THO) *XNIANIS = COVIG
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ina)
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(SHNIS=((SdX3/° T+SdX3)*¥S°)*S 7139) HNIANIS = ¢cOVIAG
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OVIGdS*NIJVIIJ = ¢€T)S
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CT YGADUVE(EVIAD
uv = 2txX1sad
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Nd 1S -= NddJs
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2x 130/72 UX G/T XK 1aG=@ = ED eXTaG/ IX 13d/2TXtTaG = 2d
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N
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(Tel )GADNV=L = T1130 F
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S/C THN) * (VWOTS )S (TIEAD av =¢
N= I SANILNOD 2 GADNV )4I MS 2 8 OG 8 1°05¢41) 41 dv = dvWOIS N)3A.uW = S OW =N
°3d0530 GATWSD 38 LSAW SAEXYND SNILNONHSNS *HLONAWDYY YVIADILadWd Sil iv
A JO SN IVA SHEL SI BASUND NI GSANYUNLAY AN WA JHL
uO4 ANWA NSBAIO Vv itv SANIMWMSVOD SHL S3LV TOdy¥siNI DW * cA * ex
GAD Ga /Qa
°415N3 D9V SHL NOIFLONAA STHL ZEASNOWWOD 43 Ad /NOWWO)D
NHOS SNOWWOD
2) 2) >)
100
(S139 )dx3 Sdx4 eX TAGxXdGVWOILS $130 CX HIAZ( (EF dCASCT+I)d cA) =eXxO CI )dgx(T+I)czx = 2X 130 TAN®2Z = I ¢ OG FDI DS (SOMME sy OSVIGdS*NISOVIG = €1T)S
iH)
€STSAG*™=SHNIS)*XNIHNIS = OVIddS €LddtS=1XG)*NIDVIG = (T)a
TOVIGdZ*L = NIOVIG
CSHNI S=€ Sd XS/7° T#SdKA) *G® *S TSO) *XNIHVIS = TOVEIG
(€SHNIS*UX739)7°T = NIHNIS
(Sax 37° leSdX4a)¥*5 ° = SHNIS
(€SW390}d xa = SdX3
TX 130k GVWOTS = S73d
(( T)dexe(N dex) (TAN) & (VND ITS SEV = dVWOIS T
Nd tS = NddIS
TdtS = tddls
Sade (a) VEO CHU OVMMONS ys ai
Tx 90 7(( T) de Ae CCA CA) xd (T }dex=(2azxX = 1x 1aG
T+N = TAN TeN = TWN dOW =N
42x 40 NOTL9NAS ¢¥ SV déABPBSNIIdS Sita 3 dOW © ddA * dex STASNOWNOD S /SSNHOF/NOWWOD 8 /3INHIPF JNOWWOD COST )dZA © (OS5ST}dexX NOISN3AWIG (807)S © (CST)S NOISN3AWIG ( VWOIS © Nd IS * Id IS )VATYND 3JZNIiNnousns
101
(1)8e 61 O9 CNY GCA*TD +°CTWNIGZAKZD + (CN) AGCAKED Ndd Ws TWN 1S3C/NN T4G/N135C = ££) TAN VTSAIG/NISU/NNISU= = 2d NTSC0C/NN 14 0/(N 139+NN1530) = TD €2=N)dex@(N)GZX = NN1SG CcmN) dcoX= (TAN ACK = TWN1350 (TWN) dgx=(N)dex = NTSC (EVA CZA*XED 44°C) dCA%ZD +01 )dZAXTD = TddIs CXTWG/ZUXTWI/IXISI= = £5 2X WAGs 1x Wad/2 TX 14G Ae) TX13S0/c2UKVSGU/¢ 1X13aGt!cEXTs0)=" = 19 (Ti déxe(f£)déX = 21x 130 (c}dexe(e)dzZxX = €X 130 2 Si IS GPP) sis NdM134
CT+HVGI DG x€aHVdT)5 ®(4VdI)G = €AVYI)Y
TelavN = AVE!I N‘¢e = I ¥ OG
(CC TWN) G *IVIGdS=CKGSNddIs)*xVIOVIG = (N)E CETWNIS *¥OVEGdS=(d719)7°t = NIOVIDG
20
-
i]
cOVIG = 1T9VIGQ 2
2G = UG OVIGGS*NIDVII = ¢1)S (ST3SG=SHNIS )KNIHNIS = SVIGdS (C1=1)G *)VIIGS=IxXd=7Zxd) xNIDVII = (1)9 ((l=1)s xOVIGdS@Z9VIG+lIVIIJ)7°1 NIOVIG
€SHNIS=((SdK3/7° 14S5SGKX353)*S°)*S 14G) ¥NIHNIS = €OVIG-
(SHNIS*¢ X150)7°T NIHNIS (S dX 3/°T=Sdxa)*S° = SHNIS
9
&
102
7 ON BSG Ch peer Cele
€ SdX3/° T=SdXd}*S° = SHNIS
€(SdxX3/7° tTeS4X3)*G° =
SdX3*ISdXS3 = SdX32 2GHNIS
(2 130% DVWOTIS)dX3 = SdX3
