1. Vincenzo Carbone Models for turbulence
Dipartimento di Fisica,
Università della Calabria
Models for turbulence
Vincenzo Carbone
Dipartimento di Fisica, Università della Calabria
Rende (CS) – Italy
Khalkidiki, Grece 2003

2. Outline of talk Why we need a model to describe turbulence?
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Outline of talk
Why we need a model to describe turbulence?
Two kind of models introduced here: (a) shell models; (b) low-dimensional Galerkin approximation.
We are interested not just to investigate properties of simplified models “per se”, rather we are interested to understand to what extend simplified models can mimic the gross features of REAL turbulent flows.
Biological or social complex phenomena can be described by simplified toy
models which are just “caricature” of reality, derived from turbulence models
Second approach
First approach
Cannot write equations, just collect
experimental data and try to write
toy models which can reproduce
observations
Write equations (if any exists!) of
the phenomena and simplifies
that equations to toy model
Khalkidiki, Grece 2003

3. Acknowledgments Pierluigi Veltri, Annick Pouquet, Angelo Vulpiani,
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Acknowledgments
Pierluigi Veltri, Annick Pouquet, Angelo Vulpiani,
Guido Boffetta, Helène Politano
Paolo Giuliani (PhD thesis on MHD shell model)
Fabio Lepreti (PhD thesis on solar flares)
Luca Sorriso (PhD thesis on solar wind turbulence)
Roberto Bruno, Vanni Antoni and the whole crew
in Padova for experiments on laboratory and
solar wind plasmas
Khalkidiki, Grece 2003

4. Turbulence: Solar wind as a wind tunnel
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Turbulence: Solar wind as a wind tunnel
Results from Helios 2
In situ measurements of high amplitude fluctuations
for all fields (velocity, magnetic, temperature…)
A unique possibility to measure low-frequency
turbulence in plasmas over a wide range of scales.
Khalkidiki, Grece 2003

5. Turbulence in plasmas: laboratory
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Turbulence in plasmas: laboratory
Data from
RFX (Padua)
Italy
Plasma generated for nuclear fusion, confined in a reversed field pinch configuration. High amplitude fluctuations of magnetic field, measurements (time series) at the edge of plasma column, where the toroidal field changes sign.
Khalkidiki, Grece 2003

6. Turbulence: numerical simulations
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Turbulence: numerical simulations
High resolution direct numerical simulations of
MHD equations. Mainly in 2D configurations.
R  1600
Space collocation
points
Fluctuations BOTH in
space and time
Khalkidiki, Grece 2003

7. Turbulence: Solar atmosphere
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Turbulence: Solar atmosphere
Solar flares: dissipative bursts
within turbulent environment ?
Turbulent convection observed
on the photosphere (granular
dynamics), superimposed to
global oscillations acoustic modes
Khalkidiki, Grece 2003

8. “Turbulence”: different examples
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
“Turbulence”: different examples
Strong defect turbulence in
Nematic Liquid Crystal films
Density fluctuations
in the early universe
originate massive
objects
The Jupiter’s
atmosphere
Khalkidiki, Grece 2003

9. Main features of turbulent flows
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Main features of turbulent flows
1) Randomness in space and time
2) Turbulent structures on all scales
3) Unpredictability and instability
to very small perturbations
Khalkidiki, Grece 2003

10. What’s the problem Turbulence is the result of nonlinear dynamics
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
What’s the problem
Turbulence is the result of nonlinear dynamics
Incompressible
Navier-Stokes equation
u  velocity field
P  pressure
n  kinematic viscosity
Nonlinear
Dissipative
Hydromagnetic flows: the same
“structure” of NS equations z+ z-
Elsasser
variables
Nonlinear interactions happens only between fluctuations propagating in opposite direction with respect to the magnetic field.
Khalkidiki, Grece 2003

11. Fourier analysis 3D 2D Consider a periodic Divergenceless condition
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Fourier analysis
Consider a periodic
box of size L, Fourier analysis
Divergenceless condition 3D 2D
Khalkidiki, Grece 2003

12. Equation for Fourier modes
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Equation for Fourier modes
The evolution of the field for a single wave vector is related to fields
of ALL other wave vectors (convolution term) for which k = p + q.
Infinite number of modes involved in
nonlinear interactions for inviscid flows
Khalkidiki, Grece 2003

13. Why models for turbulence?
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Why models for turbulence?
Input
In the limit of high R, assuming a Kolmogorov spectrum E(k) ~ k-5/3 dissipation takes place at scale:
Transfer
Output
the # of equations to be solved is proportional to
For space plasmas: R ~
At these values it is not possible to have an inertial range extended for more than one decade. No possibility to verify asymptotic scaling laws, statistics...
Typical values at present reached by
high resolution direct simulations
R ~
Khalkidiki, Grece 2003

14. Two kind of approximations
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Two kind of approximations
1) To investigate dynamics of large-scales and
dynamics due to invariants of the motion:
2) To investigate scaling laws, statistical properties
and dynamics related to the energy cascade:
Khalkidiki, Grece 2003

15. Fluid flows become turbulent as Re  
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Fluid flows become turbulent as Re  
Osborne Reynolds noted that as Re increases
a fluid flow bifurcates toward a turbulent regime
Flow past a cylinder
viscosity n.
U is the inflow speed,
L is the size of flow
Look here L U
Khalkidiki, Grece 2003

16. Landau vs. Ruelle & Takens
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Landau vs. Ruelle & Takens
Landau:
turbulence appears at the end of an infinite serie of Hopf bifurcations, each adding an incommensurable frequency to the flow
The more frequencies 
The more stochasticity
2) Ruelle & Takens:
incommensurable frequencies cannot coexist, the motion becomes rapidly aperiodic and turbulence suddenly will appear, just after three (or four) bifurcations.
We can understand
what “attractor” means,
but what about strangeness?
The system lies on a subspace of the phase space: a “strange attractor”.
Khalkidiki, Grece 2003

17. The realm of experiments
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
The realm of experiments
PRL, 1975
Khalkidiki, Grece 2003

18. Gollub & Swinney, 1975 Incommensurable frequencies cannot coexist
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Gollub & Swinney, 1975
Incommensurable
frequencies
cannot coexist
Khalkidiki, Grece 2003

19. E.N. Lorenz (1963) Even if the phase space has infinite
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
E.N. Lorenz (1963)
Even if the phase space has infinite
dimensions, the system lies on a subspace (strange attractor).
 THE SYSTEM CAN BE DESCRIBED BY ONLY A SMALL SET OF VARIABLES
The presence of a strange attractor simplifies the description of turbulence
Edward Lorenz in 1963: a Galerkin approximation with only three modes to get a simplified model of convective rolls in the atmosphere.
The trajectories of this system, for certain settings, never settle down to a fixed point, never approach a stable limit cycle, yet never diverge to infinity.
Butterfly effect:
Extreme sensitivity to
every small fluctuations
in the initial conditions.
Khalkidiki, Grece 2003

20. Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Simplified models
“The idea was that, although a hydrodynamical system has a very large number of degree of freedom, technically speaking infinitely many, most of them will be inactive at the onset of turbulence, leaving only few interacting active modes, which nevertheless can generate a complex and unpredictable evolution.”
Bohr, Jensen, Paladin & Vulpiani,
Dynamical system approach to turbulence
Cambridge Univ. Press.
Dissipation in a complex system, is responsible for the elimination of many degree of freedoms, reducing the system to very few dimensions
Coullet, Eckmann & Koch, J. Stat. Phys. 25, 1 (1981).
Khalkidiki, Grece 2003

