(SU ‘ =: Away Gat 5 Gag Rea, WE TR 76-3 A, AE . oe Storm Surge Simulation in Transformed Coordinates VOLUME I Theory and Application by John J. Wanstrath, Robert E. Whitaker, Robert O. Reid, and Andrew C. Vastano TECHNICAL REPORT NO. 76-3 NOVEMBER 1976 WHO? DOCUMENT | COLLECTION / ae Approved for public release; distribution unlimited. Prepared for U.S. ARMY, CORPS OF ENGINEERS COASTAL ENGINEERING RESEARCH CENTER CB Kingman Building 4S 6 Fort Belvoir, Va. 22060 ies tu "76-3 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: National Technical Information Service ATTN: Operations Division 5285 Port Royal Road Springfield, Virginia 22151 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. I oOo x oT a o Oo oO s==0 — — ro Oo m Oo oOo MO J = SECURITY CLASSIFICATION OF THIS PAGE (When Data Entered) READ INSTRUCTIONS REPORT DOCUMENTATION PAGE BEFORE COMPLETING FORM 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 A&M Research Foundation Roleo ope) Jel 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 Wiel, 1, Sexe) Q 176 15. SECURITY CLASS. (of thie report) UNCLASSIFIED 15a. DECLASSIFICATION/ DOWNGRADING SCHEDULE 16. DISTRIBUTION STATEMENT (of this Report) Approved for public release, distribution unlimited. DISTRIBUTION STATEMENT (of the abstract entered in Block 20, if different from Report) . SUPPLEMENTARY NOTES . 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 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. 20, (Continued DD , Phe 1473 = EDITION OF 1 NOV 65 IS OBSOLETE UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE (When Data Entered) UNCLASSTELED SECURITY CLASSIFICATION OF THIS PAGE(When Data Entered) 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. : UNCLASSIFIED SES SECURITY CLASSIFICATION OF THIS PAGE(When Data Entered) PREFACE This report is published to provide coastal engineers with the results of a study to develop an operational 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). Volume I of the report was prepared by John J. Wanstrath (who also authored Volume II), Robert E. Whitaker, Robert OQ. 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. ee OHN H. COUSINS Colonel, Corps of Engineers Commander and Director WAL IV VI APPENDIX A CONTENTS CONVERSION FACTORS, U.S. CUSTOMARY TO METRIC (ST) INTRODUCTION. 1. Background. Bo Ooijoetnves. CONFORMAL MAPPING . 1. Development . 2. Applications. 3. Limitations . THE STORM SURGE EQUATIONS IN THE SHELF COORDINATE SYSTEM. SINUS Eig oes aiaitA tae CFv e LUTON ACH ga or Stretched Shelf Coordinate System . Storm Surge Equations . Puen rabies Numerical Algorithm . Boundary Conditions . ; Wind and Pressure Fields. ABWN eH SIMULATION OF THE FREE WAVE IN AN ANNULUS . 1. Problem Statement . 2. Results . STORM SURGE SIMULATION. 1. Hurricane Carla . 2. Hurricane Camille . 3. Hurricane Gracie. SUMMARY AND CONCLUSIONS . LITERATURE CITED. CONFORMAL MAPPING COEFFICIENTS FOR THE GULF COAST REGION FROM ATCHAFALAYA BAY TO APALACHEE BAY . CONFORMAL MAPPING COEFFICIENTS FOR THE EAST COAST REGION FROM CAPE KENNEDY TO PAMLICO SOUND. CONFORMAL MAPPING COEFFICIENTS FOR THE EAST COAST REGION FROM PAMLICO SOUND TO PENOBSCOT BAY . CONFORMAL MAPPING COEFFICIENTS FOR THE GULF COAST REGION FROM LAGUNA MADRE TO MARSH ISLAND . CONFORMAL MAPPING COEFFICIENTS FOR THE GULF COAST REGION FROM MATAGORDA BAY TO TIMBALIER BAY . 138 142 147 151 CONTENTS APPENDIX-Continued F NUMERICAL ANALOGS OF SURGE EQUATIONS. G MODEL VERIFICATION. H WIND DEFORMATION PROCEDURE. I SYMBOLS AND DEFINITIONS . TABLES 1 Convergence table for the gulf coast to Apalachee Bay . 2 Convergence table for the east coast to Pamlico Sound . 3 Convergence table for the east coast to Penoboscot Bay. 4 Convergence table for the gulf coast to Marsh Island. 5 Convergence table for the gulf coast to Timbalier Bay . cial 1 Conformal 2 Coastline 3 Coastline 4 Coastline 5 Coastline 6 Coastline FIGURES mapping planes. and seaward boundary curve, and seaward boundary curve, and seaward boundary curve, and seaward boundary curve, and seaward boundary curve, region region region region region of Atchafalya Bay of Cape Kennedy of Pamlico Sound of Laguna Madre of Matagorda Bay western gulf coast. central gulf coast. eastern gulf coast. lower upper east coast. east coast. 7 Transform-generated coastline and seaward boundary curve after one iteration, eastern gulf coast. 8 Transform-generated coastline and seaward boundary curve after 20 iterations, eastern gulf coast. 9 Transform-generated coastline and seaward boundary curve after 40 iterations, eastern gulf coast. S) Page ILS) 163 31 33 34 SS) 36 10 Wal 13 14 15 16 17 18 NY) 20 ll 22 23 24 25 26 CONTENTS FIGURES-Continued Transform-generated coastline and seaward 80 iterations, eastern gulf coast. Transform-generated coastline and seaward 160 iterations, eastern gulf coast Transform-generated coastline and seaward 120 iterations, lower east coast Transform- generated coastline and seaward 135 iterations, upper east coast Transform-generated coastline and seaward 80 iterations, western gulf coast. Trnasform-generated coastline and seaward 80 iterations, central gulf coast. Transform-generated coastline and seaward one iteration, western gulf coast. Transform-generated coastline and seaward ‘40 iterations, western gulf coast. Transform-generated coastline and seaward one iteration, central gulf coast. Transform-generated coastline and seaward 40 iterations, central gulf coast. The "curvilinearity" variance spectra for of the five mapped regions boundary curve boundary curve boundary curve boundary curve boundary curve boundary curve boundary curve boundary curve boundary curve boundary curve after after after after after after after after after after the final transform Coastline and seaward boundary curve, lower east coast. Coastline and seaward boundary curve, central and eastern gulf coasts. Orthogonal curvilinear grid system. Shelf coordinate system for Hurricane Carla surge simulation. Stretched shelf coordinate system for Hurricane Carla surge simulation Functional relationships for Hurricane Carla surge simulation . Page 29 30 Si 38 39 40 41 42 43 44 46 47 48 Sl 53 54 55 27 28 29 30 Sil S 55) 34 SS 36 37 38 59 40 4] 42 43 44 CONTENTS F IGURES-Cont inued Functional relationships for Hurricane Carla surge simulation . Shelf coordinate system for Hurricane Camille surge simulation . Stretched shelf coordinate system for Hurricane Camille surge Simulation . Functional relationships for Hurricane Camille surge simulation . Functional relationships for Hurricane Camille surge Simulation . Shelf coordinate system for Hurricane Gracie surge Simulation . Stretched shelf coordinate system for Hurricane Gracie surge simulation . Functional relationships for Hurricane Gracie surge Simulation . Functional relationships for Hurricane Gracie surge Simulation . Scheme for computed variables . Hurricane Carla symmetric and deformed wind fields. Hurricane Carla symmetric and deformed wind fields. Hurricane Carla symmetric and deformed wind fields. Hurricane Carla winds at selected points. Hurricane Carla atmospheric pressure. Observed and computed water levels using a deformed wind and a symmetric wind for Hurricane Carla . Hurricane Camille symmetric and deformed wind fields. Hurricane Camille symmetric and deformed wind fields. Page 56 s/f 58 59 60 61 62 63 83 84 45 46 54 55 56 S7/ 58 59 60 61 62 63 64 CONTENTS F IGURES-Cont inued Hurricane Camille symmetric and deformed wind fields. Hurricane Camille winds at selected points. Hurricane Camille atmospheric pressure. Hurricane Gracie symmetric winds. Hurricane Gracie atmospheric pressure . The annulus in polar coordinates. Rectilinear grid representing the annulus . Polar grid representing the annulus . Computing grid for polar system representation of the annulus. Computed water surface topography in the rectilinear grid system . Computed water surface topography in the polar grid system . hydrographs for the polar grid and rectilinear grid . Hydrographs for the polar grid and rectilinear grid . Hydrographs for the polar grid and rectilinear grid Observed and computed water levels for Hurricane Carla. Computed water surface topography for Hurricane Carla . Computed high water of the coastal surges from Hurricane Carla corrected for the astronomical tide. Computed water velocity at selected grid points for turricane Carla. Values of alongshore current as a function of time at selected grid points. Observed and computed water levels for Hurricane Camille. Page 85 86 87 88 89 il 9B 94 95 98 99 100 : 101 102 104 105 106 107 109 65 66 67 68 69 70 71 72 18 74 nS 76 HY 78 VW 80 81 Computed water hours. Computed water 26 hours Computed water 27 hours . Computed water 27.5 hours . Computed water 28 hours . Computed water 28.5 hours Computed water ZIM ours|: Computed water 30 hours . Computed water 32 hours . surface surface surface surface surface surface surface surface surface topography a, , Pee pean ey Hest ae topography topography CONTENTS F IGURES-Continued for for for for for for topography for for topography for Hurricane Hurricane Hurricane Hurricane Hurricane Hurricane Hurricane Hurricane Hurricane Computed water velocity at selected grid points Hurricane Camille. Camille Camille Camille Camille Camille Camille Camille Camille Camille for at 24 at at at at at at at at Computed high water of the coastal surges east of the delta from Hurricane Camille corrected for the astronomical tide . Cartesian grid for Hurricane Gracie storm surge simulation. Observed and computed water levels for Hurricane Gracie in the curvilinear and rectilinear grid system. Computed water surface topography for Hurricane Gracie in the curvilinear grid. Computed water surface topography for Hurricane Gracie in the rectilinear grid . Computed water velocity at selected grid points for Hurricane Gracie in the curvilinear grid . Computed high water of the coastal surges from Hurricane Gracie corrected for the astronomical tide in the curvilinear and rectilinear grids. APES 116 IALY/ HAS) 120 2a WAS 124 WAS 126 128 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 3/9 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 I. Theory and Application by John J. Wanstrath, Robert &. Whittaker, Robert O. Retd, and Andrew C. Vastano I. INTRODUCTION 1. Background. Storm surges are transient fluctuations in the sea level induced by atmospheric disturbances, notably those due to extra-tropical storms and hurricanes and to a less frequent extent pressure jumps associated with line squalls. The rise of the water and circulation caused by a hurricane can be considerable and is of special practi- cal importance with respect to loss of lives and property not only adjacent to the coast but also well inland. Statistical studies of hurricanes of record provide a means of predicting the surge height along an open coast. The empirical formulas developed from these studies relate the maximum expected surge height to meteorological parameters and effective coastal bathymetry (Donn, 1958). However, all such studies do not provide the time history or even the time scale of the forcing function which is necessary as input for bay- response studies (Reid and Bodine, 1968). More recently, time-dependent models based upon the physics of the storm surge phenomena have been developed to study the generation or modification of the surge as it leaves deep water and moves over the Continental Slope and Shelf. These models, like the one proposed herein, involve the vertically integrated equations of motion and mass continuity. The greatest difficulty in utilizing these models has been the manner in which the shoreline has been portrayed and the application of realistic boundary conditions at the specified shore. Jelesnianski (1967, 1972) takes the shoreline as a vertical plane of infinite height, thereby facilitating the mathematical representa- tion of the shore boundary. More general portrayal of the shoreline is achieved by the schemes of Miyazaki (1963) and Platzman (1963) in which the coastline is represented as a series of straight-line seg- ments connected at right angles. Specifying the shoreline in this stairstep manner results in greater numerical programing complexity for the shore boundary condition in that the algorithm must possess the ability to search for and substantiate the location of land. A more serious objection is that this approximation may inject spurious oscillations into the calculations. This adverse effect is more than academic since the concentration of energy is near its maximum at the coast, and further, this is precisely the region from which water level observations are usually available for model verification. Clearly, these difficulties are avoided or minimized if a curvi- linear coordinate system is employed which will map the interior region bounded by the actual coastline, the seaward boundary (taken as the 180-meter depth contour) and two lateral straight lines. Thus, the transformation is such that the coast, as well as the 180-meter isobath, is mapped as a straight line in the image plane. In order to minimize the associated modification of the differential equations governing the storm surge, an orthogonal curvilinear system is employed in the present study, the desired mapping being carried out via a conformal transformation. 