Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 331 / 4 / a

The zero streamfunction gives zero velocity. The temperature ${\theta }_0=-z$ obeys both prescribed temperature values, has no horizontal gradient, and has zero Laplacian. Thus it satisfies both equations with ${\psi }_0=0$:This is the conductive state of Darcy-Bénard convection.