q'qq

Modelling Tides in the PersianGulf using Dynamic Nestittg

Hashem Saberi |tajafi, B.Sc (Hons, Mashhad-Iran), M'Sc(Brunel-IJK)

Thes'is subm'itted for the degree ofDoctor of Ph'ilosophy

TN

Appli,ed Mathematicsat

The Untuersi,ty of Adelai'de

(Facutty of Mathemat'ical and Computer Sciences)

Department of Applied Mathematics

February 14, L997

Acknowledgements

I wish to use this opportunity to express my sincere thanks to my supelvisol, As-

soc. P¡ofessor. John Noye, for his support, encouragement and criticism thr-oughout

the development and writing up of this thesis. The thesis would not have been

possible without his help and continuing guidance.

I wish to sincereiy thank m)i co-supervisor Dr. Michael Teubner for his continual

interest, inspiring discussions and enthusiasm throughout the wolk. I also r'vould

like to thank him for his final reading of my thesis and providing me rvith man¡''

valuable suggestions and comments.I also wish to thank Dr. Peter Bills my first co-supervisor during the fir-st )'ear'

of my studies. I appreciate very much the encouragement and expelt guidance that

I received from him during my studies.

I would like to thank all members of the tidal modelling group in the Departrnent

of Applied Mathematics, for their assistance and friendship.i also would like to thank the Head of the Applied Mathematics Depar-tment,

Dr. Peter Gitl for his help and friendship.I wish to express my gratitude to my wife, and to my two daughtels for- their'

company, encouragement and assistance during the period of rnv studies.

Finally i would like to acknowledge the financial assistance of the h'anian Min-

istry of Culture and Higher Education during the peliod of my studies in the for-m

of a Guilan University Scholarship.

Contents

Signed Statement

Acknowledgements

Abstract

1 Introduction1.1 Description of the Persian GulfI.2 Importance of this study1.3 Summary of the thesis

2 Mathematical Formulation2.L Tides and tidal prediction

2.1.1 Tidal prediction

lv

1

1

3

.)

2.22.32.4

2.52.6

Tidal dynamics in shallow seas .

ForcesDepth-averaged Cartesian tidal equations2.4.L The continuity equation2.4.2 Physical meaning of each term .

2.4.3 The momentum equation2.4.4 Physical meaning of each term .

2.4.5 Boundary and initial conditionsDepth-averaged spherical tidai equationsEquations of continuity and momentum .

2.6.L Viscosity coefficients

Ð

b

b

6

7

B

9o

10

10

11

T2

72

13

L4I416

16

17

77

18

20

27

3 Finite Difference Techniques3.1 Introduction

3.1.1 Explicit and implicit schemes

3.1.2 Arakawa C spatiai grid .

3.1.3 The temporal grid3.1.4 The finite difference code .

Discrete formulation in Cartesian coordinates3.2.1 Discrete formulation of the governing equations3.2.2 Boundary conditions

¡)où.L

3.3

3.2.3 Initial conditionsDiscrete formulation in spherical coordinates

3.3.1 Computational grid3.3.2 Discrete formulation3.3.3 Computational algorithm

3.4 Programming scheme

4 A Depth-averaged Cartesian Model of the Persian GulfIntroductionTides in the Persian Gulf .

Review of Cartesian models of the Persian GuifDepth-averaged Cartesian model and its calibration4.4.L Generating the mean sea level depths

Presentation of the resultsSummary and discussion

6 A Depth-averaged Spherical Model of the Persian GulfIntroductionReview of spherical models of the Persian GulfDepth-averaged spherical tidal model . . .' .

Calibrating the modelPresentation of the results and discussion .

6 Nested Spherical Model Development6.I Introduction6.2 Review of the literature

Method of nesting6.3.1 Spatial and temporal refinements6.3.2 The grid matching technique6.3.3 Initial and boundary conditionsComputational algorithms and automatic systems

6.4.1 Algorithm for solving the governing equations

6.4.2 Transferring algorithm6.4.3 Back transferring algorithm6.4.4 Stability criteria

6.5 Numerical tests6.5.1 Automaticsystems6.5.2 Test without Coriolis6.5.3 Test with Corioiis6.5.4 Summary and conclusion .

7 The INST Model of the Persian Gulf7.I Introduction7.2 Numerical model7.3 Presentation of the results

28

28

29

30oôùz

33

4.t4.24.3

4.4

3636

ot40

47

50

53

60

6161

61

64

65

66

4.54.6

5.1

5.25.3

5.4b.Ð

6.3

6.4

7474

74

77

78

78

80

81

81

82

83

86

88

90

91

93

96

9898

98

101

Ù

7.3.17.3.2l.J.tj

7.3.4i.ó.Ð7.3.67.3.7

Sea surface elevations .

Elevation contours at a given time.Depth-averaged tidal current ellipses

The diurnal ellipses 01 and K1The semi-diurnal ellipses M2 and Sz . .

Depth-averaged velocity vectors at a given timeDepth-averaged velocity residuals

.102

.110

. L74

. TT7

. Lr7

. L20

. r24

126

131

I Summary and conclusion

Bibliography

ur

a

a

a

a

a

a

a

a

o

a

ERRATA

The references Bills (1991) and Bills (1992) are the same; they indicate different years ofpublication of the same document.

p. 7: references to Figure I should read Figure 2.1.

p.50: reference to Mitchell (1985) should be changed to Bills (1992).

p.52: the definition of q should read "is the absolute difference in amplitude (cm) andphase (in degrees) between the observed . ..."

p.88, 2"d paragraph: change "do not contaminated" to "do not contaminate".

p.88: delete equation and following sentence after ".... in which case C"/Cr would be

I3

p.97 ,2"d paragraph: change "do not contaminated" to "do not contaminate".

p.l 14, caption for Figure 7.18: should read ".... of the fifth hour of a five hour period".

p.I28,2"d paragraph: change "do not contaminated" to "do not contaminate".

p.130, 3nd to last line: delete "is developed".

Abstract

The Persian Gulf is a centre for world oil production, both on-shore and ofi-shore.

The coastai regions of Iran and the Arabian countries on the Gulf support industrialactivities and produce petroleum products which are shipped out through the Gulf.As a result of oil transport activities and consequentiai occasional oil spill events,

the hydrodynamics of the Gulf waters is of great importance.The tides of the Persian Gulf are very complex and while vast areas are very

shallow and populated by tiny islands and mangroves, a third of the Gulf is morethan 40 rn deep. What is needed is a modeling approach that can incorporate thedisparate bathymetry of the different regions more appropriateiy.

The aim of this thesis is to develop and apply mathematical models to examinetidal movement in the Gulf, especially around the Iranian coast which has largelybeen neglected in the past. To do this a new technique for improving numericalmodels of tides in coastai regions has been developed and applied to the Persian

Gulf.In -this approach a fine grid has been superimposed over important regions of

a coarse grid model in order to improve the accuracy of the model predictions inthese areas. Having completed one sweep of the coarse model with a large timestep, the information from that model is used as a boundary condition to solve thetidal equations with a number of smalier time steps within the fine-grid regions.These time steps are an integral fraction of the coarse time step, and are chosen tomaintain the same average Courant number of both coarse and fine-grid models attheir interface. Because of this, there is no abrupt change in phase-speed of modelledtidal waves, with associated refraction and reflection, at the junction of the coarse

and fine grids.The thesis discusses the development of and application to the Gulf of three

different numerical tidal models, namely a uniform grid Cartesian model, a uniformgrid spherical model and a spherical modei containing dynamically nested regionswith finer spatial and temporal grids. The following are used in this approach:

o The depth-averaged Cartesian and spherical equations of tidal motions.

o An explicit finite-difference scheme based on the "Arakawa C" spatial grid

o Caiibration against measurements at forty observational stations, in the Per-sian Gulf.

IV

o Dynamical nesting in space and time in shallow waters and other regions ofinterest such as areas containing large numbers of oil wells.

The results from the three rnodels have been compared with the Admiralty chart,

5081, and tidal constants computed from records from tidal stations. There is good

agreement between computed and observed values. In particular, results from the

dynamically nested model match recorded values better than those obtained fromboth the Cartesian and spherical models.

v

Chapter 1

Introduction

1-.1 Description of the Persian GulfThe Persian Gulf has the following specifications:

o Latitude: 24o to 30o north latitude.

o Longitude: 48o to 57o east longitude'

o Surface: 239000 km2.

o Length: 1000 krn.

r Depth: the Gulf basin lies on a shallow continental platform with a mean

depth of about 32n¿.

¡ Width: the width varies from a maximum of 340km to a minimum of 60knz.

Figure 1.1 shows the position of the Guif with the countries that surround it. It is asemienclosed marginal sea oriented in the north-west and south-east direction. It is

connected to the Gulf of Oman in the Indian ocean through the Strait of Hormuz at

its eastern end. The maximum depth in the Strait of Hormuz is aimost 90nz whichis much smaller than the typical depth of the Gulf of Oman (1000rn).

The Gutf is relatively young, having been formed 3-4 miiiion years ago (see Lehr,1984). A wide area of mud bottom is found in the northern and eastern parts of theGulf, with sand bottom predominant in the southern and western part. The bottomof the Strait of Hormuz is essentially rocky. The land south of the Gulf consists ofarid, flat desert stretching hundreds of miles.

The Tigris-Euphrates-Karun delta at the head of the Gulf provides a modestamount of freshwater discharge, known as the Shatt-al-Arab/Arvand river, into thebasin (see Chao et al., L992).

1

\

H !Àtrq Cl

j^tí i-.

h,lAi

-rn

ç

¡\I

YAZD

lB¡m

SÊi.4Ni\

Á rl Ê.1 rlt)l

FARS

"lrire't

\\

ri

Àft

\

Fand¡t,c

ú lÃO lìlirn-Et*rÈf**ïJ"."".l_*ro l$Ü ùÀ1Ér

*è

æ

0eå úûì i$? (E¡,r{ì¡Sf C-0ô

Figure t 1: A map ol the persian Gut! and. netghbouring countries

2

t.2 Importance of this studyThis study intends to provide an analysis of the water movement in the Persian Gulfdue to astronomical tides, especially in shallow, complicated coastal areas. Thisinformation will help oil companies, oceanographers, meteorologists and engineersas well as assisting mariners to navigate in these waters. Few numerical modelsfor the Gulf are available in Iran and the Iranian government is interested to havemore work performed on the Gulf. These studies are of great importance for thecountry in terms of economic, scientific and environmental factors. For example,oil pollution represents a serious threat to the total marine environment in theGuif. Research work in this respect should be developed in order to eliminate or tominimize the environmental damage caused by oil pollution and chemical discharge.Since poilutants are usually most concentrated in the shallow coastal areas, priorityshould be given to study these regions. Another economic importance of the studyis fisheries. There are three main fisheries of current economic value in the Gulf,pelagic or midwater fisheries, demersal fisheries and the shrimp or prawn fishery(see Crisp, 1976).

The aim of this thesis is to develop and appiy mathematical models to examinetidal movement in the Gulf, especially around the Iranian coast which has iargelybeen neglected in the past. This may help the marine science division in Iran,which is interested in gaining comprehensive knowledge about all natural phenomenawithin the Persian Guif. Furthermore, it may provide ne'cessary basic informationfor engineers to solve the problems of the coastai areas. Developing a suitablenumetical hydrodynamic model for the Gulf will help the country and the interestedresearchers to carry on more works in the coastal areas.

1-.3 Summary of the thesis

This thesis deals with three numerical tidal models, namely a Cartesian, a sphericaland a dynamically nested tidal model of the Persian Gulf. The main reason fordeveloping the Cartesian numerical model is to compare the results with that fromthe spherical-coordinate model, to determine the effect of considering the earth'scurvature in the computations.

To improve the predictions obtained from the application of a depth-averagedsphericai-coordinate model of the Persian Gulf, a new model is developed in Chapter6, with the advantage of having a fine grid in areas where high resolution is required,without using the fine grid everywhere with very small time steps. Results of usingthis model for the Persian Gulf are presented.

From the application of a depth-averaged spherical coordinate model of the Per-sian Gulf, close agreement between observed and computed tides was obtained, butto solve coastal engineering problems and to provide relevant answers for the otherquestions in the shallow and complicated coastal a eas, it seems to be necessary torefine the grid to improve the predictions there.

3

In this approach the aim is to dynamically superimpose a fine grid over certainregions of a coarse grid modei in order to improve the accuracy of model predictionsin areas of particular importance. Having completed one s\l/eep of the coarse modeiwith a large time step, the information from that modei is used as a boundarycondition to solve the tidal equations with a number of smaller time steps withinthe fine-grid regions. These time steps are an integral fraction of the coarse timestep, and are chosen small enough to maintain the same average Courant number ofboth coarse and fine-grid models. Because the Courant number is approximately thesame each side of the interface of the coarse and fine grid, there is no abrupt changein phase-speed of modelled tidal waves, with associated refraction and reflection.

4

Chapter 2

Mathematical Formulation

In this chapter the equations describing barotropic tidal motion are presented fordepth-averaged flows in Cartesian and spherical coordinates. In order to explain thecharacteristics of tidal dynamics in shallow seas, the physical parameters of each

equation along with the physical meanings of various terms are given.

2.L Tides and tidal predictionThe periodic rise and fall of the sea level is known as "the tide" . Most of the earth'soceans have tides with average period of L2.4 å, but some places have tides witha period of 24.8 h, for example, the southern coast of Western Australia and thePersian Gulf. Tide heights vary significantly; for example, on the Mediterraneancoast the tides are almost unnoticeable, whereas at the head of the Bay of Fundy a

tidal range of over 12 m has been reported, which is one of the largest tidal ranges

in the world (Greenberg, 1982).

2.L.L Tidal prediction

For prediction of tidal heights at a particular iocation, the rise and fall of the waterneeds to be recorded as a function of time. The observed tide-height consists of thesum of a number of simple harmonic or partial tides of different periods, phases andamplitudes. The period of each of the partial tides corresponds to the period of therelative astronomical motions between the earth, the sun and the moon. These con-stituents are usually classified into three categories; with periods of approximatelya half day (semi-diurnai), one day (diurnal) and longer. However, the observedtide-height must be analyze, to obtain its simple harmonic constituents, which maythen be used for predicting future tides. The usual and most useful method, whichresolves the observed tide into its simple harmonic constituents, is the harmonicmethod. Each constituent can be represented by a sinusoidal function. The am-plitude and phase describes the contribution of the constituent to the total tide.For any future time the predicted tidal height can be obtained by adding ail theconstituents for that place.

5

The basic equation of tidal heights (see Pugh, 1987, pp.19 and i12) is

((ú) : D f¡"¡ cos(a¡t *V"¡ I pj - T)k

j=l(2.1)

where

aj,'lj are the amplitude and phase for constituent j, respectively,uj is the angular frequency of the constituent,

f¡,tt¡ are the correction factors for the amplitude and phase, respectively, andV.¡ is the phase of the constituent j at Greenwich at the start of the epoch.

In this formula, the k major astronomical constituents are used.

The four major constituents are Ot, I{r, M2 and,Sz. Table 2.1 (Pond and Pickard,1993) lists the important constituents, and gives the Darwinian symbols and periodin solar hou¡s.

Species and name Symbol Period (solar hours)

Semi-diurnal:Principal lunarPrincipal solar

Larger lunar ellipticLuni-solar semi-diurnal

M2Sz

N2

K2

12.42

12.0012.66rr.97

Diurnal:Luni-solar diurnal

Principal lunar diurnalPrincipal solar diurnal

Larger lunar elliptic

KtOtPr

Qt

23.9325.82

24.07

26.87

Long period:Lunar fortnightlyLunar monthly

Solar semi-annual

M¡M^S"o

327.9661.34383

Table 2.1: Showing the characteristics of the principal tide consti,tuents. (SourcePond and Pickard ,1993, p.260)

2.2 Tidal dynamics in shallow seas

The shape and configuration of coastal basins influence the tidal motion whichoccurs in them. As the tide moves in the deep ocean a series of waves is produced.These \ /aves enter a partially enclosed basin, like a gulf, at its mouth and propagate

6

through the basin. As a consequence, the water level rises and falls as the water

enters or leaves.The most common type of tide is the serni-diurnal, in which two high waters and

two low waters occur each day, Figure 1 (b,c). The times of occurrence of high or low

water are not the same each day; they occur with about 50 minutes delay from one

day to the next. Another type of tide is the diurnal tide, in which there is one highwater and one low water in each lunar day (24.8 å), Figure 1 (u). The third type oftide is the mired tide which is a combination of the diurnal and semi-diurnal tides;essentialiy, they appear as semi-diurnal tides in which successive high waters have

different heights, and consecutive low waters have different depths as in Figure 1 (c).

For these reasons, mixed tides are sometimes called serni-diurnal unequal tides. Itshouid be pointed out that the magnitude of each type depends on the topographyof the basin and the location of the region with respect to the earth's surface.

(c)(o) (b)

HW HW

OLWLW

HHW

h9 LLW

HW

LHW

0 LW 24h

rrT24h

Figure 2.L: Showing: (a) diurnal, (b) semi-diurnø\, (c) semi-diurnal unequal tidesHW : high water, LW : low water, HHW : higher high water, LL\ / : lower lowwater, LHW : lower high water, HLW : higher low water. (Source: Pond andPickard, 1993, p.266)

2.3 Forces

Dynamical oceanography is concerned with the forces acting on the ocean watersand investigates the resultant motions. The important forces are:

o Gravity: In Section 2.1 the gravitational effects on the production of tideshave been described. This is the main force exerted on a body of water.

nI

o Wind stress: This force is generated by the wind over the surface of thewater.

r Atmospheric pressure: In some cases the effect of atmospheric pressure can

be ignored because it has a negligible effect, whereas in some other cases, likestorm-surge predictions, it is one of the main parameters.

o Coriolis force: This occurs as a result of the rotation of the earth around itsaxis and gives clockwise rotation to flows in large water bodies in the NorthernHemisphere and anti-clockwise rotation in the Southern Hemisphere.

o Bottom friction: This frictional force has the effect of retarding the flow.

o Viscosity: Viscosity is basically due to adhesion and molecuiar mobility of afluid. It opposes the motion of the fluid.

2.4 Depth-averaged Cartesian tidal equations

The depth-averaged partial differentiai equations for tidal flow in shallow water can

be obtained by integrating the three-dimensional equations over the water depth,from the sea bottom to the sea surface. These equations are based on the general

hydrodynamic continuity and momentum equations (Kowalik and Murty, 1993, andWeiyan, 1992). They are presented in this section, along with the physical meaningof each term.

Thle equations of motion used in depth-averaged hydrodynamic models are ob-

tained from the two fundamental iaws of physics:

(f) Mass conservation law

In a control volume V with closed surface S of a fluid, if mass is not generated ordestroyed in V, then mass is conserved. In other words:The time rate of increase of total mass inside the control uolume V : The netamount of mass fl,owi,ng out of the control uolurne V across S. For an incompressiblefluid this law mathematically can be written as

v'g:o

V is the gradient operator,q is the vector velocity field (rns-1).

(2) Momentum conservation law

The conservation of momentum law is based on Newton's second law, which states

(2.2)

herew

8

that: if a resultant force F acts on a body of mass rn then, the rate of i'ncrease

of li.near momenturn is equal to the force F. For a viscous incompressible fluid the

mathematical equation representing this law is

ry:-lvp ¡vy2qtF (2.3)Dtpwhere

p is the fluid density (kgm-3),,ú is time (s),V2 is the Laplacian operator,p is the pressure (N*-'),u is the coefficient of viscosity (*t-').

2.4.L The continuity equation

The equation that resuits from appiying the mass conservation law to tidal motionsis often called the continuity equation. The depth-averaged form of this equation(Bills, 1991, p.15) is:

ô.ç + Aqu) * ô(!r) :0.ot - o* ös (2'4)

where

trra(U,U

H

2.4.2 Physical meaning of each termThe terms in the continuity equation have the following meanings

is the time (s),are the horizontai Cartesian spatial coordinatet (-),is the elevation of the sea surface above mean sea level (rn),are components of vertically averaged velocity in the r and y directions,respectively (nzs-1),is the total depth (å + (), where å, is the mean sea depth (rn).

(") #rb) a(fl")\ / dr/ \ ô(Hu\lcl

-\ / o!

is the rate of change of the surface elevation,

is the gradient of z component of volumetric flow between surface and seafloor,

is the gradient of y component of above.

I

2.4.3 The momentum equation

For non-stratified well mixed coastal flows involving tides, winds and atmosphericthe depth-averaged momentumequation along the r-direction (Bills, 1991, p.15) is:

ðu 0u 0u , Ae I?p"

= + uï+u]-fu:-s- (2.5)0t *0r' 0y ¿' "0x p0x

and along the y-direction (ibid., p.15) is:

ôu 0u 0u , . 0e 10p";; -r ø-Tt/;--T ju:-Y-æ dr' 0y "09 pïy

1

HU(uto)'+(un)2-Ca u2 I u2,

(2.6)

+N"#-N"#+4",ou,o (rto)'* (rto)' ^1_ uuEu u2+u

The symbols used in these equations are:

t, rrA, (, H,u, u have the same meaning as in the continuity equation,

Í is the Coriolis parameter, and equals 2fl sin þ where f^l isthe earth's angular velocity, (r-t), and / is north latitude(positive northward),

g is the acceleration due to gravity (*t-'),p is the density of sea water, assumed uniform (: 1027 kg*-"),po- is the atmospheric pressure (N^-'),Nr, No are the eddy viscosity coefficients in the x-direction and y-

direction, respectively (*t-')po is the density of air, assumed uniform (: \.22Slrg*-t),cro is a dimensionless drag coefficient which depends on the wind

speed according to an empirical formula (Bills, 1991, p.13 ),?.tto¡'uto are the vertically averaged air velocities at a distance of.I}m

above the sea surface (rns-1), andCa is the dimensionless coefficient of quadratic friction.

2.4.4 Physical meaning of each termThe physical meaning of each term in the r-momentum equation is described below;the terms in the y-momentum equation are similar.

10

Ø*

(z) "# + r#

(a\ o%\ / rÕI

t5) !þ\/pOT

$) N,# + N,#(7) frcrcurc (uto)'* (rto)'

(8) cuþut@ +7

the time rate of change of ø componentof veiocity at a point, also called the Io-

cal variation of momentum with time,

the advective terms. The terms are non-linear and are sometimes called spatialacceleration terms,

the Coriolis term, a function of the ro-

tation of the earth,

the gravity term, which involves theforce due to the gradient of the surfaceelevation in the r-direction,

the atmospheric pressure gradient, inthe r-direction,

horizontal eddy viscosity terms,

the r component of wind stress, actingon the surface of the sea,

the r component of friction, acting onthe bottom surface.

(3) /,

2.4.5- Boundary and initial conditions

The solution of the partial differential equations (2.7)-(2.9) requires the definition ofboundary conditions, one condition per highest degree of derivative; for example, a

first order derivative in time requires one initial condition; a second order derivativein space needs two boundary conditions.

Assumptions

1. When solving the continuity and momentum equations in coastal regions, twotypes of boundary conditions usually appear:(a) If the entire coastal boundary consists of a shoreline with no river inflows,veiocities normal to the boundary are set at zercl in other words, it is assumedthat the model coastline is impermeable so there can be no flow through theboundaries.(b) If the boundary is an interface between two water bodies, it is called anopen boundary; the water level elevation ( is defined at all points on theopen boundary.

2. A no-slip condition is applied at the bottom of the sea.

11

3. The total depth is initially set to be the depth below mean sea level and thesea is considered stationary.

2.5 Depth-averaged spherical tidal equations

When fluid motion is considered over a sufficiently iarge region, the earth's curva-ture is important and should be taken into account. In this case, the equations ofcontinuity and momentum should be written in spherical polar coordinates. Thespherical polar coordinates, which will be referred to as spherical coordinates forsimplicity, are À (latitude) and / (longitude) and the radial coordinate, which has

direction opposite that of gravity.

2.6 Equations of continuity and momeritum

The differential equations which describe depth-averaged tidal motion are presented

below. These equations have been derived from the full three-dimensionai sphericaltidal equations (Semtner, 1986, p.188) and are the same as those used by Matthews(1995, pp.37-39).

Continuity equation

Ae , 1 ð(Hu), 1 ð(Hucosfl)_,.,At- Rr""a A^ - Rroró Aó -"' (2.7)

Latitudinal momentum equation

u ðu u 0u uutan$+ RAó

1

2ulsinS:

pRcosS ôÀ

2tanS Ôu

R2 cos S ôÀ

Ct

H(rto)'* (rto)'- u2 * u2,

0u---dt

+

+

+

(2.8)R

ðp" o.h 02u o,h ô , 0u.+

R, ,.rt ó ô^, + R, ,., ó Aó\cosa M)Rcos/ô)

(L - tan2 $)u

Longitudinal momentum equation

12

q A( I 0p" ah Ô2u ah 0 , ,0u,EAó- pR Aó

+ R'r"r'óAN - R'*tóAó\cosQ Aó)(I - tanz þ)u

where

^, ó are longitude (positive eastward) and latitude (positive

northward), respectively (deg),R is the radius of the Earth, assumed uniform (: 6.32 x 106 m),a,h is the coefrcient of horizontal eddy viscosity (*"-').

The relationship between dh,, N, and /f, is discussed in detail beiow.

2.6.L Viscosity coefficients

The horizontal eddy viscosity coefficients in the u-direction, ÀI", and in the g-direction, Iú,, are formulated as in Greenberg (1983), namely

¡{, - i, o, uo:lnr¡rl

0u u 0u uôu-l- I

--ôt ' ,Rcosgô) ' RAó-ry i2uesin/ - (2.e)

(uro)'* (rto)' -Ca

-1)H- u2 * u2,

+

+

and (2.10)

where Au and Ay are the eiement sizes (nz) and ø is the depth-averaged reduced hor-izontal eddy viscosity coefficient (s-1). The formulathat Greenberg (1983) providesfor eddy viscosity coeff.cients, is a simplification of Schwiderski's (1980a) generalformula in spherical coordinates. This formula has also been used by Bills (1991,p.16a) and Matthews (1995, p.35).

In this thesis, the horizontal eddy viscosity coeficient in spherical coordinates is

denoted as arl. It is formulated as

(on)^: |f ^n6 t p,cosS), latitudinat (2.11)

and

(on)ø : |t rn6 * p cos /), longitud,inal (2.12)

where tr.1 : EA) and L6 : RLó,A) and L,þ arethe element sizes (rad), a is thereduced eddy viscosity coefrcient, and p is a longitudinal grading parameter. Thisparameter varies by latitude; Schwiderski(l980, p.184) uses a value of ¡; - 1 ford : 60" to ó - -60o, which includes the region covered by the Persian Gulf.

