61

* E-mail: mehdizadeh@hormozgan.ac.ir

A Mathematical Model for Indian Ocean Circulation in Spherical Coordinate

Ghazi Mirsaeid, Mojgan1; Mohammad, Mehdizadeh Mehdi1*;

Bannazadeh, Mohammad Reza2

1- Department of Physical Oceanography, Faculty of Marine Science and Technology,

Hormozgan University, Hormozgan, IR Iran

2- Erfan University

Received: February 2015 Accepted: October 2015

© 2015 Journal of the Persian Gulf. All rights reserved.

Abstract In recent years, the Indian Ocean (IO) has been discovered to have a much larger impact on

climate variability than previously thought. This paper reviews processes in which the IO is, or

appears to be, actively involved. We begin the mathematical model with a pattern for summer

monsoon winds. Three dimensional temperature and velocity fields are calculated analytically

for the ocean forced by wind stress and surface heat flux. A basic thermal state involving a

balance of lateral and vertical heat diffusion is assumed. The wind stress is chosen such that a

tropical mass transport gyre is generated. An effect of nonlinear heat advection is calculated by a

perturbation method. The zero order temperature field gives a rough overall representation of

oceanic thermocline. A baroclinic eastward flow in the upper part, with a westward return flow

below is associated with this field.This circulation is closed through thin up and downwelling

layers at the sides. Superimposed, there is a barotropic wind driven circulation, with a transport

field of the type described by Munk. The interior temperature field to the next order is affected,

not only by interior heat advection but also by heat advection in the Ekman layer, in the up and

downwelling layers and in the main western boundary current.

Keywords: Spherical coordinate, Heat advection, Circulation, Indian Ocean

1. Introduction

Several review papers have been published

previously on IO circulation. Schott and McCreary

[2001] reviewed the current state of knowledge on the

monsoon circulation, both with regard to recent

observations and to the hypotheses put forward for

their interpretation. They summarize, and try to

reconcile, classical concepts and interpretations with

the newly available observations. Their focus is on the

monsoon circulation north of about 10 S. Schott, et al.

(2009) reviewed climate phenomena and processes in

which the IO is or appears to be actively involved.

We begin with an update of the IO mean

circulation and monsoon system. It is followed by

reviews of ocean/atmosphere phenomenon at intra-

seasonal, inter-annual, and longer time scales. Much

Journal of the Persian Gulf

(Marine Science)/Vol. 6/No. 22/December 2015/17/61-78

Ghazi Mirsaeid et al. / A Mathemtical Model for Indian Ocean Circulation in Spherical Coordinate

62

of above reviews address the two important types of

inter-annual variability in the IO, El Nin˜o–Southern

Oscillation (ENSO) and the recently identified

Indian Ocean Dipole (IOD).

Muni et al., (2014) studied Tropical Indian Ocean

Surface and Subsurface Temperature Fluctuations in a

Climate Change Scenario. The feature and evolution

mechanisms of the tropical Indian Ocean temperature

between pre-warming and warming periods and also

during IOD events that co-occurred with El Niño are

studied using Simple Ocean Data Assimilation

(SODA) data set. During the positive IOD with co-

occurred El Nino years in a climate change period

(1970-2008) surface warm temperatures are extended

from the Sumatra region to off the African coast in

pre-monsoon season. But, the subsurface temperature

shows different pattern, the warming is more in the

central Arabian Sea and south Bay of Bengal area.

Our review is covering the description and

understanding of physical oceanography concepts by

mathematical method. In later experiments, the

temptation is to improve oceanic investigation by

numerical models by bringing in new effects, such as

bottom topography, nonlinear equation of state, etc.

The repetitions of experiments over a range of

parameter values which give a better insight into the

nature of simpler problem are often neglected. Thus,

there seems to be some gaps in our basic theoretical

knowledge which needs to be filled, by a combination

of analytical theory and relatively simple numerical

models.

The problems studied here involve the calculation

of the three dimensional temperature and velocity

field in a bounded, rectangular ocean on the real

earth, forced by a meridionally varying wind stress

and surface heat flux. The motion and driving forces

are taken to be independent of time. In this paper, the

equations of motion are approximated and

simplified. After averaging the equations of motion,

the Reynold stresses are defined. Phenomenological

transfer coefficients associated with the horizontal

and vertical eddy motion are assumed to be

constants, with the same values for the eddy

viscosity and the eddy diffusion assumed. The eddy

coefficients in the real ocean most likely vary with

position; but since the main purpose of this paper is

the investigation of the general behavior of a model

ocean, the use of constant eddy coefficients is

considered to be sufficient for the present. We note

that, at present, no theory exists which predicts the

variable eddy coefficients in the ocean. Salinity is

represented through an equivalent temperature effect.

