Abstract: In this paper a meteorological model is described as a non-linear matrix differential equation. It can be solved by an analitical non-modal matrix solution, for linear part without $\beta$--effect. The Helmholtz wind field reconstruction is made by using the Moore-Penrose inverse. The solution of such matrix differential equation can be written in such a way that it depends essentially on a particular matrix solution and its convolution with the non-linearity as a forcing term. The knowlegde of an explicit non-modal formula for this later matrix solution, allows performance of the integration. The reconstruction the wind fields leads to the solution of an algebraic linear equation, which corresponds to the discretized Poisson equation for finding the zonal wind and meridional wind from vorticity and divergence. A least-square solution is proposed by using the Moore-Penrose matrix inverse to calculate this later.