# Mathematical modeling of Eulerian currents induced by wind

## Transcript Of Mathematical modeling of Eulerian currents induced by wind

Advances in Fluid Mechanics X 349

Mathematical modeling of Eulerian currents induced by wind and waves at the sea surface

M. Rahman1 & D. Bhatta2 1Faculty of Computer Science, Dalhousie University, Canada 2The University of Texas Pan-American, USA

Abstract

This paper deals with the study of mathematical modeling of Eulerian currents in ocean circulations. The wave-wave interaction of four progressive waves traveling with four wave numbers and four frequencies are elegantly described by Komen et al. (J. Phys. Oceanogra., 14 (1984), 1271–1285). Therefore, detailed investigations are avoided in this present paper. We shall rather devote our study to the description of the analytic solutions of the Eulerian currents present in the ocean circulation. This study contains the mathematical descriptions of nonlinear wave interactions, wind and wave induced surface currents, unsteady Eulerian currents in one-dimension, and steady two-dimensional Eulerian currents in ocean circulations. A variety of solutions that satisfy the governing equations with their initial and boundary conditions are obtained. A Laplace transform method in conjunction with the convolution concept is used as a solution technique and the accuracy of the solution is conﬁrmed by using the powerful separation of variables method. Some of the solutions are graphically illustrated in non-dimensional forms and the physical meaning is described. Keywords: Eulerian currents, Lagrangian currents, Ekman spirals, mathematical modeling, ocean circulations, surface currents, nonlinear waves, wave energy, wind stress, steady currents, unsteady currents, Laplace transforms, integral transforms, convolution, separation of variables method.

1 Introduction

The ocean current is deﬁned as a continuous, directed movement of ocean water generated by the forces acting upon this mean ﬂow, such as breaking waves, wind, Coriolis effect, temperature and salinity differences and tides caused by the

WIT Transactions on Engineering Sciences, Vol 82, © 2014 WIT Press www.witpress.com, ISSN 1743-3533 (on-line) doi:10.2495/AFM140301

350 Advances in Fluid Mechanics X

gravitational pull of the Moon and the Sun. The current’s direction and strength are inﬂuenced by the depth of the ocean, the shoreline conﬁgurations and interaction with other currents. A deep current is any ocean current at a depth of greater than 100 meters. Ocean currents can ﬂow for great distances, and together they create the great ﬂow of the global conveyor belt which plays a dominant part in determining the climate of many of the Earth’s regions. Perhaps the most striking example is the Gulf Stream, which makes northwest Europe much more temperate than any other region at the same latitude.

In oceanography and in ﬂuid dynamics in general, our observations can be made in two ways: Lagrangian measurements and Eulerian measurements. Lagrangian measurements involve following a parcel of ﬂuid as it moves. For example, we could measure temperatures from a weather balloon or from a free ﬂoating buoy. In other way to make the measurements is to have an observation site geographically ﬁxed. For example, we can measure temperature at a ﬁxed weather station or from an anchored buoy in the ocean. Measurements made in this manner are known as Eulerian measurements. These descriptions can very easily be extended to encompass the current measurements in ocean circulations. Eulerian currents play a very important role in the study of ice-ﬂoe drift in ocean circulations. So we shall concentrate our study in the mathematical modeling of Eulerian currents in ocean circulations.

Measurements of ocean current are collected using a variety of methods. One popular way to measure ocean currents is to determine the water’s velocity at one ﬁxed place in the ocean. This type of measurement is called Eulerian, in honor of the Swiss mathematician Leonhard Euler. This is typically accomplished using an electro-mechanical current meter. Surface ocean currents are generally winddriven and develop their typical clockwise spirals in the northern hemisphere and counter-clockwise rotation in the southern hemisphere because of the imposed wind stress. In wind-driven currents, the Ekman [2] spiral effect results in the currents ﬂowing at an angle to the driving wind. The areas of surface ocean currents move somewhat with the season; this is most notable in equatorial currents. Due to physical importance of Eulerian currents we present here their fundamental aspects using sophisticated mathematical models.