CIS dX3/°T=1SdX5)*5° = IGHNIS (1190 *40UNDIS)AXS = TSAX3 (l]=1)déxe(I)d2gx = S73aG etn (le) icicxe = cals (T=I)dzx=i = I173G ¥ i ty © Ca nlel i *yoe ea (t=l) dex )4I Niel JIN ILNOD 2 2*2*2 ( te( I) dex )4I NOUkh = hk @ OM jf
[hig (Uw O@S!O Mis) Shu
S/(T#N)* (VWOIS)SHVY =%dVWOIS
€ClydeCXx=tNJdex = §$ : GOW =N
*Hd3SITVS GSAIIV)D 36 LSNMwW VATUND SANILNOYENS
MN AY S&S SQ sini siRuy si *x wa anavn NSATS EV AUG QuVa Vas
VAAUND NI SHL SGLV WOdGuSsiNI elas ©
dOW *¢
COST) dZA *
(24425
VWOTS
é
G3NaNL3SH ANIWA SHI NOILONOS STHL acx /SVAZNOWWOD & SAINHOF/SNOWWOD (051)a@ NOISNAWIG (0Sl)4acéx% NITSNAWIG 4 JVAAGND NOI LONAS GNA Nanisga so) = (cis
2 > 2
103
+
y OL OO ¢
( T)} dcx
(SdX 37° I+SdX3)%5° = ¢CQGHSOD (2 1W30*d0NITS)dX3 = SdX3 (TS dX 34 PT +ESdx3}eS° = TGHSOD
CTIS0*d VNOIS)dXS = ISdKS (TI )dexe(I}dex = S130 dh = (I}dex = 2730 (tel )acexXsei = Tad 4 tT OL O93 e¢ =TI CANTO si Osa IE OeyG Guo upeays sii N = I
SNNILNOS ra d=(I}d¢ex }SI NS = lca OG
GS Ut Cb CWE UY sul S/ET#N)XCVNDIS) SAY = dY¥WOIS (ET ddex=(N)dex = S
dOW =N
SUBS (
*ysi Wva JarW> 38 ISNN VATHND SNILNOYNENS
*X LV XG/AG 30 ANTVA SHE SI ANIA NSAID V IV
1130*¢(1}G=(1}dZA) }4SHNISZ (CCGHNIS*ET@1 2G *+TGHNISk*(1I)3 }
¢
VAGYND NI GQS3NYUNLSe ANIWA SHL °X AOA
AXYGG GHYVMVSS BHL SSIVIINSYS4S510 NOILONNSA STHL
dzX /¥A/NOWWOD @ /GNHOfF/NOWWOD . (0S1)3 NIISNAWIG (0ST)dz2x NOISN3WIO
dOW * ddA *
COST }dcZA *
41 © GNDIS © £ YVAGHND NOTLONIS GN 3
Nanta3y
Tara e ST30/€ 2WI*( CTel d= (le ld cA)S
=V AAYND
3 3 3
104
TOV¥VIG/°T = NIOVIG (SHNIS=(SdX S/ ° 34S dX53)*G°*S TISG)*NIHNIS = ITOVIG (SHNIS*TX13G)/7° 1 = NIHNIS (SdX37° T=SaGx3)*%S*° = SHNIS (S130}dx3 = SdX3 TXT3G*dVNDIS = S173ad COT IVA DYUV=(NIVADEV ISOC TEN) KC VWDIS)S37 = dVWOIS T Nd tS = NddIS T3735 = IddIS G OL 0D (°O0°LTSYNODIS}! STI IX 130/ (( 1) deéxe(é}dzZx)}) = IXG CT)VADSVe(2Z)VADSW = IXT3G L+N TdN T=N TWN dJOW = VADYW SO NOITLONN4A V SV d2éX==3NI1dS SiI4 9D dOWw © ddA © dex SVASNOWWOD VADIUV 47 Ad /NOWWOD S “SSNHOfS/NOWWOD 3 /3NHIF/NOWWOD (O51 )dZ2A © (051) d2x NOI SNAWIG €807)S * €05T)3 NOISN3EWIG ( OGT)VADS¥V NOIISNSWIG ( VWWOIS © NdWS * TdTIS )WAcYND SNIiNaysns
ONS
Nanise
I = Tl! STSO/((CT#L Ga Tel kdcA de ( (1 T
YG=e(1)dZA) )+SHNIS/( ZOASOD*( T=1)a stTIHSOI*€I)a ) =VWAGYND AVWOISZ (SdX a4 PT =SdGX3)kS*° = SHNIS
SdX3*ISdGX3 = SdX3a
105
(2 ydzZxK xe D +¢(2)dzgX*2ZD +01 )dCKRED = TddIs 2X1V390 72 1X73 0/1xX13G= [25 2X WG/ IX 130 7Z1x13G eS) UX TV4G/ ZIXT3EGZ ( 1X TAG+7Z1X 7350 J= 5D) CT} ADS Ve (E)VADNY = ZIXIZG €2)VADdVeE(E)VAIUY = |@xTad 9 OL 3D (Z*DS°*N) JI Naniay (T4+HVGI)3S x(4VEI)S =(HVGI)3 = €xAVdI)S I=I1dN = HVGI VC e hf wv OG (CTWN)S *£9VIGdS=2XG=Ndd1S)*xNI9vIG = (N)J3 CC INNIS *xOVIGdS=I9VIG)7°l = NIOVIG ZDv1d = t9VIG AG = Ty6G) OvIddS*¥NIOVIG = €1)S (S732 =SHNIS})*NIANIS = S¥IGdS (€T=1)3 *9VIGdS=1XG=2xG)*¥NIOVIG = (193 C€T=1)S *OVIGAS=ZOvVIG+I9VIG)/°T = NIOVIOG (SHNIS &¢ (SGX 3/° 14S dK 5}kS° )kS13G)*NIHNIS = ZOVIG (SHNIS *2X730)4°T = NIHNIS (SdX3/°T=S3X5}%S° = SHNIS ($139 } xa SdXxa ex WaG*xdVWOIS = S730 2x WI/( (1 aAexe(t+I)agx) =2xa CT )VADSV=(T+1)VAR4s = Z2xX13G cwMoe =& k SG AG € OL 3D (€2°DS°N)} GI OVIGdSXNIOVIG = (T)S (STAG=*SHNIS)*NIHNIS = OSVIGdS (Tl ddIS=IXG)*NIDVIG = ¢1)3
