This paper presents the results of an ensemble data assimilation methodology over the Wasserstein space for high-dimensional nonlinear dynamical systems, focusing on the chaotic Lorenz-96 model and a two-layer quasi-geostrophic model of atmospheric circulation. Unlike Euclidean data assimilation, this approach is equipped with a Riemannian geometry and formulates data assimilation as a Wasserstein barycenter between the forecast probability distribution and the normalized likelihood function. The methodology does not rely on any Gaussian assumptions and can intrinsically treat systematic model and observation errors. To cope with the computational cost of the Wasserstein distance, the paper examines the efficiency of the entropic regularization. Comparisons with the standard particle and stochastic ensemble Kalman filters demonstrate that under systematic errors the presented methodology could extend the forecast skills of nonlinear dynamical systems.