er Carer Lee VEZ I ARS Sy A ct S22 (GGG TECHNICAL REPORT CERC-85-5 AN INVESTIGATION OF HURRICANE-INDUCED BEEN Snecaae FORERUNNER SURGE IN THE GULF OF MEXICO by Mahunnop Bunpapong, Robert O. Reid Robert E. Whitaker Department of Oceanography Texas A&M University College Station, Texas 77843 DOCUMENT LIBRARY Woods Hole Oceanographic Institution September 1985 Final Report Approved For Public Release; Distribution Unlimited Prepared for DEPARTMENT OF THE ARMY US Army Corps of Engineers Washington, DC 20314-1000 Under Contract No. DACW39-82-K-0001 Monitored by Coastal Engineering Research Center US Army Engineer Waterways Experiment Station PO Box 631, Vicksburg, Mississippi 39180-0631 U.S, Amy Gah, or Gh. “ech S&F. CLAC -SS- S~ aye ma S22 (9 TECHNICAL REPORT CERC-85-5 AN INVESTIGATION OF HURRICANE-INDUCED SEENON Gera FORERUNNER SURGE IN THE GULF OF MEXICO by Mahunnop Bunpapong, Robert O. Reid Robert E. Whitaker Department of Oceanography Texas A&M University College Station, Texas 77843 DOCUMENT LIBRARY Woods Hole Oceanographic Institution September 1985 Final Report Approved For Public Release; Distribution Unlimited Prepared for DEPARTMENT OF THE ARMY US Army Corps of Engineers Washington, DC 20314-1000 Under Contract No. DACW39-82-K-0001 Monitored by Coastal Engineering Research Center US Army Engineer Waterways Experiment Station PO Box 631, Vicksburg, Mississippi 39180-0631 Destroy this report when no longer needed. Do not return it to the originator. 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. The 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. NN QU 0 0301 0091247? 3 Unclassified OE A SECURITY CLASSIFICATION OF THIS PAGE (When Data Entered) READ INSTRUCTIONS REPORT DOCUMENTATION PAGE 1. REPORT NUMBER 2. GOVT ACCESSION NO.| 3. RECIPIENT’S CATALOG NUMBER Technical Report CERC-85-5 4. TITLE (and Subtitle) 5. TYPE OF REPORT & PERIOD COVERED Final report AN INVESTIGATION OF HURRICANE-INDUCED FORERUNNER SURGE IN THE GULF OF MEXICO 6. PERFORMING ORG. REPORT NUMBER 8. CONTRACT OR GRANT NUMBER(a) Contract No. DACW39-82-K- 0001 7. AUTHOR(a) Mahunnop Bunpapong, Robert O. Reid, Robert E. Whitaker 10. PROGRAM ELEMENT, PROJECT, TASK AREA & WORK UNIT NUMBERS 12. REPORT DATE September 1985 13. NUMBER OF PAGES 217 15. SECURITY CLASS. (of thia report) 9. PERFORMING ORGANIZATION NAME AND ADDRESS Department of Oceanography Texas A&M University College Station, Texas 77843 CONTROLLING OFFICE NAME AND ADDRESS DEPARTMENT OF THE ARMY US Army Corps of Engineers Washington, DC 20314-1000 - MONITORING AGENCY NAME & ADDRESS(if different from Controlling Office) US Army Engineer Waterways Experiment Station Coastal Engineering Research Center PO Box 631, Vicksburg, Mississippi 39180-0631 Unclassified DECL ASSIFICATION/ DOWNGRADING SCHEDULE 15a. - 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 Available from National Technical Information Service, 5285 Port Royal Road, Springfield, Virginia 22161. . KEY WORDS (Continue on reverse side if necessary and identify by block number) Storm surges--Mathematical Models (LC) Storm surges-~Gulf of Mexico (LC) . ABSTRACT (Continue om reverse sides if neceseary and identify by block number) A system of coupled, normal mode equations describing a two-layer ocean basin of variable depth was derived from the quasi-hydrostatic equations of motion using a general form of the method of Veronis and Stommel (1956). A finite difference, time marching, numerical model for the normal mode equa- tions, employing an Alternating Direction Implicit (ADI) scheme, on a space staggered grid has been developed. The model is quasi-linear and allows for (Continued) FORM DD . jan 73 1473 = EDrTion OF 1 NOV 65 1S OBSOLETE Unclassified SECURITY CLASSIFICATION OF THIS PASE (When Data Entered) Unclassified ae SECURITY CLASSIFICATION OF THIS PAGE(When Data Entered) 20. ABSTRACT (Continued). variable bathymetry and variable Coriolis parameter. The model domain in- cludes the Gulf of Mexico and the Cayman Sea with a resolution of 15". A no- flow condition is taken at all solid boundaries and the inverted barometer term is used to stipulate barotropic height anomalies on the open boundaries. Hurricanes Carla (1961) and Allen (1980) are used as historical storms to verify the model by comparing numerical and observed hydrographs. A parametric study utilizing three forward speeds, two radii to maximum winds, and five paths characterizing Gulf hurricanes is presented. The results of the study show that volume transports through Florida and Yucatan Straits consisted of in-phase (both in or both out) and out-of-phase com- ponents. The in-phase volume transport excites a volume mode, nec , in the Gulf of Mexico having periods of about 28 h and 3.4 days. The ne mode of oscillation can produce a forerunner surge for many storm tracks. Although all the model storms generated ne ,; not all had an associated positive forerunner. The hurricane path and evolution play important roles in gener- ating a forerunner. The out-of-phase volume transport through the ports was found to produce a Gulf-wide quasi-geostrophic tilt mode of about 6.5-day period. Surges on the shelf including the forerunner are primarily a barotropic response. The quasi-linear model transmitted only a fraction of the baroclinic energy onto the shelf. Results of a limited area model (no wind in deep water) showed that either wind or pressure forcing generates ng - The limited area model simulations also demonstrate clearly that the use of a simple inverted barometer boundary condition at the shelf break will always underestimate the peak surge at the coast. Unclassified ——— ee SECURITY CLASSIFICATION OF THIS PAGE(When Data Entered) FOREWORD This study was conducted by the Department of Oceanography at Texas A&M University, Reference 85-2-T. The research was conducted through Texas A&M Research Foundation Project 4667. This study was motivated by the fact that existing storm surge models which are generally restricted to a limited reach of the continental shelf do not adequately simulate the forerunner surge. In the Gulf of Mexico the forerunner stage of a surge event (which can have an amplitude as large as 1 m) can precede the peak surge by more than 24 hours. It is common practice, in employing coastal surge models for design purposes, to start computations with an initial non-zero constant water level field over the shelf in recognition of the influence of forerunners. This procedure makes no allowance for initial flows or gradient thereof which must exist during the forerunner stage. On the other hand, starting local coastal models well in advance of the arrival of the hurricane at the shelf break cannot simulate the forerunner surge since the latter is ex- cited by mechanisms acting over the adjacent deep basin seaward of the coastal zone being modeled. The objectives of this research were to determine for the Gulf of Mexico the space and time scales of forerunners, examine possible excita- tion processes, and evaluate the role of baroclinic modes in surge events at the coast and shelf break. A numerical approach was used in this study which modeled hurricane-forced responses of the barotropic and first baro- clinic modes over a computing domain representing the Northwest Cayman Sea and Gulf of Mexico. The inclusion of the baroclinic response adds a degree of realism lacking in previous studies on hurricane effects in the Gulf of Mexico. It should be stressed that the model includes only wind and atmospheric pressure forcing. Tidal forcing was not included. Moreover, the model is quasi-linear (only the bottom and interface stresses are nonlinear). The model was verified by simulating the Gulf's response to hurricanes Carla and Allen. Although the focus of the study was on forerunner surge events, the results also provide insight into fundamental concepts concerning the large- scale, low-frequency free modes in the Gulf of Mexico and Cayman Sea. This study also constitutes the doctoral dissertation research of one of the authors, Mahunnop Bunpapong. stat PREFACE This study was carried out under Contract No. DACW39-82-K0001 with the Coastal Engineering Research Center (CERC) of the US Army Engineer Waterways Experiment Station (WES). The study was funded by the Office, Chief of Engi- neers, under two work units: the Hurricane Surge Prototype Data Collection Work Unit, Coastal Flooding and Storm Protection Program, and the Nearshore Waves and Currents Work Unit, Wave and Coastal Flooding Program. Partial support was also provided by the Texas Engineering Experiment Station (TEES), Texas A&M University System, through a TEES Research Fellow grant to one of the authors, Professor Robert O. Reid. The authors appreciate the advice and encouragement of Dr. Robert W. Whalin, former Chief, CERC, Dr. William Wood, Chief, Engineering Development Division, and Mr. Andrew W. Garcia, Prototype Measurement and Analysis Branch. Mr. Charles C. Calhoun, Jr., was Acting Chief, CERC, during publica- tion of this report. The authors wish to acknowledge also the help of Dr. Ignacio Galindo, Director, Instituto de Geofisica (Mexico City), and Mr. Milton Rutstein, National Ocean Service, National Oceanic and Atmospheric Administration, in obtaining serial water level data. COL Tilford C. Creel, CE, and COL Robert C. Lee, CE, were Commanders and Directors of WES during the conduct of the study. COL Allen F. Grum, USA, was Director of WES during the publication of this report. Mr. Fred R. Brown and Dr. Whalin were Technical Directors. iii PORNWORD 5 9 5 ooo 6 oO 6 DRIQWNGD 6 5 o 6.90 0 9 0 TIES ONY IEW 6 56 oo ILLS Ol SUNS 5 5 5 0 5 0 O CHAPTER I INTRODUCTION . 1. Background ..... Zo Opi@CEMNVES oo 6 6 o 34 IPm@@achtTa 5 56 5 0 s CHAPTER II THEORETICAL AND 1. Theoretical model . a) Basic equations b) Normal Mode Equati 2. Numerical model... a) Grid system... b) Numerical integrat TABLE OF CONTENTS NUMERICAL MODEL ons . ion scheme c) Surface, interface, and bottom stress d) The coupling terms e) Initial and boundary conditions . 3. Wind and pressure forcing ..... a) Analytical wind model ...... b) Analytical pressure model... . CHAPTER III SIMULATIONS OF HURRICANES OF RECORD 1. Selected hurricanes of 2. Meteorological data . record . iv 37 43 a) Surface wind for hurricane Carla b) Surface wind for hurricane Allen Sig GAC CENCE CEM 6 56 6 6 0.600 4 Go 4. Simulation procedure ....... 5. Results of Carla simulation... . 6. Results of Allen simulation... . CHAPTER IV PARAMETRIC STUDY ...... 1. Selection of paths ........ 2. Hypothetical storms ....... =. 3. Simulation procedure ....... 4. Results of parametric simulations . a) General results ........ b) Simulated hydrographs ..... 5- Long period variation of ng... . a) The 3.4 day volume mode .... b) The 6.5 day tilt mode ..... 6. Results from related simulations . a) Variation of pressure drop... b) Barotropic model ........ c) Radiation boundary condition .. d) Limited area model ....... CHAPTER V SUMMARY AND CONCLUSIONS ... IAW AAS 56 6 50606 606065600 0 6 101 101 103 104 106 106 lll 156 156 161 167 167 171 175 179 191 199 FIGURE 10 Akal 12 LIST OF FIGURES Computing lattice showing locations where U, V, w are evaluated. Positive U is toward the east (increasing index I). Positive V is to the north (increasing index J) Grid system for the Gulf of Mexico and the Cayman Sea. The grid increments are 15 in latitude and longitude. .. Model coastline and sample contours of the digitized bathyme Gisy ee DePiEnSmanye me Ger Slam cient otelircitoNon NCH NoNn Inner nto Schematic of one dimensional channel with stair-step depth profile. Capital letters A through E indicate locations where the gradients [$.(H,/D) ] are defined. .. Track of hurricane Carla from 1200 GMT 6 September to 1800 GMT Vi September P9602 3 3 8 8 ew Time sequence of Carla central pressure 4 - 13 September, Wes, Caieeere hin Ge Ellog UWOGZYs 0 6 6 6 6 050 605600 00 Track of hurricane Allen from 0000 GMT 7 August to 1200 GMT 12 August 1980. Lallteuncey plien) io neue leadetamismhesultetkaten im scibeaioh ihe) © Koja te Time sequence of Allen central pressure 4 - 10 August, 1980 (after Lawrence and Pelissier, 1981). ....... - Schematic showing azimuthal distribution of radial sections where wind profiles were fitted. ........ Hurricane Carla surface wind (m/s) at 0000 GMT 7 September 1961, as obtained from Holland's model. The solid circles indicate the observed wind speed from the Surbace sGhamtiy. vce lenin wl ol tel) ov Ge, few WoL com Sepiliel isl eye ten Ie Hurricane Carla surface wind (m/s) at 1200 GMT 9 September 1961, as obtained from Holland's model. The solid circles indicate the observed wind speed from the Na SWIAESA EEIECo 56 0056 0000050056000060000 Locations of tide gauge stations used in this study. KW is Key West, NP is Naples, SP is St. Petersburg, CK is Cedar Key, AP is Apaiachicola, PE is Pensacola, GI is Grand Isle, GV is Galveston, PI is Port Isabel, MD is Madero, VC is Veracruz, CM is Carmen, PG is Progreso. The solid circles indicate those stations used in the simulation of hurricane Carla. Both the solid and open vi Page - 18 19 20 29 38 40 41 42 45 47 48 FIGURE 13 14 15 16 17 18 19 20 21 22 Page circles are those used in the simulations of hurricane Allen and also as the model tide gauge stations in the joeiceiaerone Seen 2 ooo Gg oo Bl Observed water level at Galveston during hurricane Carla, 1961. The arrow indicates the time at which Carla entered the Gulf through Yucatan Strait. The datum is CEMCGE WEEN Wey CENEEISo 6 6 610 6 Oo oO OO oO OOO HY Observed water level at Port Isabel during hurricane Carla, 1961. The arrow indicates the time at which Carla entered the Gulf through Yucatan Strait. The datum is CANES mMeein Wow WENEASc 6 clo 0 0 6 6 66 0 6 6 616 6 6 ole) BS Observed water level at Key West during hurricane Carla, 1961. The arrow indicates the time at which Carla entered the Gulf through Yucatan Strait. The datum is gauge mean low water. Sis Wath ae aaa oho Geroe ato Ghana roe aba Observed water level at Galveston during hurricane Allen, 1980. The arrow indicates the time at which Allen entered the Gulf through Yucatan Strait. The datum is gauge mean low water. ai a coULiGhioae hs) cals) uclgiciaele\ ie) Cchte cba sh io oML sen TO Observed water level at Port Isabel during hurricane Allen, 1980. The arrow indicates the time at which Allen entered the Gulf through Yucatan Strait. The datum is gauge mean low water. COS Oe OP OP MOT SO CCL Gt Ore Cen cui a) Observed water level at Key West during hurricane Allen, 1980. The arrow indicates the time at which Allen entered the Gulf through Yucatan Strait. The datum is gauge mean low water. BY Celene ieiy foley Yotkaal main tet hal Me lejucet comes cent ner mt ONy Radial profile of the inflow angles computed from (78). . 61 Computed (solid) and observed (dashed) hydrographs at Key West during hurricane Carla 6 - 13 September 1961. The ckvetin SbS msein SGa Hewyatlo 61d 6/6 6 6 0° 6 6 66 5. 69 610 6 | GA Computed (solid) and observed (dashed) hydrographs at St. Petersburg during hurricane Carla 6 - 13 September 1961. TS Cevetin IS Mee SEE Newells 6 6 6 56 66610 56 60000 6 8 Computed (solid) and observed (dashed) hydrographs at Pensacola during hurricane Carla 6 - 13 September 1961. Gina Ceyetin SS meen Gea Teyails 6 6650660606000 005 WH vil FIGURE Page 2s) Computed (solid) and observed (dashed) hydrographs at Grand Isle during hurricane Carla 6 - 13 September 1961. Gis Ceteihn US fGen GG Uealls oo 6 oo co oO pot coo 8 OD 24 Computed (solid) and observed (dashed) hydrographs at Galveston during hurricane Carla 6 - 13 as ase 1961. The datum iS mean sea level. ...... oo 0 6 ota oo 25 Computed (solid) and observed (dashed) hydrographs at Port Isabel during hurricane Carla 6 - 13 September 1961. Tox Geet 6 Mee SE Tewele 6 6560000 ooo ODO OW 26 Computed (solid) and observed (dashed) hydrographs at Veracruz during hurricane Carla 6 - 13 September 1961. The datum 12s mean sea level. © 2 2. 2 5 5. © 5 2 2. 5 2. . 