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Statistical averaging, smoothing and filtering

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If you have a question about this talk, please contact Dr William Béthune.

Theory of turbulence is based on ensemble averaging, but this type of averaging is not practical in most laboratory measurements, astronomical observations and numerical simulations as it requires very large number of independent realisations of random fields involved. An alternative, widely used in many applications, and indeed the only one possible in astronomy, is the smoothing or filtering of random fields (in fact the output of any astronomical instrument is a smoothed or filtered physical field). However, spatial (or temporal) smoothing or filtering do not satisfy the Reynolds rules of averaging. Similarly, modern numerical simulations of turbulent flows are performed in spatial and temporal domains of a relatively small size and ensemble averaging is out of question. Thus, analyses of such numerical simulations employ various averages (such as plane averages) that satisfy the Reynolds rules. However, such averages can be physically inappropriate, e.g., when the mean quantities do not need to be independent of any spatial coordinates. We discuss the approach to spatial smoothing and filtering that allows one to avoid this difficulty and to derive consistently averaged quantities and equations that retain a full 3D structure.

This talk is part of the DAMTP Astrophysics Seminars series.

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