evat 554 ocean-atmosphere dynamics gyre-scale ocean circulation lecture 16 (reference: peixoto &...
TRANSCRIPT
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