13

Chapter 3

Finite Difference Techniques

3.1- IntroductionThe finite difference technique is the most commonly used method in depth-averagednumerical tidal models, for example, Heaps (1969, 1973), Blumberg and Mellor(1977), Leendertse (1967, 1970), Bills (1992), Bills and Noye (1984, 1986, 1987,

1992) and Casulli and Cheng (1992).

The basic idea of the finite difference scheme is to replace derivatives by finite dif-ference approximations. In this section the most commonly used forward, centredand backward difference approximations for derivatives of a function of one vari-able are described. These types of approximations are used in discrete formulationof the-depth-averaged tidal equations in Section 3.3.2.

Assume u is a function of one variable r, and suppose its derivatives are single-valued, and continuous functions of r. Using a Taylor's series expansion for u(r¿ tAr), about the point oi, where ti: ro I i\,r,, i: I,2,3r'", gives:

u(x¿ +Aø) : u(x¿) i (^ù(#),, + fi{a')'(#),,+ }{a')' r#¡,,+ .. ., (3.1)

If u(r¿) is denoted z¿ then u(r¿+ Ar) : z¿11, where Ar is the grid spacing in ther-direction. Using this notation in Equation (3.1) leads to

1ri+t:u¿ + (ar)( #r,* jr"a'rffi)n + fi{n' )"(ffi>, +'.. (3.2)

Also consider the similar expansion for u¿-1, that is,

,tL¿-L:z¿ - (az)( #r,* jr"a'rffi;, - fi{l, )'(ffi1, +.-. (3.8)

Using the above equations the required approximations for the first derivative canbe found as follows:

(1) Forward difference

I4

Rearranging Equation (3.2) gives

. . .d2u.(Az)( --)¿d:r'

(3.4)

(3.8)

ui+t - ui 1

2t

and from this equation the most commonly used forward difference approximationis found, namely

,du,,_u¿+t-ui. (B.b)\ dr)i

: A"

The remaining terms on the right hand side of Equation (3.4) represent the trun-cation error. Using the usual mathematical notation for truncation error, that is,

O(L,r), in Equation (3.a) leads to

r9l, : ui+t-- 1t'i + o(aæ). (3.6)\¿*)t- Ar rv\swl'

The term (A,r)(d2uldaz)¿lQ\ in Equation (3.4), which is proportional to Ar, is

the largest term in the truncation error. That is, the error in the approximationdepends on Ar to the fi.rst power, so this forward difference approximation for(duldx),; is said to be first-order convergent.

(2) Centred difference

If Equation (3.3) is subtracted from (3.2) and the result r,earranged,

,dur. + o(L,r)2d,r)'- 'ffi t O(Lx)' (3'7)

is obtained. This yields the commonly used centred difference approximation

L,r

Equation (3.7) shows that the error in the approximation (3.8) depends on Ar to thesecond power, so this centred difference approximation for (duldr); is second-order convergent.

(3) Backward difference

Rearranging Equation (3.3) and using the notation for the truncation error yieldsthe expression

, du , LIi - u;-t(ï)n:T+o(Ar)dr /).r

(3.e)

This equation gives the most commonly used backward difference approximation

.du. 'u; -'tt;-t(*),=Ë. (3.10)

As Equation (3.9) suggests, the error in this approximation depends on Ar to thefirst power, so this backward difference approximation for (duldæ); is first-orderconvergent.

du

d,

15

The commonly used approximation for the second derivative is one which involves

central-differencing, and is obtained as follows:Adding equations (3.2) and (3.3), rearranging and using the notation for truncationerror to obtain the expression

r#1,:%#vt*o(a,x)2, (3.11)

so this centred difference approximation to the second derivative is

,d'u, ui+t - 2u¿ I u¿-t/_ì N_ (3.12)\ ¿rz ), - (Ar),

3.L.1 Explicit and implicit schemes

A fi.nite difference scheme which expresses a single unknown value at the new timelevel directiy in terms of known values at the present time level, is called an explicitscheme. A scheme which calculates the unknown values at the new time level bysolving a set of simultaneous equations is known as a implicit scheme. In thelatter scheme the values at a new time level depend also on the unknown values

at the new time level. The former method is computationally simple, and needs

less storage than the latter. However, the expiicit technique must satisfy a stabilitycondition, whereas the impiicit technique may not. Implicit methods are generally

very computationally stable and therefore may use large time steps. However, itmust be kept in mind that making the time steps very large may cause the loss ofaccuraty of the approximate solution.

3.1.2 Arakawa C spatial grid

The frnite difference representation of horizontal space derivatives depends on thedistribution of the dependent variables on a grid. Various types of grids have been

developed and used by different authors for different purposes. Arakawa and Lamb(L977) investigate four different spatially staggered grids which are used to differ-ence the shallow water equations. They consider the space discretisation error fordifferent distributions of the dependent variables on a grid, and suggest the use ofthe layout shown in Figure 3.1. This iayout is known as the "Arakawa C" gridor the "Richardson lattice". It has been used by many authors, for example, Leen-

dertse (1967, 1970), Flather and Heaps (1975), Blumberg (1977), Davies and Flather(1977,1978), Greenberg (1931), Stevens and Noye (1984), Bills and Noye (1984,

1987,1992), Noye and Bills (1990,1992), and Bills (1992). Some of the advantagesof using the "Arakawa C" grid are:

o It allows spatial derivatives to be conveniently approximated in second-ordercentral difference form.

o Land boundaries can be simply represented, by means of straight-line segmentsparallel to the coordinate axes.

16

u u

a

Figure 3.I: The Aralcawa C grid or the Ri,chardson lattice.

3.1-.3 The temporal grid

In the following a grid staggered in time as well as in space is used (see Figure 3.2).

The general finite difference scheme used is three-level in time (see Noye and Bills,1990, L992, and Bills, 1992), that is, each timestep involves three timelevels: theentry level, the half time step level and the full time step level (see, Bills, 1992,

p.22). Figure 3.2 shows the time levels included in a computational time step. As

this figure shows, in each time step one calculation of ( is carried out for each set ofcalculations of u and u. This calculation uses ("-rl2, un and un to provide a timecentred (n+r12. To obtain ("+t th" values (n, un*r and u'*1 are used.

3.t.4 The finite difference code

A number of finite difference codes exist for solving the depth-averaged tidal equa-

tions, for example:

1. In Cartesian coordinates: Blumberg and Mellor (1980), Lardner et al. (1982),

Murty and EI-Sabh (1984), Casulli (1990), Bills (1992), Casulli and Cheng(1992), and the Proudman Oceanographic Laboratory model (Proctor et al. 1992)

2. In spherical coordinates: Davies and Flather (1977, 1978), Evans-Roberts(1979), Chilicka et al. (1983), Mitchell and Noye (1983), Verboom et ai. (1992).

The finite difference code described in Biils (1992) has been chosen because it has

been thoroughiy tested and has been used for some years by the tidal modellinggroup of the Department of Applied Mathematics at the University of Adelaide.

a

17

The programs, written in Fortran 77, have been converted, by the author, fromthe original Cartesian coordinates to spherical coordinates. The code has also been

modified to produce the nested spherical model of Chapter 6. The basic idea of thecode along with some necessary definitions will be given here, to allow an explana-tion of the development of the nested spherical model later. The original idea and

definitions come from Bills (1992, pp.17-70).

n+I n*I)a tn+Isun+I

n+rl2 (n+rl2

LLN ,un)n

n-Il2 ( n-I12

Figure 3.2: The time leuels in each time step

3.2 Discrete formulation in Cartesian coordinatesIn this section, the governing equations are discretised on the "Arakawa C" spacegrid and the time-staggered grid.

The grid element, the basic unit of the grid, consists of three grid points, whichare the free surface elevation (, and horizontal velocity components u and u, as

shown in Figure 3.3.This figure shows that the spacing between different grid points in the ¿-direction

and gr-direction are Aæ and Ag respectively, and between similar grid points it is

2Aø and 2A,y. Therefore the area of the element is 4AøAy.

18

Lrt-

aÇn U¿

Lyr

U¿

Figure 3.3: The ith gri,d element showing locations of grid poi,nts e¿, u¿ and u;. L.rand L,y are the grid step size. The dashed lines show the boundaries of the element.

Figure 3.4 shows the numbering system of the ith computational element and

its eight neighbours, used in the code. The elements above and below the ith are

indexed as ilth and i2th elements, respectively. The immediate elements to theright and left side of the ith, are specified i * 1 and i - L, respectively. If the modelneeds to refer to the next neighbours, other than the fi.rst eight neighbours of theith element, this indexing system can be generalized to a 25-element block by usingindices in, izz, etc. The i11th element is located above the ilth element and i22this below the izth element, and so on. This 25-eiement indexing system is used inChapter 6. It should be noted that the calculations in each time step are carriedout by passing from one element of each model row to the next from left to right,and proceeds down the rows from top to bottom.

19

r t- r+ +

I

+ ++'h*1

t

,h-7 ?'1

Ir

t-

t- t-

++i-r

t

+

I

+->i+I

II'

t- t-

+'iz-l

Ix2 'izII

I

Figure 3.4: The ith computational element and its eight irnmediate neighbours, show-

ing the i,ndeúng system used to lable the elements. The dashed lines represent ele-

ment boundaries (see Bills, 1992, p.20)

3.2.L Discrete formulation of the governing equations

The finite difference equations which approximate the depth-averaged tidai equa-

tions in Cartesian coordinates, that is, equations (2.4), (2.5) and (2.6), are presented

in this section. These equations are similar to those used by Bilis (1992, pp.24-70)for three-dimensional tidal equations, but the complete form of the approximationsto the depth-averaged equations is given in this thesis.

The depth-averaged continuity equation (2.a) is written as

a( 0@u) 0(Hu)(3.13)

+

I

+ +

0t 0r ôy

Consider the finite difference expression of the above equation; centring it about thetime level n, in discrete time, and the ( grid point, in discrete (horizontal plane)space (see Figure 3.3), gives

I ae\" _ _lô(Hu)\" _ Ja(¡/u)\"tdi,,:-tË1,,-tËlJ,, (3'14)

20

The difference approximations of the terms in this equation are:

(i)ae...-öt

1 (¡.n+t¡z - ¡.n-r12\

Aú\5' 5? ),

(ii )

n+r/2

ui- {f"}7",

(3.15)

(3.16)

(3.20)

)_

(iii )

J a@ù\" - + l@: * Hi*,),T - (Hi_, + Hi) u?_,),\ a" I r,- +L,*

I a(¡/u)l\avl

n1

4La l@i + Hi) ,T,, - (Hi, + Hi) uil . (3. i7)

Expression (3.15) is second order convergent, because it is a centred difference intime, and (3.16) and (3.17) are second order convergent in space because the dif-ference uT - uT-t is centred about Ç.

By substituting equations (3.15), (3.16) and (3.17) into equation (3.14), we ob-tain the explicit expression for the sea surface elevation at time levei (n + ll2),namely

çn+ttz = (¿n-Ltz #fg, + Hi+,) u? - (Hi-, + Hi) u1-l (8.1s)

* f çri + Hi,) ,i, - @i, + Hn ui) ,4LY t

which-is second order convergent in space and time.The total depth H : e * h, at time (n + Llz)Lt, is approximated as

H"+'rlz - (0"*'/' + ho, (g.1g)

where fI¿ is computed at the same position in the horizontal plane as Ç.The finite difference expression for the ir-momentumequation (2.5), using centred

differences about u¿ in space and about the (n +I12) levelin time,is based on theequation

ôuðt

j",*,''' * {*,#}",*,'''

+/2

u2 +u2',{+"Note that the wind stress and the atmospheric pressure are ignored

The terms of this are differenced as foilows:

2l

(i)

{#}

n+Ll2

u;

n+t/2 IAf

,T+ 1( "?)(3.21)

(3.22)

ur

which is second order in time, because this is a centred-difference in time.

(ii)( a"_\"* '

= u,n*, {y\" - r ,,?*L (,uT+t - uT_r) ,

\" a*l ,, t u'¡''-' t *,1,, = +L*uo"'' \

which is approximately second order in space'

(iii)

{,H\",.,'t' = {u}i",{HY,,= nit (uT, + uf,,¡1* ui + ui*,) (uT, - u7) , {z.zz)

which is quasi second order in space.

(iu) r

{f r}T,, = i ff' * /,+r) (rf, * rT,+r r ui -t ui*1) ,

in which the Coriolis coefficient /¿ is given by f¿:Z0sin'$¿-

(3.24)

(")

((;¡{+tlz - ,n"+r/2) ,,

a(A"

i2A,r

(3.25)

(3.27)

which has a second order spatial discretisation error.The depth-averaged horizontal eddy viscosity coefficients are given in equation

(2.10). They are approximated as foliows:At the u; grid point and the (n + ll2) time level

{N,}i:'/' : t})lzn,l {H}::'/' (3.26)

(ø)(aø)(u7*'r' + Hîlr'/')

2

and

Ì{Ns riltztal {H}i:'t'

= (o)(as)(u7*'P + uîí'/')

nIu¿

I It

2

The approximations (3.26) and (3.27) are used in the terms (vi) and (vii), respec-tively.("i)

{*,#}",*,''' ñ {N,}::'/'{#}",, (3.2s)

", # (u:.'t' + Hïí''') ("7*, - 2uT + uT-,) ,

22

which is approximateiy second order in space.

(vii)

{No}î!'t'

= 6{r:.1t2 ¡ ¡¡?lrtt2

which is approximately second order in space.

(viii)

Ca !ñ2Ca

Hi+rlz * i+1

{*,#}",*,'''ô2u

æ 1'17

(3.2e)

(3.30)

(3.31)

(3.32)

¡ u?,) ,)(,i- 2uT

1

E"n* Ll2

n*l,

@Ð' + # þ, * uT,+r * ui -r ur*r)'

Substituting all the approximations (3.22)-(3.29) into (3.21), gives the correspondingfinite difference approximation

*Qr,-"7)

X

+

+

:

+

+

"7, - "L)

"T - & (ri-r+ ri,+, t ui + ui*r) (rT, - rL)

f; * f¿+t) (ri, + r?,+, * ry + u7*r) - t# (e¿+{+t/z - ¡ ,.+t/2)

*uT*' (uT*, - uT-,)

át (ui, + uT,+, t ui + ui*) (

I fr, * Í,+r) (ui * ui,+t + ui + ui*r)

-t*;(Ç+{+'t'- i.',+tt2)

# (rf.'/' + Hîlr''') ("T*, - 2uT + "T-,)

* {r:.'/' + Hïï''') ("?, - 2uT ¡ "T,)

2Ca(uÐ' +* (r* * u?,+t * ui t rï*r)'uT+t

(u7*'r' + nîÅ't')

Values of u at the new time levei (rz + 1) are given by making ui+r the subject ofthe formula (3.31), yielding

uT+I D;

Aú-:E

aL

1:

+

+

+

(

t8AraLt

(n7*'t' + nîÅ't") ("T*, - 2ui + ui-r)

- 2"7 * rT,)] ,'') ("?,8av(u7*'r' + HiT:

23

where

+

D, : ft * n* (uT*, - uT-,)

zCaLt

(n7*'r' + nîlr'/")1_L_

'16t ^:2\ui ) (rf, * uT,,+, i ui + ui*r)2

(3.33)

(3.34)

This is second orde¡ in space, and is approximately second order in time.

At the u¿ point and the (n + ll2) time level, the depth-averaged y-momentumequation (2.9), namely

. {,#\":,''' * r,ur**'''+\":,'''

ôu

At

ô2u

æ¡ú,

u

It

+

n+t/2

U¿

Ju'?Ñ(I

)

-

o¡

ì.¡¡"

*'Y*\

)

n+rl2 l2nj

u.-g +

ui

Ca

is differenced in the same \4/ay as the c-momentum equation. To make it easier forthe other users of this model, the finite difference expression for the y-momentumequation (2.6) given in detail, because indices are different since the list runs in theø-direction and not in the gr-direction. The averaging is different so, the values of /are obtained differently, and updated values of un*L are used in place of u".

The approximations for the terms of (3.3a) follow:(i)

/ ^ : n*ll2

\#I ",-'- = h(u:*' -,i), (8.3b)

which is second order in time, because this is a centred difference in time.(ii)

{r3}"*' =unt,/-[y\" - ]

I ot ) u, \a* J, - 16; ("T*' + u1il t uT'+' + "T:-\) (ui*' -'i-') '

which is approximately second order in space. Note that the values of u at tii"t"'l(n + 1) are no'w available (the z-momentum equation is solved first).(iii)

{,H}":,''' =,n*, {H\ = ù,r*, (,î_,1), (3 32)

which is approximately second order in space.

24

(i")

Once again the values of u at the level n * 1 are now available. In this expression

f¿ :20sin d¿.(")

{f u}î!'t'= } tf, -r f¿z) (uT*' + uTjl * uTr+' + "i{}r) . (3.38)

n+Ll2ae

ôy(3.3e)

vl

which has a second order spatial discretisation error("i)

n+Ll2

{N,}::'/'{y-Y"{-"#} (3.40)

which is approximately second order in space.

(vii)

n+L12

h @r.'t' + Hî,*''') (ui*, - 2ri + ,i-,) ,

ru {Nr}?,!'/'{#}",

= * {rf.rtz ¡ ¡vn+r/t) @i, - 2ri + ,T) ,

02u

aytNy (3.41)

ui

1

H'

which is approximately second order in space

(viii)

Cu u2+u2l2Inl

(3.42)

(ui)'+fr þt.' +uTI tuTl'+,Tï-',)'

Note that updated values of u"*L are used in (3.42), which is different from the use

of u' in (3.31), used in the ø-momentum equation.Substituting approximations(3.35)-(3.42) into (3.34) yields the approximation

X

*þ,- - rî) 1("7*' + "Tjl + uir J- "Tr-r) ('i*, -'î-')+ (3.43)

16Ar

+ &,r*' @i, - uT,)

I t,t, * fn) (ui+' + uijl + uTz+' + rTr\)

-t* (eo"*'t' - Çr"*'t')

25

+

+

o,

8Aso,

84"

(u7*'r' + uî|'t') @i, - 2ui + uî2)

(u7*'t' * Hir+ttz¡ (uT*, - zui + ui-1)

uT+' : r;'lrf - & (uT*' + uTI * ui2 t uïr-r) (,7*, - ui-,) (8.44)

Lt.-5 (/; t f;z) (uT*' + uïjl + r?l' + ui,i!,) - t# (eo"*'r' - Cor^*'/')

+ # frr.'t' + Hîl'/') {,i, - 2uT t uî,)

+ # @r-t/2 ¡ ¡¡ft+t'') (ri*, - 2uî * ,1,)1

@Ð' +fr ("t*' + uTjl + uTl' + uTf-',)'

(3.45)

2Ct

(n7*'t" * Hir+rlz¡n*Lu

Values of u at the new time level (n + 1) are therefore given by making uf+l thesubject of the formula (3.43), giving

where

AtDs 1 + @i' - uir)

4La

2CaLt+

(n7*tt' + nir*'t')(uT)' +* þt-'' + uTI + uTl' + uTl-',)'

This is second order in space, and is approximately second order in time.

It should be pointed out that in equation (3.18) the value of. H",, which is as thefuil time level, is required. Values at the full time level are found using

(¿nr' : c;' # t("t+t/2 + Hi¡'t') @i+L + ui) (3.46)

(n;1,r, + H?+L/2) @?!i + "l_,)]Aút&K'r+Lt2 + Hi+t/z¡ @i'+r + ui)

(Hrr*'t, + Hi+ttr) @i*, + rî)l .

which is second order convergent in space and time.The total depth H : e j h, at time (rz + 1)A¿, is approximated as

Hi*t - Ç¿"+' + h¿,, (3.47)

Note that, in the above ç'n+tlz are calculated for ali i, then the pair u,¿n+!,, ts'n*r

are calculated for all i, and finaily Ç"+1 calculated for all i.

26

3.2.2 Boundary conditions

In finite-difference models the land-water boundaries of any shape are approximatedby grid line segments, parallel to the u and y axes. The boundary conditions used

for the depth-averaged Cartesian tidal equations are similar to those commonly used

in the literature, for example, Flather and Heaps (1975), Greenberg (1983), Noye et

al. (1982), and Noye and Bills (1992). These conditions are:

1. Assuming the model coastline is "solid", that is, there is no flow across thecoastal boundary, the following condition of impermeability is set:

(u,r) . rù :0 on the coastline, (3.48)

where rz is the normal vector to the coastline.

The coastal line segments pass through the velocity grid points u or a on thestaggered grid (see Figure 3.2), therefore

(i) if the coast is parallel to the ø-axis,

u:0atugridpoints;(ii) if the coast is parallel to the y-axis,

u :0 at u grid points-

2. The open boundaries of the model pass through elevation points and gridpointsat which the velocity is tangential to these boundaries. At the position ó onthe open boundary of the model, the sea surface elevation (6 is specified forall time by means of ten major tidal constituents, using

10

(¿ : D f¡a¡cos(a¡t *Vo¡ * tt¡ - l¡) (3.4e)j=L

Parameters in this formula are defi.ned in Section z.I.L. The ten tidal compo-nents provided by Proctor et al. (1992), see Section 4.4.

3. There is some discharge from the Arvandrood/Shatt-al-Arab, at the head ofthe Persian Gulf (Figure 1.1), into the basin. This may require a boundarycondition, other than u : 0, but data was not avaiiable and the discharge 'ñ/as

assumed to be negligible compared with other effects. It should be noted thatin recent years, reservoirs and dams in Iraq, Syria, and Turkey have reducedthe river discharge (Chao, et al. 1992) so that assuming a zero velocity in thisregion is considered to be a reasonable approximation.

27

3.2.3 Initial conditionsThe initial conditions for the depth-averaged tidal equations are similar to those

used by Flather and Heaps (i975), Greenberg (19S3), Noye and Bills (1992), Bills(199i), and Matthews (1995).A common method for setting up initial conditions is to start from a zero condition.That is, for all points in the interior of the model, set

C=u:?):0 attimeú:0 (3.50)

which means that the fluid is initially at rest. This assumes that the dependenceof the solution on the initial conditions will rapidly decay with time (Kowalik andMurty, 1993, p.112).

Using zero initial conditions (sometimes referred to as a "cold" start) produces

starting transients. The model must run until appropriate flow conditions are sim-ulated everywhere within the model area, and so a few tide cycies are required inorder for the transients to die out. In this way an acceptable "steady" periodicsolution, is obtained (see Figure 3.5.)

Y

Acceptable "steady" periodic solution.

x

hitial trqnsients

Figure 3.5: The initial transients and periodic response of the wl,ues.

3.3 Discrete formulation in spherical coordinatesThe spherical coordinate grid has been widely used in the computational oceanog-raphy field, for example, Mitchell and Noye (1983), Hunter (1984), Fang and Yang(1988), Verboom et al. (1992) and Davies and Lawrence (1994).

The explicit finite difference technique used to discretize the spherical tidal equa-tions of (2.7), (2.8) and (2.9) on the "Arakawa C" grid in space. is analogous tothat used in Section 3.2.3 to discretize the Cartesian tidal equations. This is nowpresented.

28

3.3.1 Computational grid

Figure 3.6 shows the discretised locations of Ç¿, u¿ aîd u¿ in the ith spherical grid

element. Here AÀ and A@ are the grid step sizes between grid points in the / and

À directions, respectively.The elements are indexed in a manner analogous to that described for the Carte-

sian model in Section 3.3. Figure 3.7 shows the type indexing system of the ithcomputational element and the eight neighbours.

Aì"

A0

Figure 3.6: Theith spheri,cal grid element showing discretised locations of (¿, u¿ and

u¿, whlere L,À and A,S are grid si'zes(degrees).

Figure 3.7: The ith spherical computational element and its eight immediate nei.gh-

bours, showing the index system used to label the elements. The solid l'ines representelernent boundaries.

rÊ- :'i

Ç,

I

____>

u I

ai

+*lr+L

I

/ir- | LL

*-

++¡- 1

II

+-t

+t+7

+ir+l

1

29

3.3.2 Discrete formulationThe finite difference equations used to approximate the depth-averaged spherical

tidal equations, (2.7)-(2.9), are presented in this section. They are approximated ina similar manner as depth-averaged Cartesian equations (see Section 3.2.1), for this

reason the approximations are not presented in detail'The depth-averaged continuity equation (2.7) is written as

ae

at ,Bcos / AÀ Rcosþ Aó

0(Hu) I 0(Hu)(3.51)

1

The latitude / is evaluated at grid points e¿, u¿, and u¿. These values will be denoted

as óe¿, óu¿ and do;¡ respectively. The latitude at the Ç grid point is the same as the

latitude of the u¿ grid point, since both are horizontal grid points located in a plane

parallel to the equatorial plane.It should be pointed out that in spherical coordinates the arc lengths are 2.R(AÀ)cos {

and 2RL$, which correspond to the horizontal grid spacings 2Ar and 2L,y respec-

tively in Cartesian coordinates.The fi.nite difference approximation of equation (3.51), centred about the time

level n in discrete time and the ( grid point in discrete (horizontal plane) space, is:

fL1

.R cos þ

I 0(Hu cos {)e¡

.R cos / Aó

nae

ðt(3.52)

(;

By- substituting the required approximations into equation (3.51) the explicitexpression for the sea surface elevation at timelevel (n +112) is obtained as follows:

ç.n+Llz = ç.n-t/z (9.b3)

= - 1' , l(ni + Hî*,) uT - (Hî-, + ui) ui-,)

4&L^cos /1, L\

*^rlu''r * nr,),r,-@h+ Hnu?)

