CHAPTER 4. Stationary currents - ttu.eemsi.ttu.ee/~elken/DO4.pdfDynamical Oceanography Lecture notes...

Dynamical Oceanography

Lecture notes by Jüri Elken Page 30

CHAPTER 4. Stationary currents Contrary to the wave motions where oscillating fluid particles transfer motion phase and radiate wave energy, currents transport water but do not radiate energy apart from the fluid particles. Wave processes are fully absent at idealized stationary currents. 4.1. Streamfunction Considering currents and long waves, vertical velocity often does not depend on depth

0=∂∂

z

w. Such flow is horizontally non-divergent and the continuity equation (2.24) is written

as

0=∂∂

+∂∂

y

v

x

u.

(4.1)

Two-dimensional non-divergence (4.1) is satisfied if velocity components are expressed by the streamfunction ( )yx,ψψ = [m2/s] in a form

xv

yu

∂∂

=∂∂

−=ψψ

; .

(4.2)

Streamfunction can be determined with an accuracy of a constant addend. Introducing the two-dimensional gradient operator

jy

ixyxh

rr

∂∂

+∂∂

=

∂∂

∂∂

=∇ , ,

(4.3)

horizontal velocity vector ( )vuvh ,=

r is perpendicular to the streamfunction gradient since

0=∂∂

∂∂

+∂∂

∂∂

−=∇⋅yxxy

v hh

ψψψψψ

r .

(4.4)

Horizontal velocity vector can be expressed as a vector product of the vertical unit vector kr

and the streamfunction gradient

ψhh kv ∇×=rr

. (4.5)

Let us show that the mass transport ABQ [m2/s] of a unit thickness layer between the points A and B is equal to the streamfunction difference at these points. Transport through the arbitrary section element ds is dsnvh

rr⋅ where n

r is a ds normal vector. The full transport is

∫ ⋅=B

A

hAB dsnvQrr

. If angle between the tangent of ds and x -axis is α then αsindsdx = ,

αcosdsdy = and normal vector is ( ) ( )( ) ( )αα cos,sin,cos,,cos −== ynxnnr

. Line integral is



( ) .cossin BA

B

A

B

A

B

A

B

A

B

A

hAB ddxx

dyy

dxvdyudsvudsnvQ ψψψψψ

αα −=−=∂∂

−∂∂

−=−=−=⋅= ∫∫∫ ∫∫rr

4.2. Geostrophic motion a) relations for pressure gradients In the equations of motion we assume:

1) flow is stationary 0=∂∂t

;

2) turbulent effects are negligible outside the boundary layers, therefore µ = ν = 0 ;

3) advection is small (Rossby number 1<<=fL

URo ), therefore 0=∇⋅ vv

rr.

Equations for the x - and y -components of momentum (2.20) and (2.21) will take the form

x

pfv

∂∂

−=−ρ1

,

(4.6)

y

pfu

∂∂

−=ρ1

,

(4.7)

where geostrophic balance holds between the Coriolis’ force and pressure gradient force. Using the f - plane approximation const=f and assuming 0ρρ = at the horizontal pressure

gradient terms, we obtain geostrophic streamfunction

pfg

0

1

ρψ = .

(4.8)

In the Northern Hemisphere 0>f and geostrophic flow is directed along the streamlines

from the pressure gradient force pm hp −∇=r

to the right (Fig. 4.1), in the Southern

Hemisphere ( 0<f ) to the left.

p

p

2

1

p >p2 1

x

ym

v

p

H

mp v

L

mpv

Figure 4.1. In the Northern Hemisphere, geostrophic flow along the isobars is directed to the right from the pressure gradient force. In the anticyclone (H) the rotation is clockwise, in the cyclone (L) counterclockwise.


Lecture notes by Jüri Elken 2

Geostrophic equations (4.6) and (4.7) can be written in a vector form

pvkf hh ∇−=×ρ1rr

,

(4.9)

where velocity vector is derived as

pkf

v hh ∇×=rr

0

1

ρ .

