Wind-driven circulation
Thermohaline circulation
Ocean Circulation
Joe LaCasce
Section for Meteorology and Oceanography
November 3, 2015
Joe LaCasce Section for Meteorology and Oceanography
Ocean Circulation
Wind-driven circulation
Thermohaline circulation
Ocean forcing
The ocean is driven primarily by:
• Wind : forcing at the surface transfers momentum to the ocean,
via waves and turbulent motion
• Heating : the sun warms the low latitudes more than the high
latitudes, creating a large scale density gradient at the surface
• Evaporation/precipitation : fresh water removal and input at the
surface can also affect surface density
Joe LaCasce Section for Meteorology and Oceanography
Ocean Circulation
Wind-driven circulation
Thermohaline circulation
Background
Nansen, icebergs, and Ekman (1905)
Joe LaCasce Section for Meteorology and Oceanography
Ocean Circulation
Wind-driven circulation
Thermohaline circulation
Applied stress
1 ∂
d~u
=
~τ
dt
ρc ∂z
Joe LaCasce Section for Meteorology and Oceanography
Ocean Circulation
Wind-driven circulation
Thermohaline circulation
Planetary boundary layer equations
Add stress to geostrophic relations:
1
ρc
1
fu = −
ρc
−fv = −
∂
p+
∂x
∂
p+
∂y
∂ τx
∂z ρc
∂ τy
∂z ρc
Rewrite using the geostrophic velocities:
∂ τx
∂z ρc
∂ τy
fu = fug +
∂z ρc
−fv = −fvg +
Joe LaCasce Section for Meteorology and Oceanography
Ocean Circulation
Wind-driven circulation
Thermohaline circulation
Planetary boundary layer equations
Collecting terms on the LHS:
∂
∂z
∂
fua =
∂z
−fva =
τx
ρc
τy
ρc
where ua = u − ug and va = v − vg are the ageostrophic velocities,
forced by the wind. These vary with depth.
Joe LaCasce Section for Meteorology and Oceanography
Ocean Circulation
Wind-driven circulation
Thermohaline circulation
Ekman transport
Integrate vertically over the depth of the layer:
Z
0
−
fva dz ≡ −fVE =
−δE
Z
0
fua dz ≡ fUE =
−δE
Joe LaCasce Section for Meteorology and Oceanography
1 wx
τ
ρc
1 wy
τ
ρc
Ocean Circulation
Wind-driven circulation
Thermohaline circulation
Ekman transport
→ Net ageostrophic transport is 90◦ to the right of the wind
Transport is 90◦ to the left in the southern hemisphere
Accounts for the ice drift witnessed by Nansen
Joe LaCasce Section for Meteorology and Oceanography
Ocean Circulation
Wind-driven circulation
Thermohaline circulation
Ekman pumping
Joe LaCasce Section for Meteorology and Oceanography
Ocean Circulation
Wind-driven circulation
Thermohaline circulation
Vorticity equation
How does Ekman pumping affect the flow at depth? We can
cross-differentiate the equations and subtract them:
∂
1
[fu = −
∂x
ρc
∂
1
− [−fv = −
∂y
ρc
∂
p+
∂y
∂
p+
∂x
∂ τy
]
∂z ρc
∂ τx
]
∂z ρc
This eliminates the pressure terms, leaving a vorticity equation:
f(
∂u ∂v
df
1 ∂ ∂τ y
∂τ x
+
)+v
=
(
−
)
∂x
∂y
dy
ρc ∂z ∂x
∂y
Joe LaCasce Section for Meteorology and Oceanography
Ocean Circulation
Wind-driven circulation
Thermohaline circulation
Vorticity equation
From incompressibility:
∂
∂
∂
u+
v =− w
∂x
∂y
∂z
So the vorticity equation becomes:
βv = f
∂w
1 ∂ ∂τ y
∂τ x
+
(
−
)
∂z
ρc ∂z ∂x
∂y
where β = df /dy .
Joe LaCasce Section for Meteorology and Oceanography
Ocean Circulation
Wind-driven circulation
Thermohaline circulation
Vorticity equation at depth
Below the boundary layer, there is no stress:
∂w
∂z
Integrate from the bottom (z = −H) to the base of the Ekman
layer (z = −δE ). Assume the bottom is flat, so w (−H) = 0:
βv = f
Z
−δE
βv dz ≡ βVI = fw (−δE ) − fw (−H) = fw (−δE )
−H
So Ekman pumping drives meridional transport:
βVI = fw (−δE )
Joe LaCasce Section for Meteorology and Oceanography
Ocean Circulation
Wind-driven circulation
Thermohaline circulation
Vorticity equation in the Ekman layer
In the surface Ekman layer, can also integrate vertically. Assume
the ocean surface is also flat (w (0) = 0).