21. Chaotic dynamics from Navier-Stokes equations
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Chaotic dynamics from Navier-Stokes equations
Let us add an external forcing
term to restore turbulence
1) Stochastic behaviour
(randomness)
2) No predictability
Chaotic dynamics in a deterministic system
poor man’s NS equation
U. Frisch
Khalkidiki, Grece 2003

22. Sensitivity to initial conditions
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Sensitivity to initial conditions
A transformation
leads to the tent map
Numbers written in binary format
Iterates of the tent map
lead to the “Bernoulli shift”
A small uncertainty surely will grows in time !
No predictability in finite times
Sensitivity of flow to every small perturbations
Khalkidiki, Grece 2003

23. Chaotic dynamic leads to stochasticity
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Chaotic dynamic leads to stochasticity
Apply the map n times
As a consequence of the chaoticity, the trajectory of a SINGLE orbit covers ALL the allowed phase space
Ergodic theorem: Let f(x) an integrable function, and let f(Tn(x0)) calculated over all iterates of the map. Then for almost all x0
The ensemble is
generated by the
dynamics, from
the uniform
measure in [0,1].
Khalkidiki, Grece 2003

24. Dynamics vs. statistics
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Dynamics vs. statistics
1) Stochastic behaviour: the
dynamics is unpredictable
both in space and time.
2) Predictability is introduced
at a statistical level (via the
ergodic theorem and the
properties of chaos !).
The measured velocity field
is a stochastic field with
gaussian statistics.
3) On every scale details of the
plots are different but
statistical properties seems
to be the same (apparent
self-similarity).
Atmospheric flow
While the details of turbulent motions are
extremely sensitive to triggering disturbances, statistical properties are not (otherwise there would be little significance
in the averages!)
Khalkidiki, Grece 2003

25. How to build up shell models (1)
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
How to build up shell models (1)
1) Introduce a logarithmic spacing of
the wave vectors space (shells);
The intershell ratio in general
is set equal to  = 2.
In this way we can investigate
properties of turbulence at
very high Reynolds numbers.
We are not interested in the
dynamics of each wave vector mode
of Fourier expansion, rather in
the gross properties of dynamics
at small scales.
Khalkidiki, Grece 2003

26. How to build up shell models (2)
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
How to build up shell models (2)
2) Assign to each shell ONLY two dynamical variables;
In this way we ruled out the possibility
to investigate BOTH spatial and temporal
properties of turbulence.
These fields take into account
the averaged effects of velocity
modes between kn and kn+1, that
is fluctuations across eddies at
the scale rn ~ kn-1
To compare with properties of real
flows remember that shell fields represent
usual increments at a given scale
un(t)  u(x+r) – u(x)
For example the 2-th order moment
is related to the usual spectrum
time
space
Khalkidiki, Grece 2003

27. Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Measurements
In situ satellite measurements of velocity and magnetic field,
the sample is transported with the solar wind velocity
Satellite frame
SW frame
Taylor’s hypothesis: The time dependence of u(x,t)
comes from the spatial argument of u’
The time variation of u at a fixed spatial
location (supersonic VSW), are reinterpreted
as being a spatial variation of u’.
Khalkidiki, Grece 2003

28. How to build up shell models (3)
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
How to build up shell models (3)
3) Write a nonlinear equations with couplings
among variables belonging to local shells;
Different shell models have
been built up with different
coupling terms
4) Fix the coupling coefficients Mij imposing
the conservation of ideal invariants.
Khalkidiki, Grece 2003

29. Invariants Invariants of the dynamics in absence of dissipation and
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Invariants
Invariants of the dynamics
in absence of dissipation and
forcing:
1) total energy
2) cross-helicity
3) magnetic helicity 2D 3D
In absence of magnetic field only two
invariants: kinetic energy and kinetic helicity.
Hk(t) disappears in presence of magnetic field 3D 2D
Khalkidiki, Grece 2003

30. Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
GOY shell model
The model conserves also a “surrogate” of magnetic helicity
Conserved
quantities
There is the possibility to introduce
“2D” and “3D” shell models.
Gledzer, Ohkitamni &
Yamada (1973, 1989) for
the hydrodynamic case.
Positive definite: 2D case
Non positive definite: 3D case
Khalkidiki, Grece 2003

31. Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Phase invariance
A phase invariance is present in shell models, and this constraints the possible set of stationary correlation functions with a nonzero mean value
GOY shell model is invariant under
With the constraint
Owing to this phase invariance the only quadratic
form with a mean value different from zero is
Other constrants exists for high order correlations
Modified shell model
This simplifies the spectrum
of correlations.
Constraint
Khalkidiki, Grece 2003

32. “Old” MHD shell model -1 Gloaguen, Leorat, Pouquet, & Grappin (1986)
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
“Old” MHD shell model -1
Gloaguen, Leorat, Pouquet,
& Grappin (1986)
Real variables, only nearest
shells involved, one free
parameter.
Conserved quantities
Desnyansky & Novikov (1974)
for the hydrodynamic analog
Main investigations:
1) Transition to chaos in N-mode models (Gloaguen et al., 1986)
2) Time intermittency (Carbone, 1994)
Khalkidiki, Grece 2003

33. “Old” MHD shell model -2 NOT dynamical models. Introduced in order to
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
“Old” MHD shell model -2
NOT dynamical models.
Introduced in order to
investigate spectral properties
of turbulence, and competitions
between the nonlinear
energy cascade and some linear
instabilities (reconnection,..)
Obtained in the framework of closure approximations
EDQNM, Direct Interaction Approximation
Main investigations:
1) The first model of development of turbulence in solar surges
2) Spectral properties of anisotropic MHD turbulence
Anticipated results of high resolution numerical simulations
Khalkidiki, Grece 2003

34. Properties 3D model: “dynamo action”
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Properties 3D model: “dynamo action”
Constant forcing acting on large-scale:
f4+ = f4- = (1 + i) 10-3
ONLY velocity field is injected
Numerical simulations with:
N = 24 shells; viscosity = 10-8
Time evolution of magnetic energy
The Kolmogorov spectrum is a fixed
point of the system
K-2/3
E(kn) = <|un|2> / kn
Starting from a seed the
magnetic energy increases
towards a kind of
equipartition with kinetic
energy.
time
Khalkidiki, Grece 2003

35. Properties 2D: “anti-dynamo”
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Properties 2D: “anti-dynamo”
The 2D model shows a kind of “anti-dynamo” action:
A seed of magnetic field cannot increase.
The spectrum expected for 2D kinetic
situation due to a cascade of 2D hydrodynamical invariant
K-4/3
From the shell
model we have:
H(t) cannot decreases
H(t) – H(0) is bounded
Convergence for large t only when the
magnetic energy is zero.
Khalkidiki, Grece 2003

36. “Turbulent dynamo” and “anti-dynamo”?
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
“Turbulent dynamo” and “anti-dynamo”?
What “turbulent dynamo action” means in the shell model
There exists some
“invariant subspaces”
which can act like
“attractors”
for all solutions (stable
subspaces).
The fluid subspace is
stable (in 2D case) or
unstable (in 3D case).
Magnetic energy 3D
Magnetic energy 2D
We will come back to
this point in the following
Khalkidiki, Grece 2003