(4.10)

Atmospheric pressure is directly measured at meteorological stations. The data can be converted to the equal heights and directly used in the relations given above. b) thermal wind relations In the ocean and marginal seas, absolute pressure differences cannot be directly measured since it would require accuracy better than 1 cm for determination of sea level and absolute depth. Instead, relative geostrophic currents are determined from the density distribution based on the equation of state ( )pST ,,ρρ = and the measured profiles of temperature and salinity. For the derivation of relative velocity, find from the geostrophic equations (4.6), (4.7) the velocity components

y

p

fu

∂∂

−=0

1

ρ ,

(4.11)

x

p

fv

∂∂

=0

1

ρ .

(4.12)

Differentiating (4.11), (4.12) by z , we replace the pressure terms from the hydrostatic equation (1.7)

ρgz

p−=

∂∂

⇒ x

gxz

p

∂∂

−=∂∂

∂ ρ2

, y

gyz

p

∂∂

−=∂∂

∂ ρ2

(4.13)

and obtain

yf

g

z

u

∂∂

=∂∂ ρ

ρ0

,

(4.14)

xf

g

z

v

∂∂

−=∂∂ ρ

ρ0

,

(4.15)

or in a vector form



ρρ h

h kf

g

z

v∇×=

∂

∂ rr

0

.

(4.16)

Relations (4.14), (4.15) or (4.16) are thermal wind relations containing both the geostrophic and hydrostatic balance. The name originates from atmospheric physics where meridional air temperature gradient between the pole and the equator causes zonal flow. c) dynamic method for ocean currents Motion on a level 2z relative to the level 1z is determined by the integrals

( ) ( ) ∫∂∂

=−2

1012

z

z

dzyf

gzuzu ρ

ρ ,

(4.17)

( ) ( ) ∫∂∂

−=−2

1012

z

z

dzxf

gzvzv ρ

ρ .

(4.18)

Here 2z lies generally higher than 1z since vertical axis is directed upward from the

undisturbed sea surface. At some deep ocean level RR hzz −==1 (level of no motion) we

may assume that velocities are small ( ) ( ) 0≈≈ RR zvzu . Then we may define a function

( ) ∫=Φ=Φz

zR

dzf

gzyx ρ

ρ0

,, .

(4.19)

Due to (4.17), (4.18) this is a geostrophic streamfunction

yug ∂

Φ∂−= ,

xvg ∂

Φ∂= .

(4.20)

Calculation of currents by (4.19), (4.20) using available density distribution is called dynamic method. In practical use, first the dynamic height is calculated

( ) ( ) ( )( )∫ −=z

z

R

R

dzSTSTzzD 0,,0,,1

, 000

ρρρ

,

(4.21)

where ( )0,,STρ is density calculated on the basis of observed temperature and salinity

profiles taking 0=p , and ( ) const0,, 00 =STρ reference density for the region of interest.

When using pressure as a vertical coordinate (CTD probes measure directly pressure, not depth) then equivalent formula for dynamic height can be used

( ) ∫=Rp

p

R dpg

ppD α1

, ,

(4.22)



where ( ) ( )0,,

1

0,,

1

00 STST ρρα −= is a specific volume anomaly. Due to the hydrostatic

balance, (4.22) is a distance between the isobars p and Rp . In the ocean usually 0T = 150C, 0S = 35 PSU are taken that yield ( )0,35,15ρ = 1025.97 kg/m3.

Historically the unit for dynamic height has been called “dynamic meter” since the formulae have been written slightly in a different form. Spatial distribution of dynamic heights is called dynamic topography. From the dynamic topography, geostrophic streamfunction is found by (4.19) and further the geostrophic currents are found by (4.20). Most of the ocean circulation schemes are obtained by the dynamic method. Note that f -plane approximation ( const=f ) has been assumed above. For the larger ocean regions where f varies with latitude, special treatment for the streamfunction is needed, especially when calculating the mass transports from the geostrophic currents. Zonal flow is determined from the dynamic height as

y

fD

f

g

y

D

f

gD

f

g

yug ∂

∂+

∂∂

−=

∂∂

−=2

.

