39.9.6 Overturning streamfunction in the $(\phi ,\theta )$ plane

Another choice for the vertical coordinate is $s = \theta$, with $\theta$ the potential temperature. This choice was discussed by Bryan and Sarmiento (1985). Taking $s = \theta$ makes sense only when $\theta$ surfaces are monotonic. Assuming such, the overturning streamfunction from equation (39.142) is given by

$$\psi(\phi,\theta,t) = -a \, \cos\phi \int d\lambda \int^{z(\lambda,\phi,\theta,t)}_{-H(\lambda,\phi)} d z' \; v(\lambda,\phi,z',t).$$ (39.151)

Derivatives of the streamfunction yield

$$-\partial_{\theta} \psi a \, \cos\phi \, \int d\lambda \; z_{\theta} \, v$$ = (39.152)

$$a^{-1} \, \partial_{\phi} \psi a \, \cos\phi \, \int d\lambda \; (z_{t} + z_{\theta} \, \dot{\theta}),$$ = (39.153)

where z_t is the time tendency for the height of a $\theta$surface. Note that $z_{\theta} > 0$ for a stably stratified fluid dominated by temperature. For a fluid with $\theta$ the only active tracer, $\dot{\theta} = 0$ results when the flow is adiabatic. In this case, and assuming a steady state, the meridional streamfunction $\psi$ is dependent only on the potential temperature: $\partial_{\phi}\psi = 0$.