2 Mathematical formulation of wave-induced surface currents

In this section we shall ﬁrst formulate the equations of the Eulerian current in twodimensions. Given wind ﬁeld U10 at 10 meters reference height, the wave balance equation can be written as (see Longuet-Higgins [3])

dE(f, θ) = (Sin − Sds)(1 − fi) + Snl + Sice, (1) dt

where ddt =

∂ ∂t

+

Cg .∇

denotes total differentiation when traveling with the

group velocity. Here E(f, θ) is the two-dimensional wave spectrum which is a

function of frequency f , (cycles/sec, Hertz), and direction θ, time t and position x,

and where Cg is the group velocity, and ∇ is a gradient operator. Also we deﬁne

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that Sin is the input of energy due to wind, Sds is the energy dissipation due to white capping, Sn is the nonlinear transfer between spectral components due to wave-wave interactions, Sice is the change in energy due to wave interactions with ice ﬂoes, and fi is the fraction of the area of the ocean covered by ice.

The accepted formulation for wind input Sin, as parameterized in the WAM model of Hasselmann [4], Hasselmann [5] and Hasselmann et al. [6], suggests that

Sin should be represented by

Sin = βωE(f, θ),

(2)

where β is a non-dimensional function of sea state maturity and ω = 2πf

represents the angular (radial) frequency, which is related to the wave number k through the deep water dispersion relation ω2 = gk for deep ocean and ω2 = gk tanh kh for ﬁnite depth ocean. Description of all other parameters can

be found in the work of Rahman [7], and so will not be repeated here.

The associated Stokes drift is the mean velocity following a ﬂuid particle, and therefore, by deﬁnition, is a Lagrangian property. Let us consider that Us is the Lagrangian velocity of a particle at initial position (x, c, t = 0), and the Stokes

drift at time t is simply given by

Us(x, t) = 4π

f Ke2kcE(f, θ)df dθ

(3)

following Jenkins [8]. The vertical Lagrangian coordinate c corresponds to the usual vertical Eulerian coordinate z at the initial time t = 0. The quasi-Eulerian current UE satisﬁes the following partial differential equation as described by Jenkins [8]:

∂UE + f × UE = ∂

∂t

∂c

ν ∂UE ∂c

− f × Us

− 2π df f KSds2kN e2kcdθ,

(4)

where ice ﬂoes are not assumed to be present. The quasi-Eulerian current UE

can be thought of as being the Eulerian mean current Ue with reference to a

Lagrangian coordinate system and so UE = UL − Us = Ue, where UL is

the Lagrangian mean current. Other variables of (4) are ν, the eddy viscosity and

f = (0, 0, λ), the Coriolis acceleration |f | = 2Ω sin φ, where Ω is the Earth’s

angular velocity and φ is the latitude. The vector K is deﬁned as K = (cos( π2 −

θ

),

s

i

n

(

π 2

−

θ)),

which

is

related

to

wave

number

k

by

k

=

kK

=

k[sin θ, cos θ].

The integral representation of (4) represents the generation of UE from the waves

through the wave dispersion Sds. The coefﬁcient N represents the momentum

transfer from waves to current, and it can be assumed that N = 1. Finally, the

partial derivative term ∂∂c ν ∂∂UcE represents the vertical transport of momentum

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by viscous shear stress. The boundary condition at the sea surface is

ν ∂UE |c=0 = τ − 2π df f KSindθ,

(5)

∂c

ρw

where τ = |τ | = τx2 + τy2 = ρaCD|U120| is the wind stress on the water surface

, ρa is the air density, ρw is the water density, and CD is the drag coefﬁcient. In two dimensions, Ue = (u, v, 0), where u and v are the horizontal components of the Eulerian currents in x − y plane. Also we deﬁne Stokes drift components in two-dimensions as Us = (us, vs, 0).

We now avoid detailed description of the development of other terms leading to the ﬁnal equations of the Eulerian currents because of the page limitation. Interested reader is referred to the works of Hasselmann [5], Jenkins [8], Komen et al. [9] and Perrie and Hu [10]. We write the governing equations of the Eulerian currents explicitly with their boundary conditions and initial conditions as follows (considering usual vertical coordinate z):

∂u − λv = ∂ ν ∂u + λvs − Sdx

∂t

∂z ∂z

∂v + λu = ∂ ν ∂v − λus − Sdy

(6)

∂t

∂z ∂z

The boundary conditions can be written as: Surface boundary conditions:

at z = 0 : ν ∂u = τx − Sxw ∂z ρw

ν ∂v = τy − Syw.