At the solid boundaries, the normal, baroclinic,

vertically averaged flow must vanish; hence,

boundary layers must form. Actually, this circulation

is closed principally through thin upwelling and

downwelling layers next to the walls. The barotropic

part is found from the mass transport equation.

Since, bottom stresses have a negligible effect

relative to that of lateral friction in our case, the

linear mass transport equation is of the type given by

Munk. The solution to this equation can be found

analytically and analytic theory can also be given

through a perturbation calculation. The zero order

temperature field gives a rough overall representation

of oceanic thermocline. The first order calculation of

temperature includes, not only the advective effects

of the interior flow, but equally important effects of

thermal advection in the Ekman surface layer and in

the barotropic and baroclinic side layers.

2. Data and method

2.1. Basic Equations

Consider a rectangular oceanic basin in spherical

coordinate (Fig. 1). At the upper boundary a zonal

wind stress and a normal heat flux (counted positive

when directed upward) are applied, i.e.

(1)

(2) At the meridional walls, λ=0, 1, it is required that

the horizontal velocity and the normal heat flux

vanish, while the total velocity and the vertical heat

τλ = −τ0φ sin(kπφ)Q = −Q0cos π2 φ

flux vanish

vertical velo

and heat flu

conditions (

The dens

to allow the

is assumed

diffusion ca

coefficients

isotropic, al

the horizon

diffusion ar

turbulent edIn accord

problem, thexpressing hydrostatic divergence λdirection:ϕdirection:1R2 ∂∂z (R2w)

at the botto

ocity vanish

ux are specifi

(1), (2).

Fig. 1: Geome

sity variation

e Boussinesq

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ddy fluxes. dance with the following

the horizbalance, incbalance, resp

ρ DuDt − uvtanR−Rcρ DvDt + u2tR+ 1Rcosφ ∂∂φ(v

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0 DuDt − uvtanφRAH ∇H2 U − 1R2 ∂∂

pz ≅ gαTw∂z + 1Rcosφ∂DTDt = AH∇H2 T2= 1Rcosφ ∂∂φAV ∂U∂z = τλ(φ)∂V∂z = w = 0U = V = w =U = V = 1RcosR ∂U∂φ = V = 1R

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φ − 2Ωsinφv −∂U∂φ + 1Rcosφ∂V∂λ

vcosφ∂φ + 1Rco+ Av ∂2T∂z2φ cosφ ∂∂φ +), AV ∂T∂z = −Qatz∂T∂z = 0 atφ ∂T∂λ = 0 at∂T∂φ = 0 atφ

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+ 1R2cos2φ ( ∂∂Q(φ), atz = 0 z = −Htλ = 0,1φ = 0,1

er 2015/17/61

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(

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(

(14

(14

(14

(14

(14

ansion:

non-dimens

− 1Rcosφ∂p∂λ +(∂2U∂z2 )

∂∂λ)z = 0

1-78

(7)

that

, ∅,

the

the

dtl’s

ting

the

(8)

(9)

(10)

(11)

(12)

(13)

4-1)

4-2)

4-3)

4-4)

4-5)

sion

+

Ghazi Mirsaeid et al. / A Mathemtical Model for Indian Ocean Circulation in Spherical Coordinate

64

(15)

The scale depth D (thermocline depth) is set by

balancing the horizontal and vertical heat diffusion,

assuming the same typical temperature variation, in

both directions. The thermal boundary condition and

the thermal wind relation then determines the

amplitudes ∆T andU . The relations take the form: A ∆ = A ∆ ⇒ D = R (16) Q = A ∆ ⇒∆T = = ( ) (17) = ∆ ⇒U = (18)

The non-dimensionalized equations take the

corresponding form:

(19)

(20)

(21)

(22)

(23) Where the nondimensional numbers appearing in

these equations are:

(24)

(25)

(26)

(27)

The Rossby and Ekman numbers defined above

are very small for the real oceans, and it seems

natural to try to find a solution in terms of a series

developed from these. Using R E as the expansion

parameter, we can write like the following series

approximations:

(28)

This makes the zero-order thermal balance diffusive.

