Wind Driven Circulation III Closed Gyre Circulation QuasiGeostrophic

Vorticity Equation In physical oceanography, we deal mostly with the vertical component of vorticity

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

Considering the case of constant . For a shallow layer of water (depth H<<L),

Using the Continuity Equation For a layer of thickness H, consider a material column

Alternative derivation of Sverdrup Relation Construct vorticity equation from geostrophic balance (1) (2) Integrating

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

A more general form of the Sverdrup relation Ekman layer at bottom Spatially changing

Define Ep is the work done to pile up the water above the bottom

It can be shown that Therefore, the balance of forces can be written as

Define Baroclinic geostrophic transport Barotropic geostrophic transport Ekman transport in the surface layer Ekman

Following Sverdrup’s approach, we take Surface stress curl Bottom topography effect Vanish if the

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Wind Driven Circulation III • • Closed Gyre Circulation Quasi-Geostrophic Vorticity Equation Westward intensification Stommel Model Munk Model Inertia boundary layer Numerical results Observations

Vorticity Equation 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

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

Considering the case of constant . 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 vorticity

Using the Continuity Equation For a layer of thickness H, consider a material column We get or Potential Vorticity Equation

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

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

A more general form of the Sverdrup relation Ekman layer at bottom Spatially changing sea surface height and bottom topography z. B and pressure p. B. Assume atmospheric pressure p ≈0. Let , Integrating Similarly over the vertical column, we have

Define Ep is the work done to pile up the water above the bottom is the total potential energy of sea water with S=35, T=0 o. C function of bottom topography only is the potential energy anomaly

It can be shown that Therefore, the balance of forces can be written as

Define Baroclinic geostrophic transport Barotropic geostrophic transport Ekman transport in the surface layer Ekman transport in the bottom layer Therefore, we have

Following Sverdrup’s approach, we take Surface stress curl Bottom topography effect Vanish if the bottom is flat Or flow follows topographic contour Net baroclinic geostrophic transport is zero