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In this talk, we study the asymptotic behavior for the incompressible anisotropic Navier-Stokes equations with the non-slip boundary condition in a half space of when the vertical viscosity goes to zero.First, by multi-scale analysis we formally de duce an asymptotic expansion of the solution to the problem, which shows that the boundary layer appears in the zero-th order term of the expansion of tangential ve locity field and satisfies a nonlinear parabolic-elliptic coupled system.The solution to the Navier-Stokes equations is constructed through a composite asymptotic ex pansion involving a solution of degenerate incompressible Navier-Sto kes equations and the boundary layer, plus an error term.Secondly, we study the well-posedness of the nonlinear boundary value problems for degenerate incompre-ssible Navier Stokes equations and the boundary layer equations, and then rigorous-sly justify the asymptotic expansion by using the energy method.We obtain the convergence re sults of the vanishing vertical viscosity limit, that is, the solution of the incompressible anisotropic Navier-Stokes equations goes to the solution of degenerate incompressible Navier-Stokes equations away from the boundary and to the boundary layer near the boundary, for a short time independent of the vertical.