A hydrodynamic model for wind-driven and tidal ... - ScienceDirect

A hydrodynamic model for wind-driven and tidal circulation in the Arabian Gulf A. H. Al-Rabeb,

N. Gunay,

and H. M. Cekirge

Water Resources and Environment Division, Research Institute, Petroleum and Minerals, Dhahran, Saudi Arabia

King Fahd University of

The verticallhorizontal splitting (VHS) model is extended and adapted to simulate tidal and winddriven circulation in the Arabian Gulf. The extensions include the use of finite differences for the depth-averaged equations, the use of the depth following coordinate in the vertical, and relating the bottom friction to the bottom velocity. The VHS model is adapted to the Arabian Gulf. Two implementations are presented. The first, HYDROI, simulates wind-driven andlor tidal circulation in the Arabian Gulf. A two-block grid system in Cartesian coordinates is used with aj?ner grid on the western coast of the gulf. The second, HYDRO2, simulates wind-driven andlor tidal circulation in any userdefined area in the Arabian Gulf HYDR02 enables the user to study in detail a speci’c region of interest in the gulf. Both implementations carry out three-dimensional computation when the conventional two-dimensional hypotheses are locally inadequate and when specified by the user. Elsewhere, two-dimensional computation is employed. Implementation and testing details including specification of appropriate eddy viscosity profile, wind drag coefficient, and Chezy coefficient are discussed. A case study is presented to illustrate the model implementation and capabilities. Keywords:

tidal and wind-driven

circulation,

hydrodynamics

Introduction The advent of offshore oil exploration, production, and transportation in the Arabian Gulf and elsewhere has highlighted the need for information on tidal and windinduced vertical current profiles.’ Such information is needed to study pollutant and sediment transport and to determine stresses on offshore structures. Full three-dimensional models using multiple layer or grid box techniques are, however, expensive to implement. On the other hand, the two-dimensional depthaveraged models are well established and require significantly less computational effort. However, various difficulties arise as a result of using the two-dimensional depth-averaged models. For example, in these models, bottom stress is related to the depth-averaged current, whereas in fact bottom stress is physically determined by bottom current. Moreover, two-dimensional depth-averaged models cannot provide information on the vertical profile of the current. To overcome these deticiencies, a number of approaches have been proposed (see, for example, Refs. 2-4). Lardner and CekirgeS developed the vertical/horizontal splitAddress reprint requests to Dr. Al-Rabeh at the Water Resources and Environment Division, Research Institute, King Fahd University of Petroleum and Minerals, KFUPM No. 2014, Dhahran 31261, Saudi Arabia. Received 13 June 1989; accepted 25 January 1990

410

Appt. Math. Modelling,

1990, Vol. 14, August

of the Arabian Gulf

ting (VHS) model in which the depth-averaged twodimensional model is used to compute the total flow in a region. A current velocity profile in the vertical is then extracted by solving the hydrodynamic equations in the vertical. The vertical structure of currents has been considered by Tee,6 Prandle,’ Davies,2,8-12 Cekirge et al.,13 and Lardner and Cekirge.5 See also Refs. 14-16. The VHS model provides a significant reduction in computational effort compared with full three-dimensional models. Additional savings can be made if threedimensional computation is carried out when the conventional two-dimensional hypotheses are locally inadequate, for example, in cases of sharp depth variation or strong wind forcing, and when required by the user to carry out a detailed study of a particular region of interest. Elsewhere, two-dimensional computation is employed. Although the VHS model cannot be regarded as fully three-dimensional, it does provide a computationally feasible and reasonably accurate alternative to the full three-dimensional models5 A variety of computational methods have been used to obtain numerical solutions of the depth-averaged equations, including finite difference schemes (see Refs. 17-22), finite elements (see Refs. 23-28), harmonic analysis in time plus finite elements in space (see Refs. 29-31), and the method of characteristics (see Refs. 32-35). A review is given by Liu and Leendertse.36 Three-dimensional models mostly use finite differences in the horizontal coupled with a layered or grid