(7)

∂z ρw

Bottom boundary conditions:

⎧

⎫

⎪⎨

⎪⎬

at z = ⎪⎩ −∞ (inﬁnite depth); u = 0 ⎪⎭ .

(8)

−h (ﬁnite depth); v = 0

The initial conditions are assumed to be:

at t = 0 : u = 0; v = 0.

(9)

Throughout this investigation, we shall assume that u and v are functions of the vertical coordinate z and the time t. And they do not depend upon the horizontal coordinates x and y. We shall start out our investigation with a very simple problem of Eulerian currents in ocean circulation.

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3 Unsteady Eulerian current in one-dimension

This section deals with the unsteady Eulerian current in one-dimension. We discuss here the simple analytical solutions with simple boundary conditions and initial condition for the cases of deep water, ﬁnite depth water and shallow water. The method of Laplace transforms (together with the separation of variables method) are employed in obtaining the solutions. The solutions are described below.

Case I: Deep ocean

We shall consider ﬁrst the deep ocean Eulerian current in one dimension. In this case, the Coriolis force does not play any part. We assume that the wave dispersion term is negligible. We also assume that the wave input term is negligible and the eddy viscosity ν is constant. Equation (6) with its boundary conditions (7) and (8), and initial condition (9) can be written is a simple form as follows:

∂u ∂2u

∂t = ν ∂z2

(10)

z = 0 : ∂u = τx

(11)

∂z νρw

z = −∞ : u = 0

(12)

and t = 0 : u = 0.

(13)

Solution

The Laplace transform method will be suitable for this mathematical model. We deﬁne the Laplace transform of u(z, t) as L(u(z, t)) = 0∞ u(z, t)e−stdt such that L{ ∂∂ut } = sL{u(z, t)}. Equations (10) through (12) can be transformed as follows:

d2

s

dz2 (L{u}) − ( ν )(L{u}) = 0

(14)

z = 0 : d (L{u}) = τx ( 1 )

(15)

dz

νρw s

z = −∞ : L{u} = 0.

(16)

Using the transformed boundary conditions we obtain the solution as follows:

L{u} =

τ√x ρw ν

exp( √s/νz) ss

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(17)

354 Advances in Fluid Mechanics X

The Laplace inverse formula gives the solution as follows (see Abramowitz and Stegun [11]):

u(z, t) = τ√x

t

z2

z

z

2 exp(− ) + ( √ )erf c(− √ ) . (18)

ρw ν

π

4νt

ν

2 νt

non-dimensional U, i.e., U( , t’ )

= 0.0 = -0.24

deep water

= -0.64

1.82

= -2.0

1.56

1.3

1.04

0.78

0.52

0.26

0

0

2

4

6

8

10

t’

Figure 1: Eulerian currents in deep water at various depth.

deep water

non-dimensional U( , t’)

2

1.6

1.2

0.8

0.4

0

0

-0.5

-1

z

-1.5

2 1.6 1.2 0.8 0.4 0 10 6 8time 4 2 -2

Figure 2: Eulerian currents in deep water as a function of z and t.

If we assume the non-dimensional variables as η = 2√zνt which happens to be the similarity variable, and the non-dimensional time as t = 4νt/L2 where L is

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some typical length (may be wavelength), then we can write the equation (18) in

non-dimensional form as follows:

√1 U (η, t ) = t [ √

exp(−η2) + η × erf c(−η)].

(19)

π

where

U (η, t ) = u(z, t)/ Lτx , ρw ν

and the range of η is −∞ < η ≤ 0 and t > 0. At the sea surface z = 0 (i.e. in non-dimensional variable η = 0), the Eulerian

current is given by

u(0, t) = 2τx

t,

(20)

ρw

πν

and in non-dimensional form Eqn(20) simply is

U (0, t ) = t .

(21)

π

Case II: Finite depth ocean

In this case the bottom boundary condition at z = −h where h is the depth of the ocean is given by (8), and the Laplace transform solution of (14) can be assumed in the following manner:

L{u} = A cosh( s z) + B sinh( s z).

(22)

ν

ν

Using the sea surface and sea bottom boundary conditions, (22) can be written as

L{u} =

τ√x

sinh( √

νs (z + h)) .

