39.9.3 Overturning streamfunction

The question now is how to choose the reference values $\phi_{o}$ and s_o to simplify the computation of the overturning streamfunction. In general, evaluation of the vertical transport term is more difficult than the meridional transport term. One is therefore motivated to focus on equation (39.140) when computing the streamfunction, rather than equation (39.139). For choosing the reference value s_o of the generalized vertical coordinate, it is useful to consider the water budget and how it closes. In the rigid lid, the water budget is closed within the ocean domain. As such, so long as the value of s_o corresponds to a value anywhere completely outside the ocean domain, the vertical transport term $\int_{\phi_{o}}^{\phi} d\phi' \; {\mathcal W}(\phi',s_{o},t)$ will vanish. For the free surface, however, the possibility of surface water fluxes allows for an open water budget above the ocean surface. Since there is no attempt here to account for water cycling through the rock beneath the ocean, one can assume all water transport in rock vanishes. Hence, by taking s_o to be some value completely beneath the ocean bottom, the vertical transport term can again be dropped with the free surface. As a consequence, a general expression for the overturning streamfunction, valid for both the free surface and rigid lid, is given just by the meridional transport term

$$\psi(\phi, s,t) = -\int_{s_{o}}^{s} ds' \; {\mathcal V}(\phi,s',t) = -a \, \cos\phi \int_{s_{o}}^{s} ds' \int d\lambda \; z_{s} \,v.$$ (39.138)

In practice, of course, it is not necessary to evaluate the integral anyplace beneath the ocean bottom, since the water velocity vanishes there. On the bottom, the generalized vertical coordinate takes on the non-constant value $s(\lambda,\phi,z=-H,t)$. As such, when integrating just to the ocean bottom, it is necessary to perform the vertical integral first, and then the zonal integral

 

$$\psi(\phi,s,t) = -a \, \cos\phi \int d\lambda \int_{s(\lambda,\ph... ... \cos\phi \int d\lambda \int_{-H(\lambda,\phi)}^{z(\lambda,\phi,s,t)} dz' \; v,$$ (39.139)

where $z_{s} \, ds = dz$ was used to reach the final expression. Note that $z(\lambda,\phi,s,t)$ is the depth of the smooth surface whose generalized vertical coordinate has the value s.

For the rigid lid, the expression (39.142) for the streamfunction can be brought to a more familiar form by noting that the volume of water passing northward across any latitude must balance the volume flowing southward. Therefore,

 

$$\int d\lambda \int_{-H(\lambda,\phi)}^{0} dz' \; v(\lambda,\phi,z',t) = 0.$$ (39.140)

The result leads to the familiar expression for the rigid lid overturning streamfunction

 

$$\psi_{rl}(\phi,s,t) = a \, \cos\phi \int d\lambda \int^{s(\lambda... ...s'} \,v) = a \, \cos\phi \int d\lambda \int^{0}_{z(\lambda,\phi,s,t)} dz' \; v.$$ (39.141)

Again, this expression is valid only for the rigid lid, since the balance given by equation (39.143) is only valid in this case. For the free surface, the more general expression (39.142) must be used. The differences between these two expressions will be further discussed in the next section.

In the ocean interior, the no-normal flow condition implies that the overturning streamfunction is a constant along the side and bottom boundaries (for the relevant arguments, see Section 6.5 in which the boundary conditions for the barotropic streamfunction are derived). For the rigid lid, the absence of fresh water input to the ocean surface also implies that its overturning streamfunction is a constant at the ocean surface. The choice $\psi(\phi_{o},s_{o},t) = 0$ means that the rid lid overturning streamfunction is zero along all the boundaries. For the free surface, however, the overturning streamfunction need not be a constant on the ocean surface, due to the presence of surface water fluxes, whereas it remains zero on the sides/bottom just as for the rigid lid.