+ ^j# þT +,r,) Hi,

which is second order convergent in both space and time.The total depth, H : ( ! h, at time (n + Il2)Lt is approximated as explained

in Section 3.2.I.The finite difference approximation for the latitudinal momentumequation (2.8),

using centred differences about u¿ in space and (n +I12) in time, is obtained by

I}u\"*t'' - r! u }u\'*t" 1l au|"*'t' I

\at J,, Ã t cosþ0rJ, *;\'ã¿} ,, - i.{"'tan$}i+Lt(t's+¡

- {r,}î!,,,: -#{* x}",*,'''

30

+

+

1

R'

iR'

Cu

(an)¡ ô2u

cos2 Ó A^2

{

n+712

ui

1I-'Rz

("n)o o

"", óãó

nlLl2

ui

. ôu.ìGosavlJ

(o¿)ø(i -tan2 S)u)nrrlz 2 lþn)^tan/ôului - Ar l-;* d -ãiJ

n+Ll2

ur

1

HUu2 +u2

n+7/2

Substituting all required approximations into equation (3.5a) gives the followingformula for calculating the values of u at the new time level (n * 1), that is,

uT+' : ,;'l,T - #e Qi +,T,+, * oy +,y*,) (,7, - ,L) (8.55)

+ ofi"f (rT, + u1,+t * ui + ui*1) tanþu,

Lt.^+ , (/, * f+r) (,f, * ui,+t * u7 +,i*,) - #ñ Gilr''' - Ç*'t')

+ 1çon)^\?"!'t' ("7*, -2ui + ui-r)

+ {(on)a}?"!'/' ffi{uT, - 2uT + uT") + {(on)ø}?,!'t' tt(t- tan2 ó,,)uT

1 ( oo ) ^ )ïl''' ?*^^W@?*, -,t, )],

where

Dó

+ {ru}i!,/, : -*{ffi\":,''' -

31

t-uT-uT+ ) (3.56)

âj (rt * ul,+t * ui -r ui*,) tanþ',

(n7*'r' + nîí't')(uÐ' +* trt * uï,+t + uT + oï*r)'

The formula for uf;+r \s second order in space and is approximately second order intime.

At the u¿ point and the (n+L12) time level, the longitudinai momentumequation(2.9), namely

{X\":,'' + +{...u;X\":,''' . i{.#\":,''' + f,{,' tun 6\**'/(s sr)

2CbLt+

+

+

r !(an)¡azr¡n+rtz*l !¿z \cos2óA^'J,, ' .R2 l.

Lu {ø^t rr1 - tan2 ó),}::', * #{ e# X\".*,'''

(on)o ocos S 0þn*rffi,\".*'''

u2 +u2

is differenced. in the same way as the latitudinal momentum equation (2.8)' that is,

values of u at the new time level (n + 1) are given by the following formuia, which

is

t,{+, 1n+t/z)Ju, )

,T+'

lion)r)î,*' '' noø# ,"r, o-(rh, - 2ui + ui-1)

(3.58)

+

+ {(an)ø}î!'t2 tt 4Rr(Ló)2

{(or)o}|!V' L}O-

tan2 ó,,)uT

{(,n)^}T,*''' #kffi.*?+,, -"t-, )]

u?'11 -2ui *uil,an þ,,

_+

in which

+

ln¿/¡t

+ *oa¿ (

2CuLt

,T, - rîr)

+

D) (3.5e)

(n7*'t' + Hî2+r/2)(,?)' + i @r.' + uTJ' + rTï' + uTl-'r)'

This is second order in space and is approximateiy second order in time

3.3.3 Computational algorithmThe governing equations are solved on the basis of the following algorithm:

1. Input model data and set initial data. Time ú : 0 (that is, n : 0, and

t: nLt,also (¿ å : (oo : u?: u? :0',H? : h¿ and' on the open boundaryHe:h+i'l).

32

2. Update model time to level (n + ll2),, so ú : (n + Ll2)Lt.Update open boundary elevation to time level (n + 112).

Solve the continuity equation (2.4) in Cartesian or (2.7) in spherical coordinates to find en+rlz using Hn+tlz.

3. Update model time to ievel (n * 1). Time t : (n + 1)Aú.

Solve the momentum equations (2.5) in Cartesian or (2.8) in spherical coordi-nates and (2.6) or (2.9) for u'*1 and u'+1, using (-n+112.

4. Update open boundary elevation to time level (n + i).Solve equation Q.\ or (2.7) for ('+1; using vn*t,, pntL and H"+t.

5. Return to step 2 and continue until the period of the simulation is completed.

3.4 Programming scheme

In this section some important features of the programming scheme for solving thefinite difference equations are described, based on the schemes developed by Bills(1992). This procedure will be modified for the development of the nesting modelof the Persian Gulf, in Chapter 6, so it is necessary to present the basic definitionsof the scheme here. These are:

1. The element classification:Regarding the elevation gridpoint of each grid element the following definitions(Bills, 1992, p.36) are given:

o If the elevation point lies on land, the grid element is called "type 0" andcoded as '0';

o If the elevation point lies on water within the model, the grid element is

called "typ" 1" and coded as '1';

o If the grid elevation point lies on an open boundary, the grid element is

called "typ" 2" and coded as '2;

¡ If the elevation point lies on water outside the model region, the gridelement is called "typ" 4" and coded as'4'.

2. Labelling conventionThe immediate family of eight neighbour codes of the central computationalelement is labelled as in Figure 3.8.

.)J

r t- r

IVCI t{c2 ¡\rC3

t- t-

¡rc8 ?, tvc4

t- t- t-

tvcT lvc6 tvcs

Figure 3.8: The labelling conuention for the eight immediate nei,ghbours of the

central element i. (Source: Bills, 1991, p.97)

As Figure 3.8 shows, the neighbours a e labelled clockwise from the top-lefteiement of the 9-element block. It should be noted that, at each computationthe finite difference equations use a selection of neighbours. For example, forthe finite difference approximation of the continuity equation, the elements

labelled, NCz, NC+,IúCo and /úG will be used.

3. One-dimensional a raysFor the purpose of computation, several permanent one-dimensional arrays are

created. A simple example of a study region is presented below in Figure 3.9,

to illustrate how the codes are stored and retrieved, and how the arrays are

constructed from the neighbourhood. A similar process may be used where

the elevation point of the computational element lies on an open boundary.

By using the definitions of the four types of elements, the array correspondingto Figure 3.9 is shown in Figure 3.10. This array is called the type matrix ofthe model region.

r

34

ç8

6'8 ernîxi 6utsn'uoz6a.t ern to sepor pooqrnoqqîÌeu eUl to ñ,nttn eUJ :1I't arn3r¡

'suerD puDI erv suoz7at peq)?Dq e?t¿ 'l' - nq unoqs erD sez¡x?oPn eq¡ ' þu6ts I

aqy q6nolt¡t) Tapout eql {o sezrupunoq uado aqy a"t'o s7uau,6es euq pe??op eql puv

seuvpunoq IDlrIDæ ato syuaufias euzl pt'log 'uoúar lepour o 6ur,moqg :6'g ern3rg

+

0 0 0 0 0 0 0

0 I I T II 0

0 I I T ZZ 0

0 Z Z Z Z v v

0 v v v v v v

SN++ + +

+ €.+aI

o++

I

+

+a. ¡orol¡ o ùa a'L oro*o l¡

++ + +

I

+

Chapter 4

A Depth-averaged CartesianModel of the Persian Gulf

4.L IntroductionIn this chapter the deveiopment of a Cartesian depth-averaged model of the Persian

Gulf is described. The main reason for developing this model is to compare the

results from it with those from some other works who used Cartesian coordinates,

and also with the spherical coordinate model in chapter 5, to frnd out the effect of

considering the earth's curvature.Figure 4.1 shows the locations of some places inside the Gulf which wiil be

referred to later.

K

IRAQ

SAUDI ARABIA

Bushefu IRAN

Bandar

Gulf of Oman

GuIf of Salwa QATUNITED

EMIRATES

UNITED ARAB EMIRATES

Figure 4.L: The locati,on of some places in the Persian Gulf.

aof

O

OSafanþ

tr

BAHRAIN

TDoha Duba¡y

Ah¡ Zabi

PERSIANGULF

36

4.2 Tides in the Persian GulfExisting information on tides in the Persian Gulf is based primarily on the Admiraltychart, which has been constructed from observations available at a number of sites

along the coast and a limited number of points within the Gulf. By their very nature,

the Admiralty chart only provides approximations to the true tidal conditions withinthe Gulf. However, they form an initial basis for understanding tidal movement inthe Persian Gulf and for comparison with mathematical models developed withinthis thesis and by other researchers whose work is reviewed in Section 4.3.

In Figure 4.2 the locations of 84 tidal observation stations in the Gulf are given,

along with four stations in the Gulf of Oman. The density of observation stations is

high along the coasts of Saudi Arabia, Qatar and the lJnited Arab Emirates. This

map, together with corresponding sets of harmonic constituents for Ot, Kt,, M2 and

52, was supplied by the National Tidal Facility at the Flinders University of South

Australia, based on an electronic record of constants originally produced by theBritish Admiralty. From the analysis of the tidal heights of the stations in the Gulf,

the Hydrographic Department of the British Admiralty published comprehensive

charts of co-amplitudes and co-phase lines for the principal constituents Or, Kt, Mz

and ^92. Figures 4.3-4.6 show the Admiralty chart, 5081, reproduced by the Danish

Hydraulic Institute.

47' 48' 49' 50' 5 1' 52' 53' 54' 55' 56' 57' 58' 59' 60' 6 1', 62'31'

30'

29',

28',

27'

26

25'

24'.

23'

Figure 4.2: The locations of the 84 tidal obseruation stations in the Persian Gulf(Source: National Tidal Faci,lity at the Flinders Uni,uersity oJ South Australia.)

31 '

30'

29',

28'

27'

26'

25'

24',

23'.

47' 48' 49' 50' 51' 52' 53' 54' 55' 56' 57' 58' 59' 60' 61', 62'

ÓJ

The tides in the Persian Gulf are complex and consist of different types as 'rr'as

shown in Figure 2.2. The tidal ranges are large throughout: over three meters at

Shatt A1-Arab/Arvand Rood at the head of the Guif, and over a meter everywhere

else (Lehr, 1984, p.4). Tidal energy enters the Gulf through the Strait of Hormuz

and progresses up the Iranian coast and down the Arabian side as a Kelvin \ryave.

The resultant of these two opposite Kelvin waves forms a set of amphidromic points(see Figures 4.3-4.6). The Or and .I{1 components have a single amphidromic point,which is set at the centre of the Gulf (see Figures 4.3 and 4.4, respectively). The

M2 and,9z constituents have two amphidromic points, located at almost a quarter

and three-quarters up the Gulf from the Strait of Hormuz (see Figures 4.5 and 4.6,

respectively). It should be noted that if the incident and reflected \/aves have the

same amplitude, then the amphidromic points will be situated at the points equi-

distant from both coasts. However, as Figures 4.3-4.6 show, they are closer to theArabian side, because the reflected wave is weaker, and because the speed of thewave is a function of depth, and the Arabian side is shallower than Iranian coast.

The principal semi-diurnal Mz and ^92 and diurnal Kt and Or constituents are thedominant tidal motions in the Gulf.

Figure 4.3: The co-ti,dal chart for the 01 harmonic constituent. Dashed, lines are

amplitude (m) and solid li,nes are phase (degrees) contours. The tirne zone i,s 0.100

(Source: British Adrniralty, reproduced by Danish Hydraulic Institute.)

01 ( ZoNE -0¿OO)

50.

38

K1 (ZONE -O¿OO)

55.

Figure 4.4: The co-tidal chart for the K1 harmoni,c constituent. Dashed lines are

ampli,tud,e (m) and solid lines are phase (degrees) contours. The time zone is 0100

(Source: British Admi,ralty, reproduced by Danish Hydraulic Insti'tute.)

50"

50"

Figure 4.5: The co-tidal chart for the Mz harmonic constituent. Dashed lines are

amplitude (m) and solid lines are phase (degrees) contours. The time zone is 0/¡00(Source: Briti,sh Admiralty, reproduced by Danish Hydraulic Institute.)

M2 (ZONe -0400)

50"

39

52 (ZONe -0400)

50. 55"

Figure 4.6: The co-tidal chart for the 52 hørrnoni,c consti,tuent. Dashed lines are

amplitude (m) and solid lines are phase (degrees) contours. The ti,me zone is 0/¡00

(Source: British Admiralty, reproduced by Danish Hydrauli,c Institute.)

4.3 Review of Cartesian models of the PersranGulf

Several depth-averaged Cartesian models of tides in the Persian Gulf already exist.The most important aspects of these models are included in Table 4.1. It shouldbe pointed out that in Table 4.1 the aertical/horizontal splitting ((VHS)) scheme

is a three-dimensional algorithm, described by Lardner and Cekirge (1988). Inthis approach, the surface elevation and depth-averaged velocity components are

computed by using one of the existing two-dimensional schemes. These values are

then used to compute the vertical velocity profiles by solving the linearized three-dimensional momentum equations.

40

.A'uthors year Depthaveraged

(DA)or3D

Nume-rical

techniqueused

Bottomfriction

Regionmodelled

Numberof

activeor

non-activegrid

element

Elementdimen-

sion

Timesüep

Von Trepka 1968 DA Hansen quadratic law whole Gulf N/A 14 x 14 km2 N/A

LardnerBelenand

Cekirge

1982

DAsem¡-

implicitfinite

difference

Chezyclln(c2H ! cs)

with c1 - 25,C2=Ca=l

whole Gulfand

Arabian shores N/A

20x20krn2and

LO x l\km2

12 min

Lardner,Cekirge,

andGunay

1986DA N/A

Qhezylr¿ form

c2=cg=I

whole Gulfand

Gulf of Salwa30x44= 1320

20x2Okrn2and

5Xbkm-N/A

Chu,Barker

andAkbar

1988DA seml-

implicit

Equal Chezy

46-1gg*à 7"""

whole Gulf31 x45

= 1395

19.05x19.05 km2 N/A

El-Sabhand

Murty

1988DA N/A

quadraticlaw

whole Gulf60x 38

- 2280

15 x 15 km2 6min

Bashir,Khaliq,

andAl-Hawaj

r989DA Hansen

quadraticlaw

whole GulfN/A

16.67 x18 km2 N/A

Lardner,Al-Rabeh,

Gunayand

Cekirge

19893D

Crank-N

Chezyln formcL = 25,

c2=cg=l

west coast ofthe Gulf

30x44- 1320

and12 x35= 4445

2Ox2Okrn2and

bXbkm-N/A

Al-Rabeh,Gunay,

andCekirge

19903DlVHS

semt-implicit/

ChezyJn. formcL=25

C2=Cg=L

whole Gulfand

west coast ofthe Gulf

N/A20x2okrn2

andlox 1o km2

N/A

Al-Rabehand

Gunay

I OOt

3DlVHS

semr-implicit

andCrank-N

(in vertical)

Chezyln formct P2,ca

SafaniyaL2xL2- 144

2x2krn2 72min

Proctor,Elliott

andFlather

1992DA

Proudma,nOceano-graphic

Laboratory

quadraticlaw

whole GulfN/A 9xg km2 N/A

Venkateshand

Murty

1994DA N/A

quadraticlaw

whole Gulf190x 106

= 2O1,4O

4.74x4.74krn2

6 min

Table 4.L: A sun'¿rno,rA of some Cartesian models of the Gulf. N/A rnel,ns not auailable.

Locøti,ons of Safaniya o,ndthe Gulf of Salwa are shown i'n Fi'gure 1.1.

4I

Von Trepka (1968) investigates the tides in the Persian Gulf by using the h;'dro-

dynamical-numerical (I1¡/) method of Hansen (1962); the model is non-linear-.

Cartesian coordinates with l4krn xl4km elements are used and the model is ini-

tially used to predict M2,the dominant tidal constituent in the Gulf, ignoling non-

linear interaction. Von Trepka is satisfied with the predictions for the arnplitude of

M2, eyêrr in the complicated and very shallow coastline of the Bahlain alea. The

average difference between his predictions and the observed arnplitudes is not mole

than2.T cm and 18 min [i.e. 8.7"] for the phases (von Trepka, 1968, p.61)' Separate

predictions for each of I{1, 01 and 52 are then performed but the lesults ale not as

good, since the amplitude error of these constituents is large.

Evidently noniinear effects cannot be ignored if the minor tides ale to be taken

into account, and so an experiment is conducted wherein a Per-sian Gulf rnodel

comprised of.42tcm x42km elements is forced using seven constituents over-a peliod

of 2g days. Rather than perfolming a harmonic analysis of lesults, r'on Tlepka

compares his results with predictions of high and low watel given in the Gelmau

and English Tide Tables (ibid., p.62). At Kuwait Harbol th.e mean deviation rvas

about 20 min [i.e. 10"]. The maximal deviation was not more than 60 ntìn [i.e.

30"]. In most cases the difference in height was less than 10on. Only the low

waters were computed to be too high, with a maximum diffelence of 36cnu (ibid..

p.62). In conclusion, von Trepka recommends using a finer grid model and more than

seven constituents, and employing harmonic analysis of results in or-del to obtain

prediction of the minor tides.Figures 4.7 and 4.8 show the amplitude (nz) and phase (degrees) contouls re-

spectively, for the Mz tidal constituent of sea sulfa,ce elevation pledictecl b5r \i¡¡Trepka

o''.,

)ô o'

o

.c,/

ao

t.' li .to

t0t2,o

to

ora

a

)

Figure 4.7: The amplitude (cnt) contours for the dttz tidal constituet'¿t. oJ sea surJace

eLeuation, pred,icted by Vort Tt'epka (1968).

42

(

oI

0

\

c

\]-

o

f.ro

t

o

?

Figure 4.8: The phase (degrees) contours for the M2 ti,dal constituent of sea surface

eleuation, predicted by Von Trepkø (1968).

Lardner et al. (1982) developed a model by using the finite difference scheme

of Leendertse (1967) to solve the depth-averaged equations. They constructed a

three-block model consisting of two blocks using a coarse grid and one using a finegrid, to cover the shaliow coastal areas) such as the area of Bahrain and the west

coast of Qatar. They produced the co-tidal maps for .I{1, M2 and Sz. They believethat their computed amplitudes f.or M2 are slightly too small (Le Provost, 1984,

p.32 says by L0%) in a region of the Saudi Arabian coast on the northwest coast of

Qatar and Abu Zabi, (south Figure 4.1), and is siightly too large on a region of thesouthern Iranian coastline around Jazireh-ye Hendorabi; otherwise the computedamplitudes are in good agreement with observation (ibid., p.a38). The computedphases are also too low on the north-eastern coast of Qatar, and are somewhat highin the western half of the Gulf (ibid., p.a38). However, they conclude that because

of the method of computation (the amplitude values are computed as the averageof all the maximum values of tidal eievation ( and the phase values are computedfrom the time of the last maximum of (), the computed values of tidal phase varyirregularly. That is, they are not as reliable as amplitudes when compared with theobserved values, although their shape of the contours are in reasonable agreementwith the shape of the co-tidal lines shown in Figures 4.3-4.6.

In a later paper, Lardner et al. (1986) solved the depth-averaged equations usinga different technique, namely, the method of characteristics. They applied the modelto the Gulf, to predict the amplitudes and phases of constituents M2 and .I11, bycovering the region with eiements of approximately 20 lcm x 20 km mesh-size. Theypresented charts for the results of the amplitudes and phases of the M2 and K1

43

constituents by applying a tidal analysis to the output data and comparing the

results with co-tidal charts, and indicated that phase contours of both M2 and K1

are in very good agreement (the observed and computed amplitudes at the eight

tid.al stations has been presented in the paper). They noted that the amplitude

contours generally are as good, with the exception that the amplitude contours of

bolh Mz and .K1 are smaller than those of the Admiralty chart, at the head of the

Guif, and the contour of M2 is too small in the Bahrain region and too large at the

bottom of the Gulf of Salwa (ibid., p.1076). They considered that the discrepancies

in certain a eas occurred because of the rather coarse mesh used on the coastline

(ibid., p.1076). Figures 4.9 and 4.10 indicate the amplitude (rn) and phase (degrees)

respectively, for the Mz tidal constituent of sea surface elevation predicted.

aao7

Figure 4.9: The amplitude (m) contours for the Mz tidal constituent of sea surface

eleuat'ion, predicted by Lardner et al. (1986).

o3

ô1

03 0?o3

o4

o4

o

3

44

315"

15.

34 5"

?=z'

'65'

45"

195

r65'

r65'

Figure 4.I0: The phase (degrees) contours for the Mz tidal constituent of sea surface

eleuation, predi,cted by Lardner et al. (1986).

Chu et al. (1983) constructed a Cartesian depth-averaged model and studied thetidal transport in the Gulf by using a coarse spatiai resolution of 19.05km. Theyused the velocity and depth information calculated from this hydrodynamic modelto provide data to develop an Eulerian-Lagrangian transport model to study theflow and transport characteristics of the Gulf. The two-dimensional shallow waterequations have been solved by using the Leendertse (1967) semi-implicit finite differ-ence scheme. The data for the open boundary of the model and also for comparisonwere supplied by Admiralty tide tables and tidal stream tables, 1968. The dataat Didamar Station in the Strait of Hormuz was used as the boundary condition,and for comparison with their model predictions they used the tide data of the fourconstituents M2, Sr, Kt and Or at 26 stations within the Gulf. They have notprovided a table for the predictions and observed values for amplitudes and phases

at the stations, but two plots of correlations of the tide ranges and phases wereprovided. They believed that the model results match fairly well with those givenby the Admiralty Tide Table, except at a few shoreline stations along the Arabiancoast, and considered that the coarse resolution of the coastline and bathymetry ofthe model may have caused the discrepancies at these stations.

Bashir et al. (1989) developed a two dimensional explicit finite difference modelto reproduce the M2 constituent of the tide in the Guif. A difference between thismethod and schemes of, for example, Lardner et al. (1982) and von Trepka (1968)is that, in this model at the open boundary (strait of Hormuz), radiation typeconditions are applied (Beckers and Neves, 1985 and Flather, 1976), in addition to

45

prescribing the tidal height (ibid., p296).

Lardner et al. (1989) described a three-dimensional hydrodynamic model for the

Gulf. This paper was a continuation of Lardner et al. (1982, 1986), Cekirge et

al. (1986) and Lardner and Cekirge (1987) and (1983), in which tidal and wind-driven

currents were studied using the method of characteristics and the VHS technique. Inthis paper, they discussed the selection of appropriate values for the hydrodynamicalparameters and presented an interpolation algorithm to compute velocity profiles at

non-gridpoint positions.Al-Rabeh et al. (i990) described a hydrodynamic modei for wind-driven and

tidal ci¡culation of the Gulf. They extended the VHS model of Lardner and Cekirge

(1988), and in particular they solved the depth-averaged equations by a finite dif-ference scheme, based on that of Leendertse (1967), rather than solving them bythe method of characteristics, which was done by Lardner and Cekirge (1988). Al-Rabeh et al. (1990) present two models, namely HYDRO1 and HYDRO2. Bothmodels compute wind-driven and/or tidal circulation. HYDRO1 is for the whole

Gulf and the HYDRO2 can be used for any particular area in the Guif. The models

are applied to compute the tidal heights at selected stations on the Saudi coastline.

Proctor et al. (1992) used the Proudman Oceanography Laboratory (PLO) depth-

averaged tide-surge model for the whole Gulf, to provide environmental data on

tides, currents and particle trajectories during the Gulf war. In this model the gridcells are approximately 9 km x 9lcm. The open boundary of the model was located

across the eastern end of the Strait of Hormuz. Along the open boundary ten tidalcomponents, Sr, Or, Pr, Kr, pz, Nz, Mz, Lz, S, were specified.

Venkatesh and Murty (1994) developed a numerical model to investigate the

movement of the 1991 oil spills in the Gulf. As a resuit of the 1991 Gulf \ryar, morethan 200 million gallons (ibid., p.2) of oil were spilled into the Gulf, causing the

world's worst oil spill. The first spills occurred at ihe Al-Ahmadi oil terminal, offthe Kuwaiti coast (see Figure 4.1). In this paper the simulations of the movementof oil from the Al-Ahmadi and Al-Bakr spills are described. The numerical model is

used to compute the currents due to tides, winds and bathymetric influences. Theopen boundary of the model is set at the strait of Hormuz. This tidal model is ahigher resolution version of El-Sabh and Murty (1988), that is, 4.74 km x 4.74, km.

Some other numerical models of surges, and pollutant transport, also have been

developed and applied to the Gulf. Table 4.2 summarizes some of these models.

46

Authors year Application Regionmodelled

Murty and EI-Sabh 1984 storm surges whole the Gulf

Galt, Payton,

Torgrimson and Watabatashi

1984 oii spill

oil spill

whole the Gulf and

Kuwait area

El-Sabh and Murty 1988 oil spill whole the Gulf

Lardner, Lehr,Fraga and Sarhan

1988 pollutant transport whole the Gulf

Cekirge, Al-Rabehand Gunay

1989 oil spill whole the Gulf

Proctor, EIIiott

and Fiather

t992 oil spill Kuwait area

(AI-Ahmadi)

Spaulding, AndersonIsaji and Howlett

1993 oil spill AI-Ahmadi

Venkatesh and Murty 7994 oil spill Al-Ahmadi

Table 4.2: A surnnxarA of some other numerical models of surges, and pollutanttransport of the Guff.

4.4 Depth-averaged Cartesian model and its cal-ibration

The Cartesian depth-averaged model of the Persian Gulf (which assumed the earth

is locally flat) used the elements of 8.7 lcm xg.7 krn mesh-size, that is,2L,t : 8.7 lcm

and ZL.g : 9.7 km and a time step of Lt : 200sec.

It was assumed that the Mercator projection of the spherical surface formed the

Cartesian plan used. The British Admiralty maps 2837 and 2847 were used for the

model. The value of Aø in the east-west direction ',t/as assumed to be that of the

central point in the Gulf. The Cartesian grid contains 79 x 118 elements; 3199 activeelements are used in the computations. The open boundary of the model is set at

the eastern end of the Gulf (see dashed line in Figure 4.1), and the number of type2 elements on the open boundary, is 12, so that the number of elements within theGulf is 3187. Dashed line in Figure 4.1 shows the open boundary of the model,which is iocated at the eastern end of the Gulf. The first element from the left is

labelled element number one and the second is number two and so on. The top

47

left and. bottom right corner of the grid have a iatitude of 30.37" N and 23.87" N,

respectivelyThe ten tidal components Qr, Or, Pt, Kr¡ þ2, Nz', Mz,, Lz, ^92 and K2, provided

by Proctor et al. (1992), were used to produce open boundary tide-height input data.

úUt. a.B(a) shows the tide-height constants of the four major constituents at the

elements on the open boundary and Table 4.3(b) indicates the amplitude and phase

of the six minor constituents at the elements of the open boundary of the model Gulf.

Tables a.3(a) and a.3(b) show that the constants were considered to vary linearly

across the GuIf of Oman. The model was run for a simulated time of 32 days; the

fi.rst three days of data were considered to include initial transients, so the tidal

movement within the Gulf was represented by the last 29 days. Predictions for the

29 days can be used to determine tidal constants for 25 astronomical constituents

(Easton, 1977, p.16), although the four major constituents 01, Kt, M, and ,92 can

be calculated from only 15 days of data (Easton,1977,, P.17). A criterion for using

the minimum length of data required for analysis has been given in Pugh( 1987,

p.113).

Boundaryelement

O1amÞ phase

Ktamp phase

M2amÞ pha,se

S2

amp phase

1t?