(23)

ρw ν

s s cosh(

s ν

h

)

The Laplace inverse (see Abramowitz and Stegun [11]) in conjunction with

convolution integral, we obtain the solution in non-dimensional form with the

independent variables z

=

z h

and

t

=

νt h2

as

U (z , t ) = u(z, t) Λh

8∞

1

(2n − 1)πz

= (1 + z ) − ( π2 ) (2n − 1)2 cos( 2 )

n=1

× exp[− (2n − 1)2π2t ]. (24) 4

where Λ = ρτwxν . This solution is valid in the range (−1 ≤ z ≤ 0) and t > 0 and can be veriﬁed easily by using the separation of variables method. The solution at

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356 Advances in Fluid Mechanics X

the surface z = 0 is

8∞

1

(2n − 1)2π2t

U (0, t ) = [1 − ( π2 ) (2n − 1)2 exp(−

4

)]. (25)

n=1

non-dimensional U

z = 0.0 z = -0.2 z = -0.5 z = -0.9

finite depth

1

0.75

0.5

0.25

0

0 0.3 0.6 0.9 1.2 1.5 1.8

t

Figure 3: Eulerian currents in ﬁnite depth water at various depth.

finite depth

non-dimensional U

1

0.75

0.5

0.25

0

0 -0.2

-0.4

z -0.6

-0.8 -1 0

0.5 1

1.5 2 time

1 0.75 0.5 0.25 0 2.5 3

Figure 4: Eulerian currents in ﬁnite depth water as a function of z and t.

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non-dimensional U

z = 0.0 shallow water

z = -0.2

z = -0.5

1

z = -0.9

0.8

0.6

0.4

0.2

0

0.5

1

1.5

2

2.5

t

Figure 5: Eulerian currents in shallow water at various depth.

shallow water

non-dimensionl U

1

0.75

0.5

0.25

0 0

-0.2 -0.4

z -0.6

-0.8 -1

0.5 1

1 0.75 0.5 0.25 0 2 2.5 3 1.5 time

Figure 6: Eulerian currents in shallow water as a function of z and t.

Case III: Shallow water ocean

In this case the Laplace transform solution should be modiﬁed by taking into

consideration that h → 0 or (z + h) → 0. Therefore, the solution simply is (to the

order of O(h3)):

2νt

u(z, t) = Λ(z + h)[1 − exp(− h2 )]

(26)

and at the sea surface z = 0, this yields

2νt

u(0, t) = Λh[1 − exp(− h2 )].

(27)

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In non-dimensional forms Eqns. (26) and (27) which are valid in the range (−1 ≤ z ≤ 0) and t > 0, can simply be written, respectively, as

U (z , t ) = (1 + z )[1 − exp(−2t )]

(28)

U (0, t ) = [1 − exp(−2t )]

(29)

non-dimensional U( 0, t’ )

z=0

1.25

1

Deep Water

0.75

Finite Depth

Shallow Water

0.5

0.25

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6

time

Figure 7: Comparison of Eulerian currents in deep water, ﬁnite depth and shallow water at the sea surface.

4 Results and conclusions

This is a classical and fundamental theoretical work. The analytical solutions are presented in a rigorous manner to get the elementary understanding of the Eulerian currents in ocean circulations. Mathematical formulations and their solutions are evidently sophisticated. Some graphical solutions are displayed to have a little idea about the Eulerian currents in ocean circulations. The analytical solutions of the Eulerian currents for deep water and ﬁnite depth ocean circulations are extremely sophisticated. They are obtained by using very high level mathematical techniques such the Laplace transform method and the separation of variables method. These solutions perfectly satisﬁed the given governing equation, the boundary conditions and the initial condition. Graphical illustrations in non-dimensional forms are presented in Figures 1–7 to see the behavior of the solutions. Figures 1 and 2 present the solutions for the deep water case, Figures 3 and 4 depict the solutions in ﬁnite depth case and Figures 5 and 6 deal with the shallow water case. Comparisons between the Eulerian currents at the sea surface z = 0 are presented. Figure 7 displays the comparison between the surface currents at the deep water, ﬁnite depth water and the shallow water depth. It can be easily seen that they

