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A classic Discontinuous Galerkin Method with low-order polynomials and explicit as well as semi-implicit time-stepping is applied to an atmospheric model employing the Euler equations on the β plane. The method, which was initially proposed without regard for the source terms and their balance with the pressure gradient that dominates atmospheric dynamics, needs to be adapted to be able to keep the combined geostrophic and hydrostatic balance in three spatial dimensions. This is achieved inside the discretisation through a polynomial mapping of both source and flux terms without imposing filters between time steps. After introduction and verification of this balancing, the realistic development of barotropic and baroclinic waves in the model is demonstrated, including the formation of a retrograde Rossby wave pattern. A prerequesite is the numerical solution of the thermal wind equation to construct geostrophically balanced initial states in z coordinates with arbitrary prescribed zonal wind profile, offering a new set of test cases for atmospheric models employing z coordinates. The resulting simulations demonstrate that the balanced low-order Discontinuous Galerkin discretisation with polynomial degrees down to k = 1 can be a viable option for atmospheric modelling. [ABSTRACT FROM AUTHOR]