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

Consider the vorticity balance of an homogeneous fluid (=constant) on an f-plane

If f is not constant, then

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

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

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: (geostrophy) By definition 

Taking into the equations, we have

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

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

Non-dmensional 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)  (satistfying western boundary condition) Example: Let , . Over a rectangular basin (x=0,1; y=0,1)

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.

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

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

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