Explicit solution of atmospheric Ekman flows with some types of Eddy viscosity

In this paper, we study the standard problem of the wind in the steady atmospheric Ekman layer with classical boundary conditions. We consider the system with varying eddy viscosity coefficients that are small perturbation of a constant. We derive the explicit solution by using a different argument in the previous works. For two layers, the eddy viscosity is constant in the upper layer, while is only continuous with height in the lower layer, we transform the system to a first order Riccati equation with a suitable initial value and derive the solution for piecewise-constant eddy viscosity.