106
lay (by (© (Sesh Oe qo J Ope GL OBO 6 oat Norell —)) shi Nie—sel SANTIiLi NOD c OBC (CHO) yoy yeh WOT IC s I 4 aya | Ars (i VWs} CIE)» 21 S/( TEN} *X(VWITS) SHV = dVWOIS (TIVADEVFCN) VADYVY = S dOW =N °340S35H GSTIVOI SE LSIW VACHND SNELANOYSNS "YLONS WAG SVINITI¥VG Sil IV K 43D SNIVA SHL SI VAbUND NI UGSANOUNLAd AN WA SHt *ALONA WV SHL YOS ANIWA NSAID gv iv ANY3qd GavMV3S BH SALVIIDHRINI NOILIONAS STHi- dOW © ddA * dex SVASNOWWOS VADaGV /VAV/ZNOWWOD a SANHIF/NOWWOD (GsT)3 NOISNIAWIO (OStIdcdA © (CS T)dz2K NOISNSIWIG ( 0ST} VADISI NIISNGWIG ( EY © VWSTS © 1 )WAaMd! "N@IeEONAS GNA Ndnisga Se = (e)) S) Po = (1b le! 9g bo Ob. gS) (N}déx KID #0 TWN) G2KX*29 4+(22N)dcdxeeD = Ndd Is INN TSACG/4NNTSAG/NISG = £9 IWNTAG/NVAI/NNISG@= = 23
NTSG/ NN 1302 (N TSG#NN150) >= T3 € CEN) VADYAVEEN) VADYUV = NN1TSG (Ze@N)VADUVNECTWN)VAIYF = TWN13G
CTWNI VA DGVeE(N)VADGV = N13BG
2) 2) 2)
107
S OL OF (*0°%171"VWOIS) SI IX 130/ (( T)dZA=(2)dzZaA) = 1xQ CU ADSv=(2) VAIN = ITX130 L+N = IdN T=N = TWN dOW =N VADYUW 4O NOILINNS VW SV dzZA==4NI1dS Sild 9 dOW * d2A * 4exX /SVA/ZNOWNOD VA av /¢ Av /NOWWOD S /SNAOF/NOWWOD 4 /4NHIP/NOWNOD (80v)S * (051)4 NOISN3WIG (OST)dZA * (0531)d2X NOISN4WIG ( UST) VAI4¥¥ NIISNAWIG ( WWSIS * NdtS * Td1S )WAEaND ANI LNOASNS ONS NYNiay I = VI S130/¢ €130x( (=I a= (lel )dzx 6 + [1S30x€¢1)d=(1)d2X) )+S4NIS/(Z0HNI S(T") 3 +TDHVIS*(I)3 ) =VAvEND
(SdX 37 Pb e@Sax53)kS° = SHNIS
SadxXS3*xISdXS = SdX3
(SdX3/7° T=SdX3)*S* = COGHNIS
(€2130*xGVWOIS )dX3- = SAKS
CISdXS/° T#1SS XS} *S° = TOHNIS
(1 73G*dVWDIS)dxX3 = TSdXA
CT#I) VA ONVE CL YVADUY = S713ad
de(IyVADHV = C130 (Tel)VADGVeEL = 1130 9
tT OL OD
e= TI
108
I=IdN = »>VGI MOG e& i (KG) (CINN) 4S *9VIGdS=@=2XG=Ndd1S)*NIOVIG = (N)4 NIOVIG
(CTANDS xOVIIDS@IDIVIII/S ET =
eJVII = 19VIAd exG = xa
OSV¥ILGdS*xNIDSVIG = (1I}S (S TSG=2SHNIS JXNIHNIS = OVIGGS CC TelL}S *3VIO0dS=IxXG=cxd)*xNIOVIG = (1I)}4
(€teI dS eOVIIDGdS#=Z29VIIF+tTSOVIG)/° = NI
OVIG
(SHNIS =€¢S dx 34 °14+SdX5)}*S°)*STAG)*NIHNIS = cCOVIG
€SHNIS*2X130)/°T = NIHNIS (SdX3/7° T™SdX5)*S*° = SHNIS (S730)dx3 = SdX4 eX WGxdVWOIS = S130 2X WHG4( C1 DdCASBCT+1}0G2A) =2XO CI}VADYVECT+I1) VADNVY = 2X 140 Inv‘é¢e = I ¢ OG £€ OL OD (2°03°N) AI OVIGdS*NIOVIG = (T}S (STSG="SHNIS)*NEANIS = SVIGdIS (TddIS=IXGJ*NIOVIG = (TIS IOVIGs*T = NIOVIG (SHNIS@=(SdX3/°T4+SdxK5)*S°*S BOGV*NEIHNIS = IDVIG (SHNIS5*41X130)/7°T = NIHNIS (Sd 34 *Te=SdX 5)*S5° = SHNIS (€S7t3G}4dxX3 = SdxX3a TX 13G*edVWOTIS = S390 COTPVADYVE EN) VADSVIZ EC TEND XC VWOIS)SAV = dGVWOIS Nd 1S = NddIS
td1IS = [dad1S
& 2
109
°34y0438 GatIIvD 3G LSNW VAENND SNIANGwINS *HLONA WY HV INI TLYVd Sil 1Vv
A SO SNTIWA SHI SI yuOsd ANTWA N3BAIO Vv iv
WASYUND NI GSNYNLSY AN WA SHL °419N3 19e8V AHL ANG@ GHvMVSS SHi SHLVIWIdGYSINI NOTLOINAS STHIL
dOW © dcA * dex SVA/NOWWOD
VWADUW SVAVSNOWRWOD
4d SANHIFSNONNODD
(GST)43 NOISN3WII9
(OST¥dZA © (05S T)d2x NIJISNAWIG