0 1990 Butterworth-Heinemann

Hydrodynamic

box model in the vertical. Leendertse and Liu18,‘9 used finite differences, while Davies and Owen” and Daviess,9 used a Galerkin method with various basis functions. A comparison of the formulation and computational requirements of the finite difference and spectral models is given in Ref. 38. Nihoul and Jamart and HeapsI review a number of recent three-dimensional models. See also a comprehensive review of wind-driven circulations by Heaps. I4 In this paper the vertical/horizontal splitting (VHS) model proposed by Lardner and Cekirge’ is used. In the VHS model the depth-averaged equations are first solved at each time step; then the surface heights and depth-averaged velocities computed from this part of the algorithm are used as input to the horizontal momentum equations, which are solved at each horizontal grid point for the vertical velocity profiles. The VHS model proposed by Lardner and Cekirge is extended and adapted to the Arabian Gulf to compute tidal and wind-driven circulation. The extensions include the use of finite differences to solve the depth-averaged equations instead of the method of characteristics proposed in Ref. 5, the use of the depth following coordinate in the vertical, and relating the bottom friction to the bottom velocity instead of the depth-averaged velocity. The details of the mathematical model are discussed in the next section. The simulation algorithm is presented next. Two model implementations are presented in the following section. The first, HYDROl, computes wind-driven and/or tidal circulation for the whole Arabian Gulf. The second, HYDR02, computes wind-driven and/or tidal circulation in any user-defined area in the Arabian Gulf. Both implementations compute dynamic vertical velocitv nroliles whenever necessary or required by the user.‘Model calibration and testing are discussed next, and a case study is presented.

The mathematical

au

dx+&+z=O

model

aw

(1)

~+~(UU)+;(Uu)+~(Uw) 86 =f”-gg-pz+i)?:

of the Arabian

Gulf: A. H. Al-Rabeh

et al.

= P =

pa+ gp(5 -

(3)

4

(4)

where X, y, and z are Cartesian coordinates with the zaxis pointing vertically upward and the xy-plane being the undisturbed position of the water surface; t is time; and U, u, and w are the components of the fluid velocity in the x-, y-, and z-directions, respectively. The other quantities are as follows: h(x, Y) &, Y, 1) f f.J cp g N(x, y, z) FLl P

depth of the water below the xy-plane height of the free surface above the xyplane Coriolis parameter (= 2LRsin cp) angular velocity of the earth latitude acceleration due to gravity vertical eddy viscosity fluid density (assumed to be uniform) atmospheric pressure fluid pressure

Boundary and initial conditions In addition to the differential equations (l)-(3) there are a number of kinematic and dynamic initial and boundary conditions. On the water surface, z = 5,

On the sea bottom, z = -h,

where (7;, 7;) are the components of shear stress on the water surface and (T$, ~5) are the components of bottom friction stress. For surface shear stress it is usual to take

The basic equations It is common in models of tidal and wind-driven currents to make a number of simplifying assumptions in the equations of fluid dynamics. The fluid is assumed to be incompressible and of uniform density, horizontal eddy viscosity is neglected, and the vertical momentum equation is approximated by the hydrostatic pressure equation. If we utilize these assumptions, the equations of continuity, horizontal momentum, and pressure reduce to

au

modeling

1 ap

a

(% 7;) = W&Wf

+ WWX,

W,)

at z = -h

(7;,Tyb) = -$v'iFi?(u,u)

( ) N-&

(2)

(8)

where C is the Chezy coefficient. This coefficient depends on the nature of the bottom and also to some extent on the depth of water. An expression of the following type is used by Leendertse” for these coefficients: C = C, In (C2H + C,)

au

(7)

where pa is the density of air, (W,, WY) are the components of wind velocity, and y is a dimensionless drag coefficient. Several of the forms proposed for y are discussed and compared by Mathison and Johansen. For bottom friction we take

(9)

where C,, C,, and C3 are constants. formulation, see Refs. 41 and 42.