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Mathematical modeling of Eulerian currents induced by wind and waves at the sea surface

M. Rahman1 & D. Bhatta2 1Faculty of Computer Science, Dalhousie University, Canada 2The University of Texas Pan-American, USA

Abstract

This paper deals with the study of mathematical modeling of Eulerian currents in ocean circulations. The wave-wave interaction of four progressive waves traveling with four wave numbers and four frequencies are elegantly described by Komen et al. (J. Phys. Oceanogra., 14 (1984), 1271–1285). Therefore, detailed investigations are avoided in this present paper. We shall rather devote our study to the description of the analytic solutions of the Eulerian currents present in the ocean circulation. This study contains the mathematical descriptions of nonlinear wave interactions, wind and wave induced surface currents, unsteady Eulerian currents in one-dimension, and steady two-dimensional Eulerian currents in ocean circulations. A variety of solutions that satisfy the governing equations with their initial and boundary conditions are obtained. A Laplace transform method in conjunction with the convolution concept is used as a solution technique and the accuracy of the solution is conﬁrmed by using the powerful separation of variables method. Some of the solutions are graphically illustrated in non-dimensional forms and the physical meaning is described. Keywords: Eulerian currents, Lagrangian currents, Ekman spirals, mathematical modeling, ocean circulations, surface currents, nonlinear waves, wave energy, wind stress, steady currents, unsteady currents, Laplace transforms, integral transforms, convolution, separation of variables method.

1 Introduction

The ocean current is deﬁned as a continuous, directed movement of ocean water generated by the forces acting upon this mean ﬂow, such as breaking waves, wind, Coriolis effect, temperature and salinity differences and tides caused by the

WIT Transactions on Engineering Sciences, Vol 82, © 2014 WIT Press www.witpress.com, ISSN 1743-3533 (on-line) doi:10.2495/AFM140301

350 Advances in Fluid Mechanics X

gravitational pull of the Moon and the Sun. The current’s direction and strength are inﬂuenced by the depth of the ocean, the shoreline conﬁgurations and interaction with other currents. A deep current is any ocean current at a depth of greater than 100 meters. Ocean currents can ﬂow for great distances, and together they create the great ﬂow of the global conveyor belt which plays a dominant part in determining the climate of many of the Earth’s regions. Perhaps the most striking example is the Gulf Stream, which makes northwest Europe much more temperate than any other region at the same latitude.

In oceanography and in ﬂuid dynamics in general, our observations can be made in two ways: Lagrangian measurements and Eulerian measurements. Lagrangian measurements involve following a parcel of ﬂuid as it moves. For example, we could measure temperatures from a weather balloon or from a free ﬂoating buoy. In other way to make the measurements is to have an observation site geographically ﬁxed. For example, we can measure temperature at a ﬁxed weather station or from an anchored buoy in the ocean. Measurements made in this manner are known as Eulerian measurements. These descriptions can very easily be extended to encompass the current measurements in ocean circulations. Eulerian currents play a very important role in the study of ice-ﬂoe drift in ocean circulations. So we shall concentrate our study in the mathematical modeling of Eulerian currents in ocean circulations.

Measurements of ocean current are collected using a variety of methods. One popular way to measure ocean currents is to determine the water’s velocity at one ﬁxed place in the ocean. This type of measurement is called Eulerian, in honor of the Swiss mathematician Leonhard Euler. This is typically accomplished using an electro-mechanical current meter. Surface ocean currents are generally winddriven and develop their typical clockwise spirals in the northern hemisphere and counter-clockwise rotation in the southern hemisphere because of the imposed wind stress. In wind-driven currents, the Ekman [2] spiral effect results in the currents ﬂowing at an angle to the driving wind. The areas of surface ocean currents move somewhat with the season; this is most notable in equatorial currents. Due to physical importance of Eulerian currents we present here their fundamental aspects using sophisticated mathematical models.