2. Objectives. The objectives are twofold: (a) to develop a method which will conformally map the interior region bounded by two arbitrarily shaped curves each beginning and ending on parallel straight lines drawn between the two curves; (b) to develop and verify a numerical model of a hurricane-induced surge on the Continental Shelf employing the transformed equations of motion in the orthogonal curvilinear coordinate system. II. CONFORMAL MAPPING 1. Development. The selection of appropriate conformal mapping and error functions is of principal concern in achieving the first stated objective. We desire to map a region R of the Z-plane into a rectangle in the c- plane (Figure 1), in which the seaward boundary and coastline curves are specifically transformed into the image plane as constant values of n . Furthermore, the curves in the negative x region are re- quired to be the mirror image of those in the positive x region. As an artifice to assure that the lateral boundaries of the mapped region represent straight parallel lines normal to the shoreline and the seaward boundary, symmetry about one of these boundaries (x = 0) is imposed and the whole range in x (-A to A) is considered to be one wavelength of a periodic function. Thus only the range 0 < x < i in the top panel of Figure 1 corresponds to the real region of the shelf. The conformal mapping relation is taken in the form: 62 186)" (1) where Z=x+ iy, (2) SSS iP ali) (3) and te veil. (4) An appropriate form of the transformation for the mapping considered in Figure 1 is: N PG) oP. 9 Qe |) (P. cos me > @ sim ake) - (5) n=! where k= /) , (6) and i is half the horizontal extent of the region in the Z, or ¢f plane. The coefficients P einl @. , MEO ooo, EheSs Win (emMereUl . complex and independent of € or n. Let P eA iBL 5 (7) *aueTd-2 94} FO UOTZEL TOTLIOJUT sy. OJUT sUeTd-Z 9Y4R FO UOTZOL LOTLejUT oy} BuTWIOFSueI, IOZ souevtd B3utddew TewLoszuo) ‘T emn3sty 14 and Q, = Cc. + iD, 5 (8) where the coefficients A, , B, , C, , and D, are real quantities. However, the symmetry condition about x = 0 (or € = 0) requires that all A, and D, be zero. The resulting relations for x and y in terms of & and pn are: N “(E.m) = E85 © ) (B_ sinh nkn + C| cosh nkn) sin nke , (9) n=1 and N yom) 2 Bo Gy se ) (B_ cosh nkn + C. sinh nkn) cos nké . (10) n=1 The condition that the range of x and € be the same requires that C, = 1. The remaining N values of C, and N+l values of B, are determined by matching the coastal and seaward boundary curves at n = + 6, respectively, 8 also being a parameter to be determined. The coordinates X,Y of the given coast or seaward boundary curves are specified parametrically in terms of arc length measured along each curve from some fixed point. The functions NSU AY ote. X° , and Y° are stngle-valued functions of this parameter where the superscript s or c represents the seaward boundary or coastline curve, respectively, This property is essential since a Fourier series-type representation is employed in determining the coefficients. The problem is to determine 8 , Bo , and the set of coefficients, By and Cy, , for a given N such that equations (9) and (10) give a best fit to the given curves, in the sense of minimizing an appropriate mean square error function, Since the specified curves are not known directly in terms of €& , but, rather in terms of arc length, the bicurve fitting equations require an iterative process starting from some initial estimate of arc length in terms of €& for each curve. It is found that a convergent iterative procedure results if one chooses the following as the i+l approximation of the coefficients: fo aR SA oe a Ne) a B Soil k eG) deo FW) ale Ip (11) (0) @) fo ae. a Ag 4 B Senile iwe@) dee FIO@) dell » (12) (0) 2X 0 0 Bee Cie te NG ee GI) » (13) and itl Cag = CEA VN OLN CTR ONT COR) lex, (14) where i=1,2... denotes the iteration number, A is the transform generated arc length and Me tee ae Woe enna ei i 2 || stiles mg = nae cosh2 nkg-_, (15) (We ewe (Wet?) ma o6 = sinh 2nkg ; (16) n 4 Cll ol eS ea: pula Sy! = TnMarvele oD itl et 2 i+] QF = 5 sinh* nkg + 5 cosh~ nkg > (17) : itl] UMN A go 6 Se ee ff CCM) = S) sim ake de n r x a 5 A 4 SWE? OG (AM Nere}yscaninkegdem z oO in cosh afege = TN , 7 fwo?* ¢ y°(a7) cos nké dé Yo 6 aN é + ce f Y°(A") cos nké dé] , (18) (0) cosh nikeuie Coin : Cra! = [Wie (X (A) - &) sin nké dé n A Xi ° NY , We i Oe (Ap) 92) esanlinikey de} @) z sinh aig oN ; - [wo?* + y°(a*) cos nké dé Yo @ AX A - We S Y°(A*) cos nké dé] . (19) Po) The W terms are weighting factors used to distribute the error pro- portionally between the components of the transform generated and specified curves. It can be shown that the above relations result from minimizing the following error function: E. au wot poeitl i wot poeitl i+l Xx X y y ts woot poeitl woot pooitl f (20) x x y We where X Bal 1 i i Be Soe [DS ate.-) Sa (EOE ag (21) -r r ee See [OS ate,-8)) - yh (e,-8) 2 de, (22) -X r wi Be or [ox * ,+8)) See (Gee te (23) -r and r ae = + [ovat e,+8)) 3 Ee ale (24) -X The transform generated arc length and weighting factors are deter- mined relative to the results of the previous iteration by: Ss ‘Si Wit = SERENE (25) S S W = E/E , 26 : y/ (26) Cc C= tee Be (27) Cc CS W =E/E, 28 y . (28) where E is the average value of equations (21) through (24). The arc length functions are given by: gt AE" 8) S90 | ic? 4 Gor az (29) (@) and + moe + ls A(E',B) = ¥° | 1c 276 ae, (30) (0) Or OM ge < A and y° and yo are prorating factors taken such that A(A,n) is exactly the known length of the given curve. Two levels of approximation are involved in the error function. The ith approximation of these quantities is involved in the interpolation of the given curves. This approximation is mandatory in deriving a deterministic form for these quantities relative to the i+l level. The iterative procedure is initiated by setting all coefficients to zero, all weighting factors to unity, and equations (29) and (30) are given by yS&t and yet , respectively. The first estimates of 8 , BS and the Fourier-type coefficients are determined from equations (11) through (14), respectively. Hav- ing the first approximation for these quantities, the transform- generated arc length for each curve and weight factors may be deter- mined thus permitting a second approximation. This procedure is repeated for given N until 8 , B, and the transform coefficients have converged as indicated by successive values of the error func- tion. The error function approaches a constant value governed by N and the inherent errors in estimating the various integrals by numerical methods. The trapezoidal rule employing M values of € at equal increments over the interval 0 < € Xipouuoy ade) outysn3sny °4S Oll LOATY 600IT “O'N YURUUPARS qyrey)Q = :worly OZ! 23 O06 \ *3S5e09 3Se9 toddn OSE Aeg oosqoued Ol *(wo230q) dAInS ALepunog pxemess pue (doi) sutzTise0oj) °9 aLn3Ty aL? O€2 O6I Osi Oll OL O£ OF wy sog = SzTUn QOL punos (eve) | ODT [WP Aeqg Aeg ayeodesoygr~y dLEME LIQ Ovi WyLOL Max Osi 24 the coast and the 180-meter contour were digitized from an overlay of bathymetric charts for the regions concerned. The smooth curve is the result of applying the numerical spline interpolation routine to these points. In reading the discrete positions, both the coastline and the 180-meter isobath have been smoothed subjectively to suppress variations with scale lengths less than the grid scale ultimately employed in the surge calculations. For example, narrow entrances to bays are replaced by a fictitious coastline across the entrance, and the cusp-shaped features are smoothed. A method was adopted which minimizes the amount of computer time required to solve equations (9) and (10) by hastening the convergence of the iterative procedure. The method consisted of increasing N at selected iteration intervals. The iterative procedure used to determine the transformation coefficients was terminated if the con- vergence criterion was satisfied or if the available computer storage was exceeded because of increasing N , The convergence criterion was a mean variance of less than 1 square kilometer between the transform-generated curves and that specified, However, for develop- ment purposes, the iterative procedure for the western and central gulf coast and lower east coast areas was continued beyond this criterion to obtain a better fit, It is convenient to discuss the mapping of the western and central gulf coast regions at the end of this section since additional testing of the solution to the mapping equations was performed with these areas. Figure 4 shows the shelf region for the eastern gulf coast where the Mississippi Delta is shown as the shaded area. Considera- tion of the numerical time step for the surge algorithm was the prin- ciple reason why the coastline to be mapped did not follow the delta. The reduction in the time step if the actual coastline had been followed would, probably, have been at least tenfold with respect to the one used in the simulation of the storm surge induced by Hurri- cane Camille (Section V). Since the delta, or really the levee adjacent to the Mississippi River, has a controlling influence on the circulation and surge caused by the hurricane, this geographical feature was included in the surge model as a wall protruding from the coast. The fit of the transform-generated coast and seaward boundary — curves with respect to that specified after 1, 20, 40, 80, and 160 iterations is shown in Figures 7 through 11, respectively. The suc- cessively better agreement of the mapped curves with respect to that specified is obvious from the convergence table for the gulf coast region of Atchafalaya Bay to Apalachee Bay (Table 1). The table shows the variance and convergence behavior of selected transforma- tion coefficients as the iteration and N increases. The coast variance in X , the third column of Table 1, is calculated by: M : var PS Gio) ae. P (31) jel 25 *3se09 FINS utoyses ‘(poysep) petzyroeds yey pue (PTToOs) UOT}EL9}L QUO 19qFe OAAIND ALepunog pleMess pUe SUT[ISBOD pa}eLoUsd-W1OFsueLy, OO€ 092 O22 Osi Ov! 00! 09 "2 9an3Tty 02 Oe- jO9 02! Os! 26 *yseo0d FINS uszoyseo ‘(poysep), potztoeds yey pue (Pptfos) SUOTIBIAII QZ 193Fe BAIND ALepuNog piemvas puke JUTTISBOD pe .elouss -WiojSuely, OO£ 092 O2e os! Ovi ool 09 °Qg o1n3T4 02 All "Zse80D F[ns utejses ‘(poysep) petzroeds yey pue (ptfos) SUOT}ELI9}L Qp La}Fe dAIND ALeEpunog pileMedsSsS puke SUTTSPOD pojeToUas-WtOFSUeL] Ove OOE& O9z Oze os! 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Appendix A contains the transformation coefficients used to produce Figure 11, The successful application of the mapping equations to the other shelf regions was accomplished. Tables 2 through 5 clearly indicate the successive convergence of the mapped curves with respect to that specified for the lower and upper east coast and the western and central gulf coast, respectively. Figures 12 through 15 show the fit of the transform-generated curves and that specified after the last iteration for the above regions, Appendixes B and C contain the mapping coefficients used to produce Figures 12 and 13, respec- tively. The bottom part of Tables 4 and 5 show the results of addi- tional testing of the conformal mapping equations (9) and (10) with the western and central gulf coast regions. Another, less general, solution of equations (9) and (10) is possible if one minimizes the least square error function defined only in terms of the Y inte- grals. This solution for the transformation coefficients, hereafter referred to as an alternate solution, may be obtained from equations (13) through (19) with we = We = 0. The testing procedure was to continue the iterative process as outlined in the previous section with the initial approximation for the coefficients being those values determined from the 80th iteration. The alternate solution as applied to the western and central gulf coast regions was stable and, moreover, provided a better fit with N = 110 than the more general one. The fit of the mapped curves with respect to that specified after 1 and 40 (or, 81 and 120) iterations with the alter- nate solution is shown in Figures 16 and 17, respectively, for the western gulf coast and Figures 18 and 19, respectively, for the cen- tral gulf coast. Appendixes D and E contain the transformation co- efficients used to produce Figures 17 and 19, respectively. In test- ing with the other three shelf regions, the alternate solution was nonconvergent in that the successive values of the error function do not decrease or approach a constant as outlined in the previous section. 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Lake Borgne * e a < = a ea a ee 9 | ke co co i= | = Oo wa (120.3 km) S* (km) 14.8 km AS AT* 17.0 min Stretched shelf coordinate system (S*,T*) for Hurricane Camille surge simulation. Figure 29. 3 wWa4skS d2eUTPIOODS FTaYsS oS oo! OS | O02 OS2 OOE OSE OUPITIINY LOF O00 | UUb *S 07 3 “UOTJeTNUTS os8INs oT Tue) sutwLoysuer} sdtysuotzepat ~euorqouny d (wy) 3SBOD Buoye sdue4siq S O08 0 OL goog HOS 00h QUE u0d OU | WY C°OTb > «S$ > Wy 967°S (¢7-O1)SST-0 E° LES Wy £°06P “og emnsty 0 0 O01 i@p) ct Le 1a) cot ooz $. (o) Q G cH OOf — ee Q (o) 5 00h € ke =) (3) ct (or) O05 ¥ n ct 9) 5 009 DOL *S (wy) 38205 BuoTe ad5ue4STp 59 u wa3sds a},euTpLoOS J TaysS *UuOT}ETNUTS odains oT [TwWe) guedtiiny roy yf 02 U BuTWIOFsuUeT, sdtysuoTJeToL [BUOTIIUNY “TE ainst 4 Perit ae 9 Beore waqysks aqeutpioos Fyays payoie14S u (uy) Atepunog vas WOLF soUeISTP S§ (uTW) aWti [oAeI. «ll, o8| OS! Oz | U6 og OE | 06 O81 oL2 OSE OE- SI- 4 *& ct 4 ie8) S o o Looe ct Ee co) S a. S| 3 OE 60 200 Cape Fear Charleston Savannah St. Augustine eee: GEESE OE ©, 22% 170 Wilmington 110 i:]1) 6 ‘ <—— Track of Hurri- cane Gracie so = -- = ---- 20 380 320 260 230 170 140 so 20 -10 Shelf coordinate system (£,n) for Hurricane Gracie surge simulation. Figure 32. *UOTJETNUTS o3Ins eToOeIH QueITLINY IOF (41‘¥S) Weysks oJeuTpLOOD FTays paeydyetys ‘¢¢ omn3IZ uTW p2°L = «LV Ul TTL S 260 Ww ie es (wy ¢°SSP) Ss I PEOSEBHBEEEOEEEO BASE SGBECEEEGEU ieay ode) uojsaTs1euyg 62 3 WazSkS 9JeUTPIOOD FToysS OS 001 OS! O02 OS2 DOE OSE *UOTIE[NUTS o8ins atToeID suBoTIInH OF »S OF 3 BuTUtOoFsuRPL] sdTysuoiqepat ~Teuotjyouny “pes oan3ry 4. d (Wy) JZSBOD SuojTe adueYSTp S O0S| OSZ| uoo| OSL OUS O82 0 00h IL Ge (9 {-O1)S6L°0- = 4 Os Swag = VW Wyss 79S 009 OOL WY 8°7S6 > xS > WY S° O08 S10 01 = 9 (gz-ONIrl-0 = 4 ( LLS8 = V 1 8°72S6 < « O06 oud! O01 | wozsks 32eUTPLOOD FTaYys payoiedas xS (wy) 3sv0D Buoye aoueqstp 63 u woisks 9}eutTpLoo. FTeysS il- 9] *UuOTIE[NUTS o8ins 9tToOeID auedtiiny Io} yl O23 U Butwszossuery sdtysuotjefat TeuotzouNy “se ainsty Le@@e = 3) suoype a Wa3SkS a}eUTpPLOOD FTsys paydze14S (wy) ALepunog eas UOLF JOURISTP S (uTW) OWT} [OAPI yf Oz! oo) O08 og Oh Oz 0} OE og O6 O2! 64 2. Storm Surge Equations. The vertically integrated form of the quasi-linear long wave equations in a Cartesian system are well known (e.g., Welander, 1961). The appropriate forms for storm surge computation in the stretched shelf coordinate system are as follows: dQ HORS Ti) Br 2D OR as iN i Aite fy Qe ts Fu aS* (H Hp) i Tox Tox > (40) dQ T* gD 3 i . yr See ey pre BS) = Gs 9 ore ~ (en and SHE ual ates 13 \ ara ap a as (FQcx) tp Ser Glee)! = (0), (42) where Q is the volume transport per unit width, t is the wind stress divided by the water density, o is the bottom resistance stress divided by water density, f is the Coriolis parameter, D is depth of water, H is the sea surface elevation relative to mean sea level, and Hp is the hydrostatic elevation of the sea surface corresponding to the atmospheric pressure anomaly. The term F “is a variable scale factor associated with the conformal part of the orthogonal curvilinear coordinate system. Specifically, F= (()2+ @7, (43) where x and y are given by the transformation equations. The terms wu and vy , are additional scale factors that relate to the transformation of &€ to S* and n to T* , respectively, and are given by: 3E aS Me ee ae ve (44) ie and 3S Gy y = -t goon eb M (45) chs) oT oT* n Furthermore, F*yv corresponds to the Jacobian of the transforma- tion in the sense that Nea fF Fi dss cite; (46) Re 65 where A is the area of a closed region R in the x,y plane whose mapped image is R* . The kinematic wind-stress components T., and lated to their x,y component counterparts Cr 5 point by: Tre are re- Fo) at a given uae = cos 6 + fe sin 6 , (47) and Tp. = “Tt, Sin @ + Sy cos 6 , (48) where d = (Oy/OE Om=mtan Gama) (49) (50) where p, is the air density, pw is the water density, Cy is nondimensional drag coefficient, and W,9 is the windspeed at an elevation of 10 meters above the water surface. The value of K (Reid and Bodine, 1968) is taken as; Ky » see Wag, < oO moses Ke -l Ky, + [1.0 - 7.0/Wy9]? Ko , if Wyo > 7.0 m-s MMe Ky ds lei sa0°S and kK as 2.5 % 1076, The values of K, and Ky used by Whitaker, Reid, and Vastano (1973) for Lake Okeechobee were found to be too large for the shelf surge computations investigated in this study. The form of the bed resistance terms are: K,Q Gigs ae Ses 9 oo and Ke T* = D2 Qn ’ (53) 0, 66 where 2 2435 OQ lle, 7 Grell” 6 (54) and Ky is a nondimensional drag coefficient taken as 2.5 x 1073. 