4

5

b78q

1011

L2

0.2350.239o.243o.247o.2510.2550.2590.263o.2670.271o.275o.279

330.3330.6330.8331.1

331.6331.8331.9

332.4332.6332.8

0.3900.3950.400o.4040.4090.4130.4180.423o.424o.4320.437o.447

350.0349.8349.6349.4349.3349.0348.9348.7348.5348.4348.2348.0

o.7590.763o.767o.771o.7760.780o.7840.789o.7940.799o.8030.808

L70.4169.9169.3168.8168.3167.8t67.3766.7166.3165.8165.3164.8

o.2920.2930.295o.2960.2980.3000.3020.3040.3060.3090.3110.314

204.2203.0201.9200.8199.6198.5L97.4196.3195.3194.2t93.2r ot t

Table a.3(a): Amplitude (cm) and phase (degree GMf ) of the four major constituents

at the elements of the open boundary of the Persi,an Gulf model. See also Table 1.3(b)

for the rernaining constituents used as the forcing døta of the model. GMT rneans

Greenwich Mean Ti,me.

The distribution of the observational stations used for caiibration is shown in

Figure 4.I2, and Table 4.4 lists their latitude and longitude. Among the list of 84

stations there are 44 stations which have not been used in calibration because they

have one or more of the following disadvantages; they are

(1) "p

rivers,(2) in very shallow regions, or(3) very close together.

It should be noted that in the situation where two stations are in close proximit¡the station nearer to an elevation grid point has been selected.

48

tsoundaryelement

Qtamp phase amp phase

Pl MUzamp phase

Nzamp phase

LzamÞ phase

K2amp phase

1tó

45

67

8I1011

12

0.0290.0300.0300.0310.0320.0320.0330.0330.0330.0340.0340.035

317.0

317.5317.8318.0318.2318.53t7.7318.9319.1319.3319.5

0.1300.1320.1320.1350.1360.1380.1390.1410.L42o.L440.146o.r47

350.O349.8349.6349.4349.3349.0348.9348.7348.5348.4348.2348.0

0.0310.0310.0310.0320.0320.0320.0320.0330.0330.0330.0330.033

r09.4108.9108.3107.8107.3r06.8106.3r05.7105.2104.8104.3103.8

0.1670.1680.1690.1700.171o.L720.1 73o.t740.1 75

o.176o.t770.178

153.4152.9152.3151.8151.3150.8150.3149.8149.2148.8148.3L47.8

0.0310.0320.0320.0320.0320.0320.0320.0330.0330.0330.0330.033

190.4189.9189.3189.8188.2187.8t87.2186.7L86.2185.8185.3184.8

0.0780.0780.0790.0790.0790.0800.0810.0810.0820.0820.0830.084

204.2203.0201.9200.8199.6198.5197.4196.3195.3L94.2193.3r92.2

Table .B(b): Amplitude (cm) and phase (degree GMT) of the sir minor consti'tuents

at the elements of the open boundT,ra or the Persian Gulf model.

Figure 4.L1: The distributi,on of the l0 stations in the Gulf used for calibration

7

I

11

l9

22.

28

29

p.

PERSIAN GULF

€)IRAN

/;\

o

49

Lat.(oN)position

Long.(oE)observationstation

50020/50o45'51016'/48031'4go3gr

4go 431

560L71

4go54l50021

530231

55 0 54/54053/56033'66022t550 18t500471õ4031 /

560 15''aoo29t

5ro22l540331510 13/5603/50071

51000/51035'55035/50051'51 030t

55026151031f550 17t5305t52034154030/520191540 10/

48290 16'2go54t28015,28o2-õl2go7l270291270L31270LLt26051t260391

260 481

26048t260 4ll26033t260zgt260221

260161

260221260 15t260121

260 3t260 3'25054t26091

250681250531

250591250541

25035125028t25034t25o25125 018,

25016124053t240461240261

240281240201

t

45

Õ

7

891011

1213

t415

16L71819202Lt¿)

2324

262728to3031

343536ðt383940

KHARK ISLANDBUSHEHRLAVARRAS-AL.KHAFJIRAS AL MISHAABASALUBERRI - DAWHAT ABU ABANDAR ABBASRAS AL QULAAHYTARUT BAYJAZIRAT SHAIKH SHUAIJAZIRÐH YE LAVANHENJANBANDAR LINGEHLITTLE QUOIN ISLANDKHOR KAWIJEZIREH YE TUNBHBAHRAIN APPROACH BEAJEZIREH YE FORURKHASAB BAYZELLAqJABAL FUWAIRTJAZIREH YE SIRRIRUIVAIS INSHOREKHOR KHWAIRQURAYAHRAS ASHAIRIQRAS LAFFANUMM AL QAYWAYNZEKRITSUMAISMAHAJMANAD DAWHAHDUBAYYZARQAARDHANAUMM AL NARDALMA, JAZIRATKHAWR ZUBAYYAH

Tabie 4.4: Map positi,on of the l0 selected stati,ons used in calibrating the model

of the Persian Gulf.

4.4.L Generating the mean sea level depths

The bathymetry for the region has been taken from the British Admiralty charts

2837 and 2847. Because the bathymetry on the charts was so sparse, depths at grid

points were estimated by averaging over the region covered by the grid box sur-

rounding each grid. point. Since chart datum is Indian Spring Low Water (ISLW),and depths should be relative to Mean Sea Level (MSL), so MSL must be gen-

erated.. By using an iterative process suggested by Mitchell (1985), MSL can be

determined accurately. Having the I S LW depth field as a zeroth. estimate for M S Levery where inside the model region and on the boundary line, and starting a 29 day

simulation to predict tide heights at each elevation point of Type 1 in the model, a

tidai analysis of the predictions provides a first estimate for M S L as,

MSL\ : ISLW * (O, I Kti Mzl,Sr)(t), (4.1)

50

where the quantities in the bracket are amplitud,es and superscript "(r)" means

values are obtained after the first iteration'The MSL(') u, . second estimate is computed in a similar mannel' by using

the M S L(1) as the first estimate for the MS.L field. The general mathematical

procedure can be written as

MSL:,li1 MSL(i) : ISLW + lim (O, * Kt I Mz* ^92¡(;). Ø-2)

This process generates MSL accurateiy. Figure 4.12 shows a bathymetric contour

plot of the resulting MSL depth field of the Persian Gulf region. As this figure

.ho*., the Arabian coasts are shaliower than the Iranian coasts. The maximum

depth of the Gulf is about 90 m, which appears in the Strait of Hormuz; the average

depth of the Gulf is 32m.

Figure 4.12: Bathymetric contour plot in meters of the Persi,an Gulf.

It should be noted that, along the open boundary, no such calculation is required.,

because the added amplitude, ur" pro,rided by the forcing data. The value of M S LQ)

was taken as a reasonable estimate for M S L because it was aimost the same as

M S LQ) , to within 0.2m.

çæ

15

E

45015

\75 \;oz5

Þ15

Þ15

\50æ

15.15

fil=

ô ç30

51

The calibration of the depth-averaged model of the Persian Gulf was carried out

as follows:Before applying a model to a particular coastal region, the parameters in the model,

namely, the bottom friction, C6, and the reduced eddy viscosity coeffi.cient, ø, in the

model are calibrated by comparing the model solutions against observations available

at stations in the particular region. Most of the models are calibrated by an iterative

trial and error scheme, that is, some initial values are assigned to the parameters

of the model, which are then adjusted after several runs, by considering the error

measure between model solutions and observations. The tide-height observations at

stations listed in Table 4.4were used in the process of the calibration the model.

The following error measure formulae were used in the calibration procedure

(Bills, 1991, pp. 161-162).

(1) Absolute difference ellor: For constituent c, this error measure is

(4.3)P",toc"r ./V

.-tF"" - N "u=,

whereê

Pc,"

(cm) is

¡\IO","

/. 'r' d .,:ir e Jø'

is the absolute difference in amplitude (cnz)nbetween the phobserved (in degrees) and predicted results averaged over allinterior caiibration stations for tidal constituent c,

is the number of appropriate calibration stations,is the observed amplitude (cnz) or phase (degrees) of con-

stituent c at station s, and

is the predicted amplitude (cnz) or phase (degrees) of con-

stituent c at stations s.

(2) Average absolute error: The overall average absolute amplitude error eø

14

^-l

S./¿"c (4.4)Ça

4

where 6", is the average absolute amplitude error defined for constituent c. Anoverall average absolute phase error ep (degrees) is defined in a similar way.

(3) Absolute complex difference error: An overali average absolute complex

difference between observed and predicted tide-height, proposed by Mitchell (1985),

is

, : fiå å fi¡-t"exp(iS.)1"," - l¡t,enp(iþ,)1.,"1 (4.5)

where

52

€ is the absolute complex difference (cm) between observed and

predicted tide-heights averaged over all interior calibration

stations for the four constituents Ot, Kt, M2 and 52,

llro,óof.,, are the observed amplitude (crn) and phase (degrees),Iespec-'

tively, of constituent c at station s, and

lpr,órf",, are the predicted amplitude (cnz) and phase (degrees), re-

spectively, of constituent c at station s'

This error measure takes lengths of the differences between complex vectors given

by observed and predicted tide-heights and averages them over ali interior calibration

stations and the four constituents.The absoiute complex difference error is used to compa e the results of differ-

ent runs of the various values of bottom friction, C6, and reduced horizontal eddy

viscosity coefficient, a.

For the Persian Gulf a series of 16 simulations were carried out for different values

of Ca, namely 0.0010, 0.0015, 0.0020 and 0.0025 with various values of ø, that is,

0.005b, 0.01, 0.03 and 0.04, and an analysis of tide-heights was done for the selected

tidal stations listed in Table 4.4. Table 4.5 shows the error measure e resulting from

16 runs, when the third estimate lor M S L was used. The optimal values for C¿ and

ø were found to be 0.0015 and 0.03sec-1, respectively, with associated error vaiue

of e : 5.376 cm. Il should be noted that Proctor et al. (1992) used C¡ : 0.0015 and

El-sabh and, Murty (igss) chosed ca:0.0025. The optimal values have been used

in all runs of the programs.

Caa 0.0055 0.01 0.03 0.04

0.0010

0.00i5

0.0020

0.0025

5.958 5.874 5.853 5.861

5.632 5.412 5.376 5.379

5.653 5.592 5.516 5.446

5.831 5.822 5.811 5.810

Table 4.5: Error e (cm), for uarious cornbi,nations of the ualues of quadrati'c friction,C6, and, reduced edd'y coeffi'c'ient a'

4.6 Presentation of the results

In this section the tidal constants computed from the model and observations of tidalelevation are presented in Table 4.6. For comparison also the contour charts of tidalelevation for the four major tidal constituents, that is, diurnal tidal constituents

01 and Kr and the semi-diurnal constituents Mz and Sz a e presented in Figures

4.L3-4.20. The model used the values C¡ : 0'0015 and ¿: 0'03'

Tabie 4.6 summarizes the results of model predicted amplitudes and phases,

of. Ot, Kr, M, and ,52 components with observation values at forty tidai stations

53

around the Gulf, identifred in Table 4.4. The averaged complex difference error for

the amplitude and phase of sea surface elevation is computed using equation 4.3, and

includ.ed in Tabie 4.6. This error is e : 5.376 cm. The agreement between predicted

and observed values is generally good, with amplitudes being in better agreement.

Table 4.6: Comparison of computed (Model) and obserued (Obs.) results for amplitude

(cm) and phase (degrees) for the four major tidal constituents at selected stations in the

Persi,an Gulf.

1

station277.O263.O

247.2233.0

24L.8227.O

214.9209.0

266.2256.0

259.0

130.8115.0

291.3273.O

22.411.0

292.1274.O

345.6325.0

101.694.0

115.692.O

28.428.0

73.974.O

14.9359.0

6.7350.0

57.L64.0

31 .1

10.0

103.685.0

52.O62.7

39.336.4

40.2

42.349.7

24.9

t5.725.O

4t.95r.2

36.044.O

tL2.4100.0

39.148.0

45.054.O

37.830.0

27.433.0

88.973.0

60.6ó9.7

89.376.4

81.068.9

62.259.1

bb. r63.0

35.145.r

265.624a.O

166.3149.0

t22.7110.0

82.368.0

273.8256.0

294.5276.O

33.r19.0

49.237.0

200.5197.0

56.8,36.0

113.895.0

334.9335.0

351.4332.0

20t.7204.O

258.2245.O

185.8185.0

190.9193.0

234.0229.O

74.354.0

258.0

11.016.8

14.312.8

t4.4

t4.917.7

5.811.6

8.0

13.9t7.lt2.216.0

43.636.0

12.516.0

ró.217.o

14.31.2.O

10.4t2.035.625.0

23.022.6

35.227.4

32.526.3

24.620.tt2.t20.o

10.1L5.2

329.8312.0

207.7196.0

I75.2160.0

131.3117.0

336.8320.0

335.0

57.0

12 5.1107.0

239.1229.O

].t3.793.0

170.0159.0

20.910.0

20.L6.0

246.3247.0

300.3282.O

237.422t.0

236.O227.O

276.9266.0

L22.7102.0

314.2294.0

ModelObs.

ModelObs.

ModelObs.

ModelObs.

ModelObs.

ModelObs.

ModelObs.

ModelObs.

ModelObs.

ModelObs.

ModelObs.

ModelObs.

ModelObs.

ModelObs.

ModelObs.

ModelObs.

ModelObs.

ModelObs.

ModelObs.

ModelObs.

L7.62A.7

t7.125.7

15.420.4

11.918.0

L5.22t.6

14.92t.o

t2.l11.9

8.614.0

26.420.7

4.910.0

4.39.0

14.515.0

to.716.0

22.920.o

75.42t.924.O20.4

2t.215.8

16.418.9

3.57.O

14.1

229.92t5.O

206.6L92.O

20L.4189.0

t79.4171.0

22t.O

237.022t.O

100.989.0

245.8227.O

0.13.0

238.1218.0

300.4281.0

80.655.0

82.865.0

358.214.0

44.340.0

344.535 5.0

342.8349.0

24.838.0

28E.1268.0

61.341.0

3L.242.9

3t.738.8

25.930.7

18.5

34.4

28.038.0

20.923.8

11.017.o

42.433.8

10.118.0

6.8L2.O

24.429.O

21.930.0

38.829.O

24.932.6

42.t28.7

38.126.2

26.L29.3vo

9.0

28.9

1

,

o

4

5

b

7

8

I

10

11

t2

13

I4

15

16

T7

18

19

20

54

ph1

amp phstation ampI97.3196.0

195.1196.0

42.O44.O

25L.3235.0

71.057.O

207.6210.0

L87.2169.0

90.472.O

22.O37.O

227.8234.O

27t.2257.O

37.0

24t.3248.O-42.7

51.0

256.5254.O

2a4.3266.O

100.295.0

342.6

110.9115.0

303.9289.0

30.922.O

4.62.r

6.813.1

13.614.s

10.918.0

26.220.o

6.04.0

4.L11 .3

8.511 .0

17.615.0

1.74.3

13.911 .0

13.613.0

L4.610.8

TL.711.0

/.b4.0

5.1

4.0

2.O

8.0

t2.L5.8

16.415.0

242.3239.0

259.5273.0

87.288.0

294.8279.0

120.5102.0

261.9250.0

238.0223.O

L43.2128.0

7t.576.0

270.1267.O

300.12a4.0

77.567.O

289.7301.0

98.682.O

299.3304.0

353.6337.0

lto.795.0

L2.O26.0

140.5722.O

358.0343.0

30.822.O

1.2

L4.720.1

15.324.5

tt.215.0

19.721.0

3.07.O

b.J7.O

18.025.0

11 .1

20.0

a.2

26.834.0

8.177.O

28.836.2

8.819.0

24.735.8

29.140.4

14.022.O

47.2

2L.l30.0

10.8352.0

t2L.7r 16.0

66.854.0

95.477.O

39.920.0

19.829.O

113.1104.0

36.516.0

71.853.0

74.O

101.584.0

81.464.0

90.2103.0

90.772.O

123.5110.O

109.095.0

105.390.0

169.3151.0

120.8r01.0

136.8120.0

75.867.0

11.34.9

36.742.4

.J/,O39.0

4L.251 .0

60.862.O

22.416.0

31.64t.729.038.0

43.645.0

/.ò16.2oÉ Ò

34.0

35.143.0

38.0

32.432.O

L7.7LL.2

L5.46.3

r4.025.O

t4.244.637.O

348.7345.0

47.536.0oo0.0

44.631.0

327.7308.0

15.528.0

24.99.0

328.7301.0

18.40.0

23.933.0

19.54.0

atÉ

t7.o53.8bU.U

37.628.0

75.171.0

80.462.O

80.762.O

t25.2114.0

83.067.0

97.O81.0

2l

,a

24

26

26

28

10

30

31

33

34

35

36

aa

38

39

40

Obs.

ModelObs.

ModelObs.

ModelObs.

ModelObs.

ModelObs.

ModelObs.

ModelObs.

ModeIObs.

ModelObs.

ModelObs.

ModelObs.

ModelObs.

ModelObs.

ModelObs.

ModelObs.

ModelObs.

ModelObs.

t7.816.0

L.71.8

7.89.4

10.4L7.6

b.b8.0

11.516.0

1.95.0oo

o.t

9.712.0

9.8L7.0

2.12.4

14.815.0

8.315.0

16.016.4

8.015.0

14.416.3

16.818.5

7.O

12.0

18.924.2

12.619.0

ModelObs.

ModelObs.

Model

differences

Table 4.6(continred): Compo,rison of computed (Model) and obserued (Obs') results foramplitud,e (cm) and, phase (degrees) for the four rnajor tidal constituents at selected sta-

ti,ons in the Persian Gulf (see Table 4.1(o) forthe rernT,ining stati'ons).

The model was applied to predict the surface elevations for the principal diurnal

Or, K, and semi-diurnal Mz and,Sz constituents. These components are dominant

tidal constituents in the Persian Gulf. The contour charts of tide height predicted by

the model are presented in Figures 4.13-4.20. These contours show lines of constant

amplitude (nz) and lines of constant phase (degrees). The co-phase lines are given

in terms of time zorle - 0400, to agree with the Admiraity chart 5081 (Figures 4.3-

4.6). As the results show, two amphidromic points appear for the semi-diurnal

55

constituents M2 and ,92, and one for the diurnal tide' 01 and K1. The bigger values

of amplitudes are seen around the the Strait of Hormuz, and in particular the largest

valuels located at Bandar Abbas (Station 9 in Figure 4.11), and at the head of the

Gulf. It should be noted that as Figures 4.13-4.16 show, the basic pattern of the Or

tide is similar to that of the K1 tide, because the period of the 01 is close to the

period of Kt, but the magnitude of the Or is neariy 20% lower. The same remarks

can be made for the similarity of the Mz and Sz tides (Figures 4.17 and 4-19, and

Figures 4.18 and 4.20). In this case the amplitude of the ,52 is approximately 30%

of the M2 values.

Comparing the resulting contours with the Admiralty chart, Figures 4.3-4.6,

suggest that the locations of the amphidromic points and the basic pattern of the

"ontorrrr, predicted by the model are similar to those of the chart, but differ in

certain important aspects. For example, modeled results for Or and Mz phases

show that in the Strait of Hormuz, near the open boundary, the tidal constituents

are moving slower than Admiralty chart, while the 1{r constituent appears to be

moving at the same speed, the reason may be a boundary effect.

o'f

Éo

o-

! ,/,.--t-

o,l

o.Zf-

Figure 4.13: Amplitude contours (m) for the 01 tidal constituent of sea surface

eleuation in the Persian Gulf obtained from the Cartesian coordinate model.

bb

3oo 27o

q0,20

tto

oE

tr

21o

6o

o),t0

Figure 4.I4: Phase contours (d,egrees GMT) for the ot tidal constituent of sea sur-

faie eleuation in the Persian Gulf obtained from the Cartesian coordinate model.

Figure 4.15: Arnplitud,e contours (m) for the K¡ tidal constituent of sea surface

eliuation in the Persian Gulf obtained frorn the Cartesian coordinate rnodel-

o.?-

o,o-z

0.3

o,

e.l

o

3

o

-¡

O-?, o-l

n

57

3ia

36o

¡80

t26 9o

t6o

2too É

tSo

270

2to

3oo

29o

Figure 4.16: Phase contours (degrees GMT) for the Kt tido'l constituent of sea

,r.i¡o", eleuation in the Persian Gulf obtained frorn the Cartesiøn coordinate model.

fL-I'ro.

o,¿l

o.3

o ô.2

o.3!.0É

e

o.1

oi o.L¿.\

o,.l(:

Figure 4.12: Amplitude contours (m) for the Mz tidal constituent of sea surface

eliuation in the Persian Gulf obtained from the Cartesian coordinate model-

58

3196c 303

2totBo

tlo5oo

t2o 33o36t

1o 60 3o

-/îo

E

Figure 4.18: Phase contours (degrees GMT) for the Mz tidal constituent of sea

surface eleuation in the Persian Gulf obtained from the Cartesian coordinate model.

o.z

o-l

oE¡

o-3o2

o.l

¿

Figure 4.19: Amplitude contours (m,) for the Sz tidal constituent of sea surfaceeleuation in thc Persian Gu[ olttained frorn the Cartesian coordinate modeL.

59

27o2to

,5o

24coIl6o 3o

6o

o

3cc

360o

t2o ¿ìo

t5o oJonoloó

3oo

Figure 4.20: Phase contours (degrees GMf ) for the 52 tidal consti'tuent of sea surface

eleuøtion in the Persian Gulfobtained from the Cartesian coordinate n¿odel.

4.6 Summary and discussion

A depth-averaged model of tides using Cartesian coordinates has been developed

and applied to the Persian Gulf. The amplitudes and phases of the four major con-

stituents, Or, Kr, M2 and.S2, have been reproduced and compared with Admiraltychart 5081. The locations of the amphidromic systems predicted by the model are

almost the same, for all constituents, as those shown on the Admiralty chart, and

the basic patterns of cotidal and co-phase lines are similar, but computed phases

f.or Ot and Mz show that in the Strait of Hormuz the tidal constituents are moving

slower than the same constituents in the Admiralty chart. The predicted amplitudes

at the selåcted stations are generaliy in better agreement than the predicted phases

(see Table 6). It is expected to improve the performance of the model by using a

spherical coordinate grid, since the Persian Gulf is too large to be approximated by

a piane surface. Also the grid may be refined in certain regions, because the spatial

variations in tides are very large in the Gulf, especially near the amphidromic re-

gions, and because the coastline generally has a complicated shape and very variable

bottom topography.

60

Chapter 5

A Depth-averaged SphericalModel of the Persian Gulf

5.l- IntroductionThe depth-averaged Cartesian model of tides in the Persian Gulf was developed in

Chapter 4. The amplitudes and phases of the four major constituents, 01, Ii1, M2

and ,S2, have been reproduced and compared with Admiralty chart 5081. Although

there was generally good agreement between observed and computed tides from

this model, the results can be improved by using a spherical coordinate grid since

the Persian Gulf is large. Some researchers, for example Le Provost (t984), have

already noticed that using spherical coordinates can improve the agreement between

modelled and measured tide heights. This chapter discusses the development of aspherical depth-averaged model of the Persian Guif.

5.2 Review of spherical models of the Persian GulfA number of spherical coordinate models of tides in the Persian Gulf have already

been developed. In this section these works are discussed; Table 5.1 summarizes the

important features of these models.

Evans-Roberts (1979) provided an extensive survey of tides in the Persian Gulf.

He d.eveloped a spherical coordinate model with a mesh-size of 10' latitude and

10' longitude. He used a low-cost method to reproduce the Mz and.[11, the most

important tidal components in the Gulf. This method simulates the Mz constituents

and an artificial diurnal tide (AM1) with the amplitude and phase of Kl but period

of exactly twice M2. This produces a tide cycle which will repeat at intervals of

approximately 24.8 hours. This is an economical method because the main features

of the tides in the Gulf can be studied without the need to generate and analyse

large quantities of resuits, but it should be pointed out that the other constituents

such as Sz and 01 cannot be studied by this method; they can be approximated

using the averaged ratios formula calculated for tidal amplitudes (see Bills, 1992,

61

p.I42). The co-tidal map of M2 produced by his model is presented in Figure 5.1.

The amplitudes generated f.or Mz are lower than those shown on the Admiralty chart

5081, but the phase contour lines are in good agreement. However, he believed that

using the fairly coarse grid he used reduced the acculacy.

Table 5.I: A surnnxtlA of some spherical models of the Gulf. N/A means not auøi'lable

Ti,me steps are N/A.

Figure 5.L: Amplitude contours (dashed lines,(m)) and phøse contours (solid lines,

(degrees))for the M2 tidal consti,tuent oJ sea surface eleuation in the Persian Gulf,

produced by Euans-Roberts (1979, p.46).

I

.A'uthors Year Depthaveraged

(DA)or3D

Nume-rical

techniqueused

Bottomfriction

used

Regionmodelled

Numberof

activeor

non-activegrid

element

Elementdimen-

sion

Evans-Roberts L979 DA N/A N/A whole Gulf N/A 10 x10

Hunter 1984 DA explicitfinite-difrerence

quadratic law Kuwait waters N/A 2.5t x 2.5

Elahiand

Ashrafi

1992aDA

explicitfinite-difference

quadratic law whole Gulf 39x54= 2IO6

18x 18 km2

Elahiand

Ashrafi

1992bDA

explicitfinite-difference

quadratic law whole Gulf 39x54= 2106

18x 18km2

62

Hunter (1gS4) developed a depth-averaged spherical-coordinate numerical model

to investigate the M2 and, Kr tidal motions for Kuwaiti waters, the first time that

this region has been modeled in detail. The depth-averaged velocities produced by

the tidal model have been used as an input for developing a stratified model. An

explicit finite difference scheme has been used to solve the depth-averaged equa-

tions -using a combination of forward and centered difference approximations. The

non-linear effects due to the advective terms in the equations were included. He

examined separateiy the two tidal constituents, M2 and K1, as was done by Von

Trepka (1963). Because they were run separated the model neglected the non-linear

interactions between Mz and, Kt. When Von Trepka was trying to reproduce the

Or and 52 components, he noted that the results were poor: the amplitude of the

water elevations were always too large. He came to the conclusion that since the

Persian Gutf is an extremely shallow sea the complete non-linear effects can not be

neglected (ibid., p.61). In order to take into account the complete non-linear interac-

tion between the different constituents, he prescribed a real prediction of sea surface

eievations at the open boundary of the model by including seven constituents.

