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61
* E-mail: [email protected]
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
d that the
an be descri
s, and that th
llowing the u
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re considere
ddy fluxes. dance with the following
the horizbalance, incbalance, resp
ρ DuDt − uvtanR−Rcρ DvDt + u2tR+ 1Rcosφ ∂∂φ(v
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63
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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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ome:
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− 2Ωcosφw =Vλ tanφ + Av(
1sφ∂u∂λ = 0
+ 1R2cos2φ ( ∂∂Q(φ), atz = 0 z = −Htλ = 0,1φ = 0,1
er 2015/17/61
cribed such t
ents in the λ,
e, ρ denotes
g denotes
or and Prand
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mplifications,
(
(
(
(
(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
0.58694
0.58694
0.58694
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0.60061
0.60061
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0.614270.62794
0.62794
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0.66895
0.05 0.1 0.15 0.2 0.25
−2
−3
−1
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0.2
0.3
0.4
0.5
0.6
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
e derivatives
and the Ekma
. The stress
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heat capacity
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Persian Gulf
67
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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
onsider the
e T . The
t Z=0 is ind
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he homogen
ϕ ∂∂ϕ cosϕ∂T1∂φ
Jo
functions of Tection).
sport
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non-homoge
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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
, (b) Vertical a
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
d there is an
as a diffusio
So, as is sh
top Ekman
, the Ekman
orcing function
ry current ad
ider the w
i Mirsaeid et a
of
ing the resul
ransport warm
ases in the
upward adve
on of heat d
hown in Figu
layer is pos
n layer is a
n at the top Ekm
dvection
western boun
F
cos nal. / A Mathem
an
lting equation
m water. Sin
tropical gy
ection of war
down from th
ure 7, Forcin
sitive. As see
source of he
man layer
ndary curre
Fig.8: The T2 fi
− 12 π∅mtical Model f
70
nd
ns
nce
yre
rm
the
ng
en
eat
ent
adve
defin
W
cond
Th
W
subs
deco
the r
W
The
boun
posit
the w
field in a meridi
∂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
ridional section
ivided former ion to
gure7a orcing rature t hand
siomvadstca
(Marine Scien
for (a) λ=0, (b)
ide of equatiother. Since,
meridional adertical advecdvective fortrong effect. an see that a
nce)/Vol. 6/No
) λ=0.5, (c) λ=1
on (75). Thosthe vertical
dvection, the ction. Figurecing. InterioConsidering positive adv
o. 22/Decembe
1
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e 7c gives thor vertical ag equation (7vection term c
er 2015/17/61
s counteract eis stronger tt will be duehe total inteadvection ha5) for T11, corresponds t
1-78
each than e to erior s a one to a
heastroT1evewaFiglowthe
barleswa9a,dow
at sink in theong, unbalan1 is negativeerywhere. Thater from equgure 6, the cow latitudes. Ae Ekman layeThe westerrotropic. Sinss. T13 is poarming of the, 9b and wnwelling
Ghazi
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uator to higheorrected bounAs seen fromer is a source rn boundarnce, is smsitive everywe water every9c. The bnext to th
Fig
i Mirsaeid et a
calculation anheat sink at
effect is a colayer will trer latitudes. Andary conditiom Figure 8, at
of Heat (T2>ry current mall, the trawhere so theywhere, as s
baroclinic ue meridiona
.11: The solutio
al. / A Mathem
nd because oft low latitudeooling of watransport warAs is shown bon is positive t low latitude
>0). advection
ansport is alnet effect is
een in Figurpwelling anal boundari
on to problems
mtical Model f
74
f a es. ter rm by at
es,
is so s a res nd ies
prodeastewestthe eeast whicidentposit
4.3. W
If correthere
T1+T2+T3 for
for Indian Oc
duces a positern sides. i.etern boundareastern. Thisto west. Fi
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
he three wiher (T11+T1due to interior
=0.5 and (c) λ=1
on in Spherica
of at the ffects are evng effects ara diffusion oows negativ
At λ=0, for values of Is
rming there.
ure
ind induced 2+T13), at lr advection (s
1
al Coordinate
western andvident at there evident atof heat fromve isothermsλ=1, it is
sotherms are
temperaturelow latitudessee Fig.11).
e
d e t
m s s e
e s
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
iseq
ms T1+T2+T3+T
(Marine Scien
.5. Total Fiel
The total fis reasonable.quator.
T4 for (a) λ=0, (
nce)/Vol. 6/No
d
ield, T0+T1R At λ=0the i
(b) λ=0.5 and (c
o. 22/Decembe
R0E-1, showisotherms are
c) λ=1
er 2015/17/61
wn in Figure. e shallow at
1-78
13, the
T0
wa
the
iso
bar
Co
pro
dim
a m
flu
tem
the
Ek
bou
bar
we
ver
flo
wh
Figure 14,
0+T1R0E-1. I
ater is warmer
e tropical reg
otherms to t
roclinic curre
Fig.14: T
onclusions
We formulat
oblem invol
mentional tem
meridionally
ux. We deriv
mperature fie
e effect of ge
kman layer,
undary curre
roclinic upw
e computed
rtically avera
ow and fina
hich is the ge
Ghazi
shows the
In the tropica
r than the we
gion there is
the north, w
ent in the upp
Total solution T
ted a mathem
lving the c
mperature an
varying win
ved by this
eld into four
eostrophic co
the effect o
ent advection
welling and do
velocity fiel
aged flow, t
ally, the diff
ostrophic flo
i Mirsaeid et a
Fig. 13
surface tem
al region, the
estern surface
a downwar
we will hav
er layer at low
T0+T1R0E-1 at
matical mode
calculation
nd velocity fi
nd stress and
model, the
r subproblem
onvection, th
of the barot
n, and the ef
ownwelling.
ld by first c
then the top
ference betw
ow.
al. / A Mathem
: The solution t
mperature fiel
eastern surfa
water, since
d slope of th
e a westwa
w-latitudes.
surface.
l and studied
of the thr
ields forced b
d surface he
e first interi
ms as follow
he effect of to
tropic weste
ffect of coast
In this mode
computing th
p Ekman lay
ween the tw
mtical Model f
76
to problems T0
ld,
ace
in
the
ard
d a
ree
by
eat
ior
ws:
op
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yer
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GundcirM10
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Bram
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of
Oc
Anna
In
ea
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for Indian Oc
0+T1R0E-1 for
erences
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