3. Numerical Algorithm. The numerical analogs of equations (40), (41), and (42) are based upon centered difference approximations of all terms (see App. F). The algorithm treats the time dependency explicitly and employs a computing lattice as shown in Figure 36 in which the transports, Q., and Qrsx are computed at the same location but are staggered in S time and space with respect to tne water level anomaly. This scheme facilitates a simple representation of the Coriolis and bottom stress terms in the difference equations. The surge model allows for vari- able bathymetry and arbitrary coastlines which are represented as a series of connecting straight line segments situated along lines of constant S* or T* . For the simulation of the Hurricane Camille storm surge, the arbitrary coastline feature of the algorithm is mandatory to delineate the delta. However, in the more usual appli- cation, the coast is a straight line in the computing grid. The difference equations for Qcx, Qrx, and H at interior points of the computing grid are given by: i OL Cipal) os (Gi Go Ne Gal (Gr 2 G2 Maye) } (55) S Qre(isdomtl) = (G; G3 - f at GiiG 2 & hey) » (56) and H(i,j,nt+l) = H(i,j,n-1) f At IP((aleri 59) Qo. Gt1,j,n) ~ F-1,j) Qo. G-1,j,n) BGI) u(a) AS* Es Giewital) Qpa G2, j+1,n) a (ab, 9} JL) Qpx G,j-1,n) i 0G) aT oe (ines als ' 5) 5 Gicl m iinchices Gymesss wag SP 5 WY 5 eine! teams coordinates, respectively, and At is the numerical time step. The quantities G, , G» , and G3 are given by: G) = 142K, at QGi,j,n-1)/D7 , (58) 67 I AIO WT EGE TS Pil feel aay | Figure 36. Scheme for computed variables. 68 Go = Qo. G,j.n-1) ap Ge [NE Qpe 4, j,n-1) i give 1D) see : aa 2 u(i) ss) AS* lil (Gaisell) 5in)) H,(i+1,j,n) - H(i-l,j,n) + HyG-1,3.n)) + 2 At Tox ; (59) G3 = Qrx Gs j.n-1) ae aerAN Qo, 4,j,n-1) 4 g At D Ate Zi Gah vj) FA Ge) AT* HG, 51.0) Hp (4, j+1,n) - H(i,j-l,n) + Hy@i.j-1.0)| i 2D IME Tipe > (60) where D = A D(i+1,j,n) + Y(t Ik 5 3 50) at D) (Gis aerdl 5101) ty DG,j-1.)| * (61) In order to maintain numerical stability, the time step must satisfy the condition: Fuv AS* AT*) 5 At <[ Ane ye (62) where s G, = [(@m AS3)2 & Gy areyzy" , (63) and the minimum value of the right-hand side of relation (26) is implied. Thus, a search of the grid is required for a proper selection of At . However, the conditions along the seaward boundary usually limit the time step because of the deeper water. The values of »p , v=, F , and 6 given by equations (44), (45), (46) and (49), are required at the appropriate computational grid points. A numerical spline under tension is employed to interpolate & given the IM wallues oF S (S*) . Wne scale Accor, mM 5 1S determined by: P ua) = BGs) eb (64) 69 where €(0) and €(IM+1) are ascertained by expanding the computing grid one increment of AS* to the left and right, respectively. In a similar fashion, the JM values of 1 may be determined and the scale factor v given by: vie) = MUD = alien) 2 (ST 2 ed) J =& 2, e806 JME. Since the computing grid may not be extended landward or toward deeper water, the approximation at j = 1 and j = JM is made that: wa) = In@) = a@ylfare (66) and v(JM) = [n(JM) - n(JM-1)]/AT* . (67) The area in the x,y plane of each grid block, including those from expanding the grid in the & direction, is determined by numer- ical procedure from the following relation: Me ei s Lote 2 am | A(@i,j) = | 2D lea. (68) 0g OE Tee oS J 1-1 i 5 1,2 IM+1 , 1,2 1,2.¢ JM The scale factor associated with the conformal mapping is determined from equation (46) at intersectional points, excluding the coast and seaward boundary, by the expression: z I. Be) < Mas) ee A(t jel) = AGieil 7) & Atl, je) | - >J AINSI icant GN) aol Gin) (69) wee ioe oo IM. iT} ie) v W j e JME, and for grid points along j = 1 or JM by the approximation: 70 ie NCD NC EDL ||s FL.3) = la-ag* at Gp) wG) : (70) th eZ scteh MLM ied The angle, 6 , relating the orientation of the stretched shelf coordinate system to the x,y plane is determined at grid points, excluding j = 1 and JM, by the following smoothing procedure: ated, | a ft rita é : oa Tae (53) - =|6(4-1,3) + dG gen) = d@an.aye d(i,j+1) + 48.3] (71) =r iN j= 2,3 °° JM-1, where e! is given by equation (44). Along the sea and coastline boundaries we have e(i,1) = |e (i-1,1) + 20'(4,2) + g! Gsugi) s 40" Gi,1)| &® 4 21,2 ose ing | and aC ,an = de" Gta 3s eA NS 8 Galote Asta f= 1g) 392) iM 7%) 4. Boundary Conditions. A wall condition is employed in the surge simulation along the coast while the surface elevation anomaly is placed in equilibrium with the atmospheric pressure along the seaward boundary. Vanishing normal derivatives of transport are specified on the lateral open boundaries. This condition is used by Jelesnianski (1965, 1966) and Forristall (1974), although it may not be the most desirable (Reid, O75) < In the x,y plane the coast is curved making the wall (coast) and lateral boundary conditions difficult to apply in a rectilinear erid system. However, the stretched shelf coordinate system repre- sents the coast as: Qe sO. (74) or, the analog is simply, Ta Qra(i,JM,n+1) = 0, (75) 1 S158 ++- TM, Thus, total reflection is assured at the boundary. The flux, Qgox, along the coastline is calculated from equation (55) with D taken as the two point average of the local fluid depths along the coast. The local depths ranged from 1.0 to 2.3 meters, depending on the sea- ward bottom slope and surrounding elevations. The water elevations along the coast are computed from the continuity relation as given by equation (57) with the following substitution: F(i,JM+1) Qo, (i,JMt1,n-1) = -F(i,JM-1) Q.,(4,JM-1,n-1) , (76) 1S Aya ooo EZ This is an artifact consistent with total reflection. For the simulation of the Hurricane Camille storm surge, the normal routine of the surge program was interrupted at those grid points representing the protruding Mississippi Delta. Along this part of the coast (the solid heavy line in Figures 28 and 29), the normal flux was set to zero and the tangential flux was determined from either equation (55) or equation (56) with the Coriolis term vanishing. The continuity relation is altered depending on the orientation of the boundary to be consistent with total reflection. The open deep sea boundary condition is: H@, l5n+l))) = H,G,1,n+1) i (7) io 24 coo MME and H(i,2,n+l) = [HG@=1,1,n+1) * H@i+l,1,n+1) + H(i-1,3,n+1) + H(i+1,3,n+1) ]/4 , (78) 1 8 O95 999 NEP Specifying the seaward boundary in this sawtooth form obviates the calculations of Qcx and Qrx along the boundary since they are not required for computations at interior points. The lateral open boundary condition requires the normal gradient of the S*-directed transport to vanish, 2 9Qox Sem Tee (79) This condition implies that along the left side of the grid, Qox 1,5 ,ntl) = Qo (S,j,nt1) , (80) ==) Ove and along the right side, Qox(IM,j.m+1) = Q.,(IM-2,j,n+1) , (81) Te Sy OO HM ae The S*-directed transports for even j and i=2 or IM-1 are deter- mined by the average of the two neighboring interior values. Addi- tionally, the T*-directed transports are required along i=2 or IM-1 and are calculated from equation (56) where D is again the average of the local fluid depth along the boundary, Special computations are required at seaward corner points (2,2) and (3,1). Simultaneous equations are solved for Qpx(2,2,n+1) and Qox(3,1,n+1) with the approximation that Qypx(3,1,n+l1) is the aver- age of the two neighboring interior values. Similar expressions are used at the right-hand seaward corner points. Other conditions at the seaward and lateral boundaries were experimented with for the Hurricane Carla simulations. These in- cluded radiational conditions of the type discussed by Reid and Bodine (1968) and Reid (1975). The main differences, as anticipated, were close to the lateral boundaries where the "flow through" condi- tion in equation (79) gave a more realistic rendition of the long- shore flow. The basic numerical model was used successfully by Alvarez (1973). Further testing of the algorithm, where in some cases an analytical solution was possible for comparison purposes, is pre- sented in Appendix G. 5. Wind and Pressure Fields. Of major importance in simulating the storm surge is the accurate portrayal of the hurricane wind field on the computing grid. More- over, the representation of the wind must be time-dependent. The Hydrometeorological Section of the National Weather Service (NWS) provided charts of the surface winds at 10 meters above mean >) sea level and of the barometric pressure for Hurricanes Carla and Camille. One method for representing the wind field is to digitize the above charts for the hurricane in question. Usually the wind and pressure data depicting the storm are sampled in time at particular grid points and interpolated to provide the necessary input to the surge model. However, such input data are tedious and laborious to obtain and do not guarantee better quality of the input than can be obtained by analytical representations of the forcing fields. It is this latter approach which is used in this study. For a given hurricane, the following parameters are sampled in time from the NWS charts: position, forward speed, central pressure, radius vector (relative to the storm movement) to maximum winds, and the maximum winds. With an analytical representation of the surface wind and pressure fields the above parameters need only to be inter- polated in time. The wind field representation prior to consideration of the land influence as given by Jelesnianski (1965) is employed in this study. The x,y-wind components for a stationary storm are: V apoksiie. 27 iat Woe = Tp [ (x x) sin 9 (y Yn cos ] and F(r,) > (82) VR Uy = z, | + (x-x, ) cos > - (y-y,) sin ¢ | where 2 45 i 3 2 (eee) © Goa de 1° (83) Vp is the maximum wind, 9» is the ingress angle, and F(r,) is BYO I/2 « (r,,/R,) ge ig. S RB or (R/T) if ry > R- Distance from the storm center Oy,» Yn to the region of maximum winds is RX é The translation of the storm provides an alteration in the wind field which is carried out following the method of Jelesnianski (1965). This involves the vector addition to the above field of a supplemen- tal velocity whose direction is parallel to that of the storm and whose magnitude depends only upon xy, . The resulting modified values of V_ andV_ are used to compute the wind-stress components: y 74 a ! ak Ge 2 2) Wo. xX y X and (84) 45 Kk Ge ow) ¥ y x y y This constitutes the symmetric wind-stress field. ‘The stress compo- nents in the stretched shelf coordinate system are determined by applying equation (84) in equations (47) and (48). The surface atmospheric pressure field associated with the hurri- cane is given by: “Ra! Th Peps @ Pye ; (85) where P, is the central pressure and P, is the far field pressure. The term Hp in equations (40) and (41) 1s equivalent to (= EUR Se The symmetric surface stress field does not reflect the influ- ence of land. Without modeling this influence the analytically determined wind field near the coast would not be consistent with that from the NWS. A systematic procedure (App. H) is employed to alter the symmetric wind field such that it portrays the effect of land. These winds are referred to as deformed. Mr. Thomas Reid, Texas A&M University, Department of Oceanography, is responsible for the basic development of this wind model (unpublished manuscript). The pressure field was not altered in the nearshore region to conform with the wind field, Figure 37(a) represents the symmetric Hurricane Carla wind field at 1600 Greenwich mean time (G.m.t.), 11 September 1961 (approxi- mately 4 hours before the storm crossed the coast), Isovels are shown in meters per second. The analytically deformed wind field at this time is shown in Figure 37(b). The influence of land on the wind field is evidence even at a time when the storm is approximately 55 kilometers from the coast. Figures 38 and 39 show the symmetric (a) and deformed (b) winds at 2000 G.m.t.,11 September, and 0000 G.m.t., 12 September, respectively, The rotation and reduction of the wind vector in the nearshore region to reflect the land influence is illustrated in comparing Figure 40(a) for the symmetric case with Figure 40(b) for the deformed wind. These figures correspond to the contours shown in Figures 38(a) and 38(b), respectively, The arrows represent the wind vector placed such that the tail is at the com- putational point. The pressure fields, as determined from equation (85), for Hurricane Carla at the above times are shown in Figure 41(a,b,c) where the contours are in millibars, Although the pattern of isovels reflecting the land influence is in good agreement with 75 Time = 46.0 (hrs.) (b) SS SIN 24.0 Figure 37. Hurricane Carla symmetric (a) and deformed (b) wind fields at 1600 G.m.t., 11 September 1961; isovels in meters per second. 76 Time = 50.0 (hrs.) Figure 38. Hurricane Carla symmetric (a) and deformed (b) wind fields at 2000 G.m.t., 11 September 1961; isovels in meters per second. Ue Time = 54.0 (hrs.) Figure 39. Hurricane Carla symmetric (a) and deformed (b) wind fields at 0000 G.m.t., 12 September 1961; isovels in meters per second. 78 Time = 50.0 (hrs.) (a) (b) Figure 40, Hurricane Carla winds at selected points at 2000 G.m.t., 11 September 1961 for the symmetric (a) and deformed (b) representations. 79 Time = 46.0 (hrs.) (b) 9 Time = 54.0 (hrs.) (c) Figure 41. Hurricane Carla atmospheric pressure on 11 Sep- tember 1961 at 1600 (a), at 2900 (b), and 0000 G.m.t., 12 September (c); contours in millibars. 