The nonlinear heat advection appears in the first order

correction, while the dynamics remain linear.

We can rewrite σ as bellow:

(29)

(30)

The number σ∗ gives the ratio of the Ekman wind

drift to the thermal wind transport over the

thermocline depth. Under the above assumption the

zero-order problem is described by:

(31)

(32)

(33)

(34)

(35)

With the nonhomogeneous boundary conditions:

(36)

(37)

Both prescribed at z=0, and with homogeneous

boundary conditions:

(38)

(39)

UV = Ut U′V′ ,W = UtDR .W′ Z = DZ′P = f0UtR. P,T = ∆T. T′

R0 UCOSφ ∂U∂λ + V∂U∂λ +W∂U∂Z − UVtanφ − Vsinφ+DWR cosφ = E∇2U − E ∂U∂φ + 1COSφ∂V∂λ tanφ− 1COSφ ∂P∂λ

R0 UCOSφ ∂V∂λ + V ∂V∂φ +W∂V∂Z + U2tanφ + Usinφ =E(∇2V + 2COSφ ∂V∂λ tanφ − ∂P∂φ ∂P∂Z ≅ T ∂W∂Z + 1COSφ ∂V∂φCOSφ + 1COSφ∂U∂λ = 0

R0 UCOSφ ∂T∂λ + V ∂T∂φ +W∂T∂Z = E(∇2T)R0 = UtRf0 = gαAHf02E = AHf0R2 = AVf0D2

σ = τ0f0gα AHAV 1 2 [ratioofwindforcing tothermalforcing]δ = DH = AVAH 1 2 RH[relativethermoclinedepth]

T = T0 + R0E−1T1 + (R0E−1)2T2 + ⋯,

σ = τ0f0gα AHAV 1 2 = τ0f0gα RD = τf0R2AHf0DUtR =τf0DUt AHf0R = τ f0⁄DUt E−1 σ∗ = τ f0⁄DUt

−V0sinφ + DW0R cosφ = E(∇2U0 − ∂V0∂φ + 1COSφ ∂V0∂λ tanφ) − 1COSφ∂P0∂λ

U0sinφ = E(∇2V0 + 2COSφ∂U0∂λ tanφ) − ∂P0∂φ∂P0∂Z ≅ T0∂W0∂Z + 1COSφ ∂(V0cosφ)∂φ + 1COSφ∂U0∂λ = 0∇2T0 ≅ 0E∂U0∂Z = σ∗φsinπφ∂T0∂Z = cos πφ2U0 = V0 = W0 = ∂T0∂Z = 0 atz = −HD = −1δ

U0 = V0 = 1RCOSφ∂T0∂λ = 0 atλ = 0, 1

Journal of the Persian Gulf (Marine Science)/Vol. 6/No. 22/December 2015/17/61-78

65

(40)

(41)

3. Results

3.1. The Zero Order Solution

The solution to the heat diffusion equation with

the top boundary conditions is independent of λ. We

can write:

(42)

With boundary condition:

(43)

(44)

(45)

The solution becomes:

(46)

The first term, horizontally the solution varies

ascos ; and the vertical variation is of the form Ae ⁄ + Be ⁄ ; Second term is a summation

which is due to the assumed spheroid of the earth’s

surface. c (n,m) is obtained by substituting (46) into

(42), Fourier decomposing the equation in the ϕ and

Z directions and solving the resultant relations for c (n,m). The T solution is depicted in Figure 2.

3.2. Velocity Field

The discussion of the zero order velocity field

starts with the surface regime.

(47) Since the forcing function is independent of λ, the

velocity in the Ekman layer is independent of λ. By

equation (33), we have = T , since T is

independent of λ so = 0. Then, the equation of

motion can be written as follows:

(48)

(49)

(50)

(51)

Fig. 2: The T field in a meridional section.

