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

Abstract

We introduce a scalar elliptic equation defined on a boundary layer given by Π2 × [0, ztop], where Π2 is a two dimensional torus, with an eddy vertical viscosity of order zα, α ∈ [0, 1], an homogeneous boundary condition at z = 0, and a Robin condition at z = ztop. We show the existence of weak solutions to this boundary problem, distinguishing the cases 0 ≤ α < 1 and α = 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.