37. Dynamical alignment Alfvènic state: fixed point of MHD shell model.
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Dynamical alignment
Alfvènic state: fixed point of MHD shell model.
Strong correlations between velocity and
magnetic fields for each shell.
Alfvènic state is a “strong” attractor for the model. The system falls
on it, for different kind of constant forcing.
Time evolution of velocity and magnetic
field for the mode n = 7, with constant
forcing terms.
The fixed point is destabilized when
we use a Langevin equation for the
external forcing term, with a correlation
time τ (eddy-turnover time)
Khalkidiki, Grece 2003

38. Properties: spectrum and flux
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Properties: spectrum and flux
Numerical simulations with: N = 26 shells; viscosity = 0.5 ∙ 10-9
Kolmogorov fixed point of the system.
Inertial and dissipative ranges + intermediate range visible in shell models
Flux: an exact relationship which takes
the role of the Kolmogorov’s “4/5”-law
Khalkidiki, Grece 2003

39. Properties: spectrum and flux
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Properties: spectrum and flux
Numerical simulations with: N = 26 shells; viscosity = 0.5 ∙ 10-9
Kolmogorov fixed point of the system
Inertial and dissipative ranges + intermediate range visible in shell models
Flux: an exact relationship which takes
the role of the Kolmogorov’s “4/5”-law
Khalkidiki, Grece 2003

40. Evolution of magnetic field spectrum
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Evolution of magnetic field spectrum
trace of magnetic field spectral matrix 10 -5 -4 -3 -2 -1 1 2 3 4 5 6 7
1/f
the spectral break moves to lower frequency with
increasing distance from the sun
-0.89
1/f5/3
-1.06
-1.07
This was interpreted as an evidence that non-linear interactions are at work producing a turbulent cascade process
-1.72
power density
-1.67
0.3AU
-1.70
0.7AU
0.9AU
frequency
Khalkidiki, Grece 2003

41. Observations of the Kraichnan’s scaling
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Observations of the Kraichnan’s scaling
Old observations of
magnetic turbulence in
the solar wind seems to
show that a Kraichnan’s
scaling law is visible at
intermediate scales.
k-3/2
Khalkidiki, Grece 2003

42. Properties: Time intermittency
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Properties: Time intermittency
Velocity field
Magnetic field
n = 1
n = 9
Khalkidiki, Grece 2003

43. Fluctuations in plasmas
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Fluctuations in plasmas
Small scale: STRUCTURES
Increasing
scales
Large scale: random signal
Velocity increments at 3 different scales
in the solar wind: Δur = u(t + r) – u(t)
Khalkidiki, Grece 2003

44. Phenomenology: fluid-like
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Phenomenology: fluid-like
Let us consider the dissipation rate for both
pseudo-energies (stochastic quantities equality in law!)
The characteristic time (eddy-turnover time) is the
time of life of turbulent eddies
The energy transfer rate is
scaling invariant only when
h = 1/3
Kolmogorov
scaling
q-th order
moments
r 1/k
Khalkidiki, Grece 2003

45. Phenomenology: magnetically dominated
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Phenomenology: magnetically dominated
In this case there is a physical time, the Alfvèn time, which represents the
sweeping of Alfvenic fluctuations due to the large-scale magnetic field
Since the Alfvèn time in some
case is LESSER than the eddy-
turnover time, the cascade is
effectively realized in a time T:
The energy transfer rate is
scaling invariant only when
h = 1/4
Kraichnan scaling
q-th order
moments
r 1/k
Khalkidiki, Grece 2003

46. Why high-order moments?
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Why high-order moments?
Let x a stochastic variable distributed according to a Probability Density
Function (pdf) p(x), the n-th order moment is
Characteristic function
Through the inverse transform the pdf can be written in terms of moments, and moments can be obtained through the knowledge of pdf
Gaussian process: the 2-th order moment suffices to fully
determine pdf. High-order moments are uniquely defined from
the 2-th order (in this sense energy spectra are interesting!)
Khalkidiki, Grece 2003

47. Anomalous scaling laws
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Anomalous scaling laws
Δur  un
The “structure functions” in the model
ζq = q/3  Kolmogorov scaling
kn ~ 1/r
Scaling exponents obtained in the range where the flux scales as kn-1
A departure from the Kolmogorov law must be attributed to time intermittency in the shell model.
The departure from the Kolmogorov law measures the “amount” of intermittency
Fields play the same role  the same “amount” of intermittency
Khalkidiki, Grece 2003

48. Inertial range in real experiments?
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Inertial range in real experiments?
A linear range is visible only in the slow solar wind
Magnetic field in
the solar wind.
Helios data.
Khalkidiki, Grece 2003

49. Extended self-similarity
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Extended self-similarity
The m-th order structure function (m = 3 or m = 4) plays the role of a generalized scale
In this case we can measure only the RELATIVE scaling exponents
The range of self-similarity extends over all the range covered by the measurements, BEYOND the “inertial” range
For fluid flows, scaling exponents obtained through ESS coincides with scaling exponents measured in the inertial range.
Just a way to get scaling exponents
Khalkidiki, Grece 2003

50. Departure from the Kolmogorov’s laws
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Departure from the Kolmogorov’s laws
Solar wind: Intermittency is stronger for magnetic field than for velocity field. Scaling for velocity field coincide with fluid flows
Fluid flows: Intermittency
is stronger for passive scalar
Sharp variations of magnetic field
Khalkidiki, Grece 2003

51. Magnetic turbulence in laboratory plasma
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Magnetic turbulence in laboratory plasma
The departure from the linear scale increases going towards the wall
Turbulence more intermittent near the external wall
Similar to edge
turbulence in
fluid flows
r/a  normalized distance
Khalkidiki, Grece 2003

52. Numerical simulations
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Numerical simulations
Incompressible MHD equations in 2D configurations
High resolutions points.
Averages in both space and time.
No Taylor hypothesis when
we are dealing with simulations
Intermittency is different for different fields.
In particular magnetic field more intermittent than velocity field
Khalkidiki, Grece 2003

53. Comparison with velocity in fluid flows
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Comparison with velocity in fluid flows
A collection of data from laboratory fluid flows (black symbols) and solar
wind velocity (white symbols).
Differences only for high order moments.
Not fully
reliable !
Khalkidiki, Grece 2003

54. Probability distribution functions
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Probability distribution functions
Fluctuations are stochastic
variables, so the structure functions are defined in terms of pdfs:
For a gaussian pdf
The kurtosis increases
as the scale becomes
smaller
Fluctuations at small scales increasingly
depart from a GAUSSIAN
Anomalous scaling exponents implies that
pdfs have also anomalous scalings
Khalkidiki, Grece 2003

55. About self-similarity
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
About self-similarity
Let the scaling
law holds for differences
And let us consider the
normalized variables
Then by changing the scale r  r, it can be
shown that, if h = cost. pdfs at two scales are related
i.e. the pdfs of normalized fields increments at different scales collapse on the same shape
(self-similarity)
Khalkidiki, Grece 2003

56. Departure from self-similarity
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Departure from self-similarity
Experimental evidences in atmospheric fluid flows
PDFs are not Gaussians
PDFs changes with scale
Small scales
No global self-similarity!
Inertial range
Large scales
Khalkidiki, Grece 2003

