Class Meeting Nov. 26, 2:00pm-4:45pm Research I, 162

Geostrophic Approximations Hydrostatic primitive equations Geostrophic approximation (diagnostic) In between (approximately geostrophic models) prognostic inertial-gravity waves filtered Planetary geostrophic approximation Quasi-geostrophic approximation ……

Planetary Geostrophic Approximation Momentum equation where Continuity Equation Vorticity Equation where Potential Vorticity Equation Basic condition L must be large, L~R, basin-scale movement

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

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

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

Quasi-Geostrophic Approximation 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

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

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

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

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

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.,

Note that U has not been decided yet.

Non-dmensional vorticity equation If we choose we have Sverdrup relation Define the following non-dimensional parameters , 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  <<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 an eastern boundary layer?