numerical simulation of local wind-driven currents in

NUMERICAL SIMULATION OF LOCAL WIND-DRIVEN CURRENTS IN ARABIAN GULF By

ABDUL-HALEEM ALI HUSSAIN AL-MUHYI

& MARIANO A. ESTOQUE (Dissertation Adviser )

Preprint based on the Doctoral Dissertation Submitted to the Department of Meteorology and Oceanography College Of Science UNIVERSITY OF THE PHILIPPINES Diliman, Quezon City April 1998

50

52'

54"

2

58

30

28

Sl
~lP)

and the velocity distribution (26) is expressed in terms of known magnitudes and the mean depth U(x,y). The substitution of (26) and(27) in the bed shear expression 'tb

au gives . f or az

I p = v --

~ = 0.18~(~

't'b Ip:

)u-os ~

------(28)

Finally, the use of (26 ) in the integral of the convective terms u Ou/Ox, v 8v/f)y along the depth gives (Koutitas J 988):

8

1

au /

Ou

n

- Ju-dz=U--+f

0.2U

a '

+-)--

au

ax ax "40 ax 1 e av "V ( 0.2V+!!_ ) ~"V -Jv-dz=V~+ h -"

h-h

ay

By

40

-----(29)

ay

The 2D horizontal integrated model improved with respect to the horizontal momentum dispersion

and the bed friction for wind-generated circulation, becomes on the basis of (28) & (29):

BU +UoU +VoU +(o.2U+ a"JoU +(o.2V+ ot ax cy 40 ox

a.v)au

ay

40

=

-(0.18 u f(r, ·J-o.5 J av +UoV +VBV +(o.2U +5-)oV +(o. v + ay)oV = 01 ax oy, 40& 4oay ac v r7: fy - jU + r; (o h~I ~ )1-o.5 7:sy)

- g at;+ JV+ r,x

ox

ph

----(30)

r,x ph

h~p

2

-g

ph-

. 18

ph

----(31)

Equations (21), (30) and (31) represent the wind-generated circulation model of Koutitas (1988). This model was extended by Rivera (1997) to include the horizontal momentum diffusion. Hence, equations (30) and (31) are now written:

axJoU --+ ( 0.2V+- a,,JoU --= 40 ox 40 Oy

r

eo eo e: -+U-+V-+l0.2U+8t ax cy -gas_+

ax

fV

+

r,x

ph

-(0.18U h J(r·p)-o.5

av_+UoV +VoV +(0.2U+~)?V

ax

at

-g at; -

Oy

JU+

ay

Tsy

ph

40

ax

i-s,.

ph

]+A;,(a2u + a2u)

+(0.2V+

ox

2

0y2

--(32)

ay)avay =

40

-(0.18 vh J'(r·p "j-o.5 r$;,ph )+ Ah(a2v + a2v) ax2 012

--(33)

1

Equations (32) and (33) defined the current acceleration in the horizontal x.y direction respectively, together they signify the conservation of momentum. The first term of the LHS of these equations are known as local acceleration terms and the following terms represent changes in the fluid acceleration due to advection of momentum. The additional advective terms involving the stress variable "a" are correction imposed on advection to include non-uniformity on the current profile. On the RHS of these equations, the first terms are the horizontal elevation, second terms arc the Coriolis terms, third terms are the sea-surface elevation and bottom friction effect and the last term is the horizontal momentum diffusion. The wind stress is exerted along the direction of the wind and its magnitude is usually specified by the quadratic Drag low written as

9 1'.w=

Pa Cd w

IWI

Pa is the air density and Cd is the dimensionless empirical Drag coefficient varying linearly with

!iV I

the wind. w is the wind vector and

is the magnitude of wind vector. The equation above in

x and y component can be written 'tsx = Pa

Cd u

'tsy = Pa Cd v

In this study the computed value for

IWJ 1Wi

the drag coefficient Cd, is that proposed Cc1 =(0.8 +0.065

I w I) x 10-

by Wu (1982)

3

The forcing function for tide-driven circulation is provided by the semi-diurnal oscillationofthe sea-level at the opening of Arabian Gulf. In this model, only the semi-diurnal tide constituent is considered since it is much stronger than any other in the Gulf

Following mathematical principles, any continuous function which is given at every point in the interval can be represented by an infinite series of sin and cosine function. The series is called Fourier series. The truncated Fourier series ((1

J

·-~[ ·. (2n .1

=AVE+~

Asm pzt)+B,cos

(2n. Jl pzt

..i

s

Where is the tidal elevation, N is the number of harmonic, P is the period of data, and t is the time . The truncated Fourier series given by this equation is then used to force the model at the eastern open boundary simulating periodic flooding and ebbing at the Arabian Gulf 3. The finite difference formulation of the model A rectangular integration domain is used in this model. The grid network consists of 39 points along x and 37 points along y. The grid spacing is 20 km for both x and y making it a horizontal square grid system. 3.1 TI1e finite difference formulation of the mixed Layer model All predicted variables i.e., horizontal wind velocity (u, v) along (x, y) axis, mixed layer potential temperature ( eM ), depth of mixed layer (ZT ), and the surface temperature (To), are determined at each grid point. The vertical velocity (w ) and the atmospheric pressure (p ) are also determined at each grid point. The space derivatives in the advection terms are calculated using the upstream difference analogue. For other space derivatives, the centered method of finite differencing scheme is used.

3.2 The finite difference formulation of coastal circulation model equations. Equation (21), (32), and (33) presented in sec.(2.2) are solved by using numerical technique. This usually done by finite difference approximation. The numerical solution of these equations

10

Fig. (3) where u, v, and surface elevations are staggered in space relative to each other. 4. Initial and boundary conditions. 4.1 For mixed layer . The values of the variables, the horizontal velocity (u, v), geostrophic wind ( ug . Vg ), stable layer temperature 8H, mixed layer potential temperature 9M, surface temperature on land and Guff 80, stable layer height ZR? mixed layer height ZT, and surface layer height Zs, are prescribed initially, based on real cases taken from synoptic data and set to be equal at alJ points of the domain. At the inflow lateral boundaries (upper and left boundaries), all the predicted variables are set to be equal to the adjacent.

c..

'IJ

U;.Jj

•

+

V;J.J

Fig.

~.

Location of variables in the staggered grid used in the present study. (Rivera 1997)

inner grid points. The boundary conditions at the other two lateral boundaries set all the predicted variables equal to their respective predicted values at the adjacent inner grid points. 4.2 For Coastal Circulation Model For the evaluation of time dependent variables u, v and t; , an initial state must be prescribed at time = 0. A very common initialization technique is the "cold start" which means the sea is initially at rest without any surf ace disturbance. =u =v= O at t. = 0 In this model, the computational domain has four boundaries. 111e open boundary is localed at the eastern part of the domain. The term "open" implies a sea. boundary where the solution is unknown and must be assumed or extrapolated from the interior solution. In U1e domain u, v, and



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