Turbulence

Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Models for turbulence
Vincenzo Carbone
Dipartimento di Fisica, Università della Calabria
Rende (CS) – Italy
[email protected]
Khalkidiki, Grece 2003
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Outline of talk
1) Why we need a model to describe turbulence?
2) Two kind of models introduced here: (a) shell models; (b)
low-dimensional Galerkin approximation.
3) 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
First approach
Write equations (if any exists!) of
the phenomena and simplifies
that equations to toy model
Khalkidiki, Grece 2003
Second approach
Cannot write equations, just collect
experimental data and try to write
toy models which can reproduce
observations
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
Turbulence: Solar wind as a wind tunnel
900
Fast Wind  49:12-51:12; 75:12-77:12; 105:12-107:12
Slow Wind  46:00-48:00; 72:00-74:00; 99:12-101:12
600
300
1.0
0.9AU
0.7AU
0.3AU
0.5
0.0
40
Results from Helios 2
50
60
70
80
90
100
Heliocentric Distance [AU]
Solar Wind Speed [km/sec]
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
110
Helios 2: day of 1976
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
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Turbulence in plasmas: laboratory
0
-50
Br
Data from
RFX (Padua)
Italy
-100
-150
0
50
100
150
200
250
time(s)
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
300
350
400
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 10242 collocation
points
Fluctuations BOTH in
space and time
Khalkidiki, Grece 2003
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Turbulence: Solar atmosphere
Solar flares: dissipative bursts
within turbulent environment ?
Khalkidiki, Grece 2003
Turbulent convection observed
on the photosphere (granular
dynamics), superimposed to
global oscillations acoustic modes
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Density fluctuations
in the early universe
originate massive
objects
“Turbulence”: different examples
Strong defect
turbulence in
Nematic Liquid
Crystal films
The Jupiter’s
atmosphere
Khalkidiki, Grece 2003
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
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
What’s the problem
Turbulence is the result of nonlinear dynamics



u 
 u    u  P  2u
t
Nonlinear
Dissipative
 
u  0
Incompressible
Navier-Stokes equation
u  velocity field
P  pressure
  kinematic viscosity
Hydromagnetic flows: the same

z


2 
 z   z  P  z “structure” of NS equations
t
z+
z- Elsasser
    
variables
z  u  b  u  B / 4


Nonlinear interactions happens only between fluctuations propagating in
opposite direction with respect to the magnetic field.
Khalkidiki, Grece 2003
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Fourier analysis
Consider a periodic
box of size L, Fourier analysis




ik  x
z ( x , t )  
z
(
k
,
t
)
e


Divergenceless condition
  
 

z (k , t )   z (k , t )e (k )
 1, 2
k
  




3

ik  x
z (k , t )  L  z ( x , t )e dx k  e (k )  0 
 2 
k
n
L

  ik  B0   ik  
e1 (k )    ; e2 (k )    e1 (k )
k  B0
k
  
 

z ( k , t )  z ( k , t )e ( k )
  
2D
k  e (k )  0
Khalkidiki, Grece 2003
3D
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.

    

z (k , t )
 
2 
  M  (k , p, q ) z ( p, t ) z (q, t )  k z (k , t )
 
t
p  q k

z (k, t )
 M  (k) z  (p, t ) z (k  p, t )
t
p
k k

M  (k)  ik       2 
k




Infinite number of modes involved in
nonlinear interactions for inviscid flows
Khalkidiki, Grece 2003
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
lD  LR 3 / 4
Output
the # of equations to be solved is proportional to
For space plasmas:
R ~ 108 - 1015
Typical values at present reached by
high resolution direct simulations
R ~ 103 - 105
Khalkidiki, Grece 2003
N  ( L / lD )3  R 9 / 4
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...
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Two kind of approximations

    

z (k , t )
 
2 
  M  (k , p, q ) z ( p, t ) z (q, t )  k z (k , t )
 
t
p  q k

1) To investigate dynamics of large-scales and
dynamics due to invariants of the motion:

    
z (k , t )
 

M
(

k
,
p
,
q
)
z
(
p
,
t
)
z



 (q , t )
t
Finite
number
of modes

2) To investigate scaling laws, statistical properties
and dynamics related to the energy cascade:
z  (k n , t )
 ik n  M ij z  (k n i , t ) z  (k n  j , t )  k n2 z  (k n , t )
t
i,j
Khalkidiki, Grece 2003
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 .
U is the inflow speed,
L is the size of flow
Khalkidiki, Grece 2003
U
L
Re 
UL