1R ∂U0∂φ = V0 = 1R∂T0∂φ = 0 atφ = 0,1

W0 = ∂V0∂Z = 0at z = 0

∂2T0∂Z2 + 1cosϕ ∂∂ϕ (cosϕ ∂T0∂φ) = 0∂T0∂Z = cos π2 φatZ = 0∂T0∂φ = 0atφ = 0,1 ∂T0∂Z = 0atz = −HD = −1δ

T0 = cos πϕ2 . coshπ2 Z + 1δπ2 sinh π2δ +

c0(n,m)∞m≠1

∞n=1 cos π − 12 πφ cos(m − 1) πδZ

τλ = E∂U0∂Z = −σ∗φsinkπφ

−v0sinφ = E∂2U0∂Z2−u0sinφ = E∂2V0∂Z2 − ∂P0∂φ∂P0∂Z ≅ T0∂W0∂Z + 1COSφ∂(V0cosφ)∂φ = 0φ(latitude)

Z(d

epth

)

0.573270.57327

0.57327

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0.60061

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0.05 0.1 0.15 0.2 0.25

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0.3

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Ghazi Mirsaeid et al. / A Mathemtical Model for Indian Ocean Circulation in Spherical Coordinate

66

Integrating the above equations and using the

stress condition yields:

(52)

(53)

Where, v is the meridional Ekman transport and W is the velocity at the bottom Ekman layer. By

combining equations (52) and (53), we will have:

(54) or

(55)

Near the ocean bottom, a second Ekman layer

exists. Here, the velocities are of order 1 at most. A

second velocity field which extends from z=0 to is

superimposed from the Ekman layer and splits up into

barotropic and baroclinic parts:

(56)

(57)

We can use the Munk model and Ekman layer

solution to calculate barotropic velocity components.

By definition, we have:

(58)

(59)

(60)

If we neglect the correction part due to the side

layer in the Munk model then,

(61)

Since V is uniform over the depth of Z so:

(62)

Similarly:

(63)

(64) The associated velocity W varies linearly with

depth, from W at z=0 to zero at the bottom then for W = W (1 + δz)Then:

(65) The baroclinic velocity is the remaining part of the

velocity field below the Ekman layer. The baroclinic

current U , V has zero transport, and the vertical

velocity w vanishes at both z=0 andz = −1 δ⁄ .

Since, we are interested in the velocity in the interior

flow then, we start with the geostrophic balance:

(66)

(67)

(68) Taking a Z derivative of equations (65) and (66)

and using equations (67) and (46) gives:

(69)

And

(70)

then

(71)

Similarly:

(72)

Using the equation of incompressibility gives:

(73) Now, since W vanishes at the top and the bottom,

then:

(74)

−vEsinφ = τλ WE = 1COSφ∂(VEcosφ)∂φ

WE = −1COSφ∂τλcosφ∂φWE = σ∗COSφ . ∂(φsinkπφcosφ)∂φ

U0 = Ub + Us V0 = Vb + Vs W0 = Wb +Ws Where (Us, Vs)0−1 δ⁄ dz = 0

U0−1 δ⁄ dz = Mλ = ∂Ψ∂φ

Mφ = 1COSφ∂Ψ∂λ = V00−1 δ⁄ dz + VE

Ψ = −1COSφ ∂τλ∂φ COSφf(λ)andVE = −τλSinφ

Vb0−1 δ⁄ dz = VM − VE = −1COS2φ∂τλ∂φ COSφ + τλSinφ

Vb = δ( −1COS2φ ∂τλ∂φ COSφ + τλSinφ)U0

−1 δ⁄ dz = Mλ = ∂Ψ∂φ = ∂∂φ ( 1COSφ∂τλ∂φ COSφ)

Ub = δ( 1COSφ ∂τλ∂φ COSφ)

Wb = 1COSφσ∗ ∂(τλCOSφ)∂φ (1 + δz)

Usinφ = − ∂P∂φ−Vsinφ = −1cosφ ∂P∂λ∂P∂z = T0∂U∂z = − 1sinφ ∂T0∂φ = sin πφ 2⁄sinφ . coshπ 2 z + 1 δ⁄sinh π 2δ +1sinφ C(n,m)(n − 1 2⁄ )∞

m=1∞n=1 π cos(m − 1)πδzsin(n − 1 2⁄ )πφ

∂V∂z = ∂2P∂λ ∂z = ∂T0∂λ = 0Us = 2π sinπϕ2sinϕ sinh π2δ sinh π2 Z + 1δ − 2δπ cosh π2δ − 1+ c0(n,m) n − 12sin[(m − 1)πϕ] sin n − 12 πφ sin[(m − 1)πδZ] + (1 + (−1)m)(m − 1)π∞

m≠1∞n=1+ 1sin[ϕ] c0(n, 1) n − 12 π sin n − 12 πφ z + 12δ∞

n=1Vs = 0∂Ws∂z = 0Ws = 0

The wind− and the

stress curl a

in Figure 3

kinematical

divided by h

dimensions

Fig. 3: (a) Varvelocity(− )

