Reconstruction of Diffusion Coefficients and Power Exponents from Single Lagrangian Trajectories

Abstract

Using discrete wavelets, a novel technique is developed to estimate turbulent diffusion coefficients and power exponents from single Lagrangian particle trajectories. The technique differs from the classical approach (Davis (1991)’s technique) because averaging over a statistical ensemble of the mean square displacement (<X2>) is replaced by averaging along a single Lagrangian trajectory X(t) = {X(t), Y(t)}. Metzler et al. (2014) have demonstrated that for an ergodic (for example, normal diffusion) flow, the mean square displacement is <X2 > = , where (T,s) = 1/(T − s) − X(t))2 dt, T and s are observational and lag times but for weak non-ergodic (such as super-diffusion and sub-diffusion) flows <X2> = , where is some additional averaging. Numerical calculations for surface drifters in the Black Sea and isobaric RAFOS floats deployed at mid depths in the California Current system demonstrated that the reconstructed diffusion coefficients were smaller than those calculated by Davis (1991)’s technique. This difference is caused by the choice of the Lagrangian mean. The technique proposed here is applied to the analysis of Lagrangian motions in the Black Sea (horizontal diffusion coefficients varied from 105 to 106 cm2/s) and for the sub-diffusion of two RAFOS floats in the California Current system where power exponents varied from 0.65 to 0.72. RAFOS float motions were found to be strongly non-ergodic and non-Gaussian.

Hybrid Model for Time Series of Complex Structure with ARIMA Components

“…As a rule, time series of empirical data have a complex non-stationary structure and contain local features of various forms. The methods for the time series analysis include deterministic [11], stochastic [12][13][14] approaches and their various combinations [15][16][17][18][19]. Traditional methods for data time series modeling and analysis (AR models, ARMA [20,21], exponential smoothing [22], stochastic approximation [13], etc.)…”

“…do not allow us to describe the time series of complex structure adequately [23]. At present, hybrid approaches [16,17,19,[23][24][25][26][27][28] are widely applied. They make it possible to improve the efficiency of the procedure of data analysis in case of its complicated structure.…”

“…They make it possible to improve the efficiency of the procedure of data analysis in case of its complicated structure. For example, in [19], on the basis of wavelet decomposition, a technique was developed to estimate the coefficients of turbulent diffusion and power exponents from single Lagrangian trajectories of particles. Wavelet transform is a flexible tool and was applied in the paper [29] to study the relationship between vegetation and climate in India.…”