€ 0ST) VADHYY NIISNAWIG
é
( LI * WWOTIS i )VASHYND NOI LONNS
ONS NYnlay °o0 = (2)4 Co eS Gil ep tT OL 2D
(N}AGCAxXTD F¢TWNIGZA%ZD +(C=N)dcARED = NaddTIS
TWN TS0/NNTSG4N13G = £9 LTWNTVSO/N139/NN 1802 = 23d NTSG7NN1W3G/ (NTSC#+NNT3G) = U3 (2@]N) ¥ ADHVE(NIVADYY = NN IAG
(Z=N)VADUVECLWN} VAIN = TWN 180 ( TWN) VA DHVE(N)VADYUW = NISZG (EVGZAKED +€ 2) dZAK2D +¢T)ACAxXTD = Idd TS
ZX 1W90/Z1xX WAG/IX Wd] = £9
Exdadg/ IXVaG/eTXI13G = cd
Ux Wd /ZEXTVSO/( ExXTaG+2TxtdI)= = 19 (Tt }vA DYVe(e VADUV = 2IX 130
€2)VADIUVeE(E)VADSVY = eX 130 9 OL OD ¢€2°D3°N) SI S Nanisd
(t+ nVGEI da eC AVAIDS e(AVSI}4S = CAVET}S WZ
2 2) 2)
110
Nanisd °¢e435 =HNIS (AY GXd=(X} dXxS=S Xe=A (XHNIS NVOTLONAS QN3 NYUNL AY I= IU! ST3G4€ 2 T50*€ (TI bd= (T=1) d2A)6 + T13G0*(€1T)4=C1)deA) }+SANIS/(€(COANIS*( 1l#1} 5 +TQHNIS*(I}3 } =VAGUNAD
€SdXd/7°T=SAXK3}*¥S® = SHNIS (SdK a4 PE @=SdxXa)kS° = SHNIS SdXsaxTSdx3 = Sdk3Z €SdX3Az7® T=SdxX3)*S*° = ZGHNIS (2 IAGe dWWOIS)GdXS3 = SAX €TSdX353/7°T=1SdX3)*S* = TGHNIS (C(t T3G*dV¥WOI1S)axX3 = ITSdkKS C T=] ) VA D6V= C(I }WVASDdV = S130 4e°(I)VADY4V = 21hG (tl }vVADGV=i = 1 HA iS (aut {aj} e= tI {Je @S) (iA ars) - CalajO, Je Seo f CSS CSpot Yi N = I
SNNITLNOD
£*2*2Z ( L=(T}VADHV )AI NeW SB if & (BG)
@ = (hl CUOweyO Sys 2)y S/CT=N)x(VWOTIS)SEGY = GdVWOIS CTIVADHTS(NIVADYY = S
GOW =N
QN3
Nani3s
°2/S =HSOD
(A) GXS340K% ¥dXS =S Xm= A
€X)HSOD NOI LINAS GN3
I2
APPENDIX B
FORTRAN Listing of Program GRID
3
°09E =VGWV 1x ®US/ °09=d VWIS
(vie *L°Sse *S*vle)ivVWwaod SN *GN*L4950%LH1SS*SS 190% VLIWAN * IXWAN * XVWNO TSS) OVSy
SiNIGd 30 YSEWAN *SHId3aSCG 40 Y3SEGWAN *S3ENNIW NI aviS 1 Virsa JO SNIVA *WN NI avWLS S SO ANIA LSHYIS *WN NI YvtS S VWiT3G0 40 3NIWA *S3NI7 VL SO YSBEWNN S$2+SSNIT FIX 30 YAEWNN *S355309 30 H3SEGANNN QV3a
XVWN° SX *OdZESD0D°EG03 /SV1YXYOS/7 NOWWID
LEWZ8* L6d7E8°VLSExAxX /OS/NOWWOD
AS®XS SAXS/NOWWOD
A*X®WL3° IX SAKLAIX CNOWWOD
GOW SdcAtdcx 4VA4 NOWWODD
OWS CAS SCX SAAS NOWWOD
(SIDJTIEESID*(S)8* (S)¥ NOISNSWIG
(OOTIAS © (O00T}xXS NIISNSWIG
(SES¢LMyZ © (SE®ZHIA © (ST*20)xX © (STIVID © (2e)IX NOISN3SWIG (OOOT}IEALVY © (OOTIEXLV * (OO0O0T)IVALVY © (O00T)YXLVY NOISNSwWI1d (OOOCTIGZA © (O00TIdexX NIISNSWIAa
€OOOECA * (OGOT)2X NIISNSWIG
(G0¢}I209 * (002)80D NIISNAWIG
VTYVD SNVWOIIYYNH eOsa GIYo
®°Sa55b9
SHL SO LNdGING HLIM ( WHOSNOD WVdOOXd } ALATVIWOD SI NOIOSY 3HL 30
ONIdGdVW WWHYOSNOD BHI LVHL G3SWNSSV SE 41 °3949NSS WYYDOdd OL LNdGNI 803 ViVG GIYS SNELNDWOD SHI SSNIWHS1350 *adl¥S wyveS0ud
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114
(T=9N)/VOGNV IX =1X 13G tet =9N T=EN
AYGG VSS UNV SNITLSVOD GSLVYHAN]AD WYHOSSNVWYL ANIWYSL9G
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c=IXWON =CIXWNN T@=IXWAN =TT XWON Tev LAWN TL AWIN eck =9 T= VWOIS
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N
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( AWSGeATSG + X 13G*exX130 )LYOSkdvyWIS =(I1)¢2x
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(I}GALWVeC(LTI)GALVY }SGV =A1SG (1LVEXLVECIIaxXLv SAVY =x1S0
