39.9.10 Discrete vertical-meridional streamfunction

For the $(\phi ,z)$ plane, a straightforward discretization of equation (39.142)

$$\psi(\phi,z,t) = -a \, \cos\phi \int d\lambda \int_{-H(\lambda,\phi)}^{z} dz' \; v,$$ (39.155)

leads to the discretized overturning streamfunction valid for either the rigid lid or free surface

$$vmsf_{jrow,k} = -\csuj \; \sum_{i=2}^{imt-1} \sum_{m=k}^{kmt(i,jrow)} \dxui \; dzt_{m} \; u_{i,m,j,2,\tau}.$$ (39.156)

Note that the switch in the limits on the sum, relative to the integral, accounts for k increasing downward, whereas z increases upwards.

The vertical integration procedes from the ocean bottom, which is the bottom of the bottom-most velocity cell, upwards to the top of the particular grid cell of interest. As such, the vertical placement of the meridional streamfunction is at the interface of the cells in which the velocity is located. These points are labeled by zw_kin the model, where k=1,km. To aid in visualization, the top of the ocean at z=0 is plotted as well. For the rigid lid, the streamfunction value is zero at z=0; for the free surface, it represents the zonally averaged fresh water flux crossing the ocean-atmosphere interface.