80 those provided by NWS, the problem of specifying the proper deforma- tion is not completely resolved. The observed high winds which remain along the coast after the storm has proceeded inland are attained by setting R, to be the distance the storm center is from that coastal area of persistent high winds. This procedure was also followed in modeling the winds from Hurricane Camille. However, R, in the pressure expression was not allowed to increase in this manner. The hydrographs at selected grid points along the coast are presented in Figure 42 illustrating the improved agreement with respect to the observed water levels in using a deformed wind forcing function rather than a symmetric one, The observed water levels, corrected for the astronomical tide, are indicated by squares. The computed water levels have been raised for comparison purposes to correspond to the local sea level at the start of the computations (1800 G.m.t., 9 September). The solid line represents the computed surge employing the deformed wind and the dashline shows the results with a symmetric wind. The surge computations were formed using identical boundary conditions. The symmetric (a) and deformed (b) winds for Hurricane Camille at 0000 G.m.t. 18 August 1969 (approximately 4 hours before the storm crossed the coast), 0400 G.m.t. and 0800 G.m.t. on the same date are shown in Figures 43 through 45. The rotation and reduction of the wind stress for this storm are different from that of Hurricane Carla because of the differences in the coastal configuration. Figure 46 (a,b) shows the wind vectors for the symmetric and deformed representation of Hurricane Camille at the time the storm proceeded inland. The pressure fields at the above times are presented in Figure 47(a,b,c). The wind fields associated with Hurricane Gracie were not deformed because the documentation was not amenable to the deformation procedure. Also, Rh was not increased in the manner specified above after the storm crossed the coast. The symmetric winds at 1200 G.m.t., 29 September 1959, 1600 G.m.t. (the time the storm proceeded inland), and 2000 G.m.t. on the same date are shown in Figure 48, Figure 49 presents the pressure fields at the above times. 8l 3 © Pog & = @ (Io) ) (= Re 42 PLEASURE PIER Galveston, Texas oe 32 y 7 2 oll MUD BAYOU Texas 1S BO 4 SABINE PASS Texas 6 12 18 24 30 36 42 48 54 60 66 Time (hours) Observed (squares) and computed water levels using a deformed wind (solid) and a symmetric wind (dashed) for Hurricane Carla. 82 Time = 24.0 (hrs.) (a) (b) 2, “Oo SS Ww / ~ © Z \ (jo) N [a : Figure 43. Hurricane Camille symmetric (a) and deformed (b) wind fields at 0000 G.m.t., 18 August 1969; isovels in meters per second. 83 Time = 28.0 (hrs.) Figure 44. Hurricane Camille symmetric (a) and deformed (b) wind fields at 0400 G.m.t., 18 August 1969; isovels in meters per second. 84 (b) Figure 45. Hurricane Camille symmetric (a) and deformed (b) wind fields at 0800 G.m.t., 18 August 1969; isovels in meters per second. 85 7 oa 50.0 m-s~2 Figure 46. Hurricane Camille winds at selected points at 0400 G.m.t., 18 August 1969 for the symmetric (a) and deformed (b) representations. 86 Time = 24.0 (hrs.) (a) | Time = 28.0 (hrs.) (b) (c) Figure 47. Hurricane Camille atmospheric pressure on 18 August 1969 at 0000 (a), at 0400 (b), and at 0800 G.m.t. (c); contours in millibars. : 87 Time = 24.0 (hrs.) Figure 48. Hurricane Gracie symmetric winds on 29 September 1959 at 1200 (a), at 1600 (b), and at 2000 G.m.t. (c); isovels in meters per second. 88 Time = 24.0 (hrs.) (a) (b) 9 Time = 32.0 (hrs.) (c) Figure 49. Hurricane Gracie atmospheric pressure oi 29 September 1959 at 1200 (a), at 1600 (b), and at 2000 G.m.t. (c); contours in millibars. 89 IV. SIMULATION OF THE FREE WAVE IN AN ANNULUS 1. Problem Statement. If the wind stress, bottom friction, atmospheric pressure, and the rotational effect of the earth are neglected, then the equations in Section III admit a simple analytic solution for free gravity waves of long wavelength. A study using curved boundaries will demonstrate the superiority of modeling the long gravity wave in orthogonal curvilinear coordinates over those models which employ rectilinear coordinates. Basically, the study involves comparing the numerical solution of a free-standing long wave in a 90° section of an annulus with an inner radius r, of 393 kilometers and an outer radius ry, equal to 2r, (Figure 50). The annulus is bounded on all sides by a vertical wall. The depth of the basin relative to the mean water level is assumed everywhere to be 40 meters. The analytical solution for the free-wave oscillations in a section of an annulus may be obtained with some modifications from Lamb (art. 191, 1932). The boundary conditions require that: oH L Oh he ap seh) Ele! see ae) (86) or and a7 = 0, at @ = 0° and @ = 0/2 . (87) The analytical solution for the given initial conditions is: H(r,6,t) = yi y A J (ge 32) = Talkan?}) ie) | ehh oe m,n n> m,n Y'(k r) 2% m,n cos né cos Gamat ‘ (88) m= 0,1 3 n = 0,2 where omn i Spi ved, (2) and for given n , kh 2 is the mth root of 90 Land 393 km T} Fisure 50. The annulus in polar coordinates (rT,€). 91 Ji(kr2) Y¥ (kr) - J) (kry) Yi(kra) = 0. (90) The terms J and Y are Bessel functions of the first and second kind, respectively. The order is given by the integer subscript and the superscript refers to differentiation with respect to the argu- ment. The term A in equation (88) is an arbitrary constant repre- senting the initial amplitude. The first azimuthal mode is for n= 2 and the lowest value of k that satisfies equation (90) is approximately 1.340/r,; meters-!. The period of oscillation for this mode (m = 0 , n = 2) is 25.85 hours. The numerical solution of the free wave in the quarter annulus is sought by performing the integration in two different computing grids. In one case the grid is rectilinear (Figure 51), and in the other a polar system is used. In the rectilinear system the outer and inner radii of the annulus (the light line in Figure 51) is simu- lated in a stairstep fashion. Considering a limitation on computer time and storage, an acceptable rendition of the curved boundaries is present in the Cartesian grid. Proper representation of the quarter-annulus requires a transport point at corners on the outer and tnner boundartes. Consequently, the rectilinear boundary is not symmetric about 1/4. The locations of nine hydrograph positions are indicated by small boxes. Although the computing grid is 43 by 43, only 1,052 points are used to represent the annulus. In this grid system, AS* and AT* are just Ax and Ay , respectively, with Ax = Ay = 19.65 kilometers. The maximum allowable time step as determined from equation (62) is approximately 700 seconds where F = u = v=1. Finally, the analog form of the long wave equations is obtained by setting those terms in equations (55) through (57) to zero which are neglected, and setting F , uy, and v to one. The transports, Qox 5 Qr* » are now aliases for Q. and Qe 5 respectively. The numerical algorithm for the rectilinear grid system must be capable of determining if a computational point is interior, exterior, or on the boundary of the annulus. For exterior points, no calculations are performed. Grid points on the stairstep boundary require special attention subject to the condition of a wall. Furthermore, the numerical program must identify and apply the appro- priate wall condition depending on whether the point in question is located at a boundary corner (an H computation) or on a segment of the boundary. Clearly, extensive programing and computer time is required to accomplish this task. The other computing mesh which is used for obtaining the numerical solution of the free wave in the annulus is the polar (or stretched shelf coordinate) grid system (Figure 52). The transformation to the computing grid as shown in Figure 53 is accomplished V2 fa) & TT Se 4 HUBe Se Neseeceseos | BOBUSSEEBEBSEQGH08E, OO BESS GOESRBeebesat LG Figure 51, Rectilinear grid representing the annulus; location of hydrographs shown by (0). 35 Figure 52. Polar grid representing the annulus; location of hydrographs shown by (0). 94 95 ORDER ESSER C lessee esseals LE | —— ie (CG) AS* 19.65 km ANI Computing grid for polar system representation of the annulus. Figure 53. 1 sie o gan eG s 2) ea Nigel Gian) = Ee Ce es (91) ay SS 2N oo 1) AB” 5 Iden yt Sacred ey 2m where AS* = 21.05 kilometers and AT* = 19.65 kilometers. Approxi- mately the same number of computational points are employed in this system as that of Figure 51. The actual area of the annulus is well represented in both computing grids. Although the inner and outer boundaries are straight-line segments in the numerical grid sense, the portrayal of the annulus is certainly more representative than that shown in Figure 51. The positions of nine hydrographs in Figure 52 (denoted by large dots) correspond to the hydrograph posi- tions in the rectilinear grid system. For the polar system, the maximum numerical time step is approximately 600 seconds. As a practical note, the stretched shelf coordinate system, although re- quiring less sophisticated programing and allowing better physical and mathematical portrayal ot the boundaries, generally results in a smaller numerical time step than a rectilinear system of comparable scale. 2. Results. For comparison purposes, a time step of 540 seconds is used in both grid systems for the numerical solution. As noted above, the polar grid program should compile and execute faster on the computer than the rectilinear grid program. On the IBM 360/65, the polar grid program required 0.47 minute to compile and 0.39 minute per 100 iterations. For the rectilinear system, the program consumed 0.59 minute to compile and 0.61 minute per 100 iterations. The initial surface elevation condition which is imposed on both grid systems is that d0H/d0 is a constant. For the rectilinear grid system, aes 4 =1 fe il H@,37,0) = 1 - | tan G3) é (92) where only the i,j indices appropriate for the annulus are employed in the above relation. For the polar grid system, the same initial surface condition is determined as: 96 Goa sO Soils 6e (sll) 5 (93) (4 I NO . WN 2b.\b The initial condition of constant azimuthal slope and no radial vari- ation of H along a line of constant 9 implies that there are several modes of seiching present; however, the dominate mode is for m= 0 andn-= 2. This is indicated by Figures 54 and 55 which show the simulated surface topography at 27 hours in the rectilinear and polar grid systems, respectively. The solution in the polar grid is more representative of the analytical solution than that shown in the Cartesian system. The hydrograph at 6 = 0° and r = r, is shown in Figure 56 (a) where the water elevation as determined from the polar grid is the solid line and that from the rectilinear mesh is the dashline. Figure 56 (b,c) shows the hydrographs along 6 = 0° forr=r (the average radius of the annulus) and r = ro, respectively. The three hydrographs along 6 = m/2 and the same r positions are shown in Figure 57 (a,b,c) and along the nodal line by Figure 58 (a,b,c). The average period of oscillation as determined from the hydrographs of Figures 56 and 57 is approximately 26 hours in the polar system and 28 hours in the rectilinear grid. The error in the period of oscillation (about 8 percent) for the rectilinear system is most evident in the figures by noting the lag of the dashline with respect to the solid line. The longer period of oscillation is directly related to the stairstep boundary. Effectively, the length of the basin is increased by the reflections introduced by these boundaries. This distortion is more than academic since many recti- linear grid models of enclosed irregular bays require adjustments to reproduce the fundamental seiching mode. The analytical solution at any point is a smooth function of time. The solid lines in the various hydrographs portray this feature better than the dashlines which are contaminated by high- frequency spurious oscillations. The nodal line in the polar grid solution as evidenced by Figure 58 (a,b,c) remains fixed at 6 = 1/4 which agrees with the theory. This is not observed in the recti- linear system. Although the hydrographs in the rectilinear grid are not positioned exactly on 6 = 1/4 (actually, about 44°), the Magnitude of the oscillation about 1/4 is approximately 2 to 4 times larger than expected. SiG Figure 54, Computed water surface topography in 0.0l-meter contour increments at 27 hours in the rectilinear grid system. 98 Figure 55. Computed water surface topography in 0.01-meter contour increments at 27 hours in the polar grid system. 99 (a) W a 10 s ] B [ 1 r 205 - aq E a E 0.0) |= : (b) 1 l @ L i eK a a it 3 -.10 O n 10 7 om a m y rf ae V 0) 10 20 30 40 50 60 Time (hours) Figure 56. Hydrographs along ®@ = 0° and vy = rq (a), 5) Uo) ain 4 (C)) Tor ene polar gail (solid) and rectilinear grid (dashed). 100 “eros 8 ° ° qa ae ae sdorra coy Figure 57. (a) (b) (c) Time (hrs.) Hydrographs along 6 = 7/2 and r = rj (Qo a (OD) incl sey (CE) sie NS jorollebe grid (solid) and rectilinear grid (dashed). 10l J 3 TS (a) Va-o & EO Mebet fl Of AA Dporroadtem Time (hrs.) Figure 58. Hydrographs along 6 = 1/4 and r = 19) (a), Ym (b) and rp (c) for the polar grid (solid) and rectilinear grid (dashed). 