2 Mathematical formulation of wave-induced surface currents

In this section we shall ﬁrst formulate the equations of the Eulerian current in twodimensions. Given wind ﬁeld U10 at 10 meters reference height, the wave balance equation can be written as (see Longuet-Higgins [3])

dE(f, θ) = (Sin − Sds)(1 − fi) + Snl + Sice, (1) dt

where ddt =

∂ ∂t

+

Cg .∇

denotes total differentiation when traveling with the

group velocity. Here E(f, θ) is the two-dimensional wave spectrum which is a

function of frequency f , (cycles/sec, Hertz), and direction θ, time t and position x,

and where Cg is the group velocity, and ∇ is a gradient operator. Also we deﬁne

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that Sin is the input of energy due to wind, Sds is the energy dissipation due to white capping, Sn is the nonlinear transfer between spectral components due to wave-wave interactions, Sice is the change in energy due to wave interactions with ice ﬂoes, and fi is the fraction of the area of the ocean covered by ice.

The accepted formulation for wind input Sin, as parameterized in the WAM model of Hasselmann [4], Hasselmann [5] and Hasselmann et al. [6], suggests that

Sin should be represented by

Sin = βωE(f, θ),

(2)

where β is a non-dimensional function of sea state maturity and ω = 2πf

represents the angular (radial) frequency, which is related to the wave number k through the deep water dispersion relation ω2 = gk for deep ocean and ω2 = gk tanh kh for ﬁnite depth ocean. Description of all other parameters can

be found in the work of Rahman [7], and so will not be repeated here.

The associated Stokes drift is the mean velocity following a ﬂuid particle, and therefore, by deﬁnition, is a Lagrangian property. Let us consider that Us is the Lagrangian velocity of a particle at initial position (x, c, t = 0), and the Stokes

drift at time t is simply given by

Us(x, t) = 4π

f Ke2kcE(f, θ)df dθ

(3)

following Jenkins [8]. The vertical Lagrangian coordinate c corresponds to the usual vertical Eulerian coordinate z at the initial time t = 0. The quasi-Eulerian current UE satisﬁes the following partial differential equation as described by Jenkins [8]:

∂UE + f × UE = ∂

∂t

∂c

ν ∂UE ∂c

− f × Us

− 2π df f KSds2kN e2kcdθ,

(4)

where ice ﬂoes are not assumed to be present. The quasi-Eulerian current UE

can be thought of as being the Eulerian mean current Ue with reference to a

Lagrangian coordinate system and so UE = UL − Us = Ue, where UL is

the Lagrangian mean current. Other variables of (4) are ν, the eddy viscosity and

f = (0, 0, λ), the Coriolis acceleration |f | = 2Ω sin φ, where Ω is the Earth’s

angular velocity and φ is the latitude. The vector K is deﬁned as K = (cos( π2 −

θ

),

s

i

n

(

π 2

−

θ)),

which

is

related

to

wave

number

k

by

k

=

kK

=

k[sin θ, cos θ].

The integral representation of (4) represents the generation of UE from the waves

through the wave dispersion Sds. The coefﬁcient N represents the momentum

transfer from waves to current, and it can be assumed that N = 1. Finally, the

partial derivative term ∂∂c ν ∂∂UcE represents the vertical transport of momentum

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352 Advances in Fluid Mechanics X

by viscous shear stress. The boundary condition at the sea surface is

ν ∂UE |c=0 = τ − 2π df f KSindθ,

(5)

∂c

ρw

where τ = |τ | = τx2 + τy2 = ρaCD|U120| is the wind stress on the water surface

, ρa is the air density, ρw is the water density, and CD is the drag coefﬁcient. In two dimensions, Ue = (u, v, 0), where u and v are the horizontal components of the Eulerian currents in x − y plane. Also we deﬁne Stokes drift components in two-dimensions as Us = (us, vs, 0).

We now avoid detailed description of the development of other terms leading to the ﬁnal equations of the Eulerian currents because of the page limitation. Interested reader is referred to the works of Hasselmann [5], Jenkins [8], Komen et al. [9] and Perrie and Hu [10]. We write the governing equations of the Eulerian currents explicitly with their boundary conditions and initial conditions as follows (considering usual vertical coordinate z):

∂u − λv = ∂ ν ∂u + λvs − Sdx

∂t

∂z ∂z

∂v + λu = ∂ ν ∂v − λus − Sdy

(6)

∂t

∂z ∂z

The boundary conditions can be written as: Surface boundary conditions:

at z = 0 : ν ∂u = τx − Sxw ∂z ρw

ν ∂v = τy − Syw.