16 OL GOD ¢ HN ° 03° TIT IASI t+I =I!
IxdWd*eCtel} =¢C1)V2A YNfeN=I 16 O39
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( WN
€ UNSEN
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SA =CI}VWALV
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DA =( 1 YEALV
ox =(1)aExXLv
SSASSK FDASOX INYUL TTIW) IX75G*x(T=I1) = GIX
ONS EN=I SB OG
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116
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€(Il)dex + CI11)dex =( 11) d2ex
( ADVSOxA735G + X1590«X130 )LYuYOSkxdvVWIS =(11)d2x € CL)VWALV=(II)VALVY )59GV =A7SG € €1)WxXive(II)vxiv )Sav =x73G (1)2x + €11)2xK =(11)2x
6S OS 60 OY 6€e
66 66 62 69
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16
17
(LSVOD OL WANYON SWIL WAVAL)
SN Sear easiginSiniad
(Ltsvod a2 Tai iveiva SONVESTG@)
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°O =W3ASA °Q =VAdS*x °OQ =LSDA °O =1LS9Xx
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HNS*(N)GG9 =8
HSd*(N) 30D =¥
( L@YNX )}HNIS =HNS
( LaXINX ¥HSOD =HSD
IX>XkNX =I XINX
VLSAX*NX =LEHNX
N=NX
XVWN*T=N OF OG
VLa*xKX =VLIK
IX*aX =1KAX
°Q =WATA
°o =wnax
XVANSNX *9YZEG*9990°8I9D /¥1YOS/NOWWOD €00Z}39> *( 002) 83> NBISNAWIQ
( wid * 1X © WANGA * WONGX JINVYL JSNIINOYENS
GN3
NoUnlLagd
tdwWZ3 + VASA =VASA LadZa + LSDA =1SDA
IX + W3A3SXK =VASX
IX + 4S)X =1S9X SNNILNOD
4x(2279 ) + WHSA =VWASA de (340 ) + LSDA =LSDA ax(d=V) + VasXx =VASX 3*x(Gt+v) + LSDxX =1S9X
( IXDNX )SOD =4 ( IXANX INIS =3 4SD*(N) GID =9
135
Nunisa
C€EXQX0 SI XGADIENG Liv =3T19NV
666 O1 99 (¢€°03*°dO0y DSI
SANTLNIO
CIXOINX INIS* €HNS¥(N)D0D4HSI*(N} GOD) *(N) ANX=IXGAD =1 XGAT C1 XoNX}SOI*(4S9*%€N) DIDFHNS *¥(N) GOD )&(NJANK+IXGXG =I X0XG ( WLSAINX JHSOD =HSD
( ViSANX JHNIS =HNS
IX*(N)ANKX =I1X INX
VLAXCN)ANK =V LAANX
XVWN®T=N SO2 OF
°G =IKQAG
°Tt =1xdx9d
655 O1 QD
AXeNX = CN DANK
N =NX XVWNST=N SOT OO 632 OL 39
OOT OL JD(T°9S*%dOW}SI
XVWN* HX £34829 °9D09SED9/V1YO4/NOWNOD
(002) 4NX * €002)393 * €002)803 NIISNSWIG ( IXGAG* IXOXG*STIONVOE VLS*IX*£dOW JENVYL JANI LNOYENS QN3
Nauniay
Vl3 + OYZ3 + NNDA =WNGA
IX + WNGX =wnax
SANILNOD
3e(3+9 ) + ANIA =NNAGA
3*x(8+vV) + wnax =wnax
( IXANX }SOD =4
( IXONX JNIS =3
666
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ct
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ROOK KC ROI RK a aR RO OR RI ROK RO RR RR RR OK CK 2K 2 ok OK ake ake ak ok >
*Z2NYHVD NUTLONIA SHL
WWvd OL AYVWSSSDSN SI 41 SANNMD 3SH1i NOD SINIOd 30 NOTIViNndwod
AWALIV SOS
*d3i1IwO HO G3ISI9NSdS 30 AWW JAGND 3BHL JO SONS
OMi 3SH4h Lv S3dD7S FHL *SANWVA TYNOTLINISA 4D SONAINOSS V HONOYHL NOISNSL YSQGNN SNI WdS AXOLV IDSGHYS4NI NV SINdWOD GOL AYVSSIDSN SH¥S13NdNId J4i SAININGILSI 3BNILNODAYSNS STHL
2 Ok FR ORK a aK IK RCI RK I Ife IR I RR I IC a I Kk aR aR aK ak kak a aI IC RR fe Kk ka a i
ot
S /¢eNAOF SNOWWOD
yu ZINHOS SNOWWOD
DN © ZA * ex SIASNOWWDD
(908)S * (CO8)8 NOISNAWIOD (OGCOT)2A * (0990T)2K NOISNSWIG
( VWW9SIS © NoWS * tdIs JITAdAD JANI LNOYSENS GN&3 NYONLay ANNI LNOD ot (° T= }ea (WVGK DX D*G =TA
Gi Op CG {GGI@ CNW fy CEO LOECEGlO CMS Z~ en
Dee CWNIX) kEtY =A @ =IA
( *Te> )Sav =m
( wnax Sav =z
( 2*A%v* TASAENNIX YSASND ANILNOVENS GNA
OLONOFOROTOLOROE OO
137
SATAVATIH3SQO GNIODSS SH1L OL TWNOTLHIdDYdD SSN IVA SNIVLINOD Y
ee iNdino NOI
SAN WA JLNTOSEV NI ®t AISLIWEx¥Ndddy SI GVNDIS 4OS AJNIVA GYVGNVLS v¥ PATIVNYSINI GINEFWwusl3qg 4ad TM SAAT LVATINSG ANI OGGNS SHL SJATLVISN SI VWWOIS GI e* LIN 43 LVANI N3SY SVH NOILVNYOSANE BATLVAT ASG SAL G34L3HM SZLVIIINE VNOIS 40 NOIS 344 *3NI7 WNODAIGG W ATHV3SN SI S3ANTD ONTLN9534
art (OeG SCS°%s)) S557 1 Si CYNDUSYSEY su CAN TIS DUN
V ATSLVANIXOYUddVY SI SAYUND ONILIWINSAG SHL (€T00° °° 3°53) ONAZ ATSYVSN SI (WADI S) SAv SI *G3SuY1S3SG0 SSANIASWID SHL SSLVOTIGNI GNV US3Z=NON SIT SIHt *HULIVS NUOISN3SL S4lL SNIVLNOD VAOIS
*G3A041S399 43N LAd GSXYINDI 3H IIIA Seslaganvavd ONIGIOH @JDVIiag HONS °NdIS GNV Id7S YOs S43 LIWIe7d DNIIDIDH#S0V Id HS INNYT4 AINO G3SSN 43S BHL GNV ATWNYSLNI JANI Wadlad
38 WIM SSNWA 4ASAAL SATLVSISN ST VNDIS ALILNGISD GHL Al
“A TSAILISdS SY *€(N)2X GNV CT )EX LV BSAHND 344i 49 SALTIVAIYHSG 1ShI3 S3di 49S SANTVA GSSES3G SHEL NIVLNOD NGIS GNV Td1S
Of sOVvewonrs Totes )) aN) USVela Jk alttaiNe) 2 gah AY Sa NW Sh S ( S9VeU1LS INSNVNdG3Sd ) N LSvaT7 LV HLONST 30 Avs NV SI &
°(>)2X Ci ONIGNOGSSYYOD SAINWA TWN3T LINAS FHL SI (aA) CA °3°I SSNTVA SHi SO SSLVNIGUYQ N 34D Avudv NV SI. ZA SSN WA AWNOILINAS 3Hi 3O 3VSSIDGV ONISVSNONI N 4D AvuUV Nv SI ex (2 °39° )} GQ3LV WdYySINI 38 OL SSANTIVA 4O ASANNN JAdL SI IA =!