Hunter (1984) also ran the Mz and K1 components separately, by using a different

bottom friction coefficient for each component. For testing the performance of the

model he compared the model results with the observed harmonic constants at five

tidal stations. From the initial runs of his model using an M2 tidal input, Hunter

found that the Admiralty chart 5081 of the Persian Gulf was possibly in error in

Kuwaiti waters (ibid., p.15).Elahi and Ashrafi. (1992a) described a spherical-coordinate model to consider

tides and. currents in the Gulf. They solved the two-dimensional shallow water

equations using an explicit finite difference scheme. Along the open boundary the

elevations of four constituents, 01, I{t, Mz and,Sz were specified. They presented

contour charts of amplitude and phase of the major components of tide-height' The

locations of the amphidromic points are in reasonable agreements with the chart

b081 and the patterns of the contour lines are aiso similar. They report that around

the coast of Bahrain the results are not encouraging'

Elahi and Ashrafi (1992a and 1992b) described a depth-averaged hydrodynamic

numerical model. They have provided a table of comparison between the predicted

and observed tidal constants at 18 stations around the Gulf (ibid., p.168) for the

four major constituents.There a e some other spherical coordinate numerical models which have already

been developed and applied to certain regions other than the Persian Gulf' Table

5.2 summarizes some of these models.

63

Authors Year Application Regionmodelled

Zahel 1973 DA/ tides world ocean

Fiather and Davies 1976 DA/ storm surges North Sea

Davies and Flather 1978 DA/ storm surges North Sea

Flather 1981 DA/ storm surges North Sea

Flather and Proctor 1983 DA/ storm surges North Sea

Chilicka, Kowalikand Wierzbicki

1983 DA/ storm surges Baltic Sea

Mitchell and Noye 1983 3D/ tides Gulf of Carpentaria

Flather 1984 DA/ storm surges North Sea

Fang and Yang 1988 DA/ tides Korea Strait

Davies and Jones 1992 3D/ tides Celtic and Irish Sea

- Verboom, Rondeand Dijk

r992 DA/ storm surges North Sea

Davies and Jones 1993 DA/ storm surges Ceitic and Irish Sea

Davies and Lawrence r994 3D/ tides Irish Sea

Table 5.2: A surnmtry of some spherical coordi,nate numerical models of other

regi,ons. DA means depth-aueraged'

5.3 Depth-averaged spherical tidal model

The spherical depth-averaged tidal model of the Persian Gulf developed for this

thesis uses an element dimension of 5' x 5' and a time step of Aú : 200sec. The top

left and bottom right corner of the grid have coordinates of 30.37o and 48.55o and

23.87'and 53.58o, north latitude and east longitude, respectively. The positive faxis and À-axis are directed north and east, respectively. The grid contains 79 x 118

elements; 3199 of which are used in the computations. The location of the open

boundary is the same of that of the Cartesian model, that is the dashed line inFigure 4.1. The same set of open boundary data were used for the Cartesian model,

is used for the spherical model (Tables 4.3(a) and a.3(b)). The model runs for 29

64

days, after ignoring the first three days of simulation which are contaminated by

initial transients. Predictions for the 29 days are saved hourly and then tidallyanalysed as described in Chapter 4 (Section 4.4).

The original Cartesian coordinate computer program (Section 3.1.4), has been

converted to spherical coordinates to implement the fi.nite difference form of the

depth-averaged spherical tidal equations, including the additional terms resulting

from the transformation into spherical coordinates.

5.4 Calibrating the model

This section discusses the approach and results of calibrating the spherical modei.

The model was applied to the Persian Guif and calibrated with tidal data at the

forty stations listed in Table 4.4. The model calibration was done using the trial-and.-error scheme, in a similar manner to that used for calibrating the Cartesian

modei in Chapter 4. in this study the bottom friction, C6 and the reduced eddy

viscosity coefficient, a) ate calibrated.Viscosity coefficients were formulated in equations (2.1I)-(2.L2) and are approx-

imated in equations (3.55) and (3.58), respectively. In some tidal models of shallow

seas the horizontal eddy viscosity terms are not inciuded, for example, Flather and

Heaps (1975), Blumberg (1977) and Lardner et al. (1982) and some others use global

constants for the reduced eddy viscosity coefficient, for example, Noye et al. (1981,

1982). Greenberg (1933) used equation (2.10); the vaiue of ¿ in his model was set to

1.64x10-3sec-1. Schwiderski (1930) included the horizontal viscosity terms in his

global-models. He found that the value of the coeficients depends linearly on both

the depth and the horizontal eiement sizes (see equations (2.L1)-(2.I2)).Bitis (1992) included the viscosity term in his Spencer Gulf model; the opti-

mal value for reduced viscosity coefficient was found to be ¿:0.0055sec-1, with a

constant coefficient of quadratic friction C¡ : 0.0025.

The objective of the calibration is to match the model solutions to the observa-

tions as closeiy as possible; to do this the mean absolute compiex difference errormeasure, equation (4.5), is used. The bathymetric data used in the Cartesian model

is for the spherical model. The calibrating experience of the Cartesian model is

used to assign the appropriate start values for the parameters, which reduces the

number of runs. For the Persian Gulf model a series of simuiations were carried

out for a range of C6 between 0.0010 and 0.0025 and different values of ø, that is,

0.0005, 0.001, 0.002 and 0.003; then an analysis of tide-heights was carried out forthe selected tidal stations listed in Table 4.4. The optimal values for Cu and ø were

found to be 0.0015 and ø:0.001sec-1, respectively, with corresponding error ofe:4.347crn. The value for Ca is the same as that of the Cartesian model, but thevalue of a for the Cartesian model of Chapter 4 was found to be ø : 0.03sec-1,

which is larger comparing with the value of a in the spherical model. Table 5.3

shows the error measure e resuiting from 16 runs, and the values of C¿ and a used inthe calibration process. As this table shows, for different values of a the error varies

bl)

slightly, indicating that the model is not very sensitive to this parameter (see Bills,

tOõ2, p.tOa). The optimai value is used in all runs of the programs. It should be

noted that Elahi and A'hrufr (1992a) chose C¿ : 0'003' and Proctor et al' (1992)

used C6 : 0.0015, for their model of the Gulf'

Ca 0.0005 0.001 0.002 0.003

0.0010

0.0015

0.0020

0.0025

4.916 4.852 4.857 4.866

4.5L2 4.347 4.348 4.349

4.553 4.517 4.516 4.488

4.879 4.860 4.851 4.840

Table 5.3: Error e (cm), for uarious combinations of ualues of quadratic fri'ction,C6, and reduced eddg coefficient a'

5.5 Presentation of the results and discussion

In this section the model predictions for tidal elevations are presented' The predic-

tions have been compared with the observational data at the forty stations shown

in Table 4.4, and they are presented in Tables 5.1(a) and 5.1(b).

Th_e tables show good agreement between predicted and observed values, espe-

cially the amplitudes.The model was applied to simulate the tidal elevations of the four major tidal

constituents, 01, Kr, Mz and ^92 constituents. The contour charts of amplitude and

phase of the major components as predicted by the model, are presented in Figures

5.2-5.9. These contours show lines of constant amplitud" (nt) and lines of constant

phase (degrees). The co-phase lines are given in GMT, time zone - 0400, to agree

with the Admiralty chart, 5081 (Figures 4.3-4.6).

66

amPstation ph274.0263.0

246.6233.0

237.1227.O

220.3209.0

266.8256.0

27t.).259.0

125.1115.0

283.3273.0

2t.911.0

285.1274.O

332.2325.0

104.894.O

106,792.0

31.628.0

85.674.O

lo.7359.0

2.7350.0

66.164.0

18.710.0

103.385.0

50.462.7

36.936.4

33.7

41.749.7

24.7ó¿, t

L7.72 5.0

43.05t.2

36.044.O

107.1100.0

39.648.0

48.854.0

33.030.0

25.633.0

82.673.0

57.469.7

85.376.8

tt.o68.9

63.259.1

54.163.0

34.245.1

262.8248.0

161.0149.0

L24.5110.0

/ u.b68.0

270.0256.0

290.6276.0

30.219.0

49.537.O

200.1197.0

48.136.0

95.395.0

331,9335.0

343.6332.O

201.9204.O

254.4245.O

185.9185.0

190.6193.0

zôJ.o229.O

62.654.0

258.0

10.416.8

t2.4L2.8

t2.5L2.3

13.5L7.7

b.b11.6

2.68.0

L2.6t7.L

1 1.4r6.0

40.036.0

12.016.0

L5.2t7.oL2.O

12-o

9.512.o

31.925.0

20.L22.6

32.427.4

30.9zc.ô

23.r20.1

70.220.o

9.1

326.2312.0

2LO.8r96.0

77L.9160.0

L27.8117.0

334.6320.0

347.6335.0

7t.657.O

tLg.7107.0

237.7229.O

105.593.0

156.9159.0

t7.510.0

2L.66.0

246.3247.O

291.6282.O

230.722L.O

234.4227.O

272.4266.0

r14.2102.O

310.8294.O

ModelObs.

ModelObs.

ModelObs.

ModelObs.

ModelObs.

ModelObs.

ModelObs.

ModelObs.

ModelObs.

ModelObs.

ModelObs.

ModelObs.

ModelObs.

ModelObs.

ModelObs.

ModelObs.

ModelObs.

ModelObs.

ModelObs.

ModelObs.

28.7

19.025.7

17.020.4

72.418.0

18.82L.6

18.42L.O

12.811 .9

9.814.0

23.820.7

6.910.0

5.99.0

rb.l15.0

tt.416.0

22.120.o

L7.62t.9

22.O20.4

19.315.8

18.618.9

3.47.O

t3.7

215.0

204.9192.0

200.6189.0

180.8L7t.O

22L.O

232.4221.0

98.389.0

238.8227.0

359.43.0

218.0

285.2281.0

63.555.0

78.565.0

to.214.0

47.240.0

344.9355.0

344.9349.0

29.338.0

283.3268.O

66.241.0

36.442.9

31.338.8

2 5.830.7

19.3zo.ó

31.642.3

31.138.0

23.8

11.217.0

42.833.8

9.718.0at12.o

28.129.O

30.0

37.6tqn

28.832.6

34.628.7

36.326.2

29.529.3

7.99.0

Jf.ð

1

,

3

4

b

6

8

o

10

11

t2

13

t4

15

16

17

18

l9

20

Table 5.1(a): Comparison oT conxputed (Model) and obserued (Obs.) results for amplitude

(cm) and phase (degrees) for the four nrzjor tidal constituents at selected stations i,n the

Persian Gutf (see Table 5.1(b) for the remaini'ng stations')

67

Iamp ph

1

amp phstation30.522.O

3.6L.2

16.320.t

2t.524.5

11.915.0

2t.32t.o3.67.O

6.47.O

20.125.0

12.020.0

4.48.2

30.734.0

10.6L7.O

33.436.2

L2.419.0

31.035.8

35.340.4

24.522.O

39.547.2

28.630,0

6.13ó2.O

L28.9116.0

69.654.0

89.277.O

39.220.0

33.629.O

120.Or04.0

31.816.0

64.353.0

70.L74.O

96.284.0

77.864.0

99.8103.O

82.472.O

L2T.L110.0

1o7.295.0

102.890.0

163.4151.0

TI2.6101.0

120.0

¡i b.u67.0

4.9

42.4

39.639.0

43.651.0

64.062.O

24.916.0

32.64r.7

38.0

47.045.0

9.3t6.2

32.434.0

4L.143.0

32.1

38.032.O

15.8It.2t6.7b.,J

32.425.0

30.8L4.2

47.437.O

t97.4196.0

180.2196.0

5L.444.O

247.2235.0

72.457.O

209.6210.0

184.6169.0

87.772.O

34.337.O

224.7234.O

247.2267.O

31.537.O

240.r248.O'40.4

51.0

252.4254.O

278.4266.O

86.495.0

333.0323.0

105.2115.0

300.0289.0

28.922.O

4.92.r7.O

13.1

13.514.6

9.9r8.0

24.320.o

6.34.O

5.311.370

11.0

16.915.0

2.54.3

12.411 .0

14.113.0

10.8

L2.511.0

b.b4.0

5.94.O

L2.78.0

t2.55.8

L7.215.0

242.O239.0

262.8273.0

92.888.0

293.6279.O

117.0102.0

253.5250.0

234.9223.O

t42.7128.0

76.0

272.5267.O

294.9284.0

85.267.O

290.4301.0

94.182.0

300.8304.0

349.4337.0

107.095.0

.Jb.b

26.O

137.8L22.O

354.4343.0

2L

22

24

25

26

2a

29

30

31

33

34

35

36

38

39

40

ModelObs.

ModelObs.

ModelObs.

ModelObs.

ModelObs.

ModelObs.

ModelObs.

ModelObs.

ModelObs.

ModelObs.

ModelObs.

ModelObs.

ModelObs.

ModelObs.

ModelObs.

ModelObs.

ModelObs.

ModelObs.

ModelObs.

ModelObs.

17.916.0n,1.8

7.89.4

t3.7t7.6

8.0

L3.216.0

5.0

3.0ö. I

9.812.0

9.617.0

2.82.4

15.615.0

9.015.0

L7.O

L6.4

9.415.0

t6.716.3

L8.718.5

16.312.0

20.524.2

16.1r9.0

351.7345.0

45.036.0

11.50.0

43.231.0

322.8308.0

18.62A.O

23.19.0

324.8301.0

L2.80.0

36.233.0

14.84.0

30.017.o

53.850.0

34.628.O

80.271.0

73.462.O

/b.b62.0

7t9.4114.0

78.167.O

96.281.0

difierences4.347

Table 5.1(b): Comparison of computed (Model) and obserued (Obs.) results for amplitude

(cm) and phase (degrees) for the four major tidal constituents at selected stations in the

Persian Gulf (see Table 5.1(a) for the remaini,ng stations.)

68

Figure 5.2: Amptitude contours (m) for the Ot tidal const'ituent of sea surface ele-

uation in the Persian Gulf obtained from the spherical coordinate model.

Figure 5.3: Pho.se contours (degrees GMT) for the Ot tidal constituent oJ sea surface

eleuation in the Persi,an GuIf obtained from the spheri,cal coordinate model.

69

04

03

T02

0.4

Figure 5.4: Ampti,tude contours (m) for the Kt tidal constituent of sea surJace ele'

uation in the Persian GuIf obtained from the spherical coordinate model.

180

150

180

tr :

Figure 5.5: Phase contours (degrees GMf ) for the Kt tidal constituent of sea surface

eleuation in the Persian Gulf obtained frorn th,e spheri,cal coordinate model.

70

tr

0.8

o2

03

0.4

03

Figure 5.6: Amplitude contours (m) for the Mz tidal constituent of sea surface ele-

uation in the Persian Gulf obtained from the spherical coordínate model.

Figure 5.7: Phase contours (degrees GMf) forthe Mz ti,dal constituent of sea surface

eleuation in the Persian Gulf obtained, from the spherical coordinate model.

71

0.2

-0.1

0.1 E

02 \0.3

01

Figure 5.8: Ampli.tude contours (m) for the 52 tidt,l constituent of sea surface ele-

uation i,n the Persian Gulf obtained from the spherical coordi,nate model.

Figure 5.9: Phase contours (degrees GMT) for the Sz tid,al constituent of sea surface

eleuation in the Persian Gulf obtained from the spherical coordinate model.

120 SOI

330

2102f0

18030150

@

6o 3¡

3m

270

2q

120 90

72

The main features of the results and their comparison with the Admiralty chart

are described in the following:

1. In the results

o Two amphidromic points appeared for the semi-diurnal constituents, M2

and 52, and one for the diurnal components 01 and 1(1'

o In general, the high amplitude values occur around the Strait of Hormuz,

especially at Bandar Abbas (Station 9 in Figure 4.i1) and the head ofthe Gulf.

¡ The basic pattern of the 01 constituent is similar to that for K1, because

the period of the 01 is close to the period of. Ky but the magnitude ofthe Or is almost 20To lower'

¡ The ^92 tide has a similar pattern to the M2 for the same reasons' but inthis case the amplitude of the ,Sz is approximately 30% of the M2 valtes.

2. When compared with the Admiralty chart

r The location of the amphidromic points predicted by the model are sim-

ilar to the Admiralty chart 5081.

o The modeled results for Or and M2 phases indicate that in the Straitof Hormuz, near the open boundary the phases are moving mote slowlythan those shown in the chart and they move anticlockwise through the

- Gulf.

o The predicted phases for the -I(1 constituent in the Strait of Hormuz are

almost the same as those observed in the chart and they move clockwise

in the Gulf.

o The predicted phases for the ,Sz constituent in the Strait of Hormuz ap-

pears to be slightly behind those shown in the chart and they move clock-

wise through the Gulf.

t,f

Chapter 6

Nested Spherical ModelDevelopment

6.1- IntroductionIn this chapter a model that involves interactive nesting in space and time referred

to as the INST( INterpolated in Space and Time) model, is deveioped and used

to compute the changes in tide heights and tidal currents in the Persian Gulf. The

coastal regions of the Persian Gulf are shaliow and complicated, suggesting that a

high resolution grid is required to satisfactorily approximate these a eas. It should

be noted that a very fine grid over the whole Gulf is too computationally expensive,

because the Gutf is very large and increasing the number of grid points causes a

dramatic increase in the CPU time require to obtain solutions.In this approach a fine grid has been superimposed over important regions of the

coarse grid model. Having completed one sweep of the coarse model with a large

time step, the information from that model is used as an open boundary conditionto solve the tidal equations with a number of smaller time steps within the fine-gridregions. These time steps are an integral fraction of the coarse time step, and are

chosen small enough to maintain the same average Courant-Friedrich-Lewy (CFL)number of both coarse and fine-grid models. Because of this, there is no large change

in phase-speed of modelled tidal waves, with associated refraction and reflection, at

the junction of the coarse and fine grids. This produces a smoother variation ofwave properties each side of the interface between the fine and coarse grids.

6.2 Review of the literatureThe idea of increasing resolution in a particular subregion of a model domain withoutrunning the fine grid model over the whole region has been considered by otherresearchers, and some nested modeis for oceanic appiications have been developed,

although most of the models developed are in the area of meteorology. These nested

models fall into two categories, one-way nested or passive, and two-way nested or

74

interactive mod.els, both of which will be described in this section. It should be

pointed out that the INST model which is developed and described in this chapter,

is a two-way model and invoives interpolation in time as well as in space'

1. Meteorologicai models:

Philtips and Shukla (i973) deveioped a nested grid system for weather predic-

tion. They pointed out that there are two basic strategies which they refer

to as one-way interaction and two-way interaction. In the one-\¡/ay model the

predictions are first made on a coarse grid, independent from the fine-grid

calculations. Boundary values for the fine-grid region are obtained from what

they have called a"history tape" of the coarse-grid predictions. The coarse-

grid predictions do not use any information from the frne-grid predictions. In

the two-way model the time integration proceeds simultaneousiy for the fine

and coarse resolution mesh areas so that the two mesh areas interact dynam-

icaily with each other. In this v/ay some of the computed values from the

fine-grid are used in the coarse-grid. From the results of the numerical tests,

they have found that in all cases the error for the two-way was smaller than

for the one-way interaction and it was concluded that for the integration using

the nested grid system, two-way interaction is more favorable.

Kurihara et al. (1979) constructed a numerical technique to develop a two-way

movable nested grid for meteorological applications. The grid points were not

staggered and all meteorological variables have been defined at the same point.

For this nested scheme, two types of dynamic interfaces were defined, the

ihput dynamic interface at which the coarse grid model provides the boundarydata for the fine grid model, and the feedback dynamic interface at which the

predicted fine grid values were used to update the coarse grid region. In order

to reduce the refraction and reflection of the waves at the interface between the

two different mesh grids, they intentionally separated the input interface fromthe feedback dynamic interface. The main reason which supported their idea

of separating interfaces \ryas, since the fine grid boundary values are externallyprovided by interpolation from the coarse grid, so these values should not be

used in the feedback procedure. In fact by separating the feedback interface

the values which were internaliy predicted used in the feedback scheme. Inthe INST model developed in this thesis the wave refraction and reflection isreduced, because of the nesting in time which allows the Courant number forboth grids to be approximately the same on each side of the interface.

Zhang et al. (1986) presented a two-way nesting procedure. Like the others

they also believed that by adding the nesting capability to a model a sub-

stantial improvement in the horizontal resolution is achieved, without using

a f.ne grid mesh throughout the whole model domain. They used the idea ofseparated input interface from the feedback interface described by Kuriharaet al. (1979) to reduce the interface noise.

75

2. Oceanographic modeis:

Ramming (1976) used the well-known basic model of the North Sea developed

by Hansen (1956), and developed a nested North Sea model. He noted thatto obtain a better approximation, it is necessary to refi.ne the spatial grid incoastal regions as it may not be possible to obtain a good approximation ofislands, flats, sands, coastal configurations and depth distribution from the

model which uses a coarse grid. He believed that from the classical numerical

models of the North Sea which already exist, very good results have been

obtained, especially for the reproduction of storm surges and tides. He believed

that this model with a grid size of 37 km could handle the problems concerning

Iarge areas and provides good topographical approximation. But he noted thatto give ans\ryers to the more special questions of oceanography and neighboring

sciences a new refined-grid model would have to be developed. In his nested

model the refinement factor is always an odd number, because of using the

Arakawa C-type staggered spatial grid. Ramming uses a refinement factor ofthree, and so a one-third model has been developed.

In order to provide more detailed tidal current information in some sensitive

regions of the Persian Guif, Lardner et al. (1982) developed a two-dimensional

hydrodynamical tidal modei for the Gulf. The horizontal eddy viscosity was

considered to be zero. The model consisted of three blocks. The deep areas

were covered by a coarse grid, which formed the flrst block; the shallow coastal

areas were covered by a finer grid, which formed the second block; and the

area of Ddahran, Bahrain and the west coast of Qatar ',4/ere covered by a super-

fine grid, which formed the third block. The three blocks were fitted together

to form a single array. They used the finite difference scheme of Leendertse

(1967) modified to study each of the above-mentioned regions of the Gulf.The only open boundary of the model was set at the Strait of Hormuz, which

was contained in the first block. The tide heights at the open boundary ofthe second block at each time step were provided by the first block and the

open boundary data for the third block, which was entirely located within the

second block, was specified from the computed tide heights of the second block.In this model no spatial averaging of fine grid values was used for updatingthe values of grid points of the coarse region. This model may be classified as

a two-way model. They reproduced the amplitudes and phases of three majorconstituents in the Gulf, namely K1,, M2 and,9z The pattern of co-amplitudeand co-phase lines of the computed constituents were in reasonable agreement

with the Admiralty chart 5081.

Greenberg (1933) uses two two-dimensional nonlinear numerical models tostudy the residual barotropic circulation generated by tides and winds in theBay of Fundy and Gulf of Maine (ibid., p.885). A multi-grid nested model was

used to examine the Bay and a coarse grid for the Gulf. In terms of nestingmodels this is classified as a two-way model. In this model the same time steps

are used in both grids and no spatial averaging of fine grid values has been

76

used for updating the values of grid points of the coarse region.

In order to assess certain implementation aspects of the three-dimensionai

hydrodynamic model of the Persian Gulf deveioped by Lardner and Cekirge

(1987, 1988), discussed in Section 4.3, Lardner et al. (1989) developed a two

block model. In this model the Gulf ruas covered by a coarse grid block and the

coastal areas of Saudi Arabia by fine grid block. This model is the extension

of the model developed by Lardner et al. (1982) to three dimensions. It should

be noted that in this model the advective terms were neglected.

In order to increase horizontal resolution in a subregion of the model domain,

Spall and Holland (1991) developed an interactive nested model for oceanic

application. They also classified the nested models as passive and interactive

models. The refinement factor in the model is three. The feedback interfaces

are separated from their dynamic interfaces and in order to give maximum

information from the fi.ne grid to the coarse grid they put the feedback interface

as close as possible to the dynamic interface. They also note that, this agrees

with the approach of Kurihara et al. (1979) and Zhang et al. (1986) in which

the feedback interface is separated from the dynamical interface. In this model

the coarse grid uses the same time step as the fine grid (Spall and Hoiland,

1991, p.209).

In order to investigate the covering (wetting) and uncovering (drying) of sand

bars and shallow coastal flats that occur extensively throughout Northern

Spencer Gulf in South Australia, Bills (1992) covered the region by a finergrid, that is, the element of the fine grid is one-third that of the coarse-grid

element of the Spencer Gulf model. The open boundary forcing of the fine

model has been taken from the solution field of the coarse model. This may

be classified as a one-\May model because the two models are not connected

interactively.

6.3 Method of nesting

The spherical INST model described in this chapter uses the governing equations de-

scribing barotropic tidal motion in spherical coordinates, namely, equations (2.10)-(2.I2). The code which has been adopted for use in the INST model procedure, is

a considerably modified version of the spherical code, which was used in Chapter 5.

The INST model involves embedding a fine grid in space and time to cover a

region within a coarse grid. The open boundaries of the fine grid modei pass throughelevation points and grid points of the coarse model. At the position b on the open

boundary of the coarse model the sea surface elevations (6 is specified for a1l timeby means of ten major tidai constituents, using the equation (2-1)'

The elevations on the open boundary of the fi.ne grid (interface between the twogrids) are prescribed from the fine model, using the transferring algorithm describedin Section 6.4.2

77

The grid matching technique, the levels in a time step of the coarse grid and the

corresponding levels in time steps of the fine grid are now described.

6.3.L Spatial and temporal refinements

The grid element, the basic unit of the grid, consists of three grid points, which are

the free surface elevation (, and the horizontal velocity components u and u. They

have the same meaning and iocations as indicated in Figure 3.3, with the spatial

refinement factor being an odd integer rn which is defrned by

-: *þ, (6.i)AÀ¡,

where a)" and aÀ¡ are the coarse and fine grid lengths, respectively.

In the present model the refinement factor is 3 so the scheme is called a one-

third-refinement model. In the INST model refinement in time as well as space is

carried out, so the coarse spatial model uses the large time steps and the fine spatial

model the small time steps.

The general finite difference scheme used for both grids is three-level ìn time(Noye and Bills, 1992); Figure 6.1 shows the time levels in one time step of the

coarse grid, and the corresponding levels in two equivalent time steps of the finegrid. It should be noted that the temporal refinement factor is defined by the

integer p given byAt"p: Ë,

(6'2)

and in this model p is chosen to be 2.

6.3.2 The grid matching technique

The grid matching technique used in this approach, which embeds the fine gridwithin the coarse grid, is shown in Figure 6.2. The model uses nine fine grid elements

embedded within a single coarse grid eiement in such a way that the elevation pointwithin the coarse grid will coincide with the elevation point of the top left fi.ne gridelement. Each Ç, u¿ and u¿ coârs€ grid point will coincide with a fine grid point ofa similar type.

