We propose a dynamical model for the wind speed which is a Markov diffusion process with Weibull marginal distribution. It presents several advantages, namely nice modeling features both in terms of marginal probability density function and temporal correlation. The characteristics can be interpreted in terms of shape and scale parameters of a Weibull law which is convenient for practitioners to analyze the results. We calibrate the parameters with the maximum quasi-likelihood method and use the model to generate and forecast the wind speed process. We have tested the model with wind speed dataset provided by the National Renewable Energy Laboratory. The model fits very well with the data. Besides, we obtain a very good performance in point and probabilistic forecasting in the short-term in comparison to benchmarks.