evat 554 ocean-atmosphere dynamics gyre-scale ocean circulation lecture 16 (reference: peixoto &...
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![Page 1: EVAT 554 OCEAN-ATMOSPHERE DYNAMICS GYRE-SCALE OCEAN CIRCULATION LECTURE 16 (Reference: Peixoto & Oort, Chapter 8,10)](https://reader036.vdocuments.mx/reader036/viewer/2022070413/5697bff41a28abf838cbcfd3/html5/thumbnails/1.jpg)
EVAT 554OCEAN-ATMOSPHERE
DYNAMICS
GYRE-SCALE OCEAN CIRCULATION
LECTURE 16
(Reference: Peixoto & Oort, Chapter 8,10)
![Page 2: EVAT 554 OCEAN-ATMOSPHERE DYNAMICS GYRE-SCALE OCEAN CIRCULATION LECTURE 16 (Reference: Peixoto & Oort, Chapter 8,10)](https://reader036.vdocuments.mx/reader036/viewer/2022070413/5697bff41a28abf838cbcfd3/html5/thumbnails/2.jpg)
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!)
![Page 3: EVAT 554 OCEAN-ATMOSPHERE DYNAMICS GYRE-SCALE OCEAN CIRCULATION LECTURE 16 (Reference: Peixoto & Oort, Chapter 8,10)](https://reader036.vdocuments.mx/reader036/viewer/2022070413/5697bff41a28abf838cbcfd3/html5/thumbnails/3.jpg)
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
![Page 4: EVAT 554 OCEAN-ATMOSPHERE DYNAMICS GYRE-SCALE OCEAN CIRCULATION LECTURE 16 (Reference: Peixoto & Oort, Chapter 8,10)](https://reader036.vdocuments.mx/reader036/viewer/2022070413/5697bff41a28abf838cbcfd3/html5/thumbnails/4.jpg)
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
![Page 5: EVAT 554 OCEAN-ATMOSPHERE DYNAMICS GYRE-SCALE OCEAN CIRCULATION LECTURE 16 (Reference: Peixoto & Oort, Chapter 8,10)](https://reader036.vdocuments.mx/reader036/viewer/2022070413/5697bff41a28abf838cbcfd3/html5/thumbnails/5.jpg)
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
![Page 6: EVAT 554 OCEAN-ATMOSPHERE DYNAMICS GYRE-SCALE OCEAN CIRCULATION LECTURE 16 (Reference: Peixoto & Oort, Chapter 8,10)](https://reader036.vdocuments.mx/reader036/viewer/2022070413/5697bff41a28abf838cbcfd3/html5/thumbnails/6.jpg)
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
![Page 7: EVAT 554 OCEAN-ATMOSPHERE DYNAMICS GYRE-SCALE OCEAN CIRCULATION LECTURE 16 (Reference: Peixoto & Oort, Chapter 8,10)](https://reader036.vdocuments.mx/reader036/viewer/2022070413/5697bff41a28abf838cbcfd3/html5/thumbnails/7.jpg)
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
![Page 8: EVAT 554 OCEAN-ATMOSPHERE DYNAMICS GYRE-SCALE OCEAN CIRCULATION LECTURE 16 (Reference: Peixoto & Oort, Chapter 8,10)](https://reader036.vdocuments.mx/reader036/viewer/2022070413/5697bff41a28abf838cbcfd3/html5/thumbnails/8.jpg)
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
![Page 9: EVAT 554 OCEAN-ATMOSPHERE DYNAMICS GYRE-SCALE OCEAN CIRCULATION LECTURE 16 (Reference: Peixoto & Oort, Chapter 8,10)](https://reader036.vdocuments.mx/reader036/viewer/2022070413/5697bff41a28abf838cbcfd3/html5/thumbnails/9.jpg)
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
![Page 10: EVAT 554 OCEAN-ATMOSPHERE DYNAMICS GYRE-SCALE OCEAN CIRCULATION LECTURE 16 (Reference: Peixoto & Oort, Chapter 8,10)](https://reader036.vdocuments.mx/reader036/viewer/2022070413/5697bff41a28abf838cbcfd3/html5/thumbnails/10.jpg)
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
![Page 11: EVAT 554 OCEAN-ATMOSPHERE DYNAMICS GYRE-SCALE OCEAN CIRCULATION LECTURE 16 (Reference: Peixoto & Oort, Chapter 8,10)](https://reader036.vdocuments.mx/reader036/viewer/2022070413/5697bff41a28abf838cbcfd3/html5/thumbnails/11.jpg)
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(
![Page 12: EVAT 554 OCEAN-ATMOSPHERE DYNAMICS GYRE-SCALE OCEAN CIRCULATION LECTURE 16 (Reference: Peixoto & Oort, Chapter 8,10)](https://reader036.vdocuments.mx/reader036/viewer/2022070413/5697bff41a28abf838cbcfd3/html5/thumbnails/12.jpg)
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(
![Page 13: EVAT 554 OCEAN-ATMOSPHERE DYNAMICS GYRE-SCALE OCEAN CIRCULATION LECTURE 16 (Reference: Peixoto & Oort, Chapter 8,10)](https://reader036.vdocuments.mx/reader036/viewer/2022070413/5697bff41a28abf838cbcfd3/html5/thumbnails/13.jpg)
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(
![Page 14: EVAT 554 OCEAN-ATMOSPHERE DYNAMICS GYRE-SCALE OCEAN CIRCULATION LECTURE 16 (Reference: Peixoto & Oort, Chapter 8,10)](https://reader036.vdocuments.mx/reader036/viewer/2022070413/5697bff41a28abf838cbcfd3/html5/thumbnails/14.jpg)
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
![Page 15: EVAT 554 OCEAN-ATMOSPHERE DYNAMICS GYRE-SCALE OCEAN CIRCULATION LECTURE 16 (Reference: Peixoto & Oort, Chapter 8,10)](https://reader036.vdocuments.mx/reader036/viewer/2022070413/5697bff41a28abf838cbcfd3/html5/thumbnails/15.jpg)
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
![Page 16: EVAT 554 OCEAN-ATMOSPHERE DYNAMICS GYRE-SCALE OCEAN CIRCULATION LECTURE 16 (Reference: Peixoto & Oort, Chapter 8,10)](https://reader036.vdocuments.mx/reader036/viewer/2022070413/5697bff41a28abf838cbcfd3/html5/thumbnails/16.jpg)
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
![Page 17: EVAT 554 OCEAN-ATMOSPHERE DYNAMICS GYRE-SCALE OCEAN CIRCULATION LECTURE 16 (Reference: Peixoto & Oort, Chapter 8,10)](https://reader036.vdocuments.mx/reader036/viewer/2022070413/5697bff41a28abf838cbcfd3/html5/thumbnails/17.jpg)
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
![Page 18: EVAT 554 OCEAN-ATMOSPHERE DYNAMICS GYRE-SCALE OCEAN CIRCULATION LECTURE 16 (Reference: Peixoto & Oort, Chapter 8,10)](https://reader036.vdocuments.mx/reader036/viewer/2022070413/5697bff41a28abf838cbcfd3/html5/thumbnails/18.jpg)
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