The better definition provided by the fine grid aliows the coast line to be modelledmore accurately, or rapidly changing bathymetry to be incorporated in offshore

areas.

78

n+r n+1uu u7*' o7*'n+I

n+Il2

n+I

n+314

n+Il2

c c

n+314(

n+Il2 n+l12U¡u f

l4

ncu n

cU a?nn

unf

n-Il4

n+Ll4

n - Ll4

n-Ll2

(

(:-'t' n-Ll2n-Ll2f¡

n-t12Íu u

Figure 6.1: (Left) Showing the time leuels in each ti,me step of the coarse gri,d and

(Right) Showing the correspondi,ng time leuels in each time step of the f,ne gri'd.

Sub-indi,ces c and f stand for coarse and fine, respectiuely.

l(, +

+

lu¡+ + + -+

rl

t t t

Figure 6.2: The ith coarse grid element showing matching of the coz,rse and finegrids. Values lo, C¿, u¿ and, ui o,re located i,n the appropriate places on the coarse

and fi,ne grid. For simpli,city grids are drawn in a rectangular coordinate system.

r

+

t

+?,

ut +

+ +

It

+

79

6.3.3 Initial and boundary conditions

The initial conditions used for both coarse and fine models are

e :u:1):0 attime ú:0. (6.3)

which means that the fluid is initially at rest. They are the same as those described

in Section 3.2.3.The boundary conditions used for the coarse and the fi.ne grid of the INST model

are the same as those explained in Section 3.2.2, but the forcing data for the fine

model are specified as follows:The elevations at the open boundary of the fine grid model (interface between two

grids ) are prescribed from the coarse grid model using the transferring algorithmexplained in below.

At the end of each time step of the coarse grid model, the elevations at the open

boundary points on the fine grid, which coincide with ( grid points of the coarse

grid, wili be specified at the levels of time used in the fine grid by linear interpolationin time using the ( value of the corresponding coarse grid point. For example (¡rand (¡a in Figure 6.3 use Ç1 and ("2. Data for the other fine grid elevation points,for example (¡2, e ¡s, e ¡s and (¡o in Figure 6.3, are obtained by interpolation in space

from (¡r and (¡e.

(q- Çt, ("2 - e,^

t-

a ac1

I

Figure 6.3: Showing sfu f,ne gri,d elements on the open boundary of the fine grid,

embedded withi,n two coarse gri,d elements. The correspondi,ng ( grid points of the

fi,ne and coarse grids are circled. The (",, Lt,., and l)ci o,re the grid poi,nts of the coo,rse

and (¡,, u¡, and, aÍi o,re of the fi,ne rnodel.

cz

l r1u

uc2

te¡u J-.' \Ja ?.r Ja

I ulu

uJt

l rÍ, I uÍ,

uc1

te¡"

l rJ,

+Çr, T,¡" uÍt

lrJn

80

6.4 computational algorithms and automatic sys-

tems

In INST there are three computational algorithms, namely the algorithm for solv-

ing the governing equations, the transferring algorithm and the back transferring

algorithm. In this section these algorithms are described.

6.4.t Algorithm for solving the governing equations

The discretized form of the spherical tidal equations, (3.52), (3.54), and (3.57). are

soived on the basis of the following mathematical algorithms. In these algorithms,

and through the rest of this chapter, sub indices c and / which have been used with(, u and u, indicate the coarse and fine grid, respectively.

Based on Figure 6.1 the algorithms are:

(u) For the coarse model:

1. Solve the continuity equation (3.52) to flnd e!*t/', using horizontalvelocity components ul and ui at time levei n, elevations at timelevel (n - ll2) and the total depth H, at time level (n - Il2).

2. Solve the momentumequations (3.54) and (3.57) for u"*r and u!+1,

using (n+rl2 and u! and ui'

(b) For the fine model: The algorithm for the fine model requires more steps

than the coarse model, depending on the temporal reflnement factor de-

fined in equation (6.3). The following illustrates the situation where the

temporal refrnement factor P:2.1. Solve equation (3.52) for çi+tl+ using u!,, ui, (i-tta and certain

values of ç+r¡z and (:-r12, which coincide with the fine grid open

boundary points.

2. Solve equations (3.54) and (3.57) fot u7+tlz and' ,l+t12, usingfn+tl4sJ

3. Soive equation (3.52) ror Ç+sl+ using ul*tl', ,?*tl', eî*tln und

certain values of ç+r¡z and ç"-t¡2, which coincide with the finegrid open boundary points.

4. Solve equations (3.54) and (3.57) for u]+r and u!+I, using (î*t'n.

81

Fine / embedded grid

_l-nlo

+b

+c i +

e

IInterface betvveen fine and coarse grid

f+

Coarse grid

8+

Hh +Figure 6.4: Embeddi,ng the fine gri,d within the coarse grid. The larger bars and

crosses are the ouerlapping grid points. The coordinates of the coarse gri,d are shown

by capi,tal letters and for the fine grid by lower case letters.

6.4.2 Transferring algorithmFigure 6.4 illustrates embedding the fine grid within the coarse grid. In this figurethe iarger bars and crosses are the overlapping grid points. The coordinates of thecoarse grid are shown by capital letters and for the fine grid, by lower case letters.By using the figure and considering that the temporal refrnement factor is p - 2,

the transferring algorithm is described as follows:

1. After completing a sïveep of the coarse model, that is, at the end of time leveln I I, and having the values of the overlapping coarse grid points, namely

Ç¿"r'/' , (o"+t/' and (¡¡"+1/2 , calcrtlate (o"rr/a, çdn+r/4 and (r,"+r/a, by usingthe following interpolation in time:

eo"+Ll+ : (JeA"+rl2 + ço"-rl2¡¡4. (6.4)

Values of Ç+tla' and Ç+r/a can be calculated similarly

ea"+Ll¿ : (zeDn+L/2 + çp"-Ll2¡¡4, (6.5)

82

and(t"+t/n : (J(¡¡"+rl2 + Ç¡"-rl2¡¡4, (6.6)

2. Compute the values of elevations at the same overlapping fine grid points for

the next half time level, using

(o"+sla : (5(An+r l, _ (o"-tlr) I 4,

çon+z/a : (5(D"+rl, _ eo"-tlr) I 4,

and

çnn+s/+ : (5eH"+11, _ Çr"-rlr) I 4.

3. Use linear interpolation in space to calculate the elevations at the other grid

elements on the interface between the fine and coarse grids, namely ei*'ln,(i*t/n ,, (i*'ln , (:*tln and so on. The formula for the grid point ó at the timeieveln*1/4is

eî*'/n : (zç+rt+ + (i*'/n)lg (6.10)

and a similar formula can be used for the grid point /, using (n+rl4 and (f+l/a.For grid point c the equation is

(:*'tn : (e:*'t' +zç!+un¡¡1. (6.11)

A similar formula applies for grid point g, using the data from ¿ and å. Ifthe elevation at e is just inside the grid, that is, the coastal boundary passes

through the velocity of this grid, then (n+rlL must be extrapoiated from ø and

d, that is' çn+tt+ : ( Ç+rta _ Ç,+rll)l:-. (6.12)

Similar formulae at each grid point can be used to obtain the values of ( at

thelevelntSl4.

6.4.3 Back transferring algorithm

At the end of each time step of simulation of the fine grid model, a spatially averaging

procedure is used to estimate the corresponding coarse grid velocity points from thefine grid velocities. These averages are transferred back to the coarse grid model

to provide a set of updated values for specific u and u veÌocity grid points withinthe coarse grid model to initiate the next time level within the overall model. Thefollowing example describes how this algorithm works.

f.b./)

(6.8)

(6.e)

83

r(q-e,, ("2 - Ç,^

Figure 6.5: The ei.ghteen fine grid elements embedded within two coarse grid ele-

rnents. Circledu¡ ueloci,ties a,re used, in the aueraging procedureto updateuc2 coz,rse

grid point.

Figure 6.5 illustrates a schematic of the embedded 18 frne grids within two coarse

grid elements away from the interface between two grids. In this figure the frne-grid

velocity points, circled u¡ velocities, are used in the averaging algorithm to update

the u"" velocity, which is specified by l. This algorithm estimates the u", velocity by

obtaining an average from the corresponding fine-grid velocities, using the 9-pointweighting function identified in matrix A, where

A: 1116

1

2

1

2

4

2

1

2

1

The averaging equation used in this scheme is

'ucz : (r¡, * 2u¡n * u¡u (6.13)

+ 2u¡n l4u¡ro * 2u¡r,

+ uf,.u l2u¡ru I u¡rr)f 76.

When a coarse-grid velocity point is on the open boundary of the flne grid the

-1 -2 -1363

uJt

| ,i,

* C¡" u c1

l rJ,

L¿' t.fg u J, Ur.

+r,lJ5 u c2

r(.r. Ut^

l, ju

| \Íz u¡,J-t'Ç"f¿ uJ,

l rJ,l, ",

te¡, i¡n -Lt, t.fro ?.¿Jro

lu ",

J-.t \ Jtt 1.1 Jtt

)

J-2' \Ín LLJn

l rJ*

' \Íts L[Jß*e Íu-uju

l rJrn

J-.' t,frs U/rs

| ,!r,

' \Írc LrÍrc-L/' \Jn LrJn

L.' \Jß UJß

lrj*

weight matrix becomes

rl8

84

Figure 6.6 illustrates the locations of two coarse-grid velocity points, u", and Lt'c2 on

the open boundary of the fine grid. The u., is updated by using six fine-grid velocity

points above it.The averaging equation used is

'u,"t : (-r¡, - 2uJ" - uÍs + 3u¡, I 6u¡, * 3u¡") 18. (6.14)

There are some remarks about averaging system which should be mentioned here.

To find a good weighting function for the averaging procedure, the other types of the

weighting matrix (Abramowitz and Stegun, 1964) have been tested, when a coarse

velocity grid point is on the open boundary of the flne grid; finaliy the one described

in Section 6.4.3 was selected and used in the model, because the results that are

shown in Figures 6.10, 6.12 and 6.13, are better than the results obtained from using

the other weighting functions. Other weighting functions have been tested including(u)

rl12

For Figure 6.6 the averaging is then

't!,cL : @n I2u¡, * u¡" (6.15)

+ 2u¡r, * 4u¡r. *2u¡r)f 12.

Note that this equation uses three grid points on the open boundary and three of

the one row up.(b)

42Considering Figure 6.6 the averaging equation is

'¿.r,"t: (2u¡r, * 4uJrn l2u¡r)f 8 (6.16)

This equation uses three grid points on the open boundary.As suggested by Kurihara et al. (1979), using values on the boundary leads to

instabiiities. Each of these weighting function was tested and the most satisfactory

solutions were found using equation (6.15).

lti;)

rl}l2

85

+

I uÍt

+ +

lufz I Dfe

+uJn

luͿ

+ Ur-

I ufs

+ uÍu

luÍa

+

, ur,

+ +

l, j, l rÍn

+u Jro

luÍro

+ ãÍ,, +u Írz

I uÍr, l rf*

ü Ír"

| ,Ír"

+

l rJrn

+ü fru

l rÍru

+ u +Jrc u c2

lrÍru l rf*

+ u Íre

l rÍ,.,

r--

Uc1 ac2

Figure 6.6: The two coarse-gri,d uelocity points, u", and L¿c2, on the open boundarg

of the f,ne grid. The u., is updated wi,th si,r fine-gri'd uelocity grid points'

6.4.4 Stability criteriaThe stability of the explicit fi.nite difference scheme is assured provided that the

usual Courant-Friedrich-Lewy (CFL) stability condition is met. This implies thatthe Courant number in each grid must be less than or equal to unity. The Courant

number in two-dimensional Cartesian form is

cr: LttF lL", (6.17)

where

As : ((Ar)-, * (Ly¡-z¡-trz ,

(see Flather and Heaps, 1975, Benque et a1.,1982, Murty and El-Sabh, 1984, and

Al-Rabeh et al., 1990).

The CFL condition is

(A,r\2 + lAu)2 < 1\--l ' \-¿t -

-)

l-

86

(6.18)

where h*o, is the maximum depth of the modeled region'Equation (6.17) is used to obtain the optimum time step, that is,

A¿<L,rA,y

Jsñ (A,")' + (Ly)'

If Aø : La, equations (6.17) and (6.18) imply that

(6.1e)

(6.20)

and

(6.21)

By anaiogy of the Cartesian results, the Courant number in spherical form can be

written as

or

I '¡ -'' - (BA).* dXAd)If A) : Ad then

(ÊA) cos {)2 + (RLó)2 < L, (6.22)

(AÀ cos ó) + (Ad)' S t (6.23)

cos2/-¡1. (6.24)

To satisfy the stability condition in both fine and coarse grids it is required thatthe maximum Courant numbers in both grids be approximately the same, that is,

,@.x (cos /)l + 7l R\^.(cos /)" æ 1[stu tt r (cos $)2, + LIR\^f(cos /)y,(6.25)

whereå." is the maximum depth in the coarse model,ä¡ is the maximum depth in the fine model,Lt", Lt¡ are the time steps for the coarse and fine grids, respectively.

AÀ"Using * :

l^r, and choosing p as the temporal refinement factor, it can be seen

thatplm N tltu fi, , (6.26)

SO

h¡ lh" (6.27)pNrnIn the INST model of the Persian Gulf by considering m : 3, h":80rn and

h¡ :20rn the value of p using equation (6.26) is p = 1.5, in other words Lt" x tntt.The largest reasonabL p - 2 has been chosen to minimize the simulation time, thati.,

AÚ':2Ltr' (6'28)

87

I

Wave refraction as well as reflection may occur due to a change in phase speed

at the interface of the two grids of different size (see Noye, 1984, p.298 and Roache,

Ig72, p.298). When a tidal wave passes through the interface, it is reflected or

distorted.. In order to minimize these effects and to assure a smooth variation on

both sides of the boundary and over the boundary, it is necessary that the Courant

number be approximately the same on each side of the interface.

It should be noted that, if the Courant number are almost equal in both sides of

the interface the transition between the models are made properiy, that is, refraction

and reflection do not contaminated the computed patterns even near the boundary,

so the objectives due to changing wave speeds would met properly. In other words

when two grids are combined to form an embedded model, the solution obtained

from the model is very similar to that of the fine grid.At the interface between the fine and coarse grids å¡ N h". Then the ratio of

the Courant number numbers C" and C¡ in two grids will be

C" A,"AÀC ¡ A¿JAÀ"

and this causes less distortion compared with using the same time step in both the

fine and coarse regions, in which case C"f C¡ would be 1/3.

C" 2

c¡-3'and this causes less distortion compared with using the same time step in both fine

and ccjarse region.

6.5 Numerical tests

In this section the application of the model in a rectangular bay is described. The

aim of this application is to test the INST model. A rectangular bay of Figure 6.7

was chosen for this application and has been used for a fully coarse grid model, a

fully fine grid modei, and a nested model using the INST model. The contour plots

of surface elevations obtained from the models are presented and compared witheach other in order to verify the expectation that the contours resulting from the

INST model are between the results of the fine and the coarse model and closer to

that of fine model.The coarse grid model of the region of Figure 6.7 uses a uniform grid of 0.25"

latitude by 0.25" longitude. The time step AÚ" : 400s is chosen for the coa se model'

The depth is specified to be a constant 10rn. In this figure the coastal region of the

shallow water which needs higher resolution is covered by a uniform grid of 0.0833"

Iatitude by 0.0833" longitude, that is, m:3, and the time step is Al¡ : 200s. The

coastal boundaries of the embedded region coincide with those of the coarse model(solid lines passing through velocity grid points) and the open boundary of the finegrid model passes through those ( grid points of the coarse model which are labelled

p2m 3'

88

by u + sign. The u and u velocities which are inside the model and specified by -and l, respectively, will be updated by the back transferring (averaging) algorithm.

It should be noted that the coarse model is run until it hits the fine grid boundary.

r-IIIII

-T- --- ---- --- r--- --- - -- --lr-Il'II

I'lII

t'I't'I't'

IIr-

+ + + +

I I

IIìIIII

IIIIII

--!

+

II

--1

ooa

II---r I-T--

I

I

III

I

I

I

II

IIII--TII

III--rII

t-"

III¡r-I

II-l-I-r-I-t'

I I

I---1I

IIIIIIIIIIrIIIIII

+ +IJIIIrIII

i+

IIIII

I

IIr-----

¡I

--lIIr---

...oel oooI

o aaoaaI

þ

III

I

I¡IIIIIttt

Figure 6.7: A rectangular cqastal model with embedded grids, coastal boundaries

(solid lines), the open boundary of the col,rse region (dotted li,nes), updated u and uuelocities (- ,l insi,de), and showing the open boundary of the Ji,ne model or interfacebetween the grids (through the I signs).

89

6.5.1 Automatic sYstems

As explained in Section 3.5, the region covered by the Noye and Bilis (1992) model

is labåIled using four types of elements and each element is assigned one of the four

integers 0, I,2 and 4. The numbering system in INST has been modified by adding

three new types of elements, namely type 8, type 6 and type 5. The averaging

scheme uses type 8 and type 6 elements and the repiacing procedure is involved

with the type 5. It should be pointed out that, instead of creating the replacing

algorithm, the averaging system could be extended and used to update the other

grid points, that is, those grids points which have already not been updated, no

ãiff"r".r"" appears, but in terms of programming procedure it was easier to develop

and use the replacing scheme, especially for complicated coastal boundaries such as

those in the Persian Gulf. For this reason a new type of element, namely type 5, has

been defined. In the input matrix of the coarse model those type 1 elements which

must be replaced. by new values computed using fine grid values, are replaced by the

type 5. A similar substitution should be done for the corresponding elements in the

input matrix of the fine grid. Figures 6.8 and 6.9 show the exact positions of the

type 5 elements in the coarse and the fine models, respectively. These elements are

used for the connection procedure of the two models. The open boundary of the fine

model, that is, the interface between the coarse and the fine grids must be specified

by using number 8. Figure 6.8 shows the position of the interface of the two models

in Figure 6.7, in the input matrix of the coarse model. In the input matrix of the

fine model the number 8 must be put exactly on the interface, but just over those

grid. points which have the same ( points as the coarse grid, then between each two

8's thé number 6 must be placed. The number of type 6's between each two type

8's is equal to m- 1. Figure 6.9 shows the location of these numbers in the type

matrix of the fine grid. The reader may refer to Figure 6.7 to compare the positions

of the relevant grids in the figures.

0

0

0

0

0

5888850

00000005555550

58118508811880

171110222220

010204444440

Figure 6.8: The input matrir of the coo,rse model associated with Fi,gure 6.7, showing

the elements labelled I on the interface between the fine and the coa,rse rnodel and

the elements labelled 5 for use in the replacement scherne'

90

00000000 0000000 000000000000000000000000000001 1 1 1 1 1 111 1 1 11 1 111 100015 1 151 151151151 1510001 t7t1 1 11 1 1 1 1 1 1 1 1 1 100017tL11M-t7171111100000000000000000

1

1

1

1

T

1

6

+

L

1

1

tr

1

7

I4

1

t7

t1

1

6

4

t1

1

L

1

t6

4

b

4444444

I4444444

6

4444444

I4444444

b

4

44

44

44

Ib

6

8

6

6

8

4

1

T

1

1

1

1

o

4

1

1

L

1

1

1

6

4

8

b

b

Ib

6

I4

6

4

44

444

4

6

4444444

6

4444444

510110110510110110860440

Figure 6.9: The input matrb of the f,ne model of Figure 6.7, showing the elements

numbered 8 and 6 on the interface, the open boundary of the fine grid and elements

labelled 5 for use in the replacement scheme.

6.5.2 Test without Coriolis

A set of fuli non-Iinear equations in spherical coordinates, namely equations (3.50),

(3.59) and (3.76), including all terms but the Coriolis force, are solved in a constant-

depth rectangular bay. The Coriolis force excluded to verify that at ali points across

the model bay the tidal elevations are the same, and the tidal currents do not vary.

In order to analyze the results obtained from the INST model, they are compared

with the corresponding results predicted by the fine and the coarse models of the

same computational domain. Different orientations of the interface between thethree models also have been examined, to find out the exact behavior of the resultsthrough the embedded modelled region, compared with the fine and the coarse

resuits.The results from all four orientations (open boundary opening towards the north,

south, east, and west) for each of the three models were the same for each orientation,so that only the results for the,southerfV facing boundary (Figure 6.7) are presented

below. (i. pa¿; n.¡),,.,'v.

Figure 6.10 iilustrates the contour plots of the sea surface elevations resultingfrom the three models of the computational region of Figure 6.7, namely (a) is thefine, (b) is the INST and (c) is the coarse model.

The results shown in Figure 6.10 indicate that the elevations predicted by INSTiie between the fine grid elevations and the coarse grid elevations. In addition, allthe contours are linear, a result of Coriolis being excluded.

91

0.1

0.3

0.2

0 .0

1

0.5

0

(a) Fine

(ó)rNST

210 0.5 1.5

1

¡'/ 0.5

0 0.5 1 1.5

1(c) C oarse

0,5

000.5 1 1.5

Figure 6.L0: Comparison of sea surface eleuation contours for the rectangular bay ofFigure 6.7 using (a) the fi,ne grid, (b) the INST, and (c) the coarse grid, models.The dotted line on the right is the open boundary of the bag in each case. TheCoriolis term is not included.

02

2

I

I

I

I

I

I

I

i

0.1i

I

0.2

0.0

-0 -00.3

03

2

300.2

1

0.0

-0.1-0.2

-0 3

I

0

92

6.5.3 Test with Coriolis

This section presents the results of testing the INST model when Coriolis is in-cluded. This test has been carried out to ensure that models work independent oforientation, and also to examine whether the INST results be between the coarse

and the fine grid results, and are possibly closer to the fine results.The rectangular bay illustrated in Figure 6.7 was modelied using four different

orientations. Figure 6.11 shows two of these, namely the orientation with the open

boundary facing south and that facing north. The three models, fine, INST, andcoarse, were run with each of these orientations, and the results are shown in Figures

6.12 and 6.13. Both sets of results show that elevations resulting from applying theINST model lie between those resulting from the coarse and fi,ne grid models. Inaddition, with the change in orientation, the change in Coriolis produced resultsthat were mirror images of each other.

NÂ

I

open boundary of the coarse grid

oooooo o

Interface Coarse grid

aoao aoo

aao aaa

Coarse grid Interface

ooooooopen boundary ofthe coarse grid

Fine / embedded grid

(a) (b)

Figure 6.11: .4 rectangular coastal mod,el with embedded grids, showing (a) southerly,and (b) northerly facing open boundaries.

Fine / embedded grid o

oo

93

1

00.5 1 1.5

0.5

0

0,5

0

2

(a) Fine

(b)rNST

1

¡/

0' 0.5 1 1.5 2

(c) Coarse

0.5

00 0.5 1 1.5 2

Figure 6.12: 7åe INST contours of sea surface eleuation f1r the rectangular bay

model of Figure 6.11(a), compared with the contours for the sarne region, couered

by fi,ne and coarse grids. The Coriolis term is included,. The dotted line is the open

boundary of the bay.

1

ljil

¡

I

0.3

0.2

0.1

-0,3

05

0.4

4.0.0'

t/Ã5r

',/

.iijj

tt

ìlt)

0.í

00

0

I

I

¡ìtI

o2

I!

-0.3-0.;7 -o.z

-0,3

0.4

UU

0.5

0.3

_,-0.2'-'-.-.-.-'-

01

-0.2-0.1

94

-0.3 -0.2 -0_l

0.5

0,4II

I(I

01

ll

t:riiì

0.3

-02\

1

0.5

0

(a) Fine

(ó)rNST

0 0.5 1 1.5 2

1

jv <- 0.5

00- 0.5 1 1.5 2

1(c) Coarse

0.5

00 0.5 1 1.5 2

Figure 6.13: ?å,e INST contours of sea surface eleuati,on for the rectangular bay

model of Figure 6.11(b), compared wi,th the contours for the sarne reg'ion, couered by

fine and coa,rse grids. The Coriolis term is included and the open boundary (dottedline) is on the left.

I

- 0t

0.5

I

I

-0 I-0. t

02\

00-03

II¡

0.4

0_3

II

I

II¡

I

¡

0.5

=--..---.0.2

-0.2 4.1

.40.3

- 01

00{.3

II

1

)

\

tI

95

Figure 6.14 shows the easterly and westerly facing open boundaries of the mod-

elled bay. The results from using these orientations were basically the same as those

found using the northerly and southerly orientations, and so they are not repeated

here.

NÀ

I

a

' openbounda4'

of theO coarse grid

^ .. aCoarse grrcl

a

o

a

a

(a)

a

Fine /embedded grid

Intelace

aaaaaa

a

a

a

a

aopen boundary 1of the

coarse grid O c Fine /' embedded grid

O

o

o

o

o

Coarse grid

Inretface

aaaaaa

a

a

(b)

Figure 6.14: A rectangull,r coastal model with embedded, grids, showing easterly, (a),and westerly, (b), facing open bound,aries.

6.5.4 Summary and conclusion

The INST model was tested in a rectangular bay with different orientations of theinterface between the fine and the coarse grids, that is, southerly, northerly, easterlyand westerly facing open boundaries. The coarse, fine and the INST models havebeen applied to each orientation. Tests have been conducted to ensure the modelswork independently of the orientation of the open boundary, and to make sure thatthe INST results are between the coarse and the fine results, being closer to the

96

fine. The contour plots of the surface elevations obtained from the three modelswere compared with each other, and good agreement between the results throughthe embedded regions were shown. It should also be pointed out that the resultingcontours from the southerly facing boundary model are the exact mirror image ofthose of the northerly model and similar results were found for the easterly andwesterly models.

It should be noted that, because the Courant number has been aimost equal inboth sides of the interface the transition between the models has bee',', made p¡operly,that is, refraction and reflection do not contaminatedlhe computed patterns evennear the boundary, so the objectives due to changing wave speeds are met properly.In other words when two grids are combined to form an embedded model, thesolution obtained from the modei is very similar to that of the fine grid.

97

Chapter 7

The II\ST Model of the PersianGulf

7.t IntroductionThis chapter discusses the application of the INST model to the Persian Gulf. From

the appiication of a depth-averaged sphericai coordinate model of the Persian Gulf,described in Chapter 5, close agreement between observed and computed tides was

obtained, but to provide reievant information in shallow, complicated coastal areas,

especially around the Iranian coast which has largely been neglected in the past, thegrid was refined to improve the predictions in those regions.

Figure 7.1 shows the coastal regions of the coarse model of the Persian Gulfwhich are covered by the fine grid, in order to have a higher resolution within theshallow waters and reproduce the shape of the coastline more accurately. These

regions of the Persian Gulf are important because of the shipping activities into and

from islands and the coastai ports and the oil wells in coastal waters.

