We propose a nonlinear, quasi-geostrophic, baroclinic model of Jovian atmospheric dynamics, in which vertical variations of velocity are represented by a truncated sum over a complete set of orthogonal functions obtained by a separation of variables of the linearized quasi-geostrophic potential vorticity equation. A set of equations for the time variation of the mode amplitudes in the nonlinear case is then derived. We show that for a planet with a neutrally stable, fluid interior instead of a solid lower boundary, the baroclinic mode represents motions in the interior, and is not affected by the baroclinic modes. One consequence of this is that a normal-mode model with one baroclinic mode is dynamically equivalent to a one layer model with solid lower topography. We also show that for motions in Jupiter's cloudy lower troposphere, the stratosphere behaves nearly as a rigid lid, so that the normal-mode model is applicable to Jupiter. We test the accuracy of the normal-mode model for Jupiter using two simple problem forced, vertically propagating Rossby waves, using two and three baroclinic modes and baroclinic instability, using two baroclinic modes. We find that the normal-road model provide qualitatively correct results, even with only a very limited number of vertical degrees of freedom.