Class 8. Oceans II. Ekman pumping/suction Wind-driven ocean flow Equations with wind-stress.
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Transcript of Class 8. Oceans II. Ekman pumping/suction Wind-driven ocean flow Equations with wind-stress.
Class 8. Oceans II
Ekman pumping/suction
Wind-driven ocean flow
Equations with wind-stress
€
x - component 0 = −1
ρ
∂p
∂x+ fv +
1
ρ ref
∂τx
∂z
€
y - component 0 = −1ρ
∂p∂y
−fu +1
ρ ref
∂τ y
∂z
Wind-driven ocean flow
Equations with wind-stress
€
x - component 0 = −1
ρ
∂p
∂x+ fv +
1
ρ ref
∂τx
∂z
Split velocity in geostrophic ('g') and ageostrophic parts ('ag')
€
v = vg + vag
€
vg =1
ρf
∂p
∂x
€
−fvag =1
ρ ref
∂τx
∂z
€
fuag =1
ρ ref
∂τy
∂z
e.g.
Ekman transport
€
MEk = ρ ref−δ
0
∫ uagdz
€
MEk = ρ ref uag,vag( )−δ
0
∫ dz =1
fτy,−τx( )
Ekman pumping (downwards)/suction
X wind into the screen
Ekman pumping (downwards)/suction
tropics
midlatitudes
elevated sea level heightin convergence area
Ekman pumping/suction due to wind stress
Ekman pumping/suction
Explanation
€
∂w∂z
= −∂u ag
∂x+
∂vag
∂y
⎛
⎝ ⎜
⎞
⎠ ⎟ mass conservation
€
w zsfc( ) −w zEkman,bottom( ) = −∂u ag
∂x+
∂vag
∂y
⎛
⎝ ⎜
⎞
⎠ ⎟dz
zEkman ,bottom
zsfc
∫
0
Ekman pumping/suction
€
w zEkman,bottom( ) =∂u ag
∂x+
∂vag
∂y
⎛
⎝ ⎜
⎞
⎠ ⎟dz
zEkman ,bottom
zsfc
∫
€
w zEkman,bottom( ) =1
ρ ref
∂∂x
τ y,sfc
f−
∂∂y
τ x,sfc
f
⎛
⎝ ⎜
⎞
⎠ ⎟
€
−fvag =1
ρ ref
∂τx
∂z
€
fuag =1
ρ ref
∂τy
∂z
Ekman pumping/suction
Example
€
w zEkman,bottom( ) =1
ρ ref
∂∂x
τ y,sfc
f−
∂∂y
τ x,sfc
f
⎛
⎝ ⎜
⎞
⎠ ⎟
€
= 11000
2×10−1
2×106( ) 1×10−4
( )
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥=10−6 m /s
= 32 m/year
Ekman pumping/suction from wind stress climatology
€
w zEkman,bottom( ) =1
ρ ref
∂∂x
τ y,sfc
f−
∂∂y
τ x,sfc
f
⎛
⎝ ⎜
⎞
⎠ ⎟
downward
upward
f=0
The equatorial strip is a region of upwelling, because the trade winds on either side of the equator drive fluid away from the equator in the surface Ekman layer, and do demand a supply of fluid from below (p205)
Wind-driven ocean flow
Eliminate pressure by cross differentiating (rref=cst)
€
take∂
∂y : 0 = −
1ρ ref
∂p∂x
+fv +1
ρ ref
∂τ x
∂z
€
take∂