57. Plasmas and shell model: the same property
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Plasmas and shell model: the same property
The full line
corresponds
to a fit made
by using a
multifractal
model to
describe the
scaling of
Pdfs.
In the
following I
describe this
model.
Khalkidiki, Grece 2003

58. A multifractal model for pdfs
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
A multifractal model for pdfs
According to the multifractal model (scaling exponents h(x) depend on the position) the PDF of a field du at scale r can be described as a superposition of Gaussians
each describing the statistics in different regions of volume S(h)
each of different variance (h)
each weighted by the occurrence of S(h)
This is achieved introducing the distribution L(s) and computing the convolution with a Gaussian G
the sum of gaussians of different width 
(blue) gives the resulting “stretched” PDF (red)
Khalkidiki, Grece 2003

59. Evidence: conditioned pdfs
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Evidence: conditioned pdfs
In 2D numerical simulations, we have calculated the pdfs of
fluctuations, CONDITIONED to a given value of the energy flux e(x,r)
–0.1 < e+ < 0.1, s = 0.4
0.9 < e+ < 1.0, s = 0.9
At each scale they collapse to a GAUSSIAN with different values of s.
Khalkidiki, Grece 2003

60. A model for the weight of each gaussian
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
A model for the weight of each gaussian
Width (variance) of the Log-normal distribution
When l²= 0, L(s) is a d-function centered in s0 so that: Gaussian P(du)
As l² increases, L(s) is wider then more and more Gaussians of different width are summed and the tails of P(du) become higher
The parameter ² can be used to characterize the scaling of the shape of the PDFs, that is the intermittency of the field!
Khalkidiki, Grece 2003

61. Scaling of ² and relevant parameters
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Scaling of ² and relevant parameters
The parameter l² is found to behave as a power-law of the scale
To characterize intermittency, only two parameters are needed, namely:
l²max, the maximum value of the parameter l² within its scaling range, represents the strength of intermittency
(the intermittency level at the bottom of the energy cascade)
b, the ‘slope’ of the power-law, representing the efficiency of the non-linear cascade
(measures how fast energy is concentrated on structures at smaller and smaller scales)
2d MHD
Khalkidiki, Grece 2003

62. Scaling of ² for solar wind turbulence
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Scaling of ² for solar wind turbulence
Magnetic field
solid symbols: fast streams
open symbols: slow streams
Velocity field
magnetic field is more intermittent than velocity
Khalkidiki, Grece 2003

63. Scaling of ² for numerical simulations
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Scaling of ² for numerical simulations
l²max (b) = 1.1
l²max (v) = 0.8
b (v) = 0.5
b (b) = 0.8
Khalkidiki, Grece 2003

64. Turbulence: “structures”+background ?
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Turbulence: “structures”+background ?
A description of turbulence: “coherent” structures
present on ALL scales within the sea of a gaussian
background. They contain most of the energy
of the flow and play an important dynamical role.
Examples from Jupiter’s atmosphere
Need for space AND scale analysis
Khalkidiki, Grece 2003

65. Orthogonal Wavelets decomposition
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Orthogonal Wavelets decomposition
Let us consider a signal f(x) made by N = 2m samples (being Dx = 1), and
build up a set of functions starting from a “mother” wavelet
Then we generates from this a
set of analysing wavelets by
DILATIONS and TRANSLATIONS
Scale Position
Khalkidiki, Grece 2003

66. Local Intermittency Measure
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Local Intermittency Measure
The energy content, at each scale, is not uniformly distributed in space
L.i.m. greater than a threshold
means that at a given scale and
position the energy content is greater
than the average at that scale L L
l.i.m. larger than threshold
l.i.m. smaller than threshold
Complete signal
Gaussian background
Structures
Khalkidiki, Grece 2003

67. In the solar wind Point process The sequence of intermittent
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
In the solar wind
The sequence
of intermittent
events generates
a point process.
Statistical
properties of the
process gives
information
on the underlying
physics which
generated the
point process.
Point process
Khalkidiki, Grece 2003

68. Waiting times between “structures”
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Waiting times between “structures”
The times
between events
are distributed
according to a
power law
Pdf(Δt) ~ Δt -β
The turbulent
energy cascade
generates
intermittent
“coherent”
events at small scales.
Solar wind
Interesting! the underlying cascade process is NOT POISSONIAN, that is the intermittent (more energetic) bursts are NOT INDEPENDENT (memory)
Khalkidiki, Grece 2003

69. Power law distribution for waiting times
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Power law distribution for waiting times
Fluid flow
Laboratory plasma
Turbulent flows share this characteristic.
Power law is generated through the chaotic dynamics
and must be reproduced by models for turbulence.
Khalkidiki, Grece 2003

70. Waiting times in the MHD shell model
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Waiting times in the MHD shell model
Time intermittency
in the shell model
is able to capture
also that property
of real turbulence
Chaotic dynamics
generates non
poissonian events
Khalkidiki, Grece 2003

71. What kind of intermittent structures ?
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
What kind of intermittent structures ?
Minimum variance analysis around isolated structure allows to identify them
Solar Wind: shock
(compressive structures)
Solar wind: tangential discontinuity (current sheet)
Khalkidiki, Grece 2003

72. Magnetic structures in laboratory plasmas
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Magnetic structures in laboratory plasmas
RFX edge magnetic turbulence:
current sheets
Current sheets are naturally produced as coherent, intermittent structures by nonlinear interactions
Khalkidiki, Grece 2003

73. Intermittent structures in laboratory plasmas
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Intermittent structures in laboratory plasmas
RFX edge turbulence
of electrical potential
Structures are potential holes
Khalkidiki, Grece 2003

74. Dynamics of intermittent structures
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Dynamics of intermittent structures
Relationship between intermittent structures of edge turbulence and disruptions of the plasma columns at the center of RFX
Time evolution of
floating potential
Minima are related
to disruptions
Appearence of
intermittent
structures in the
electrostatic
turbulence at the
edge of the plasma
columns
(vertical lines)
We don’t have explanation for this!
Khalkidiki, Grece 2003

75. Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Statistical flares
Ratio of EIT full Sun images in Fe XII 195A to Fe IX/X 171A.
Temperature distribution in the Sun's corona:
- dark areas cooler regions
- bright areas hotter regions
Dissipation of (turbulent?) magnetic energy
Khalkidiki, Grece 2003

76. Solar flares are impulsive events
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Solar flares are impulsive events
Hard X-ray ( > 20 keV):
Intermittent spikes
Duration 1-2 s,
Emax ~ 1027 erg
Numerous smaller spikes down to 1024 erg (detection limit)
Time series of flare events
X-ray corona: superposition of a very large number of flares
Nano flares
Khalkidiki, Grece 2003

77. Power law statistics of bursts
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Power law statistics of bursts
Total energy, peak energy
and (more or less!) lifetime
of individual bursts seems
to be distributed according
to power laws.
Khalkidiki, Grece 2003

78. The Parker’s conjecture (1988)
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
The Parker’s conjecture (1988)
Nanoflares correspond to dissipation of many small current sheets, forming in the bipolar regions as a consequence of the continous shuffling and intermixing of the footpoints of the field in the photospheric convection.
Current sheets: tangential discontinuity which become increasingly severe with the continuing winding and interweaving eventually producing intense magnetic dissipation in association with magnetic reconnection.
Khalkidiki, Grece 2003