7.2 Numerical modelThe coarse grid section of the INST model uses the same uniform grid (5' latitudeand 5' longitude) and a time step of A"f : 200s as the spherical depth-averagedtidal model of the Persian Gulf described in Chapter 5. The fine grid covers theshaded area of Figure 7.1 and uses a uniform grid which is (5/3)' square. The finegrid has been generated on the basis of the grid matching technique, described inSection 6.3.2. Using this technique the coastal boundaries of the coa se and the finegrids have different shapes; Figures 7.2 and 7.3 illustrate this, in particular aroundthe heád of the Gulf and in the Bahrain area.

98

E

Figure 7.1: The shallow coastal regions of the coarse model of the Persian Gulf which

are couered by the fine grid of the INST model.

The embedded fine grid contains 8383 grid elements that lie within the embedded

regions. The time step used, as described in Section6.4.4, is half of that of the coarse

model, that is, (A¿)¡ : 100s.

The bathymetry for the embedded fine grid has been developed for the 8383

elevation grid points, from the bathymetry of the coarse model using bivariate in-terpolation, three and four point formulae. The four point formula interpolates thevalue of a function / at a point (*,A), which lies in a grid square with vertices of(rt,At), (xz,At), (xz,,Az) and (r1, Uz) (see Figure 7.4), that is,

f @,a) : dtÍ(ût,at) -f azf (xz,vt) * asf (nz,az) * a+f (tt,az)

where ai, i : I,2,3,4, are the coefficients of the interpolation function described inAbramowitz et aI. (1964, p.882) and Press et al. (1989, p.117), they are combinationsof the the functions described in Section 6.4.2.

The three points formula defi.ned similarly (see Abramowitz et al.1964, p.882).The value C¿ : 0.0015 is used for the quadratic friction coefficient, for both the

coarse and fine grids.

99

(o)

( )b

Figure 7.2: The coastal boundary near the head of the Persian Gulf , (a), øs couered,

by the co(rrse grid and (b), as couered by the f,ne grid in the INST model.

The reduced eddy viscosity coefficient for the fine model was chosen to be threetimes that of the coarse model, because the refinement factor in this model is rn : 3

and because ø depends on A), equations 2.11-2.12. In this way the horizontal eddyviscosity did not change abruptly at the interface between the fine and coarse grids.

100

)J

!lj--I

LJ

:l_(

1-l

-L

Ll

(b)(o)

o (x , y)

Figure 7.3: The coastal boundary of the Baltrain coast'in.the Persian Gulf, (a), as

couered by the coe,rse grid and (b), as couered by the fine grid in the INST model'

( xr, !r)(4,tr)

(x. v)' t''l'

Figure 7.4: The labeling of points used in the four point i,nterpolation formula.

7.3 Presentation of the results

In this section model predictions for tidal elevations and depth-averaged tidal cur-

rents are presented. The main interpretations of the predictions are by means ofcontour charts of tidal elevations and ellipse plots of the depth-averaged currents forthe diurnal constituents 01 and K1 and the semi-diurnalconstituents M2 and,92.

(x, v)' 2 'l'

101

The model was run for a simulated time of 32 days, the first three days ofpredictions are ignored (as initial transients). Predictions for the remaining 29 days

were saved hourly and were tidally analysed.

7.3.L Sea surface elevations

In this section the tidal constants computed from the INST model are comparedwith the observations of tidal elevation at the forty stations identified in Table 4.4.

They are also compared with the predictions from the Cartesian model of Chapter4 and the spherical model of Chapter 5. Table 7.1 summarizes the results of thethree models. The absolute complex differences error, described in Chapter 4, forthe computed values are obtained and are included in Table 7.1. The errors forthe Cartesian, the spherical and the INST models a e € : 5.3 cm, e : 4'3 cm ande : 3.75cm, respectively. The errors show that a steady improvement is achieved

using the later models. The computed values obtained from the INST model have

only small deviations from the observed values; the absolute difference in phase

generally is less than 12o and for the amplitudes in most of the stations the differences

are less than 5 cm. The computed results compare favourably with the observations;

at 90% of the stations both the amplitude and phase for results from the INSTmodel are better than the same results obtaining from the other two models.

The co-amplitude and co-phase lines for the principal diurnal Ot, Kt and semi-

diurnal M2 and,92 constituents have been predicted by the INST model. They are

presented in Figures 7.5-7.I2 and compared with Admiralty chart 5081 (see Figures4.3-4.6). The Admiralty chart has been constructed from observations available ata number of locations along the coast and a limited number of points within theGulf. The chart only provides approximations to the true tidal conditions withinthe Guif. However, the Admiralty chart at a location may be very accurate (if muchdata was available) or very inaccurate (if little data was available).

The location of the amphidromic points predicted by the model are similar tothe Admiralty chart but differ in certain important aspects. Modeled results for M2phases suggest that the tidal constituent is moving more slowly through the Straitof Hormuz than shown on the Admiralty chart, while the Kl constituent appears tobe moving at the same speed, this is probably a function of the boundary conditionin the Strait of Hormuz.

It should be mentioned that Hunter (1984), using an M2 tidal input, notes that"the Admiralty co-tidal chart of the Gulf was considerably in error in Kuwait wa-ters". The INST results confirm Hunter's comment, as the location of the am-phidromic point in the Kuwait area is not exactly the same as that shown in theAdmiralty chart. AIso the INST-modeled predictions for the M2 amplitude in thisregion are lower than those of the chart in the same region, the same being true forOr, K, and ,92 (see Figures 7.5-7.I2 and 4.3-4.6). In the case of phases, as Figures7.10 and 4.5 show, the constituents in this area are not moving with the same speed

as that shown in the Admiralty chart.

102

phstation ph215.0220.6

229.9

L92.O20t.4204.9206.6

189.0197.3200.620r.4

171.0176.5180.8179.4

22L.O

226.O

233.7

221.O

228.O232.4237.O

89.0ova

98.3100.9

227.O

235.0238.8245.8

3.00.1

359.40.1

218.0

227.3238.1

281.0284.5286.2300.4

55.062.363.580.6

65.079.078.582.8

42.936.7JO.4

3t.238.8acÉ

3t.730.726.825.825.9

25.3

19.318.5

30.631.634.4

38.032.631.128.0

23.8

23.520.9

L7.O13.5tL.211.0

33.840.042.842.4

18.0tt.49.710.1

L2.Oot8.26.8

29.O29.428.124.4

30.024.523.727.9

263.0270.5274.O

277.O

233.0242.2246.6247.2

227.0

237.724L.8

209.02t8.2220.32L4.9

256.0260.3266.8266.2

259.0265.027L.1

1 15.0t22.5L25.1130.8

273.O284.5283.3297.3

11.018.427.922.4

274.O243.2285.1292.L

32 5.0330.1

345.6

94.0LO2.5

104.8101.6

92.O

106.0106.7115.6

36.436.036.939.3

36.3

40.2

49.741.94L.742.3

26.824.724.9

25.O

20.otl.t75.7

5L.241.043.04t.944.O38.536.036.0

100.0106.0to7.lr12.4

48.041.539.639.1

62.756.050.452.O

54.050.048.845.0

30.0

33.037.8

33.027.525.627.4

248.O254.7262.8265.6

149.0156.2161.0166.3

110.0119.0L24.5122.7

68.069.0/U.Cr

82.3

256.0270.0270.O

273.8

276.O288.2290.6294.5

19.028.330.2

'37.O

49.049.549.2

197.0200.0200.1200.5

36.049.048.156.8

95.095.õ95.3

113.8

335.0

331.9

332.0340.0343.6351.4

16.8L2.810.411 .0

L2.a13.172.414.3

t2.3L2.6L2.5L4.4

t7.7I ö.ó

13.514.9

11.6

D.O

5.8

8.03.52.6a2

t7.L14.5t2.613.9

16.0

7r.4t2.236.0aoo

40.043.6

16.014.0t2.o12.5

17.o16.0L5.2t5.2

L2.0L2.O

t2.o14.3

t2.o10.59.510.4

312.0320.3326.2329.8

196.0206.3210.8207.7

160.0166.3t71.9t75.2

117.0128.0r27.8131.3

320.0330.2334.6336.8

335.0345.2347.6

57.O68.67t.672.2

107.0L2O.L

119.7L25.l229.O235.6

239.0

93.0101.2r05.5113.7

159.0157.3156.9170.0

10.0rb.517.520.0

6.020.021.620.1

11

t2

)

c

4

6

8

I

10

13

Obs.INSTsph.Car.

Obs.INSTsPh.Car.

Obs.INSTsPh.Car.

Obs.INSTsph.Car.

Obs.INSTsph.Car.

Obs.INSTsph.Car.

Obs.INSTsph.Car.

Obs.INSTsph.Car.

Obs.INSTsph.Car.

Obs.INSTsPh.Car.

Obs.INSTsPh.Car.

Obs.INSTsPh.Car.

Obs.INSTsph.Car.

28.724.222.7L7.6

25.720.619.0L7.r

20.418.217.0L5.4

r8.0t4.4L2.41 1.9

21.620.418.815.2

2t.o19.518.414.9

11.9r1.212.812.l14.012.o9.88.6

20.723.023.826.4

10.07.26.94.9

9.06.55.94.3

15.0rb.bL5.714.5

16.012.67L.4LO.7

Table 7.L: Compt,rison of cornputed (INST), spherical (tph.), Cartesian (Car.) and ob-

serued (Obs.) results for amplitude (cm) and phase (degrees GMT) for the four majortidal constituents at selected stations in the Persian Gulf. (see Table /r.4 fo, the names

and locations of each of the obseruati,on stati,ons.)

103

1

ampstation amp204.0202.O201.920t.7245.0250.0254.4258.2

185.0185.0185.9185.8

193.0t92.O190.6190.9

229.O

233.5234.O

54.060.262.674.3

258.0271.0274.3

196.0196.5L97.4197.3

196.0185.6180.2195.1

235.0243.2247.225L.3

67.O70.072.47L.O

210.0210.0209.6207.6

44.O48.551.442.0

25.O30.031.935.6

22.620.020.L23.0

27.431.032.435.2

25.328.530.932.5

20.L2t.2

24.6

20.013.5LO.2

L2.7

7ó.211 .59.110.1

22.O26.028.930.9

2.14.O

4.94.6

13.18.57.O

6.8

14.514.013.513.6

18.011.49.910.9

20.0ôâE

24.325.2

247.0246.5246.3246.3

282.0290.029t.6300.3

22r.O226.O230.723r.4227.O230.0234.4236.O

266.O270.O272.4276.9

102.0110.5r14.2t22.7294.0304.2310.83L4.2

239.O240.Ot^t

^242.3

273.O264.6262.8259.5

88.090.092.887.2

279.O289,0293.6294.8

102.0114.0117.0120.5

250.0251.5253.525t.9

l5

t7

18

T4

16

19

20

21

22

24

25

26

Obs.INSTsph.Car.

Obs.INSTsPh.Car.

Obs.INSTsPh.Car.

Obs.INSTsph.Car.

Obs.INSTsph.Car.

Obs.INSTsph.Car.

Obs.INSTsph.Car.

Obs.INSTsph.Car.

Obs.INSTsPh.Car.

Obs.lNSTsPh.Car,

Obs.INSTsPh.Car.

Obs.INSTsph.Car.

Obs.INSTsph.Car.

20.022.O

22.1tto

2t.918.517.615.4

20.422.O22.O

24.O

r5.8t7.619.321..2

18.919.018.6L6.4

7.O

5.53.43.5

15.0

t4.1

16.0L7.O

t7.9L7.E

1.82.O,t

1.7

9.48.27.87.4

17.615.8L3.710.4

8.06.8b.ó5.6

16.015.0L3.21 1.5

14.0t2.5to.2

358.2

40.046.047.244.3

355.0351.0344.9344.6

349.0345.6344.9342.8

38.028.529.324.8

268.0280.3283.3

288.

41.054.056.261.3

345.0349.4357.7348.7

36.042.045.047.5

308.0318.2322.8

28.O

22.118.615.5

0.010.01 1.5oo

31.040.243.244.6

29.O36.0.51.þ

38.8

32.630.128.824.9

28.732.534.642.L

26.2

36.338.1

29.329.O

29.526.7

9.O

8.57.97.9

,J /,)

30.3

28.9

22.O28.530.530.8

L.23.03.6

20.LL7.316.314.7

24.520.021.515.3

15.0L2.21 1.9LI.22t.o2L.O

2r.319.7

28.029.531.628.4

74.O

80.085.673.9

359.0LO.2r0.714.9

350.01.3

6.7

64.O

65.0t)rr.l57.1

10.0t7.Ot8.731.1

85.0100.2103.3103.6

352.05.36.1

10.8

116.0L24.O

128.9r2t.754.0ôõ.169.666.8

77.O

86.589.295.4

20.o.JÒ. D

39.239.9

29.O

31.533.619.8

73.O

80.182.688.9

59.758.457.460.6

76.883.585.389.3

68.975.OI I.õ

81.0

59.160.063.262.2

63.055.054.156.1

45.136.534.235.1

67.O72.676.O

I ð.ð

4.9to.7t2.311 .3

42.433.53t.236.7

39:039.039.637.6

51 .045.743.64I.2

62.O

62.564.O60.8

Table 7.1(cont.): Comparison ol conxputed (INST), spherical (tph.), Cartesian (Car.) and

obserued (Obs.) results for amplitude (cm) and phase (degrees GMT) for the four majortidal constituents at selected stati,ons i.n the Persian Gulf.

104

Observationstation

Oramp ph amp

K1ph

M2amp ph

Szamp ph

28

oo

30

31

32

?a

34

35

36

J¡

38

39

Obs.INSTsph.Car.

Obs.INSTsph.Car.

Obs.INSTsph.Car.

Obs.INSTsph.Car,

Obs.INSTsPh.Car.

Obs.INSTsph.Car.

Obs.INSTsph.Car.

Obs.INSTsPh.Car.

Obs.INSTsPh.Car.

Obs.lNSTsPh.Car.

Obs.INSTsph.Car.

Obs.INSTsph.Car.

Obs.INSTsph.Car.

b.u

1.9

3.03.0DO

t2.oLl.29.89.7

t7.o11 .3

9.69.8

2.42.52.82.1

15.015.315.614.8

15.07r.29.08.3

16.416.517.O16.0

15.011.59.48.0

16.316.0rb. /

14.4

18.518.5La.716.8

t2.o14.616.37.0

24.222.O

20.518.9

9.020.423.124.9

301.0318.6324.8328.7

0.010.012.818.4

33.034.636.223.9

4.010.514.819.5

17.o

30.033.5

50.0ó2.553.853.8

28.0

34.637.6

71.077.580.275.1

62.O69.073.880.4

62.0

/b.b80.7

114.0116.0119.4125.2

67.074.278.183.0

7.04.83.63.0

7.O

6.56.45.3

25.022.520.L18.0

20.0L4.3t2.o11 .1

8.26.24.420

34.0óz.o30.726.8

17.o12.410.68.1

36.234.033.428.8

19.014.3L2.4ea

35.8

31.024.7

40.4

cca

29.7

22.O24.024.514.0

47.241.539.5aaa

104.0tL7.2120.0113.1

16.0taa31.836.5

53.060.364.371.8

74.O

I t.c70.16L.2

84.092.596.2

101.5

64.074.L77.881.4

103.0100.399.890.2

72.080.482.490.7

110.0119.0L2I.LL23.5

95.0105.3L07.2109.0

90.0oot

102.8105.3

151.0160.4163.4169.3

101.0109.4112.5120.8

16.0

24.922.4

4t.734.832.63r.638.026.524.329.O

45.046.347.O

43.6

t6.2It.79.3

34.033.232.4óÒ..J

43.041.541.135.1

33.5o<D

38.0

32.O36.038.032.4

Ll.213.515.8Lt.(

6.315.5t6.715.4

25.O29.432.414.0

r4.229.O

30.831.5

169.O

180.2184.6r87.2

72.O

83.587.790.4

37.O

36.4tÀ t

22.O

234.O230.5228.7227.8

257.O25L.4247.227L.2

37.Oóó. z31.53t.2

244.O242.5240.L24L.3

51.04L.540.442.7

254.O253.0252.4256.5

266.O274.5278.4244.3

95.089.086.4

100.2

330.0333.0342.6

115.0100.3105.2110.9

4.05.4þ.ó6.0

I 1.37.25.34.1

11.08.7ao

8.5

15.015.816.9r /.b

4.33.42.5t.7

11.011.5t2.413.9

13.013.6r4.113.6

10.811 .5

t4.611.0t2.o12.51r.7

4.O6.06.67.6

4.O5.45.95.1

8.011.0t2.72.O

5.8It.272.5t2.t

223.O

234.9238.0

128.0140.0t42.7t43.2

76.0/ b.,Jtt.z7t.5

267.O270.O272.6270.t

284.O290.2294.9300.1

67.O

80.185.2(t.J

301.0293.5290.4289.7

82.092.O

94.L98.6

304.0300.0300.8299.3

337.0atE a

349.4353.6

95.0102.6107.0110.7

26.Oee o

36.6t2.o

t22.0

137.8140.5

Table 7.1(cont.): Compari,son oÍ conxputed (INST), splterical (tph.), Cartesian (Car.) andobserued (Obs.) results for amplitude (crn) and phase (degrees GMf) for the four majortidal constituents at selected stati,ons in the Persian Gulf.

105

Observationstation

O1amp ph

K1amp ph

M2amp ph

Sz

amp ph

40 Obs.INSTsph.Car.

19.0L7.816.1L2.6

E1.092.495.297.O

30.030.028.62L.I

120.0128.O

136.8

37.O44.O

47.444.6

289.0296.2300.0303.9

15.016.0L7.2L6.4

343.0350.1354.4358.0

differences J./b 4.3 5.3

Table 7.1(cont.): Compørison of cornputed (INST), splteri,cal (tph.), Cartesian (Car.) d,nd

obserued (Obs.) results for amplitude (cm) and phase (degrees GMT) for the four majortidal constituents at selected stati,ons in the Persian Gulf.

As Figures 7.5 and 7.7 show, the basic pattern of amplitude contours for the01 constituent is simiiar to that for /fr; this is the same for phase contours (see

Figures 7.6 and 7.8) because the period of the 01 is close to the period of the ,I(rconstituent, but the magnitude of the Or is almost 22% Iower. The Sz tide has asimilar pattern to the M2 for the same reasons, but in this case the amplitude ofthe Sz is approximately 35% of the Mz values. The results show that the largervalues of the constituents are observed around the Strait of Hormuz, especially thebiggest value appearing at Bandar Abbas and the head of the Gulf. In general thepattern of the model results are similar with the Admiralty chart. It should also

be noted that the model predictions show reasonable agreement with the patternsof previous models (for example Von Trepka (1968), Figures 4.7-4.8,, Lardner etal. (1986), Figures 4.9-4.10).

0.2

0.15

0.1

o

r 0.15

I

0.1

0.2

o :

Figure 7.5: Amplitude contours (m) for the Ot ti.dal constituents of sea surfaceeleaøtion from the INST model.

106

270

300 24)

330 150

60 90120 @

90120

15{)

-l

Figure 7.6: Phase contours (degrees GMT) for the Or tid,o,l constituents of sea surfaceeleuation from the INST model.

Figure 7.7: Ampli,tud,e contours (m) for the Kt tidal constituent of sea surface ele-

uation from the INST model.

t07

I

90

\

Figure 7.8: Phase contours (degrees GMf ) for the K1 tidal.constituent of sea surface

eleuation from the INST model.

Figure 7.9: Ampli,tude contours (m) for the Mz tidal constituent of sea surface ele-

uation from the INST model.

108

Figure 7.10: Phase contours (degrees GMf) for the M2 tidal consti,tuent of sea

surface eleuation frorn the INST model.

Figure 7.11: Amplitude contours (m) for the Sz tid,al consti,tuents of sea surfaceeleuation from the INST model.

03

109

1æ 90 60I

30

?oo -.-10

270

2fi210

E tr

180 tr

15030

t20 60

90

Figure 7.12: Phase contours (degrees GMf) for the S" tidal constituents of sea

surface eleuation frorn the INST model.

7.3.2 Elevation contours at a given time

Snapshots of sea surface elevation predicted by the model at the end of each of thefirst six hours of a l2hr tidal period are presented in Figures 7.13-7.18. As Figure7.13 shows, at the beginning on the head of the Gulf the value for surface elevation

is 0.8n2 above MSL and in the Strait of Hormuz it is 0.6n2 below MSL. At the middleof the Gulf this figure shows that the value for sea surface elevation is 0.2m above

the MSL. For the next hours the tidal wave moves into the Gulf. The elevationincreases at the middle of the Gulf and decreases at the head (Figures 7.16-7.18).In particular the position of the zero elevation has moved from the middle of the Gulfto the head of the Gulf (see Figures 7.13 and 7.18, respectively). Figure 7.18 also

shows a minimum value of 0.6rn below MSL for the surface elevation at the head.

As the process suggests, over the next six hours the sea surface elevation increases

at the head and decreases in the Strait of Hormuz until the elevation reaches amaximum vaiue at the head of the Gulf, then the process is repeated.

110

-0.6-0.4-o.20.00.20.40.60.8

ABcDE

G

Figure 7.L3 Contours of sea surface eleuation (m) in the Persian Gulf at the begin-

ni,ng hour of a fiue hour period.

Figure 7.14: Contours of sea surface eleuation (m) in the Persian Gulf at the end,

of the first hour of ø f,ue hour period.

-06-o4-020002040608

À

cD

F

11i

-0-0-0

00000

64202468

B

DEFGH

Figure 7.15: Contours of sea surface eleuation (m) in the Persi,an Gulf at the end

of the second hour of a fiue hour period.

Figure 7.16: Contours of sea surface eleaation (m) in the Persian Gulf at the endof the third hour of a fiue hour period.

0.60.40.20.00.20.40.60.8

A

cD

FG

TL2

Figure 7.17 shows that near Jazireh-ye Lavan (see Figure 4.1) along the Iraniancoast, the value for surface elevation is 0.6m below MSL, which gradually increases

along the Iranian coast to reach a maximum value of 0.8rn above MSL near Lavar(Station 4 in Table .a). The value for surface elevation also keeps rising along the

Qatar coast (see Figure 4.1) . It is at its highest value of 0.6r¿ above MSL at the

left corner of the United Arab Emirates.Figure 7.17 also indicates that the surface elevation is relatively high, 0.6m, at

Bushehr Port in the Iranian side (Station 3 in Table 4.4) and diminishes throughthe head of the Gulf; near Kuwait the elevation is zero.

Figure 7.18 shows a minimum value of 0.6rn at Bandar Abbas near the Strait ofHormuz (station 9 in Table 4.4) and Jazireh-ye Lavan (Station 13 in Table 4.4), at

the sixth hour of a five hour period, and gradually increases along the Iranian coast

and along the Qatar coast (see Figure 4.1). This flgure shows that in the Bahrainarea the surface elevation diminishes to a value of zero (along the Gulf of Salwa).

Figure 7.17: Contours of sea surface eleuati,on (m) i,n the Persian Gulf at the endof the fourth hour of a fiue hour period.

-0-0-0

00000

4 zB

0:D4tF6:G8:H

113

-0-0-0

00000

Ã.À4:B0:D4zF6:G8:H

Figure 7.18: Contours of sea surface eleuati,on (m) in the Persi,an Gulf at the end

of the fuè¡'hour of a fiue hour peri'od'

7.3.3 Depth-averaged tidal current ellipses

In this section the model predicted tidal current ellipses for the Or, Kt, M2 and

,52 constituents are presented in Figures 7.I9-7.22,, respectively. The current ellipse

plots show the behaviour of the depth-averaged current over a complete tidal cycle.

A comparison is made between predictions from the INST and the spherical

model of Chapter 5 (coarse model of the INST). As no published reports of ob-

servational data on tidal currents exist for the Gulf no comparisons can be made

between observed and simulated currents. The discussions below are based on the

good comparisons for the tidal elevations. The tidal ellipses predicted by the INSTmodel are similar to those of the spherical model for each constituent, see for exam-

ple Figures 7.19(a) and 7.19(b). The current patterns between the two diurnal tidalellipses, Or and I{1 are similar in each model, see for example Figures 7.19(a) and

7.20(a). Aiso the similarity in current pattern is clearly noted for the semi-diurnal

tidal ellipses, see for example Figures 7.zL(a) and 7.22(a).

Figures 7.79-7.22 indicate that the ellipses for all constituents have longitudinaland circular movement. The circular movement generaily appears in the Strait ofHormuz, at the head region and near the United Arab Emirates coast. Figures

7.2I and 7.22 show a group of clockwise rotating ellipses for the M2 and ^92 tidalconstituents near Abu Zabi and Dubayy (see Figure 4.1) and another group ofanticlockwise rotating ellipses at the head of the GuIf. The circular movement

suggests that the particle paths are not totally periodic, that is, they do not returnto same initiai point.

r74

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C.

e)

\

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q\\\\\\\\:--:

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o\\\\\\\\\\--

f,r\\\\\::r:\\\\\\\\\ tr\\\\\-\ -:./z//\.\

:\E

t

\\rr"\\qqnlqqqa

irtot l0

lt

00qqøao..///

I0

o 0

000c090ao

laoeI le t t ,tr

f-

(o)

\\\ \ \ \ \

\\\\\ \ \ \ \ Lms-r\\\\\\\\l\\\\\\\\\\\\\\rr\\\\

r:\\\\\\\\

\\\\\\\\\\r\\\\\\\\

-rr\\\\\\\\---\\\\\\\\\---'\\\\\\\\-\\\

\\\\\\\\\\\t\\\\\\lr:r

o\\R

F-l

l-,l

\\\\\\\\\\

l///t.E

- : ./ t./-\

\\rjqlqsq

ttoøI l0sltt4

tr

l:

- - azzt,/_ _ _ __.?,2./

â ø A a c - - ¿ ¿ ¿?/

00Qaoo..¿/./00oe9ta//

0et?or//t/e ,. / / o

//t//

(ó)

Figure 7.19: Ti,dal cument ellipses (-t-t) for the Ot ti,d.al constituent of d,epth-

o,uero,ged curcent for the (øi INST and (b) coo,rse tidal models of the Persian Gulf.Note that the scale is d,ifferent to that used in the later figures.

