Alternative derivation of Sverdrup RelationConstruct vorticity equation from geostrophic balance

(1)

(2)

Integrating over the whole ocean depth, we have

Assume =constant

where is the entrainment rate from the surface Ekman layer

The Sverdrup transport is the total of geostrophic and Ekman transport.The indirectly driven Vg may be much larger than VE.

at 45oN

For a rotating solid object, the vorticity is two times of its angular velocity

Vorticity

In physical oceanography, we deal mostly with the vertical component of vorticity, which is notated as

Relative vorticity is vorticity relative to rotating earth

Absolute vorticity is the vorticity relative to an inertia frame of reference (e.g., the sun)

Planetary vorticity is the part of absolute vorticty associated with Earth rotation f=2sin, which is only dependent on latitude.

Absolute vorticity =Relative vorticity + Planetary Vorticity

FIGURE 7.12Vorticity. (a) Positive and (b) negative vorticity. The (right) hand shows the direction of the vorticity by the direction of the thumb (upward for positive, downward for negative). From Talley et al. (2011, DPO)

Vorticity Equation

, From horizontal momentum equation,

(1)

(2)

Taking , we have

Shallow Water EquationConstant and uniform density , incompressible

Aspect ratio hydrostatic

Integrating the hydrostatic equation

Use the boundary condition at the sea surface, z=, p=0

The horizontal pressure gradient is independent of z

Therefore, it is consistent to assume that the horizontal velocities remain to be independent of z if they are so initially.

Assume that at the bottom of the sea, i.e., z=-hB

At the sea surface, z=,

Let be the total depth of water, we have

The system of the shallow water equations

Vorticity Equation for the Shallow Water System

>> x, y

No twisting No baroclinicityMain mechanism:Vortex stretching

For a layer of thickness H, consider a material column

We get

or Potential Vorticity Equation

Potential Vorticity Conservation

Alternative derivation of Sverdrup RelationConstruct vorticity equation from geostrophic balance

(1)

(2)

Integrating over the whole ocean depth, we have

Assume =constant

where is the entrainment rate from the surface Ekman layer

The Sverdrup transport is the total of geostrophic and Ekman transport.The indirectly driven Vg may be much larger than VE.

at 45oN

In the ocean’s interior, for large-scale movement, we have the differential form of the Sverdrup relation

i.e., <<f

FIGURE 7.13Sverdrup balance circulation (Northern Hemisphere). Westerly and trade winds force Ekman transport, creating Ekman pumping and suction and hence Sverdrup transport. See also Figure S7.12.From Talley et al(2011, PDO)

Have vorticity input

No vorticity dissipationCannot hold everywhere in a closed basin

Friction has to be important somewhere within the basin

For most of the basin

Question

What causes the strong narrow currents on the west side of the ocean basin?

The westward Intensification

Westerly winds in north, easterly winds in south

Ekman effect drives the water to the center,

Increase sea level generates anticyclonic geostrophic currents

Internal friction (or bottom Ekman layer) generate downslope cross-isobaric flow, which balance the wind-driven Ekman transport

From Knauss, 1997

Vorticity Balance on a f-Plane

Vorticity Equation:

Vorticity generated by wind stress is consumed by local friction

Stommel’s ModelRectangular ocean of constant depth

Surface stress is zonal and varies with latitude onlySteady ocean state

Simple friction term as a drag to current

Vorticity balance: Sverdrup balance +friction

Flow patterns in this ocean for three conditions:(1) non-rotating ocean (f=0)(2) f-plane approximation (f=constant)(3) -plane approximation (f=fo+y)

f-plane -plane

Wind stress () + friction () =0

Negative vorticity generation Positive vorticity generation

The effect

Generate negative vorticity Generate positive vorticity

In the west, water flows northward

Wind stress () + Planetary vorticity () + Friction () = 0

In the east, water flows southward

Wind stress () + Planetary vorticity () + Friction () = 0

Friction (W) > Friction (E)

Non-rotation Ocean, f=0

If f is not constant, then

F is dissipation of vorticity due to friction