79. Self-Organized Criticality (P. Bak et al., 1987)
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Self-Organized Criticality (P. Bak et al., 1987)
A paradigm for complex dissipative systems exibiting bursts, is invoked as a model to describe ALSO turbulence.
Self-organized state  critical state (at the border line of chaos) reached by the system apparently without tuning parameters.
Critical state  attractor, robust with respect to variations of parameters and with respect to randomness.
Khalkidiki, Grece 2003

80. Sandpile model Avalanches of all size i.e. FRACTAL PROCESS
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Sandpile model
SANDPILE IS THE PROTOTIPE OF SOC
Sandpile profile is the critical state.
Perturbed with one single sand grain added at a random position.
When the local slope exceedes a critical value the sand in excess is redistributed to nearest sites generating an avalanche whose dimension L is that of the marginally stable region.
Lack of any typical length
Avalanches of all size i.e.
FRACTAL PROCESS
Size, lifetimes and number of sand grains in each avalanche are power law distributed.
Khalkidiki, Grece 2003

81. Sand Pile Model for Solar Flares
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Sand Pile Model for Solar Flares
The coronal magnetic field spontaneously evolves in a self-organized state (critical profile).
Perturbations: convective random motion at footpoints of magnetic loops. Avalanche: reconnection event
Cellular Automata model for reconnection :
Vector field Bi on a 3D lattice
Local slope dBi = Bi -Sj wj Bi+j
When | dBi| > some treshold: instability at position i :
Field readjusted in the nearby positions so that the grid point i becomes stable
The readjustment can destabilize nearby points producing an avalanche (flare)
Power peak, total energy and duration are power law distributed.
Khalkidiki, Grece 2003

82. Waiting times between solar flares
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Waiting times between solar flares
Sand piles cannot describe all observed features of solar flares (Boffetta, Carbone, Giuliani, Veltri, Vulpiani, 1999)
Intermittency in sand piles is produced by isolated and completely random singularities  Poisson process  pdf of waiting times must be exponential (see inset in figure)
On the contrary flares from the GOES dataset show asymptotic POWER LAW DISTRIBUTION
P(Δt)  Δt -b
b  2.38 0.03
Khalkidiki, Grece 2003

83. The origin of power law distribution for waiting times
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
The origin of power law distribution for waiting times
The waiting time sequence forms a “temporal point process”,
statistically self-similar
A rescaling gives the same
statistical properties
This suggests to try to fit the WTD with a Lèvy distribution whose characteristic function is
The parameter 0 <   2, for  = 2 one recovers the definition of a Gaussian.
Khalkidiki, Grece 2003

84. The Lèvy function For large t this function behaves like a power law
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
The Lèvy function
For large t this function behaves like a power law
P(t)  t-(1 + )
WTD is a Lèvy function. A fit
on the GOES flares gives the
non trivial value   1.38  0.06
Stable distribution, obtained through the Central Limit Theorem by relaxing the hypothesis of finite variance
 The underlying process has long (infinite) correlation, and is a non Poissonian point process.
Khalkidiki, Grece 2003

85. Parker’s conjecture modified
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Parker’s conjecture modified
Nanoflares correspond to dissipation of many small current sheets, forming in the nonlinear cascade occuring inside coronal magnetic structure as a consequence of the power input in the form of Alfven waves due to footpoint motion.
Current sheets: coherent intermittent small scale structures of MHD turbulence
Khalkidiki, Grece 2003

86. Dissipative bursts in shell model
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Dissipative bursts in shell model
The energy
dissipation rate is intermittent in time.
Energy is
dissipated
through impulsive
isolated events
(bursts).
Khalkidiki, Grece 2003

87. Multifractal structure of dissipation
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Multifractal structure of dissipation
Coarse-grained dissipation
has been generated from
simulations
Moments of dissipation
have a scaling law
Khalkidiki, Grece 2003

88. Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Singularity spectrum
Spectrum of singularities  described by the function f(), which represents the fractal dimension of the space where dissipation related to a singularity .
From saddle-point we get
Inverse transform
As p is varied we select different
singularities from an entire
(continuous) spectrum
Khalkidiki, Grece 2003

89. Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Inside bursts
Through a threshold process we can identify and isolate each dissipative bursts to make statistics
Khalkidiki, Grece 2003

90. Some different statistics
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Some different statistics
Let us define some statistics on impulsive events
1) Total energy of
bursts
2) Time duration
3) Energy of peak
In all cases we
found power laws,
the scaling exponents
depend on threshold.
Khalkidiki, Grece 2003

91. The waiting times The time between two bursts is t, and
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
The waiting times
The time between
two bursts is t, and
let us calculate the
pdf p(t).
WE FOUND A
POWER LAW
Even dissipative
bursts are NOT
INDEPENDENT
Khalkidiki, Grece 2003

92. Statistics of dissipative events
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Statistics of dissipative events
Pdfs of normalized fluctuations of energy released in the MHD shell model, are the same as normalized fluctuations of solar flares energy flux.
shell model
flares
Khalkidiki, Grece 2003

93. Could SOC describes turbulent cascade?*
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Could SOC describes turbulent cascade?*
Simulations of Kadanoff SOC model: Rescaled energy fluctuations at
different scales and waiting times at the smallest scale
Kadanoff sand pile
Dissipative Kadanoff sand pile
PDFs are non gaussian and collapse to a single PDF (fractal)
Esponential distribution for waiting times:
 avalanches are INDEPENDENT events
* Apart for the 4/5-law
Khalkidiki, Grece 2003

94. Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
The Running sandpile
Need for a correct definition of time scale (not often discussed in literature). In the sand pile no way to define a timescale (no time series). Avalanches must be considered as a collection of instantaneous events.
The Running sandpile:
in each temporal step (properly defined in this model) the system is continuously fed with a finite deposition rate Jin and the unstable sites are simultaneously updated
 the energy dissipated can be properly followed step by step, so that time series are obtained.
Khalkidiki, Grece 2003

95. Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Running sandpile
Simulations of running sandpile: Rescaled energy fluctuations at different scales and waiting times at the smallest scale
Running sand pile with two different deposition rates.
Low Jin  non gaussian pdfs and exponential distribution for waiting times
High Jin  gaussian pdfs and power law for waiting times
Khalkidiki, Grece 2003

96. 1/f spectra in the Running Sandpile
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
1/f spectra in the Running Sandpile
From the running sandpile model we can get continuous time series.
From time series obtained in this way (for example of total “dissipated” energy) we can easily get power spectra.
Unless in the classical SOC model, 1/f spectra are visible, at large scales, but only for high values of Jin.
f-1
This is an interesting property, with profound consequences.
The 1/f spectrum is ubiquitous in nature
Khalkidiki, Grece 2003

97. Anomalous transport in laboratory plasmas
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Anomalous transport in laboratory plasmas
Diffusion causes loss of particles, energy, …
Perhaps the main cause of disruption of magnetic confinement needed to achieve nuclear fusion.
In general turbulent fluctuations of electric field enhance loss,
the transport is called “anomaloues” since it is due to turbulence.
Anomalous transport  A problem with language:
Plasma physics: Transport driven by turbulent fluctuations
Physics of fluids: Transport with non-Gaussian features
Khalkidiki, Grece 2003

98. Fluxes of particles in Tokamak
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Fluxes of particles in Tokamak
The generation of BARRIERS for transport is a way to enhance confinement in plasmas.
 We need models of turbulent fluctuations
Khalkidiki, Grece 2003

