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 X[0, z_{top}], where Pi_2 is a two dimensional torus, with an eddy vertical 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 <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.