Appl. Math. Modelling,

For an alternative

1990, Vol. 14, August

411

Hydrodynamic

modeling

of the Arabian

Gulf: A. H. Al-Rabeh

consists of two parts. In the first part the surface elevation and depth-averaged velocity components are obtained from the two-dimensional depth-averaged hydrodynamic equations. In the second part the vertical velocity profiles are computed by solving the one-dimensional time-dependent vertical momentum equations, which form a coupled parabolic system of equations. These equations use the depth-averaged velocity components computed above as input. The shallow-water approximation forms the basis for the depth-averaged hydrodynamic equations. According to this approximation, equations (l)-(3) are replaced by their averaged versions over the vertical dimension. This leads to a two-dimensional system of equations rather than the full three-dimensional system given by equations (l)-(3).** In integrating the model equations we introduce the depth-averaged horizontal velocity components U, and i?, defined as follows:

On coastal boundaries the component of the current velocity along the outward normal to the boundary is set at zero. On open boundaries the most appropriate condition, especially for shallow seas, is to specify water elevation.43 Thus for the open boundary we express the elevation in a standard harmonic form: 5 = 20 + i fiA;COS (Vi + Vi + ait + gi)

(10)

I

where the mean value amplitude of constituent i the phase lag behind the equilibrium constituent i on the Greenwich meridian nodal factors accounting for the 18.6-year period variation in amplitude and phase of constituent the phase of the equilibrium constituent i at Greenwich at t = 0 speed of constituent i number of tidal constituents being considered

-&I Ai gi

fi,

ui

vi

(+i

n

Other types of open boundary conditions have been reviewed and tested by many, including Blumberg and Kantha.

u dz

(11)

vdz

(12)

and u=-

The verticallhorizontal

et al.

splitting model

1 i: H I

-h

where H = 5 + h is the total water depth. The governing equations may then be written as follows:

The VHS model was developed as an alternative to the full three-dimensional hydrodynamic model given by equations (l)-(4). The VHS simulation algorithm

(13) ;(HZ)

+ ;(HTiiS)

+ ;(Hliij)

+ gH$

~~H~~+~(HTTii)+~(Hl;o)+gH~=

= fHE

- fg

+ v

-fHu-E5+7-: P

where the following notation is used: z=-

H

u2 dz

(16)

uv dz

(17)

v=dz

(18)

-h

I

jjijz- 1 H

-h 5

ijij=-- 1 H

which represent the mean squares and the mean product of the velocity components averaged over the water column. In the VHS model used, the mean squares and products of velocity components are replaced by squares

412

Appl. Math. Modelling,

(15)

P

and products of the mean velocities, -iz = (U)2, uu = uv, and

i

1

ay

(14)

1990, Vol. 14, August

namely, iz = (E)2

This approximation is reasonable, since the advective terms are usually small. Another approximation that is usually made in the solution of the depth-averaged equations is to replace the velocities appearing in the bottom friction expression given in equation (8) with the depth-averaged velocities. In the VHS model employed in this paper the actual bottom velocity obtained from the three-dimensional computation is used whenever available; otherwise, the depth-averaged velocity is used. By neglecting the advective terms, which in most cases are sma11,45equations (2) and (3) reduce to

au a NE --at az

( >

-fv=

_gi!_;g

(19)

Hydrodynamic

au a --at a2

i$!

( ) a2

+fu=

_g!!_;f

(20)

Equations (19) and (20) form a coupled parabolic system, and these equations can be solved by a finite difference scheme as proposed by Lardner and Cekirge5 and adopted in this paper. It would be feasible to retain the advective terms in equations (19) and (20) (see Ref. 46); however, in this study the approximation used in (19) and (20), which is adequate for most cases, will be considered.39 The simulation

algorithm

The proposed simulation scheme consists of two parts. In the first part the system of hyperbolic equations for U, u, and l given by equations (13)-(15) is solved by using a finite difference scheme. In the second part a transformed version of the parabolic system of equations given by equations (19) and (20) is solved by using a generalized Crank-Nicolson finite difference scheme. The depth-averaged equations A modified Leendertse’s finite difference scheme” is used to solve the system of hyperbolic equations given by equations (13)-(15). The scheme is modified so that the actual bottom velocities are used in the bottom friction expression whenever available from the three-dimensional computation. An Arakawar Cgrid is used. The quantities 5, h, H, C, w,, and wy are computed or specified at the grid points (m, n), the velocity component U is computed at the points (m & l/2, n), and the component U is computed at the points (m, n + l/2). The time step is divided into two halves, each of length 7. In the first half-step, equations (13) and (14) are solved semi-implicitly for 5 and U; then equation (15) is solved explicitly for U. In the second half-step, equations (13) and (15) are solved semi-implicitly for 5 and 0; then equation (14) is solved explicitly for U. Details of the algorithm are discussed in Ref. 47.

