a cos 0

Substituting Eqns. (4.169) and (4.170) into Eqn. (4.171) gives the vorticity equation cos 0h2v2 = a sin 0 we (4.172)

This equation corresponds to Eqn. (4.35) which is formulated in a j-plane, and it can be rewritten as h2U2- Vh (f /h2) = f We/h2 (4.173)

Equations (4.169) and (4.170) state that h contours are streamlines, and Eqn. (4.173) states that potential vorticity changes along streamlines due to Ekman pumping. Substituting this relation into Eqn. (4.169) and integrating in the zonal direction gives h2 = h2e + D2 (4.174)

where h2e is the constant layer thickness along the eastern boundary and

is the layer thickness (square) deviation from that at the eastern boundary. Note that the thickness of each layer has to be constant along the eastern boundary of the model basin in order to satisfy the boundary condition that there is no geostrophic flow across the boundary. Within the formulation of an ideal-fluid thermocline, we cannot determine h2e. In fact, h2e is an external parameter of the ventilated thermocline model, and different values of h2e would give rise to different solutions. The basic philosophy of the ideal-fluid thermocline is to assume that h2e is set up by some external processes, such as the thermohaline circulation, not directly simulated in the model. When h2 is known, both u2 and v2 can be determined geostrophically.

As water moves southward, crossing latitude 01, the second layer continues to flow southward under the first layer. South of the outcropping line both layers are in motion; the momentum equations for the second layer have the same form as Eqns. (4.169) and (4.170), although now h = hi + h2; thus we have

a cos 0 dk

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