∂x : 0 = −
1ρ ref
∂p∂y
−fu +1
ρ ref
∂τ y
∂z
Wind-driven ocean flow
Eliminate pressure by cross differentiating (rref=cst, b=df/dy)
€
take∂
∂y : 0 = −
1ρ ref
∂p∂x
+fv +1
ρ ref
∂τ x
∂z
€
take∂
∂x : 0 = −
1ρ ref
∂p∂y
−fu +1
ρ ref
∂τ y
∂z
€
f∂u
∂x+
∂v
∂y
⎛
⎝ ⎜
⎞
⎠ ⎟+ βv +
1
ρ ref
∂
∂z
∂τx
∂y−
∂τy
∂x
⎛
⎝ ⎜
⎞
⎠ ⎟= 0
€
−f∂w∂z
+βv+1
ρ ref
∂∂z
∂τ x
∂y−
∂τ y
∂x
⎛
⎝ ⎜
⎞
⎠ ⎟= 0
b≈2x10-11 m-1s-1
Interior ocean flow structure
€
−f∂w∂z
+βv+1
ρ ref
∂∂z
∂τ x
∂y−
∂τ y
∂x
⎛
⎝ ⎜
⎞
⎠ ⎟= 0
Below Ekman layer:
€
βv = f∂w∂z
Interior ocean flow structure
€
−f∂w∂z
+βv+1
ρ ref
∂∂z
∂τ x
∂y−
∂τ y
∂x
⎛
⎝ ⎜
⎞
⎠ ⎟= 0
Below Ekman layer:
€
βv = f∂w∂z
wEk>0 v>0 (weak
northward flow)
Interior ocean flow structure
€
−f∂w∂z
+βv+1
ρ ref
∂∂z
∂τ x
∂y−
∂τ y
∂x
⎛
⎝ ⎜
⎞
⎠ ⎟= 0
Below Ekman layer:
€
βv = f∂w∂z
wEk<0 v<0 (weak
southward flow)
The Sverdrup relation
€
−f∂w∂z
+βv+1
ρ ref
∂∂z
∂τ x
∂y−
∂τ y
∂x
⎛
⎝ ⎜
⎞
⎠ ⎟= 0
€
β vdz−D
0
∫ ≡ βV =1
ρ ref
∂τ y
∂x−
∂τ x
∂y
⎛
⎝ ⎜
⎞
⎠ ⎟z=0
apply integration between a 'very large' depth(*) and the surface where w=0
The Sverdrup relation explains how the depth integrated meridional transport (y-direction) is related to the wind stress
(*) ocean should be deep enough to prevent bottom friction acting on the flow
The observed ocean circulation (from NOAA)
equatorial countercurrentgyres
The wind stress
trade-winds
westerlies
easterlies
€
β vdz−D
0
∫ ≡ βV =1
ρ ref
∂τ y
∂x−
∂τ x
∂y
⎛
⎝ ⎜
⎞
⎠ ⎟z=0
€
I) V = 0 where ∂τ x
∂y= 0
€
β vdz−D
0
∫ ≡ βV =1
ρ ref
∂τ y
∂x−
∂τ x
∂y
⎛
⎝ ⎜
⎞
⎠ ⎟z=0
€
II) V < 0 where ∂τ x
∂y> 0
€
β vdz−D
0
∫ ≡ βV =1
ρ ref
∂τ y
∂x−
∂τ x
∂y
⎛
⎝ ⎜
⎞
⎠ ⎟z=0
€
II) V > 0 where ∂τ x
∂y< 0
€
w zEkman,bottom( ) =1
ρ ref
∂∂x
τ y,sfc
f−
∂∂y
τ x,sfc
f
⎛
⎝ ⎜
⎞
⎠ ⎟
€
III) downwelling where ∂τ x / f
∂y> 0
€
w zEkman,bottom( ) =1
ρ ref
∂∂x
τ y,sfc
f−
∂∂y
τ x,sfc
f
⎛
⎝ ⎜
⎞
⎠ ⎟
€
III) upwelling where ∂τ x / f
∂y< 0
€
MEk = ρ ref uag,vag( )−δ
0
∫ dz =1
fτy,−τx( ) Ekman layer:
deflection to the right of the wind stress
deflection to the left of the wind stress (southern hemisphere)
Ekman pumping/suction due to wind stress
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