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Plasma turbulence in the tokamak scrape-off layer (SOL) region, where magnetic field lines are open and intersect the reactor inner walls, determines the heat loads on the limiter or divertor targets. This is one of the most crucial issues on the way towards a fusion reactor. Since SOL plasma is colder compared to the tokamak core, it is reasonable to use a fluid approximation to describe its dynamics. In particular the drift-reduced Braginskii equations are chosen to study the SOL plasma turbulence. To further simplify the drift-reduced Braginskii equations, the Boussinesq approximation is also applied in a number of numerical codes. This approximation consists in considering the plasma density constant in the evaluation of the divergence of the polarisation current, which simplifies substantially the solution of the Poisson equation necessary to evaluate the electric potential. In this study a new formulation of the drift-reduced Braginskii equations is presented together with a new numerical implementation that allow us to relax the Boussinesq approximation in the plasma turbulent code GBS. We show the energy conservation properties of the new system of equations. Also we compare the results of three-dimensional turbulent simulations with and without the Boussinesq approximation.