In order to find an efficient spectral representation for the variables in a non-linear partial differential equation, one may require that the expanding functions should satisfy the equation, not exactly as in the linear case, but instead as well as possible in the least square sense. This variational approach is here applied to the non-linear potential vorticity equation for a zonal flow and finite amplitude perturbations in a beta-plane channel. The solutions are required to be neutral. For zonal wave numbers 2–10 primary modes are found for which the zonal wind has a typical jet-like profile. Its form varies very little with the zonal scale and it is shown that a short wave will have the same meridional profile as a much longer wave. The longest waves are quasi-stationary as is usually observed in the atmosphere. Shorter waves move east with phase velocities which are higher than those obtained with the Rossby formula. They agree, however, well with observations. For higher orthogonal modes a mathematical modification is made in the equation for the primary modes so that a complete Sturm-Liouville orthogonal system is obtained for all modes. These higher modes also show little dependence on zonal wave number and one may therefore question if it is necessary to have separate expansion functions in the meridional direction for each zonal wave number. For the beta-plane model this necessity was not confirmed in an expansion of the meridional profile of the perturbation Jacobian, where a comparison also was made with Fourier expansions. In this case the functions determined here are found to give a much faster convergence.DOI: 10.1034/j.1600-0870.1996.00003.x