Spatial extrema of advected scalars

Scalar fields satisfying the stationary advection–diffusion equation with no source or sink terms cannot have strong local extrema. This can be deduced from the elliptical nature of the equation. Here, however, an alternative, original and more physically motivated proof is offered. It highlights the positive role of diffusion in preventing extrema and the inability of advection to create them. Application is made to the theory of energy transfer by species interdiffusion and some anomalous numerical solutions from the literature are identified.