Abstract
This work presents an analysis of ocean wave data including rogue waves. A stochastic approach based on the theory of Markov processes is applied. With this analysis we achieve a characterization of the scale-dependent complexity of ocean waves by means of a Fokker–Planck equation, providing stochastic information on multi-scale processes. In particular, we show evidence of Markov properties for increment processes, which means that a three-point closure for the complexity of the wave structures seems to be valid. Furthermore, we estimate the parameters of the Fokker–Planck equation by parameter-free data analysis. The resulting Fokker–Planck equations are verified by numerical reconstruction. This work presents a new approach where the coherent structure of rogue waves seems to be integrated into the fundamental statistics of complex wave states.
Highlights
Rogue or freak waves are exceptionally large waves which at present are the subjects of intensive studies in a number of different fields [1,2,3,4,5,6,7,8,9]
The present approach is described. It is shown how a Markovian process for the pdf of the surface elevation can be derived from the data, how Kramers–Moyal coefficients can be estimated, and how a Fokker–Planck equation governing conditional probability densities can be derived that contains a dependence on the time scale of the process, by which the analysis allows insight into the temporal multi-scale nature of irregular ocean waves, i.e. a natural seaway
In terms of physical characteristics of the wave field in the neighborhood of a breather state, one would expect significant local changes of wavelength and wave frequency. This is confirmed for the large waves given in the present data set: from the empirical mode decomposition (EMD) results it turns out that the wave dynamics in the phases of rogue events is mainly captured by higher intrinsic mode functions (IMF), here say C3 to C8, i.e. IMFs corresponding to slightly shifted underlying numbers of waves
Summary
Rogue or freak waves are exceptionally large waves which at present are the subjects of intensive studies in a number of different fields [1,2,3,4,5,6,7,8,9]. The approach that we apply was originally introduced by Friedrich and Peinke [17] and since has been successfully applied in a large variety of fields, such as turbulence [17, 51], economics [18,19,20], biology [21, 22, 38], and many more; see [40] It is based on identifying and exploiting Markov properties for the evolution of probability density functions, and starts out with measured data, to identify coefficients of fundamental stochastic differential equations, e.g. Fokker–Planck equations. It is shown how a Markovian process for the pdf of the surface elevation can be derived from the data, how Kramers–Moyal coefficients can be estimated, and how a Fokker–Planck equation governing conditional probability densities can be derived that contains a dependence on the time scale of the process, by which the analysis allows insight into the temporal multi-scale nature of irregular ocean waves, i.e. a natural seaway
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