Inverse scattering problem in turbulent magnetic fluctuations


We apply a particular form of the inverse scattering theory to magnetic turbulence in a plasma. In thepresent note we develop the theory, formulate the magnetic turbulence problem in terms of its electrodynamic turbulent response functionand reduce it to the solution of a special form of the famous Gel’fand-Levitan-Marchenko equation of quantum mechanical scattering theory. The latter applies to transmission and reflection in an active medium. Turbulence theory does not refer to such quantities and therefore requires a somewhat different formulation. We reduce the theory to the measurement of the magnetic fluctuation spectrum alone. This allows for providing a form of the Gel’fand-Levitan-Marchenko equation suitable for application in magnetic turbulence where fluctuation spectra are obtained. The inverse theory in this form enables obtaining information about the turbulent response function of the medium. This is of vital interest in magnetohydrodynamic turbulence theory. The theory is developed until presentation of the equations in a form that is directly applicable to observations as input from measurements. Solution of the final integral equation should be done by standard methods based on iteration. Such methods result in Neumann seriesrespectively fractional chains which can be treated numerically. At this stage we stop. We point on the possibility of treating Kolmogorov turbulence as an example in this way, however, do not attempt it leaving it for a separate investigation because the use of a theoretical power law in a bounded range of frequencies leads to severe mathematical problems and requires purely numerical work. Investing more efforts would make sense only if using more extended observations. One particular aspect of the present inverse theory of turbulence is that its structure naturally leads to spatial information which is obtained from the temporal information that is inherent to the observation of time series without any need to refer to the Taylor assumption. This is a consequence of Maxwell’s equations which couple space and time evolution, as well as of the inversion procedure which takes advantage of a particular mapping from time to space domains. Though the theory is developed for homogeneous stationary non-flowing media, its extension to turbulent flows, anisotropy and non-stationarity is obvious.

Further Information
  author = {Treumann, R. A. and Baumjohann, W. and Narita, Y.},
  doi = {10.5194/angeo-34-673-2016},
  journal = {Ann Geophys},
  language = {en},
  note = {published},
  pages = {673-689},
  title = {Inverse scattering problem in turbulent magnetic fluctuations},
  url = {},
  volume = {34},
  year = {2016},
%O Journal Article
%A Treumann, R. A.
%A Baumjohann, W.
%A Narita, Y.
%R 10.5194/angeo-34-673-2016
%J Ann Geophys
%G en
%O published
%P 673-689
%T Inverse scattering problem in turbulent magnetic fluctuations
%V 34
%D 2016