The last two terms in Eqn. (4.114) can be neglected compared with other terms; thus, we have a geostrophic balance in the cross-stream direction; while Eqn. (4.115) indicates that momentum balance in the downstream direction must be ageostrophic. In this case the system is in a semi-geostrophic balance. Semi-geostrophy allows us to obtain useful analytical solution within strong boundary currents. The corresponding boundary conditions are f = 0 at n = 0, f ^ fI (0, y) at n ^^ (4.116)

where the subscript I denotes the interior solution discussed above. Dropping the small terms in Eqn. (4.114) and integrating from the western boundary leads to h2 - hi, f = (4.117)

where hw is the layer thickness at the western wall. Letting n ^^ and using boundary conditions from Eqn. (4.116), we obtain h2w = hf - 2f fi (0, y) ^ h2w = h2e - 2ptx (0) (4.118)

Cross-differentiating Eqns. (4.114, 4.115) leads to fn +(h ^ = 0

Integrating it once gives

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