The authors argue that the concept of snapshot attractors and of their natural probability distributions are the only available tools by means of which mathematically sound statements can be made about averages, variances, etc., for a given time instant in a changing climate. A basic advantage of the snapshot approach, which relies on the use of an ensemble, is that the natural distribution and thus any statistics based on it are independent of the particular ensemble used, provided it is initiated in the past earlier than a convergence time. To illustrate these concepts, a tutorial presentation is given within the framework of a low-order model in which the temperature contrast parameter over a hemisphere decreases linearly in time. Furthermore, the averages and variances obtained from the snapshot attractor approach are demonstrated to strongly differ from the traditional 30-yr temporal averages and variances taken along single realizations. The authors also claim that internal variability can be quantified by the natural distribution since it characterizes the chaotic motion represented by the snapshot attractor. This experience suggests that snapshot-attractor-based calculations might be appropriate to be evaluated in any large-scale climate model, and that the application of 30-yr temporal averages taken along single realizations should be complemented with this more appealing tool for the characterization of climate changes, which seems to be practically feasible with moderate ensemble sizes.

We study the teleconnection between El Niño–Southern Oscillation (ENSO) and the Indian summer monsoon (IM) in large ensemble simulations, the Max Planck Institute Earth System Model (MPI-ESM), and the Community Earth System Model (CESM1). We characterize ENSO by the June–August Niño-3 box-average SST and the IM by the June–September average precipitation over India, and define their teleconnection in a changing climate as an ensemble-wise correlation. To test robustness, we also consider somewhat different variables that can characterize ENSO and the IM. We utilize ensembles converged to the system’s snapshot attractor for analyzing possible changes in the teleconnection. Our main finding is that the teleconnection strength is typically increasing on the long term in view of appropriately revised ensemble-wise indices. Indices involving a more western part of the Pacific reveal, furthermore, a short-term but rather strong increase in strength followed by some decrease at the turn of the century. Using the station-based Southern Oscillation index (SOI) as opposed to area-based indices leads to the identification of somewhat more erratic trends, but the turn-of-the-century “bump” is well detectable with it. All this is in contrast, if not in contradiction, to the discussion in the literature of a weakening teleconnection in the late twentieth century. We show here that this discrepancy can be due to any of three reasons: 1) ensemble-wise and temporal correlation coefficients used in the literature are different quantities; 2) the temporal moving correlation has a high statistical variability but possibly also persistence; or 3) MPI-ESM does not represent the Earth system faithfully.