(4.23)

We see that artificial currents may appear in (4.23) when density/pressure gradients are

missing 0=∂∂

y

D. This problem can be overcome by introducing ( )ybDD +=* , where

correction term ( )yb is found by minimizing the errors (second term on the right side of 4.23) over the area of interest. Determination of the level of no motion is the most problematic item in the dynamic method. In the open ocean usually =Rh 1500 m is taken but in several oceanic regions the velocities on that level can be quite remarkable. Let us show that uncertainty of the zero level originates from the unknown sea level elevation ξ relative to undisturbed sea surface 0=z that cannot be practically measured, except for some strong oceanic currents using satellite altimetry. Integrating the hydrostatic equation (1.7) from the depth z to the free surface ξ we obtain

∫∫ −=∂∂ ξξ

ρzz

dzgdzz

p or ∫∫ −−=−

0

0 z

z dzgdzgpp ρρξ

ξ where atmospheric pressure is

ξppa = . Density can be taken constant in the thin surface layer therefore ξρρξ

0

0

gdzg =∫ .

Absolute pressure zpp = at depth z is obtained as

dzggppz

a ∫++=0

0 ρξρ .

(4.24)



Expression (4.24) can be only theoretically used for calculating absolute currents by (4.6) and (4.7). Pressure changes due to sea level and density are of the same order and quite often smaller than 0.1 dbar, that is less than 10 cm for the sea level deviation. Due to the dynamical features of current formation, including the geostrophic adjustment, the sea level tends to take opposite deviation to the deviation of upper isopycnal surfaces. Then the surface pressure gradient is formed by the free surface deviation. By depth this pressure gradient is reduced since the density gradient is oriented against the sea level gradient. In a cyclonic eddy (Fig. 4.2) the sea level is lowered in the center, corresponding to the lower pressure as compared to the eddy periphery. In the water column, the isopycnals are lifted or the water density is increased. Pressure gradient is reduced by depth reaching const=p at the

level of no motion Rz . In an anticyclonic eddy, the sea level is lifted and isopycnals are lowered in the center.

z0

zR

ρ

ξ

Figure 4.2. Deviations of sea level ξ and isopycnals ρ in the cyclonic (left) and anticyclonic (right) eddy. Pressure gradients and currents are absent on the level of no motion

Rz . 4.3. Drift currents, Ekman spiral a) general solution In the boundary layers, turbulent friction plays an important role. Consider frictional effects in a steady flow in the case when horizontal flow gradients can be neglected. In the equations of motion we assume:

1) temporal and horizontal changes are negligibly small 0=∂∂

=∂∂

=∂∂

y

v

x

v

t

vrrr

; then in case of

constant depth nstco=H also 0=w due to the continuity equation; 2) horizontal pressure gradient is constant const=∇ ph and related to the geostrophic

velocity pvkf hg ∇−=×0

1

ρrr

;

3) density and vertical viscosity coefficient are const0 == ρρ , const=ν .



At these assumptions, horizontal momentum equations (2.20), (2.21) are reduced to the balance between the Coriolis force, pressure gradient force and turbulent friction force

2

2

0

1

z

u

x

pfv

∂

∂+

∂∂

−=− νρ

,

(4.25)

2

2

0

1

z

v

y

pfu

∂

∂+

∂∂

−= νρ

.

(4.26)

Subtracting the geostrophic velocity

x

pfvg ∂

∂−=−

0

1

ρ ,

(4.27)

y

pfug ∂

∂−=

0

1

ρ

(4.28)

we obtain

( ) 02

2

=−+∂

∂gvvf

z

uν ,

(4.29)

( ) 02

2

=−−∂

∂guuf

z

vν .

(4.30)

Multiplying (4.30) by 1−=i and adding to (4.29), we obtain an equation for the complex velocity ivu +=χ

gififz

χχχ

ν −=−∂

∂2

2

,

(4.31)

where ggg ivu +=χ . Linear ordinary differential equation of the second order (4.31) has

characteristic equation 02 =− λr where ( )ifi

f+== 1

2ννλ . General solution of (4.31)

is

( ) ( )[ ] ( )[ ] gg ivuziBziAz +++−++= γγχ 1exp1exp , (4.32)

where

νγ

2

f= .