Z
0
0
∂w
1 ∂ ∂τ y
∂τ x
+
(
−
) dz
ρc ∂z ∂x
∂y
−δE ∂z
∂τ wx
1 ∂τ wy
−
)
= −fw (−δE ) + (
ρc ∂x
∂y
Z
βv dz = βVE =
−δE
f
The term βVE is smaller than the others, so:
fw (−δE ) =
1 ∂τ wy
∂τ wx
(
−
)
ρc ∂x
∂y
Joe LaCasce Section for Meteorology and Oceanography
Ocean Circulation
Wind-driven circulation
Thermohaline circulation
Sverdrup relation
Combine the boundary and interior equations:
βVI =
1 ∂τ wy
∂τ wx
1
(
−
) = k̂ · ∇ × ~τ w
ρc ∂x
∂y
ρc
This is the Sverdrup balance
Joe LaCasce Section for Meteorology and Oceanography
Ocean Circulation
Wind-driven circulation
Thermohaline circulation
Sverdrup relation
For example, if the wind blows east:
βV = −
Joe LaCasce Section for Meteorology and Oceanography
∂ wx
τ
∂y
Ocean Circulation
Wind-driven circulation
Thermohaline circulation
Boundary currents
Imagine a negative wind stress curl over a basin:
But how does the fluid return north?
Joe LaCasce Section for Meteorology and Oceanography
Ocean Circulation
Wind-driven circulation
Thermohaline circulation
Gulf Stream
Benjamin Franklin (1770)
Joe LaCasce Section for Meteorology and Oceanography
Ocean Circulation
Wind-driven circulation
Thermohaline circulation
Boundary currents
Actually two possibilities:
Joe LaCasce Section for Meteorology and Oceanography
Ocean Circulation
Wind-driven circulation
Thermohaline circulation
Boundary currents
Problem solved by Stommel (1948)
Joe LaCasce Section for Meteorology and Oceanography
Ocean Circulation
Wind-driven circulation
Thermohaline circulation
Stommel’s Gulf Stream
Stommel realized need additional dynamics to allow a return flow.
The simplest is linear bottom friction. In the interior ocean,
geostrophy is replaced by:
1
ρc
1
fu = −
ρc
−fv = −
∂
p − ru
∂x
∂
p − rv
∂y
Friction breaks geostrophy, allowing ageostrophic return flow.
Joe LaCasce Section for Meteorology and Oceanography
Ocean Circulation
Wind-driven circulation
Thermohaline circulation
Stommel’s Gulf Stream
Now the Sverdrup relation is:
1 ∂τ wy
∂τ wx
βV = (
−
)−r
ρc ∂x
∂y
Z
−δE
(
H
∂u
∂v
−
) dz
∂x
∂y
For simplicity, assume the velocities are depth-independent:
βHv =
1 ∂τ wy
∂τ wx
∂v
∂u
(
−
) − rH(
−
)
ρc ∂x
∂y
∂x
∂y
Also have assumed δE H
Joe LaCasce Section for Meteorology and Oceanography
Ocean Circulation
Wind-driven circulation
Thermohaline circulation
Stommel’s Gulf Stream
Break velocity into two parts: v = vI + vB , with:
βvI =
1 ∂τ wy
∂τ wx
(
−
)
ρc H ∂x
∂y
in the interior, and:
βvB = −r (
∂vB
∂uB
−
)
∂x
∂y
in the boundary layer. The boundary current is narrow so:
βvB ≈ −r
Joe LaCasce Section for Meteorology and Oceanography
∂vB
∂x
Ocean Circulation
Wind-driven circulation
Thermohaline circulation
Stommel’s Gulf Stream
Now consider the shear in the boundary current:
Joe LaCasce Section for Meteorology and Oceanography
Ocean Circulation
Wind-driven circulation
Thermohaline circulation
Stommel’s Gulf Stream
βvB ≈ −r
∂vB
∂x
• Western boundary current
r
∂vB
<0
∂x
→
βvB > 0
→
βvB < 0
• Eastern boundary current
r
∂vB
>0
∂x
Joe LaCasce Section for Meteorology and Oceanography
Ocean Circulation
Wind-driven circulation
Thermohaline circulation
Stommel’s Gulf Stream
Works the other way too. If the interior flow is northwards, the
boundary currents go south:
West : r
∂vB
>0
∂x
Joe LaCasce Section for Meteorology and Oceanography
→
βvB < 0
Ocean Circulation
Wind-driven circulation
Thermohaline circulation
The mid-latitude gyres
Joe LaCasce Section for Meteorology and Oceanography
Ocean Circulation
Wind-driven circulation
Thermohaline circulation
Observed wind stress
Joe LaCasce Section for Meteorology and Oceanography
Ocean Circulation
Wind-driven circulation
Thermohaline circulation
Ekman pumping
Joe LaCasce Section for Meteorology and Oceanography
Ocean Circulation
Wind-driven circulation
Thermohaline circulation
The Pacific gyres
Joe LaCasce Section for Meteorology and Oceanography
Ocean Circulation
Wind-driven circulation
Thermohaline circulation
Observations: Pacific
Joe LaCasce Section for Meteorology and Oceanography
Ocean Circulation
Wind-driven circulation
Thermohaline circulation
Observations: Atlantic
Joe LaCasce Section for Meteorology and Oceanography
Ocean Circulation
Wind-driven circulation
Thermohaline circulation
Thermohaline circulation
The ocean is also heated by incoming shortwave radiation and
cooled by outgoing longwave radiation and evaporation