99. “SOC-Paradigm” for Turbulent Transport
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
“SOC-Paradigm” for Turbulent Transport
Plasma confined in toroidal devices is dominated by anomalous transport (on machine scales) driven by fluctuations (on microscopic scales).
SOC apparently solves the paradox.
The marginally unstable profile of plasma is continuously perturbed by driving gradients (sand grains  microscopic level).
1/f spectrum obtained for the floating potential at the edge of RFX (Padua).
Note: The SOC mechanism continuously can sustain active bursty transport (avalanches  macroscopic level), and relaxes back to the linearly least unstable profile. The dominant scale for the transport is the system scale.
Khalkidiki, Grece 2003

100. Waiting times between transport events
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Waiting times between transport events
Bursts of density fluctuations at the edge of plasma revealed through both microwave reflectometry and electrostatic probes.
Power laws for waiting times: The SOC-PARADIGM does not describe all features of observations.
Khalkidiki, Grece 2003

101. A modifications of sandpile model
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
A modifications of sandpile model
Sanchez, Newman, Carreras, PRL 88, (2002)
Introduces correlated
input to reproduce power laws in waiting times.
Quite trivial!
Note 1: Power laws
with scaling exponents
greater than 3 corresponds
to gaussian processes NOT
to Poisson processes
(the central limit theorem is
actually not broken)
Note 2: Correlated input  (necessarily!) correlated output (SOC is a linear model)
Khalkidiki, Grece 2003

102. A “different” sandpile model
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
A “different” sandpile model
Gruzinov, Diamond, Rosenbluth, PRL 89, (2002)
Two unstable ranges with different rules for grains toppling. When the
second range is unstable the height of the pile is lowered at a level of the
first range (coupling between internal and external structures).
 Formation of pedestal region with bursty transport
Modification of output
Khalkidiki, Grece 2003

103. Charged particles diffusion
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Charged particles diffusion
Typical problem:
Lagrangian evolution of particles in a given fluid flow. Chaotic behaviour is assured by non-integrability.  Anomalous transport ALSO in very simple “laminar” fluid flows!
Anomalous diffusion is not a trivial problem!
Diffusion is anomalous (non-Gaussian) when the central limit
theorem is broken.
This leads to very restrictive conditions
Khalkidiki, Grece 2003

104. 3D velocity field from shell model
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
3D velocity field from shell model
Using a shell model (in the wave vectors space) it is
possible to build up a model for a turbulent field
(in the physical space)
Introduce a wave vector with a
given amplitude kn = k0 2n and
random directions.
Use an “inverse transform”
on a shell model (with random
coefficients Cn) to get a
velocity and magnetic field.
(e.g. P. Kalliopi & L.V.)
Khalkidiki, Grece 2003

105. A different approach Perhaps there is no need to run a shell model
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
A different approach
Perhaps there is no need to run a shell model
+ the equations of motion for a test particle.
A simple model for turbulence with coherent structures at all dynamical scales:
Amplitudes an and bn are related
to energy spectrum.
Wave vectors have random directions and amplitudes kn = 2n k0
Time evolution is related to the eddy-turnover time.
Reproduce characteristics of pair diffusion, …
Khalkidiki, Grece 2003

106. Barrier for transport in plasmas
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Barrier for transport in plasmas
Since turbulent fluctuations causes losses, barriers
are tentatively generated with a simple equation in mind:
No turbulent fluctuations  No anomalous transport
For example:
Shear flows are able to decorrelate turbulent eddies and to kill fluctuations.
Mechanism: stretching and distortion of eddies because
different points inside an eddy have different speeds.
The eddy loses coherence, the eddy turnover time decreases  turbulent intensity decreases.
Low  High mode confinement transition have been observed in real experiments (a lot of money to generate a shear flow in a tokamak!!).
Khalkidiki, Grece 2003

107. Confining turbulence ? E X B drift
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Confining turbulence ?
In astrophysics, turbulent fluctuations are useful
since they CONFINE cosmic rays within the galaxy
E X B drift
Test-particle simulations in
electrostatic turbulence
2D slab geometry B0 = (0,0,B)
A simple model for electrostatic turbulence with coherent
structures at all dynamical scales
Khalkidiki, Grece 2003

108. A barrier for the transport
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
A barrier for the transport
A barrier has been generated by randomizing the phases of the field ONLY within a narrow strip at the border of the integration domain.
Q(x,y) = strain2 – vorticity2
Khalkidiki, Grece 2003

109. Diffusive properties De ~ 10-3  = 0.5 De ~ 1  0.1  ~ 0.68
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Diffusive properties
De ~ 10-3
Correlated phases
(weak superdiffusion)
 = 0.5
De ~ 1  0.1
 ~ 0.68
Random phases
Khalkidiki, Grece 2003

110. Reduction of particle flux
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Reduction of particle flux
When the barrier is active we observe a
reduction of the flux of particles
No barrier
Cumulative number of
particles as a function
of time which escape
from the integration
Region.
Different curves refers
to different values of
the amplitude of the
barrier.
Khalkidiki, Grece 2003

111. Symmetrization of particle flux-1
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Symmetrization of particle flux-1
When the barrier is active we observe
a symmetrization of the flux.
barrier
Particle flux through the line N+ N-
N  # of particles which cross the line
Khalkidiki, Grece 2003

112. Simmetrization of particle flux-2
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Simmetrization of particle flux-2
# of crossings for each particle of a line near the border
Without barrier particles
leave the integration region
after some few crossings. The
flux is mainly directed from the
center towards the border.
With the barrier active, particles
are trapped and make a standard
diffusive motion inside the
integration region.
The flux is symmetric, each
particle makes multiple crossings
of the line in both directions.
Khalkidiki, Grece 2003

113. Experiments at Castor tokamak (Prague)
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Experiments at Castor tokamak (Prague)
A barrier have been generated by biasing the electric field
with a weak perturbation on the border (low amount of money!!)
Principle of control: perturbate rather than kill turbulence!.
Control Ring in Castor
before
Poloidal Angle (°)
during
Pascal Devynck et al., 2003
Perhaps crazy people taking more seriously than ourself our “continuous playing” in the realm of tokamak plasma physicists
Time [µs]
poloidal mode number
Khalkidiki, Grece 2003

114. Fluxes are reduced and symmetrized
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Fluxes are reduced and symmetrized
Particle Flux during « open loop »
PDF of the Particle Flux
The positive bursts (towards the wall) still exist but a backward flux (towards the plasma) is created.
Khalkidiki, Grece 2003

115. Galerkin approximation
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Galerkin approximation
Models can be obtained by retaining only a finite number
of interacting modes in the convolution sum.
For example in 2D MHD
The convolution sum involves an infinite set of wave vectors
Rugged invariants of motion:
they remain invariant in time for
each triad of interacting wave
vectors which satisfy the condition
k = p + q
Khalkidiki, Grece 2003

116. Simplified N-modes models
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Simplified N-modes models
Simplified models can be obtained by retaining only a
finite number of interacting modes in the convolution sum.
Among the infinite modes which satisfy k = p + q, retain only wave vectors which lye within a region of width N
The result is a “Pandora’s box” of different N-modes models whose dynamics exactly conserve the rugged invariants.
Khalkidiki, Grece 2003