(4.33)



Here 21 aiaA += and 21 bibB += are complex constants that have to be determined from the boundary conditions. Using ααα sincosexp ii += , the solution (4.32) can be written for the velocity components ( ) ( ) ( ) g

zz uzbzbezazaezu +++−= − γγγγ γγ sincossincos 2121 , (4.34)

( ) ( ) ( ) g

zz vzbzbezazaezv +−++= − γγγγ γγ sincossincos 1212 . (4.35)

b) surface Ekman layer

Surface layer motion is forced by the wind stress ( )yx τττρτ

,0

*

==r

r

[m2/s2] that is balanced by

the tangential stress due to the viscous friction appearing in the shear flow

∂∂

∂∂

z

v

z

uνν , .

Here τρτrr

0* = is tangential stress of wind per unit surface (N/m2) that depends on the wind

vector 10Ur

(10 m above the sea surface) approximately by the quadratic formula

101010* UUc a

rrrρτ = , (4.36)

where 2.1≈aρ kg/m3 is air density and 10c is a drag coefficient which numerical values lie

in a quite wide range, 3410 105103 −− ⋅÷⋅=c .

With 0=z on the sea surface, the boundary conditions are

xz

uτν =

∂∂

, yz

vτν =

∂∂

at 0=z .

(4.37)

If geostrophic currents are missing 0== gg vu then in the ocean of infinite depth

0=u , 0=v if −∞→z . (4.38)

By the condition (4.38) the coefficients of the general solution (4.34), (4.35) are 021 == bb .

When the wind is directed along y -axis ( 0=xτ ) then 21 aa = and γν

τ

21ya = . The drift

current is given by the specific solution

( ) ( )zzzu zy γγγν

τ γ sincose2

−= ,

(4.39)

( ) ( )zzzv zy γγγν

τ γ sincose2

+= ,

(4.40)



that represents the Ekman spiral (Fig. 4.3). Speed of the drift current decays exponentially with depth whereas on the Ekman depth

fhE

νπ

γπ 2

==

(4.41)

the value is only 4.3% of the surface speed. At the same time, drift current vector rotates clockwise by depth in the Northern Hemisphere and at Ehz −= the decreased current is opposite to the surface current. At the surface 0=z the components of the drift current with wind stress ( )yττ ,0=

r are

fhvu

E

yπτ== 00 ,

(4.42)

which means that surface drift current is deflected by 450 to the right from the wind direction. This is valid for the Northern Hemisphere. In the Southern Hemisphere 0<f and the drift

current is deflected to the left. In that case f has to be taken in the definition (4.41) of the

Ekman depth. Volume transports through the section of unit cross-length, calculated from the solution (4.39), (4.40) are

fdzuU yτ

== ∫∞−

0

, ∫∞−

==0

0dzvV .

(4.43)

This means that volume transport of drift currents (Ekman transport) is perpendicular to the wind stress and deflected to the right in the Northern Hemisphere.

Figure 4.3. Ekman drift currents as function of depth: (a) perspective view of current vectors, (b) projections of current vectors from different depths to the plane 0=z .



c) bottom Ekman layer Let us consider next the frictional flow of a rotating fluid above the ground floor (above bottom in the sea, above land in the atmosphere). Placing 0=z on the ground, viscous non-slip condition is written

0=u , 0=v at 0=z . (4.44) At infinite height above the ground the flow is geostrophic

guu = , gvv = if ∞→z . (4.45)

In the general solution (4.34), (4.35) the coefficients are 021 == aa due to the boundary

condition (4.45). When the geostrophic flow is directed along the x -axis ( 0=gv ) then

gub −=1 , 02 =b . The solution for the flow is (Fig. 4.4)

( ) ( )zuzu z

g γγ cose1 −−= , (4.46)

( ) zuzv z

g γγ sine−= . (4.47)

Figure 4.4. Ekman flow near the ground as a function of height: (a) projections of flow

vectors to the plane 0=z , (b, c) vertical profiles of velocity components, above γπ

=Eh the

flow approaches to the geostrophic flow ( )0,gg uu =r

.



