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 on which they are based converge to the analytic solutions of the underlying differential equations for small time steps and grid sizes. In this paper, the authors investigate the time step sensitivity of three nonlinear atmospheric models of different levels of complexity: the Lorenz equations, a quasigeostrophic (QG) model, and a global weather prediction system (NOGAPS). It is illustrated here how, for chaotic systems, numerical convergence cannot be guaranteed forever. The time of decoupling of solutions for different time steps follows a logarithmic rule (as a function of time step) similar for the three models. In regimes that are not fully chaotic, the Lorenz equations are used to illustrate how different time steps may lead to different model climates and even different regimes. A simple model of truncation error growth in chaotic systems is proposed. This model decomposes the error onto its stable and unstable components and reproduces well the short- and medium-term behavior of the QG model truncation error growth, with an initial period of slow growth (a plateau) before the exponential growth phase. Experiments with NOGAPS suggest that truncation error can be a substantial component of total forecast error of the model. Ensemble simulations with NOGAPS show that using different time steps may be a simple and natural way of introducing an important component of model error in ensemble design.