modeling

of the Arabian

depth-following We introduce

Gulf: A. H. Al-Rabeh

coordinates,

et al.

or sigma coordinates.48

o-1- z+h H

(22)

so that u = 0 at the bottom and (+ = 1 at the free surface. This results in a simple transformation of equations (19) and (20). A fixed number of grid points is used. The finite difference scheme used for the solution of the transformed equations (19) and (20) is a generalization of the Crank-Nicholson scheme. The details of the algorithm are described in Refs. 5 and 47. Model implementation The hydrodynamic model described in the previous sections is used to simulate wind-driven and/or tidal circulation in the Arabian Gulf. Two implementations are presented. The first, called HYDROl, is designed for the whole gulf with emphasis given to the part of the gulf adjacent to the Saudi Arabian coast. The second, called HYDR02, is designed for any user-defined area in the Arabian Gulf. In both implementations, three-dimensional computation is carried out when the conventional two-dimensional hypotheses are locally inadequate, for example, in cases of sharp depth variation or strong wind forcing, and when required by the user to carry out a detailed study of a particular region of interest. Elsewhere, two-dimensional computation is employed. The Arabian Gulf is represented by a two-block vari-

-------

r-

Stability The finite difference algorithm used imposes the usual Courant-Friedrich-Lewy stability requirement; thus the linearized version of the algorithm is stable, provided that

where H* is the maximum value of H. Note that this is the only stability requirement for the model, since the method used for vertical integration is unconditionally stable. The vertical momentum equations The use of a fixed finite difference grid in the vertical may lead to a reduced accuracy for shallow areas (see, for example, Refs. 9 and 38). This problem can be overcome by transforming equations (19) and (20) into

L--.-

-

SW

FOINT

Figure 1.

Appl.

a

OPEN

The grid system for the Arabian

Math. Modelling,

BOUNDARY

POINT

Gulf: Block 1

1990, Vol. 14, August

413

Hydrodynamic

modelling

of the Arabian

Gulf: A. H. Al-Rabeh

et al.

Table 1. Amplitudes and phases of the significant tidal constituent at the Strait of Hormuz Tidal constituent

Amplitude

(cm)

71.4 25.2 17.4 26.2 2.8 18.1 9.0 6.7 2.1

* SER POINT * OPEN BOUNIIRRY POINT

Figure 2. The grid system for the west coast of the Arabian Gulf: Block 2

able grid in Cartesian coordinates with an equal grid spacing in the x- and y-directions. Figures 1 and 2 show the grid systems of Block 1 and Block 2, respectively. Block 1 covers the whole gulf with a coarse mesh of approximately 20-km grid spacing. Block 2 covers the coastal waters of the gulf from Kuwait in the north to the Gulf of Salwa in the south with a finer mesh of approximately lo-km grid spacing. For HYDR02, any region lying in Block 1 or 2 can be selected. The lowest astronomical tide (LAT) is read from a navigational chart of the gulf.49 An average constant correction h,,,, is added to LAT to compute the mean sea level (MSL). The time step is selected to satisfy the numerical stability condition given by equation (21). and open boundary

conditions

Both models start computations assuming that the water velocity is zero everywhere. The water heights, however, are given a small constant initial value. The models run for 74 hours of real time to reach conditions that are reproduced from one tidal period to another independent of initial conditions. The wind-driven flow computations, on the other hand, start at time zero. The tidal oscillations at the Strait of Hormuz are taken as the only driving force for tidal flow computations and prescribed as open boundary conditions. The tidal heights are computed by harmonic analysis, equation (lo), using the most significant tidal constituents at the Strait of Hormuz (see Table 1). Since the open boundary across the Strait of Hormuz is very

414

Appl.

Math.

Modelling,

319.0 357.0 289.0 79.0 181.0 69.0 79.0 357.0 260.0

narrow, the tidal constants are assumed to be the same for all the open boundary points. The water heights in Block 1 provide the open boundary conditions for Block 2 using an appropriate interpolation scheme. HYDROl provides HYDR02 with the water heights at the grid points neighboring the open boundaries of the region. HYDR02 interpolates these heights linearly in space and time to obtain the water heights at the open boundaries.

