Time-step sensitivity of nonlinear atmospheric models: numerical convergence, truncation error growth and ensemble design

Computational models based on discrete dynamical equations are a successful way of approaching the problem of predicting or forecasting the future evolution of dynamical systems. For linear and mildly nonlinear models, the solutions of the numerical algorithms in which they are based, converge to the analytic solutions of the underlying differential equations, for small time-steps and grid-sizes. In this paper, we investigate the time-step sensitivity of three non-linear atmospheric models of different levels of complexity: the Lorenz equations, a quasi-geostrophic (QG) model and a global weather prediction system (NOGAPS). We show that for chaotic systems, numerical convergence cannot be guaranteed forever. The decoupling of solutions for different time-steps follows a logarithmic rule similar for the three models. In regimes that are not fully chaotic, the Lorenz equations are used to show that different time-steps may lead to different model climates and even different regimes. We propose a simple model of truncation error growth in chaotic systems that reproduces the behavior of the QG model error growth for different timesteps. Experiments with NOGAPS suggest that truncation error is a substantial component of total forecast error of the model. Ensemble simulations with NOGAPS show that using different time-steps is a simple and natural way of introducing an important component of model error in ensemble design.