668 27 Computed (solid) and observed (dashed) hydrographs at Carmen during hurricane Carla 6 - 13 September 1961. The Get 18 Wem SGA UeyEGlls 6 66 0 oo OOo OO OKT KK OH 28 Computed (solid) and observed (dashed) hydrographs at Progreso during hurricane Carla 6 - 13 September 1961. Tne CeNetin sto mae SGA Uewealls 6 6 6000000000000 7) 29 Location of Port Isabel tide gauge station which is sheltered by South Padre Island. .....-+.-.+.+-.+.+.-.+-. 72 30 Computed barotropic height anomaly field (meters) for hurricane Carla at 1200 GMT 10 September 1961. ...... 74 31 Computed baroclinic height anomaly field (meters) for hurricane Carla at 1200 GMT 10 September 1961. ....-.-. 75 32 Computed surface current field (cm/s) for hurricane Carla at 1200 GMT 10 September 1961. ....-.+ +++ +++ +e «76 33 Time sequences of the average water levels in the Gulf of Mexico, ng for hurricane Allen. The solid line is computed by averaging water levels from every grid point in the Gulf at each time step. The dashed line is computed from the continuity equation. The datum is mean Seae Weveln se Wess Le’ SS abe emer ciate Moten ben tents ch! TG 34 Computed (solid) and observed (dashed) hydrographs at Key West during hurricane Allen 7 - 12 August 1980. The Gen SG meem Gea Weyeale 6:6 6100 6 0 050606 0 46 6 6 VY 35 Computed (solid) and observed (dashed) hydrographs at viii FIGURE Page Naples during hurricane Allen 7 - 12 August 1980. The okKetin) ebS IneeIn Gey Weyento Gg o 6 6 ob 0 0 06 5 0 610 0 oo, HO) 36 Computed (solid) and observed (dashed) hydrographs at St. Petersburg during hurricane Allen 7 - 12 August 1980. the datumyussemeanseamlevelyy a. puis) nce icles «Mey mreinneire oe im Oe 37 Computed (solid) and observed (dashed) hydrographs at Cedar Key during hurricane Allen 7 - 12 August 1980. The dGatumaiis.mean. sea levels «45 sp tite ees sete 6 6) ohemoum « «4 82 38 Computed (solid) and observed (dashed) hydrographs at Apalachicola during hurricane Allen 7 - 12 August 1980. theeGatumsd's imeaninsea -levelhin.sss5 so +2 > aM, /ot + £kxM, + 0, 9H,V(hy+ho) = 10 (1) > and for the lower layer, > 7° > > P2 dho/ at cP WONG) 0, (4) where M is the mass transport per unit width, f is the Coriolis parameter, k is the vertical unit vector, p is the water density, g is the gravitational acceleration, H is the mean depth, H+h is the instantaneous depth, F is the external forcing and dissipation defined as: Fy — Ts ae Ti -H,VP,, (5) > > Fo _ Ti > Th HoVP.- Here . is the stress vector where subscripts s, i, and b stand for surface, interface, and bottom, respectively. The atmospheric pressure at the sea surface is Pi: The normal mode form of the equations can be derived from these primitive equations by a method similar to that employed by Veronis and Stommel (1956). To transform (1) and (2) into normal mode form, we multiply (1) and (2) by a and (3) and (4) by 6 and add the corresponding momentum and mass conservation equations, respectively, to obtain > Sed OM/odt + £kxM + g{V[aH, (ph, +0 ;h2)+BHo (phy +poh>)] > -[ (ph, +07 hz) V (aH, )+(p 7h, +ogho)V(BHo) J} = G, (6) > > d¢/ot + aV-M, + BV-Mp = O, (7) where + 4 > M = aM) G2 BM>, @ = ap hy + Begs, (8) > ne 4 G Ee QF) + BF5. Note that a and $6 are non-dimensional and, for the case of variable depth, may depend upon x and y (this is the generalization of the Veronis and Stommel analysis). The constraint imposed on a and 6 to make the elevation anomalies in (6) and (7) proportional, is aH, (p, hy +pyho) G2 BH (pyhy +poho) = T¢, (9) where T, a factor of proportionality, is an equivalent depth to be determined. Since ¢ = ap hy + Bpoho, then (9) will be valid for all combinations of hy and ho slie Ta, and (10a) aH, + BH5 " aH, (p/P) + BH> rg. (1Ob) It can be readily shown that (6) and (7) can be written in the form > a) > oM/ot + £kxM + gI'V¢d - B = G, (11) 2 ¢ dd/dot + VM-C = O, (12) where => B = oI (p,h,Va + PohoVB) , and (13) > > r= M, °Va + Mo °VB. For the case of constant layer depth H,, Hj, the a and 8 are => constant and B, C vanish. In general these terms produce coupling of the modes in the presence of bottom topography. Eqs. (10a,b) can be written in matrix form as H,-T Ho a 0 a (14) (p3/ 2) Hy Ho-T. B 0 and hence the eigenvalues of [ are the roots of the characteristic equation r? - Dr + eHjHp = 0, (15) where D is the total depth at rest (H,+H>), and e is the relative density difference, (p2-P1)/P2- The two roots of (15), which correspond to the equivalent depths for external and internal modes, are T, = D[l-e(H,H>/D2) + O(e%)] , and (16) Ty = e(HyHp/D)[1+e(H,H5/D?) + O(e)]. Since Hy H>/D2 < 1 and e << 1 for the general ocean basin, then ne = D, and (17) are the equivalent depths for external and internal modes as obtained by Veronis and Stommel (1956). In general these may depend on x,y. The ratio of a and ~ can be determined from (10a,b) as B/a Sa io Hy /H> SS (Pp, /P2)H,/T = H5- (18) Substituting (17) in (18), the ratio (a/$) for each mode is (B/a), = 1 - e(Hy/D) - e*(H?H5/D>) , and (19) (B/a); = -H,/H> + e(H,/D) + e*(HTHS/D>). Note that ae and Be are of like sign, but a; and Bs are of opposite sign. Ger Bor a; are chosen to be positive and f; to be negative. 10 Veronis and Stommel (1956) took a, = a; = 1 for a constant depth basin. However, the presence of coupling terms B and C in (11) and (12) due to varying depth in the present study precludes an arbitrary choice of a. Notice that both a and $ are functions of x and y. The procedure used to determine the individual values of a and 6 for both modes is based on energy considerations. In essence the energy equations derived from the primitive equations and the normal mode equations must be consistent. The two energy equations formed from the primitive equations and the normal mode equations, respectively, are > Q(E,tED)/at + VJ = S, (20) where Ex = 39(p7hy* + 2pyhyhy + ph”), Ep = 90(M,2/p,Hy) + (M$/p9H2), (21) > > > J = g[(hy+hy)My + 1/p2(pyhy+eoh>)Mo], > 3° > 9 SS bila @nbh, 7 MycPay abhi and > ELE /Beo we oe Se Ss. (22) Equation (22) contains not only the additional term T, but the expressions for the kinetic energy per unit area, E,, the potential > energy per unit area, the energy flux per unit width, J, and the Eps net energy supply per unit width per unit time, S, are also different from those given in (21). The added term, T, which defines the 11 energy transfer between modes, and the other terms in (22) are defined as follows: SoS SS T = 1/po[(Mg°B,/T,) +e (M; °B,/T; )+9(6,C,te eC; )]- (23) E, = 1/pl(w2/r,) + e(Mt/T,) 1, (24a) En = 1/p29(43 + 87), (24b) > > > > > where G, and G; are defined in (31). If (20) and (22) are to be consistent, their corresponding terms must be equal and T must be zero. Inserting (8a) to (8c) in (24a) to (24d) it can be shown that the necessary conditions that E, as well as S in (20) and (22) are consistent for arbitrary hh, ho, My and Mo are that ae2/T, + €a;2/Ty = 1/Hy (2/01), (25a) Bet/Ne + By2/T, = I/Hp, (25b) a6./T, + ¢a;8;/T; = 0. (25c) It is also necessary, for consistency in Ep and a that ae + ea;* = py/Pos (26a) POS Gn? (26b) abe + ea; By Sot. (26c) 12 The requirement that T must be zero can be examined by substituting (13) in (23) and rearranging to yield, > 2 2 > > aS pbb Ga cen)! se (Daleilehptnlehlby) NING Se CON sn) pohoM>-V(B,2+€B; 2). (27) Daudwe. e aL Using conditions (26a) to (26c), it can be shown that T vanishes provided that the ratio pj/p, is constant. Therefore, (25) and (26) assure that (20) and (22) are entirely consistent with T equal to zero and all other terms are identical. These two sets of equations will also be used to determine a and 6. From (25a),(25b) and (26a),(26b) we find ae? = pol Q/0y Hy [(Hy-Fy)/(TeTy) 1, (28) Be? = Te/Ho[(Ho-Ty)/(Te-T;)], and a3* = pol y/eoqHy[(Te-Hy)/(Te-Ty)], (29) 8,27 = [,/eHo[(T-Ha)/(T,-T,))- It is not difficult to show that the ratios ($/a), and (8/a); obtained from (28) and (29) are the same as those in (19). If we substitute (16) in (28) and (29), we obtain the relation for a and B,accurate to order e, in terms of H,, Hp and D as follows: a, = 1+ ge - de(Hy/D)? + O(c), (30a) 1- $e(H,/D)? + O(e%), (30b) As) o u 13 i = Ho/D{ lege + e[(HyHp/D2)-$(Hy/D)] + 0(e%)}(30c) i=} i) Bs -H,/D{1+e[ (HyH>/D2)-4(H5/D)] + O(e*)} (30d) Note that positive roots are chosen except for §;. Using a and £6 from (30) it can be verified that (25c) and (26c) are correct to the order e. b) Normal Mode Equations To express Eqs.(11) and (12) with all dependent variables in terms of modes, (8) is rewritten for each mode as: > > > Me = GM) + BEM > > > Mj; = a;M) + BiM2 %e = G—P hy + Bepgh2 (31) On Case) 08 DAG > > Ge = a.F) + BF? > > G; S a;Fy + ByFo- It readily can be shown from (31) that Po r(B.o3 - Bi %_) Pohp = I(-aeb; + 25 %Q) (32) > = 4 M, = §r(BQM; - BiMQ) 14 > > > Mo = r(-a.M, ot a;M.) ’ where r = (a;68, - a,6,;) 1. Using (30a) to (30d), the factor r is (p1/09)2/2 to order e and al [6 + (Hy/D)$,](P10) 2’ Im 8 +, ho = [(eH5/D Saal) Oat (H>/D) ¢,] (1 P2) (33) > > > M, = (H,/D)M, + Mz, > > > Mo = (H5/D)M, = M;. Using (31) and (32) the coupling terms defined by (13) can be approximated by, > Be = rgP,e¢,V(H,/D), > B, = —9Iy¢_V(Hj/D), (34) > Ce = reM;°V(H,/D), > Cc; = -M,°V(H,/D). These forms show clearly that the external mode is influenced by the internal mode and vice versa when H,/D is variable. It can be shown by substituting (34) in (23) that the energy transfer between modes, T, is zero as required. Finally, the forcing terms G can be approximated by using (5), (8), and (31) as: 15 > > G. = Ts = Th = DVP, + Oe), (35) > > > G; = (Hj/D)t, + (Hy/D)T, - Ty + Oe). > > The kinematic counterparts of M, ¢, 7 and P, are defined as > > > > Of = Maer Q; = M,;/op, (36) > > > > We ES a/Pn ae & vay > > uy = T3/P» -VP./p = gVb, The variable b in (36) is the barometric pressure deficit expressed in terms of an equivalent head of water (sometimes referred to as the inverted barometric effect). The normal mode equations (11) and (12), therefore, take the form > >> 0,/at + £kxQ, + gDV¥, - egD¥jV(H)/D) = i = H nun ! Hy o + a 5 we (37) > > dw,/at + V-O, - €Q;°V(H)/D) i] fo) and > 7° > > > a0, /at + £kxO; + gM yVWy + OF yeV(Hy/D) = (Ho/D)T, + (H\/D)T, - Ty, (38) > > dy; /ot + Wola + Q,°V(H,/D) = 0, where [; is given in (17). 16 2. Numerical model a) Grid system A space-staggered computational mesh is employed in the numerical analogs of the normal mode equations (37) and (38). The grid spacing is taken as 15' x 15' in latitude (A®) and longitude (AX). The modal transport per unit width, represented by U, V for east and north components, respectively, is located at the mid points of the appropriate sides of each grid block. The modal water level anomaly, yw, is defined at the middle of each grid block. Figure 1 illustrates locations of these variables whose positions are identified by I,J and time is indexed by n. The grid spacing is taken as the distance between the same variables, thus consecutive U, y values are at half increments of I and consecutive V, w values are at half increments of J. The depth is stored at every variable location. The grid system used in this study is the same as that used in the Gulf of Mexico Tidal Model (Reid and Whitaker, 1981). It nominally covers 70 x 50 grid blocks for the Gulf of Mexico and the northwest portion of the Caribbean Sea (Cayman Sea) as shown in Fig. 2. The depths of the Gulf of Mexico were digitized from bathymetry charts prepared by Buffler et al. (1984). The Cayman Sea depths were taken from available hydrographic charts. Depth over most of the Cayman, especially near the open boundaries to the east and south, was taken as 4000 m. Figure 3 shows the computer plotted contours of 17 *(f xeput Bbuyseerout) y}IOU 9YyR OF SF A AATI}FSOd “(I XeputT Hbuyseerduyt) ysee 2u} PIEMOZ ST N SATAITSOd “paqzenzteas are f ‘A ‘M BTeYyM SUOFAeDOT HhuyMoys 92d723eT Hut {nduiod I+I I t-I Pa cise Socuseesentt Cocuseesent bef tiff | pfefete fetid” ioe fee gi 8) Ei | °T °6ta 18 88 T8202 8= 050282 0—0-0-02828-02 0-0 -929=8= 0-8-8 8-8 6-8-8: 0-8 8 -0-0-8= 8-020 -8- 0-0-8288 8 e- 8-8-8. 5— 8-8-8 0- 6288 -0= 6-0-8 - 6-885 0- 8-0-0: 0-028: 8-8- 650205 828-3- 24 ! | ] D Aaa | 1 | ' | HV f 1 | = | \ ) Ey GUSEBaBEEa pf NUD BLS OUaSeeSeaSeuSsaoeab' BSD. RSS SbesoODw AS SOOSMS0: SC DEEG0SRE2S2S2E2See008z Jp ne eg TALL mi Nl 2820287828 0-0- 82! rr) 9) lo) wo fo) Ye) o n < t+ pe) fe) N N 2 2 0 19 70 The grid increments are 15 in Grid system for the Gulf of Mexico and the Cayman Sea. Fig. 2. latitude and longitude. *szajeul ut syjded -Xryeudy}eq peztIHtp ey} JO sanoquod aTdues pue suTT,SeoOD TepoN SB 48 68 16 6 S6 46 "¢ ‘BT 20 the digitized depth field. Special care was observed when comparing the digitized depth fields to the bathymetry chart, especially in the shelf break regions and modifications were made where necessary. b) Numerical integration scheme The multioperational alternating direction implicit algorithm developed by Leendertse (1967) was adopted for time integration of the finite difference equations. The following notation is used in the discussion. The spatial average of a field variable X is written as X"(1,5) = $[x(1-$,3-$) +x (144, 3-4) +x(1-4, 4d) 4x(34d,34+4)], (39) where X"(I,J) = X(Ao+IAA, $0+JAG,to+nAt). Time and space derivatives are depicted by the standard centered differences, ax/at = (1/At) (x™*1(1,3)-x®-1(1,35)], (40) @X/8A = (cos®/Ar) [X™(1+5,3)-X"(1-4,)]. A spherical coordinate system is employed in representing the gradient terms on a level surface, i.e., VX = (1/a8(3)) {8X/3b a + (5) aX/IA b} (41) > where X is any scalar field variable, a is a unit vector along lines => of constant $, b is a unit vector along lines of constant A, and 6(J) = cos($ot+tJAb), where $. is the reference latitude (18°N). 21 The cycle of calculation is separated into two operations. During the first-half cycle, at odd time steps, w and U are computed implicitly along lines of constant latitude, followed by an explicit computation of the V field. For the second half-cycle, the computations proceed along lines of constant longitude with y, V updated implicitly and U computed explicitly. In each half-cycle, the external mode computations are executed first and, after completing the entire computing domain, are repeated for the internal mode. The implicit formulation of the finite difference analogs of the external mode momentum and mass conservation equations, respectively, are, for odd time steps, 5 n+l 1 n+l 1 n+l 4 —Yy (1-3, I) ¥,(1-9,9) + U,(1,9) + Vy (1t9, I) Ve (15,9) = UN(1,J)+2Att(J)VE(I,J) + AtFg+ xj, and (42) n+l 4 n+l n+l 4 -v,(J)U,(1-5,9) + We(1,I)+ vy (J)U,(1+5,I) = We(1,J) - vySyVE(1,I) + Ex, (43) and for even time steps Ts 1 1 n+l cl snaes 1 1 “Vy (15-9) We (15-3) OC IH) + V,(1,J) + Vy (11 I4+5) Q(T, I+5) O(S+5) = Vis NACE CUS (LI) xy esate (44) 1 n+l TL n+l 1 n+l Snailk —vy(IFVE(L IH) + vQCE, I) + vy (T+dVQ (1, 544) SUS > OA Gia) Bao (45) 22 where Ye(1,J) = (At/Ad){gD(I,J)/a0(J)}, Yy(1,J5) = (At/Ae){gD(1,5)/ae(5)}, Vy(J) = (At/Ad) {1/ae(I)}, vy(J) = (At/AS) {1/ae(J)},. 