39.9.5 Overturning streamfunction in the $(\phi ,z)$ plane

For those running z-level models such as MOM, the simplest choice for vertical coordinate is s=z, where z is the time independent height of the constant depth surfaces. Derivatives of the streamfunction yield the zonally integrated Eulerian meridional and vertical transport

$$-\partial_{z} \psi(\phi,z,t) a \, \cos\phi \, \int d\lambda \; v(\lambda,\phi,z,t)$$ = (39.148)

$$a^{-1} \, \partial_{\phi} \psi(\phi,z,t) a \, \cos\phi \, \int d\lambda \; w(\lambda,\phi,z,t),$$ = (39.149)

and the streamfunction from equation (39.142) is given by

$$\psi(\phi,z,t) -a \, \cos\phi \int d\lambda \int_{-H(\phi,\lambda)}^{z} dz' \; v.$$ = (39.150)