Abstract
Ambit fields were introduced by a need for sophisticated stochastic models in turbulence. We present in this chapter a detailed study of the statistical theory of homogeneous turbulence in view of volatility modulated Volterra processes and ambit fields. After a review of the statistical theory due to Kolmogorov-Obukhov, with a particular emphasis on scaling laws, we discuss ambit fields and various subclasses and their relevance to turbulence. In particular, the concept of correlators is shown to be nicely analysed in the framework of ambit fields. Furthermore, energy dissipation goes hand in hand with power variation, and one can resort to the general theory presented in Chap. 3 to analyse this central object in turbulence. An empirical study of Helium gas jet flow is presented where some striking results in the context of ambit stochastics can be observed. Next we discuss purely temporal, purely spatial and one-dimensional turbulence to some detail using specific cases of ambit fields and volatility modulated Volterra processes. We are grateful to Jrgen Schmiegel (University of Aarhus) for co-authoring this chapter with us.
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Notes
- 1.
The reader should not confuse an exponentiated trawl field with the concept of exponential trawl function discussed in Sect. 8.3.2, the latter referring to a specific definition of the ambit/trawl set.
- 2.
Recall that we have reserved the name cumulant for the logarithm of the characteristic function.
- 3.
For spectral functions in turbulence, see Monin and Yaglom (1975, Chapter 6). The spectral density function is defined as \(\frac {1}{2}\) times the sum of the diagonal elements of the Fourier transform of the covariance matrix R.
- 4.
The data consist of 20 million one-point measurements of the longitudinal component of the wind velocity in the atmospheric boundary layer, 35 m above ground. The measurements were performed using a hot-wire anemometer and sampled at 5kHz. The time series can be assumed to be stationary. See Hedevang and Schmiegel (2014) and Dhruva (2000) for details.
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Barndorff-Nielsen, O.E., Benth, F.E., Veraart, A.E.D. (2018). Turbulence Modelling. In: Ambit Stochastics. Probability Theory and Stochastic Modelling, vol 88. Springer, Cham. https://doi.org/10.1007/978-3-319-94129-5_9
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