Fig. 4: The nogyre the transp

d stressτ , th

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. The stress

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Ghazi Mirsaeid et al. / A Mathemtical Model for Indian Ocean Circulation in Spherical Coordinate

68

3.3. the first-order temperature: Tranformation to an

interior problem

The first order correction T to the temperature

field satisfies below equation.

(75)

Since T is independent of λ this equation takes on

the simpler form:

(76)

With the right-hand side, a function of z and ϕ

only. The boundary conditions are homogeneous,

i.e.,

(77)

For better understanding, we suppose the total

problem as the sum of four sub-problems:

(78)

(79)

And all boundary conditions are homogeneous. This

solution reflects the effect of the interior solution.

(80)

(81)

With the boundary condition (81) and other

homogeneous conditions,T indicates the effect of

the Ekman layer advection.

(82)

With the boundary condition:

(83)

And other conditions are homogeneous. T reflects

the effect of western boundary current advection.

(84)

With the condition:

(85)

and other conditions are homogeneous. T Indicates

the effect of baroclinic upwelling and downwelling

nearλ = 0and1 .

3.3.1. The Interior Advection Solution

The interior advection solution T has forcing

which is independent of λ and the governing

equation can be written as:

(86)

Where T , V and W are defined as equation (47)

and (87) to (88), respectively.

(87)

(88) The solution to equation (49) is:

(89)

By substituting equation (89) into equation (86),

Fourier decomposing in the ∅ and Z directions and

solving the resulting equations, c (n,m) are obtained.

According to equation (86), the nonlinear interior

temperature advection to first order is due to the

meridional and vertical (barotropic) motions. Figure 6a

indicates the meridional temperature advection and

figure 6b shows the vertical temperature advection in

the interior, i.e. the right hand side of equation (86).

Considering equation (86) forT , one can see a

positive advection term. Since T is negative, the net

effect is a cooling of the water.

∇2 1 = 0∅ 0 + 0 0∅ + 0 ∂T0∂z∂2T1∂Z2 + 1cosϕ ∂∂ϕ cosϕ∂T1∂φ + 1cos2∅∂2T1∂λ2 =V0 ∂T0∂φ +W0 ∂T0∂z

∂T1∂Z = 0atz = 0andz = −1δ ∂T1∂λ = 0atλ = 0andλ = 1 ∂T1∂φ = 0atφ = 0,1

T1 = T11 + T12 + T13 + T14 1. ∇2T11 = V0 ∂T0∂∅ + w0 ∂T0∂z

2. ∇2T12 = 0∂T1∂z = σ∗ ∅ ∅∅ 0∅ = 0

3. ∇2 13 = 0

∂T13∂λ = σ∗δ B(n,m) cos n − 12π∅ cos(m − 1)πδz at λ = 0

4. ∇2 14 = 0∂T14∂λ = D(n,m) cos n − 12 π∅co s(m − 1)πδzat λ = 0 and λ = 1

∂2T11∂Z2 + 1cosϕ ∂∂ϕ cosϕ ∂T11∂φ = V0 ∂T0∂φ +W0 ∂T0∂z0 = ∗ 1 + − 20 = ∗( 1 + − 2 )(1 + )

T11 = σ∗δ c11(n,m)∞m=1

∞n=1 cos n − 12 πφ cos[(m − 1)πδZ]

Fig. 6: The in(meridional pl

3.3.2. The E

Next, co

temperature

condition at

equation is

With th

∂2T12∂Z2 + 1cosϕ

nterior forcing lus vertical adve

Ekman Trans

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e T . The

t Z=0 is ind

as follows:

he homogen

ϕ ∂∂ϕ cosϕ∂T1∂φ

Jo

functions of Tection).