117. Models vs. Simulations Main advantages :
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Models vs. Simulations
Main advantages :
rugged invariants are conserved in absence of dissipation, true dissipationless runs.
Main disadvantages :
higher computational times (N2 vs. N log N)
Example:
N = 25
Khalkidiki, Grece 2003

118. 2D example: inverse cascade
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
2D example: inverse cascade
Coarse-grained
energy averaged
over circular shells
of amplitudes
m = (kx2 + ky2)1/2
t = 0
Equipartition between
kinetic and magnetic
energy at small scales
and dominance of
magnetic energy at
largest scale
Note: the occurrence of an inverse cascade of magnetic helicity in shell models is yet controversial
Khalkidiki, Grece 2003

119. 2D example: self-similarity in the decay
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
2D example: self-similarity in the decay
In the inviscid limit, constant quantities
Kinetic and magnetic enstrophy
decay in time, but their ratio
tends to a fixed value.
In the limit μ  0 and N  ,
we found Δ  1.
Equipartition between kinetic and magnetic energy on
small scales in the inviscid case.
Khalkidiki, Grece 2003

120. Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Do your own model !
If you have some free time to spent, and you want see
chaotic trajectories on your screen, you could investigate
time behaviour of some N-mode models (N ≥ 5).
You can find nice sequences of bifurcations, transitions
to chaos, very beautiful attractors, etc…
(for fluid flows see e.g Franceschini & Tebaldi, 1979; J. Lee, 1987; …)
Some triads which satisfy ki= kj + km
N = 7
N = 3
N = 5
Etc..
k1 = (1,1)
k2 = (2,-1)
k3 = (3,0)
k4 = (1,2)
k5 = (0,1)
k6 = (1,0)
k7 = (1,-2)
k1 = (1,1)
k2 = (2,-1)
k3 = (3,0)
k1 = (1,1)
k2 = (2,-1)
k3 = (3,0)
k4 = (1,2)
k5 = (0,1)
No chaos here
Khalkidiki, Grece 2003

121. A triad-interaction model
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
A triad-interaction model
The most basic model to investigate nonlinear
interactions in 2D MHD
k1 = (1,1)
k2 = (2,-1)
k3 = (3,0)
Vi(t) = Re[v(ki,t)]
Bi(t) = Re[b(ki,t)]
Only real fields
No chaos here!
How a simple model can be interesting without chaos?
Khalkidiki, Grece 2003

122. Free decay: asymptotic states
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Free decay: asymptotic states
Analysis of a wide serie of different numerical simulations on free decay 2D MHD reported by Ting, Mattheus and Montgomery (1986).
Starting from any
initial condition,
the system evolves
towards a curve
in the parameter
space (A/E, 2Hc/E)
μ = 0.01
The 3-modes real model seems
to reproduce these results.
E : energy
Hc : cross-helicity
A : magnetic helicity
Khalkidiki, Grece 2003

123. Selective decay and dynamical alignment
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Selective decay and dynamical alignment
Extreme points of the curve represents decay of rugged
invariants with respect to total energy.
Variational principle
Selective decay (SD) due to inverse cascade
(large-scale magnetic field)
Laboratory
experiments
Dynamical alignment (DA) due to approximately equal
Decay of energies of alfvènic fluctuations
(alignment between velocity and magnetic field) DA
Astrophysics SD
The curve “… does not represent the locus
of the extrema of anything over its entire
range of variation”. (Ting et al., 1986) DA
Khalkidiki, Grece 2003

124. Time-invariant subspaces
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Time-invariant subspaces
Fluid equations are characterized by the presence of time-invariant subspaces, which are interesting
for the dynamics of the system.
A point in the phase space S,
evolves according to a
time-translation operator
Let I  S a subspace of S, and let Φ(0)  I a vector of I.
The subspace I is invariant in time if, for each vector Φ(0),
the time evolution is able to maintain the vector Φ(t) on I.
Example: the fluid subspace of MHD
Φ(0)={v(k,0),b(k,0)} such that b(k,0) = 0.
From MHD equations b(k,t) = 0 for each time.
Khalkidiki, Grece 2003

125. Subspaces in the 3-modes model
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Subspaces in the 3-modes model
Fluid
Alfvènic
(fixed point)
Cross-helicity = 0
(B1,B2,V3)
Subspaces due to symmetries
can be generalized to the true
MHD equation to any N-order
truncation
Subspaces
(B1,V2,B3)
(V1,B2,B3)
Khalkidiki, Grece 2003

126. Example (B1,V2,B3) Cross-helicity = 0
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Example (B1,V2,B3)
Example of invariant subspace
Cross-helicity = 0
When μ = 0 two invariants  the motion is bounded on a line given by the intersection of the circle E with the cylinder A.
The system reduces to a Duffin’g equation without forcing term. Solution in terms of elliptic function dn
Khalkidiki, Grece 2003

127. Stable and unstable subspaces
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Stable and unstable subspaces
Stability of subspaces are investigated according to
time evolution of distance from a given subspace.
Example:
Let Φ(0) and Γ(0) such that
Eext « Ein at t = 0.
D= √Eext is the distance
of the point from
a given subspace.
Let us investigate the time evolution
of both Ein and Eext
Khalkidiki, Grece 2003

128. Subspace (V1,V2,V3) dissipation = 0.0 dissipation = 0.01 Stable
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Subspace (V1,V2,V3)
dissipation = 0.0
dissipation = 0.01
Stable
(no dynamo effect)
Selective dissipation
Attractor
Khalkidiki, Grece 2003

129. Subspace (B1,B2,V3) dissipation = 0.0 dissipation = 0.01 Stable
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Subspace (B1,B2,V3)
dissipation = 0.0
dissipation = 0.01
Stable
(Magnetic field on
the largest scales)
Selective dissipation
Attractor
Khalkidiki, Grece 2003

130. Subspace (B1,V2,B3) dissipation = 0.0 dissipation = 0.01
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Subspace (B1,V2,B3)
dissipation = 0.0
dissipation = 0.01
Unstable (inverse cascade
at work from k3)
The subspace repels all
nearest trajectories.
Khalkidiki, Grece 2003

131. Subspace (V1,B2,B3) dissipation = 0.0 dissipation = 0.01
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Subspace (V1,B2,B3)
dissipation = 0.0
dissipation = 0.01
Unstable (inverse cascade
at work from k2 and k3)
The subspace repels all
trajectories
Khalkidiki, Grece 2003

132. Attractors and “repellers”
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Attractors and “repellers”
Vi = Bi
Only one wave vector survive
(V1,V2,V3)
(V1,B2,B3)
(B1,B2,V3)
(B1,V2,B3)
Attractors drive the
system towards
“Repellers” drive the
system towards the whole
Vi = - Bi
Khalkidiki, Grece 2003

133. Do you remember?
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
What “turbulent dynamo action” means in the shell model
There exists some
“invariant subspaces” which can act like “attractors”
for all solutions (stable
subspaces).
The fluid subspace is stable (in 2D case) or unstable (in 3D case).
Magnetic energy 3D
Magnetic energy 2D
The structure of stable
and unstable time-invariant
subspaces of real MHD are
reproduced in the GOY
Shell model
Khalkidiki, Grece 2003

134. Models for low-β plasmas
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Models for low-β plasmas
When L z
a/L << 1
β << 1 y a x
Laboratory plasmas
Coronal loops
Khalkidiki, Grece 2003

