One of the most intriguing facets of the climate system is that it exhibits variability across all temporal and spatial scales; pronounced examples are temperature and precipitation. The structure of this variability, however, is not arbitrary. Over certain spatial and temporal ranges, it can be described by scaling relationships in the form of power laws in probability density distributions and autocorrelation functions. These scaling relationships can be quantified by scaling exponents which measure how the variability changes across scales and how the intensity changes with frequency of occurrence. Scaling determines the relative magnitudes and persistence of natural climate fluctuations. Here, we review various scaling mechanisms and their relevance for the climate system. We show observational evidence of scaling and discuss the application of scaling properties and methods in trend detection, climate sensitivity analyses, and climate prediction.

In this paper, a new method, detrended partial-cross-correlation analysis (DPCCA), is proposed. Based on detrended cross-correlation analysis (DCCA), this method is improved by including partial-correlation technique, which can be applied to quantify the relations of two non-stationary signals (with influences of other signals removed) on different time scales. We illustrate the advantages of this method by performing two numerical tests. Test I shows the advantages of DPCCA in handling non-stationary signals, while Test II reveals the “intrinsic” relations between two considered time series with potential influences of other unconsidered signals removed. To further show the utility of DPCCA in natural complex systems, we provide new evidence on the winter-time Pacific Decadal Oscillation (PDO) and the winter-time Nino3 Sea Surface Temperature Anomaly (Nino3-SSTA) affecting the Summer Rainfall over the middle-lower reaches of the Yangtze River (SRYR). By applying DPCCA, better significant correlations between SRYR and Nino3-SSTA on time scales of 6 ~ 8 years are found over the period 1951 ~ 2012, while significant correlations between SRYR and PDO on time scales of 35 years arise. With these physically explainable results, we have confidence that DPCCA is an useful method in addressing complex systems.

Long term memory (LTM) in climate variability is studied by means of fractional integral techniques. By using a recently developed model, Fractional Integral Statistical Model (FISM), we in this report proposed a new method, with which one can estimate the long-lasting influences of historical climate states on the present time quantitatively, and further extract the influence as climate memory signals. To show the usability of this method, two examples, the Northern Hemisphere monthly Temperature Anomalies (NHTA) and the Pacific Decadal Oscillation index (PDO), are analyzed in this study. We find the climate memory signals indeed can be extracted and the whole variations can be further decomposed into two parts: the cumulative climate memory (CCM) and the weather-scale excitation (WSE). The stronger LTM is, the larger proportion the climate memory signals will account for in the whole variations. With the climate memory signals extracted, one can at least determine on what basis the considered time series will continue to change. Therefore, this report provides a new perspective on climate prediction.

Slowing variability in climate system is an important source of climate predictability. However, it is still challenging for current dynamical models to fully capture the variability as well as its impacts on future climate. In this study, instead of simulating the internal multi-scale oscillations in dynamical models, we discussed the effects of internal variability in terms of climate memory. By decomposing climate state x(t) at a certain time point t into memory part M(t) and non-memory part \(\varepsilon (t)\), climate memory effects from the past 30 years on climate prediction are quantified. For variables with strong climate memory, high variance (over \(20\%\)) in x(t) is explained by the memory part M(t), and the effects of climate memory are non-negligible for most climate variables, but the precipitation. Regarding of multi-steps climate prediction, a power law decay of the explained variance was found, indicating long-lasting climate memory effects. The explained variances by climate memory can remain to be higher than 10% for more than 10 time steps. Accordingly, past climate conditions can affect both short (monthly) and long-term (interannual, decadal, or even multidecadal) climate predictions. With the memory part M(t) precisely calculated from Fractional Integral Statistical Model, one only needs to focus on the non-memory part \(\varepsilon (t)\), which is an important quantity that determines climate predictive skills.