39.9.7 Overturning streamfunction in the $(\phi ,\rho ^{(0)})$ plane

Another choice is to set $s = \rho^{(0)}$, where $\rho^{(0)}$ is the potential density using the ocean surface as the reference pressure. This choice, as with the choice $s = \theta$, is adequate only when the ocean stratification is monotonic in $\rho^{(0)}$. The discussion with $s = \theta$ follows through here, with the material derivative now given by

$$\frac{D \rho^{(0)}}{Dt} \frac{\partial \rho^{(0)} }{\partial \theta} \, \dot{\theta} + \frac{\partial \rho^{(0)} }{\partial S} \, \dot{S},$$ = (39.154)

with S the salinity.