135. Reduced MHD equations Incompressible 2D MHD in perpendicular variables
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Reduced MHD equations
Incompressible 2D MHD in perpendicular variables
Alfven wave propagation along background magnetic field
Total energy and cross-helicity survive.
Only two time invariants in ideal RMHD
Khalkidiki, Grece 2003

136. Simplified models The cylinder has been divided in
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Simplified models
The cylinder has been divided in
Nsez “planes” at fixed zn.
A Galerkin approximation with
N-modes of 2D MHD on each “plane”, and a finite difference scheme to solve the propagation in the perpendicular direction.
Periodic boundaries conditions at
z = 0 and z = L to simulate toroidal
situations. Simulations with
Nsez = 256 and N = 18.
Khalkidiki, Grece 2003

137. The Galerkin truncature model
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
The Galerkin truncature model
Both magnetic and kinetic
energies accumulates at m = 1. for all z.
Equipartition between energies.
Inverse cascade without
conservation of A ?
Actually A is quasi-invariant in the model
No inverse cascade, but a
kind of self-organization due to the fact that ΔA/A « 1 ?
Khalkidiki, Grece 2003

138. Self-organization in RMHD
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Self-organization in RMHD
R = 14
R = 21
Magnetic energy
on the wave vectors
plane (m,n)
A kind of self-organization also in the vertical direction.
Depending on the aspect ratio the spectrum is dominated by some few modes (the higher R the more modes are present in the spectrum).
Khalkidiki, Grece 2003

139. Quasi-single helicity states in RFX
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Quasi-single helicity states in RFX
Quasi-single helicity states observed in laboratory plasmas in some situation (example RFX).
Time evolution of some modes
Spectrum for m = 1
Characterized by: a) the mode m = 1 in the transverse plane; b) a few dominant
modes in the toroidal direction, depending on the aspect ratio (the higher R the
more modes are present in the spectrum).
Khalkidiki, Grece 2003

140. Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
A Hybrid Shell Model
RMHD equations in the wave vector space perpendicular to B0 :
A shell model in the wave vector space perpendicular to B0 can be derived:
(Hybrid : the space dependence along B0 is kept)
Khalkidiki, Grece 2003

141. Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Boundary Conditions
Space dependence along B0 allows to chose boundary conditions:
Total reflection is imposed at the upper boundary
A random gaussian motion with autocorrelation time tc = 300 s is imposed at the lower boundary only on the largest scales
The level of velocity fluctuations at lower boundary is of the order of photospheric motions
dv ~ cA ~ 1 Km/s
Model parameters: L ~ Km, R ~ 6, cA ~ Km/s
Khalkidiki, Grece 2003

142. Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Energy balance
Stored Energy
After a transient a statistical equilibrium is reached between incoming flux, outcoming flux and dissipation
Energy flux
The level of fluctuations inside the loop is considerably higher than that imposed at the lower loop boundary
Dissipated Power
Dissipated power displays a sequence of spikes
Khalkidiki, Grece 2003

143. Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Energy spectra
Magnetic Energy
Kinetic Energy
A Kolmogorov spectrum is formed mainly on magnetic energy
Magnetic energy dominates with respect to kinetic energy
Khalkidiki, Grece 2003

144. Statistical analysis of dissipated power
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Statistical analysis of dissipated power
Power Peak
Burst duration
Burst Energy
Waiting time
Power laws are recovered on Power peak, burst duration, burst energy and waiting time distributions
The obtained energy range correspond to nanoflare energy range
Khalkidiki, Grece 2003

145. Low-dimensional models for coherent structures
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Low-dimensional models for coherent structures
In many turbulent flows one observes coherent
structures on large-scales. In these cases the basic
features of the system can be described by few variables
Proper Orthogonal Decomposition (POD) is a tool that
allows one to build up, from numerical simulations or
direct spatio-temporal experiments, a low-dimensional
system which models the spatially coherent structures.
Khalkidiki, Grece 2003

146. Proper Orthogonal Decomposition
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Proper Orthogonal Decomposition
The field is decomposed as:
The functions which describe the base are NOT GIVEN A PRIORI (empirical eigenfunctions).
We want to find a basis  that is OPTIMAL for the data set in the sense that a finite dimensional representation of the field u(r,t) describes typical members of the ensemble better than representations in ANY other base
This is achieved through a maximization of the average of the proiection of u on 
An inner product is defined
Khalkidiki, Grece 2003

147. Empirical eigenfunctions
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Empirical eigenfunctions
The maximum is reached through a variational
method thus obtaining the integral equation
whose kernel is the averaged autocorrelation function.
Very huge computational efforts !
In the framework of POD, j represents the energy associated to j -th mode.
They are ordered as j > j+1
lower modes contain more energy.
Khalkidiki, Grece 2003

148. Low-dimensional models
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Low-dimensional models
Through empirical eigenfunctions, we can reconstruct
the field using only a finite number N of modes
In this way we capture the maximum allowed for energy
with respect to any other truncature with N modes.
Low-dimensional models can be build up through a Galerkin
approximation of equations which governes the flow
The coupling coefficients
depend on the empirical eigenfunctions
Khalkidiki, Grece 2003

149. Turbulent convection – Time behavior
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Turbulent convection – Time behavior
We analysed line of sight velocity field of solar photosphere
from telescope THEMIS (on July, 1, 1999).
32 images of width 30” x 30” (1” = 725 km) sampled every 1.25 minute)
j = 0,1 aperiodic behaviour  convective overshooting
j = 2,3 oscillatory behaviour T about 5 min  5 minutes oscillations
The behaviour of other modes is not well defined  both behaviors
Khalkidiki, Grece 2003

150. Turbulent convection – Spatial behavior
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Turbulent convection – Spatial behavior
0 1 spatial pattern similar to granulation pattern
Spatial scale about 700 km. Modes j = 0, 1 are mainly due to a granular contribution.
2,3 largest structures and low contrasts (with exceptions of definite and isolate regions). These eigenfunctions are associated to oscillatory phenomena characterized by a period of 5 minutes.
Khalkidiki, Grece 2003

151. Reconstruction of velocity field
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Reconstruction of velocity field
The velocity field has been reconstructed using only
J = 0, 1
J = 2, 3
Khalkidiki, Grece 2003

152. Playing with POD POD have been used to describe spatio-temporal
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Playing with POD
POD have been used to describe spatio-temporal
behaviour of the 11-years solar cycle
Daily observations ( ) of green coronal
emission line nm. For every day 72 values
of intensities from 0 to 355 degrees of position angle
Angle
Time
Khalkidiki, Grece 2003

153. 3 POD modes Original Reconstruction with 1 POD mode periodicities
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
3 POD modes
Original
Reconstruction with 1 POD mode
periodicities
Reconstruction with 2 POD modes
+ migration
Reconstruction with 3 POD modes
+ stochasticity
Khalkidiki, Grece 2003

154. Conclusions Became a “Plasma Physicist” Acknowledge Loukas
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Conclusions
Became a “Plasma Physicist”
Acknowledge Loukas
Vlahos and the local
organizing committee
Deadline for applications:
September 28, 2003
Khalkidiki, Grece 2003

155. Let sand piles evolve … Vincenzo Carbone Dipartimento di Fisica,
Università della Calabria
Let sand piles evolve …
Khalkidiki, Grece 2003

156. Don’t care about… Avalanches or Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Don’t care about…
Avalanches or
Khalkidiki, Grece 2003