LONGITUDE

Initial

Phase angle (degree)

1990,

Vol.

14, August

Basic model parameters Wind drag coefficient. A number of formulas have been proposed for the dimensionless drag coefficient y. Donelan 50 has shown that y depends on the state of development of the surface waves. A wave-forecasting model would therefore appear to be a requirement for the moper determination of wind stress for a winddriven circulation model. The drag coefficient proposed by Lystad and Martinsens is used in the model:

y = a + blwl

for 14 < IwI

y=c

for Iw( 2 lw,l

(23)

where w is the surface wind speed. Extensive experimentation to optimize the deviation between model output and observed results indicates that the most appropriate values are a = 0.00089, b = 0.00013, c = 0.0028, and lwll = 15 m/s. Chezy coefficients. It is known that the magnitude of the bottom stress significantly influences the magnitude and direction of surface currents. This is especially significant for shallow seas such as the Arabian Gulf.8 The model computes the values of the Chezy coefficient C at each grid point from the given values of depth using (9). Extensive tests indicate that C, = 25, C2 = C3 = 1 give the best fit between computed and observed values of water height and current velocity in the gulf. Vertical eddy viscosity coefficients. A number of formulations for vertical eddy viscosity have been proposed.8*14,52-54It is clear, however, that an uncertainty exists in defining vertical eddy viscosity for three-di-

Hydrodynamic

viscosity profile that is similar to what has been proposed by Davies,’ Davies and Furness5 and Heaps and Jones.54 Therefore it is proposed that (see Figure 3)

mensional sea models of tide and wind-driven motions. As has been observed by Davies8 the vertical eddy viscosity profile significantly influences the current speed and direction. In our model we adopt a vertical eddy

forHh,szsH for h2 5 z 5 H - h, for 0 I z 5 h2

N = N, - (N, - N,)(z - H + h*)lh, N = N, N = N2 + (N, - NJzlh,

N,, = 0.3043. 1O-4 W3 N, = 0.0375 (aH + p)w N2 = 0.0132hl=hZ= 10m (Y= 0.25 p=1 The model has a top layer of depth h, in which N increases from No to Ni and a bottom layer of depth h2 in which N decreases from N, to N2. In between

Model testing and calibration Hydrodynamic models require extensive testing before they can generally be applied with confidence to the solution of coastal and offshore problems. Therefore after the completion of the basic computational code development, several sample problems are used to test the model.

Surface

t --I

H

Application to the Arabian Gulf: Tides The tides in the Arabian Gulf have an interesting structure, the diurnal components having a single amphidromic point and the semidiurnal components having two such points. In addition, the open boundary across the Strait of Hormuz is very narrow, so the boundary values can be specified reasonably accurately. The models are applied to the computation of tidal amplitudes, phase angles, and water heights at selected stations on the Saudi coastline. The comparisons of the computed and observed tidal amplitudes and phase

NI

Sea bed

Figure 3.

(24)

the two layers, N is uniform over depth. Note further that in the expression for NZ the bottom velocity is used whenever available; otherwise, the depth-averaged velocities are used. The proposed formulation seems appropriate for shallow seas with a relatively strong tidal current regime, such as the Arabian Gulf.

where

_ 1

modelling of the Arabian Gulf: A. H. Al-Rabeh et al.

Eddy viscosity profile

Table 2. Computed and measured tidal amplitudes stations (computed values are printed first)

m AMP

and phase angles at selected tidal

S2

01

K,

AMP

PHS

AMP

PHS

AMP

PHS

km)

PHS (deg)

km)

(deg)

km)

(deg)

(cm)

(deg)

Abu Ali

49.3 45.5

133.7 131.5

22.9 15.5

143.8 192.3

16.7 13.7

280.3 272.2

25.3 17.7

303.6 322.9

Abu Safah

56.6 52.5

131.9 133.5

22.9 17.9

157.0 190.0

6.0 8.7

301.7 275.2

10.1 9.4

9.4 331.7

Lawhah

10.8 8.3

139.5 145.0

5.0 4.1

155.8 208.9

13.9 20.1

266.4 252.9

21.9 29.8

285.5 299.3

Pas Tanura

66.5 60.9

144.7 130.2

29.1 20.7

166.6 188.0

10.8 12.3

297.8 280.5

15.8 14.3

9.7 340.1

Safaniya

21.2 21.1

1.7 347.4

8.2 5.9

43.2 46.3

29.9 25.2

267.1 258.2

47.2 37.3

293.8 306.6

7.3 7.2

247.3 345.1

2.2 1.1

66.4 34.9

26.9 22.1

262.5 256.8

42.4 33.8

285.8 303.5

Station

Zuluf

Appl.