(7)

∂z ρw

Bottom boundary conditions:

⎧

⎫

⎪⎨

⎪⎬

at z = ⎪⎩ −∞ (inﬁnite depth); u = 0 ⎪⎭ .

(8)

−h (ﬁnite depth); v = 0

The initial conditions are assumed to be:

at t = 0 : u = 0; v = 0.

(9)

Throughout this investigation, we shall assume that u and v are functions of the vertical coordinate z and the time t. And they do not depend upon the horizontal coordinates x and y. We shall start out our investigation with a very simple problem of Eulerian currents in ocean circulation.

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Advances in Fluid Mechanics X 353

3 Unsteady Eulerian current in one-dimension

This section deals with the unsteady Eulerian current in one-dimension. We discuss here the simple analytical solutions with simple boundary conditions and initial condition for the cases of deep water, ﬁnite depth water and shallow water. The method of Laplace transforms (together with the separation of variables method) are employed in obtaining the solutions. The solutions are described below.

Case I: Deep ocean

We shall consider ﬁrst the deep ocean Eulerian current in one dimension. In this case, the Coriolis force does not play any part. We assume that the wave dispersion term is negligible. We also assume that the wave input term is negligible and the eddy viscosity ν is constant. Equation (6) with its boundary conditions (7) and (8), and initial condition (9) can be written is a simple form as follows:

∂u ∂2u

∂t = ν ∂z2

(10)

z = 0 : ∂u = τx

(11)

∂z νρw

z = −∞ : u = 0

(12)

and t = 0 : u = 0.

(13)

Solution

The Laplace transform method will be suitable for this mathematical model. We deﬁne the Laplace transform of u(z, t) as L(u(z, t)) = 0∞ u(z, t)e−stdt such that L{ ∂∂ut } = sL{u(z, t)}. Equations (10) through (12) can be transformed as follows:

d2

s

dz2 (L{u}) − ( ν )(L{u}) = 0

(14)

z = 0 : d (L{u}) = τx ( 1 )

(15)

dz

νρw s

z = −∞ : L{u} = 0.

(16)

Using the transformed boundary conditions we obtain the solution as follows:

L{u} =

τ√x ρw ν

exp( √s/νz) ss

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(17)

354 Advances in Fluid Mechanics X

The Laplace inverse formula gives the solution as follows (see Abramowitz and Stegun [11]):

u(z, t) = τ√x

t

z2

z

z

2 exp(− ) + ( √ )erf c(− √ ) . (18)

ρw ν

π

4νt

ν

2 νt

non-dimensional U, i.e., U( , t’ )

= 0.0 = -0.24

deep water

= -0.64

1.82

= -2.0

1.56

1.3

1.04

0.78

0.52

0.26

0

0

2

4

6

8

10

t’

Figure 1: Eulerian currents in deep water at various depth.

deep water

non-dimensional U( , t’)

2

1.6

1.2

0.8

0.4

0

0

-0.5

-1

z

-1.5

2 1.6 1.2 0.8 0.4 0 10 6 8time 4 2 -2

Figure 2: Eulerian currents in deep water as a function of z and t.

If we assume the non-dimensional variables as η = 2√zνt which happens to be the similarity variable, and the non-dimensional time as t = 4νt/L2 where L is

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Advances in Fluid Mechanics X 355

some typical length (may be wavelength), then we can write the equation (18) in

non-dimensional form as follows:

√1 U (η, t ) = t [ √

exp(−η2) + η × erf c(−η)].

(19)

π

where

U (η, t ) = u(z, t)/ Lτx , ρw ν

and the range of η is −∞ < η ≤ 0 and t > 0. At the sea surface z = 0 (i.e. in non-dimensional variable η = 0), the Eulerian

current is given by

u(0, t) = 2τx

t,

(20)

ρw

πν

and in non-dimensional form Eqn(20) simply is

U (0, t ) = t .

(21)

π

Case II: Finite depth ocean

In this case the bottom boundary condition at z = −h where h is the depth of the ocean is given by (8), and the Laplace transform solution of (14) can be assumed in the following manner:

L{u} = A cosh( s z) + B sinh( s z).

(22)

ν

ν

Using the sea surface and sea bottom boundary conditions, (22) can be written as

L{u} =

τ√x

sinh( √

νs (z + h)) .

