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Home > Seminars and Workshops > Past seminars, PhD & HDR defense > 2013 > Séminaires 2013 > Le lundi 4 février à 14h

Le lundi 4 février à 14h

Sara Moradi (Dept of Applied Physics, Chalmers University of Technology)- Application of fractional kinetics in turbulent fusion plasmas

Lieu : Palaiseau, salle de conférences du CPHT

Résumé :

The problem of finding a proper kinetic description for dynamical
systems with chaotic behavior is one of the main problems in physics
today. Over the past two decades it has become obvious that behaviors
much more complex than standard diffusion can occur in dynamical
Hamiltonian chaotic systems. Thus, kinetic descriptions which arise as
a consequence of averaging over the well-known Gaussian and Poissonian
statistics (for diffusion in space and temporal measures, respectively)
seem to fall short in describing the apparent randomness of dynamical
chaotic systems. Turbulent random processes and related turbulent (i.e.
anomalous) diffusion phenomena are examples of such complex processes
which are found to be ubiquitous in nature.

Lévy statistics describing fractal processes (Lévy index α) lie at the
heart of complex processes such as turbulent diffusion. Lévy statistics
can be generated by random processes that are scale-invariant. This
means that a trajectory will possess many scales, but no one scale will
be characteristic and dominate the process. Geometrically this implies
the fractal property that a trajectory, viewed at different
resolutions, will look self-similar.

Indeed, self-similar analysis of fluctuation measurements by Langmuir
probes at different types of fusion devices such as the linear device,
spherical Tokamak, reversed field pinch, stellarator and several other
tokamaks have provided evidence to support the idea that density and
potential fluctuations are distributed according to Lévy statistics.
where Probability Distribution Function (PDF) of the turbulence induced
fluxes at the edge of W7-AS stelerator are shown to exhibit power law
characteristics in contrast to exponential decay expected from Gaussian
statistics. Furthermore, the experimental evidence of the wave-number
spectrum characterized by power laws over a wide range of wave-numbers
can be directly linked to the values of Lévy index α of the PDFs of the
underlying turbulent processes.

Fractional derivatives offers the possibility to generalize the
Gaussian statistics to Lévy statistics which can be introduced into the
Langevin equation thus yielding a fractional Fokker-Planck (FP) kinetic
description. In our work we quantified the effects of the fractional
derivative in the FP equation in terms of a modified dispersion
relation for density gradient driven linear plasma drift waves where we
have considered a case with constant external magnetic field and a
shear-less slab geometry. In order to calculate an equilibrium PDF we
used a model based on the motion of a charged Lévy particle in a
constant external magnetic field obeying non-Gaussian, Lévy stable
distributions. The fractional derivative is represented with the
Fourier transform containing a fractional exponent. A deviation from
Maxwellian equilibrium is considered by assuming that the fractional
exponent is 2-ε where 0<ε<<2. By an expansion around ε=0, we have found a relation between ε and the eigen values (growth-rate γ and real frequency ω) of the plasma drift modes through the quasi-neutrality condition. We have shown that a deviation from the Maxwellian distribution function alters the dispersion relation for the density gradient drift waves such that the growth rates are substantially increased and thereby may cause enhanced levels of transport.


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Tutelles : CNRS Ecole Polytechnique Sorbonne Université Université Paris Sud Observatoire de Paris Convention : CEA
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