sport

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non-homoge

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neous and

12φ = 0

Journal of the P

T field in a m

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Persian Gulf

69

meridional secti

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The solutio

The coeff

ubstituting e

T12∂Z = σ∗ A12 = σ∗ δz22 ++σ∗ c12

nce)/Vol. 6/No

onal advection,

on to equatio

ficients cequation (92)

An cos(n − 12)π+ z An cos(n,m) cos n

o. 22/Decembe

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on (50) is:

(n,m) are

) into (90), b

π∅atZ =s n − 12 π∅

− 12 π∅cos(m

er 2015/17/61

advection, (c) t

(

(

obtained

by invoking

= 0

m− 1)πδz

1-78

total

(91)

(92)

by

the

ort

for

T

wa

Ek

fun

fro(T

3.3

cosVE

thogonally

r c (n,m). The Ekman

decreases

an

ater as well

kman layer.

nction at the

om Figure 8,0) .

Fig. 7: Fo

3.3. Boundar

Now, consi

s(m − 1)πδz

∂T12∂∅ < 0

Ghazi

property

and by solvi

layer will tr

as ∅ increa

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as a diffusio

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top Ekman

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m water. Sin

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Fig.8: The T2 fi

− 12 π∅mtical Model f

70

nd

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the w

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∂2T13∂Z2∂T13∂λcos(T13 =cos(mcos

for Indian Oc

ection soluti

ning T is as

With the ho

ditions:

he solution to

Where C (n,tituting equa

omposing in

resulting equ

We have

heat adve

ndary curren

is small and

tive everywh

water everyw

ional section.

3 + 1cosϕ ∂∂ϕ= δ ∞

m=1∞n=1m − 1) πδz

= σ∗δ λ − λ22m − 1)πδz +n − 12 πφ co

ean Circulatio

ion,T . Th

s follows:

omogeneous

o equation (93

, m, k) and Bation (95) int

the ∅ and Z

ations for the< 0 at the ed

cted to hi

nt diffuses i

d the transp

here. So, the

where, as seen

cosϕ ∂T13∂φσ∗B(n,m)cos

at λ = 0B(n∞

m=1∞n=1+ σ∗δ m

∞n=1os(m − 1)πδz

on in Spherica

he governin

and below

3) is:

B(n,m)are

to equation (

Z directions

em.

dge of bound

igher latitud

into the inte

port is also s

net effect is

n in Figure 9

+ 1cos2φ ∂2T1∂λ2s n − 12 πφ

n,m)cos n −C13(n,m, k∞

m=1z cos(k − 1)π

al Coordinate

ng equation

(93)

w boundary

(94)

(95)

obtained by

(93), Fourier

and solving

dary current.

des by the

erior. Since,

small, T is warming of

9a, b and c.

132 = 0

− 12 πφk)π

e

n

)

y

)

)

y

r

g

.

e

,

s

f

Jo

Fig.9:T

Journal of the P

field in a mer

Persian Gulf

71

ridional section

(Marine Scien

n. (a) λ=0, (b) λ

nce)/Vol. 6/No

λ=0.5, (c) λ=1

o. 22/Decembeer 2015/17/61

1-78

Ghazi Mirsaeid et al. / A Mathemtical Model for Indian Ocean Circulation in Spherical Coordinate

72

3.3.4. Coastal up and downwelling

The coastal upwelling induces the solutionT

The equation governing T is

(96)

With the homogenous and below boundary

conditions:

(97)

The solution to equation (96) is:

(98)

Where the values of D(n,m)and C (n,m, k)are

obtained by substituting equation (98) into equation

(96), using the orthogonally property of cos n − πφ

and cos m − πδz and solving the resulting equation

for C (n,m, k). The baroclinic upwelling and downwelling next to

the meridional boundaries produce a positive value

of at the western and eastern sides (Figure. 10),

i.e., cooling effects at the western boundary and

heating at the eastern boundary. We find a diffusion

heat from east to west.

4. Results and Discussion

To start the discussion of our results, let us

determine the nondimensional coefficients E, R, δ

andσ∗ and define a reasonable range for each,

respectively. Let us choose the typical ocean depth,

H, to be 4000 m and the relative thermocline depth,

δ, to beπ 30⁄ (≅ 0.105). Since, δ = , we will

haveD ≅ 420m, which seems acceptable. From

relation, we have Q = (A ∆T D⁄ ) where Q ,the

amplitude of the kinematic heat flux stands. If we let

corresponding to a heat flux of 150calcm day , we find A ≈ 3.6cm s . From

relation (27), we have where R,

the radius of the earth, is 6400 km. we then have

which is large but reasonable.