102 V. STORM SURGE SIMULATION i, Jnitrerclceine (Cereller. Hurricane Carla was an immense, slow-moving, and meandering storm which struck the gulf coast south of Port Lavaca, Texas, at 2000 G.m.t. on 11 September 1961. The radius to maximum winds was 50 kilometers. The atmospheric pressure drop across the storm was 75 miltlibars. The maximum sustained winds were approximately 51 meters per second. For numerical stability a time step of 180 seconds was used in the computations. The surge simulation was performed for a 66-hour period from 1800 G.m.t. on September to 1200 G.m.t on 12 September. The boundary conditions are those given in Section III. The wind fields are analytically deformed in a manner previously discussed. The ob- served (squares) and computed (solid line) water levels at selected grid points along the coast are shown in Figure 59. The observed water levels have been corrected for the astronomical tide. At the start of the computations, the water surface along the coast from Sabine Pass to the Matagorda Bay area was elevated by approximately 1 meter. Consequently, for comparison the computed water levels have been raised to correspond to the local sea level condition. The water surface topography in 0.5-meter contour increments at 46, 50, and 54 hours are shown in Figure 60. The computed maximum coastal surge profile corrected for the astronomical tide is presented in Figure 61. Indicated on the figure are the observed high-crest values from tide gages and water marks. The depression of the water surface to the left of the storm track (as viewed from the sea) and the low-peak surge in the Aransas Pass area results from the offshore wind and from simulating the coast as a wall. It should be emphasized that the surge model does not consider the coastal flooding from the shelf surge nor includes the influence of the increased communication between the semienclosed bays and the open sea. While the computations agree fairly well with those observed for the peak coastal surges, especially for the area to the right of storm landfall, and for the Galveston hydrography, there is some discrepancy for regions far from the storm track such as Sabine Pass. This might be attributed to the problem of proper deformation of the wind field (both speed and direction) to reflect the influence of land. Considerable erosion of beaches and adjacent offshore areas along the Texas coast occurred during Hurricane Carla (U.S. Army Engineer District, Galveston 1962). Figure 62 shows the water velocity V = Qgxi + Qr*j)/D in centimeters per second at selected grid points for the same times as that in Figure 60. Of practical importance in assessing the erosion potential and transport of the material is the alongshore current, Qcox/D - In shallow water, the current is uniform from the surface to the sea bed. Values of the current in centimeters per second as a function of time at selected grid points are presented 103 Boren cs orm Roce = ) (3 PLEASURE PIER Galveston, Texas MUD BAYOU Texas SABINE PASS 6 12 18 24 30 36 a2 aS 9 &4 60 66 Time (hours) Figure 59. Observed (squares) and computed (solid) water levels for Hurricane Carla. 104 / Time = 46 hours Se (b) / Time = 50 hours au / Time = 54 hours S Figure 60. Computed water surface topography in 0.5-meter contour increments for Hurricane Carla at 46 (a), 50 (b), and 54 hours (c). 105 *(peut{[tepun) se8e3 apti pue SyteU L9qeM YSTY [T]TTIS Woz SLaqaW UL SqSeT LaqeM YBIY paAtesqgo oae SUT[ISPOD ay Sutjuosaidet aAind Wo1{OG ayy 3noge poeynqitistp ssequnu aul “ePT2 [RoaTWOUOIZASe 842 TOF P9JIOLOD KIL) suPITIINH WOTZ SoBins Te Seo aya FO 191eM Y3ty peanduoy —— ayey Netsvoptey) -qug aye] autqes mt OS -4uy ' EES) qiodaeiy eg - ep.loseiey Ves iy “19 9INn3T.] UOLIeADTA La ey {UW ) 106 (a) / - iy ! / a My p ue2yemes / / Time = 50 hours S s ~ Ate (c) e e a eee ie CeCe - ~ ie e sy oe Op 77130 cms-> ~X ~ Cae ak af —> me Re, ~ 8 S ~ = OF XN = ie \ ~ = a 7 \ N ~~ =~ 7 Time = 54 hours ~S ~ ag, ~< hos XN oN XN ~ Figure 62. Computed water velocity in centimeters per second at selected grid points for Hurricane Carla at 46 (a), 50 (b), and 54 hours (c). Vector scale in centimeters per second. 107 in Figure 63. Also given is total transport in 10°© cubic meters per second as a function of time across the transect lines indicated in the figure. 2. Hurricane Camille. Hurricane Camille traversed the extensive low-lying marsh area of the Mississippi Delta, moving over the shallow waters of Breton and Chandeleur Sounds for 4 hours prior to landfall near Bay St. Louis, Mississippi, at approximately 0400 G.m.t., 18 August 1969. The radius to maximum winds was about 37 kilometers. The atmospheric pressure drop across the storm was 90 millibars. The maximum sus- tained winds were approximately 60 meters per second with the highest wind gusts estimated at 90 meters per second. The surge simulation extended over a 48-hour period beginning at 0000 G.m.t. on 17 August. A time step of 60 seconds was selected for numerical stability. The boundary conditions are the same as those used in the Carla simulation. The Camille winds are analytically deformed as shown in Figures 43 through 45. The bottom drag coeffi- cient was increased to 5 x 1073 at computational grid points in the area bounded by 6 < i < 11 and 14 < j < JM to simulate the larger resistance associated with the low flooded marsh lying northeast of the simulated Mississippi Delta (see Figures 28 and 65). Figure 64 shows the observed and computed water levels at se- lected grid points along the coast. At Pascagoula, Mississippi, the water level increased from 2.2 to 4.5 meters in a 2-hour period indicating a strong convergence. In fact, the simulated hydrography at the approximate location of the maximum surge (Bay St. Louis) showed a similar convergence with the sea level rising from 2.9 to 7.0 meters in the same time period (figure not presented). The water surface topography in l-meter contour increments at selected times from 24 to 36 hours illustrating the surge development is shown in Figures 65 through 73. The effect of the delta, modeled as a wall protruding from the coast, is visible in these figures by the presence of high water along the eastern side of the wall. The track of the storm with respect to the delta is such that the circulation is inhibited in this area resulting in the setup. The water velocity, V , at the selected times and grid points as shown in Figure 74 are in accord with these observations. The computed maximum coastal surge envelope east of the delta presented in Figure 75 shows that the calculated peak surges are approximately that observed in the Gulfport to Pascagoula area (U.S. Army Engineer District, New Orleans, 1970). A test run employing a constant wind-stress coefficient, K , of 3 x 10°© with the same boundary conditions produced a similar maximum surge envelope. There is qualitative agreement between the observed and predicted surge envelopes. This is shown by the rela- tively low water elevation of 2.7 meters near Hopedale, Louisiana, 108 -~ Aq pa yeoTput St puoses zad sio}eu sTqno ut yszodsuer. aLoyssuote [eOL *squtod pt13 pe deTes ze owt JO uoTzoUNZ B Se puodes zed sxe aultques UT * a/**d ‘yuaLind e1oyssuotTe FO sanTeA “29 9in3Tt4 ctv- ez- ze- e'- f gz'- 6t'- S¥- ac'- g0'- Te'- v7'- zr- ce - wy - 99-499 - SHE fC - O9- @9- Wr Swe 7 - 09 - @8 - S8- Swe €l- Sse - 7 - 97 - €-WE SE - Ov - S9- 9S - CAE Gio ho GS Gyo salir Gio ee Ceo MES Weis ez- (6 - 9 - 2S - I-WE G68 Me Go Wo Ate Se ee eas ES Cee pon AS ee 9S 0S 99 79 9S 15 99 7 "S 0s 9” 79 (834) aWIL OveT (914) SWIL ztet (©14) SWIL 922} cz- it- t- st- iz'- 90'- or'- tr- | COR RE E% = O59 = {( w- U6 - SS - Ly - SWE oy - 1S - BL - 69- S-HE G@ Mo Mie Sue Shh Te - SE - 9S - cy - C-WE "9 - 9S - 08 - OL - C-wE Il 66- CIT- %II- €-He gt - f2- OS - = I-We 02 - 67 - 2 - 99 - IT-We Sp- 66 - 61I- 601- I-k ae Paes ae ee Se Se a ee + Game OS IRE, CY 7 «OS ” ~«6tY ms Os wy Co \ San, (814) WIL ote (824) aWIL ever (814) FNL Oz=t Ors Toan q Qe ae PS s oe tw MP XS & net" Fe 2 wzet ore Mes goer S-HE la =, “yy =o Ps ainojuod yjdap @ PT om wel Sets) Pe 0 Saose sas -- weRewe om nw es ao ¥ cK (eres woooeon” T-he = e SS : Si eC >» 310da034 ar a = ; y SS oN) EZ Aeg ep1ioseiey SS wae v —+ aye] eutars pea Aug uo asaatey 109 i) ©) fe) er (ce = @) T= [eel ROodcpo = (3) PASCAGOULA Mississippi DAUPHIN ISLAND Alabama Figure 64. PENSACOLA Florida Time (hours) Observed (squares) and computed (solid) water levels for Hurricane Camille. 110 *sinoy pz 3 OTT twe) sueoTIIny Io} spunos tddtsstsstw pue ‘iInajTespuey) ‘uojJatg FO Bole oY. UT SjUdUaTIUL InojUOd Jayau-g‘{ ut Aydexr30do0q adezans 19zeM paynduo) Inojuod yydep 1979U-Q] “ -- x x “i utydred aT Tqow ENG SS eB[noseoseg Aeg “InaTapuey) INO}UOD L3}0Ul-Q°T ut Ayder80d03 ade TXOTTG “sanoy 92 3e oaf[rwe) “uojorg JO Pole 9yQ FANS 191eM paynduoy 2 ‘ a Off sae 4 > v “99 din3Ty 12 *sinoy £Z 3e aT[twe) suPedtriny IoJ spunos tddisstsstw pue ‘IneTepuey) ‘uojJetg FO vate oy} ut S}USWALIIUT InojUOD IazaeU-g°], ut Aydersodo} asoezans tajem poqynduoyj -79 ouns8ty Inoju0. yydap LezeU-gT A \ am } Lew i eG Aeg C : f = a ox et (a eee e[noseosed keg, CSI) wr 2eg = OWI, 113 “sanoy $°/z 1@ OT [TwWe) suedTLIny LoF spunos tddtsstsstw pue ‘inapTepuey) “‘uojoeLlg JO Bate oy. UT SJUdWaTIUT InojUOD LajoUl-Q°T ut Ayder3odo0q sdezuns 1ezem paynduojD -gg sinsTy gd i ie \ SS \ ~— =e wy 0°02 2 1 te =>, Jay Inojzu0d yydep 1979uI-gT a7 ee So Si tin Be EE -]_ usydteg keg oT T GOW Lo C8 \, BInoBeosed™ gtrg S14) G22 = UN], 114 *sanoy 8Z }#e OT[[TWeD oUedTIINyH OF spunos tddtsstsstw pue ‘aneTopuey) ‘uo.erg Bore oY} UL S}UsWeLSUT InojUOD IszeW-Q°T ut AyderZ0d0} odezains 19}em pajndwoj) -69 oan3Ty Inojuo0d yydep LazeUW-gy{ 7 \ a \ Se ee : Wf "J utydjeg Uw —<——— — A Aeg SF — eTtqon D> a 2 np Se i Keo FLOdZ [Ng B[noseosed z TXO[Tq (‘S1y) O8e = IU], 5 *sanoy $°8z 3e OTT We) suedTIINH OF spunos tddtsstsstw pue ‘inejTopuey) *uoj.erg Fo evore OY} UT sqUdWETOUT INO}JUOD T9}OU-O°T UT kydexs0doj oovezans 193M poinduoj “OL eansty Ce ‘ 116 ! S See ' guBx0g., SS ») . \ inojuo0d yydep 1a}0ul-gT ae j fees 2 a a p -" ie Va shig i ae \Se2 Ss sseSc" eee “IT utydteq ise ed Ss keg we 2G ) aTTqon (FeV ba \ ay NSS todg np. \ A cea\Se OUST) gee = WNL e[noseosed Txo[ Tq = S\ \ *sanoy 6Z 28 OTT TWe) suedTIINyH oF spunos tddtsstsstw pue ‘AnoeTepueyD) ‘uojorg FO VOLS OY} UT SjUOWOeTDUT InN0jJUOD I9JZOW-Q°~T UT AYydersodoj sdezZans 104eMm po qndwo) Inoju09 yydep 1a j0u-9gT oS -- — —~ — "I urydred Ras fez = - owt] ‘TL omnsty 117 *sInoy 0g 28 OT[TWe) oueoTIINY 10x spunos tddtsstsstW pue ‘ineToepuey) ‘uo, FO vole 9Y} UT SjUSWeTDUT InojUOD I9ejZOW-Q°T ut Aydersodo, sd¥FINs 193M peqanduoy °7Z san3sty ———4 wy 0°02 Inojuo0s yydep I90Ul-gT eee - ¢ oS oes 2 a wens “I utydteg ~ ‘S—=0dz tng e[no3eoseg ow 9 SIU) O'OF = UU, 118 “sanoy 7¢ ye oT True) aueotIInyH 10} spunog tddtsstsstw pue ‘mopopuey) ‘uojJoIg FO Bale ayy UL SJUDWALIUT INO\UOD ATOJOW-Q°,T UL AYdesso0do0} odezZIns 19}eM pajnduoy) wy 0°02 Inoju0s yydap 13 9U-gT "Lutudred Z 1149 LY, Se “i kegN$ om a EO qy rn) “CZ amnBty eC =r = Z ul B[noseoseg LxXOT Tg ( Siu) 0 OS = OWN 119 24 hours Time Ss 28 hour Time > io) wo bef ee e PAr ® rig t fA 4\ ae ,\ \ \ & a ' \ \ \ hours cq iS) laine Computed water velocity in centimeters per second at Preure 74. selected grid points for Hurricane Camille at 24 (a), (e) 8 (b), and 32 hours 5 120 *SLa}aW UT S}S9LD 19}eM y3ty [TIS paAtesqo oie aUTTASeOD OY BuTqUasaitdat aAind WOo13OqG dy YNoqe peynqt1ISTp sioqunu sy, ‘“aptz [eoTwoU -O1}SE 94} TOF p9.IeTIOD S[[ we) sUeITINY WOLF eILOpP ay JO Sea SAAIND [eYSBOD 942 FO 19qeM YyStYy pa ndwo) | ! | J ] "GZ omnsty (a) DH veremgsyn oe Ne) Zzorewn 12 in the marsh area south of Lake Borgne, and a high water mark of 4.8 meters along the eastern side of the Mississippi River levee. A part of the discrepancy between the observed and computed surges re- sults from not considering the flooding of low coastal areas and the open communication of Mississippi Sound with Lake Borgne which is, in turn, connected with Lake Pontchartrain. The tide gage at Shell Beach located on the southern end of Lake Borgne crested at 3.4 meters. Another test simulation was conducted with the hurricane track moved 18 kilometers to the east. The results showed greater seaward circulation in Chandeleur Sound reducing the setup along the levee by approximately 1.5 meters. The location of the peak surge was shifted to the east but the surge magnitude was not reduced. 3. Hurricane Gracie. Hurricane Gracie crossed the coast near Beaufort, South Carolina, at 1600 G.m.t. 29 September 1959. The radius to maximum winds was 25 kilometers. The atmospheric pressure in the eye was 950 millibars. The maximum sustained winds were approximately 50 meters per second. A Cartesian grid for simulating the surge caused by Gracie is shown in Figure 76. The reach and grid spacing of this system are comparable to the curvilinear grid shown in Figure 32. The thin dashlines are a part of the transform-generated coast and seaward boundary curve (180-meter depth contour). The surge computations are ‘performed with both grid systems using the same boundary conditions and forcing functions. The stairstep boundary representing the coast is the heavy solid line in Figure 76. Another difference between the models other than the coastline representation is that the Cartesian system's boundary, j = 1, extends farther seaward into deeper water than that of the curvilinear system. For numerical stability a time step of 150 seconds was used in the computations with the curvilinear grid. The time step in the rectilinear system was taken as 75 seconds. The surge simulation was performed for a 48-hour period with zero hour corresponding to 1200 G.m.t. on 28 September. The observed and computed water levels along the coast in the curvilinear (solid) and rectilinear (dashed) systems are shown in Figure 77. The inserts on Figure 76 show the simulated hydrograph locations. Figures 78 and 79 present the water surface topography in the curvilinear and rectilinear grids, respectively, at the same times and for the same contour increments. The depression of the water surface to the left of the storm track exhibited in both figures is Similar to that from the Carla simulation. At the Savannah River entrance, the observed water level corrected for the astronomical tide is depressed after 27 hours reflecting the offshore-directed wind. The effect on the coastal surges and water surface topographies 122 "7S OAINBIY UT UBATS PlLs Te9UTTTAIND 3yz FO squtod , JOUIOD 91e SOUTTYSep 94} UO SOQ “Splts (@) Leourp~raqns pue (Q) sesut[tqde1 9yQ ut ysvod oy. BuoTe SuoTiesOT yderBoLpAy paje[NUTsS MOYS SjLOSU,T ‘*INOJUOD YIdep LazoU-QgT pue SUT[ISEOD poLeLOUIS-WAIOFSUeT] OY} OLE SOUT[TYSep f4sv0d OY. FO UuOTReIUBSoIddI daqsateys ey ST SUTT PTTOS AAvOY OUL “*UOTIY[NUTS 98A4Ns WIOJS 9TIeINn DUPITLIINY LOF plaid uetsajziry) atoely ouRdTIInyY JO yorer] La (7g 91nBTF) ptis IeaUuT[IALNS JO qutod 1aut10)7 Iva, ode) wy O°7T ‘\ UuOJBUTUT IM y) a2uRt4ua q OSU] uozsoT sey) LOATY YeuUUBARS VY 4Loesuy “OVE ain3ty 123 BDoreowodcartht HOcp = (3) 0 | SAVANNAH RIVER 00909 South Carolina “curvilinear = iG i i=18, j=JM \ Rectilinear -2t— bs AIA, Je2i ne re rt ea tsa | Sonal de te erie 3 Lea (nee wr | eee ay eae) . ; Curvilinear 9 CHARLESTON 4=30, j=JM South Carolina Rectilinear O 4=24, j=17 WILMINGTON Curvilinear North Carolina i=50, j=JM Rectilincar 0 6 12 18 24 30 36 42 48 Time (hours) Figure 77. Observed (squares) and computed water levels for Hurricane Gracie in the curvilinear (solid) and rectilinear (dashline) grid systems. 