6,U2(1,5) = ubcr+$,5) - u2¢1-4,3), 6yVE(I,I) = VE(I,I+4)0(I+G) - VE(I,I-G)O(I-Z), a = radius of the Earth, £(J) = 2Qsin($ot+JAe), F = forcing and friction terms, x,& = coupling terms. The forcing and coupling terms will be discussed in the next section. The coupling terms in (42) and (43) or (44) and (45) are of opposite mode relative to the other terms. Upon replacing the total depth, D, by the equivalent depth, eH,H>/D, interchanging modes of all field variables and coupling terms, and using the proper forcing terms, (42) through (45) are also representations of the implicit formulation for the internal mode computation. The explicit coding of the external mode momentum equation at odd time steps is 23 n+1 =nt+1 = Ve(IJ) = va(1,3) - Zot £(5){U, (1,5) + UB(1,I)} “ry (I,J) dyve(1, 5) + xq + AtFe. (46) The even time step counterpart of (46) is n+1 =n+l = Ue(1,J) ULL) eAEEE CI) Wa(I,d)) HevaCia))! —ECinaSseAGtna) @ seq <8 (Ne ec (47) Likewise, the explicit coding, at odd and even time steps, respectively, for the internal mode computation is obtained by replacing the total depth by the equivalent depth, interchanging modes of all field variables and coupling terms, and employing the proper forcing terms in (46) and (47). Eqs. (42) and (43) or (44) and (45) form a system of linear algebraic equations in the collective I or J, depending upon time step, of y and either U or V at time level n+l. The coefficient Matrix is tridiagonal for which there exists a double sweep solution algorithm for inversion, provided that boundary conditions on U or V or some combination of conditions on U or V and wy are supplied at each end of the array of variables. c) Surface, interface, and bottom stress The forcing term F in (42) and (44) consists of the surface stress, the bottom stress and either the atmospheric pressure force due to a surface pressure deficit for the external mode forcing or the interface stress for the internal mode forcing, respectively. That is 24 > > F S Ut oO Th + gDVb, or e s (48) Ho > Hy > > Fis Dos) a (Dior by tae The stress terms are presented in the form > 27° T = «|W|W. (49) => For the surface stress, W is the wind speed at an elevation of 10 m above the water surface. Reid and Bodine (1968) considered x as a function of wind speed in the form lA = > fs C5) for |w| (50) IV = +9 > ra Sy ko (1-W./ |W) for |w| where Ky and Ko are taken as 1.1 x 1076 and 2.5 x 10-6, respectively, and W,. is a critical wind speed which is taken as 7.0 m/s. The coefficient x is related to the drag coefficient, Cp by the relation kK = (P5/ Pw)Cp (51) where Pa is air density and p, is water density. For large wind speed, x approaches the limiting value of 3.6 x 10~© which corresponds to a drag coefficient of about 3.0 x 107° if the density ratio between air and water is assumed to be 1.2 x 1073. Equation (50) was used by Wanstrath (1975) in his simulation of storm surge in transformed coordinates while Miyasaki (1963) used a constant 3.2 x 1076 for x in his computation of storm surge for > hurricane Carla. The choice of x for intense winds (|W|2 50 m/s) is 25 controversial. However, taking into account that wind speeds associated with hurricanes are not steady or uniform, x is taken as a function of wind speed. The surface wind stresses are computed at every time step using linear interpolated positions of the hurricane center which are given at 6 h intervals, except for the case of hurricane Carla simulation which will be discussed later. A constant inflow angle of 20° is used to rotate the surface stress vector before the components are computed. For the bottom stress, W is the depth averaged lower layer current velocity, i.e. > > 3 where [vo | is the magnitude of the depth averaged lower layer current. A constant value of 2.5 x 10-3 is assumed for the coefficient «. The lower layer current, hal is given in terms of the modal velocities by the relation > “> > Vo = (1/D)V, = (1/H5)V5 4 (53) > > where V, and V; are the external and internal velocity vectors, respectively. The bottom stress for the external mode computation at odd time steps is coded as Sil sul Ty = 2.5 x 10-SAt{[(1/D)Ug(I,J) - (1/Hy)U, (1,5) I? =n-1 =n-1 4 -1 + [(1/D)VECT,3) — (1/Hg)Vy (2,3) ]?}2{-(a/ay Uy 1, 5)}. (54) Upon substituting (53) in (52) both U, and U; from the previous time 26 step are used in v5 while only the previous value of U; is used in Tose This implies that the coefficient of U, on the left hand side of (42) has to be changed from a constant value of 1 to 1 - 2.5 x 1073(At/D). However, the tridiagonal form of the coefficient matrix is retained. At even time steps, coding for the bottom stress is obtained by directly interchanging U and V to obtain -3 n-l n-1 2) Tp, = 2-5 x 10° “At{[(1/D)V,(1,J) - (1/H5)V;(1,J)] =n-1 =n-1 15,4 n-1 + [(1/D)U,(1,5) - (1/H9)U; (1,5) ]°}“{-(1/Hg)V5(1,5)}. (55) The bottom stress for the internal mode computation is depicted as -3 n-l n-1l 2 Dey ao xe LOR ALUN (L/D) UNG dy onl /H> URL a) =n-1l =n-1 i n-1 + [(1/D)V,(I,J) - (1/H2)V; (1,5) ]7}4{(1/D)U,(1,5)} (56) = n-1 n-l 2 Dy = 205) x Os ACID) VAGh DE (@/EO VC, oy =n-1 =n-1 2 -1 + [A/D)UR(T,J) - (1/4905 (1,9) 732 {pve} (57) for odd and even time steps, respectively. > The velocity vector W in the interface stress is the internal mode volume transport per unit width multiplied by the appropriate depth > > W = (D/H,H2)Q; (58) A constant value of 2.0 x 107> is assumed for the friction coefficient, xk. The finite difference form of this stress is, for odd time steps, 27 Hi I a =1 =n-1 # on-1 — =n-1 = 2.0 x 107°{[u, (1,5) ]2+[v, (1,3) ]2}2{u, (1,5)+¥, (1,3)}, and (59) | I = 1 x 4, = 2.0 x 10->{[vP(1,5) ]2+[uB(1,5) ]2}2{vB(r,5)+U2(1,5)}, for even time steps. d) The coupling terms The coupling terms are coded based on the premise that the energy transferred between modes due to coupling must be balanced. The energy equations are formed by multiplying (37a,b) and (38a,b) by > > Q,/D, Wer Q;/T;, and evir respectively, and thus the coupling terms appear as = ~egy;0,V(H,/D) (60a) > —egyv0, V(H,/D) (60b) > egwv,Q;V(H,/D) (60c) 2 oD e9V.0,V(H,/D) (60a) For simplicity, consider a one dimensional channel in cartesian coordinates as sketched in figure 4. The solid boundary is at point 1 while there are three possible conditions at point 7; i) a solid boundary, ii) an open port, and iii) an interior point where the interface intersects the bottom. Since the depth, and hence the ratio H,/D, are defined at every field variable location the gradients [#:(H,/D)] are defined at midpoints between U and y locations (refer to points A, B, C, ... in Fig. 4). As an example, 28 Sooo coe Moo Seo oe Fig. 4. Schematic of one dimensional channel with stair-step depth profile. Capital letters A through E indicate locations where the gradients [$(H,/D)] are defined. 29 [$5(H)/D)], = (Hy/D)7 = (Hy/D)¢, (61) where numeral subscripts indicate the points where the ratios are computed. For the external mode computations, the continuity and momentum equations are alternately applied at points 2 through 6, while the internal mode computations start at point 4 in the same Sequence. The summation of the coupling terms for one complete operation, after eliminating like terms, is -eg[Ue]3 F{L¥,],[$(H,/D) 1p} tea vile ${lU.],[$:(H,/D) Ja} (62) -eg vel ${lU; ],($(H)/D) Jp + (0,1, [4;¢H/D) J} -eg ele S{lU,],[$(4,/D) ]q}- The internal mode computations start at point 4 as discussed earlier. Hence the first and third terms in (62) vanish. The residual terms depend upon the condition at point 7. If point 7 is a solid boundary, then [U,]7 = [U,]7 = 0 and the Foe energy transferred between modes is zero. The second possibility that point 7 is an open boundary requires that either the depths at point 6 and 7 are set equal or the gradient, [$.(4,/D) 1, is set to zero. In fact the two choices imply one another. It is more rational to set the gradient H,/D to zero since all these gradients are computed only for the purpose of evaluating the coupling terms. If the interface intersects the bottom at point 7, then the external mode computations 30 are continued until a solid boundary or an open port is encountered, while the internal mode computations stop at point 6. It immediately follows that the last term in (62) vanishes. The coupling term obtained by applying the external momentum equation at point 7 is -eglUs]> F{l¥y]_l$(H,/D) 1a}, (63) which exactly balances the only remaining term in (62). Note that the second part of the coupling term in (63) is omitted since it involves ~;, which is zero. It can be shown from (59a,b) that all the coupling terms are zero beyond point 7. As a result, the net transfer of energy between modes is zero. In summary, with this form of coding, employing the average of the products u[$5(H,/D)] or viS(Hy/D)], the only constraint needed to fulfill the energy requirement is that the gradient of (H,/D) along the points just inside the open boundary be zero. Therefore, the coupling terms, X;, and §;, in the implicit computations of the external mode momentum and mass conservation equations, (41) and (43), are coded as follows: Xi = er, ¥{VR(I-J, 5) ((Hy/D(1,5) )-(Hy/D(1-G,J))] +y2(1, J+%) [ (Hy /D(14+4, J) )-(Hy/D(1,J)) 1}. (64a) t, = de{v, {uR(1-$,5) [ (Hy/D(I,J))-(Hy/D(1-$,35))] +uR(145,5) [(H,/D(1+9,J))-(Hy/D(1,J)) 1} ty {VP CT, J+5) 0(S+5) [(Hy/D( 1,349) )- (Hy /D(1,J)) ] +v2(1, 3-4) 6(I-$) [(Hy/D(I,J) )-(Hy/D(I,I-%)) ]}}. (64b) For the explicit calculations at odd time steps, x,, in (46) becomes Syl Xz = eryz{vP (1,549) 0 (3+9) [ (Hy/D(1, J+9) )-(Hy/D(I,5))] +y2 (1-5, 5) (I-$) [ (Hy /D(I,J))-(H,/D(1,I-4)) J}. (65) Note that the values of all the field variables at previous time steps are used in the expressions of the coupling terms. This implies that the codes for these terms at even time steps are the same as those at odd time steps except for the sequence in which they are applied. Eqs. (65), (64b), and (64a) are the codes employed for implicit computations of V and w and for explicit computations of U, respectively. Sequential coding of coupling terms for the implicit and explicit internal mode computations at odd time steps, are X_ = -gleH,Ho/D(1,J) Ju, 4{y2(1-4,3) [(H,/D(I,J) )-(H,/D(1-4,J))] e 2) x2 Ye 1 1 Fe +y2(1,5+4) [(H,/D(I+5,J))-(H,/D(1,5)) J}, (66a) te = d{v, (u8¢1-$,5) [(H,/D(1,J))-(Hy/D(1-§,J))] +U2(1+3, 5) [ (Hy /D(1+5,5))-(Hy/D(1,J)) 1} tv {VB (1,549) (545) [ (Hy/D(1,J+g))- (Hy /D(1,5))] +v9(1, 5-3) 0(I-$) [ (Hy /D(1,J))-(H,/D(1,J-$))]}}, and (66b) Xe = ~GLeHyHy/D(1,I) ]uyg{v9(1,I+Z) 0 (I+) [ (Hy /D(1,I+9))) -(H,/D(1,3)) ]+v2(1-$,5) 6(I-4) [ (Hy /D(I, J) )- (Hy, /D(1, 3-4) ) }} . (67) The same considerations for the coding at even time steps as discussed above also apply. It is noteworthy that the numerical coding of the coupling terms employed is the only possible form, on the basis of energy transfer, 32 for the spatial-staggered grid system used in this study. Furthermore, the energy transfer is balanced globally, but not locally. In other words, a balance is obtained when considering the entire domain but not at any individual grid block. e) Initial and boundary conditions The model was taken initially at rest. The initial positions of the storms were in the Cayman Sea for the hurricanes of record and for all except one synthetic storm employed in the parametric study which will be discussed in Chapter IV. Specified volume transport or height anomalies are employed as boundary conditions. Along the solid coastal boundaries the normal component of volume transport is specified as zero. At the open grid elements water levels are placed in equilibrium with the inverted barometric pressure for the external mode and zero for the internal mode. 3. Wind and pressure forcing a) Analytical wind model Practically, there are two methods of portraying hurricane wind fields on the computing grid. One method is to digitize the surface charts that are available. The data are sampled at stipulated time intervals and grid points and then interpolated in space and time to provide the necessary information to the model. This method requires detailed surface charts of wind throughout the simulation period. In addition, this is a laborious technique to apply, especially for this 33 study which requires these data over long time periods. The alternative, which is used in this study, is to derive the required forcing fields from a parametric analytical model. Schwerdt et al. (1979) developed a model for surface wind fields associated with hurricanes for the Gulf and East Coasts of the United States which is commonly known as National Weather Service Model, NWS-23. This model was employed to reconstruct radial wind profiles for hurricane Carla. Comparisons between the model results and the observed profiles revealed that this model does not satisfactorily depict the observed Carla winds in the far field. It is remarked that effect of the storm forward speed is ignored In NWS-23 there are two variables, the maximum wind, V,, and the radius to maximum wind, R, that determine the wind speed for a stationary storm. Therefore, in order to obtain good comparison of the winds in the far field, either V, or R, or both have to be changed and this inevitably deteriorates the winds near the center of the storm. Holland (1980) proposed a new model that has two parameters, C and k that independently define the location of the maximum winds and the shape of the wind profile, respectively. The model thus allows the adjustment of wind speed in the far field through the parameter k, without changing R. The gradient wind profile is given as 3 Wy = [Ck(P, - Py)exp(-C/r*)/park + (grt)?]* - gre, (68) where Wg is the gradient wind at a distant r from the center, f is the Coriolis parameter, Pa is the the air density (assumed constant), 34 Po is the central pressure, and P, is the ambient pressure. Holland determined C and k by fitting the pressure profile P =P, + (Py - Py)exp(-C/r*). (69) S However, he pointed out that this approach would underestimate the Peak winds and that wind observations, if available, should be used directly. For hurricane Carla, there were detailed surface charts available. Equation (68) was modified for the direct approach as follows: First, C was eliminated by considering that the cyclostrophic wind 3 We S lexGy = P.)exp(-C/r*)/p.r*] ; (70) was a more appropriate representation of the maximum wind. Upon taking dw./dr and setting it equal to zero at r=R, we obtain the relation R=cl/Kk orc=R, (71) Since detailed surface charts allow a good estimate of the maximum wind and radius of maximum wind, the pressure drop (P,-P.) was represented in terms of the maximum wind by substituting (71) in (70) to obtain 1 Wy = [(k/pge) (Pp - Po) 12. (72) Using (71) and (72), (68) reduces to 7 Wy = [(R/r)*wexp(1-(R/r)*) +($r£)?]? - dre. (73) 35 It should be remarked that for k=l, this wind model is equivalent to the NWS-23 model. Another important point to note is that the azimuthal variation of k automatically results in an asymmetry of the wind field due to the translation of the storm. From the NWS-23 model, this asymmetry has to be taken into account by augmenting the maximum wind speed with the forward speed of the storm and the cosine of the angle that depends upon the location of the Maximum wind relative to the storm path. b) Analytical pressure model The pressure profiles obtained from (69) using k obtained from fitting the velocity profiles were not in good agreement with the observed pressure profiles. This confirms Holland's remark as Mentioned earlier. Therefore, the pressure profile P= PL + (P,-P.)e R/T), (74) which is equivalent to (69) with k=l, was employed in this study. 36 CHAPTER III SIMULATIONS OF HURRICANES OF RECORD 1. Selected hurricanes of record Eligible hurricanes of record for the purposes of this study were considered as those storms for which the historical information required by the numerical model was available. This includes the time series of the central pressure, radius of maximum wind and surface charts of wind and pressure. These charts should cover the entire Gulf of Mexico and Cayman Sea from the time at which the storm center was outside the Gulf to sometime after landfall. Additional important data required includes the water level associated with these storms at stations around the Gulf, in both the United States and Mexico. Of the 26 hurricanes spanning the period 1950-1980 which were examined as potential hurricanes of record, only hurricanes Carla in 1961 and Allen in 1980 had sufficient observations of the forcing fields and the response histories. These are the hurricanes of record used for verification purposes in this study. Hurricane Carla was an exceptionally slow moving storm with an average forward speed of 13 km/h. Figure 5 shows the path indicated by serial positions of the storm's center at six hour intervals. Carla reached hurricane stage at 1200 GMT 6 September 1961 and the center entered the Gulf through Yucatan Strait at approximately 1500 GMT 7 September with a central pressure of 970 mb. As it moved to the northwest, it continously deepened and reached a minimum central 37 "1961 Jequieqjdes TT IND OO8T OF Jequejdes 9 IWD OOZT wWorJ eTIeD euedTIINYy JO yoeIL "g ‘6T4 38 pressure of 935 mb at 1200 GMT 11 September. The time sequence of the central pressure from September 4 to September 13, Fig. 6, was presented by Dunn et al. (1962). During the period 1200 GMT 9 September through 1800 GMT 12 September, the average radius of Maximum wind was approximately 40 km with a slight increase observed. Upon entering the Gulf, Carla had winds of 60 km/h. As it drifted northwest, then west northwest it increased in size with cyclonic winds observed over the entire Gulf. Maximum wind speeds of 82 km/h were observed inland as Carla approached the coast. Carla made landfall near Pass Cavallo at 2100 GMT 11 September and by 12 September it was positioned north of Waco. Hurricane Carla has the distinction of being the best documented storm in history. Hurricane Allen was a fast moving storm compared to Carla. Its average forward speed in the Cayman Sea was 35 km/h. Its forward speed decelerated to about 30 km/h as it moved west-northwest across the Gulf. The center of the storm entered Yucatan Strait at approximately 1800 GMT 7 August, 1980. The hurricane center crossed the coastline near Brownsville at 0700 GMT 10 August. Figure 7 shows the path of Allen from the Cayman Sea until landfall. The time sequence of Allen's central pressure, as shown in Fig. 8, was obtained from Lawrence and Pelissier (1981). There were three cycles of a 50 mb fluctuation in the six-day period from 4-10 August. The first minimum central pressure of 911 mb occurred early on 5 August when the storm was approximately 370 km south of Puerto Rico. The next minimum of 899 mb was measured at 1742 GMT 7 August when the eye just passed through the strait. Allen deepened for the third 39 (mb) PRESSURE 1000 930 980 970 960 950 ° 940 x e 4 5 6 7 8 9 10 mT 12 13 SEPTEMBER I96I Fig. 6. Time sequence of Carla central pressure 4 - 13 September, 1961 (after Dunn et al., 1962). 