Surface boundary layers through a scalar equation with an eddy viscosity vanishing at the ground

We introduce a scalar elliptic equation defined on a boundary layer given by $\Pi_2 \times [0, z_{top}]$, where $\Pi_2$ is a two dimensional torus, with an eddy vertical eddy viscosity of order  $z^\alpha$, $\alpha \in [0, 1]$, an homogeneous boundary condition at $z=0$, and a Robin condition at $z=z_{top}$. We show the existence of weak solutions to this boundary problem, distinguishing the cases  $0 \le \alpha <1$ and $\alpha = 1$.  Then we carry out several numerical simulations, showing the ability of our model to accurately reproduce profiles close to those predicted by the Monin-Obukhov theory, by calculating stabilizing functions.