Above the Ekman layer the flow is geostrophic. By approaching the ground, current vector rotates counterclockwise (in the Northern Hemisphere) reaching maximum deflection angle 450 just near the ground where the flow speed vanishes. Using =f 10-4 s-1, Ekman layer thickness is ca 1.4 km at =ν 10 m2 s-1 (typical for the atmospheric boundary layer) and ca 4 m at =ν 10-4 m2 s-1 (typical for the near-bottom ocean layer). In the oceanic surface layer

=ν 10-2 m2 s-1 and typical surface Ekman layer thickness is 44 m. 4.4. Ekman boundary layers in the finite basin a) combined surface and bottom layers We have seen that outside the surface and ground Ekman layers the flow is nearly geostrophic. Consider the solution of (4.25), (4.26) at constant finite depth Hz −= . General solution (4.34), (4.35) has to satisfy boundary conditions

xz

uτν =

∂∂

, yz

vτν =

∂∂

at 0=z (4.48)

and

0=u , 0=v at Hz −= . (4.49) Using the complex representations ivu +=χ , ggg ivu +=χ , yx iττσ += we obtain

g

fi

fi

zχ

νχ

νχ

−=−∂

∂2

2

(4.50)

with boundary conditions

σχ

=∂∂

=0zzv , 0=−= Hzχ .

(4.51)

The solution is written in a complex form

( )

−+

+=

H

z

H

Hzg η

ηχ

ηη

νησ

χcosh

cosh1

cosh

sinh

(4.52)

where ( ) ( )ifi +=+= 1

21

νγη .



Let us simplify the complex solution (4.52). When the Ekman depth is considerably smaller

than the water depth Hf

hE <<=ν

π2

then

guu ≈ , gvv ≈ if ( )EE hHzh −−<<<<− . (4.53)

and (4.52) can be written as a sum of surface and bottom boundary layer solutions. Assume the wind stress acting along the y -axis ( 0=xτ ) and geostrophic flow taking place along the

x -axis ( 0=gv ), then (4.52) gives

( ) ( ) ( ) ( )( )Hzuzzzu Hzg

zy +−+−= +− γγγγν

τ γγ cose1sincose2

,

( ) ( ) ( ) ( )Hzuzzzv Hzg

zy +++= +− γγγγν

τ γγ sinesincose2

,

(4.54)

that is similar to the solutions for infinitely thick layers (4.39)-(4.40) and (4.46)-(4.47). In case of thin Ekman layer ( HhE 2.0< ) the approximate solution (4.54) is visually indistinguishable from the exact solution (4.52). Let us choose in (4.54) the geostrophic velocity gu in a way that volume transport in the

whole water column along the x -axis is set to zero ( ) 00

== ∫−H

dzzuU . Accounting for the

volume transport of “pure” drift (4.43) then in case of HhE << we obtain 0=+= Huf

U gyτ

or

fHu y

g

τ−= .

(4.55)

Corresponding solution shown in Fig. 4.5 reveals that below the surface Ekman layer the (geostrophic) compensation flow is opposite to the surface layer Ekman transport. Approximate solution (4.54) allows for easy consideration of different turbulent viscosity in the surface (sν ) and bottom ( bν ) boundary layers, as known from the observations. Therefore

in Fig. 4.5 thickness of surface (Esh ) and bottom ( Ebh ) Ekman layers is different.



-5 0 5 10 15

-100

-90

-80

-70

-60

-50

-40

-30

-20

-10

0

Velocity (cm/s)D

epth

(m

)

uv

hEs

hEb

u = ug

Figure 4.5. Drift currents in a basin with finite depth where the volume transport is set to zero by appropriate choice of geostrophic flow. Parameter values are: H =100 m, sν = 41025 −⋅

m2s-1 , bν = 4105 −⋅ m2s-1 , f = 41023.1 −⋅ s-1 , Esh =20 m, Ebh =9 m, xτ =0, yτ =10-4 m2 s-2

(corresponding to wind speed ca 8 m/s), gu =0.88 cm/s, gv =0.

b) horizontal Ekman layers Next consider the steady flows outside the vertical Ekman layers where the currents are geostrophic guu = , gvv = and vertical gradients do not appear. Placing a vertical wall at

0=x then near the wall velocities depend on the x -coordinate (more specifically, on the distance to the wall) ( )xuu = , ( )xvv = . A non-slip boundary condition is valid at 0=x

00 == = xx vu , (4.56)

The mathematical problem for such flow is similar to the problem of vertical friction in a rotating fluid (4.25)-(4.26)

2

2

0

1

x

u

x

pfv

∂

∂+

∂∂

−=− µρ

,

(4.57)



2

2

0

1

x

v

y

pfu

∂

∂+

∂∂

−= µρ

.