40 Otzriny Jo yoOeIL ysnBny zt LWD OOZT 03 3SNhnw L IND OOOO Wor UeTTW eUeOTIAINY 086T a a 41 ‘JSTSST[9q Pue BdUeIMeT 3ajjJe) O86T ‘3snbny OT - F eansseazd [Ter}UeD. UaeT{TW JO Bsouenbas oeuTy, LSNONnv 006 OG6 (qu) 3yYunNss3aud OoOOol *(T86T °@ °bta 42 time when it moved toward Texas coast and reached a 909 mb low early on 9 August local time. 2. Meteorological data The Hydrometeorological Section of the U.S. Weather Bureau provided surface charts of hurricane Carla winds and pressure for the entire Gulf of Mexico. These charts were available at 6 h intervals from 1200 GMT 9 September to 1200 GMT 10 September and at 3 h intervals thereafter. Prior to this period, the surface pressure charts that covered North America, obtained from the National Climatic Center, provided observed surface winds from land stations, a bouy and ships of opportunity. Ho and Miller (1980) presented several surface wind charts for the period when Allen was in the western Gulf. The coverage of these charts is limited to the western Gulf only. Despite the lack of surface wind charts however, the time sequence of the central pressure and the available surface pressure charts provided enough information to construct surface wind fields. However, this information was not sufficient to provide the same degree of detail as obtained with hurricane Carla. a) Surface wind for hurricane Carla The hurricane Carla surface wind charts were analyzed to determine the location of the eye and R, W, and k which were required for the computation of wind fields. 43 The position of the eye and the radius of maximum wind were first determined. To account for the asymmetry of the wind fields, k was determined by fitting the observed radial wind profiles along different sectants around the eye. A total of 8 profiles was fitted for each chart. Figure 9 shows sections where these profiles were fitted. In each case, the profile was plotted by digitizing the radial distance from the center to the isovels on the observed chart. The maximum wind speed, W,, was determined from the plot at the distance R from the center. The radial wind profile was computed from (73) by substituting R, W, and assuming k=l. The resulting profile was compared to the observed and k was adjusted to obtain the best possible agreement. Prior to 1200 GMT 9 September, during which there were no regional Carla surface wind charts, the analysis to obtain R, Wm and k depended upon the surface pressure charts from the National Climatic Center. Eq. 68 was rewritten, by using (71), as k Sep! | eee W = [(R/r)*(k/p,) (Pp-Poe + (dr£)2]* - dre. (75) The time sequence of the radius of maximum wind was extrapolated back in time using its relation to the central pressure which is summarized in NWS-23. The radial distances to those points where observed winds were reported, were digitized from the surface chart. The far field pressure was determined from the first cyclonically curved isobar. Assuming p, = 1.15 Kg/m> and k=1, the wind speed computed from (75) was compared to the observed value and again k was adjusted to obtain minimum error. Once k was determined, W, was 44 Schematic showing azimuthal distribution of radial sections where wind profiles were fitted. 9) Fig. 45 computed from (72). Attempts were made to obtain a simple expression for both k and Wy as a function of the azimuthal angle (measured from the north) such that these two parameters could be internally computed. However, no simple relationship could be established due to the complexities of the patterns of azimuthal variation of k and W, which, in addition, varied irregularly for each map time. Therefore, these parameters have to be specified serially. Furthermore, linear interpolation in time to obtain these parameters for intermediate time step computations constrained the values to the same set of azimuths (relative to heading) throughout the simulation period. As a result, an a priori linear interpolation in space was employed to get k and Wm as continuous functions of azimuth. A standard set of azimuths was then selected. Finally, the surface wind fields for each map time were constructed using the interpolational routine employed in the model. Comparison between the computed and observed wind fields was made and final adjustments of k, if necessary, were decided. Figures 10 and 11 are examples of the computed wind fields at 0000 GMT 7 September and 1200 GMT 9 September, respectively. The solid circles indicate the observed wind speed from the surface charts. b) Surface wind for hurricane Allen The inadequate coverage in both space and time of surface wind charts for hurricane Allen aborted attempts to fully analyze these charts as practiced in hurricane Carla. Therefore, NWS-23 was used 46 *qyaeyd adejyins ay} wo1y peads putM peArtesqo ay} BJeOTPUT SETIITS PTTOS syL S,PUBTIOH WOIJ pauTeyqo se ‘T96T Aequajdas 4 LWD 0000 3e (S/W) PUTM BdezJINS ered auedTIINY B[ieg auBolwiny * Tapow “OT °6td 47 *4yaeyd adejzins ay} 98y, woIzZ paads putmM paArasqo ayy ayeOTpuT SaTIITS pT{os suL S,PUETTOH WOIzZ pauTeqqo se ‘T96T JEquiejdes 6 IND OOZT © (S/W) PUTM eoezINS eTIeD sueoTIINy Ble) sueOoTIIN]{ * Tepou “IT ‘6ta 48 to determine R and W, from P. which were required to construct the wind fields. The graphical relationship between the radius of Maximum wind and the central pressure was analyzed using simple linear regression to establish a functional relationship. The radius of maximum wind was then determined. The maximum wind speed for a stationary storm Wns is given as 5 Z hg = OolCY Pa) ChSya)> = Cske/) (76) Since there were no surface charts to analyze for the parameter k, a constant value of 1 was assumed. As discussed earlier, the asymmetry of the wind field was achieved by augmenting the maximum wind speed by Hy @ Tha Mos@r O82!) Gost eaaq, (77) ms where o was the angle pees track direction and the surface wind vector and To. = 0.514791 for wind speed in m/s. The track of hurricane Allen was approximately 285° (cf. Fig. 7, p. 41) relative to true north. Assuming that the maximum wind occurred at right angles to the right of the track, then o was zero at 25_ relative to true north. Eq. (78) was used to compute the maximum wind speed for each grid point (I,J) and thus the surface wind speed at all computational points can then be evaluated. 49 3. Tide gauge data A total of 9 and 13 tide gauge stations for hurricane Carla and Allen, respectively, were chosen to provide the observed response in the Gulf of Mexico. Figure 12 shows locations of these tide gauge Stations. The Tides Prediction Branch of the National Oceanic and Atmospheric Administration (NOAA) and the National Autonomous University of Mexico (UNAM) provided hourly water level for a period of several weeks before and after the hurricanes. Filtered versions (using a 40 h lowpass filter) are shown in Figs. 13-15 for some selected series obtained during hurricane Carla. These filtered plots essentially remove the tides. The arrows in the figures indicate the time at which the center of Carla entered the Gulf through Yucatan Strait. There were indications of a gradual rise of water level well before the peak surge, a possibility of forerunner surges at all 9 stations. Note also the degree of background variation unrelated to the hurricane several weeks prior to the hurricane. The same analysis as applied to recorded water level for hurricane Carla were aoe for the records obtained obtained during hurricane Allen. Figures 16-18 show the filtered data at some selected stations. The presence of an initial rise in water level is again observed at all stations. The lowpass filtering illustrated above is known to smooth out and broaden the peak surge. To properly remove tidal signals from recorded data by the harmonic method requires a suitable number of constituents to insure proper phasing. Unfortunately, only the Key 50 *Apnys Otrjewesred ay} ut suotzeq4s ebneb apt} [Tepow ey} se OSTe pue uaT[yW eUeDTIAINY JO SUOTJETNHWUTS By} UT PesSN |aSoUu aie saToIzO uedo pue pT{OS 9y} YOM “*eTIeD suedTIINY JO UuOTJeTNUTS By UT PesSN SUOT}eYS |BSOY a}eOTpuT SaToITO PTTOS eYyL °oOSsatbOig ST Od ‘UBWIeD ST WD ‘ZNADeTaA ST DA ‘OTSPeW ST GW ‘TEeqeSI 310q ST Iq ‘UOJSaATED SJ AD ‘ATSI pueiH Sz IH ‘epTooesueg st gq ‘eTOOTYyDeTedy st dy ' Asay aepay Sf yD ‘Hangqsiaajag *4S S} dS ‘SsaTden St dN ‘350M A®y ST MN “°ApPN]S Sty} UT PesN SuOT}eIS abneb apt} jo suoqzyes0y ‘zt “bya 51 *39}eM MOT Uke abNeS st wnqzep syZ °}TeI9S ueJeONZ ySnosy, JIND 2y} peteque eTIeD YOTYyM 4e suUTQ aU} S@JLOTPUT MOTIE BUL “T96T ‘TILED eueOTIINY HuTINp uoJseATeD Je STaABT Ja}yeM PaAtesqo “ET “ht Jaquiaydas ysndny 0€ gt 1 Ie SI J (UI) yUSIEy uojseaajesy) u0T}e YS efile) aueolwiny 52 *7a}eM MOT UeoW Ohne ST unjep syuLZ °4TeIWS Ue_eONA ySnory, JIND sy perejue eTIeD YyYOTyM ze suTQ 8y} SAYLOTPUT MOTI BYL “T96T ‘eTIeD auedTIINYy Hutinp Teqes] jog je STaAST 3Ja\eM PaeAtesqg ‘FT ‘hia ysndny Ayn OE St bit GI i (uw) qYSTOH [eqes] y1og uoeyS ejie) aueoliny 53 *JajeM MOT UeeW abneb st unqep ey, *3yTeI4¥S ueVeON ySno. JIND ey} pareque eTIeD yOTYM Je suUTQ ay} SAJeCOTPUT MOTIe aUL “T96T ‘eTIeD euedTIINY Hutinp sem Aay ye STeASeT JayeM peATesqg °ST °btg Jaquiaydas ysnany oe SI L1e GI 1 (wi) 34918H ysom Aay uote ys elle) aueoliny 54 *J9]eM MOT UeaU BHneb st unjep sy °*31TeIWS UePeONA ySnoIry, JIND Sy} perejue usaT Ty YOTYM 4e sUTY 8u SAROTPUT MOTIL ABUL “OST “UST[y GUeDTIAINY ButTinp uoxsaaTey ye STaAVBT JayeM peATasqg °*9T *btA ysnsny Aine le SI Lie SI (ut) yUdTOH uoysaatey :u0lVe4S ua|[V eueoliny 55 *Ja}eM MOT UeoW eHneb st wnjep syL °3tTeIqS UeZeONA YyhnoIy, J[NH sy} psreque uaT—TwW YOTUM ye suUITQ 8YU} S8QLITpPuT MOITTe PUL “°OR6T ‘UPTTW BUeDTIINY buTAnp Teqes] 310g Je STeAeT JajeM PeATesqQ °/T °btg Jaquiaydas ysnony Ie SI 1 1e SI 1 (u1) 7YUsIOH Jeqes] og :u0Neis uPs|[V eueollinyy 56 *y1eI135 ueJeoONA ySnosry JIND ayy pereque useTTW YyOTYM 3e SUT *J9a7eM MOT UPOW obneb st wnjzep oyL ay} SezeoTpuT more ayL “OS6T ‘UETTY euedFIINYy butinp Sem Kay ye sTeAeT JazemM peATesqg “gt “btd ysndny Ayne 1€ SI 1 Ie gl J an wo ga Sp co B ysom Aay :uolzeys ual[V eueollIny 57 West and Galveston stations have enough information for the actual computation of the tide. Therefore, at each station, only the mean water level during the period considered, excluding the peak surge, was removed from the raw data. Thus the recorded hydrographs which are compared to those computed contain the tidal signal, but its amplitude is small and does not unduly complicate the comparison. 4. Simulation procedure The initial model wind and pressure fields were increased in amplitude linearly from zero to their actual initial values over a 48 hour prototype time period. During this period Carla was kept stationary at 19.04°N and 85.15 W which was its position at 1200 GMT 6 September. The simulation was carried out to 0000 GMT 13 September, approximately 28 h after landfall. The simulation for hurricane Allen starts at 0000 GMT 7 August and ends at 0000 GMT 12 August. The initial position of the eye of hurricane Allen during start up: was at 20.10°N and 81.90 °W. At each time step the east and north component of wind stress and the atmospheric pressure (the inverted barometric height) were computed at each grid point. The sequence of wind and pressure computations consisted of two linear interpolations. First, the position of the eye and the other parameters at each time step were linearly interpolated from two appropriate sets of values (6 h apart). The radial distance from the eye to each individual grid point (I,J) was computed. The inverted barometric height was then computed from (74). The angle between the line joining the eye to 58 the grid point (I,J) and true north, @, was determined. The pair of azimuthal angles that embraced @ was sorted. The values of k and Wn associated with the two angles so determined were used in the linear interpolation to obtain their values at grid point (I,J). A constant inflow angle of 20° was assumed in all simulations except Carla for the decomposition of the wind speed before computing the wind stress components. The computed water level at Galveston during the early stage of Carla simulation using constant inflow angle was lower than the observed. Careful examination of the surface wind charts at 1200 GMT 9 September revealed that there was a region along the Texas shelf where the cross shelf wind reversed direction indicating a negative inflow angle. An example of an analyzed wind map given by Miyasaki (1963), also shows negative inflow angles along the Texas shelf. Therefore, the inflow angle was allowed to vary as a function of the radial distance using the empirically determined formula 01 for r < Rj, © = (78) @5-(r-R,)*0.5*A/r for r > Ri, where A is a maximum inflow angle, 20m, Ry is the radial distance at which the inflow angle was zero, and @, = [(R,/r)AZexp(1-(R,/r))+(Sr£)2] - gre, Q5 [(R,/r)% A*exp(1-(R,/r)*)+(art£)?] = $rf, 59 K = r/50 . Figure 19 shows the adopted inflow angle profile. This radially varying inflow angle was applied only from the begining of the simulation to 0000 GMT 10 September. A constant inflow angle was resumed after this period since there was no other evidence of negative inflow angles on the remaining available surface charts. Results of the simulation were sampled every 24 hour prototype time. These included digital fields and map plots of the height anomalies and currents for both modes, and surface currents whose values were retrieved from the modal currents. At the end of the simulation, the computed and observed hydrographs at the tide stations were plotted. Contours of the peak surge on the continental shelf for the entire Gulf of Mexico and for the Texas-Louisiana shelf were also plotted. 