OCWOADUOOUUDUYUUVYVUYUUYUYUUYUYYUYYHO UY UUYUUYUYNYYY 138
(TddIS=IxXQ)*NIDVIG =
(TY
IOVIG/°T = NIOVIG
(SHNIS=@( SdX 37° T#SGXS)*KS° KS TIGIENIANIS = I9VII9 (SHNIS*ITX1730)7°t = NIHNIS
€(SdX3/7 °T=S5dK3)%*S° = SHNIS
($1430) dKk3a3 = SdX4
Tx TWAUxGVNOIS = S 130
NOLLVNIWI 175 J3¢7M4393 wWHISu4Sd ONY 4 YDS NSLSAS ANOOSVIGIYL ONV SOIS INVH LHOTd dd Las
CCT )2X=CNIZXISCTEN) &(7WOIS) SAV = dvWoIS T
4OLO¢4A NIISNAL AZT IWNYONAG
Nag IS Td 1S St ep) COMO IUS Whi)
ai SSeleyx) el Selei Dus
LX 1507 ((T-)2A=C2 ICA) (T)cxe(2)e2x =
Ttn
Ten
*"G3Syasa LANA Sev VNDIS *NdTS STATS SCA SCX
Ndd1S Tdd1S S} 4dI
SN IW 3513G
= [xa Tx 130 = Idv = TWN OW =N
© (ON) N
‘S) CCHS)
QYv
VUULUY YL
139
Nant 34
(T4+HVET)Y *CAXVET)S e-(HVdIY = CHVGEI D4 v T=TIJN = AVI N°e = I 4 39
NOILAVLTILS@Vs ADVE WYO SyH3ad
CCTANYY *9VIGdS=cXG2eNddIS)*NIDVIG = (N)Y CCTANDS *OVITIGdS@#tdvIIG)7°T = NIOVIC €&
c2VWI1U = 19VIC ¢ exgd = [xa
DVIDGdS*NIDJVIGD = (1I)S
€(STAG=SHNIS )XNIHNIS = SVIGGdS
CC T#=1)8 x01 IdS=1TXIe2XI) xNIDSVID = (1I)a4
C(t=1dS *OVIGdS=Z9VIGFITOVIG)Z°T = NIOVIG CSHNI Se (SdX 37° T+SdX3)}*S°)*STAGYXNIHNIS = COVIG CSHNIS*2xX159) 7° = NIHNIS
(3 dX 37°%T=5dX5)*«S° = SHNIS (3739)49x3a = SdX3
2X WIG*dvWOIS = St3d
CX VAID/( (1) CAMO THI} CA} =HCXO CT)cxk=(l+12xK = 2X 1350 Tnve?e = I 2c OG
€ OL OD (2°D4°N?} JI
DVIBdGS*NIDVIID = ¢(€1)S (ST730=SHNIS }*X¥NIHNIS = SVIGdS
140
( LI © WWOIS * 4b YZAUND NOPLINIG
ON3 NYNi34y 2) SC 8 Ce )its} *oQ = (1)Y4 9 2
JAXYND ¥353 ANID LHOIVWYLS OD 3510 *N3SAID 4eV SSAILVATYSG ON INV SAINIOd OM AINO SAI OD
T OL 3D
(N)ZA*XLD + (TWNIZAKZD + (22N)ZAKED = NddTIS IWN TSQ/NNTISOCN1T30 = £9 UWNTSO/NIZO/NNTSI® = do NTSG/NNTSO4(NTS0+tNN1990) = 1D
(ZEN) CXS(N) ZX = NNW3G
(Z]N) cxe( TAN) 2X = TWN130
CTWN¥CX=(N)2X = NVC
(EX SAKED + (2) 2ARZD + C1) CAKTD = IddTS ex 1W0/2Tx1adstxisd]= 3)
eX V3d/1K 130721xX13G eo
UXdad/2UxX199/¢ 1x730+t 2Ix13Id)s= 1) (TI2Xe(ECX = CTX 13d
(2 )exe(E}2X = 2x 130
*SLNIOdDGN]S LV SSNIWA BOS Vivd LIGDNI NO NOLIvViWdd3inI 5 AVIWONATId w3adYO0 INOJDAS ASN N3SAISO 3Yv SSAILVAITYSCO ON SI 9D
9 OL OD (2°95°N) JAI GS
14]
SINSNDAS V AV AJLVIOdYSINI OL SHBAS HSSA gv SI * ANIGIAVY SHOW HOD NITLVIOdsYSINI 3SHL NdO4SH3d OL 30dv SI NOILINAS SHL NOTAIVNYOANT ADNS ALIA PANTVWA SADIASYSA SHL SQ35S9XKS
4 3D SANIWA STHL LV4L GN¥ (TDAYFLIVNA VADTS ONY SH*°?2A ®EX*N ALIM) ATSONOIAZS3YD G311V9 NSS38 SVH NOILINYGS SHL LVHL SSLVOTIONI ST4i T LON SI LI SI °HOLIMS YHYSDSLNI NV SI LI
(GSYONSIT SI NOSIS SLI} YOLDV4A NOISNSL 344 SNIVINOD VNDIS ° (cA*% eX} SAGIN SH LV AAYND SHL 30 SATLVAIY30 GNOD3S 344 OL WNOLLY9DdDD44 SSNTWA HLIA Avxyud NV SI 4 ®SLNICQd G3LVIOdySINI IWNIDSTHO SHi 30 S34VNITx890 INV SVWVSSIDISV SHL ONINTVINDD SAVUUW SYV CAtcxX °JAXND 344i ANIWH3I139 OL GALVTOdNSLINI SYSM HOIH® SENTIOd 4O YSEWNN J4L SI COW) =N SAGNMD SHL OLND G3ddVN 3A DL ANIVA WAXY ¥ SNIViNID (Ox00D=xK) 1
s=iNdNI NO
FR OK OK AICO I ROI IOI RK IO IR RG IOI OI KI EI OR RR Ba a RC ok keke i kee ie ke kak ok ac *SyYsisaNVaVd
AYVSSSIAN NIVLYSD SANIWNYSZLSG O1 YSIWI3a Gard 3s
GUNCHS TASND SNILNOYANS SHL °NODISNAL YSGNA SNIIdS Vv ONISN
ANIQd NSAID V-tLV SAUGND V S3FLV1WOds3SLNI NOTLINOA STIHL
BEAR I ROK RR RO aK OR ke RFE 2k 2k ae ae akc a aC 2k ak ak ak a CE 2 2K CK 2 aK aK OIC OK akc ok 2k akc 2 2c aie ak ic
yY STNAOF/SNOWWOD
DN & cA * 2K SEACNOWWOD
(CO8)4 NIISNAWIG
CO03T IZA © €0990T)2K NOISNAWITD
VYUVUYYMVUYNVYVYVUVIUIUVYVUYVOYVYYVVYUVVIVOUYY
142
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TWASXSSALNI YOS HDYVAS
