Plasma turbulence in the tokamak scrape-off layer (SOL) region, where magnetic field lines intersect the reactor inner walls, determines the heat load on the limiter or divertor targets. This is one of the most crucial issues on the way towards a fusion reactor. To tackle this problem we have improved the turbulent model in the GBS code, a new formulation of the vorticity equation that allow us to relax the Boussinesq approximation is implemented. The energy conservation properties of the new system of equations are evaluated. Also results of turbulent simulations in the SOL with and without the Boussinesq approximation are compared. In addition turbulent simulations across the last closed flux surface taking into account a cold and a hot ion regime are shown. These last simulations show a pressure gradient increase at the separatrix in the hot ion case at low resistivity.