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=2  sin , which is only dependent on latitude. Absolute vorticity =Relative vorticity + Planetary Vorticity

Vorticity Equation, From horizontal momentum equation, (1) (2) Taking, we have

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 small. Then we have the simplified vorticity equation Since the vorticity equation can be written as (ignoring friction)  +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 Assume  =constant

whereis the entrainment rate from the surface Ekman layer The Sverdrup transport is the total of geostrophic and Ekman transport. The indirectly driven V g may be much larger than V E. at 45 o N

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

If f is not constant, then

Assume geostrophic balance on  -plane approximation, i.e., (  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 barotropic

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

Posing the gyre problem Boundary conditions on a solid boundary L (1) No penetration through the wall  (2) No slip at the wall

Non-dimensional Equation, An Example Consider a homogeneous fluid on a  -plane Define the following non-dimensional variables: By definition (geostrophy) 

Define the non-dimensional parameters Rossby Number Horizontal Ekman Number Vertical Ekman Number Ekman depth Then, we have (with prime dropped) The solution

In the interior of the ocean, E h <<1 and E z <<1 (geostrophy) Near the bottom or surface, E z ≈O(1)  In the surface and bottom boundary layers, the vertical scales are redefined (shortened, a general character of a boundary layer)

Taking into the equations, we have

Non-dimensional vorticity 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.,

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

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 In dimensional terms, The Sverdrup relation is broken down., the length of the layer  <<L The non-dimensionalized distance is

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

Let, we have Re-scaling in the boundary layer: Take into As  =0,  =0. As ,  I

The solution foris,.  A=-B , ( can be the interior solution under different winds) For,,.,.,

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?