We have with we find

that and

where

From the definition of the Rossby Number, we have

We choose

.

4.1. Velocity

Because of the wind stress, a complete mass

transport gyre is generated. We reworked the Munk

model in spherical system and then showed the

streamline in (Figure. 3) for case E=2×(10-5). This

gyre results from the stress field. In the tropical gyre,

the transport between two neighboring streamlines is

9.633 (10 12) cm3/sec. The surface velocity resulting from the spherical

case increases northward. At low latitudes, the

velocity in the spherical system velocity has a

minimum of 0.9 and maximum of 1.1 (Figure. 5).

4.2. Temperature

From the zeroth order solution, we display the

temperature field which is produced by diffusive

balance. Heating over the ocean surface diffuses

throughout ocean. The zeroth order temperature field

has a smooth meridional variation, with the isotherms

starting out horizontally from the equator, as shown in

Fig. 1.

∂2T14∂Z2 + 1cosϕ ∂∂ϕ cosϕ ∂T14∂φ + 1cos2φ ∂2T14∂λ2 = 0

∂T14∂λ = D(n,m)cos n − 12 π∞m=1

∞n=1

φcos m − 12 πδz atλ = 0,1

T14 = λ D(n,m)cos n − 12 π∞m=1

∞n=1φcos m − 12 πδz + C14(n,m, k)∞

m=1∞n=1)cos n − 12 πφcos m − 12 πδz cos(k − 1)πλ

Q0 = 1.7 × 10−3cms−1δ2 = (Av AH⁄ )(R H⁄ )2

AH ≈ 8.5 × 108 cm2s−1EH = (AH f0R2⁄ ) f0 = 10−4s−1EH ≈ 2 × 10−5 Ut = (gα∆TD f0⁄ R) ≈ 1.3 cm s−1gα = 0.1 cm s−2K−1.R0 = (Ut f0R⁄ ) ≈ 2.05 × 10−5.τ0 = 2 cm2s−2

The first into four ssection. Thefirst order iindicates thefunction andadvection fo

order tempesub-problems e nonlinear inis due to thee meridional d Figure 7b sorcing functio

Jo

Fig.10: Terature field

as describenterior tempere barotropic v

temperature shows the veon in the inter

Journal of the P

field in a mer

has been died in the forature advectivelocity. Figadvection fo

ertical temperrior (the right

Persian Gulf

73

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ivided former ion to

gure7a orcing rature t hand

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meridional adertical advecdvective fortrong effect. an see that a

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1-78

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calculation anheat sink at

effect is a colayer will trer latitudes. Andary conditiom Figure 8, at

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.11: The solutio

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nd because oft low latitudeooling of watransport warAs is shown bon is positive t low latitude

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74

f a es. ter rm by at

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4.3. W

If correthere

T1+T2+T3 for

for Indian Oc

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ch indicate tical to Figutive, which in

Wind Induce

f we add thections togethe is a cooling d

(a) λ=0, (b) λ=

ean Circulatio

tive value oe., cooling efry and heatins results in aigure 10 shocooling. A

ure 10c, the ndicates war

d Temperatur

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=0.5 and (c) λ=1

on in Spherica

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ure

ind induced 2+T13), at lr advection (s

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m s s e

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4.4. Wind In

If we addto the first cooling alon

nduced and U

d the baroclinthree results

ng the western

F

Jo

Up and Down

nic upwellings we will h

n side. This is

Fig.12: the solut

Journal of the P

nwelling Effec

g and downwhave an addi

seen in Figur

tion to problem

Persian Gulf

75

ct

welling itional re. 12.

4

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1-78

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ater is warmer

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onclusions

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meridionally

ux. We deriv

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Fig. 13

surface tem

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T0+T1R0E-1 at

matical mode

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for Indian Oc

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duz M., Özrculation an

Mazandaran 0(3):459-471

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Ghazi Mirsaeid et al. / A Mathemtical Model for Indian Ocean Circulation in Spherical Coordinate

Journal of the Persian Gulf (Marine Science)/Vol. 6/No. 22/December 2015/17/61-78

Journal of the Persian Gulf

(Marine Science)/Vol. 6/No. 22/December 2015/17/61-78