124 Time = 24 hours (a) (b) wer ae Cale 4 ~ ee “a ~ WA ae 32 hours (c) Figure 78. Computed water surface topography in 0.5-meter con- tour increments for Hurricane Gracie at 24 (a), 28(b), and 32 hours (c) in the curvilinear grid. 125 Time = 240 (hrs.) bee Bae (b) paces cus manent et ee oo eine Saeed: | oocoonsno See : IMMe = O20 (hrs. ) Zar Ss (c) Figure 79. Computed water surface topography in 0.5-meter con- tour increments for Hurricane Gracie at 24 (a), 28 (b), and 32 hours (c) , in the rectilinear grid. 126 resulting from the stairstep portrayal of the coast boundary are shown through the comparison of Figures 78 and 79. The curvilinear model shows the maximum coastal water level at the time of landfall of approximately 3 meters while the Cartesian system predicts 4 meters. Furthermore, the peak surge of 4.5 meters in the Cartesian system occurred at the corner point (22, 19) over 2 hours after storm landfall. The water velocity, V , from the curvilinear system is presented in Figure 80. The figure also shows the development of a cyclonic current in the region from Cape Romain to Cape Fear resulting from the curved coastline and the extensive Frying Pan Shoals off Cape Fear. A similar current pattern is not observed in the rectilinear system. The computed high water envelopes from both grids are shown in Figure 81. The stairstep rectilinear coastline is projected upon the curvilinear coastline providing a common basis for the comparison. Both systems predict a maximum surge comparable to that of the near- est observed high water mark. The effect of the rectilinear coast- line on the surge envelope is seen in the figure by the relative water level maximums at the interior corner grid points. The pri- mary contrast of the coastal envelope of computed surges in the two different grid systems is the greater variability of that from the rectangular grid. This leads to spurious results as can be seen particular for Savannah River. 127 (b) = 1 e ; en 166 cm-s7 > > bd : 4 ace a ~ = 0 ; 7/7. Time = 28 hours ~ Time = 32 hours ™ ee - Figure 80. Computed water velocity in centimeters per second at : selected grid points for Hurricane Gracie at 24 (a), 28 (b), and 32 hours (c) in the curvilinear grid. See Figure 5 for identification of capes referred to in the text. 128 “s1o}aW UT SyLeW 107eM YBSTY [[TIS PeALEsqo oie DUTT ISBOD 9Y} BuTyUesetder aAINd WOI}0G BY JNoge poeinqt4ysTp sroqunu oul "Spti3 (eutrTysep doi) 41eaut[Tt oer pue (pttos doz) TeauT[ TAINS oy. UT opt? [BoTWOUCT{SE dy LOF pajdeLLOD eTOPIN sUeITIINH WOLF seBins Te SeOd sYyi FO 19,eM ysty pe nduoy ufewoy ede ty oe" uojseTszey) uo BuywT TM @TTFAuosyoer ‘y yeuueaes c e z 9°? "Tg ean3Tty (W) UOTIeASTS 132eM 129 VI. SUMMARY AND CONCLUSIONS The development of the conformal mapping problem was completed for transforming the interior region bounded by two arbitrarily shaped curves and two parallel lateral boundaries into a rectangle in the image plane. Specifically, the coastline and seaward boundary curve taken as the 180-meter depth contour are transformed into the image plane as constant values of n . The bicurve fitting procedure was successively tested on five regions of the Continental Shelf in the Gulf of Mexico and east coast of the United States. The distinct advantages of the orthogonal curvilinear coordinate system over rectilinear representations of the coast are the accuracy and ease of specification of the boundary conditions. For long wave calcula- tions, the numerical simulation of a free long wavelength gravity wave in a quarter annulus demonstrated the superiority of modeling the boundaries in curvilinear coordinates. A two-dimensional time-dependent numerical storm surge model employing orthogonal curvilinear coordinates is presented, Since the transformation is conformal, the associated modification of the equations is minimized. The final coordinates allow for the greatest resolution near the coast over the area of principal storm surge development and modification. A second transformation is performed independently on each of the curvilinear coordinates to provide the desired spatial resolution in the shallower region near hurricane landfall. Results of the numerical simulation in transformed coordinates of the storm surges along the gulf coast induced by Hurricanes Carla, Camille, and Gracie which crossed the east coast are presented. The surface wind fields are analytically represented. Additionally, a systematic procedure is employed to deform the symmetric wind fields of Carla and Camille in the nearshore region, The deformed Carla wind representation produces better agreement between the computed surges and that observed than a similar simulation with a symmetric wind. Although the pattern of windspeeds produced by the deformation procedure is in good agreement with those provided by NWS, the problem of specifying the proper orientation of the wind-stress vector in the nearshore region requires further study. The surge simulations indicate that the model produces results in good agreement with the observed peak surges and hydrographs, especially in the area to the right of landfall. Moreover, the results from the simulation of the Hurricane Gracie surge indicate superior rendition in the curvilinear grid compared with that of the Cartesian grid. Certainly one contributor to any discrepancy between computed and observed water levels may result from the portrayal of the coast as a wall. Proper inclusion of attached bays, lakes, and flooding of low-lying areas should be pursued, 130 LITERATURE CITED ALVAREZ, J-.A., "Numerical Prediction of Storm Surges in the Rio de la Plata Area,"' Ph.D. Dissertation, University of Buenos Aires, Buenos Aires, Argentina, 1973. DONN, W.L., "An Empirical Basis for Forecasting Storm Tides," Bulletin of the American Meteorologtcal Scotety, Vol. 39, 1958, pp. 640-647. FORRISTALL, G.Z., ''Three-dimensional Structure of Storm-Generated Currents," Journal of Geophystcal Research, Vol. 79, 1974, pp. 2721-2729. GRAHAM, H.E., and NUNN, D.E., "Meteorological Considerations Pertinent to Standard Project Hurricane, Atlantic and Gulf Coasts of the United States," Report 33, National Hurricane Research Project, U.S. Depart- ment of Commerce, Washington, D.C., 1959. JELESNIANSKI, C.P., ''An Numerical Calculation of Storm Tides Induced by a Tropical Storm Impinging on a Continental Shelf," Monthly Weather Review, Vol. 93, 1965, pp. 343-358. JELESNIANSKI, C.P., ''Numerical Computations of Storm Surges Without Bottom Stress,'' Monthly Weather Review, Vol. 94, 1966, pp. 379-394. JELESNIANSKI, C.P., ''Numerical Computations of Storm Surges with Bottom Stress," Monthly Weather Review, Vol. 95, 1967, pp. 740-756. JELESNIANSKI, C.P., "SPLASH I: Landfall Storms," TDL-46, National Oceanic and Atmospheric Administration, National Weather Service, Silver Spring, Milo, IZ. LAMB, H., Hydrodynamics, Dover, New York, 1932. MIYAZAKI, M., "A Numerical Computation of the Storm Surge of Hurricane Carla 1961 in the Gulf of Mexico,'' Technical Report No. 10, University of Chicago, Chicago, I11., 1963. PLATZMAN, G.W., "The Dynamical Prediction of Wind Tides on Lake Erie," Meteorological Monographs, Vol. 4, 1963. PLATZMAN, G.W., "Two-dimensional Free Oscillations in Natural Basins," Journal of Physteal Oceanography, Vol. 2, 1972, pp. 117-138. RAO, D.B., ''Free Gravitational Oscillations in Rotating Rectangular Basins,"' Journal of Fluid Mechanics, Vol. 25, 1966, pp. 523-555. REID, R.O., "Comment on 'Three-dimensional Structure of Storm- generated Currents' by G.Z. Forristal," Journal of Geophysical Research, Vol. 80, 1975, pp. 1184-1185. 131 REID, R.O., and VASTANO, A.C., “Orthogonal Coordinates for Analysis of Long Gravity Waves Near Islands," Proceedings, American Soctety of Civil Engtneers, Santa Barbara Spectalty Conference in Coastal Engtneer- ing, 1966, pp. 1-20. REID, R.O., and BODINE, B.R., 'Numerical Model for Storm Surges in Galveston Bay,'' Proceedings, Amertecan Soctety of Civil Engineers, Journal of Waterway and Harbours Diviston, Paper 5805, Vol. 94, No. WW1, 1968, pp. 33-57. SCHWEIKERT, D.G., "An Interpolation Curve Using a Spline under Tension," Journal of Mathemattes and Phystes, Vol. 45, 1966, pp. 312-317. U.S. ARMY ENGINEER DISTRICT, GALVESTON, "Report on Hurricane Carla 9-12 September 1961,'' Galveston, Tex., 1962. U.S. ARMY ENGINEER DISTRICT, NEW ORLEANS, ''Report on Hurricane Camille 14-22 August 1969,' New Orleans, La., 1970. WELANDER, P., "Numerical Prediction of Storm Surges," Advances tn Geophystces, Vol. 8, Academic Press, New York, 1961. WHITAKER, R.E., REID, R.O., and VASTANO, A.C., 'Drag Coefficient at Hurricane Wind Speeds as Deduced from Numerical Simulation of Dynamic Water Level Changes in Lake Okeechobee,'' Ref. 73-13-T, Department of Oceanography, Texas AGM University, College Station, Tex., 1973. 132 GULF COAST REGION FROM =) OoOMDNANFUWD& Note: APPENDIX A CONFORMAL MAPPING COEFFICIENTS FOR THE ATCHAFALAYA BAY TO APALACHEE BAY Oe19S1S531E O02 O0«1129102E 03 = 31765 B n -Oel861917E 02 —-Q2e28046S2E O02 Oec6403131E O11 —O0e2530149E O1 Oe1S31121E£ Ol 02e1498879E-01 —-0.4S531168E OO -0e1035305E Ol -00e2334446E OO 0.259073 05E-01 O0e4296697E 00 -O0«0 2997332E-01 06.4310032E-01 -0e6075356E-02 O0e1117075SE O00 0eS209380E-01 Oe 7516444E-02 —02 1997321EF-O1 -025793062E-0 3 Oc 13441 B3E-01 Oe1SS8170E-01 0. 33577S9E-02 C n -022013675E O1 Oc6239S47E Ol —-02e5958148E O11 OeSLZO0100E OL Oe22731S0E Ci 0e3512568E O1 -O00el1606539E O1 O00«l131842E O21 -0e6252772E-01 0e6906291E O00 —-00e1417943E& OO Oel326714E-01 —-Ce2609431E 00 001616244F-01 —-0064899038E-01 Oe 7586825E-01 —-00e2902S73E-01 023440024F-O01 —0e3394415E-01 0e2085055E-01 —-0053093€2E-02 O0e1572620E-01 all quantities are in x,y units. 133 B nh O20. 453654 3E- 02 —-026767012E-02 0. 5650939E-03 -0e3992960E-02 Oo 2905252E-02 —-00« 59241 94E- 03 0 e 3964610E-023 —O0o @709535E-03 024148923E-03 Oo 4497680E-03 Oe SSS896CE-02 -004054087E-02 —0e 2529798E-03 —-0e2813288E-03 023256833E-03 0. 1094875E-02 021343705E-02 —O60 ETISYSSE-04 —-0 6 2443834E-05 —O0-.1611809F£-04 Oe 4455329E—-04 —-Oc0«c 212978 3E-04 —0. 1393920E-05 —Oe 3849607E-04 021427432E-04 —O50o 2892541E-CS 021366723F-04 —00« $772998E-05 —O0« 2762842E- 0€ —-O« 7610149E-05 Oo 8511489E—-06 -0 e821928EE-06 Oo 2588446E-05 —Oe 164524 2F- OF 0-.9023091E-O6 —-0Oc 16686365£-05 0 ©.886957SE- 06 —-02e4799319E-06 Oe €261487E- OE —0 e5237882E-06 —00e 4759656E- 07 -—0 e 259881 S5E-06 Oo 3273941E-06 Oo 2930804E-07 021020638E-06 —O52 24221 O0E-06 134 C n —- 064705 735£- 02 004449189E-02 00e1541801E-02 02«6915852E- 02 022670427E-02 Oc 3155482E—-02 -021662592E-02 0e5550052E-03 —-0012243225E- 02 O02e1261 866E-02 — 02 265304 0E- 03 024554733E-03 —-00e5726672E-03 0e5050778E-04 —-0.1838352E-03 0c 1 940ES36E-—C3 —-02e1790418E-03 —0e2377543E-04 —-02e1661804E-03 00e9141980E-04 0058422 CSE-05 0e«6875678E- 04 —-02e3022245E—-04 0e517795S8E-05 —-00el2729S2E-04 00.1886662E-04 —-021514819E-05 00e1114408F-04 —- 02741 O07 C8E-05 0 e6834E80E-05 —02¢ 368 2432E-05 0e501988iE-05 -—002587597E-05 Oo 794i 880E— 06 —-062933107E-05 —-0.7180398E-07 —-009114828E—C06 OelO067656E-05 —00e4895755E- C6 0e.2304607E-06 —-068925043E—-06 0e1912503E-06 —-022104335E-06 06 373 025S5E-C6 —-001417030E-06 00659168116-07 100 102 102 103 104 105 106 107 108 109 110 111 i112 113 114 B n —0eS880160E-08 —0e4626684E-07 021153690E-0€ -02e1692220E-07 Oo 6679772E-09 ~0« 4858599E- 07 022245889E-07 Oe 4829538E- 08 022210185E-07 —0. 1637371E-07 065777583E-90S -0 0 6394433E-08 O2. 1182490E- 07? 02e6501393E-09 Oc. 2608579E-08 —0¢ 43229S56E- 08 021574734E-08 —-0e4726193E-190 022415840E-08 -O5. 1000S88E-08 Oe €742753E- OS -066728380E-0S 0e61012167E-08 —0 « 2016696E-0S O02 3236860E-09 —O« 3145593E= 0S 0 6 3353653E-09 - 0.2 3262997E-10 021631445E-09 -0.137S170E-09 0. 107299SE- 0S —021384905F-11 0. 1086742E-09 —-02e5080765E-10 021376387E-10 —-0.2277743E-10 005459146E-10 0. 537051 7E-11 0e1355941E-10 —06¢2422210E-10 Oe 75567 OSE-11 Oe 2368470E-12 02 1447254E-10 —0e5194694E- 11 -004364368E-13 —00.4752083E-11 135 C n —-061276744E-C6 001424694E—-06 00171 9058E-07 0e88805329E-07 -02e6668830E-07 02e1698348E- C7 —022062551E-07 Oe 5085036E-07 —-0628952E1E-08 0e1013160E-07 —-0¢e1960430E-07 0 08530 744E-08 -—0.62038007E-08 009465 726E- 08 -0.26092815E-08 02e198151CE-08 -0 e3446277E-08 004146546E-08 —-029092067E-909 00e1491153E-08 —00e14086€12E-08 02.1087402F-08 -001879313E-09 0091631¢S9E-09 —02e3496718E-09 004997 C83SE-09 —-021249788E-09 06440S5611E-09 —-0e1501641E-09 001637698E-09 —02¢9262S71E-10 00e1895707E-09 —-001663476E-10 02689067 0E-10 —-O00«7510520E-10 0e4346S501E—-10 -O0e7S598091E-11 004664877E-10 —-0¢2425636E-10 0e4706750E-11 —-0061434774E-10 00e20109SSE-10 OelO0SO8E6E-i1l Oe 71SO0311E-11 —0e9073677E-11 0e3138752E—-11 B n 0e4553522E-11 Oe 7602034E-12 00194454 7E-11 —-0e2436083E-11 Ool917147E-12 —02e4928470E-12 Oe 1484346E-11 —-02.1348819E-12 0.3641300E-12 —0O0« 5487838E-12 0e2406558E- 12 -027911185E-13 O00. 32179581E-12 -0 2861194 3E-13 Oc 72384 70E-13 —-021361322E-12 0e6817892E-13 —Oe 1436969E-12: 0 e 8566666 E-123 —00. 3499625E-13 —-O0«e7835616E-15 —-024394792E-13 Oo 234941SGE-12 0e4395810E-14 Oe 1976828E-13 —0216876051E-13 -021897316E-14 —-O2.€781451E-14 Oe ll74166E-13 Oo E883079E-15 Oc 2116535E-14 -O0eSS74005E-14 Oo. 1321S51E-14 0eS5106463E-15 Oe 3148336E-14 —028159113E-16€ 02450 7668E-16 —-0.1C82406E-14 0e9193219F-15 Oc 21521 37E-16 Oe 544053SE-15 —024207780E-15 021240553E-15 —-Oel1809934E-15 O50 2134241E-15 —-O42¢€41496EF- 1€ C nh —026452875E- 12 0e6237037E-11 —-00e1114412E-11 0e1234543E-11 —-00e1740079E-11 0e1649892E-11 0eS5327387E5E-13 Oel1120676E-11 -0 e5990049F-12 0e2460370E-12 —-0-e3448143E-12 004439149E-12 —00e5831951E-13 0 e2390251E-12 —-0061814728E-12 0e5573903E-13 —009745736E-13 0011 52394E-12 -001163015E-13 0eS682837E-13 —-0e5510710E-13 0e61780100E—-1 3 —-Q0e12725S9E-13 00e4233778E-13 —-0e5816127E-14 Oe 7731 244E—-14 —-021672038E-13 0e9221180E-14 00e2324292E-14 Oel 0B88460E-13 —-0e4276SE8E- 14 0e4950588E-15 —-0e3220331E-14 004241152E-14 06e2514837E-15 0e1642eE51E-14 —-02e1803004E-14 02e5420030E-15 —062248083E-15 OelO72704E-14 -0e201 9489E-15 0e2787050E-15 —00e3249225E-15 O0e2896084E-15 —0e6586600E-16 062533124E-15 136 B nh 0064553522E-11 Oo 7602034E-12 021944547E-11 —0.2436083E-11 Ocil917147E-12 —0e4928470E-12 Oo 1484346E-11 -0.61348319E-12 0.3641300F-12 —0. £487838E-12 062406558E- 1z -O5«679i11185E-13 0. 