5. Results of Carla simulation Figures 20 through 28 cher tha computed (solid) and recorded (dashed) hydrographs at the stations used in this study. The overall comparisons are fair, especially during the first half of the simulation period. There is not much activity in the southwestern portion of the Gulf (Campeche Bay) as revealed from the hydrographs from the three Mexican stations (Figs. 26 through 28). The simulated hydrograph at Key West has a slow oscillation with a period of approximately 7 days. The mean of this long period oscillation lies close to the mean of the recorded water level. This signal is also observable in the St. Petersburg computed hydrograph. However, the 60 *(gL) worz payndwod satbue MOTJUy ayy JO attjoid tetpeu ‘61 “614 oot- (UI) aoueysiq [elpey ae 0002 OO6S OOBS OOLT OO9T O09T OOFE OOET OO2T OOTS OOOT 006 O08 O0L 009 a) og (to) § ost 002 ose (Saeidap) ajsuy MoTjU] 61 et eTieD suedtziny butainp et *T@AeT Sas ueeUl ST WNRep xu °T96T Jequeydes ET asem Avy ye sydesboipAy (peysep) peArasqo pue (pt{[oOs) peynduoy °*0z Jaquiaydas 1 ot ydeidsoipky 4seM Aay Bjseg aueoliny - 9 *bTd (ul) 2UsIEH 62 eTaeD) aueotaziny butinp bangs 1ejed el O'e- al 2 o€ OF asc et ia -35 3e sydeiboipAy (peysep) peAresqo pue (pTtTos) payndwod or *TaAaT eas uesW ST WNZeEp SUL “1961 Jequeqydes Et - 9 Jaquiaydas 6 "1? °btd 8 L 9 ™ oO} GQ iy co 3 ydeiso1pAyH dingssayed 1S elle) suedoliuny 63 el *[@AeT PaS UeAU ST UNEP SeYyL °196T AJequiaqdas ET eT Ie) euesTIINYy butsinp epTooesueg ye sydesborpAy (paysep) peAresqo pue (pT{Os) peynduod °7Z laquiaydas ZI 1 oT 6 8 ydeidoipAH ejooesuad ejie) sueoltuny = 9) “Big (ur) 3USTEH 64 el erie) aueoTIIny et *[aAaeT eas ueaW ST WNIeP eUL “T96T Aequwaydas HutInp a@[S] puesy9 ye sydeszborpAy (peysep) paArasqo pue (pt{os) pe ynduoD Jaqwajydas VW ol 6 ydeisoipAH S| puesy ejieg aUuBoLIIN}Y] €I - 9 "€z °bta (wu) YUSTIOH 65 “T@AaT eas uPaUl ST WNJep eYuL *T96T Jequieydes ET - 9 eT Ie) esuedtainy buyanp uoxseatey ye sydeiboipAy (paysep) peAtesqo pue (PTTOS) pe ynduop Jaquia}das 21 VW ol 6 8 i, "ye *bta (wi) WYUsIEy ydeidoipA}] Uoysaayey epiewy auvolqinyy 66 *T@aAeT eas UPAW ST WNIeP sUL *T96T Jequajdas ET - 9 ep ze) auesTssny HuTInp Teqesr 310g 3e SydesfospAy (peysep) peAtesqo pue (ptT{oOs) paynduod “sz “bt laquiaydas a ot 6 8 ise) “~ N = (1) qUsTEH ydeisoipAyH jeqes] 310d e[ueD IUBOIIIN] 67 “T@eAeT eas ues ST UNJEP aYyL “*T96T Jequeydes ET epieD euedtziny bButanp znzsezepq ye sydeirboipAy (peysep) peAtasqo pue (PTTOs) peynduod *92 Jaquiajydas ZI ra ot 6 8 L ydeisoipAH ZnJjoela,p elle) aueoliiny (w1) WUSsIEH 68 et *TaAeT BAS Ubu ST UNJeP SUL *T96T Jequieydes ET - 9 eT ieD suedtIiny Hbutinp ueurzeDy ye sydesborpAy (peysep) padAresqo pue (pt{Tos) peynduoy “77 *btg Jaquiaydas zt VW oT 6 (Ww) qUSIEH ydeidoipAyH usulses elle) aueBoliiny 69 eI *TaAeT ees uUeAaUI ST UNEP ABUL °T96T Jaquaydes ET - 9 elie) auedtziny butanp oseaborg ye syderboipAy (paysep) pearasqo pue (pT{os) pa ynduod Jaquiaydas et im} or 6 8 L 9 ydeidso0ipAy] osaidoid eyjleg auBolIny "8c ‘ta (1) qUSTOH 70 computed water level at St. Petersburg is higher than the recorded throughout the period. This might be due to the fact that this tide gauge station is located inland and not on the open coast. The departure of the two hydrographs at Grand Isle is probably the result of resuming the 20° inflow angle which shifted the wind direction from the alongshore or slightly onshore to slightly offshore. The comparison for Galveston is better than that obtained by Miyasaki (1963), presumably because of the special care to match the observed wind fields for the northwest Gulf Coast. At Port Isabel, the discrepancy, which resembles that at Grand Isle but with a larger deviation, is possibly due to the difference in locations of the tide gauge station and the point where the computed water level was sampled. The actual site of the tide gauge station is inside a semi- enclosed embayment, as shown in Fig. 29. As a result, the effect of the offshore component of the wind, which prevailed at this station from approximately 0000 GMT 10 September onward, is limited while the southerly component produces a set up within the constricted lagoon at the gauge site. On the contrary, the computed hydrograph, sampled at half the grid size away from the digitized coastline, is subjected fully to the wind draw-down. Miyasaki (1963) obtained a similar result at this station. Despite these discrepancies, the primary interest is the comparison at Galveston where the largest surge, among all sampled stations, occurs. It should be pointed out that the maximum computed water level is not at Galveston but at the grid point (8,42), located one grid block northeast of the path of Carla at the coast close to 71 Port Isabel Tide Gauge Jo Station esas a4, Ai “3 & Fig. 29. Location of Port Isabel tide gauge station which is sheltered by south Padre Island. 72 the location of Matagorda Bay where the highest high water marks were observed for this hurricane. The agreement is good over the entire simulation period at Galveston. The effect of the reversing inflow angle in the simulation is clearly pronounced as the computed water level suddenly drops at approimately 2200 GMT 9 September. This result is somewhat out of phase with that indicated by the observations. Figures 30-32 show instantaneous fields of the barotropic and baroclinic height anomalies and surface currents at 1200 GMT 10 September. Contours of the barotropic height (Fig. 30) clearly demonstrate the inverted barometric effect around the storm center. Upwelling along the hurricane path, a feature first investigated by Leipper (1967) in the wake of hurricane Hilda in the Gulf of Mexico in 1964, is noticeable in the baroclinic height field shown in Fig. 31. It should be remembered that the negative contours of the baroclinic height anomalies correspond to upwelling of the interface. Geisler (1970), tn his linear analytic model, found that the baroclinic response to a moving hurricane consisted of both upwelling and inertio-gravity waves in the lee of the storm. Figures 31,32 strikingly portray this wake oscillation. Chang and Anthes (1978) carried out numerical experiments to investigate the character of this wake produced by various types of atmospheric forcing. They found that the wavelength is longer for a faster moving storm and, for the same forward speed of the storm, is shorter at higher lattitude. In addition, within the limit of 50 h and 1200 km time and space scales employed, the $-effect (variation of f) did not 73 *196T Jequieqdes OT IND O0ZT 3 eTIeD SueSTIINY IOJ (SJezewW) pPTeTz ATewoue AybTey otdo1z,01eq peynduod "o€ *5t4 74 75 Computed baroclinic height anomaly field (meters) for hurricane Carla at 1200 GMT 10 September 1961. Fig. 3l. *196T Jaquieydes OT iLWD. 00cT ze eT Ie) i ‘ ‘ 4 1 i) ‘ . . ° . aueoTIIny JoJ (S/ud) WO199A UNWI XEN — "002 con COCO COSC SS SN8 et teem ewe Ke NNO eee HESS See wes SO pODSSS a Sr PTetzJ JUeTIND adejzans paynduoy Vrwercrctcccce er Sseonecesoungn6gcoc ro eaery www eer em em emnw nee Sal eat Ae 1 e we eee emer ese "ce “td 76 alter the results. For an asymmetric storm moving at 18 km/h, which is comparable to the translation speed of Carla, in an idealized f- Plane basin, the wake oscillation has a wave length of approximately 420 km. The average wave length obtained from this Carla simulation is approimately 360 km. 6. Results of Allen simulation The results of the Allen simulation show a synchronous oscillatory signal with a period of approximately 28 h in all of the Simulated hydrographs. The amplitude of this oscillation is approximately 20 cm. The existence of this in phase signal in all stations around the Gulf is indicative of a Gulf-wide, Helmholtz mode superimposed on other forced modes. Consequently, an ad-hoc code to compute the average water level in the Gulf proper (designated as 7) at each time step was added. Figure 33 shows the time series of U Tet The solid line represents ng computed by averaging water levels at each grid point in the Gulf, and the dashed line is 1G computed from the continuity equation using the difference of volume transports at the Florida and Yucatan ports. The two curves are practically coincident and very well matched with the signal present in the individually computed hydrographs mentioned above. As it can be seen in Fig. 33, this mode persisted throughout the entire simulation period. The agreement between the simulated and observed hydrographs (Figs. 34-46) is comparable to that obtained in the Carla simulation. This is surprising since less information regarding the atmospheric 77 *T@aAeT eas ues ST UNMQep sy ‘uUOTIeENba AjtTNuTUOD ay} WOT pejnduod st suTT peysep eyL ‘days aut} yoke 3e JIND oy} ut Jutod ptib AqeAe wOTJ STaAeT 3a}eM HuTbereae Aq peynduod st aut, ptTOsS eyL “U®eTIY eUueOTIAINY IOJ Du ‘ooyxen JO JIND ay} UF STeAeT JazeM OHeTaAe |vyY JO Saduenbes sutL “EE “bt4 ysnany (OVLG) UdeadorpAy] WOD Ud][Y VUBOLIIN|] (W1) WUSIAY 78 *TaAeT eas ueaW ST WNIep aL “OR6T 3sNhny ZT - L uat[y euedtsziny butinp 3sem Aey ze SydeiborpAy (paysep) paaresqo pue (pt{os) peynduoD ‘pe “Sta ysnany uw) q4SIEYy ( ydeido1pAH 1sey Aey Uud|[V euBOLIuN} 719 UeTTY euUeoTIINY *TaAeT eas ues ST WNJeEP eYyL ‘OR6T 3SNbnw ZT - butanp setden ye syderboipAy (peysep) peAtesqo pue (PpTTos) pe ynduioa ysnany ydeid01psH Safden ua|[V FuUBOIIINY L "se ‘5y4 (WI) 34518 80 "OS6T ysnbny ZT - L *T@AeT eas uve! ST WNJeP 2Y4L “Big UeT IY aueoTIINy butinp bangsizejzeq *35 ye sydesboipAéy (peysep) PeAresqo Pue (PTTOS) paynduod °9€ ysngny (WI) qYUSTEH ydeidoipAyH dingstayad 1S Uuda|[V¥ FUBOIIINH 81 UeTTY eueoTIINY H5utanp *T@AeT eas ues ST WNAIep SUL “O86T asnbny zt - L Kay aeped je sydeiboipAy (peysep) pearesqo pue (ptTos) payndwod °1e ‘6t4 ysnany ydeidsoipApy Aay Iepag udaT[V euBoIIINY 82 (ut) 74dIEH *yTaAeT eas ueauU ST uNJep oUuL "°OSET SNHny zt - L UeTIY eUeoTIINY butinp eTootyoeytedy ye syderboipAy (peysep) peAresqo pue (pT{oOs) peyndwoD “gE “bta4 ysnany (1) qUSIEH ydeidsoipAyH ejooryoeredy Ud|[[V eUBOIIINY 83 °O86T 3Ssnbny ZT - L *TaAeT eas ueoU ST WNJep sUuL °6€ °bHta UaTTy eUeSTIINYy butinp eTooesueg ye syderboipAy (paysep) paazesqo pue (pt{Tos) pa ynduoD ysnsny ydeidoipAy ejooesuag ua|[V¥ FueolIaNnY (u1) UdTAH *[@AST BAS ues ST UNJEP euL °OSET 3SNHnw 72T - L UeTTY eueoTIINY Hbutinp sTsI pueiy ye syderboipAy (peysep) peArTeasqo pue (ptT{os) paynduioy ‘Op “bts ysngny ydeidoipAy sy] puesy Uds|[V AUBOIIUN}Y at Oe- One UueTTW eueoTIINYy butanp iT *TeAeT eas ues ST wN}Zep sy “O86T ASNbny ZT - L uo}SeATED Je SsuderborpAyu (peysep) peaAresqo pue (PpT{OS) peynduod “Ty °bHta4 ysndny ol ydeiso1pAH Uoysaaye Ud|[V euUBOTIIN}] (m1) qUSIEH 86 *T@AeT BAS uUeauU ST WNJep VyuL ‘“OS6T 3sNbny ZT - L U2®TIY eUeoTIANY SuTINp TeqesrI 310d ye sydeiborpAy (peysep) peaiasqo pue (pT{oOs) peynduod °7zp “bt ysndny (Ww) yYsIEH ydeido1pAyH [aqes] 310d ua][y euBolunyY 87 ueTlW eueoTriny *[TaAeT eas ueawW ST UNJep ayy, “OR6T 3sNbNy ZT - L butinp oJepew ye sydeaboipAy (peysep) peaArasqo pue (PTTOS) pe ynduod “Ep *bt4 ysnony ydeisoi1pA}] olapeyp Uuds|[V FUBOIIINH (W1) yYUsIEyH 88 U®eTTW euedTIInNYy butianp *TaAeT eas ueaWw ST WNAep PUL “OS6T 3SMHny ZT znioeiaA ye syderboipky (pausep) peAresqo pue (pt{Tos) paynduod “FF ysnsny ydeisoipkH zn1sela, ua|[V eUuBoLIINY L BT (ui) qYSTEH 89 UuaTTY eueoTIINY butanp °O86T AsNbnw 7T - L *TSAST bas ueeu ST UINJeEp sUL "Gp “Sta uowrey ye syuderboipAy (paysep) peArasqo pue (pt{os) pe ynduwo| ysnany ydeiso1pAy uaulie) uaT[V eueOIIINY (WI) WYsTEY 90 et oe- O.- 00 oe o€ oF os *TaAeT Bas UeAW ST WN ep ay, uaTTy eueotiiny butainp osaiborg ye sydesborpAy (peysep) peAresqo pue (ptTos) pe ynduod uT ot jsndny 8 ydeidoipAH osatidolg ua|[yV eueOTIIN} “OF °O86T 3snbny 7@T - L ay on ar | . (UI) WYUSIOH 91 forcing is available for hurricane Allen than for hurricane Carla. Hydrographs from the Mexican stations (Figs. 44-46, p. 89 - 91) reveal almost no response except for ng for the entire period considered. The computed water level at Progreso shows a response to the barometric pressure drop during the passage of the center of Allen between 1800 GMT 7 August to 0006 GMT 8 August. However, the observed hydrographs have no indication of this perturbation. This circumstance also occurs in the simulation of hurricane Carla (cf. Fig. 28, p. 70). It is not yet appreciated why the Mexican coast is passive to these two hurricanes. The slow oscillation found at Key West and St. Petersburg in the Carla simulation is also present. It is noticeable again at all stations from Key West to Appalachicola. The maximum computed water level occurs at grid point (4,33) which is the location of the sample point for the Port Isabel hydrograph. This simulated peak surge is of the order of one meter higher than the recorded peak. This is to be expected considering the difference in locations of the simulated and actual hydrographs (cf. Fig. 29, p. 72). In addition to reduction of the magnitude of wind speed at the actual tide station by South Padre Island, the volume responses inside the Laguna Madre will be small because of the constricted opening. These factors contribute to a higher computed water level at this station. A slower retreat in the observed hydrograph is due possibly to the trapping of water inside the embayment . 92 The contours of modal height anomalies and surface currents at 1200 GMT 9 August are shown in Figs. 47-49. The inertio-gravity waves in the baroclinic height anomaly (Fig. 48) seem to be undetectable. Examination of this field at later times indicates that only one-half wavelength of the wake oscillation is seen in Fig. 48. This agrees with the experimental results of Chang and Anthes (1978) showing that the faster the storm moves the longer the wavelength of the oscillation. The approximate half wavelength of 350 km is measured in this hurricane Allen simulation. The existence of 1G in the simulated hydrographs for Allen prompted a repeat simulation of hurricane Carla with the added code to determine ng. Figure 50 shows the resulting time sequence of 1G for Carla. The amplitude is smaller than the one associated with hurricane Allen and the averaged period is about 24 h. The 1G signal obtained from the Carla simulation once again shows up simultaneouly in the individual hydrographs at stations around the Gulf. It is important to note that there is a correlation between the 1G signal and the transport through both Florida and Yucatan Straits. Figures 51 and 52 show the time series of the volume transport through Florida Strait (FS), Yucatan Strait (YS), and the total differential volume transport (unlabelled) as obtained from the hurricane Carla and Allen simulations. The striking feature of the differential transport is the periodicity. The average periods of 24 h and 28 h estimated from Figs. 51 and 52 are exactly the same as the period of their corresponding Ng Signals. In both cases,the 1G Signal lags the net transport by 90° in phase. The first maximum of ysnbnw 6 LWD OOZT © UeTTY eUeoTIINYy OJ (SJajeW) pTetzy ATewoue yyStey 2tdoz,01eq paynduos *O86T ‘Ly °Bta 94 95 Computed baroclinic height anomaly field (meters) for hurricane Allen at 1200 GMT 9 August 48. Fig. 1980. °O861T 3snbny 6 LWD OOZT © UeTTW eUeoTAINY JOJZ (S/WI) pTeTy JUeTAND svoezJins pajnduoy ‘6p *bHtg WOLDJA UNu) KH -_ “002 Oo 0 ene vee WVSQGo5e See PCI LIN VSSSSS See OAPORINN NAS SS scoocdf OF IINUVIO006 NI Naa SS - eesseneaeneneBeee Faas Arreass ‘ ’ - - ¢ / 4 d ‘ ‘ ' . SSeS -, VN CLL < i f df CF OSD DOD SS Ce AAA ae > OS DRS NON ce cee aZ 96 ‘elie auedTsIIny Jo uotze[nuts *T@AVT eas uke! ST wWN}ep syL "OS °bta ay} WOoIJ peuteyqo “9% ‘oToKeW JO JIND ey} UT [eAeT JeqeM pebersae |9y} JO Bouenbes osutL Jaquieaydas €T el WT ol 6 8 WYSTOH ) (uw (DVLA) Udesdo1psH WOD elie) auBoliiny 97 *(JIND ay} JO yNo Jay IO ayy pue ojUT eUO) KNTJ 310d eseyd-jo-3no 0} puodseri0. Sx pue Sq JOJ ubTs ewes ey} ey 9I0N *y10dsue14 AUNTOA [ETUETaJJTP Jou ey} ST BAIN peTTeqetun syL “eTIeD sueoTsrany Jo UOT}JETHUTS ey} WoIZ pauteqqo s3te13s (SA) ueqeonA pue (Sd) eptzoTaA ybno1y4 qyaodsuei} auntoA jo saouenbes outL “1S °6t4 Jaquiaydas eT el TT ot 6 8 0 0v- 0; 0€—- S) ° x ' S gOTX XN14 S Sul ( 0°02 0'0& Oo OF efile) aueoliiny 0°0S 98 *yz0dsues3 AWNTOA TETJUBIBJJTP JOU BY ST BAINO paeTTeqetuN vy “UseTTW BUeDTIINY JO uoTJeTNWTS ayy WoT peuteqyqo s3teiqs (SA) ueqzeonx pue (Sq) epTIOTA ybnozy, ZIOdSuei} suNTOA Jo Saduanbes sutTyZ °7zsg °bHtgY ysndny ocl Our oor 06 0'8 OL 0';9 OG 0°0S— 0 Ov- O'-OE- 0-0e- SOROS 0:0 oor 0-02 OOF us[]V IGUBOLIINY 00S 99 (S41) gOIX xnIY the net periodic transport produced by hurricane Carla is approximately 5 x 10& m/s as compared to 12.5 x 10 m?/s generated by hurricane Allen. The 1G Signal reaches the first maximum of about 0.1m for hurricane Carla and 0.35 m for hurricane Allen which are in conformity with their respective net periodic transports. These results are indicative of the generation of 1G by the net differential port forcing (volume transport). 