J f 1 qi wo y h Jf2 q2 Vh 2 DV2f2

where J (g, h) = gxhy - gyhx is the nonlinear Jacobian term, q1 and q2 are potential vorticity in the upper and lower layers qi = Py + Ff - fi) (4.150a)

P0f0

is the Ekman pumping rate, f 2

g'H r where g' is the reduced gravity, H is the undisturbed layer thickness, and Xr is the Rossby radius of deformation. We have neglected relative vorticity in these equations because it is negligible for basin-scale motions. The F(fi - fi-1) terms are the contribution due to interface deformation, also called the stretching term, noting that the interface height is proportional to the streamfunction difference. The DV^f2 term is the bottom friction, and D is a small parameter.

The interfacial friction is parameterized in terms of $j. As a crude way of mimicking baroclinic instability, the interfacial friction is assumed to be linearly proportional to the velocity shear

where R is a small parameter.

Steady and frictionless flow

Assuming that R and D are of the same order and very small, Eqns. (4.149a) and (149b) are reduced to

where O(R) denotes small terms on the order of R. From these two equations we obtain the barotropic solution

px where xe = xe (y) is the eastern boundary of the model. This solution is called the barotropic solution. For simplicity, we will assume that the Ekman pumping velocity is identically zero

Ekman Pumping

Fig. 4.22 Contours of The dashed circle is r = rj, i.e., the bounding contour of the barotropic streamfunction: a the forcing is weak, so qq2 is dominated by the planetary vorticity term fy, and all the contours are open; b the forcing is stronger, so there is a region of closed contours. Flows in this region are shielded from the blocking geostrophic contours started from the "eastern" boundary (the right edge of the forcing domain where fb = 0).

Fig. 4.22 Contours of The dashed circle is r = rj, i.e., the bounding contour of the barotropic streamfunction: a the forcing is weak, so qq2 is dominated by the planetary vorticity term fy, and all the contours are open; b the forcing is stronger, so there is a region of closed contours. Flows in this region are shielded from the blocking geostrophic contours started from the "eastern" boundary (the right edge of the forcing domain where fb = 0).

outside a circle (Fig. 4.22). We further assume the fluid to be motionless at infinity; thus, the eastern half of this circle can be chosen as the effective eastern boundary of the model. For a given Ekman pumping rate distribution, wo, the barotropic streamfunction can be calculated accordingly. Note that the Jacobian term has a very useful property:

0 0

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