32179581E-12 -02861194 3E-13 Oo 723 84 70E-13 —-061361322E-12 066817892E-13 -00« 143696S9E-12 0e85666E6E-13 —0.6 2499625E-13 —00783S5S616E-15 —0e4394792E-13 Oo 234941GE-12 0e4395810E-14 Oc 1976828E-13 —021687651E-13 —02ei897316E-14 —-O-€7814Si1E-14 Oell74166E-13 Oo E88307SE-15 Oo 2116535E-14 -0e5574005E-14 Oc 132i155i1E-14 02e5106463E-15 Oc 31483365E-14 -028159113E-16€ 0e4507668E-16 —0.1C82406E-14 0e9193219E-15 Oc 21521 37E-16 02 544053SE-15 —-0024207780E-15 021240553E-15 —021809934E-15 Oe 2134241E-15 —O0c€41496EE-1€ 137 C n —-0e6452875E-1le 0e6237037E-11 —-001114412E-11 001234543E-11 —-001740079E-11 061649892E-11 0e5327387EF-13 Ooll20676E-11 -0 e5990049F-12 0e2460370E-12 —-0e3448143E-12 004439149E-12 —- 0205831 951E-13 0e2390251E-12 —-0e1814728E-12 065573903E-13 —009745736E-13 00el1152394E-12 —-0e1163015E-13 005682837E-13 —-0e5510710E-13 Gel780100E-13 —-0012725S9E-13 00e4233778E-13 —0e3816127E-14 Oe7731244E-14 —-0e16720358E-13 0e9221180E-14 0e2324292E-14 Oeil 088460E-13 —- 00e427696E8E—-14 0e4950588E-15 —0e3220331E-14 00«42411S52E-14 O0e2514837E-15 00e16428S51E-14 -0.2.1803004E-14 065420C30E-15 —-062248083E-15 Oel O72704E-14 —-0.201 9489E-15 0e2787050E-15 —003249225E-15 e2896084E-15 —0e6586600E-16 0e2533124E-15 APPENDIX B CONFORMAL MAPPING COEFFICIENTS FOR THE EAST COAST REGION FROM CAPE KENNEDY TO PAMLICO SOUND B = 99917248E 01 BS = O6«7052965E 02 4 = 3250¢90 n B Cc n n 1 —06e4282722E O02 -0.6295406E O02 2 —O0e2821791E O02 —0e623647S5E 02 3 Oe 123890S5SE 02 001391628E 02 4 Oe4424074E O2 —-O067596874E 00 S —0-4208601E£ Ol —0e2879633& O00 6 —00«¢2069723E OO —0.7998557E OO 7 0e8464327E OO OQe2104794E Oi 8 02. 2291 799E OO 0e1254503E O1 9 —0eS578854E 00 —-Ool6788B4E OF 10 —0e2837964E 00 026156636S5E OO iil Oe 448e817E OO Ce 825G192E 00 12 —-0 66342903E-02 -0e1633873E 00 13 —QO. 2921963E OO —-0.6S65279E-01 14 06538631 7E-O1 O2ei49S777E O00 15 —0e3311124E-01 046278S084E-01 16 -—0.1305802E 00 —O«et012S93EE OO 17 0 eS479731E-O1 00541 2326E-O1 18 Oe 1251435E 00 Osi 91 8038E O00 19 —-0.2086312E O00 —-O06155743i1E OO 20 Ocll79i40E O00 O.1029936E OO 21 —Ocil4S189E 00 —-00693S3477E-O01 22 0<¢2414525E-073 Oe 3330456FE- C1 Note: All quantities are in x,y units. 138 s3 B n 0.0 8573616E-01 —O50« 721242SE-01 Oe2609131E-O1 -Ooll1S37S1E-O1 -—0 o4611936E-01 Oe 4978 760E-O01 —O5o 2400358E-01 Oel476TSSE-04 Oel2si7SLE-O}l -0.2015296E-01 —O0 3106S54E-02 Oe 1246096E- O01 —-028497261iE-02 Oo 199S0S7E- 02 027288672E-03 —00¢ 4637495E-02 —00124251iF-O2 Oe 3494871E-02 —050. 2765389E-02 021414S528E-02 —-02£1092573E-02 —00« S259S57SE- 02 —O0e5194661E-O2 O. 101 854 BE-02 —-Oe1744783E-O02 -027579974E-03 —0O0« 7027SSEE—- 04 0e4199664E-03 —O0. 7979686E-03 023107551E- 03 -0.8603580E-04 Oo 2109174E-O8S —-007456832E-03 Oc 446291 4E-03 02e1S59i1920E-023 —-0.4656487E-03 —-0.1767847E—-04 02¢3108399E-02 -—QO6 2931331E-03 02eS228238E- 04 —-O021720886E-04 — 02 20S6900E-05S (0). 6 Wika eT Hie Os 02. 4905240E-04 O02. 1S7TE664E- 04 0.23167627E-04 139 Cc n 00e9928739E-01 ~00e8458531E-O1 004790343E-01 Oo 7979626E—02 —06.541 89CBE-O1 OeSSSE3SIOE-Ol —-Ocelti3sss5ie-o1 —06«3498875E-02 Oc2114é711F5-01 —-Oe1866217E-O01 —-02e5141602E—-02 Ool?i24i4E-O1 —00713S5S786E-02 0e3350719E-O02 Ov 1991S504E—-02 —004495025E-02 Oc22432E5E- C2 0e2640530E-02 -062747S52E-02 Oc 361 95S28E-02 —029984830E-03 — Ce 987S367E-03 0«e2608439E-03 06.1251 844E-02 02404 8S0B8E-03 —0 e858 336SE—03 06326084 9E—-03 Oe867TSSS6E-02 —O0e94698S1E-03 OeS6455SSE- 03 O0e2324108E-03 02161 OS@3E-03 —0¢6130417F-03 0e5053631E-03 Oo278SG015E—-03 —0 04364 226E-03 -0e3461981E-05 0e394108SE—63 —002265753E-03 Oe 637 72 C4E— 04 0 e397S5S330E—-04 0e38307732E—-04 —-0e106580C6E-C3 O0e4872639 E-04 065831S511E-—04 0640531 38E-04 100 101 102 103 104 105 106 107 108 ic9 110 1i1 112 113 114 B n —0e 656451 6E-04 —0e1748351E-04 —O«e 3665496F-04 Oo 10043 76E— 03 —0 e 6544378E-04 —O02 S$S63962E-05 0e2498129E-04 —0e4729294E-05 —Qe2e £S413S0E-04 Oe 330S452E-04 O. 2570346E- 04 —02e1S589919E-04 —-0466349276E-04 Oo £02411 7E-04 068936679E-05 —0.2S8S5104E- 04 0e1793947E-OS 0. 28843 07E—-04 —-0e228911556-04 —-0e16792i0E-04 021770000E~-04 0.2.1685304E-04 —O20e 267449SE—-04 0247G2071E-O05S 06 f7401914E-05 —O2e5577534E-05 —022972263E-05 Oo 7447946E-05 ~0.6316889SE-O5 -00135627SE-05 —-O5« 2609007E-OF 0e367S5S452E-05 —-02¢1076716E-O€ 0.£5550138E-06 —00e 419292 8E- 06 —00e1196023E-O05 —-0.834S894E-06 Oc 311S5964E-O5S —O0606 21675S5S6E-05 —0.e 1715321E-06 Oe SITO421E- OE 021282769E-06 —Oe 213C69SE-O058 021366156E-05 Oc 11289S9E—05 —O02e 1028800E-O5 140 C n —006221421E-04 ~0.69761238E-05 —00147282S5E-04 O.o9884777E— 04 —0 eS9B7T761E-04 00511 286SE—06 063470853E- 04 —009627895E-05 ~0.0466117SE-04 0 «3855478E-04 023692045E-04 —021510879E—04 —005784341E-04 0e508872 0E—04 001105513E-04 — 02 294S38SE- C4 0 e5432613E-05 002892528E—-04 —002226605E-C4 —00¢1523216E£-04 001914123E-04 001635740E-04 —0.2516 768E-04 0eSO086707E-05 0e8127184E-05 = 065464967E-05 —0«2235201E-05 007659644 E-05 — 0¢2746120E- 05 —061211409£-05 — 02 2086289E-C5 023609181 E-05 — 0« 6883366E- 06 0e6827090E-06 —0.1 706489E-06 —001206698E-05 —005846961E-06 0. 3220246E-05 —0.2048756E- 05 —001 886759E-06 0o1148447E-05 0 013475216—06 — 0. 2057237E-05 00140652S5E-CS 00123400SE-05 — 0.101 8ES1E- 05 B n -001313448E-05 Oe. 1556963E-O§5 -0 6 8436349E-08 —O5c 752393 8E—06 ~001622793E- 0€ 021139069E-05 —0« 652760 2E-06 —0 6 7496735E-06 Oe 7258587E-06 Oe S221784E- CE -0 2.961 2895E-06 Oe 200901 OF-06 0 6 3099258E-06 —06. 1067625E-906 —0. 2802620E- 06 00 2269661E-06 Oo S356683E- 07 -0 61395750E-06 —O5. 1529258E—06 Ce Z215562SE- 06 -00e 3408843E-07 —002487944E-07 -007241448E- 07 0. 6702049E-07 Oe 1388121E-07 —-0e4186880E-07 —O. 3S01641E-07 021001300E-06 -0¢4257138E-07 —0.4805924E-07 0e71350S7E-08 Oo 74476 08E- 07 -065374578E-07 -0e2760153E-07 Oe 3739260E-07 0 .61614653E-07 —0. 5449028E-07 0¢2701048E- 07 001567296E-07 —0201215744E-07 ~0 6 2149563E-07 0. 1988487E-07 Oe 3930207E-08 -0.7725944E-08 — 0. S216542E-08 0.61S577168E-07 14] C n -0 061246 324E-05 0.6 1568S67E—05 0 0667168S57E-07 —007546471 E-06 —~001113204E-C6 0 0115C6875E-05 ~0. 616 8775E—-06 —0e75°270S5E- 06 067S5807S50E-06 005233 002E- 06 —0 0e9294666E-06 00 2025572E-06 0¢3316641E-C6 —001017822E-06 — 00 2645621E—06 0 0230 74S0E-06 001065546E-06 —001389416E-C& —001394174E-06 0021681 S3E— 06 —0 e2498253E-07 —002277810E-07 —0.6482588E- 07 006857 S88E-07 0c 2008024E—07 -0e46101790E-07 —003311435E£-07 0010054S8E- 06 —0-3833354E-07 — 00471 6878E-07 021083663E-07 007493 753E-07 —0e5108661E-07 -022660881E-07 Oe 39585S93E-07 061627980E— 07 -0252043995-07 0027223 71E-07 Oel 71 SO007E—07 —0.1168 664E-—OF —00201 88EEE- 07 'Qc2019971E-07 0. 4938¢83E—08 —0067394348E- 08 —0«68195819E-08 001577923E-07 EAST COAST REGION FROM PAMLICO SOUND TO PENOBSCOT BAY Sie OMNAHWPUD Note: APPENDIX C CONFORMAL MAPPING COEFFICIENTS FOR THE 3s w it B n O«e 8453177E O1 —-O56e2573375E OZ O022456377E O1 —-0¢4028893E O1 —02e1294892E O1 Oe 7517704E OO —-05.26833593E OC 0e8799276E O00 —0.e 568954 8E 00 0264709852E 00 -O50 350S102E 00 —-00eSS0869S6E-01 Oell06679E OO —O0« 3838463E-01 —O0« 2045463E-01 -022993307E-01 0655191 73E-01 -—0 e3588944E-01 OO. 2528971E-O1l —O00« 2555390E- 01 0e2158327£-01 —06« 2883616E-O1l 142 Oe 21852463E 02 Ce1l045535E 03 37420 C n —Oe7390710E 02 Ooll466E56E 02 —-0e131672S9SE 02 Oe8089012E O1 —-0eS803839F O01 00e«2271722E O1 —-0e6068285E 00 0e8849853E 00 —-00e1988623E O00 Q023153S9S88E 00 —-0021405933E 00 —-020«16112S7E OO Oe20470SSE CO —-0e7570612E-01 0e3443124E-02 —0 e4593338E-01 Co 9201282E-01 —-00e688 24S535-01 00e5296847E-01 —-00e439624EE-O01 G0e3201135E-01 ~—0e3150356E-01 All quantities are in x,y units. B n 0c1704723E-01 —Oe 821 84 825-02 0e5179290E~ 02 -02301S571SE-02 Oc 793892S5E-02 -0 e 7356405E-02 O. 21203 O3E-02 —062383103E- 02 02.1206459E-02 —00 4140623E~ 03 0e3749S5S28E-02 O5oc 4151922E-03 —O«. 624251 BE- 03 Oe45S18004E-0 23 -02e1179293E-03 —-0e110SO91E-02 —O50 261 8267E-03 Oe 33287 O8E- O02 —-0 e 219220SE-03 Oe 1223145E—- 03 —-028201750E- 04 Oc 1276652E—-0 3 —-0. 740803 7E-904 0.402277T1E-04 —05. €5351S50E-04 02€436008E-O5 —0e7533055E-05 0. 208 7624E- 04 —-02 8%48817E-OS —O0 273 7144E—-05 02 109S4GEE-04 -0.65011S9E-05 —O50« £406307E-O5 —-0 2e2243007E-0S Oe 4238569E-05 —-0e1688697E-O05 0263522237E-05 —00. 2636244E-905 0e1468087E-0S —-Oe 1664401E—-905 00 16220326-05 —-0e1230003E-05 —0.222695T1E-—06 0e&6773S8E-07 0. T2Z02404E-06 —O0« 679648 3E- O€ 143 C n 0.61959118E-O1 —00e7724181 E-02 Oe 3032119E-02 0 01962418E-03 Co 488653 0E-62 —0e41S50771E-02 021651054 E-03 —00e11456i6E-02 0 e4586522E-03 Ool 666064E-—03 00132 8797E-C3 0e599561S5E-03 —0e6716528E- 03 0045008S59E- 03 -0°e8762798E-04 —0e1055562E-03 —002268526E-03 Oe 295065 BE— 03 -0.16891S6E-03 0ce8370787E—04 —02382705SE—- C4 006963981 3E—-04 —0c4502C0S0E-04 0e2078711E—-04 —0e5105330E-04 0.1149840E-05 —00420 369S5E-05 0. 3067S82E—04 —0208395030E-05 —062279463 E-05 Oo 9629463E-05 —004204831E-05 — Ceo 72332S0E-C5 —0e388391SE- 06 002996593E-05 — 06521 7331E-06 0<«2865336E-05 —022091638E-C5S O0.1271E€15E-O05 —061547839E-05 0c 166901 CE-05 —-0.1246244E-05 —001S536769E-06 0e4681078E-—07 0«e7948224E-06 — 0. 727 0842E- C6 1co 101 102 103 104 105 106 107 108 109 110 112 112 113 114 B n 0064362259E-06 —O5«. 1076482E-06 0e24S59019E-07 —O6 8354027E-07 —O«. €96S583E-07 GeS715553E-—-O0S —-02.10971125-O07 Oe 7904828E-07 O062604376E-07 —-00 481 7723E-07 -Oe.lO023679E-07 O52. 35211 895-07 —-O0eS5271849E-07 O22211178E-O7 —0O. 7624720E-08 0e1252196E-07 —0e 4G695075E-—08 00 €09S036E- 08 —0e6119723E-08 Oe €175922E- 0S -Oe 3125592E-08 Oo. 560471 8E-08 —0e6158700EFE-08 02e1393683E-08 Oe 2143876E- 0€ -0 e3371285E-09 —0. 201i 3834E-0S Oxo i21930R8E- OS -02e4478S535E-09 —06 26221 87E-O09 0e1349961F£-09 —0.8012752E-10 0. 1434S38E- 0S —-0e 7944631£-10 0e469S801E-09 —0 9 3936562E-09 —0.66882636E-10 O0« Z980914ZE-16 0.6826086E-10 —Oe«1151273E-0S9 OelO01L11LO2SE-0S -O56. 3327474E-10 Oc 243134S5E-1¢€ —0e¢8947284E-11 —05. S061 761E-12 —0-.1896980E-10 144 C n 0 e4967463E-06 —0014C8421E-06 0066843044E—-C7 —-0c¢109C0609E—-06 —024082S53E-07 —-0cei3E0970E-07 O0e3200026E-08 Oe /S80621E-07 0230SS97T9E-07 —CoS 74 C57SE-07 —-0e954938SE—-08 Oe3772427E—-O07 —-0654265S9E-07 0e2568847E-O07 —0269943246E-08 001546062E- 07 —-00e«6723383E-08 00e6111858E-908 —O06e725S402E-08 001 844SC0E—-08 —-023619169E-08 02605S5163E—-08 —-0e6123S51E—-08 0¢61397110E—-08 0e2388478E—-08 —-Oce48S838 7E-C9 —-0e35220485-10 - 0.05434 760E-10 —0 62261 494E-909 —-0.2368781SE-—09 0e2747322E-69 —001470834E-09 062115 7SEE—0G9 —-O0e977741S9SE—-10 0e4927398E-09 —-0.23841 8ESE-09 —00e7195172E-10 Oe47375235E-10 0e5399223E-10 —-0069696301E-10 0e-9G092571E-10 —0c«196S5071E-10 Ooi71162SE—-10 —068525016E-12 —023881352E-11 —061490SSSE-10 B n —O2 7091S71E-11 Oe 1821453E-10 —-0¢ 3987485E-11 —02 $260623E~-11 Q5e7307409E-11 Oc 94B87284E-11 —Oe 1281272E-1C O02eS417571E-11 —05. 4854271E-11 O021272S995E-11 —0.8162463E-12 0. 2438756E-11 —0201364847E-11 Oo. 7146806E-12 —-0e4318938E-12 GO.1000S46E-11 —0.« 2052629F-11 0e615S0933E-12 Oo 4740078E-12 —O5« 39357S54E-1ée2 0061734885E-12 Oo 122661 3E-12 —-021624964E-12 0. 237601 8E-13 02¢1584382E-12 -007575429E-13 —0.160i1686E-13 -0262932933E-13 Oo 14569 78E-12 —-O0211467427E-12 062950459E-13 Oe 2668 76i1E-13 —0.2074897E-12 —Oe 2806369E-13 002E75362E-12 00247861 3E-13 0. 1650332E-13 —-021226408E-14 Oo TO71394E-14 —0O.S688173E-14 —0¢ 3497994E-15 Oe 4112430E-14 —-0e3376250FE-14 —0.3620927E-14 0651592S5S7E-14 -0264390035-15S 145 C n —O0e7386E324E-11 0061942668E-10 -O0e311S7ESE-11 =—Oe 95221 16E—1 1 OeS7TO1L047E-11 O0.874S027E-11 —0201152168E-10 0047035i9E-11 —00e587 0245E-11 O0e5C34556E-12 —002376758E-12 Ce2222751E-11 —-02e1078S58E-11 Oe6671415E-12 “On S27 SMyyse = Ue2 0e1055302E-11 —-0e2062S67E~-11 Oe?%127636E-12 0<428839E9E-12 Wo SOS GOSS ie 06i294625E—-12 OcolBSSBOLE-12 —-00l192G000E-i12 0e6448195E-13 063484 010E-14 —-0.5707148E-13 —-Oelé698161E-13 —002245293E~-13 Oeil 4S98009E-12 —0e1142878E-12 0e3582161E-13 0e2435500E-13 —-0614775VY2E-13 —-00e30550S6E-13 0.31 08SE5E—-13 —022643752E-13 Ocl9S33177E-13 —~Oe2t20201E-14 Oe 864977S5SE-14 —-Co 9941 355E-14 Oe3751107E-15 Oe4258614E-14 —-Oe3143469E-14 —063506923E-14 0.5141 81SE-14 —-Oe3103656E-15 B n -O4. 1SS0099E-14 021889396E-14 —-0e1462261E-14 Oe SSSIOISE-16 -O021863618E-15 Oo 4924783E-15 —Ce€9751L17E-15 022073116E-15 OG. 2038455F-15 022677031E-15 -026432034E-15 00e4125446F- 15 -021303287E-15 —-021122934E-15 0-e4706S5S50E-16 Oo 55621126-16 —-Oo 728751 7E-1€ 0e6455897E-16 —-0e1810812E-17 —-0e25194S57E-16 —O5« 2583353F-16 0eS397427E-17 0e284996S5E-16 — Oc 435331 7E-16 00e2342945E-16 Oe 6163340E-17 -0.21038789E-16 —02254589S5E-17 Oe 2582216E-17 —-0212928745-16 146 C n —O0el1644377E-14 0621 769S3E- 14 -0 e15E2678E-14 Oc 3085169E~-15 SWS Zoos SSVS 1S) 0e5232375E-15 —0067213222E-15 Oe2703232E-15 Oe2139917E-15 Oe2874004E-15 -O02e«6162772E-15 004083402E-15 —-069925271E-16 —-O0«0l216983E-15 0e7034376E-16 024932992E-16 —0e5844C15E-16 0e6172391E-16 Ooe71 76773E-17 —002643164E-16 —0e 2060502E-16 06. 9356655E-17 0 03153878E- 16 —-00e4277073E-16 0.2463311E-16 O0«7743982E-17 —-021057159E-16 —-0e5732719F-18 0e8081919E-17 —-0e11264SEE-15 GULF COAST REGION FROM LAGUNA MADRE TO MARSH ISLAND =) ODNANE UA = Note: APPENDIX D CONFORMAL MAPPING COEFFICIENTS FOR THE 0262110953E 02 B = Ce9SO7TR64LE O02 = 360.0 B nh —004349387E 02 —O50o Z2166200E OZ 0.1242251E 02 —-0.5063650E 01 0e1671185E O01 —-O« 3597688E O1 0.1419286E O1 —Ocol1179494E 01 0.4119087E OO —O003775935E OC 0e7971597E-0O1 —0. 1080030E 00 —0e1290084E-01 —00 SS63697E-01 —O«¢ 1379531E-01 —0.3508482E-01 Oe 6604981E-02 —001430117E-01 06 93927S5S7E-02 —0.1132046E- C1 —001077687E-02 —02¢1274931E-01 All quantities are in 147 Cc n —-Oe6817682E O02 001942726E 02 —-Oell73821E O1 ~=-001447072E-01 —-001921633E O1 —066733062E OO OcelS66718E C1 —-06309135SE 00 Ocl335779E O00 004832629E~-02 0 eS321220E-01 Oe1E0S034E 00 -00215S98e8e8E 00 Oe14032SSE 00 — 0c 6387305E- 01 0 096i 8145E-01 —004127430E-01 0¢213399S5E-O1 —0.2286828E-O01 Qce23924E0E-01 —0 e2269067E-02 06632 0797E-—O02 x,y units. 68 B n Oe 4066765E-02 —0O.2 L667407E-02 0e3995597E-0 3 ~Qe 188621 9E-02 0.2907 7224E-03 —Ooe 155919 8E-02 0eGS190331E- 02 —-02114382S5E-02 O56 37S5230E-03 —-0e6712531E-02 Oo 4371286E-03 -0. 583528 e8E- 02 02.1835159E-03 —06. 251 2983E-03 —-O0210435S64E-04 Oe 1022287E-04 —0-¢ E843900E-O0S —O00e127881 3E-04 —050e 261 3426E- 04 —-02¢1440295E-04 —-0e5079823E-05 —-O0e 238344 7E- 04 022685109E-04 —-06.4311551E-04 0 «26351 92E-04 —06 2369922E-04 Oe 1060475E- 04 —02e11909G66E-04 O-. #96911 9E-05 —-029435032E-05 Oo 3592144E-905 —0-3163786E- OF 0.8069733E-06 =O. 31561 97E-05 0e2082134E-05 —-0.22780215E-05 O021130891E-OS —-OQ2 6122396E- CE —-02e1845171E-06 -02 336542 0E- 06 0«8938491E-O07 —0. 301 2036E-06 —-00« 2673406E-08 —02141408SE-06 Oo 73883 65E-07 —-0¢ 3226247E- 06 148 C n 001411975£-02 00463 9465E- 03 —0 .2636588E-03 06568 2681E—03 0676038S0E-03 —004942792E-03 Ce 3348SC8E— 03 -001035753£-03 001 781575E-03 009487823E-04 -0e5756764E-05 0.147 0€6S5E- 03 —0 0321 2969E- 04 023041208E-03 —00e24449S9E- 03 002326994E-03 —0.1970086E— 03 001541862E-03 —069325938E-04 Oe 103344 e8E- 03 —0e5277937E-04 Oo 324 66C9E- C4 —0 e2672389E-05 —06203634S5E-05 0.11422S5E—-04 002308075E-05 — 0.281 666 0E—05 0 053S6716E-05 —007342735E-06 02 30S6901E—05 —0-2802860E-05 0.5202 766E-— 05 —003204935E-05 0e2976863E-05 —003100245E-06 067375353E—06 — 0. 7168670E-G6 001450043E—05 —001119710E-05 Oc1101696E-05 —020454 2400E- 06 065235447E-06 — 0. 