Wind-driven circulation II

●Wind pattern and oceanic gyres

●Sverdrup Relation

●Vorticity Equation

Surface current measurement from ship drift

Current measurements are harder to make than T&SThe data are much sparse.

Surface current observations

Surface current observations

Drifting Buoy Data Assembly Center, Miami, Florida Atlantic Oceanographic and Meteorological Laboratory, NOAA

Annual Mean Surface CurrentPacific Ocean, 1995-2003

Drifting Buoy Data Assembly Center, Miami, Florida Atlantic Oceanographic and Meteorological Laboratory, NOAA

Schematic picture of the major surface currents of the world oceans

Note the anticyclonic circulation in the subtropics (the subtropical gyres)

Relation between surface winds and subtropical gyres

Surface winds and oceanic gyres: A more realistic view

Note that the North Equatorial Counter Current (NECC) is against the direction of prevailing wind.

Mean surface current tropical Atlantic Ocean

Note the North Equatorial Counter Current (NECC)

Sverdrup RelationConsider the following balance in an ocean of depth h of flat

bottom

∫ +=+∫=∂∂

− −

000

0

hxyx

hfMvdzfdz

xp ττρ

(1)

∫ +−=+∫−=∂∂

− −

000

0

hyxy

hfMudzfdz

yp ττρ

(2)

∫=−

0

hx udzM ρ

∫=−

0

hy vdzM ρ

zvf

xp x

∂∂+=

∂∂ τρ

zuf

yp y

∂∂+−=

∂∂ τρ

Integrating vertically from –h to 0 for both (1) and (2), we have(neglecting bottom stress and surface height change)

where

(3)

(4)

are total zonal and meridional transport of mass

sum of geostrophic and ageostropic transports

Differentiating , we have

000=

∂∂−

∂∂+−

∂∂+

∂∂−

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

yxdydfM

yM

xMf xy

yyx ττ

€ P=pdz−h0∫Define We have

€ ∂p∂xdz=∂∂xpdz−h0∫ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥=∂P∂x−h0∫

€ ∂P∂x=fMy+τx0€

∂P∂y=−fMx+τy0(3) and (4) can be written as

(5) (6)

€ ∂6()∂x−∂5()∂y

€ ∂2P∂y∂x−∂2P∂x∂y=−f∂Mx∂x+∂τy0∂x−f∂My∂y−Mydfdy−∂τx0∂y=0

000=

∂∂−

∂∂+−

∂∂+

∂∂−

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

yxdydfM

yM

xMf xy

yyx ττ

Using continuity equation 0=∂∂+

∂∂

yM

xM yx

And define

dydf=β

( )ττττβz

xyy curlk

yxM =⋅×∇=

∂∂−

∂∂= ⎟

⎠⎞⎜

⎝⎛

rr00

Vertical component of the wind stress curl

We have Sverdrup equation

€ ∂τy0∂x−∂τx0∂y=0If

€ My=0The line provides a natural boundary that separate the circulation into “gyres”

€ My=Myg+MyEis the total meridional mass transport

€ Myg=ρvgdz=1f∂p∂xdz=1f∂P∂x−h0∫−h0∫ Geostrophic transport

€ MyE=ρvEdz=−τx0f−h0∫ Ekman transport

Order of magnitude example:At 35oN, -4 s-1, β2 10-11 m-1 s-1, assume τx10-1 Nm-2 τy=0

€ curlzτ()=−∂τx0∂y≈−10−1Nm−21000km≈−10−7Nm−3

€ MyE=−τx0f≈−103kgm−1s−1

€ My=Myg+MyE=curlzτ()β≈−10−72×10−11=−5×103kgm−1s−1€ Myg=−4×103kgm−1s−1

Alternative derivation of Sverdrup Relation

xp

gfv ∂∂=

ρ

ypfu

g ∂∂−=

ρ

Construct vorticity equation from geostrophic balance

(1)

(2)

zw

fv gg ∂

∂=β

Integrating over the whole ocean depth, we have

€ f∂ug∂x+f∂vg∂y+βvg=−1ρ∂2p∂y∂x+1ρ∂2p∂x∂y=0€

∂2()∂x+∂1()∂y€ βvg=−f∂ug∂x+∂vg∂y ⎛ ⎝ ⎜ ⎞ ⎠ ⎟=f∂wg∂z

Assume ρ=constant

€ βVg=βvgdz=fwgz=0()−wgz=−h()[ ]−h0∫

∫ ==−

0

hEgg fwdzvV ββ

kf

wE

rr⋅×∇=⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

ρτ

where is the entrainment rate from the surface Ekman layer

⎟⎟⎠

⎞⎜⎜⎝

⎛=+= ρτ

β curlVVV Eg1

The Sverdrup transport is the total of geostrophic and Ekman transport.The indirectly driven Vg may be much larger than VE.

( )6tan ≈===

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

ϕβτ

βτ

LR

LfO

f

curlO

VV

E

gat 45oN

€ βVg=βvgdz=fwgz=0()−wgz=−h()[ ]−h0∫

€ wgz=0()=wE€ wgz=−h()≈0€ Vg=fβρ∂∂xτyf ⎛ ⎝ ⎜ ⎞ ⎠ ⎟−∂∂yτxf ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟=fρβ1f∂τy∂x−1f∂τx∂y+βτxf2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟€ Vg=1ρβ∂τy∂x−∂τx∂y+βτxf ⎛ ⎝ ⎜ ⎞ ⎠ ⎟=1ρβ∂τy∂x−∂τx∂y ⎛ ⎝ ⎜ ⎞ ⎠ ⎟−VE

€ βMy=∂τy0∂x−∂τx0∂y

€ f=2Ωsinφ€ dy=Rdφ( )

⎟⎠⎞⎜

⎝⎛ Ω

=R

zcurlM y

ϕ

τ

cos2then

( ) ( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛⎟⎠⎞⎜

⎝⎛ −∂

∂Ω−=∂

∂−=∂∂ τϕτϕ zz

yx curlcurly

RyM

xM tan

cos21

€ β=dfdy=d2Ωsinφ( )Rdφ=2ΩsinφR

Since , we have

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

∂

∂+

∂∂

Ω−=∂∂

yyR

xM xxx

τϕ

τϕ

tancos21

2

2

set x =0 at the eastern boundary,

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

∫ ∫∂∂+∂

∂Ω−=

0 0

2

2tan

cos21

x x

xxx dx

ydx

yRM τϕτ

ϕ

yRM x

y ∂∂

Ω= τϕcos2

€ τy≈0

Further assume€ τx≈τxy()In the trade wind and equatorial zones, the 2nd derivative term dominates:

€ Mx≈xR2Ωcosφ∂2τx∂y2€ Mx=x2Ωcosφ∂τx∂ytanφ+∂2τx∂y2R ⎛ ⎝ ⎜ ⎞ ⎠ ⎟

Mass Transport

Since 0=∂∂+

∂∂

yM

xM yx

Let y

M x ∂∂−= ψ

,

xM y ∂

∂= ψ,

( )βτψ zcurl

x=

∂∂

( )∫=0

x

z dxcurlβτψ

where ψ is stream function.

Problem: only one boundary condition can be satisfied.

1 Sverdrup (Sv) =106 m3/s