EVAT 554OCEAN-ATMOSPHERE

DYNAMICS

GYRE-SCALE OCEAN CIRCULATION

LECTURE 16

(Reference: Peixoto & Oort, Chapter 8,10)

Sverdrup Transport

cos2/

0

x

yM

What about the western boundary???

2/

2 0

20

xaxM

We are not conserving mass (note the behavior

at the western boundary!)

Sverdrup Transport

2/

2 0

20

xaxM

Problem is that we cannot satisfy two lateral boundary conditions with a solution to a first order equation

We need to take into account missing physicsBottom Friction!

cos2/

0

x

yM

Sverdrup Transport

2/

2 0

20

xaxM

Problem is that we cannot satisfy two lateral boundary conditions with a solution to a first order equation

Bottom Friction!

cos2/

0

x

yM

Stommel ‘Bottom Friction’ model

Bottom Friction!

Assume a “Rayleigh” law for frictional stresses Ruzx

vR

zy

In areas of moderate flow, this will reduce to zero bottom stress, yielding the previous result

cos2/

0

x

yM

2/

2 0

20

xaxM

2/

2 0

20

xaxM

Assume a “Rayleigh” law for frictional stresses

We might anticipate, however, that this solution

could breakdown where we know the Sverdrup solution must break

down…

Stommel ‘Bottom Friction’ model

Ruzx

vR

zy

cos2/

0

x

yM

2/

2 0

20

xaxM

We thus assume the existence of a

boundary layer of zonal width ‘’ that provides the return flow of the

interior Sverdrup transport

Stommel ‘Bottom Friction’ model

cos2/

0

x

yM

/ˆexp11

cos2/

0 xyMx

2/

2

)/ˆexp(

0

200

xa

xxM

00/)(ˆ x

0

aR

Note that these expressions satisfy the requirement of no basin-integrated meridional transport at any latitude!

Ruzx

vR

zy

We thus assume the existence of a

boundary layer of zonal width ‘’ that provides the return flow of the

interior Sverdrup transport

Stommel ‘Bottom Friction’ model

00/)(ˆ x

Note that these expressions satisfy the requirement of no basin-integrated meridional transport at any latitude!

Ruzx

vR

zy

xdyM ˆ1

0 xdxx ˆ/ˆexp11

cos2/1

0

0

)(11cos2/

0

x=0

/ˆexp11

cos2/

0 xyMx

0

aR

Useful to interpret the circulation in terms of

‘Vorticity’ (spin)

Stommel ‘Bottom Friction’ model

V

Absolute Vorticity=Planetary Vorticity( f)+Relative Vorticity (curl of velocity field)

Only friction can take away this vorticity (i.e., add negative vorticity) once it has been added

Windstress adds positive vorticity

Stommel ‘Bottom Friction’ model

zxxpf

/v

zyypf

/u

Consider the fundamental equations

xzx

xxpxf

v

yzy

yypf

uyu

Add these together,

τz

pyxf2

u)uv(

Differentiate with respect to x and y respectively

Useful to interpret the circulation in terms of

‘Vorticity’ (spin)

V

Stommel ‘Bottom Friction’ model

τz

pf2

u

Useful to interpret the circulation in terms of

‘Vorticity’ (spin)

V

Relative Vorticityu12

fzffp

τ

fzff

pfa

τ

12 Absolute Vorticity

τzff

p12

τz

pyxf2

u)uv(

Stommel ‘Bottom Friction’ model

zxxpf

/v

zyypf

/u

Consider the fundamental equations

yzx

xypyf

vv

yzy

yypf

uyu

Now, differentiate with respect to y and x respectively

Subtract second from first,

xzx

xxpxf

v

Differentiate with respect to x and y respectively

xzy

yxpf

xu

xy

yx

zyxuf

v)v(

Stommel ‘Bottom Friction’ model

Divergence Equation

If horizontal flow is non-divergent

xy

yx

z

v

Ruzx

vR

zy

Assume Rayleigh friction

Ryx

R

uvvx

R v x

R

vv

)exp(vv0

xR

/ˆexpvv0

x

0/ˆ axx

0

aR

xy

yx

zyxuf

v)v(

Stommel ‘Bottom Friction’ model

/ˆexpvv0

x

This is only the boundary layer solution

HyM /v

cos2/

0

x

yMRecall the interior (Sverdrup) solution

Assuming vertically uniform flow (an idealization),

aHx

/

0

The full solution is thus,

aHx x

//ˆexpvv 0

0

0/ˆ axx

0

aR

ax

/

0

Stommel ‘Bottom Friction’ model

0ˆv1

0 xd

We require no net meridional transport!

0ˆ/

/ˆexp0

v1

0

0

xdaH

xx

aHx

/

v 00 aH

x

/

v 00

/ˆexp11

/v 0 x

aHx

0

aR

aHx x

//ˆexpvv 0

0

Stommel ‘Bottom Friction’ model

/ˆexp11

/v 0 x

aHx

0

aR

H

fR V

2/

We can use continuity of the horizontal flow field to derive an expression

for the zonal velocity

yv/-xu/

)/ˆexp(ˆ1

H1u

20

2

xx

y

We can thus define a streamfunction:/dxdv

/dydu

)/ˆexp(ˆ1

H1 0

xx

y

Stommel ‘Bottom Friction’ model

/ˆexp11

/v 0 x

aHx

0

aR

H

fR V

2/

We can use continuity of the horizontal flow field to derive an expression

for the zonal velocity

-dv/dydu/dx

)/ˆexp(1

H1u

20

2

xx

y

We can thus define a streamfunction:/dxdv

/dydu

)/ˆexp(1

H1 0

xx

y

=0 0

In reality, Western Boundary Current Separates

Eddy-Resolving Ocean GCM

Stommel Model

Stommel ‘Bottom Friction’ model

Stommel Model obviously an idealization, but it captures the essence of westward intensification of

ocean currents