(4.58)

Horizontal friction is considered here by the horizontal viscosity coefficient µ . The solution

of (4.56)-(4.58) is similar to that of the flow above ground (4.46), (4.47). Taking const=gu

and 0=gv we obtain

( ) ( )xuxu x

g γγ cose1 −−= , (4.59)

( ) xuxv x

g γγ sine−= , (4.60)

where µ

γ2

f= and the thickness of lateral Ekman layer (Ekman width) is

fl E

µπ

γπ 2

== . (4.61)

In case of a common value µ =106 cm2 s-1 = 102 m2 s-1 the Ekman width is El =4.4 km. In the

lateral boundary layer 0≠∂∂x

u (see 4.59) and vertical velocities appear near the lateral

boundaries due to the continuity equation. c) channel circulation An idealized scheme of steady circulation in a channel of rectangular cross-section is shown in Fig. 4.6. Along-channel wind causes cross-channel Ekman transport in the upper layer. Compensating geostrophic currents have to be opposite to the surface Ekman transport in order to keep the mass continuity for the whole water column. For that, sea level must have along-channel gradient. Near the lateral boundaries viscous forces hinder the cross-channel flow and vertical velocities appear. Within this barotropic scheme, strongest idealization is made with respect to the water level distribution. Also note, that “clueing” the different boundary layer solutions is not strictly valid in the “corners” of overlapping boundary layers.



hEs

hEb

u = ug

lE lE

x

z y

u0

v0

τyξ

Figure 4.6. A scheme of steady circulation in a channel of rectangular cross-section. Wind is directed along the channel. Sea level is changing along the channel. 4.5. Sverdrup flow, influence of β -effect a) general Sverdrup relation Ekman drift current assumed infinitely wide ocean where lateral boundaries do not interfere the volume transports. Consider volume transports in a basin of constant depth const=H

∫−

=0

H

dzuU , ∫−

=0

H

dzvV .

(4.62)

Integrating the continuity equation (2.24) for the stationary circulation we obtain

0=∂∂

+∂∂

y

V

x

U

(4.63)

since vertical velocity is zero both at the bottom and the surface

( ) 00

000

=−−=∂∂

−=∂∂

+∂∂

−==

−−−∫∫∫ Hzz

HHH

wwdzz

wdz

y

vdz

x

u .

(4.64)

Volume transports of a steady flow are horizontally non-divergent (compare 4.63 and 4.1) and streamfunction ψ can be used (see also 4.2)



yU

∂∂

−=ψ

, x

V∂∂

=ψ

. (4.65)

Integrating the horizontal momentum equations (4.25), (4.26) we obtain

xbxx

PfV ττ −+

∂∂

−=− ,

(4.66)

ybyy

PfU ττ −+

∂∂

−= ,

(4.67)

where ∫−

=0

0

1

H

dzpPρ

. Wind stress ( )yx τττ ,=r

and bottom stress ( )ybxbb τττ ,=r

are

introduced by integrating the frictional terms

xbxHzz

H z

u

z

udz

z

u

zττννν −=

∂∂

−∂∂

=∂∂

∂∂

−==

−∫ 0

0

,

(4.68)

ybyHzz

H z

v

z

vdz

z

v

zττννν −=

∂∂

−∂∂

=∂∂

∂∂

−==

−∫ 0

0

.

(4.69)

By modifying (4.66), (4.67) it is important to account that Coriolis parameter ϕsin2Ω=f (1.24) varies with latitude ϕ . In the local Cartesian coordinate frame we have used so far f -

plane approximation with const0 == ff . Variation of Coriolis parameter can be presented

by a linear β -plane approximation ( β=dy

df) where

( )00 yyff −+= β .