100 CHAPTER IV PARAMETRIC STUDY The purposes of the parametric study were to obtain responses in the Gulf of Mexico to different forcing from hypothetical storms and to investigate those cases where the forerunner surges might be generated. 1. Selection of paths A total of 5 paths, designated as PATH1 to PATH5, for which the model storms would traverse the Gulf was selected. The first four paths originated in the Cayman Sea and entered the Gulf through Yucatan Strait. Sequentially, the locations where these four paths crossed land were in the vicinity of Corpus Christi (PATH1), Sabine Pass (PATH2), Burrwood (PATH3) and Apalachicola (PATH4). The last path (PATHS) started at 25°N and 81.5 W (overland in Florida) and made landfall at Corpus Christi. All of these paths were great circles as shown in Fig. 53. The first and second track (PATH1 and PATH2) were similar to those of hurricanes Carla and Allen. The third path (PATH3) was similar to hurricane Camille's (1969) track while the fourth path resembled the track of hurricane Agnes (1972). The last track (PATH5) which is rarely observed in nature, was selected as a special case. 101 *Apnqys OF13e Fajowered sy} uy peAoTdua sw10jys DFAZayQUAS AOJ Syed pajod aptas "€S “Sta 102 2. Hypothetical storms The hypothetical storms were assumed to be characterized by the NWS23 parametric model with time invariant radius of maximum wind, translational speed and central pressure deficit. Three constant forward speeds of 15, 25 and 35 km/h were adopted. The slowest one was comparable to the average forward speed of hurricane Carla while the fastest speed was approximately the translation speed of hurricane Allen when it was in the Cayman Sea. Two radii of maximum wind, 30 and 60 km, were selected to account for the scale variation of the atmospheric forcing. The radii were kept constant for the entire simulation period. The range of radius of maximum wind for Gulf Coast hurricanes of record as summarized in Schwerdt et al. (1979) was approximately 10 to 60 km. This range of radii applies to those storms making landfall between Port Isabel and Apalachicola, which covered the landfall locations of the five selected paths. The large radius is at the upper limit and the other is about the average of the observed range. The approximate dimensions of a grid block are 25 km x 27 km and, therefore, sets a lower limit on the radius of maximum wind which can be used. Storms with a radius of maximum wind smaller than the grid size would not be resolved and the associated wind field would be highly distorted. The last characteristic considered was the pressure drop (AP =P,-P.)- It is known, however, that there is a nearly linear relationship between the peak surge and the pressure deficit, other Parameters being held fixed (Jelesnianski, 1972). Therefore, a 103 constant value of 80 mb pressure drop was assumed in most of these parametric simulations. Nevertheless, one of the sensitivity tests, which will be discussed later, was designed to verify this relationship. Table 1 summarizes the possible combinations of these parameters for the five selected paths including their designated hurricane names for further discussions. Runs were made for a representative subset of these model hurricanes (those identified by an * in the last column). This subset allows one to examine responses for different paths for average hurricane parameters, as well as to determine effects of forward speed and effects of scale. 3. Simulation procedure The start up process for all parametric simulations is the same as in the simulations of hurricanes of record. The model calculations start at 0000 h on day 1 but end at different times depending on the path and forward speed. However, in all cases the computations proceeded to at least 24 h after landfall. Wind and pressure forcings are again computed at each time step. Parameterizing the pressure drop instead of maximum wind speed (including assuming k=1) simplifies the wind stress computations by eliminating linear interpolations in space (azimuth) and time to obtain k and W,. Equation (75) is used to compute the wind speed which is then augmented by the forward speed to account for the asymmetry in the wind fields. The only information required to compute the forcing is the position of the storm center which is 104 Table 1. Characteristics and designated hurricane names for the adopted hypothetical storms for parametric study Radius of Forward Path maximum speed Name wind (km) (km/h) 35 HUR2 * 30 25 HUR1 * a5) HUR3 * 1 35 HUR5S * 60 25 HUR4 * ES HUR6 * 35 HUR8 30 25 HUR7 * ab) HUR9 2 35 HURLL 60 25 HUR1O 15 HUR12 35 HUR14 30 25 HUR13* 15 HURL5 3 35 HUR17 60 25 HUR16 ALS) HUR18 35 HUR20 30 25 HUR19* 15 HUR21 4 35 HUR23* 60 25 HUR22* aS) HUR24* 35 HUR26 30 25 HUR25* aES HUR27 5 35 HUR29 60 25 HUR28 ES) HUR30 105 given at 6 h intervals. The code added to compute 7, is retained. 4. Results of parametric simulations The simulated responses of the Gulf to the synthetic storms listed in Table 1 are presented in this section. Discussions of the results are separated into three parts. First, general results common to all simulations are briefly discussed. The simulated hydrographs obtained from each simulation and other results pertaining to the development of forerunners are highlighted in the second part of the discussion. The last part contains results from related simulations. a) General results The inertio - gravity waves in the baroclinic height anomaly fields are found in all parametric runs. Figures 54 through 57 show contours of baroclinic height anomalies and the surface current fields obtained from the HUR2 and HUR3 simulations. Increasing the forward speed of the storm resulted in increasing the wavelength and decreasing the width of the wake as seen in Figs. 54 and 56. This is in agreement with the results of Chang and Anthes (1978) and Geisler (1970) regarding the wavelength and the width, respectively, of the storm generated wake. A dominant long period oscillation is found in the hydrographs from all stations on the Florida shelf. Estimated periods of this oscillation are on the order of 4 - 6 days. Marmorino (1982) analyzed sea level records from Key West to Pensacola recorded during 106 107 Computed baroclinic height anomaly (meters) field at 0000 h on day 5 generated by model 54. storm HUR2. Fig. “ZdNH W10Zs Tepou Aq pezyereueb g Aep uo y QOOO Ye (S/W9) PTetJ JueTINS adezins paynduoyd “Gg “btg wO192a wnul uu — “003 e,cae sete ep oer ee O60cd0T5000000 tyyces oadn Gg ‘ WN SIS SOS: ‘SN ~~ 2-7 oe rd oe POPS 8 9 '2'9 PO AMMSS 6 Y PODDDOSKOKG9 PPA A anna PoPPamnmez ee Peo Powe tr Awe ea , ee eee wll e CCEAORS5505 cee SSSQB4 Boe SS SSs CGOGCGPSES GCOEEGBO?CS OP AO OGL AGRPEDSS ed CO OPO 0 OCC > ‘ 108 109 Computed baroclinic height anomaly field (meters) at 0000 h on day 7 generated by model Wie, Hp storm HUR3. ‘€YNH wWI0JS Tapow Aq pazereueb 1, Aep uo y OOOO 2e (S/Wd) PTaTJ JueTIND sdezins paynduoyD *1¢ °bTg wO127a wn |e ‘ o ee ‘ rat Soon, Vase), eeze7ZtINNNure cAeanet VN Cie) CECA AC fs Oe oh Sr CRE Poe OOUDDD wet le See yw SaFOR AY o erllttty OPUCHUV. CONN NGs NEAT ‘ \ . . ‘ ’ o ’ Nou : WINNS Ss SA AZ INNS es ae Z2llIVwraw. « PITTTANN sce ee O00INWSSeeo sos DN NS ly S98Sc0n 7 ey, { gd \ \ ‘ é \ \ \ \ S e S500000GCoS cee ee ew ee 110 January - April 1978. Dominant signals were found in the 6 day band and 3 day band. However, the Gulf-wide signal, Ng: can also be observed in the simulated hydrographs from stations on the northern part of the shelf. Figures 58 and 59 show hydrographs from Apalachicola and Cedar Key obtained from the HUR1 simulation. It is obvious from these two figures that other modes of response exist on the Florida shelf. b) Simulated hydrographs The following discussions are ordered according to the selected paths starting from PATH] through PATHS. Within each path, results obtained from the storms of small radius are presented first followed by that of the large radius storms. The time sequence of the computed water levels at Galveston obtained from the simulation of HUR1 is shown in Fig. 60. The initial rise of water level which reaches a maximum of 0.22 m at approximately 1300 h on day 4 is well-defined. Resurgence after the peak surge with an oscillation period of about 6 h is also noticeable. The initial peak of water level in the Galveston hydrograph is in phase with the first maximum of the 1G Signal as shown in Fig. 61. The first extreme of NG: which is also the maximum peak, occurs approximately 24 h after the storm entered Yucatan Strait. The average period of this signal is 30 h. This mode is not only present in the hydrographs from all stations but also accounts for almost all of the response in the southwest sector of the Gulf as shown in Figs. 111 “TUNH JO uOTJeTNUTS ay} WOT pauTe}qO eTOOTYyDeTedy 3e STeaAeT Ja}eM JO |Bduaenbes sauTy ydeiso1pA}py] eyooryoeredy [ aueoliny “8S (WI) qYsIEy 112 *“TUNH JO UOTJeTNUTS ayy WOIJ peuTeyqo Asay Iepay 3e STaAeT 3J9a}eM JO BOUaNb|aS auTLZ °*6S ‘bta ) WYsTOH Ww ( ydeiudoipAjy Aay 1epag ] ouedluIny 113 *[T@AeT eas ueaul ST wnjep eYyL “TMH JO UOTIETHUTS ayy WOTZ PeuTe IO UOJSEATEHD 4e STAST Ja}eM JO B.ueNbes auTL °09 “STA Aeq (WI) q4dTEH ydeidoipA}] Uo saaTey [ eueoTIIny 114 *TaAeT eas ueau ST wnjep eyZ ‘uoTJenba Aj,tNutTWUOD Vy} wWOIJ peynduod st auTT peysep eyL ‘*dejs aut, yoea ye JIN ayy ut jutod ptib AqeAe woz STaAeT 3Ja}eM HbuTbereae Aq paynduod st auT[T PTTOS eyL “*TYNH JO uoTJeTNUTS au worly pauTeqqgo ‘9u ‘ootxeW JO JIND vy} UT STeAeT Ja}zeM abeTaAe vy} JO Saduenbas sutL “19 *bta 00 oT Oe (t1) qYSTEH oc OF (OV.LA) YdesdospAT] WOD ] ouedTIINY osc ILLS) 62 and 63. The dashed lines represent ng and the solid lines are the simulated water levels from the indicated station. The maximum surges generated by HUR2 and HUR3 imply a direct relationship between peak surge at the open coast and forward speed of the storm. Jelesnianski (1972) proposed a correction factor for the effect of storm vector motion (track direction and forward speed at landfall) which is larger for a faster moving storm provided that the landfall angles are the same. Figures 64 and 65 show the simulated hydrographs at Galveston from the HUR2 and HUR3 simulations, respectively. The initial rise of water level before the peak surges are present in both runs which are concurrent with the first maximum of their corresponding 1G signals. The maximum peaks and periods of ng as determined from Figs. 66 and 67 are 0.22 m, 28 h and 0.21 m 30 h, respectively. The time lag, dg: is approximately 17 h for the faster storm (HUR2) and about 24 h for the slower storm (HUR3). The maximum surges produced by storms of large R,., (HUR4,HUR5,HUR6) are on the order of 2 m larger than those corresponding to the small storm (HUR1,HUR2,HUR3) simulations. Except for a larger percentage of increase in the peak surge at the open coast, the results are in qualitative agreement with Jelesnianski (1972) in which a very simple bathymetry was employed. The hydrograph at Galveston from the HUR4 simulation is shown in Fig. 68. The time of the peak surge and the time at which the initial rise of water level reaches the maximum are exactly the same as in the HUR1 simulation. The resurgence oscillation is concealed 116 *[T@AeT aS ueaW ST WNJep PUL “TUNH JO uoTJeTNUWTS ay} WOIJ pauTejqo (paysep) Suk pue (pT{OsS) TeqeSI 310g ye STeAeT 318324 JO Saduenbas ouTtTL °*79 *btA (UI) yUSTOH ydeidoipAy [aqes] 310g J] ouedlnyY 117 *T@AeT Bas ueseul ST wN}Zep sYL “THUNH JO uoT AeTHuTS ey Woy peuTe yo (peysep) Yu pue (pT{Os) oserboig 3e STaAST JaqyemM JO saduenbes osutL ydeidso01pkyH osaidoig J] euedliny “€9 BTtd (U1) yYUsIEy 118 wunjZep euL *T@AeT eas uesau ST “ZUNH JO UOTJETNUTS Gy} WOIZ PaUuTe qo UOJsSeATeD Je STeAeT JaqzeM JO BoUeNbes ouTL “79 “STA keq : £ ydeidoipAyH uojysaaley c euBOTIINH (WI) YsIOH 119 unjep euL ot- 00 oT 02 oc OF *TeAeT ees ues ST “€NNH JO UOTIETNHUTS ay} WOIZ pauUTeIqO UORSBATeD 3e STSAST 19}eM JO aouenbss sutL °Ss9 °bt4 Aeqg (U1) 74918H ydeisoipAky Uoysaaley € aueoLInyY 120 Aeq ace 9 S v o1- 0) ae eee Ne ee oT oe oc 0+ (OVLE) Udesdo1psH] WOD os *[TaAaT eas ueAU ST wnjep syZ ‘uotzJenba AytTNut}UOD |YyA woIZ peynduod st euTT peysep xyuL_ °days ewtz yoea ze JTND 3yy ut juTOod ptTz6 AzaAe worz STeAeT JajemM HutTbesraae Aq peynduod st SuTT PTTOS eYL “ZUNH JO VOTIETNUTS ey} wWoITZJ peuTe qo ‘Du ‘ootxaW JO JIND ey} UT STeAeT Ja}zeM aberaAe 9y} JO saduenbas aut, °99 °btd 2 auBolliny (Ww) qYSIEH 121 *[TaAeT eas ueeuU *daqys out} yoea ye jJ[ND auq “€aNH JO uUOTZeTNHUTS "LO ‘bt ST wn}ep eyL ‘uotTJenba Ajtnut UO. |y} wWoIZJ paynduods sft seuTT peysep eyL ut juTod ptib6 AraAe woIJ STeAeT Ja}eM HbuTberaae Aq peyndwod st seut{T PTTOS eyL ayy wory pautTeqqo “9k ‘ootxaW JO JIND ey} uy STeAeT JajzemM aberaAe ayy JO Saoduenbes sutL (Ww) 3YSIEH (DVLA) Udesdoi1pAT] WOD € aUueoLIINY 122 *[@aAeT eas ueAaU ST wnjzep eYyL “PUNH JO voTJeTNUTS ey} WOT paUTe,GO UOJSeATED 3e STeAeT Ja}zeM JO BoueNbas suTL “89 “bTA WI) JUSTE ( ydeidoipAY uo ysealey p aueoliny 123 by a high peak surge but still observable. Figure 69 shows the time sequence of the Gulf-wide oscillation, 1G: generated by HUR4. The average period and 6, are 30 h and 24 h, respectively. The maximum peak is 0.42 m, about a factor of 2 larger than in the HUR1 simulation. It is interesting to note that the largest difference in the extremes of 1G from HUR1 and HUR4 occur at their first maximum. During later stages the differences reduce to a few centimeters. The presence of nq in the simulated hydrographs from the other stations is preserved. However, this oscillation is no longer coincident with the water levels at the individual stations as in the previous cases. Figures 70 and 71 clearly demonstrate the departure between the two curves at Port Isabel and Progreso. A peak surge of 6.5 m at the coast is generated by a fast moving storm of large Rnax (HURS) - The slow moving storm of the same size (HUR6) however, produces a smaller 4.71 m peak surge. Increasing the Maximum surge at the coast with increasing forward speed is the same as that obtained from simulations of small storms as discussed above. Figures 72 and 73 display the computed hydrographs at Galveston obtained from the HUR5 and HUR6 simulations, respectively. Both hydrographs have an initial rise of water level prior to the peak surge that again matches the first peak of their corresponding 1G series. The time sequence of the 1G Signal for HUR5 and HUR6 are shown in Figs. 74 and 75. The maximum peaks of 7, (0.4 m for HURS and 0.38 m for HUR6) vary slightly with the forward speed but both are approximately twice as large compared with those produced by the small storms with the same forward speed, i.e., HUR2 and HUR3, 124 *[9AeT eas ueoU ‘uoTjenba AyTNUTUOD By} WOIZ paynduiod st suTT peysep ay ‘*deqs aut, yee ye JTND yy UT Juyod ptT41b ArsAe woz sTaAeT 33}eM HuTberzaae Aq peyndwod st aut{T PTTIOS euL *puNH jo uoTJeTAUTS 24} wory peutezqo '9u ‘ooTxaW Jo JTND ayy UF ST@AeT JazeM BbeaAe vy} JO saduanbes suty, °69 °hty ST wnjep auL qUSIOH ) (uI (SvVLa) Ydesdouphy WOD - py aueoliny 125 *T@AeT eas ueaul ST wNIep SUL ay} woIy peuTeqgo (peysep) Du pue (pTTOSs) TeqesI *PpUuNnH JO uoF Aerts y10g }e ST@eAeT 3@}3emM JO seduenbes euyL “OL “Std (ut) qYsIEH ydeisoipAp] [eqes] Od p queodlliny 126 *T@AST eas Uva ST WNQJep 9yUL “PYNH JO uoTIeTNwIsS ay} WOIJ pauTejqo (paysep) Xu pue (pt{os) oserbaig }e STaAeT Ja}yeM Jo seouenbas suTL arLee bial Aeq (W) jYdIEY ydeidoipAy] osaidoid p oueolinyy 127 wnjzep eu, *T2PAXT BIS UPOU ST “GUNH JO UOTJETNUTS BY} WOIJ peUTeIqO UOJSeATED Je STeAeT Je}eM JO BoUeNbes eutL, “7L “bia Aeg 9 S ydeiso1IpAH UoysaaTey G 9UeOTIINY (W1) qYUsTIOH 128 *[T@AeT PAS UPAaU ST wnjep eUL “9NNH JO uOTIeTHUTS By} WOIJ paUTe}qO UOoJSaATeDH Je STaAST JezeM JO BOuanbes ouTL “EL “Ht4 ) 1YS198H (ui ydeidoipA}] Uojsaatey g aueoIIInY 129 *TeaAeT Pas ueAU *dajs owt} yoea ze jJIND oyQ “GUNH jO uoTJeT HUTS ST unjep syt ‘uotTzyenbe AyTnutTWUuOD ay} worzJ peyndwod st suTT peysep eyL °plL °bta UT }uTOd ptib AraAe