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## 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.