Geo un ChOwsyo lit) shi
OSNINNID 33S WONSd LH7LS SFSTIMHSH19 SGSLVNINYSL AISIVOIASYd SYSHM HOaVSS LYVIsS T°AN° Lt Al
S/C TPN) k¥C7WITS)S3SF = aVWOTS
(T)}eKe(N)CX = S VNOIS AZIIWAYXONAG
OW =N
“(NFCA =CAUMD* (NISX "19° LF BOs CT PZA SeNAdND “(T) ex vane 2 soa
SS CINSMS Nik
°SNTWWA GSlLv1IOdy¥SiNI BSHL SNIVINOD (GadO0Id"=A=) 2ANND
eelfNdiNnd NO
GSYS1WNA Suv 3SSH1 GNV TAYND 30 LNdind 3AL NDYHA AZHALTIVNA
GSLINdNI 38 GINDHS WWOIS GNV *4 S2A 2X *(OW)} N SHSILSWVAVd SHI
*T=y 4LlIM Sidt4S L ININIVINOOD
CC T442X* (42K) AVAYSLNI FHL GOS HOUVAS SH4l 1 ST II SI "WS SHi 30 XSGNT 34a D4 LT ONTLLSS JNV ONISVAYONI SSNIVWA SHL ONTYSGNO AG JQANIVD SI AINITIII 3433 *SINIOd 40
tS)
BPUUYUYUYHMYUYVYUYYYUYSOY
143
(
COOT)dZA *
(OO0OCT} deX NII SNAWIG
( VWWOIS © NdIS * Id1S J=EAYNND ANILNOUSNS ana NYAiay R 2 2y
STB0/( 2737 eC (Lely © (lel) ZAd.6 + 1750% €(1)d=€1) ZA) )FSHNISZ(ZOHNIS *(L@L aS +TGHNIS#¢(1)Y ) = ZAYND (SdxX3/°1L=SdxX3)eS* = SHNIS SdXSe1SdXa = Sox] (SdX3 7% TeS4X3)«S* = ZGHNIS (2 130* dvWDI1S)dX3 = SdXx3 (USdX3/ *T=15dK3)*5° = TOHNIS (11ad*xdvW91S)dx3 = 1SdXx3 (t=I)ex=(1)2x = $73a Le(I)2x = 2730 (teI)ZxXelL = 1730 ¥
NOTLVVId3SiNI W43455334 GNV gd 135 { OL 8 QA a ii
( LOSYUOONT SVM sellee ANGNI )
TI 45S34 QNV HOYUVSS LYvisas b OL OD ( (12x *an® 1 esos 1 *aa° (ter)]ex yar
WAUSINI
£
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493498999 SYNSNI JL AXDISHD JD
N = I
3
144
*Q3SyYSLWNNA 3e8V VWNDIS *NdIS *IdIStadcAS dcx * (dOW )N
( ddA*dex ) SAGIN NSAATIO SH1L LV SAND AHL 340 SATLVAIYS9 GNODSS 341 JDL WNIT LHId3N4 SAINIWA SNIVLNOD A
2exiNdiIO NO
°¢ S9VuOLS HOLVYDS } N LSVWVAT LV HLONAT SO AVUYW NV SI Ss
( 39VYHOLS LNINVANYSd) N LSV735T LV HLON3ST 39 Avsuv Nv ST A
SSNWA 3414 30 S3LVNITHO N 3D AVN NV SI dea
S3SNTWA AVNOILOINAS SHL 3O 3VSSIDGV ONISVAYONI N SO Avddv NV SI dex
(2 °39* ) GSLVTOdDYSINI Jd OL SANTIVA JO HSDNNN JAHL SI dOW =N
e=a=iNdNI NO
2k ok ke 2K i A KF ORR kK kk aio ki kk ei i kr a etki ok i ke ee ie fC ee eo Re ae ek ok
*VASYND NITILINIA AHL
W1V5 OL AYWSSSIDAN SI LI SAYND AHL NO SINIOd 30 NOT ivViNdwod WOiSY YO3 °Da4LTIWO 4D G3ISTID39dS 3G AVN 3AYND SHL 3O SONS OML SH1L LV¥ S3d071S 344 °SSNTWA WNOILINAS 3D SINANDAS V HONZISL NOISNSL YSGNN ANI WdS AXOLV 1O0ddy3a3lNI NV 3LNdWOD
OL ANYVSS3DIN SH3LINVY7d J4L SINIVGYSL3ATI ANILNOYSENS STHL
FOO ROO CII IO GIO GOO IOI OOOO III FOR IOI I KORO a doi a
JDOWS dc ASdGZX SAAS NOWWIDD
A ZVNHOF SNOWWOD
S S2ENHIF/NOWWID
(008)S § (008)A NOI SNAWII
YVYVYVYVYETYODYUYBVYVYVYVYYVYVYVVIVYVYVVUEYYYO
145
OVIGQdS*xNIDVIG = ¢€1)S (STAD@SHNIS)*NIHNIS = OSVIGdS CETHT IA *£OVIGdS@IxXd=Z2xXG) eNISVIT = CIDA (( TeI }¥s *OVIddS=c9OVIGt+IOVIG)4Z°T = NIOVIG
(SHNES@¢ (SdX37° 1+SAX3)}%S°)*ST30) *aNIHNIS = e29OVIAd
CSHNIS*2xX13G)4°T = NIHNIS (SdX ae US dX 35}%S5° = SHNIS (S149) dxKa = Sdx3a 2X Wdk dVWOITS = S130 EXTSI/ EEL Vd cAe(T+I)dcA) =2xa (1l)ddX=( TtI)ddgéx = cx t3aG TWNS2 = I ¢ OG Ss OL DS (S°OSION) an IVITGS*NIDVIG = (1)S
€ST3SG"SHNIS)J*NIHNIS = OVIGdS €TddIS=tTxXd)*NIOVIG = E€ET)A
tTOvIGs*t = NIOVII
(SHNIS=( SdX3/° T#SdX3)*SG° *S 1930) ¥NIANIS = 19VI90
(SHNIS*IX730)/°t = NIHNIS
(SdxX 3/7 °TesdkK3)*S° = SHNIS (S139) 4X3 = SdX3
UX WaG*kGdVWOIS = $130
COR) AEXPENDdCXIPSC TEN) *(UWOIS¥Sdv¥ = dVWWOIS N37S = NdadTS Td1IS = tddlsS
GS Ji OD €°O°11*°VAOIS) SI
TX T3040 (Td) dzgAe(2)5dcA) = IXI
(T)dexe(2 dadzgx = Ix 15d
ttn = TAIN
T=N = TWN
dOW =N
146
(O03)A NI ISNAWIG (O00T)dZA © (0001) d2X NOISNAWIID
CJHS COs: OC Ij iNelaey Navan) ve!