(4.70) Around the latitude 0ϕ we may define β in geographic terms

R0cos2 ϕ

βΩ

= ,

(4.71)

where R is radius of the Earth. Considering deep ocean we neglect bottom stress in (4.66), (4.67). Eliminating P by cross-differentiating (4.66) and (4.67), using also volume transport non-divergence (4.63) and β -plane approximation (4.70), we obtain

yxV xy

∂

∂−

∂

∂=

ττβ

(4.72)

or in the equivalent form



τψ

βr

zcurl=∂∂

x .

(4.73)

Formula (4.72) or (4.73) represents the Sverdrup relation. Its essence is based on the conservation of vorticity. In case of “free” flow the fluid particles must follow the lines of

H

f. With const=H it means 0=V or the flow is constrained to the lines of constant y .

The fluid can cross the lines of H

f only to the extent how much vorticity is externally

introduced by the curl of the wind stress. b) Sverdrup circulation in a rectangular basin Consider the Sverdrup relation producing a circulation in the rectangular ocean basin. At the basin boundaries byax ,0;,0 == normal flow component must vanish. Based on the streamfunction definition (4.65) we obtain

( ) ( ) ( ) ( ) 0,0,,,0 ==== bxxyay ψψψψ . (4.74)

Let us prescribe the wind stress in a form

0,cos0 =−= yx b

yτ

πττ

(4.75)

representing approximately the climatic distribution of zonal wind stress over the Northern Atlantic. The Sverdrup relation is then

b

y

bx

ππτψβ sin0−=

∂∂

.

(4.76)

The equation (4.76) has two possible solutions satisfying boundary condition 0=ψ at

byy == ,0 . At the same time the solution Eψ

( )b

yax

bE

πβπτ

ψ sin0 −−=

(4.77)

satisfies the eastern boundary condition ( ) 0, =yaψ but does not satisfy the western boundary

condition ( ) 0,0 =yψ and the solution Wψ

b

yx

bW

πβπτ

ψ sin0−=

(4.78)

satisfies the western boundary condition ( ) 0,0 =yψ but does not satisfy the eastern boundary condition. This way the Sverdrup relation is not able to satisfy boundary conditions for the



circulation in the closed oceanic basin. It is also not possible to distinguish which solution – either Eψ or Wψ - is “nearly right”, i.e. wrong at only one of the east-west boundaries.

In a non-dimensional basin 1,1 == ba both the possible Sverdrup circulation regimes

corresponding to a normalized wind stress βτ =0 are (as shown in Fig. 4.7)

( ) yxE ππψ sin1−−= , yxW ππψ sin−= .

(4.79)

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

Figure 4.7. Two alternatives Eψ (left) and Wψ (right) of the circulation in the unit basin

according to the Sverdrup relation. 4.6. Stommel circulation and the western boundary layer Let us complement the Sverdrup problem with linear bottom friction

rUz

uxbHz ==

∂∂

−= τν ,

(4.80)

rVz

vybHz ==

∂∂

−= τν ,

(4.81)

where r is the linear friction coefficient (see also 2.40). Replacing (4.80), (4.81) into (4.66), (4.67) and eliminating P , we obtain an equation for the streamfunction (see also 4.65)

yxxr xy

∂

∂−

∂

∂=

∂∂

+∆ττψ

βψ .

(4.82)

Let us use the same basin and wind stress as for the earlier Sverdrup problem, then

b

y

bxr

ππτψβψ sin0−=

∂∂

+∆ ,

(4.83)

with boundary conditions



0=ψ kui byyaxx ==== ,0,,0 . (4.84)

The problem (4.83), (4.84) was first formulated and solved by Henry Stommel and the corresponding pattern is called the Stommel circulation. Let us go to the non-dimensional coordinates and variables denoted by *

***, 0 ψβτ

ψ === byyxax

(4.85)

and further omit * . Instead of (4.83) we obtain an equation

yx

ππψ

ψε sin−=∂∂

+∆

(4.86)

and boundary conditions (instead of 4.84)

