Munk model Lateral friction becomes important in WBL. Within the boundary layer, let , we have , Wind stress curl is the same as in the interior, becomes negligible in the boundary layer. For the lowest order, . If we let , we have . And for , . . The general solution is Since , C1=C2=0.

Using the no-slip boundary condition at x=0, Total solution Using the no-slip boundary condition at x=0,  .to , Western boundary current

Are nonlinear terms negligible? Given The cross-stream distance from boundary to maximum velocity is The ratio between the nonlinear and dominant viscous terms is where The continuity relation is also used: Using U=O(2 cm/s), ß=O(10-13 cm-1 s-1), AH=4106 cm2/s, we have R=4. i.e., the nonlinear terms neglected are larger than the retained viscous terms, which causes an internal inconsistency within the frictional boundary layer.

Inertial Boundary Layer If >>S and M, Given a boundary layer exists in the west where Re-scaling with , we have Conservation of potential vorticity. or

The conservation equation may be integrated to yield is an arbitrary function of where This equation states that the total vorticity is constant following a specific streamline.

Let Now consider the region of large , where Take (interior stream function plus a boundary layer correction), must satisfy Now consider the region of large , where Take into equation

Retain only linear term in (and neglect some other small terms), we have Integrate once and use the boundary condition , we have

The decaying solution is of the form If , will be oscillatory and not satisfy the boundary condition. A necessary condition for the existence of a pure inertial boundary current is The decaying solution is of the form

The dimensional width of the inertial boundary layer is At those y’s where U is on shore and small, the width of the inertial current is small. As the point y0 is approached where U=0,  will shrink and finally be swallowed up within the thickness of a frictional layer.

Since equation is symmetric under transformation A similar inertial boundary layer can exist at the eastern boundary.

Inertial Currents with Small Friction In the presence of a small lateral friction, we can derive the perturbation equation as which makes the boundary layer possible only in the western ocean. Moreover, it can be shown that a inertial-vicious boundary layer can be generated in the northern part of the basin where characterized by a standing Rossby wave.

Assume the simple balance A parcel coming into the boundary layer has The effect of friction is reduced and the boundary layer is broadened.

Bryan (1963) integrates the vorticity equation with nonlinear term and lateral friction. The Reynolds number is define as And I/M ranges from 0.56 for Re=5 to 1.29 for Re=60.

Veronis (1966), nonlinear Stommel Model Both the Sverdrup and western boundary layer solutions require: 0 (f changes with latitude) Relative vorticity  is small compared to f. When f, the circulation pattern becomes more symmetric with wind vorticity input balanced by frictional effect throughout the whole basin