Evidence of Kolmogorov like scalings and multifractality in the rainfall events

TL;DR

This study analyzes half-hour rainfall data from North-East India (2001–2020) to characterize distribution, correlations, and multiscale dynamics. It finds a Log-Normal distribution as the best fit for rainfall intensities, with spatial coherence over roughly $70$ km and a temporal structure featuring a $24$ h cycle and additional multi-day scales. Spectral analysis shows a Kolmogorov-like high-frequency scaling with exponent about $-1.5$, while nonlinear methods (Wavelets, HHT) uncover multiple dominant frequencies and scale-dependent noise. Multifractal DFA reveals a persistent, multifractal rainfall regime with $H_q$ varying with $q$, indicating rich multiscale variability and extreme-event propensity. These results enhance understanding of rainfall in a complex monsoon region and have implications for regional climate modelling and risk assessment.

Abstract

In this paper we present a detailed statistical analysis related to the characterization of the spatial and temporal fluctuations present in the rainfall patterns of North-East region ($26.05^{\circ}N-26.95^{\circ}N$, $88.05^{\circ}E-94.95^{\circ}E$) of India using half hourly rainfall data over the last 20 years for the range 2001-2020. We analyze the nature of the distribution by computing the mean, second moment of the fluctuation, skewness and kurtosis of the temporal rainfall data that indicate the presence of heavy tail in the right skewed distribution a typical feature of the presence of rare events. We find that the temporal distribution of the rainfall data follow the multiplicative Log-Normal probability distribution. Further we compute the spatial and temporal correlation of the rainfall in this region indicate that the rainfall events are correlated in the spatial direction of about 70 Km. The Power spectral density of temporal rainfall shows power law behaviour with frequency with an exponent $\sim -1.5$ close to the Kolmogorov exponent ($-1.67$) exhibited for the turbulent passive scalar driven by the mean flow. Our wavelet analysis reveals the evidence of multiple frequencies in the rainfall pattern which can attributed to different short and long range factors responsible for the rainfall. We have also used the Hilbert Huang transformation to identify the frequencies corresponding fluctuating part of the rainfall time series. Using multifractal detrended fluctuation analysis, finally we establish the multifractal nature of the rainfall pattern with Hurst exponent close to $0.65$ .

Figures (14)

• Figure 1: Temporal evolution of rainfall of (a) station $26.05^{\circ}$N, $88.05^{\circ}$E, year 2010 and (b) station $26.55^{\circ}$N, $91.65^{\circ}$E, year 2020 respectively. The time-series of rainfall event for both the stations exhibits the stochastic variation accompanied by intermittent jump of the rain intensity suggesting the presence of nonlinear nature of the rainfall events.
• Figure 2: Different statistical variables averaged over twenty years (2001-2020) for the selected area under investigation. (a) mean, (b) standard deviation, (c) skewness and (d) kurtosis on the geographic map in the $\theta - \phi$ plane. Color bar represents the magnitude of different statistical quantities. These values indicate that the distribution of rainfall intensity is right skewed and possess a heavy tail.
• Figure 3: PDF of the wet half hour rainfall data. In the left panels (a)for station $26.05^{\circ}$N, $88.05^{\circ}$E, year 2005 and (c) for station $26.05^{\circ}$N, $88.05^{\circ}$E, year 2015 while in the right panel (b) for station $26.55^{\circ}$N, $91.65^{\circ}$E, year 2005 and (d) for station $26.55^{\circ}$N, $91.65^{\circ}$E, year 2015. For all the stations over all the year, Log Normal PDF (red curve) is found to be better fitted than the Gamma PDF (blue curve). The inset of each figure contains the rainfall intensity data (green square) fitted with these two distribution functions in log-log scale suggesting the better fitted Log-normal distribution.
• Figure 4: Spatial and temporal correlation plots. (a) Spatial correlation coefficients ($\rho(\theta)$) for the stations having same latitude values but varying longitudes for the year 2005. The thin graphs (from light blue to dark blue) denote the correlation coefficients for a fixed latitude and the thick black graph shows the average behaviour of all these graphs. Inset shows the exponential fitting (dotted red) to the mean graph indicating the correlation length. (b) Spatial correlation coefficients for the stations having same longitude values but varying latitudes for the year 2005. The thin graphs (from light blue to dark blue) shows the correlation coefficients variation for a fixed longitude and the thick black graph denotes the average behaviour of all these graphs. (c) Temporal autocorrelation with time lags of station $26.05^{\circ}$ N, $88.05^{\circ}$E for 20 years. The thin graphs (from light blue to dark blue) denotes the temporal autocorrelation for different years and the thick black graph depicts the average behaviour of 20 years. The periodic ripples (indicated by the black arrows) appear at a mean interval of 24 hours. Inset shows a zoomed version of the mean graph with a red shaded area indicating mean time interval of occurrence of ripples.
• Figure 5: PSD of the rainfall time series of the station $26.15^{\circ}N,91.05^{\circ}E$ for 20 years in log-log scale. The thin lines (varying from blue to yellow) corresponds to different years while the thick black line denotes the temporal average. At higher frequency range, it shows power law behaviour (red dotted line) with the exponent value ($\beta_{2}$) 1.5 whereas in the low frequency range, it follows power law behaviour (pink dotted line) with an exponent ($\beta_{1}$) 0.3. The shaded gray rectangle highlights the presence of the distinct peak at almost 24 hours.