A toy model of climatic variability with scaling behaviour

It is demonstrated that a simple deterministic model in discrete time can reproduce the scaling behaviour of hydroclimatic processes at timescales coarser than annual, a behaviour more widely known in hydrology as the Hurst phenomenon. This toy model is based on a generalised ‘chaotic tent map’, which may be considered as the compound result of a positive and a negative feedback mechanism, and involves two degrees of freedom. The model is not a realistic representation of a climatic system, but rather a radical simplification of real climatic dynamics. However, its simplicity helps understand the physical mechanisms that cause the scaling behaviour and simultaneously enables easy implementation and convenient experimentation. Application of the toy model gives traces that can resemble historical time series of hydroclimatic variables, such as temperature and river flow. In particular, such traces exhibit scaling behaviour with a Hurst coefficient greater than 0.5 and their statistical properties are similar to that of observed time series. Moreover, application demonstrates that large-scale synthetic ‘climatic’ fluctuations (like upward or downward trends) can emerge without any specific reason and their evolution is unpredictable, even when they are generated by this simple fully deterministic model with only two degrees of freedom. Thus, the model emphasises the large uncertainty associated with the scaling behaviour, rather than enhances the prediction capability, despite the simple deterministic dynamics it uses, which obviously, are only a caricature of the much more complex dynamics of the real climatic system.