wWOIJ STeAeT 3a}emM butTbersae Aq paynduios st aut— PTTOS euL au} wory peuteqyqo 49u fooTKxeW JO JIND Vy UF STeAeT Jejzem abeTaAe ayy JO saoduenbes suzL Aeq S (OV.La) UdesdorpAH WOD G aueolinyy (ur) JUSIEH 130 *T@eAaeT eas ueoSU St unjep syL ‘uoTyenba AyyFNuFjuOD ayy wor peynduwos sft euTT peysep ey, ‘des ewy} yoee ze JIND 9yy uT quTOd ptib Adena WOIJ STeAST 19}eM butbereae Aq peynduod st suTT PTTOS eYyL “9NNH JO UOTIeTHUTS ayy woTZ peuTeqIqo ‘Du ‘ootxay JO JIND ey} UT STeAeT JezeM aberaAe 3y} JO Soouenbas ouwtL “SL °‘6t4 Oe- PAA Tt 00 ee Sw ae ae aa ee eee a Oe es wee By} UT PUue (PTTOS) OOTKXeW JO JIND ey} UT TeAST JeqeM poheeAe ay} JO Saeoduanbes suit], "G6 °5T4 Keq 12 0'Oot-— 00 (WI) JYsIOH (OVLG) UderdompAT] WOD G sueoLIINY 0'ooT 160 b) The 6.5 day tilt mode Time histories of volume transport obtained from the 21 day simulation of HUR5 with rotation are presented in Fig. 96. A prominent feature is the 6.5 day period oscillation for the transports through both Florida and Yucatan Straits. The water level from this long simulation was sampled at 12 stations, six of which are the same as those given in Fig. 12. The remaining six stations are Panama City, Tampico, Coatzacoalcas, Dimas (northern coast of Cuba), Central Gulf, Eastern Gulf and Western Gulf. The hydrograph at Key West (Fig. 97) shows a strong 6.5 day mode with an average amplitude of 0.50 m. Superimposed on this long period mode is the 28 h volume mode. Careful inspection of the remaining hydrographs indicates the presence of the 6.5 day oscillation but with a much smaller amplitude as compared to that in the Key West hydrograph. Hydrographs at Galveston (Fig. 98) and Dimas (Fig. 99) show that the water levels at the two stations are 180° out-of-phase. For example, at 1600 h on day 8 the water level at Galveston is 0.05 m while at Dimas it is -0.05 m. The three deep- water hydrographs are near zero at this time (note that the Central Gulf station is approximately midway between Dimas and Galveston). This 0.1 m differential in water surface elevation is possibly a geostrophic tilt since all the forcings were set to zero well before this time. Large space and time scales and a small surface expression are indicative of a vorticity (or quasi-geostrophic tilt) mode. 161 *z10dsue1} aUNTOA TeTUSIeJJTP JOU yz ST BAIN paTTeqeTuN eyuL “GSHNH JO uoTIeTNUITS W494 Huot ey} woIrg <4 pautTeqqo s}te13S (SA) uezeonx pue (Sd) epTsoTa ybnoryy jaodsuei} auntoA jo saaduenbes sutL °*96 “bT4 Aeqd et wT ol G aueoliny 0'0F- 0: 0£- 0'0e- ooT- 00 oot 0:02 0 OF 0°0S (s/w) gOIX XN 162 *[TaAeT eas Uva ST WNIep 2UL “SUNH JO UOTIETNUTS WIA} HuoT ay} wWoIzZ peuTeyqo 4SeM Aey Je STeAeT JozeM JO adueNbas euTL 16 “bT4 Aeq 12 02 61 eI LI ot SI v1 el el im ot 6 8 L 9 c v € 2 T 0 Oe-q DU AU EO | 1 Ow- 00 vo ee eee ae ap oT ey. gQ (SF o oe 2) oe or ydeidoipAy sam Lay G aueoliiIny osc 163 *T@AeT eas ueau ST WNZep ayuL *GUNH JO uUOTIeTNUTS wa} Huot ey} wWoOITZ pauTe}GO uoSeATeD Je STeAST JAaeM JO BdoOuSNbes suTL "86 °bta (U1) WYUSTIOH ydeidoipkY Uoysaaley G aueolliny 164 jo uot zerow *TaAeT eas ueOW ST wNnje FIETHUTS w1az Huot ay} wory pautTeyqo (eqnd) seultd 3e STeAsT JaeM JO epee ee * Gn °66 °6ta4 ydeis01psH (eqn) sewid G auPolliny (wi) 14sIEH 165 A simple geostrophic calculation using an instantaneous volume transport through Florida Strait at 1600 h on day 8 of 30 x 10 m/s and an average depth across the port of 950 m yield a northward surface gradient from Cuba of 0.2 m. A similar computation for the Yucatan Strait yields a surface gradient toward Campeche Bank of only 0.07 m. The surface height field is not available for comparison. However, uSing the mean water levels at Dimas and Key West as a rough representation of the north-south surface component of gradient reveals that the tilting of water surface across Florida Strait is on the order of 0.5 m. This result is much larger than that obtained from the simplified geostrophic calculation. It is quite possible that the current speed is not uniform across the ports. An estimate of the effect of current shear on the cross-port surface gradient by numerically integrating the geostrophic equation across the port was made. This integration is given by the relation gen a) SP (Ay£U/9D543) ’ (79) where U is the transport per unit width for each grid block across the Florida Strait and Dj+t is the average depth for each block. Upon assuming uniform distribution of transport across the port, U is Q/W where Q is the total volume transport and W is the port width. Employing the same value of Q as in the previous calculation and taking the water level at Cuba (south of Key West) as zero yields a surface gradient between Key West and Cuba of 0.5 m. For the Yucatan Strait the tilting of water surface between Cuba and the east coast of Yucatan obtained from this computation is 0.26 m. Even though Q 166 is the same at both ports, Yucatan channel is much deeper than the Florida port so that the computed surface gradient is smaller. There is no hydrograph on the east coast of Yucatan for comparison in this case. 6. Results from related simulations a) Variation of pressure drop All the synthetic storms employed a constant pressure drop of 80 mb. As a sensitivity test of the model, simulations of HUR5S with a 40 mb (HURSW) and a 120 mb (HUR5S) pressure drop were made. Maximum surges of 3.59 m and 9.35 m were generated in the simulations of HURSW and HUR5S, respectively. Figure 100 displays the maximum surge as a function of pressure drop obtained from all simulations of HUR5. This result is consistent with Jelesnianski's (1972) inference that the peak surge is almost a linear function of the pressure drop. Plots of 1, obtained from the HUR5W and HUR5S simulations are shown in Figs. 101 and 102, respectively. The maximum peak and the period of this oscillation are 0.2 m and 27 h for the weak storm and 0.6 m and 27 h for the intense storm. Note that the HUR5W simulation yields a larger magnitude of 1G compared with HUR25 despite its smaller pressure drop. This implies that a major factor governing the magnitude of the 1G from these simulations is the differential wind-driven volume transport through Florida port and Yucatan Strait. 167 Peak Surge (m) ie) 40 80 120 160 Pressure Drop (mb) Fig. 100. Relationship between the maximum peak surge at the coast and the central pressure deficit obtained from three simulations of HURS with 40 mb, 80 mb and 120 mb pressure drop. 168 *TaAeT eas ueoW ST UNQep ayL_ ‘*uoTzenba AytTNuT QUOD *daqs aut] yore ye JTND ayy ut qutod ptib AqaAa woIy STSAST "(MGUNH) doip eansseid qu Qp YITM GUNH JO UOT LeTHUTS oan 0) ae on | 38u} WOIJ pajnduod st auTT peysep eyL Jajem HuTberaae Aq payndwuod st auTT PT[OS eyL au WoIJ pauTeqgo ‘9U ‘ooTKaW JO JIND oy} UT STeAeT JOa}eM eheeAe By JO Saduanbas aut Avg S (DVLa) UdesdoipsH WOD MG aueo tiny ) qWsTEH ul ( 169 *[TaABT ees ues ST wWNJep syL_ ‘uoTenba Aytnut 4UOD @Yy} WOTZ payndwod st auTtT paysep ey, ‘days aut, yoea ye J{TND ayy ut utod ptab Asana wory sTeaet qJajzem buTberaae Aq pajnduods st euTT PTTOS eyL °(SSYNH) doip einsseizd qu OZT Y43TM GUNH JO uoTIeTNHuUTS ay} woz peuteqqo !2Uu ‘ooTxaW JO JIND 9yy UT STeAeT 3a}zeM BherVAe ayy JO saduanbsas autL °zot °6t4 Aeq 9 S (DVLA) Udeido1psH WOD SG dueOLIINY (WI) 3YsTOH 170 b) Barotropic model It is important to ascertain the effects of the baroclinic response in deep water on the maximum surge at the coast since both barotropic and baroclinic modes are coupled through the gradient terms, V(H,/D). Therefore, an additional simulation of HUR5 was repeated which excluded the baroclinic mode. A maximum peak surge of 6.42 m at grid point (14,46) was obtained from this simulation. This peak is only eight tenths of a centimeter lower than that obtained from the two mode simulation of this storm. Comparisons of the simulated hydrographs from the barotropic model and the original HUR5 reveal that the baroclinic response does not produce a Significant contribution to the barotropic response on the shelf except at Key West, Naples and St. Petersburg. The solid and dashed lines in Fig. 103 and 104, respectively, represent the computed hydrographs obtained from the two mode and pure barotropic mode simulations at the indicated stations. The reason that this effect is visible on the Florida shelf rather than other locations may be due to a larger baroclinic transport (through the Florida strait) and a stronger coupling as a result of steeper slopes at the Florida shelf break. As expected, the Gulf-wide oscillation as shown in Fig. 105 obtained from this simulation is only slightly different from that obtained from the two mode run. 171 00 oT Oe *T@AeT bas UAW ST UN}ep eYyL °(PTTOS) GHNH JO Tepow Dtdo1,0oIeq |aind ey} pue (P2euUSeP) GUNH JO SUOT}ETNHUTS ay}, WOIZ peuTeyJqo Aaey IepaDd je STeAeT 1a\eM JO SeouaeNbas suTL “*E€OT *H6Ta Ae] L 9 S ———_ TS Se Sear oe OV (WI) yUSTOH ydeidoipAy Aay 1epad G aueoliiny 7/2 *[T@AIT Bas uPA ST WNJEP ayL °(PTIOS) SHNH JO Tepow OtdorJoI1eq vind ay} pue (peysep) “POT ‘Sta GUNH JO SUOTJeETNUTS ay} WOIZ peUTeygO binqsiajeg °*4S je STeASeT Ja\eM JO SaoueNbas suTy Aeq (wi) qYSIOH ydeidoipAYH dinqgstezed 4S G aueoliny 173 *T@A9T Bas ube SF wNZep eyL uot zenba AjynutT\uos @Y} WOIJ peyNduod st euT{T paysep syL *dezs ouzz yOea ye JIND ayy uF qutod pzIb AreAe wosz stanat Jezem buTbereae Aq paynduiods st euT{T PTTOS eYyL “*SHNH JO Tepow Dtdo1z01eq aand ay} Jo uoteTNUTS ey} WOIJ peuTeqo ‘Du ‘ootKeW JO JIND ey} UT STeAeT 38}eM eHereAe ayy Jo saduanbas sUTL °SOT ‘bta keq 9 S a € (uw) qYSIEH (OV.La) YdessoipsH WOD G aueoliny 174 c) Radiation boundary condition The effects of the open boundary condition employed in this study on ng was of primary concern. In order to evaluate this effect, a pure radiation boundary condition was imposed in the simulations of HUR5, HUR23 and hurricane Allen. The names of these storms modified by (R) are used in the following discussions to distinguish them from their corresponding original simulations. Figures 106 through 108 show the time histories of Ng obtained from the simulations of HUR5(R), HUR23(R) and hurricane Allen(R), respectively. Only the ng, Signal obtained by averaging water levels in the Gulf is shown in the figures (note the changes in height scale). The radiation condition at the open boundaries effectively radiates Ng Out of the Gulf. The peaks of Nge sequentially, are 0.28 m, 0.14 m and 0.22 m for HUR5(R), HUR23(R) and hurricane Allen(R). The estimated e-folding times of the damping rate are 34 h for HUR5S(R), 24 h for HUR23(R) and less than 10 h for hurricane Allen(R). The radiation condition not only drastically damps the 1G Signal but also changes their periods. Reid and Whitaker (1981) experimentally determined the damping rate of the volume mode in the GOMT model to be 2.81 x 107° s+ or an e-folding time of 2.6 days. The radiation condition employed in the GOMT model has a complex admittance coefficient in which the imaginary part governed the effective added mass Of adjoining seas. The radiation condition employed in this study is a special case where only the real part of the admittance is considered. Exclusion of the imaginary part results in a much larger damping rate of the volume mode as found in this study. However, the 175 *[TaAeT eas ueaul ST uNQep oUL ‘uotzenba AjtTnuTUOS ay} woOTJ payndwos ST 8UTT peysep eyz *deqs aut] yore je J[ND 9y3 uy qutod pT1b AreAe wWoIy STaAeT 1a7eM butbezeae Aq peynduiod st euTT PTTOS euL ‘“SaTIepunog uedo ey} 3e UOTITPUOD UOT}ETPeI YITM GHNH JO UOTIETNUTS ay} worly psuzej,go ‘Du tootxeaW JO JIND 9yy UT STeA2T Aeq L 9 S (0) fs 00 oT 072 oe OF os Jajem aberaae ay} JO Saoduenbas suTL *90T SS LP (DVLA) Udesdo1pAyy WOOD G euvollingy “Ta (Wi) 1USTEH 176 *T@AeT eas ueAa] ST wNyep au ‘uoTJenba AjtnutuOD ayy woIJ paynduood ST ®UTT Ppeusep euL ‘deqs aut} yoea ye JrNH ey} ut qutod ptab Aqeaa woIZ STaAST JajeM buTbessae Aq payndwod st auTT PTTOS eyL *SsetTiepunoq usdo ay} 3e UOTJTPUOD UOTJeTpeI YIM EZUNH JO UOT IeTHWTS ey} wory peuteyqgo 49U ‘ooTKxeW JO JIND By} UT STOeAeT JeqeM BberaAe |9Yy} JO saduenbes sutL “LOT °6Ta Aeq L 9 G 7 € O2c- otT-— 0-0 erie Ben | 1 oT OF ga (OF + 02 g OE i ot (DVLA) UdesdoupéH WOD ee aueoliny os 177 *T@eAeT ees ues ST uNJep syL_ *uoTZenbs AR{~Nuyt UO |y} woIrzZ peynduod st auTtT peysep syuL *deqjs sult, yoee ye J[ND Bsyq ut JuTod ptab Aqsae worzZ sTeAsT 3a}3eM Huybereae Aq paynduoos ST BUTT PTIOS PYL ‘*SaTIepunog uado sy} 3e UOT]TPUOD UOT}ETPeI YIM UaTTY eUeDTIINY Jo uoT}eTHUTS ay} worzy pauTeqyqo 49u ‘ooTxeW JO JIND oy} UT STeAeT Ja}eM |aberaae ay} JO saduenbes sutL “got °6ta4 ysnany el WT or 6 8 L O'2- i ol- 00 I Ta a = ae, (om § e. ga L far al (Oir<] gy] oc oF (OVLA) UdessoipAt] WOD UudsT[V FUBOIIIN]G] osc 178 initial perturbation of 1G before the maximum peak surge is retained, but is reduced on the order of 30 %. Changes in the maximum peak surge at the coast are shown in Table 3. The magnitudes of 1G (|ng|), determined at the time of the maximum peak at the coast, are shown in column 5. The differences of |ng| between each pair of simulations are shown in column 6. The Superscript plus or minus indicates whether lng | obtained from the simulations with the radiation condition is larger or smaller with respect to the former result. The differences in |ng| are clearly responsible for the changes in maximum peak surge at the coast. d) Limited area model Results obtained from the simulations of the hurricanes of record and the synthetic storms strongly indicate close correlation between the generation of a simultaneous Gulf-wide oscillation and the net periodic volume transport through the ports. It is of interest, therefore, to determine whether 1G Still exists in the absence of wind stress in the deep water, including Florida and Yucatan Straits. Therefore, simulations of HUR5, HUR23 and hurricane Carla were repeated with a modified wind field where the wind stresses are set to zero everywhere in the deep water (water depth greater than the upper layer thickness). The continental shelf regions of these modified HUR5, HUR23 and Carla resemble limited area coastal or shelf surge models. In the following discussion these models are referred to as limited area models and their designated names are followed by (L). 179 Table 3. Results of the simulations of HURS5, HUR23 and hurricane Allen with and without radiation boundary condition. PEAK PEAK — RATIO OF STORM COAST Ne PEAK n, [ne A|ngl (m) (m) (m) (m) MR RD OS SON RNPS ERG NG Re ecas TO Taga Bd) HOES ROS HURS 6.50 0.40 0.35 0.70 0.20 HURS(R) 6.31 0.28 O15 HUR23 8.52 0.23 0.10 % 0.74 0.10 HUR23(R) 8.61 0.14 0.20 ALLEN 2.40 0.35 -0.10 a 0.69 0.20 ALLEN(R) 2.65 0.22 0.10 180 The Gulf-wide oscillation is still excited in the absence of wind stresses in the deep water. The time sequence of the ng response obtained from the simulation of HURS5(L) is shown in Fig. 109. The maximum peak of this signal is 0.2 m and the average period is 28 h. Magnitudes of 1G obtained from the simulations of HURS5 and HUR5(L) differ by a factor of 2. Neglecting the deep-water wind driven flow produces a dramatic drop of the total transport through the ports for HUR5(L) as compared to HURS (not shown). The volume transport through both Florida and Yucatan Straits obtained from HUR5(L) is caused by the tilting of the water surface toward storm center due to the atmospheric pressure. Simulation of HUR23(L) yields an ng response (Fig. 110) that is almost identical to that obtained from HUR23. Comparison of the time histories of the volume transports obtained from the two runs (not shown) reveals that the net periodic volume transports are about the same (6.6 x 10° m/s) at their first maxima. Figure 111 shows the gulf wide oscillation obtained from hurricane Carla(L) simulation which is again comparable to that obtained from the original hurricane Carla result. The initial positions of HUR5 and hurricane Carla appear to be a key factor in the almost duplicate nq responses generated by the limited area model and the complete model. Both storms are initially located closer and more directly to the south of Yucatan Strait as compared to HUR5. The stronger atmospheric pressure gradient through Yucatan channel is apparently more effective in drawing water through this opening. 