1. Wind-driven circulation
●Wind pattern and oceanic gyres
●Sverdrup Relation
●Western boundary currents: the Gulf Stream
●Vorticity dynamics: Stommel and Munk models
●Rossby wave and mesoscale eddies

4. Surface current measurement from ship drift
Current measurements are harder to make than T&S
The data are much sparse.

5. More recent equipment: surface drifter
a platform designed to move with the ocean current

6. Surface current observations

8. Annual Mean Surface Current North Atlantic, 1995-2003
Drifting Buoy Data Assembly Center, Miami, Florida
Atlantic Oceanographic and Meteorological Laboratory, NOAA

9. Annual Mean Surface current
from surface drifter measurements
Indian Ocean,
Drifting Buoy Data
Assembly Center
Miami, Florida
Atlantic Oceanographic and Meteorological Laboratory
NOAA

10. Annual Mean Surface Current Pacific Ocean, 1995-2003
Drifting Buoy Data Assembly Center, Miami, Florida
Atlantic Oceanographic and Meteorological Laboratory, NOAA

11. Schematic picture of the major surface currents of the world oceans
Note the anticyclonic circulation in the subtropics (the subtropical gyres)

12. Relation between surface winds and subtropical gyres

13. Surface winds and oceanic gyres: A more realistic view
Note that the North Equatorial Counter Current (NECC) is against the direction of prevailing wind.

14. Mean surface current tropical Atlantic Ocean
Note the North Equatorial Counter Current (NECC)

15. Consider the following balance in an ocean of depth h of flat bottom
Consider the following balance in ocean of depth h ,
Sverdrup Relation
Consider the following balance in an ocean of depth h of flat bottom ,
Integrating vertically from –h to 0, we have
(neglecting bottom stress and surface height change)
(1)
(2)
where
and
Differentiating
, we have
Using continuity equation
and ,
Sverdrup relation
we have
and ,

16. More generally,
and
Since
, we have
set x =0 at the eastern boundary,

18. Mass Transport Since Let , ,  where ψ is stream function.
Problem: only one boundary condition can be satisfied.

19. Alternative derivation of Sverdrup Relation
Construct vorticity equation from geostrophic balance
(1) 
(2)
Integrating over the whole ocean depth, we have
is the entrainment rate from the Ekman layer
where
at 45oN
The Sverdrup transport is the total of geostrophic and Ekman transport.
The indirectly driven Vg may be much larger than VE.

20. In the ocean’s interior, for large-scale movement, we have the differential form of the Sverdrup relation
i.e.,
ζ<<f

21. 1 Sverdrup (Sv) =106 m3/s

23. A more general form of the Sverdrup relation
Lower Ekman layer
Spatially chaning sea surface height η and bottom topography zB and pressure pB.
Assume atmospheric pressure pη≈0.
Let ,
Integrating
over the vertical column, we have
Taking into account of these factors, the meridional transport can be derived as ,
where ,

24. Vorticity Equation , (1) (2) Taking , we have
In physical oceanography, we deal mostly with the vertical component of vorticity ,
which is notated as
From horizontal momentum equation,
(1)
(2)
Taking
, we have

25. Considering the case of constant ρ
Considering the case of constant ρ. For a shallow layer of water (depth H<<L), u and v are not function of z because the horizontal pressure gradient is not a function of z.
(In general, the vortex tilting term, is usually small.
Then we have the simplified vorticity equation
Since
the vorticity equation can be written as (ignoring friction)
ζ+f is the absolute vortivity

26. Consider the balance on an f-plane

27. If f is not constant, then

28. Assume geostrophic balance on β-plane approximation, i.e.,
(β is a constant)
Vertically integrating the vorticity equation
we have
The entrainment from bottom boundary layer
The entrainment from surface boundary layer
We have
where

29. Quasi-geostrophic vorticity equation
For
and
, we have
and
where
(Ekman transport is negligible)
Moreover,
We have
where

30. Non-dmensional equation
Non-dimensionalize all the dependent and independent variables in the quasi-geostrophic equation as
where
For example,
The non-dmensional equation
where
, nonlinearity. , ,
, bottom friction. ,
, lateral friction.

31. Interior (Sverdrup) solution
If ε<<1, εS<<1, and εM<<1, we have the interior (Sverdrup) equation:
(satistfying eastern boundary condition) 
(satistfying western boundary condition)
Example:
Let , .
Over a rectangular basin
(x=0,1; y=0,1)

32. Westward Intensification
It is apparent that the Sverdrup balance can not satisfy the mass conservation and vorticity balance for a closed basin. Therefore, it is expected that there exists a “boundary layer” where other terms in the quasi-geostrophic vorticity is important. This layer is located near the western boundary of the basin. Within the western boundary layer (WBL),
, for mass balance
The non-dimensionalized distance is
, the length of the layer δ <<L
In dimensional terms,
The Sverdrup relation is broken down.

33. The Stommel model Bottom Ekman friction becomes important in WBL.
at x=0, 1; y=0, 1.
(Since the horizontal friction is neglected, the no-slip condition can not be enforced. No-normal flow condition is used).
Interior solution
In the boundary layer, let (
), we have

34. can be the interior solution under different winds) ,
The solution for is ,
.  A=-B ,
ξ→∞, (
can be the interior solution under different winds) ,
For , , .
For , , .

36. The dynamical balance in the Stommel model
In the interior,  
Vorticity input by wind stress curl is balanced by a change in the planetary vorticity f of a fluid column.(In the northern hemisphere, clockwise wind stress curl induces equatorward flow). 
In WBL,   
Since v>0 and is maximum at the western boundary, ,
the bottom friction damps out the clockwise vorticity.
Question: Does this mechanism work in a eastern boundary layer?

37. Munk model Lateral friction becomes important in WBL.
Within the boundary layer, let
, we have ,
Wind stress curl is the same as in the interior, becomes negligible in the boundary layer.
For the lowest order,
. If we let
, we have
. And for , . .
The general solution is
Since
, C1=C2=0.

38. Using the no-slip boundary condition at x=0,
Total solution
Using the no-slip boundary condition at x=0,
(K is a constant). 
.to ,
Considering mass conservation
K=0 
Western boundary current

41. Western Boundary current: Gulf Stream

43. Gulf Stream Transport

46. Meso-scale eddies

50. Vertical structure of the ocean:
Large meridional density gradient in the upper ocean, implying significant vertical shear of the currents with strong upper ocean circulation

51. Subduction
Water mass formation by subduction occurs mainly in the subtropics.
Water from the bottom of the mixed layer is pumped downward through a convergence in the Ekman transport
Water “sinks” slowly along surfaces of constant density.

52. Sketch of water mass formation by subduction
First diagram: Convergence in the Ekman layer (surface mixed layer) forces water downward, where it moves along surfaces of constant density. The σt surface, given by the TS-combination 8°C and 34.7 salinity, is identified.
Second diagram: A TS- diagram along the surface through stations A ->D is identical to a TS-diagram taken vertically along depths A´ - D´.

54. Interaction between the Subtropical and Equatorial Ocean Circulation:
The Subtropical Cell

55. The permanent thermocline and Central Water
The depth range from below the seasonal thermocline to about 1000 m is known as the permanent or oceanic thermocline.
It is the transition zone from the warm waters of the surface layer to the cold waters of great oceanic depth
The temperature at the upper limit of the permanent thermocline depends on latitude, reaching from well above 20°C in the tropics to just above 15°C in temperate regions; at the lower limit temperatures are rather uniform around 4 - 6°C depending on the particular ocean.
The water of the permanent thermocline is named as the Central Water, which is formed by subduction in the subtropics.