## Convergence of distributions constructed from increasingly long trajectories, towards the long term distribution

**Figure 2.** Convergence of distributions constructed from increasingly long trajectories, towards the long term distribution. (a) and (b) show normalized frequency distributions of *S* (a) and *T* (b) extracted from a single 100 000 year simulation. (c) and (d) show the difference, quantified by the KS statistic, between the (a) and (b) distributions and those generated using shorter simulations, for *S* (c) and *T* (d). The distributions are constructed from either single simulations (black) or by combining data up to the given time point from multiple simulations with different ICs; red—three simulations, blue—ten simulations. Uncertainty is quantified by repeating the process 100 times using different ICs distributed around the same location on the attractor (see figure 1 and the supplementary materials available at stacks.iop.org/ERL/8/034021/mmedia); error bars do not therefore represent uncertainty in our assessment of *D* but rather the distribution of values from which any particular simulation/ensemble may be considered a draw. Lines connect the median values at each time point; errors are the 10th–90th percentiles.

**Abstract**

Can today's global climate model ensembles characterize the 21st century climate in their own 'model-worlds'? This question is at the heart of how we design and interpret climate model experiments for both science and policy support. Using a low-dimensional nonlinear system that exhibits behaviour similar to that of the atmosphere and ocean, we explore the implications of ensemble size and two methods of constructing climatic distributions, for the quantification of a model's climate. Small ensembles are shown to be misleading in non-stationary conditions analogous to externally forced climate change, and sometimes also in stationary conditions which reflect the case of an unforced climate. These results show that ensembles of several hundred members may be required to characterize a model's climate and inform robust statements about the relative roles of different sources of climate prediction uncertainty.