ONS
NyYAtay
*d = (€2)A
°G = CI}A by Ble
EN} GEAKID +€C TWN) G2ZAKZD #+0( CHEN) ACAXED = Ndd IS TWN T3G/NN 1390/N150 = E€) TAN TAG/NISG/NN1TSEG= = ¢d NIZI/NNTIBO/SONTIItENNGISI) = 19 (ZN deg X=(N)G2é@xX = NN WI (2=N) dceXe(twN)dex = TwWN1W¢G
C TWN) doxe® (NJ 4eX = N39
(EVACAKED +°¢2)d2A*2ZD + (T)deAexTD = TddIs exVaAI/?2 TX 13G/TX tWaAG= = ED
ex Wad/txwd/2tx1sd co
IX T3G/2TXK 1404 C1X 13G+21K 130 }= = 19 (l)dcxe€&)dex = ctx tad
(2 )dexe(edadzgxXx = ¢e€x139d
9 Of - aD (2°dseN) aI NYnisy (T4HHVELE)DA *xCAVEI IS =(HVEI)A = ( AVAGI)A
l=IdN = YVEI oe Sr oO TG
COINNIA *9VIGGS@eXG=Ndd1S}*NIOVIG = (N)A (CTAN)S xIVIGDS@19VIFJ4°L = NIOVIG
Zdv129 = LdVvIG
zxGQ = Ixa
mM
147
S/(T#eN)&(¢WDIS) S37 = dVvNDIS (l )agx=(N}dzx = 5 dOW =N
*SN1VA GSLVIOGHSLNI SHL SNIVINOD (GHOOD=A=)} HASAND
eeiNdinO NO
®* pASND NI GSxe¥S3LIWNO Jav 3S34L ONV EAHND 3D LNG1ND SHL NOUNS GA4SLTYNNA GSLLNAGNI 3A QINOHS WWOIS GNV A %dedA*dZXxX (dOW)N SHYSLAWVYVS SHI
*( déAtdZ2x ) SSGON 341 Lv 3A8TD AAL 3D BATIVATAHRO GNOD335 B4t GL AVNOITLYOdOdd SANWA 4HLIM AtS40 NV_LSI A *SENIOd GILVIDdYeSLNI IWNISTSO SHi 30 SSLVNIGYHO INV S¥SSID30 34Ll ININIVINID SAvV3sS¥ SHV GecA* dex °BAMND 344 JNIWH3Z1LSG OL GSLVWWdNS4LNI S3535M HOTHM SANIDd 433 ‘YSGNAN SHi SI GOW =N SAS) SHL GQLNO Gaddvw 3G O01 SANIWA 1V5N V SNIViNOD (359905=xK} +t
selNdiNI NO
VUVYVVUYGUYVYVOEYVYGBYNYOYYG
FOO IO OO CGO OO OGIO OG IOI OGIO ICO RK kK mack (#2 - 3 *Sy¥sLSWV ad AYUVSSSDIN NIvViusdD SNEWHS159 O1 HSI TH75 BRAID 3d OD QIABDHS ENAND ANILADYSNS BHL *NOFISNSL YSONN SNI TdS V ONISN 9D INIOd NSAIS V LV SASND V S3LVWdasiNI NOI LONAS SIHL D 2) FORO CO III GIGI RO IGOR I OGIO OO I IOI RRR AOI IC RGK gk a akc a ak aki ak 2k : 3
DON* dZA*dZX /SVAS NOWWOD
A /9YNHOPSNOWWID
148
+
O° 2/ANIS
(NVGAYDXS = (NVOX}AXS nv 0X
(NVGX)HNIS NO
0°2/HSOD
(NVGA)IXS + (VOX }dx3 o NVOXe= (NVGx }HSOD NO
SIad/¢ 2IWI«Kl Cll pA] CT TVSG*k CCI )A=C€1)dZA) DFSHNISS OC 2GHNISx( T=] )A +TIHNISXCIDA 9) (Sd a4 PT eSaGX3)«S° = Sdk3*TSAXS
(SdX 34° T]=SdX5d)}«S° =
(2 V30xdVWOIS )dX3 €TSdXS/° T=TSAXxX3)«S® = (T1V30*dVWSIS)dxX3 = (T=I)d2éxe(I1)d2x
do = (I) dex
(tT=1)dexer
T
yy se DS NE Caloak CBO Lf Seo Cuca pel
SANI
B 2092) 10 bei id N° TI =
e = TI (1°03°
agN3 Nanas =HNI5S =HNIS =WVTA IL ONS GN3 NY M434 =HSOD =HSOD =WVGA ILONNS GN 3 NY nisy I =TIl I )d2A )6 PAUND SHNIS = SdX3 COHNIS = SdX3 TGHNIS TSdX3 = $7130 = 2739 = 17139 OL 939 c= Tl 2x )al N = I 1NOD ex jal I 2 0a 4I)} GI
149
APPENDIX C
FORTRAN Listing of Program SSURGE
150
ft SIAC
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