Math.

Modelling,

1990,

Vol.

14, August

415

Hydrodynamic

modeling

TIDE

of the Arabian

HEIGHTS

AT

ABLJ

vs.

COMPUTED

PREDICTED DRTE

l/2/1989

:

AL1

tlSL-LRT

Gulf: A. H. Al-Rabeh PIER

TIDE

:

102

Cm

60 70 50 40 30 20 10 0

d

2

4

6

6

10

12

TIME Figure4a. Comparison at Abu Ali Pier

14

16

(Hours

1

18 20

:

DRTE

140 130 120 110 100 90 00

i :: r

HEIGHTS

22

24

150 140 130 120 110 100 90 80 70 60 50 40 30 20 10 0

60 50 40 30 20 10

HEIGHTS

DRTE

_ : 2 3

150 140 130 12011010090 00

ki ’

:

AT VS.

l/2/1999

1 3 5 7 9 ,,,,,,I,,,,,,I-TTT-~-~~@

:

13

15

17

75

19

-

PREDICTED

,-OM,B”TE,,

0 0 ‘r~~rr’rrr~rrr”r”“rri 2 4 6 E 10 TIME Figure4b. at Marjan

Comparison

12

_ -

10

6

21

23

170 160 150 140 130 120 110 100 50 60 70 60 50 40 - 30 - 20 - 10

12

TIME

14

16

16 20

Cm

TIDE

21

23

14

16

(Hours

1

16

20

22

24’

of predicted and computed tide heights

- 140 - 130 - 120 - 110 - 100 - 50 ‘.. - 80 \70 - 60 - 50 - 40 - 30 - 20 - 10 22

HEIGHTS

24’

(Hours)

1

150 140

; c”

6050 40 30 20 10

3

HT

PREDICTED : l/2/1999

DRTE

5

7

5

SAFRNIYA

312/317

VS. COMPUTED MSL-LAT

11

13

15

:

121

17

19

Cm

21

23

-

-

-..... ......

100 90 60 70 60 50 40 - 30 - 20 - 10

PREDICTED COMPUTED

of predicted and computed tide heights 0 0 ~“““~‘~‘~“~““‘~~’ 2 4 6 6 10

angles of the major tidal constituents are given in Table 2. The computed and observed water heights are presented in Figures 4a-4d. The computed values of tidal constants, phase angles, and water heights compare well with those reported in Ref. 56. The computed results confirm the fact that tidal regimes at Abu Ali and Ras Tanura are predominantly semidiurnal, whereas tides at Safaniya and Marjan are predominantly diurnal .

Appl.

19

PREDICTED _...._...... coMPuT6D

TIME

416

17

Cm

MARJAN

,,.....” ....,,

.

122

COMPUTED

-

40 30 r 20 10

:

MSL-LRT

-

0 0 “““I”““““““” 2 4 6

MSL-LRT

11

TANURA

2 -

of predicted and computed tide heights

PREDICTED

RAS COMPUTED

1 3 5 7 9 11 13 15 - I 8 I I t , I I I I I-M-T--IT-

170

K r

vs.

l/2/1999

Figure 4c. Comparison at Ras Tanura TIDE

AT

PREDICTED

150

_ : ? 5

et al.

Math. Modelling,

1990, Vol. 14, August

Figure4d. Comparison at Safaniya

Application

12

14

16

(Hours

1

18

20

22

24’

of predicted and computed tide heights

to the Arabian

Gulf: Wind

Surface currents in the Arabian Gulf are generated by wind, tides, and residual effects. For predicting 24hour trajectories, tidal effects can be disregarded, since they cancel out over this time span. This factor is significant, however, for predicting net drift for periods

Hydrodynamic

modeling

of the Arabian

Gulf: A. H. Al-Rabeh

et al.