3582571 £- C6 004244058E—06 —001289455£-06 0¢5886075E-07 100 101 102 103 104 105 ic6é 107 108 109 110 sh aval ll2 113 114 B n 0627314 74E-06 —-O« 2Z802S570E- 0€ 0,61234109E-06 —Oe S4S8712E-07 Ocl9O07TSE4E-O07 —0.38S5SS5S9C0E-07 O«1227572E-07 -022280785E-07 —-00e 4772584E-08 —-O21305931E-07 Oel1346654E-907 —-O© 2178457E-O07 0«2829028E-07 —0e 28633683E-07 02 1390158E- 07 —-O02104130S5E-07 Oe €942560E-08 -0 6 2494349E-08 —Oe 2140656E-0& 0261858878E£- 08 ~0e3792454E-08 Oe 19027S8E-08 -0 eG630992E-09 -021545032E-08 Oc 1 7S4988E- OE —-022454563E-08 Oe 1989445E-08 -022018450E-08 02¢155404356-08 -0.1189770E-C8& 0eS5607077E-09 ~—04-. 2872680E-09 —-9-e9899369E-10 Oc SSB7T29SE-10 —0.1920187E-OS 0e8736652E-10 -O5. €521299E-10 —0«¢3684938E-11 —O23i85317E-1i —02eS5854169E-10 065440021 E-10 —O50c« 784317S5SE-10 0e5485389E-10 -0. SQE31984E-10 Qe 3508252E-1C —-00335SS14E-10 149 Cc n 001002326E-06 ~00¢ 3232 7S4E-07 00108928CE-07 068281825E-07 —066229692E-07 007623788E-07 —004725345E-07 005952231E-07 —0.3917647E-07 024041 3845-07 —001214022E-07 —0.401 8400E-10 001231 370E-07 —005542695E-08 004262090E-08 0 e3628934E-08 — 02 3900482E-08 0e7090147E-08 —006023740E—08 0«839S4S6E-CB8 -0 .6924818E-08 0e609681SE-08 —042973249F-C8 001536916E-08 0o3730656E-C9 —0 e5550416E-09 00e9392092E-09 —0.6327296E-CS 007495602E-09 — 0.261 0C46E-09 0 2922899€E-10 003045806E-09 —0.3766485E- 09 005183951 £-09 — 0.3851 0S5E~-C9 0¢3454421£-09 —002178382E-09 001826045E-C9 —068295703E-10 0c 61 88589E-10 —0.67661680E-11 0.6 3384400E-11 001760203E-10 —001176308E-11 0. 77S55S7E-11 003438477E-i1 B n 02. 1969359E-16 -90 62062679F-10 Oe 1077043E-10 —-Oce 1097489E-1C 063749065E-11 —Oc 3668546E-11 —-0 e4585235FE-1le —O5o 7479945E—-12 -0«1297924E- 11 029331983E-13 —0¢«¢ 8512745E-1e2 —-0e4626ES511E-12 Oo 2742915E-12 —-021541422F-11 02e1520291E-11 —0¢194622T75F-11 021428788E-11 —O00 1423591E-11 Oe 209S5S198E-12 —-0e¢6745650E-12 O5o 2695209E-12 —-02e16877S58E-12 —O0«e 9926446EE-13 02e6907761E-13 -02e1294383E-12 Oe 170263 9E-13 001289878E-14 -0.21019788E-12 Oe S003627E-12 —0¢2¢12S53069E-12 Oe 1061 812E-12 —-O501132419E-12 O0Oe7077363E-13 —O0.e 6203187E-12 02344327 3E-i12 —0e 1978280E-13 150 C n 0e3890720E-12 0e5851155E-11 —00e1611202E-11 O00e67709S4E—-11 —0 65035324E-11 Oe 81 81160E-11 —-OeSS46S987E-11 Ge7430254E-11 —-0¢5i1S512S55E-11 0e5S351474E-11 —Oce 3465696F-11 0e3223064E-11 —O0el36S864E-11 0e8782235E-12 002639 725F-12 —00e2869286E-12 026296024E—- 12 —00l1924273E-12 026.241 6221E-12 00e1149748E-12 —-060«1547703E-12 0e40972S0E-12 —Oe3738514E-12 Ce4700165E-12 —-00e3406081E-12 Oe2911647E-12 —-00e1232917E-12 0290 33253E-13 — 0.635404 8E-14 _ 700e1082376E-14 004432039E-13 —-00e1362410E-13 0e24S50R861E-13 Oe 3025605E-14 —-0e1103780E-13 Oce376E0087E-13 GULF COAST REGION FROM =) wo MBI anFrWHN & Note: APPENDIX E CONFORMAL MAPPING COEFFICIENTS FOR THE 8 — B = e} MATAGORDA BAY TO TIMBALIER BAY O0e3266563E 02 Co6402541E 02 =. 2355 B n —00 1345455E 02 —-069308576E 01 O06 3063064E-01 0. 1601213E OO —0e5839307E OC —0.1973795E 00 —O5« 8462828E-01 005467411E-02 —005171223E-O1 —001875S02E- 01 -0.9801641E-02 —0.6387778E- 02 —0 e7022325E-02 Oo 14392 77E-02 —0.1912029F-02z -0615240S5S£-02 0.1180221E-03 —0 21692594E-03 —O« 2428762E-03 —O0. 2506050E- 0= 021094825E-03 —06 SS71678E-04 C n —O50ol2B8847E O02 —-00e1253234E 02 O0e3363035E Ol OelLO0S1622E O1 0.35S9E888E-O1 —-O4e2109445E 00 Oe 717051 3E-02 Oel255C86E CO —02e7066 019E-02 —-O0ce1100479E-O01 0 024728E2E-01 0.862 8163E-02 —0e2627833E-02 O0e4 072074 E-02 Oo.1lSOO03C7E- 02 —-0 e61732357S5SE-903 Oe 1046393E—02 OeS270761E-O03 001903720E-03 —O001241S541E—-C4 0<22S5244SS5E- 03 0e2129799E—-04 All quantities are in x,y units. 151 B n -0 6 3508286E-04 —O. 37814 96E-04 —02e1100774E~04 00114749S5E-05 —OQ. 1094296E-04 —0 6 4207996E-05 Oe 516511 3E-06 -064495374E-05 001628793E-06 — 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009077216E-26 0e1107780E-25 02 3105713E-26 —00e1015449E-26 APPENDIX F NUMERICAL ANALOGS OF SURGE EQUATIONS Admissible finite difference analogs of equations (40) and (41) which are nominally centered at i, j , n (bearing in mind the storage of variables in Figure 36), are as follows: [Qa (isd mt1) - Qo, (G,j,m-1]/2at - £[Qpe (4.5.41) + Qpe(i,j.n-1)]/2 + (gD/F(i,j)u(i)) [HG+1,j,n) - Hy (i+1,j,n) = GF on) < Gol jaja) |//ans = to, (isj,n) - (K,Q(i,j,n-1)/D7)Q,, (i,j n+) (F-1) and [Qpe(i.5.n*1) - Qp,(i,i.n-1)]/2at Oe Gi and) Oey Gai n=t)))/2 + (gD/F(i,j)v(j)) HG, j+1,n) - Hy (i,j+1,n) - H(i,j-1l,n) + H, (i, j-1,n)]/2aT* Some) COG jyn-1)/D Ore Ej net) (F-2) where Q is as defined by equation (54) and D as defined by equation (61) is the arithmetic average of the four values of D about the point i,j at which the flow is evaluated. All terms are spatially centered at i,j and all but the bottom friction terms are exactly centered at time level n . The latter involve flow com- ponents at the new level n+l and at the old level n-1 within the flow magnitude term Q . This form is known to lead to a stable algorithm (Reid and Bodine, 1968). The Coriolis terms are exactly centered at n as well as i,j . Equations (F-1) and (F-2) repre- sent two equations in the two new Q components Qcox(i,j,ntl) and Qrx(4,j,n+1) in terms of quantities at previous times and are readily solved for each component individually. The resulting explicit relations for these components are given by equation (55) where G; , Go , G3 are defined in equations (58) to (60). The centered finite difference analog of the continuity equa- tion (42) leads directly to. the explicit relation for H at the new time level given by equation (57). 155 APPENDIX G MODEL VERIFICATION Prior to the surge computations the conservative properties of the numerical model were examined. The model was applied first to seiches in an enclosed square basin 12 kilometers on a side and 5 meters deep. The number of grid increments along each side was 20. To invoke the most severe seiching mode, the water surface was initially assigned a linear slope downward from one end of the basin (j=JM) to the opposite end (j=l). The initial setup for the nonro- tating basin computations was 50 centimeters, but this was reduced to 10 centimeters when rotation was allowed to suppress unnecessary fluctuations. The time step taken for all test runs was 50 seconds. The simulation of a uninodal seiche in a nonrotating basin was considered first. With the specified time step, tne period of the severest mode is theoretically 68.57 At. Figure G-1 shows that the computed fundamental period is within one-time increment of the theoretical period. Water Elevation (m) ! 0 30 60 90 120 150 180 210 Time (nAt) Figure G-1. Computed water level at grid point (21,20) for uninodal seiche test case. The time variation of the volume, potential, kinetic, and total energies are given in Figure G-2. Each of these quantities are area- weighted sums. This evaluation scheme should exhibit small perturba- tions more readily than unweighted sums of the variables over the grid. The variation of the volume, potential, kinetic, and total energies is small, with the potential energy showing the largest deviation. 156 Energy in relative units Potential energy (P) Kinetic energy (K) Figure G-2. energy (T) 30 60 90 120 150 180 Time (ndAt) Volume, potential, kinetic, and total energies for uninodal seich test case. 157 210 Rotational effects on the seiching mode were checked by specify- ing the Coriolis parameter as 0.75 05, 1.00 og, and 1.25 o9 where oo is the angular: frequency of the fundamental mode. The computed fre- quencies, o , were compared with those obtained by Rao (1966) and Platzman (1972). Table G-1 gives the comparison with Rao's and Platzman's results. Table G-1. Comparison of computed values of c/o, with those obtained by Rao (1966) and Platzman (1972). 5/59 SS S=SSSS—EEE Average From Rao From Platzman computed 0 £/o9 (1966) (1972) 0/09 oI ee 0.75 0.769 0.76 ed 1.00 ORZS OQ. 72il ORS 4.0 WEBS 0.686 0.683 0.71 3.9 Although the errors occurring with f/o, > 1.00 are large, these values of f/o, represent stern tests of the algorithm. It is definitely encouraging that no evidence of instability is indicated ‘in the simulated hydrographs. 158 APPENDIX H WIND DEFORMATION PROCEDURE Assuming that the wind-stress components (co OD) 5 eo) are known in the stretched shelf coordinate system from the applica- tion of a symmetric wind field model, the problem in part is to identify a region for a given hurricane and coastline where it is appropriate to alter these stresses to reflect the influence of land. The proposed deformation equations are applied only at points which are located within this region. Existing charts of hurricane winds and the investigation by Graham and Nunn (1959) provide the basis for the empirical deformation formulas. In this manner, the analytical representation obviates the detailed input of a massive sequence of digitized wind field data that conforms with the observations near the coast as well as offshore. The wind-stress components, rent) and on oe are altered at a point for which Y; > Yp according to the relationship: (SYM), Tox BF. Tox I, 2 (H-1) and SYM m a ae (H-2) where m is a constant chosen to be 2 and I. is an influence factor given by: = - - R 2 = I 1 Del (Yy Yp)/R I 5 (H-3) where Rp is a constant taken as the average radius to maximum winds of the hurricane, Dg is a distortion factor, Yp is the shortest distance the point is from land, and Yj ais the distance the influence region extends from the coast relative to the point in question (Figure H-1). In general, Yy should depend on Xp , the distance the point in question is from a centrally located point along the coast (X,,y,.).- This latter point is generally assigned the coordinates of intersection of the hurricane track with the coastline. From Figure H-1, the distances Xp and Yp are given by: 1 Kye Bl @& = a) ® Gas Ye alll ; (H-4) and - Ixy pue * Gx * dy * Yq usemzeq drysuotze[et oy. BurMoYys uoTseL adUeNT FUT FO usISep Tenjdsou0) “[-H ein3sty yore} auedtiiny k Ors ot esuentauL uotsel sdueNTFUl (Hae Nyy 9 T, N (Sed CK 5 ‘\ N qa » - NX“ — “) ¢, S = 3 ~ oS os > Vave- —— 2 3 3580) Cx X) 160 Yy =x, - x)? +, - v2 (H-5) where (X5,y,) 1S a point along the coast determined by the inter- section of a line normal to the coast and passing through the point in question (X,,y,). The sign convention is that Xp is positive if the point in question is to the right of the hurricane track and negative if it lies on the left. In general, the influence region is defined by: Yr =[ Fi (Xp) + Sp] Re ; (H-6) where Fy; is the influence function of Xp, S_¢ is a shift factor which permits the influence line to be moved normal to the coast, and Re is a range factor. The influence function attempts to re- late the degree of distortion in the wind isovels relative to the tract of the hurricane and the shape of the coastline. The extent of the land influence is given by Yj , which depends on S¢ and Re . These coefficients as well as Dg are considered to be related to the distance of the hurricane from the point where it crosses the coast. Additional investigation is required to establish a generalized procedure based on physical principles for deforming the winds in a manner which faithfully reproduces the observations. Consequently, each storm is parameterized individually to accomplish the task. The relationships used for deforming the winds of Hurricane Carla are: Fr 5 | X Bs RX, |/7 > (H-7) Dee sea (H-8) S. =f s5- = tan! (0, /2R)) R, , (H-9) 1.5 peices D, < -5R,, Re = ie is (H-10) Be 7 025 (D, /R, +5)?, if D,, > SR, where RX. is taken as approximately 37 kilometers and 1 fh is 2 = 272 RS De meal(Ge sa NAC en ele: (H-11) The sign convention is that is positive if the hurricane is over water and negative if the storm is over land. For Hurricane Camille, 161 these relations are: (X46, )/1-5.5 2£ 5 < -3R, BS [2S 5 [/ed 2 He =§i, << (H-12) | x Wag » HEI SO De = 052 5 (H-13) S¢ = 0.5, (H-14) and Re = i360 (H-15) The Hurricane Gracie wind fields were not deformed because the winds are presented in digital form. Any deformation obtained would be dependent upon a subjective analysis which requires additional developmental work. Moreover, the selection of Gracie was made on the knowledge that the results would possibly be compared with those obtained by other investigators using symmetric wind fields. 162 APPENDIX I SYMBOLS AND DEFINITIONS Transformation coefficients Nondimensional wind-drag coefficient Depth of water Distortion factor for deforming the wind Distance of hurricane from the coast Maximum expected depth of water at any point within computing grid Depth of water relative to mean sea level Error function Coriolis parameter Scale factor associated with the orthogonal curvilinear coordinate system Influence function for deforming the wind Acceleration due to gravity Water level relative to mean sea level Hydrostatic elevation of the sea surface corres- ponding to the departure of the atmospheric pressure from a constant Influence factor for deforming the wind Number of computational grid points along the S* and ieecaxXes Wave number = 1/A Wave number of free wave in the annulus 163 X,Y Nondimensional bottom-drag coefficient Number of Fourier-type transformation coefficients in the series Atmospheric pressure Far field pressure Central pressure of the hurricane Volume transport per unit width in the S* and T* directions Polar coordinates Distance from the hurricane center to any point Range factor for deforming the wind Distance from the hurricane center to the region of maximum winds Shift factor for deforming the wind Distance normal to the seaward boundary and along the coast Coordinates of the stretched shelf coordinate system Long wave travel time Maximum winds of the hurricane Weighting factors for the coast and seaward boundary curves Windspeed at an elevation of 10 meters above the water surface Rectilinear coordinates 164 At Usv Coordinates of landfall of the hurricane Position of hurricane center A point in the influence region A point along the coast determined by the inter- section of a line normal to the coast and pass- ing through (x, >¥,) Coordinates of the coast and seaward boundary curves to be mapped Extent of the land influence for deforming the wind Plane of (x,y) Extent of n (+) in the c-plane Arc length Surge algorithm time increment The alongshore curvilinear coordinate of the shelf coordinate system Prorating factors for the length of the transform- generated curves Half the horizontal extent of the region to be mapped in the Z or f-plane Functions transforming & and n to the stretched shelf coordinate system (S*,T*) Ingress angle Density of air and water Total and nth contribution to the "curvilinearity" variance Bottom resistance stress divided by in nS Se p and T* directions W 165 Tox? 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