1,0,1,0,0 ===== yyxxkuiψ , (4.87)

where L

r

βε = is a non-dimensional parameter showing the relation of frictional and Coriolis

force, baL ,≈ is the length scale of the basin . With small 1<<ε we use the method of decomposition the domain into the interior part and the boundary layers. In the interior we assume negligible friction 0=ε . In that case the Sverdrup relation determines the solution of (4.86)

( ) yxcI ππψ sin−= , (4.88)

where c is an arbitrary constant. At that (4.88) satisfies 0=ψ at 1,0 == yy but does not necessarily satisfy the conditions at western and/or eastern boundaries. Consider expanded x -coordinate in the western boundary layer

l

x=λ

(4.89)

where

1<<=L

Ll w

(4.90)

is a ratio of the boundary width wL to the basin scale L .

Assume that in the boundary layer the streamfunction depends on the expanded x -coordinate (4.89)



( )yw ,λψψ = . (4.91)

The equation (4.86) obtains in the western boundary layer the following form

ylyl

www ππλψψ

ελψε

sin1

2

2

2

2

2−=

∂

∂+

∂

∂+

∂

∂

(4.92)

or

yly

ll

www ππλψψ

ελψε

sin2

2

2

2

−=∂

∂+

∂

∂+

∂

∂ .

(4.93)

The second term in (4.93) (multiplier 1<<lε is a product of two small terms) and the wind stress term (multiplier 1<<l ) are smaller than the two other terms. Therefore at l≈ε the equation (4.93) is simplified into

02

2

=∂

∂+

∂

∂

λψ

λψ ww .

(4.94)

Let us search for the solution of (4.94) as a sum

( ) ( ) ( )yyxy wIW ,,, λψλψ Φ+= (4.95)

where Iψ is general solution (4.88) for the interior region and ( )yw λΦ is the correction

function near the western boundary. At that ( )yxI ,ψ changes slowly with boundary layer

coordinate λ and 0=∂

∂

λψ I . In order to assure continuous transition Iw ψψ → far from the

western wall (at large λ ), 0→Φw at ∞→λ must hold. In that case general solution for wΦ

is

( ) ( ) λλ −=Φ eyCyw , , (4.96)

where ( )yC is an arbitrary function of y . In order to satisfy boundary conditions at

00 =→= λx we must take ( ) ( )yxyC I ,ψ−= . Then special solution of (4.95) is

( ) ( )yxe Iw ,1 ψψ λ−−= . (4.97)

Replacing ε

λx

= we obtain the solution in the original coordinates in a form

( )yxe I

x

w ,1 ψψ ε

−=

− .

(4.98)



This way solution of the Stommel problem is the sum of interior solution (4.88) and the boundary layer solution (4.98)

yexcx

wI ππψψψ ε sin1

−+−=+=

− ,

(4.99)

where 0=c has to be taken to satisfy the boundary condition at 0=x . Therefore the Stommel circulation (Fig. 4.8) is

yexx

ππψ ε sin1

−−=

− ,

(4.100)

that satisfies exactly the western boundary condition 0=ψ at 0=x and approximately the eastern boundary condition at 1=x . The solution (4.100) does not satisfy exactly also

equation (4.86), since in the equation (4.94) the term 2

2

yl w

∂

∂ ψε was omitted compared to the

original equation (4.93) of the western boundary layer using the assumption 1<<lε . This kind of approach to construct the approximate solutions is quite common in hydrodynamics.

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

Figure 4.8. Stommel circulation in the unit basin. It is quite easy to show that the “eastern boundary” solution (using the similar approach as above, but for the eastern boundary layer) does not exist. In that case the sign of exponential

coefficient (similar to ε1

) does not allow for the decay of the boundary layer part of the

solution in the interior domain. The sign of exponential coefficient is determined by the β -effect. Stommel circulation scheme explains the westward intensification of ocean currents. Indeed, the Gulf Stream and Kuroshiu appear near the western coasts of the Atlantic and Pacific Ocean.

CHAPTER 4. Stationary currents - ttu.eemsi.ttu.ee/~elken/DO4.pdfDynamical Oceanography Lecture notes...

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