39.9.4 Comments on the free surface overturning streamfunction

Consider now an alternative method for computing the free surface streamfunction. It is less concise than equation (39.142), and requires some approximations whereas equation (39.142) is exact. However, it has the virtue of exposing the fresh water contribution to the streamfunction. It is discussed here for pedagogical reasons.

The central difference from the previous result is that the reference value of s_o is taken someplace above the ocean surface rather than beneath the bottom. Now care must be taken for how to handle the upper surface boundary condition where the fresh water enters the ocean. To do so, some approximations will be made. First, assume that the region for which the water flux enters the ocean is a completely mixed region, and use a z-coordinate as the generalized vertical coordinate for which z_t = 0 and $z_{s} \, \dot{s} = w$. Next, recall from Section 4.3.2 the surface boundary condition for the free surface

$$\eta_t = w + q_{w} - \vec{u}_{h} \cdot \nabla_{h} \eta \qquad z = \eta,$$ (39.142)

where $q_{w}(\lambda,\phi,t) > 0$ means that water enters the ocean through the free surface. In order to account for the large-scale effects of fresh water in the overturning streamfunction, it is sufficient to assume that $\eta = 0$, and in turn to employ

$$w \approx -q_{w} \qquad z = \eta.$$ (39.143)

This approximate expression is the ``Natural Boundary Condition'' of Huang (1993). Taking s_o = z_o = 0, this assumption brings the zonally integrated vertical transport at the ocean surface to the form

$${\mathcal W}(\phi,z=0,t) = - a \, \cos\phi \int d\lambda \; q_{w}.$$ (39.144)

Note that a net zonally integrated fresh water input leads to transport into the ocean, and so ${\mathcal W} < 0$. The corresponding overturning streamfunction then takes the form

 

$$\psi(\phi, s,t) = a \, \cos\phi \int d\lambda \left( \int^{z=0}_{... ...mbda,\phi,s,t)} dz' \; v - \int^{\phi}_{\phi_{o}} a \, d\phi' \; q_{w} \right).$$ (39.145)

Setting the fresh water term to zero in equation (39.148) recovers the rigid lid result (39.144). Note that the small transport in the region between $z=\eta$ and z=0 has been neglected. Using

$$\int^{0}_{z(\lambda,\phi,s,t)} dz' \; v \int_{-H}^{0} dz' \; v - \int_{-H}^{z(\lambda,\phi,s,t)} dz' \; v$$ = (39.146)

in equation (39.148), and recalling the general expression (39.142) for the overturning streamfunction, results in the identity

$$\int d\lambda \int_{-H(\lambda,\phi)}^{0} dz' \; v = \int^{\phi}_{\phi_{o}} a \, d\phi' \int d\lambda \; q_{w}.$$ (39.147)

Taking the reference latitude $\phi_{o} = \phi_{south}$ leads to a simple interpretation for this result. For example, if there is a net water input through the surface in the region to the south of a particular latitude, then in the steady state, there is a net northward meridional transport in the ocean (see Figure 39.1).

  

Figure 39.1: Sketch of an ocean domain in which there is a net input of surface fresh water through the free surface in the region to the south of a particular latitude. In the steady state, the result is a net northward meridional transport of water in the interior of the ocean at this latitude.