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

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 Approximation.

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

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  1. 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

  2. Quasi-Geostrophic Approximation Quasi-geostrophic approximation has three components (1) The β-plane apporximation (2) Small surface deviation (3) Geostrophic approximation in terms of fo Basic condition

  3. Quasi-Geostrophic Approximation Potential Vorticity Quasi-Geostrophic Potential Vorticity Quasi-Geostrophic Potential Vorticity Equation Defines the evolution of geostrophic stream function ψ

  4. Quasi-Geostrophic Approximation If we ignore the surface change (or have a rigid lid), we have the absolute vorticity conservation, i.e.,

  5. What does QG momentum equation look like? Momentum equation Continuity equation is ageostrophic flow • is responsible for the divergence in the QG system • has a rotational component • is totally determined by geostrophic flow at any given instance

  6. Quasi-Geostrophic Approximation Replace the relative vorticity by its geostrophic value Approximate the horizontal velocity by geostrophic current in the advection terms Under -plane approximation, f=fo+y, we have

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

  8. Boundary Value Problem Boundary conditions on a solid boundary L (1) No penetration through the wall   (2) No slip at the wall

  9. Quasi-geostrophic vorticity equation where Boundary conditions on a solid boundary L (1) No penetration through the wall (used for the case of no horizontal diffusion) along the boundary L (2) No slip at the wall along the boundary L n is the unit vector perpendicular to the boundary L

  10. Different terms are important at different places of the basin f-plane -plane

  11. Non-dimensionalize Quasi-Geostrophic Vorticity Equation Define non-dimensional variables based on independent scales L and o The variables with primes, as well as their derivatives, have no unit and generally have magnitude in the order of 1. e.g.,

  12. Note that U has not been decided yet.

  13. Non-dmensional vorticity equation If we choose we have Sverdrup relation Define the following non-dimensional parameters , nonlinearity. , , bottom friction. , , lateral friction. ,

  14. 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)

  15. 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  <<L In dimensional terms, The Sverdrup relation is broken down.

  16. The Stommel model Bottom Ekman friction becomes important in WBL. , S<<1. 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

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

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

  19. 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 an eastern boundary layer?

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