181 *T9AVGT eVS UPD ST unjep ey ‘uofyenba AyTNuTIUOD 9y} wWoIZ peynduiod st euTT peysep euL *daqs aut} yore ze JIND ayy ut jutod pti6 AzeAe worz STeAeT 1a}zeM HbuTbereae Aq peynduiod st eUuTT PTIOS eYuL °(1)SYNH JO uOoTJeTHUTS ay} wory pauteqqo ‘Du ‘ootxaw JO JIND ey} UT STeAeT 3a}eM BhereAe ayy JO SeoueNbes suTL “60T “Std Keq (DVLA) Udesdo1pAH WOD (1)G aueoiny (Ww) qU3I98H 182 *T9AVGT Bes ueaUu ST ‘days owt yee 3e 31ND ey3 ut “(1T)EZHNH JO uoTJeTNUTS “OTT “5t4 wnjzep ey, ‘uotyenba AAtnut}uoD. ay} woIzJ payndwod st auTT peysep suL qutod pti6b Arana wo1rz ST@aAeT JaeM HutTbereae Aq paynduod st aut{T PTTOS seuL 24} worz peuteyqo !I9u ‘ootxay jo JIND dy UT STaAeT J3a}3eM BHeraAe ay} JO Saduanbas auty Keg y € 8 ¢ (DVLd) UdesdorphH WOD (1)€@ aeueoiny (ur) yYsTEH 183 *[T@aAeT Bas ueau] ST uNJep aus,°(7)eTIeD euedTIINY jo uoTIeTNHuTS dy} Woy peuTeqgo ‘2 ‘ootxepW JO JIND eu UT TeAeT JOeqyeM Pabetaae ayy JO souenbes ouTL ‘“TIt ‘6t4 Jaquiaydag zy WI or 6 8 L 9 (UI) yUSIEH (OVE) Uderso1pAH WOD (7)e[4e9 sueolIunY 184 Another important result found in the three limited area model simulations is the appreciable water surface elevation obtained at the edge of the shelf. This water level perturbation, excluding the inverted barometric height at the edge of the shelf, %, reaches a Maximum of 0.1 m- 0.3 m. In the simulations of the same storms with the actual wind field, the maximum ¥ increases to 0.7 m- 0.8m. The Maximum of # and the peak surge at the coast, locations where the two peaks occur and the ratio of the two peaks obtained from all simulations are presented in Table 4. It is important to note that the differences in the maximum peak surges at the coast between the full and limited area model simulations are comparable to the differences in the ¥ maxima. The presence of # is important to a properly posed open boundary condition for limited area coastal surge models. Generally, these models neglect # by specifying a constant water level, which is equivalent to the inverted barometric height, as the open boundary condition. As noted above, this condition might result in an underestimated maximum peak surge at the coast by a value comparable to the neglected #. Another salient feature of % pertaining to the consideration of open boundary conditions is the nonuniform distribution along the shelf break. Figs. 112 through 114 show profiles of # obtained from the original versions of HURS5, HUR23 and hurricane Carla. The nonuniformity of these profiles implies that the effect of % on the peak surge at the coast is different from one grid block to the other. This result ultimately prompted a question on specifying a constant water level along the shelf break as the 185 *4seood au ye adans yead oy} pue yeo1q JTOYS 9Y4W 3e edgins yeed oy} UeeMJZeq OT}EI BY ST ¥ Z eTqeL uT se ewes ay} o1e g§ “4 ee ee a A NS SSS SE SSS SQL GS BS? 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The time at which the maximum ¥ occurs, shown at the top of the figures, introduces the time scale as another factor to be considered. In addition, the results shown in Table 4 indicate that the magnitude of # decreases with increasing storm forward speed. Accordingly, there appears to be no simple resolution to the question of what is the proper open boundary condition for limited area coastal surge models. A possible pragmatic solution is to use the time sequence of water levels at each grid point along the shelf break obtained from an a priori simulation covering the entire basin. 190 CHAPTER V SUMMARY AND CONCLUSIONS A time-dependent, numerical, normal mode model portraying the linear (except for dissipation) physics of a two layer Gulf of Mexico has been developed, tested and verified. Comparisons of the numerical hydrographs and the known local responses produced by hurricanes Carla and Allen generally verify the barotropic surge response within the Gulf of Mexico. Those significant deviations which occur can be explained by unique gauge locations, insufficiently resolved model renditions of the coastline and bathymetry, and possible inaccuracies in the stipulated atmospheric forcing fields. The model was used to obtain the responses of the Gulf to a series of synthetic storms. These responses provided information which were used to answer basic questions on forerunner surges in the Gulf and related aspects of the surge prediction problem. These questions concerned the influence of baroclinic motions on the nearshore surge, the establishment of forerunners and the time and space scales of this initial rise in water level, and surge conditions at the shelf break. Results from the simulations of the pure barotropic and the’ two mode models of HURS (a synthetic storm with 30 km radius of maximum wind translating at 35 km/h from the Cayman Sea through Yucatan Strait and landfalls at Corpus Christi, PATH1) demonstrate the insignificant contribution of the baroclinic responses to the water levels on the shelf areas. The maximum surges in particular are 191 scarcely affected by the baroclinic deep-ocean responses in spite of the fact that such modes contain significant energy. In addition, the initial rise of water level before the peak surge (which, by definition, is the forerunner)in the Galveston hydrographs obtained from the two versions of HUR5 show no visual differences. Based on these results, the baroclinic response is not important in the forerunner surge phenomenon. The quasi-linear, coupled, normal mode model shows that the hurricane induced forerunner surge in the Gulf of Mexico is associated with a Gulf-wide oscillation of water level, 1G: The ubiquity of the 1G signal in the hydrographs from stations around the Gulf, except near Florida Strait, indicates that 1G is dominated by a volume (i.e., Helmholtz) mode. The Helmholtz mode is characterized by a relatively uniform amplitude and phase, except near the open ports where the amplitude and phase changes rapidly. The forerunner, therefore, has space scales comparable to the horizontal dimensions of the Gulf of Mexico. Examination of the transport through the ports, particularly for the 21 day simulations for PATH1, reveal large and nearly equal amplitude but out-of-phase oscillations of about 6.5 day (156 h) period. Superimposed on the 6.5 day oscillations are smaller amplitude in-phase oscillations with a period of about 28 h. The out- of-phase transport implies that when the transport is in one port it is out of the other. It is the in-phase oscillations in transport which are associated with the Helmholtz mode in the Gulf. The long period out-of-phase oscillations in transport, on the other hand, are 192 associated with a quasi-geostrophic tilt of water level across the ports as well as a tilt from southeast to northwest across the Gulf. The 6.5 day period oscillations coincide with the natural modes on the west Florida shelf (Marmorino, 1982). The excitation of the out- of-phase transport and associated quasi-geostrophic tilt mode in the present model is due to out-of-phase wind forcing at the ports (as can be produced by the cyclonic circulation in those hurricanes which traverse the Cayman Sea). Observational evidence to confirm the existence of the 6.5 day mode in the actual Gulf is lacking (e.g., the Key West gauge records do not show clear evidence for the large amplitude 6.5 day oscillations which the model simulations for hurricane Allen or HUR5 produce). However, except for locations near the ports, the water level variations associated with the 6.5 day tilt mode are small (less than 0.1 m along the northern and western coast of the Gulf). Hence the quasi-geostrophic tilt mode is probably not important with respect to forerunners in the northern and western parts of the Gulf. Close examination of the 7, time sequences for the long term simulations for PATH1 show that, in addition to the presence of the Helmholtz mode, a period of about 3.4 days is also present. But there is very little evidence of the 6.5 day tilt mode in n,. The 3.4 day mode in fact shows up in the time history of the spatial mean water level for the Cayman Sea. Some observational evidence for such a mode of oscillation exists for the Gulf (Halper, 1984, Kelly, 1985, Kirwan et al., 1984). The relative excitation of these modes during the forcing stage by hurricanes depends upon the path, the storm 193 scale and intensity, and to a lesser extent on the translational speed of the hurricane. In general, the 1G during the first few days after the hurricane enters the Gulf consists of three components. The first two components (i.e., the Helmholtz mode and the 3.4 day Gulf-Cayman mode) are associated with the volume transport through the ports. The third component is a directly forced response associated with the spatial average value of b (the inverted barometer term) over the Gulf. PATH1 tends to give the largest peak 1G: Particularly for larger radius storms (compare HUR1 and HUR4 results). Results from the tamed area model and the full model disclose that both the central pressure deficit and the wind induced transports through the ports can excite the Helmholtz mode. However, the relative importance of the two forcing fields in generating 1G depends upon the storms' paths and their evolution. The almost identical 1g obtained from the full and limited area models of hurricane Carla and HUR23 show that the atmospheric pressure gradient through Yucatan Strait was more important in the generation of 7, than the wind forcing. In contrast, the small Ng response in the absence of wind in deep water for HUR5(L) implies that in this case wind forcing was the major factor in generating 7. The average periods of the Helmholtz mode from this study are in the range of 25-32 hours. Platzman (1972) obtained a free Helmholtz mode with a period of only 21.2 hours. This difference in periods is due to the fact that ng is composed of both forced and free components. The average period of 28 hours obtained from the 21 day 194 simulation of HURS5 was estimated during the later stage of the simulation for which the forced component of ng probably has been damped out. The variation of the period of ng, from one simulation (of the original synthetic storms HUR1 through HUR25) to the other may be caused by the variation of the period of the forced component which is subjected to different forcing. There is some indication that the period of 1G is smaller for a faster moving storm. Based on the Gulf mean tidal response in the diurnal band, the GOMT model of Reid and Whitaker (1981) used a 28.5 hour Helmholtz mode. Their volume mode period was adjustable because the radiation boundary condition employed in the GOMT model effectively takes into account the added mass of the Cayman Sea (including the Atlantic Ocean) by Means of a complex ocean impedance. The 28.5 hour period used by Reid and Whitaker (1981) is close to the average period of the free component of 7g obtained from this study. It is likely that Gulf hurricanes in general elicit the Helmholtz mode and the longer period modes, but only certain storms generate a forerunner. With the definition of a forerunner as the initial rise of water level before the peak surge, storms traversing PATH4 do not generate forerunner surges but certainly excite Ng: The synthetic storm HUR25 (along PATHS) serves as another example of the situation where ng exists but with no forerunner. The presence of a forerunner in local hydrographs is PHeresorel dependent upon the path, but is also dependent on the landfall position and the time of the excitation of 1G relative to the peak surge. This would explain why every hurricane of record does not have an associated forerunner. 195 For example, a fast moving hurricane, which is generated inside the Gulf might have an associated initial peak of 1G which is nearly in phase with the primary shelf surge. A supplementary question addressed in the present study concerns the surge behavior at the shelf break. This is at least as important as the forerunner behavior for limited area models. In many applications of surge models the domain of the model is limited to a section of the continental shelf extending seaward from shore to the shelf break (about 200 m depth). A common boundary condition employed in such limited area models is to set the water level (7n) equal to the local value of b for a given time during the traverse of the hurricane through the model domain. The present study in which the whole Gulf of Mexico (and part of the Cayman Sea) is modeled, shows that the water level at the shelf break can depart significantly from b at the shelf break. Moreover, this departure (n-b) , has a behavior differing from that of 1G and generally of larger magnitude. For example, large scale (Riax = 60 km) hurricane Simulations along PATH4 yield Naaer values of (n-b) , than do those along PATH1; this is just the opposite behavior of nq for these two paths. In order to gain some further insight with respect to the shelf break condition, three hurricane runs (HUR5, HUR23 and Carla) were repeated with the winds turned off in the deep region of the Gulf and Cayman Sea (i.e., for depths greater than Hj). This is equivalent to having a limited area shelf model (including all shelves in the system) but allowing wave energy to radiate into the deep Gulf. 196 Comparison of the 7, (i.e.,at the shelf break) from these runs with their counterparts for the fully forced model shows that there remains a significant difference (of order of 0.4 m) in the peak values. The primary conclusions of this study can be summarized as follows: (1) The surge on the shelf including the forerunner is primarily a barotropic response; very little of the baroclinic energy generated within the deep water regions is transmitted onto the shelf. (2) The forerunner, when it occurs, is associated primarily with the Gulf-wide modes contained in Ng (the spatial average of the Gulf water level at a given time); this is a volume mode which exhibits periods of about 28 hours and 3.4 days. (3) Forerunners are always associated with UTel but not all hurricanes which excite 1G have an associated forerunner. Regardless of whether or not forerunners exist, Ng can, if properly phased, affect the magnitude of the peak surge. (4) Limited area shelf models (at least within the Gulf of Mexico), which employ the seaward boundary condition 7 = b (or the generalization of this which allows outward radiation of free waves) will always underestimate the peak surge at shore; the underestimate can amount to as much as 10 percent. 197 (5) A Gulf-wide quasi-geostrophic tilt mode of about 6.5 day period is found in the model superimposed on the volume mode, but verification of this from observations is lacking. The above conclusions are based on a quasi-linear, two-layer model of the Gulf of Mexico and a portion of the Cayman Sea. The model used a grid size of 15' in latitude and longitude and allowed for a variable Coriolis parameter. The lack of non-linear advection and the ad foc conditions at the open boundaries of the Cayman Sea and Florida Strait should be borne in mind, particularly with regard to conclusion (5). Long period, quasi-geostrophic (planetary) modes are known to exist in the ocean; however, their spatial structure and behavior is known to be sensitive to open boundary conditions and to non-linear phenomena such as advection of vorticity (which is not admitted in the present model). Indeed, the effect of the strong quasi-steady Loop Current within the eastern Gulf is missing in the present linear model. Regardless of these limitations, it is felt that conclusions (1) through (4) remain valid. 198 REFERENCES Buffler, R.T., S.D. Locker, W.R. Bryant, S.A. Hall and D.H. Pilger Jr., 1984: Ocean Margin Drilling Program Regional Atlas Series, Gulf of Mexico, Atlas 6, Marine Sci. /nternat., Woods Hole, 36 pp. Chang, S.W. and R.A. 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Ph.D. Dissertation, Dep. Oceanogr., Texas A&M University, 190 pp. 201