Oceanic fronts are important sites as they enhance the biological productivity and the cross isopycnal exchanges of heat and salt which influences the transformation of water masses. The ageostrophic secondary circulation (ASC) near open ocean fronts has been previously diagnosed in observations using the quasi-geostrophic (QG) and balanced model Omega equations that assume inviscid, adiabatic flow conditions. For idealized cases, previous studies using numerical models and semi-geostrophic (SG) equations typically with constant mixing coefficients show that viscous diabatic processes can influence the ASC. This study adds vertical mixing to the traditional two-dimensional QG and SG Omega equations. We diagnose the ASC due to variable mixing and constant confluence (α) in Azores front as well as in several idealized cases. The resulting ASCs indicate a strong dependency on the Burger number, Bu = NH/fL when Bu <<1, where H and L are the vertical and horizontal scale of front, N is the buoyancy frequency, and f is the Coriolis parameter. The relative importance between deformation rate and vertical mixing can be parameterized by αH²/Av, where Av is the vertical viscosity. The tilting of planetary vorticity due to along front ageostrophic shear is found to be important when α/f is O(1) especially for weak fronts, i.e., Bu <<1. The re-analyzed SeaSoar sections in Azores front indicate that the ASCs in horizontal and vertical direction are strongly modified in top 80 m.