Lvy Distributions in Plasma Diffusion across a Magnetic Field:

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Lévy Distributions in Plasma Diffusion across a
Magnetic Field Experimental Evidence
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Turbulent transport through a magnetic field
Turbulent transport in a magnetized plasma is one
of the main issue of controlled nuclear fusion. A
good insight is necessary to design future
machines properly. Plasma transport across a
confining magnetic field occurs by random motion
resulting from turbulence. It is often assumed to
imply a brownian statistics and a diffusive
process.
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• Some words about random walks
• Experimental device and results
• Concluding remarks

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Plasma turbulent transport
random walk description
Probability of plasma displacement from a volume
d3R centered at R(t) to a volume d3R centered at
R(tt) Plasma displacement probability
distribution (DPD) P(Dt)
R(tt)
D(t)
R(t)
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Gaussian statistics Fick diffusion
There is a characteristic length lc all steps
have the same order of magnitude. If the steps
are done at a regular pace, there is a
characteristic time tc and we get the Fick
scaling
D is a turbulent diffusion coefficient
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Lorentzian distribution
No characteristic length extreme events are
scarce but possible Autosimilarity
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Stable distributions
Let be xi n independant random variables whose
probability distribution is P. P is stable if P
is also the p. d. of X 1/c S xi, c being a
renormalisation number. The gaussian function is
not the only stable distribution. The stable
probability distributions are Lévy
distributions. Their Fourier transform is

• 2 P is a gaussian
• 1 P is a Cauchy function (Lorentzian)

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• Many theoretical works use Lévy distributions
• M. F. Shlesinger, J. West, B. and J. Klafter,
  Phys. Rev. Lett. 58, 1100 (1987)
• R. Balescu, Phys. Rev. E 51, 4807 (1995)
• E. Van den Eijnden and R. Balescu, Phys. Plasmas
  4, 270 (1997)
• B. Carreras, V. Lynch, and G. Zaslavsky, Phys.
  Plasmas 8, 5096 (2001)
• B. Van Milligen, R. Sánchez, and B. Carreras,
  Phys. Plasmas 11, 2272 (2004)
• R. Sánchez, B. P. Van Miligen, and B. A.
  Carreras, Phys. Plasmas 12, 056105 (2005)
• Few and indirect experimental evidences
• C. Hidalgo Plasma Phys. Control. Fusion, 37, A53
  (1995)

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• Some words about random walks
• Experimental device and results
• Concluding remarks

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The plasma machine ToriX
A toroidal machine without rotational transform.
R 0.6 m. r 0.1 m A vertical magnetic field is
added. Gas Argon (0.1 Pa) Toroidal B - field
0.25 0.36 T Vertical B - field 1.8
mT Electron density 1017 m-3 Electron
temperature 2 eV
Thanks to the small magnetic B field, we get a
homogeneous density and turbulence.
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From collective scattering
Superheterodyne signal is proportional to the
space Fourier transform of the density
fluctuations. s(t) ñ(k, t) k ks ki 770
m-1 lt k lt 1900 m-1 8 mm gt l gt 3 mm
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to statistics on plasma displacement
Hypothesis Density fluctuation and
displacement at the scale observed not
correlated The normalized time autocorrelation
function of the scattered signal is the Fourier
transform of the plasma displacement probability
distribution. CN(t) lt e-i k.D gt ? P(D, t)
e-i k.r d3D
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Time spectra and autocorrelation
BT0.36 T, k1300m-1
Time spectrum calculated from experimental data
(0.5 s time series, 2 MHz sampling rate)
Time autocorrelation function calculated from
time spectrum
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The F.T. of the probability distribution as a
function of k
We have ln C(t) for different ks The
experiments are reproductible
We get ln C(k) for different t0
10msltt0lt40ms
A Lévy model is considered C(k t0) exp
(d(t0) k)a -ln C(k t0) (d(t0) k)a
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Lévy exponents

• Model P(kt) exp-(d k)a
• decreases with t from 2 to 1.1
• P(kt) is close to a gaussian at small time and
  close to a Cauchy function at larger time.
• The parameter d, homogeneous to a distance and
  related to the width of the distribution is found
  to increase linearly with time.
• There is a self similar expansion of the plasma

BT0.36 T. Similar results for other values of BT
Submitted to Phys. of Plasmas
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Time dependence of transport behaviour
If P(Dt) is a Lévy distribution with an exponent
a, CN(tk) exp- (d(t) k)a. Then Ca(t)
expln CN(tk) (k0/k)a should not depend on k
Ca(t) exp- (d(t) k0)a. For a2, Ck2(t)
expln CN(tk) (k0/k)2 does, except at short
time, especially for large scales (small
ks). For a1.1, Ck1.1(t) expln CN(tk)
(k0/k)1.1, does not, except for small time,
especially for large scales (small ks).
a2
a1.1
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Physical interpretation
At short time (small compared to the lagrangian
velocity correlation time), D v t. The
displacement distribution is the velocity
distribution which is close to a gaussian there
is a typical turbulent velocity u.
The length of each step done at a typical
velocity u depends on the time necessary to lose
the memory of the initial velocity. If the
displacement probability distribution converges
towards a Lévy distribution, it means that the
lagrangian velocity time correlation has no
typical time (i.e. it is long tailed).
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Conclusion

• The transport of a toroidal plasma through a
  magnetic field has been studied.
• Evidence of a Lévy process have been found.
• It seems to be characterised by a turbulent
  velocity distribution close to a gaussian and by
  a long tailed time autocorrelation of the
  lagrangian (particle) velocity.

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r r(tt)- D(t) n(r, t t) d3r n(r,
t) d3r
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