115

\\\qì\\ \\\\\\\\

.\\\\\\\\\\

Ims-r

0i\\

\\\\\\\\\\\\\\\\\\\\\

Ct

!'aE

\\\\\g\\\\\\\\\ \ \ \ \ \ \ \ \\- -

s \ \ \ \ \ \ \\\- - \\\\\\\\\\\\\-\\\\\\\\\\\\ Do

rr\\\\\:rr\\\\\\\\\\ É

û0loat"//,//

--/-z/-?\. \ I

\\¡

I

II

I

lolo/

0 A O a O a - - z //.?

eeaooaa.zz/,/

12/ t t / t / /,////,tt////

t.t////to

(o)

\\\ \ \ \ \ \

\\\\\\ Lms-L+\\\\\\

rrrf \\ \rrf\\\\

:r\\\\-\\\\\\

!'

\\\\\\- r r \ \ \ \ \ \ \\--\ \ \ \ \ \ \ \\--- -\\\\\\\\\\\\

\\\\\\\\\\r\\\\\r.:r

trtE

\\\\\\\\\\\\r

0 e A O a a a - ¿ z z/7

tr

tt//,/.//t///

--r/-z/-z\ \

I

\\

I

I

\\\\

e0ooaaaa.//./,/I laotta

l0a

e//.r.O

(ó)

Figure 7.20: Ti.dal current ellipses (-t-t) for the Kt tidal consti,tuent of depth-aueraged curcent for the (ø) INST and (b) col,rse tidal rnodels of the Persi,an Gulf.

116

7.3.4 The diurnal ellipses 01 and If1Comparison of the 01 tidal eilipses (Figure 7.19) with the llr ellipses (Figure 7.20)indicates that the 111 current is stronger than the 01 current (note that the scale ofthe Or plot in Figure 7.19 is different from that in Figure 7.20). The behaviour ofboth constituents is similar everywhere within the Gulf.

The diurnal current is strongest in the Strait of Hormuz and at the middle ofthe Gulf. It is weak close to the coastal boundaries, where the water is shallower.

As Figures 7.19 and 7.20 show these currents are stronger along the Iranian coastthan the Arabian side, because the Arabian side is shallower. Through the head ofthe Gulf these currents have longitudinal movement, while from the open boundarytoward the middle of the Gulf they generally show a circular movement. In thefigures a group of anticlockwise rotating currents is seen in the south of the Gulf,toward United Arab Emirates. This continues on the western coast of Qatar. Nearthe Khark island (Station 2 in Table 4.4) on the Iranian side the current movementis longitudinal, changing to circular towards the end of the Gulf.

The coarse model predictions for the Or tidal current ellipses, Figure 7.19(b),are different at the top of the Gulf from the INST model results, Figure 7.20(b).

In general the strongest currents appear in the middle of the Gulf and also inthe Strait of Hormuz.

7.3.5 The semi-diurnal ellipses M2 and 52

Figures 7.21 and 7 .22 show the results for the M2 and.9z tidai constituents, respec-tively. - They have similar current patterns, but the M2 culrents are stronger thanthe 52, (note the change of scale for the S2). They also exhibit similar behavioureverywhere in the Gu1f.

The semi-diurnal currents are strongest from the open boundary through themiddle of the Gulf and also at the head of the Gulf. They are weak near Lavar andRas Al Qulaahy (station 10 in Table 4.4) on the Arabian side. Through the middlepart of the Gulf, near the Iranian side, and around Lavar and Ras Al Qulaahy, thesecurrents exhibit longitudinal movement, while in the other parts of the Gulf theyare more circular, such as at head of the Gulf and in the United Arab Emirate area.These figures indicate a group of anticlockwise rotating currents in the Strait ofHormuz toward the coast of Qatar and another group of clockwise rotating currentsat the head of the Gulf. The figures also show that there is a group of clockwiserotating currents near the coast of Bahrain. Around the Khark Island (Station 2in Table 4.4) the tidal current movement is circular and keeps a similar movementtoward the end of the Gulf, except along the Kuwait region which tends to belongitudinal and weaker than the currents on the opposite side.

In generai the semi-diurnal current is stronger on the Iranian side and weaker incertain regions of the middle of the Gulf compared with the currents at the head ornear the United Arab Emirate coast.

I17

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\\\\\\\\\s\\\\\\\\\\\\\\\\\\\\\\ss\\\\\ t\\\

\

"ù

\\ \ \ \I n'¿s -\\\\\ètt1ì

\\\èòtì1q\\õ t x t ì \ \ \

\\15\\\\\\

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a-õ-\ - '.E- o Itr o'\.y'-/e<\ rò

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/ ¿ /.// @ õ 6) ø 0 o /ù////'dç>6øO00

¿ ¿ ¿ / -æõç) ø ô I E

(o)

i\\ \ \ \ r \

7 rns-r\\\\\\\ì1rr\\ò rrr I

+ò

1 It\\\1\\\\\\\\\\\\\\\\\

ó \ \ \- -\\\\-I

o\,ôaa--l r -:R q

tr a.\.? - ¿ e.\ \

-----/,/-o_e3/

///-ÒAooCCt

/ /,/--øõøøO/t¿./--ãõø0

, ¿ ¿ t /-/qrù O ø

Oo.

¿ ¿ d / /,6Q) O o I E

(b)

Figure 7.22: Tidal current ellipses (-t-t) for the Sz tidal const'ituent of depth-aueraged curcent for the (ø) INST and (b) coarse tidal models of the Persian Gulf.Note the change of scale relatiue to the preuious fi,gures.

119

As Figures 7.27-7.22 indicate the strongest currents appear around Zarqa and

Ardhana (Stations 36 and 37 in Table 4.4), near Kuwaiti waters and in the Straitof Hormuz.

It should be noted that the predictions of the coarse model for the ,52 tidalellipses, Figure 7.22(b), are different at the top of the Gulf from those shown by the

INST modei, Figure 7.22(a). The differences are due to the grid refinement in thatregion. Similar differences can be seen for the M2 tidal ellipses in Figure 7.21(b),

for the coarse model, and in Figure 7'21(a), for the INST model.

7.3.6 Depth-averaged velocity vectors at a given time

This section presents snapshots of depth-averaged current predictions resulting fromthe INST model of the Persian Gulf, for the fi.rst six hours of a 12 hr tidal cycle,

Figures 7.23-7.28. For clarity, only velocity vectors at every ninth grid point on

every ninth line of the fi,ne grid data has been drawn, and velocity vectors at every

third grid point on every third line of the coarse grid data has been drawn.

Figure 7.23: Depth-aueraged tidal curcent (ms-r) uectors i,n the Persian Gulf at the

end of the fi,rst hour of a sir hour si,mulation using úåe INST model.

<-R //

\ìi'!

{

,

\\\

\{

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ú,

*

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(l, \r

vvt*\l

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- : r\. \.3 * I

-- tss<=R\\ r+1RRRR\\.\'\

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r\

/RvRR\\.\.XS,\. \ R R\ \. \. \. \. \. \.* =* l-

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a

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\+¿+++S+É+

-+-¿<-€-<-+¿-

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útt\\\,,11,\

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\\xx.\.\.RRR-I \xx,xRR\rì

\xrx\\\\ \ls*\R\\\\

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- ¿ +e ss R R \ \

\

t / /l/- azy'

+¿z<1-f-S\'\ \

\l r t /r/\4t^ò4l'l

<--ÉÉ\x.r¡

Figure 7.24 Depth-aueraged tidal current (ms-r) uectors in the Persian Gulf at the

end of the second hour of a sir hour sirnulation using ¿he INST model.

\

-àr,

ú

\.

\\\l\r

\ \trl\ú,

--l+llLJ-n

- - * \Ñ s--ss s \s<*/ ss. \s<- s-

¿<-Ss-

\ \ t',r\ I' ,/'

^

,î'\\ \ \<ô

4l

4\\ \R<.t\ \

?<<-

ù\

-+=\s.\5-ç-*r.

+\\\\\\ x x\Q.\

/

rt\ \,lj .\\

S-\.R\\\\\R:--E\\ \ \ \ \ ¡ r r

\trxn \ \ \ \ . ' ' rzl\\\\\\tl\+ttaAr\\\\\\ô44f/ /

/

. -1Inxs'

\

+

Figure 7.25: Depth-aueraged, ti,dal current (ms-L) uectors in the Persian Gulf at the

end of the thi,rd hour of a six hour si'mulation usi,ng úäe INST model.

721

*

Irl

I \l\\

\ \\ ->\\ \\

--*r\\\ s

\\ E\I tr\R+ \ \\ ->\ \ z

t I zô 4 / II /

/\ i 4\ I4, ,ll\ 4 4\

/'\ 4 ,1,\\ ô ôs.\+ tr

tr

\ r * -s.\ss-<-s \ \-rRRç\5a--.

+<s-SS.s. \ R \

\\\ \\'\ \\

{ ' r \

Ims-r

\

Figure 7.26: Depth-aueraged tidal current (ms-r) uectors in the Persian Gulf at the

end of the Jourth hour of a siæ hour si,mulation usi'ng úå.e INST rnodel.

\*\\'\\

t

E

l'

D \\ \

\

4tlt\ I t \r + I t\

rl\ìrll\\

t\

lrt

ltltll

tt'/414, tô

tr

{4

\

a X \ \\

,r.lzzZ

rrtô+

,l\ I/¡

r rr \\ç- *

/,

r/ltrrlll

/-sR\'

-:\R\.\.s+++S\RR: :

. \\\\\ \

tt4/tt/trZZZ.ZZ vz

Lms-L

ùrú

Figure 7.27: Depth-aueraged tidal cunent (ms-L) uectors in the Persi,an Gulf at the

end of the f,fth hour of a sir hour si,mulation using úå,e INST model.

r22

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'JInÐ er{} JoslJ"d Jer{}o eq+ uerl} Jâ^\olleqs eJ" suor8e¡ esâq+ esneceq 'ra¡eurs eJ" seI}IJolâ^ âq+

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eI{} +3 Pu" elPpIIII eq} ul pacnpo.rd saI+Icole^ eq+ et€clPu\ 8G'L-97;L sarn3rg sy't-su 80'0-?0'0 epnlru8eur Jo .,!\oU

l"a/v\ 3 sr erer{+ JInÐ uJeq}Jou eq} Jo "ere l€rluec eq+ uI '+seoc eq+ ser{csordde 1t

eJeq^\ +se/\/Ir{+rou eql sPJe.&\o} senur}uoc srrIJ 'uees sr ivroB: urn+eJ {eei!\ 3 '¡ng aq1¡ouor8a¡ ure+se¡Àr{+rou eq} Jo er}uec aq1 lnoq3norl{+ }eq+ pe}ou eq osle plnoqs +I '+seoc

u"ru"rl eql uer{} re.,!\oll"qs sr +seoc uelqerv er{+ pu€ re+€.&\ eq} Jo l{+dep Jo uol+cunJ

e sr ,,{1rco1e^ eq+ se 'dyurols âroru se^oru lspor uerqeJv er{} 3uo1e Jã+"^\ âq+ ellq/\À

'1seoc u"rueq aq1 tuole JInÐ aqt s+rxâ pu" srelue sseru reÌ€.ü eI{+ Jo }soru +eq+ llror{s

sarn8g er{J '(t't elq"J Irr ? uor}€+S) epl. ueru€rI eq+ ul re^"T r"eu pu€ (.,{1aa,r1cadsar

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Jo elpprrrr eq+ punoJ€ 3ur¡¡ncco sel+Itole^ 3uo:1s

rllr.&\ pe+ecrldruoc alrnb sl JInÐ eqt ul pleg luerrnc l"pll eq+ +eql uees eq uel lI

'ppolu ISNI aqy 6uzsn uoz?Dlnluxs rnoq î,zs o {o .tnoq q?r!,s eq? {o pua

eql ?D finp ums.ta¿ a!? ux sroJcen (r-"-) ?uerrnc 7opt7 pa6n.rano-q7dag :g¿'¿ arn3rg

¡ t- -^ ù\\\\\\r\t/+\\\\\\'\\

rl

'!t

SS*+-tr Rç-

\ ¿R ,1, "

\\ r

\¡ù\{9 r

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\{

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vü

E

,,*S

<-+< + 4 ¿ t t , V

??kZzsEtt¿lllZZZzz.?Zzz// Y t t Y tl

Z/Z/Z//zzzzt///øøV// ø t/ // // tt t¿ t/21 t t /üt/øø '1,/ø/

/ tz t

zo

vsu I

7.3.7 Depth-averaged velocity residuals

A residual current is defined as the part of the velocity that is left after removal ofall tidal oscillations. The purpose of producing residual plots is to present the broad

regions where model generation of non-linear constituents takes place. The residuals

are a consequence of the non-linearity of the model. So a fully linear model would

predict zero residual tidal circulation.The predicted four major tidal constituents of depth-averaged velocity fields are

used to reconstitute a time series of velocity for a 29 day period at each computa-

tional point. This series is then subtracted from the raw velocity predictions of the

numerical model at the corresponding computational points. The difference is

then averaged by dividing by the number of hourly data points (697) it represents.

The resuit which is plotted as a vector shows the contribution to the mean 29-day

circulation due to the non-linearity of the model.The residuai vectors for the velocity obtained from the INST model are shown

in Figure 7.29(a). Large residuals are predicted through the head of the Gulf and

the Strait of Hormuz. The pattern of residuals around, the northern part of the Gulfshows large ualues 'in areas whi,clt, do not coincide with the interface between the two

gri,ds; also note th,at area D is not along the fi'ne grid open boundary.

The residual plots for the coarse model (spherical model of Chapter 5) are pre-

sented in Figure 7.29(b). There are some differences between the pattern of residuals

resulting from the two models. At the head of the Gulf region, A, the coarse modelpredicts larger values compared with the predictions of the INST model in the same

area. In the north-east of Bahrain, B, large residuals are predicted by the coarse

model, while in the same region INST shows small values. Around the Strait ofHormuz, C, quite different residuals are also predicted by the models. The coarse

model shows large values and the INST model predicts smaller residuais which are

parallel to the coastline. Around the Saudi coast line, D, which is very shallow,iarge values are shown by the INST model compared with the values predicted bythe coarse model in the same region.

124

A0.05 rns-1

C

D

(o)

A0.05 ræs-1

C

D

B

J5(ó)

Figure 7.29: Depth-aueraged uelocity residuals for a 29 day period for the (ø/ INSTand (b) coarse tidal models of the Persian Gulf.

125

Chapter ISummary and concluston

This thesis discusses the development of a nested finite difference model of tidal flow

in coastal regions and the application of this model to the Persian Gulf, together withcomparisons between this model and two simpler models, a Cartesian coordinatemodel and a spherical coordinate model.

Chapter 2 discusses the characteristics of tidal dynamics in shallow seas as well

as the physical meaning of each term in the depth-averaged Cartesian and spherical

tidal equations.The basic idea of fi.nite difference techniques is presented in Chapter 3. In

this chapter the finite difference techniques used to solve the governing equations

in Cartesian and spherical coordinates are explained. An explicit finite-differencescheme based on the Arakawa C spatial grid is used to solve the tidal equations.

The Arakawa C grid is used because it allows spatiai derivatives to be convenientlyapproximated in second-order central difference form and iand boundaries can be

represented simply, by means of straight-line segments parallel to the coordinateaxes.

A grid staggered in time as well as in space is also used and these are described

in this chapter.The finite difference code described in Bilis (1992) has been chosen for solving

the governing equations. The programs, written in Fortran 77 ,, have been convertedfrom the original Cartesian coordinates to spherical coordinates. The code has also

been modified to produce the nested model of Chapter 6.

The development of a Cartesian depth-averaged model of the Persian Gulf ispresented in Chapter 4. The model used elements of 8.7km x 9.7km mesh size and

a time step of 200sec. The ten componentt Qt, Ot, Pt, K1,, pz, Nz, Mz, L2, 52 andK2 were used to produce open boundary tide-height input data.

The amplitudes and phases of the four major constituents, 01, I{r, M, and Sz

have been reproduced and compared with Admiralty chart 5081. Existing informa-tion on tides in the Persian Gulf is based primarily on the Admiralty chart. Thedata which has been used in this thesis were provided by the Admiralty chart, theNational Tidal Facility at the Flinders University of South Australia and Proctor etal. (1992). The chart has been constructed from observations available at a number

a

t26

of sites along the coast and a limited number of points within the Gulf. The Admi-ralty chart forms an initial basis for understanding tidal movement in the Persian

Gulf and for comparison with mathematical models developed within this thesis and

by other researchers whose work is reviewed in Section 4.3. The observational data,

sets of harmonic constituents for Ot,, Kr, M2 and ,92, for 84 tidal stations in theGulf was supplied by the National Tidal Facility at the Flinders University of SouthAustralia. From this iist, 40 stations have been used in the calibration process.

The amplitudes and phases of the four major constituents, 01, I{1, M2 and 52

have been reproduced and compared with Admiralty chart 5081. The locationsand the patterns of the amphidromic points predicted by the model are similar tothose shown on the Admiralty chart, and the basic patterns of co-tidal and co-phase

lines are the same, but computed phases for 01 and M2 show that in the Strait ofHormuz the tidal constituents are moving slower than the same constituents in theAdmiralty chart. The predicted amplitudes, at the selected stations, are general inbetter agreement than the predicted phases. A series of simulations were carriedout for different values of C6 and ø. The optimal values f.or Cu and ø were found tobe 0.0015 and 0.03sec-1, respectively, with associated error value of e: 5.3cm.

It was determined that the performance of the model could be to improved bytwo changes. Firstly, a spherical coordinate grid could be used, since the Persian

Gulf is too large to be approximated by a plane surface. Secondiy, the grid couldbe refined in certain regions, because the spatial variations in tides are very large inthe Gulf, especialiy near the amphidromic regions, and because the coastline has acomplicated shape in most of the regions.

The depth-averaged tidal model in spherical coordinates is developed in Chapter5 and applied to the Persian Gulf. The model uses an element dimension of 5' x 5'

and a time step of Aú : 200sec. The same set of open boundary data, used forthe Cartesian model, is used for tidal reproduction. The original Cartesian coordinate computer code has been converted to spherical coordinates to implement thefinite difference form of the Persian Gulf spherical tidai equations. A calibrationwas done in order to determine optimal values for the quadratic bottom coefficientC6, and the reduced horizontal eddy coefficient a. This was done by obtaining aminimum overall average absolute complex difference between tide-height predic-tions and observations. The optimal values for Ca and ø were found to be 0.0015and ø:0.001sec-1, respectively, with corresponding error of e:4.3cm. The tidaldata which were used in the calibration process were the same as those used in theCartesian model. The predictions at most of stations are no'¡/ closer to the observa-

tions, when they are compared with the corresponding stations used in the Cartesianmodel. The improvements in the results indicate the effect of considering the earth'scurvature in the Persian Gulf model. The contour charts of tidal elevation for thefour major tidal constituents, that is, 01 , Kt, Mz and Sz are produced. These con-tours have been compared with the Admiralty chart. In general the behaviour ofthe contours comparing with the Admiralty chart are similar to that obtained fromthe Cartesian model.

From the application of a depth-averaged spherical coordinate model of the Per-

727

sian Gulf, described in Chapter 5, reasonabiy close agreement between observed and

computed tides was obtained, but to solve coastal engineering problems and to pro-

vide reievant answers for the other questions in the shallow and complicated coastal

areas, it was necessary to refi.ne the grid to improve the predictions and have a betterapproximation in those regions. In order to achieve this a new model was proposed,

which has the advantage of having a finer spatial grid in areas where high resolutionis required together with a fine grid in time in these regions, without using the fine

grid everywhere with very small time steps. It should be noted that a very fine

grid over whole the Gulf is too computationally expensive, because the Gulf is very

large and increasing the number of grid points causes CPU time to be too excessive.

To achieve the above aims, an INterpolated in Space and Time model, the INSTmodel, is developed in Chapter 6. This model basically involves embedding a fine

grid in space and time to cover a region within a coarse grid. The general finitedifference scheme used for both grids is three-level in time. Two basic algorithms,referred to as the transferring and back-transferring algorithms, are used. The firstalgorithm is used to prescribe elevations on the open boundary of the fine model;

the second algorithm is used to provide the feedback of values on the fine grid tothe coarse grid.

To satisfy the stability condition in both fine and coarse grids also to insure wave

stability it is required that the maximum Courant numbers in both grids be approx-

imately the same. Because this condition was met in both sides of the interface

the transition between the models has been made properly, that is, refraction and

reflection do not contaminated'the computed patterns even nea the boundary.In order to test the INST model it has been applied to a rectangular bay. The

bay has been used for a fully coarse grid model, a fully fine grid model, and anested model using the INST model. A set of full non-linear equations in spherical

coordinates including ali terms but the Coriolis force, are solved in the bay. In orderto find out the exact behavior of the results through the modelled region, differentorientations of the interface between the two models also have been examined. Theresults from the INST model a e compared with the corresponding results predictedby the fine and the coarse models of the same computational domain. The contourplots of surface elevations obtained from the models are compared with each other.

The results show that the model works well, because contours resulting from theINST be between the resuits of the fine and the coarse model and are closer tothose of the fine model.

The INST model has been tested also with Coriolis included. This test has

been carried out to ensure that the model works independently of the orientation,and also to examine whether the INST results be between the coarse and the fineresuits. The contour plots of the sea surface elevations of the fine, the INST, and

the coarse models u/ere compared with each other; good agreement between theresults was found.

The application of the INST model to the Persian Gulf is discussed in Chapter 7.

The coarse grid section of the INST model uses the same uniform grid (5' latitudeand 5' longitude) and a time step o1200sec. as the spherical depth-averaged tidal

r28

model of the Persian Gulf described in Chapter 5. Regarding the grid matching

technique, the coastal boundaries of the coarse and the fi.ne grids have different

shapes. In addition over the fine grid, there are 9 times as many grid points as

for the coarse grid, so with this technique the coastline is modelled much more

realisticly.The predictions from the INST model are represented by contour charts of

the tidal elevation and ellipse plots of the depth-averaged current for the diurnal

constituents Or and K1 and the semi-diurnal constituents M2 and ^92. The tidal

results computed from the INST model are compared with the observations of

tidal eievation at forty stations throughout the Gulf. The results are also compared

with the predictions from the Cartesian model of Chapter 4 and the spherical model

of Chapter 5. The absolute complex differences error 'tras found to be e -- 3.7 cm fotthe INST mod.el Compared with the corresponding error values obtained, namely

e : 5.3 cm for the Cartesian and e : 4.3 cm for the spherical model, respectiveiy.

For 90% of the stations both the amplitude and phase for results from the INSTmodel are better than the same results obtained from the other two models. The co-

amplitude and co-phase lines for the principal diurnal Or and K1 and semi-diurnal

Mz and.,Sz constituents have been predicted by the model. They are compared withAdmiralty chart 5081. The location of the amphidromic points predicted by the

INST model and the basic pattern of amplitude and phase contours are generally

similar to the Admiralty chart. But the detaiied results show that the predictions

from ail three models deveioped in this thesis differ with the Admiralty chart incertain important aspects. For example, modeled results lor M2 phases suggest

that the tidal constituent is moving more slowiy through the Strait of Hormuz than

shown on the Admiralty chart probably because of the boundary conditions. Also

the modeled predictions for the M2 amplitude in the Kuwait area are lower than

those shown on the chart. The same is true for the Ol., K, and ,Sz constituents.

The model-predicted tidal current ellipses for the Ot, Kt, Mz and ^92

constituents

are given in this chapter. As no published reports of observational data on tidalcurrents exist for the Gulf, no comparisons can be.made between observed and

simulated currents. A comparison is made between predictions from the INSTmodei and the spherical model of Chapter 5. The diurnal ellipses predicted by

the INST model are generally similar to those of the spherical model for each

constituent. The current patterns for the tidal ellipses associated with the two

diurnal constituents, 01 and 1(1, are similarin each model. Also the similarity incurrent pattern is clearly noted for the tidal ellipse for the semi-diurnal constituents.

On comparison of the Or tidal ellipses with the llr ellipses it is apparent that .I(1

current is stronger than the 01 current. The behaviour of both constituents is similareverywhere within the Gulf. The diurnal current is strongest in the Strait of Hormuzand at the middle of the Gulf. It is weak close to the coastal boundaries, where thewater is shallower. In general the strongest current appeared at the middle of the

Guif and also in the Strait of Hormuz.The semi-diurnal ellipses M2 and ^92 show similar current patterns, but the Mz

current is stronger than the Sz. The semi-diurnal current is strongest from the open

r29

boundary through the middie of the Guif and also at the head of the GuIf. Through

the middle part of the Gulf, near the Iranian side, these currents exhibit longitudinalmovement, while in the other parts of the Gulf they are more circular, such as at the

head of the Gulf and in the United Arab Emirate region. The results also indicatea group of anticlockwise rotating currents in the Strait of Hormuz toward the coast

of Qatar and another group of clockwise rotating currents at the head of the Gulf.

The results also show that there ìs a group of clockwise rotating currents near the

coast of Bahrain. It should be noted that the predictions of the coarse model for the

,52 tidal ellipses at the top of the Gulf are different from those shown by the INSTmodel. The differences are due to the grid refinement in that region.

Snapshots of depth-averaged current predictions resulting from the INST model

of the Gulf are presented in Chapter 7. They show the behaviour of the velocity

fields for the first six hours of a 12 hr tidai cycle. The tidal current fie1d in the Gulfis quite complicated with strong velocities occurring around the middle of the Gulfnear, Bahrain, and in the Strait of Hormuz. Veiocities are weak in the middle of the

Gulf. Because the velocity is a function of depth and the Arabian coast is shaliower

than the Iranian coast, it is noted that most of the water appears to enter and exitthe Gulf along the Iranian coast.

The results presented in this chapter suggest that the velocities produced inthe middle and at the top of the head of the Gulf are relatively large, but theIargest velocities exist in the Strait of Hormuz, where the water has its maximumdepth. Along the coastai regions the velocities are smaller, because these regions

are shallower than the other parts of the Gulf.Chapter 7 also presents the residual vectors for velocity obtained from the INST

model. Large residuals are predicted through the head of the Gulf and in the Straitof Hormuz. A comparison between the residual plot for the coarse model and thatof the INST model indicate that, at the head of the Gulf, the coarse model has

predicted larger values compared with the prediction of the INST model. in thatregion. In the north-east of Bahrain, large residuals predicted by the coarse modelare not shown in the INST model. The predicted residuals by around the Strait ofHormuz are different within each model.

In summary the spherical model of the Persian Gulf works better than the Carte-sian model, and INST model predictions at the observational stations are closer tothe observational data compared with the predictions of the other two models. Alsoit should be pointed out that with the technique used in the INST model, muchmore information on tidal elevations and velocities is available in regions over theembedded grid is developedi, This provides more information to engineers and de-

velopers for coastal regions, as well as for shipping and environmental concerns near

off shore islands.

130

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