MEPR BUOY MOVEMENTS IN THE RRABIRN GULF OBSERVED vs. COMPUTED WIND-DRIVEN CURRENTS MEPR BUOY NUMBER : 23587 __.__._.____

OBSERVED

-

COMPUTED

LONGITUDE

-I

LONGITUDE

Figure 5.

Comparison

of observed

and predicted

buoy move-

Figure 7a.

Surface velocity after 6 hours in the Abu Ali region

ments

LONGITUDE

-I

’

LONGITUDE

Figure 7b. Figure 6.

Surface velocity after 12 hours in the Abu Ali region

The grid system for the Abu Ali region

Case studies of less than 24 hours. Residual currents due to density and temperature differences are long-term effects, and their contribution to the 24-hour net drift of floating objects is negligible. Consequently, drift paths of floating objects for 24 hours are presumed to be largely determined by wind-driven currents. The paths of drifting buoys under various wind conditions and in different regions in the gulf have been recorded in Ref. 57. The paths of the floating buoys as predicted by the model are plotted in Figure 5. The predicted trajectory is in broad agreement with actual trajectory as reported in Ref. 57 (see Figure 5).

HYDR02 is used to simulate tidal and wind-driven circulation in the region of Abu Ali on January 2, 1989. The region is represented by a 10 x 10 grid with a grid size of approximately 3 km (Figure 6). A northwesterly synthetic wind with a maximum speed of 10 m/s is applied. Wind direction is considered constant, and wind speed is computed as a Gaussian function of time. HYDROl is used to obtain water heights at the open boundary. For stable computations the time step must be equal to or less than 125 seconds. The time step selected for this case study was 120 seconds. In computing vertical velocity profiles, 50 grid points are used.

Appl.

Math. Modelling,

1990, Vol. 14, August

417

Hydrodynamic

modeling

of the Arabian

Gulf: A. H. Al-Rabeh

Figure 8. Figure 7c.

et al.

Time evolution of a vertical velocity profile

Surface velocity after 18 hours in the Abu Ali region

a two-block variable grid system with a finer grid on the west coast of the gulf. The second, HYDR02, computes tidal and wind-driven circulation for any userdefined area in the gulf. This enables the user to examine the hydrodynamics of a particular region of interest in more detail. Both implementations carry out three-dimensional computation when the conventional two-dimensional hypotheses are locally inadequate and when specified by the user. Elsewhere, two-dimensional computation is employed. Appropriate selection of eddy viscosity profile, wind drag coefficient, and the Chezy coefficients is made. Both implementations have been calibrated and tested extensively. Results indicate that the model developed is reasonably efflcient and accurate. Acknowledgments

LONGITUDE

‘-

-

Figure 7d.

Surface velocity after 24 hours in the Abu Ali region

Surface current velocities and vertical velocity profiles are presented in Figures 7a-7d and 8, respectively.

Conclusion The vertical/horizontal splitting (VHS) model is extended and adapted to simulate tidal and wind-driven circulation in the Arabian Gulf. The extensions include the use of finite differences to solve depth-averaged equations, the use of the depth following coordinate in the vertical, and relating the bottom friction to the bottom velocity. Two implementations of the model are made. The first, HYDROl, computes tidal and wind-driven circulation for the whole gulf. It employs

418

Appl.

Math. Modelling,

1990, Vol. 14, August

We thank the KFUPM Research Institute and Saudi Aramco for support of this research effort under King Fahd University of Petroleum and Minerals Research Institute Contract No. 24079 and the Saudi Arabian Ministry of Petroleum and Mineral Resources for their authorization to publish this work. We would like to acknowledge the comments made by two anonymous referees. References A report on the oil pollution in the Arabian Gulf. World Information System, Cambridge, Mass., 1981. Davies, A. M. On extracting vertical current profiles from vertically integrated numerical models. Coastal Engrg. 1987, 11, 445477 Jelesnianski, C. P. Bottom stress time-history in linearized equations of motion for storm surges. Month. Weather Rev. 1970, 98, 462-478 Forristall, G. Z. A two-layer model for hurrican-driven currents on an irregular grid. J. Phys. Oceanogr. 1980, 10, 1417-1438 Lardner, R. W. and Cekirge, H. M. A new algorithm for threedimensional tidal and storm surge computations. Appl. Math. Modelling 1988, 12, 471-481

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