Drives large scale flow, the thermohaline circulation
Global “over-turning” circulation superimposed on wind-driven
gyres
Important for redistribution of heat and CO2 in climate system
Hard to observe: time scales of 1000s of years and weak velocities
Joe LaCasce Section for Meteorology and Oceanography
Ocean Circulation
Wind-driven circulation
Thermohaline circulation
Buoyancy forcing
Joe LaCasce Section for Meteorology and Oceanography
Ocean Circulation
Wind-driven circulation
Thermohaline circulation
Net upward heat flux
Joe LaCasce Section for Meteorology and Oceanography
Ocean Circulation
Wind-driven circulation
Thermohaline circulation
Zonally-averaged buoyancy forcing
Joe LaCasce Section for Meteorology and Oceanography
Ocean Circulation
Wind-driven circulation
Thermohaline circulation
Thermally-driven flow
What type of flow do we expect, with warming at low latitudes and
cooling at high latitudes? From thermal wind:
∂
g ∂
ug =
ρ
∂z
ρc f ∂y
Use the equation of state, assume temperature dominates:
g ∂
gα ∂
∂
ug =
ρc (1 − α(T − Tref )) = −
T
∂z
ρc f ∂y
f ∂y
In the northern hemisphere:
∂
T < 0, f > 0
∂y
Joe LaCasce Section for Meteorology and Oceanography
→
∂
ug > 0
∂z
Ocean Circulation
Wind-driven circulation
Thermohaline circulation
Thermally-driven flow
Joe LaCasce Section for Meteorology and Oceanography
Ocean Circulation
Wind-driven circulation
Thermohaline circulation
Surface velocities in a thermally-driven box
Gjermundsen and LaCasce (2015)
Joe LaCasce Section for Meteorology and Oceanography
Ocean Circulation
Wind-driven circulation
Thermohaline circulation
Circulation schematic, Nordic Seas
C. Mauritzen (1996), Bentsen et al. (2002)
Joe LaCasce Section for Meteorology and Oceanography
Ocean Circulation
Wind-driven circulation
Thermohaline circulation
Oceanic convection
Joe LaCasce Section for Meteorology and Oceanography
Ocean Circulation
Wind-driven circulation
Thermohaline circulation
Stommel-Arons (1960)
Joe LaCasce Section for Meteorology and Oceanography
Ocean Circulation
Wind-driven circulation
Thermohaline circulation
Stommel-Arons abyssal layer
Joe LaCasce Section for Meteorology and Oceanography
Ocean Circulation
Wind-driven circulation
Thermohaline circulation
Stommel-Arons model
Same equations as for Gulf Stream:
1
ρc
1
fu = −
ρc
−fv = −
∂
p − ru
∂x
∂
p − rv
∂y
Cross-differentiate to make a vorticity equation:
βv = f
∂w
∂v
∂u
− r(
−
)
∂z
∂x
∂y
Joe LaCasce Section for Meteorology and Oceanography
Ocean Circulation
Wind-driven circulation
Thermohaline circulation
Stommel-Arons model
In the abyssal layer, assume no vertical shear:
∂
∂
u=
v =0
∂z
∂z
Integrate vorticity equation vertically, from the (flat) bottom to the
top of the abyssal layer:
βHA v = fwT − rHa (
∂v
∂u
−
)
∂x
∂y
where HA is the layer depth and wT is vertical velocity at top
Joe LaCasce Section for Meteorology and Oceanography
Ocean Circulation
Wind-driven circulation
Thermohaline circulation
Basin interior
Away from boundaries, Sverdrup balance:
βHA v = fwT
We don’t know wT . Stommel assumed this was constant and
upward in the interior: wT = W . So:
v=
fW
>0
βHA
everywhere in the interior
Joe LaCasce Section for Meteorology and Oceanography
Ocean Circulation
Wind-driven circulation
Thermohaline circulation
Basin interior
Joe LaCasce Section for Meteorology and Oceanography
Ocean Circulation
Wind-driven circulation
Thermohaline circulation
Boundary current
Flow returns in deep western boundary current
βvB = −r (
West :
∂uB
∂vB
∂vB
−
) ≈ −r
∂x
∂y
∂x
∂vB
>0
∂x
Joe LaCasce Section for Meteorology and Oceanography
→
vB < 0
Ocean Circulation
Wind-driven circulation
Thermohaline circulation
Stommel-Arons circulation
Joe LaCasce Section for Meteorology and Oceanography
Ocean Circulation
Wind-driven circulation
Thermohaline circulation
Swallow’s observation of deep western boundary current
Joe LaCasce Section for Meteorology and Oceanography
Ocean Circulation
Wind-driven circulation
Thermohaline circulation
Summary
Wind-forcing drives gyres with western boundary currents
Buoyancy forcing drives a global circulation with deep western
boundary currents and interior upwelling. Largely driven by
surface heating/cooling, although sensitive to fresh water
input (melting).
How these two interact is not well understood
Extremely important for the climate system
Joe LaCasce Section for Meteorology and Oceanography
Ocean Circulation