(23)

ρw ν

s s cosh(

s ν

h

)

The Laplace inverse (see Abramowitz and Stegun [11]) in conjunction with

convolution integral, we obtain the solution in non-dimensional form with the

independent variables z

=

z h

and

t

=

νt h2

as

U (z , t ) = u(z, t) Λh

8∞

1

(2n − 1)πz

= (1 + z ) − ( π2 ) (2n − 1)2 cos( 2 )

n=1

× exp[− (2n − 1)2π2t ]. (24) 4

where Λ = ρτwxν . This solution is valid in the range (−1 ≤ z ≤ 0) and t > 0 and can be veriﬁed easily by using the separation of variables method. The solution at

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356 Advances in Fluid Mechanics X

the surface z = 0 is

8∞

1

(2n − 1)2π2t

U (0, t ) = [1 − ( π2 ) (2n − 1)2 exp(−

4

)]. (25)

n=1

non-dimensional U

z = 0.0 z = -0.2 z = -0.5 z = -0.9

finite depth

1

0.75

0.5

0.25

0

0 0.3 0.6 0.9 1.2 1.5 1.8

t

Figure 3: Eulerian currents in ﬁnite depth water at various depth.

finite depth

non-dimensional U

1

0.75

0.5

0.25

0

0 -0.2

-0.4

z -0.6

-0.8 -1 0

0.5 1

1.5 2 time

1 0.75 0.5 0.25 0 2.5 3

Figure 4: Eulerian currents in ﬁnite depth water as a function of z and t.

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Advances in Fluid Mechanics X 357

non-dimensional U

z = 0.0 shallow water

z = -0.2

z = -0.5

1

z = -0.9

0.8

0.6

0.4

0.2

0

0.5

1

1.5

2

2.5

t

Figure 5: Eulerian currents in shallow water at various depth.

shallow water

non-dimensionl U

1

0.75

0.5

0.25

0 0

-0.2 -0.4

z -0.6

-0.8 -1

0.5 1

1 0.75 0.5 0.25 0 2 2.5 3 1.5 time

Figure 6: Eulerian currents in shallow water as a function of z and t.

Case III: Shallow water ocean

In this case the Laplace transform solution should be modiﬁed by taking into

consideration that h → 0 or (z + h) → 0. Therefore, the solution simply is (to the

order of O(h3)):

2νt

u(z, t) = Λ(z + h)[1 − exp(− h2 )]

(26)

and at the sea surface z = 0, this yields

2νt

u(0, t) = Λh[1 − exp(− h2 )].

(27)

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358 Advances in Fluid Mechanics X

In non-dimensional forms Eqns. (26) and (27) which are valid in the range (−1 ≤ z ≤ 0) and t > 0, can simply be written, respectively, as

U (z , t ) = (1 + z )[1 − exp(−2t )]

(28)

U (0, t ) = [1 − exp(−2t )]

(29)

non-dimensional U( 0, t’ )

z=0

1.25

1

Deep Water

0.75

Finite Depth

Shallow Water

0.5

0.25

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6

time

Figure 7: Comparison of Eulerian currents in deep water, ﬁnite depth and shallow water at the sea surface.

4 Results and conclusions

This is a classical and fundamental theoretical work. The analytical solutions are presented in a rigorous manner to get the elementary understanding of the Eulerian currents in ocean circulations. Mathematical formulations and their solutions are evidently sophisticated. Some graphical solutions are displayed to have a little idea about the Eulerian currents in ocean circulations. The analytical solutions of the Eulerian currents for deep water and ﬁnite depth ocean circulations are extremely sophisticated. They are obtained by using very high level mathematical techniques such the Laplace transform method and the separation of variables method. These solutions perfectly satisﬁed the given governing equation, the boundary conditions and the initial condition. Graphical illustrations in non-dimensional forms are presented in Figures 1–7 to see the behavior of the solutions. Figures 1 and 2 present the solutions for the deep water case, Figures 3 and 4 depict the solutions in ﬁnite depth case and Figures 5 and 6 deal with the shallow water case. Comparisons between the Eulerian currents at the sea surface z = 0 are presented. Figure 7 displays the comparison between the surface currents at the deep water, ﬁnite depth water and the shallow water depth. It can be easily seen that they

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