Look here
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Landau vs. Ruelle & Takens
1) 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.
The system lies on a subspace of the
phase space: a “strange attractor”.
Khalkidiki, Grece 2003
We can understand
what “attractor” means,
but what about strangeness?
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
The realm of experiments
PRL, 1975
Khalkidiki, Grece 2003
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Gollub & Swinney, 1975
Incommensurable
frequencies
cannot coexist
Khalkidiki, Grece 2003
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.
Khalkidiki, Grece 2003
Butterfly effect:
Extreme sensitivity to
every small fluctuations
in the initial conditions.
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
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Chaotic dynamics from Navier-Stokes equations


u


2
 u   u  P    u  f
t
un1  un  2un2  un  1
un1  1  2un2
un [0,1]
un1  T (un )
1
xn
Let us add an external forcing
term to restore turbulence
0
-1
0
50
100
150
200
250
300
350
400
nonlinear map
1) Stochastic behaviour
(randomness)
2) No predictability
-1
10
Chaotic dynamics in a
deterministic system
|xn-yn|
-2
10
-3
10
-4
10
-5
10
0
2
4
6
8
iteration n
Khalkidiki, Grece 2003
10
12
14
16
poor man’s NS equation
U. Frisch
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Sensitivity to initial conditions
un1  1  2u
2
n
A transformation
leads to the tent map
un  sin (xn   / 2)
xn  0,1/ 2
 2 xn
 xn1  
2(1  xn ) xn  1/ 2,1
Numbers written in binary format
xn  0.an (1)an (2)...an (i)...
an (i)  [0,1]
Iterates of the tent map
 an (i  1) an (1)  0
lead to the “Bernoulli shift” an 1 (i )  1  a (i  1) a (1)  1

n
n
Khalkidiki, Grece 2003
A small uncertainty surely will grows in time !
No predictability in finite times
Sensitivity of flow to every small perturbations
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Chaotic dynamic leads to stochasticity
xn1  T  T  ...T ( x0 )  T n ( x0 )
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
1
N
1
n
lim
f
(
T
(
x
))

f
(
x
)
dx
0


N 
N n 0
0
f
Khalkidiki, Grece 2003
TIME
 f
ENSEMBLE
The ensemble is
generated by the
dynamics, from
the uniform
measure in [0,1].
longitudinal speed (m/sec)
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Dynamics vs. statistics
1) Stochastic behaviour: the
dynamics is unpredictable
both in space and time.
Atmospheric flow
6
4
2
0
0
4
3
2
1
4000
4
1000
2000
3000
4000
5000
6000
7000
4100
4200
4300
4400
4500
4600
4700
4010
4020
4030
4040
4050
4060
4070
3
2
1
4000
Time (sec)
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
While the details of turbulent motions are
to be the same (apparent
extremely sensitive to triggering
self-similarity).
disturbances, statistical properties are not
(otherwise there would be little significance
in the averages!)
Khalkidiki, Grece 2003
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);
k n  k0n
n  1,2,..., N
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
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
How to build up shell models (2)
2) Assign to each shell ONLY two dynamical variables;

n
z (t )  un (t )  bn (t )
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
In this way we ruled out the possibility
to investigate BOTH spatial and temporal
properties of turbulence.
To compare with properties of real
flows remember that shell fields represent
usual increments at a given scale
For example the 2-th order moment
is related to the usual spectrum
u( x  r )  u( x)
2
Khalkidiki, Grece 2003

 sinkr 
 2 E (k ) 1 
dk

kr 

0
un(t)  u(x+r) – u(x)
unp  u ( x  r )  u ( x)
time
space
p
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
u ( x, t )  u ' ( x  VSW t , t )  VSW
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
  rVSW
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;
dzn (t )
 ik n  M i , j zni (t ) zn j (t )
dt
i , j  2 , 1
Different shell models have
been built up with different
coupling terms
4) Fix the coupling coefficients Mij imposing
the conservation of ideal invariants.
dun
 k n F  , un , bn  k n2un  f n
dt
dbn
 k nG , u n , bn  k n2bn  g n
dt
Khalkidiki, Grece 2003
dzn
 kn F  , zn  kn2 zn  f n
dt


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
E (t )   z ( x, t ) dx
  
H (t )   A    A dx



2D
Khalkidiki, Grece 2003

H (t )   A dx
In absence of magnetic field only two
invariants: kinetic energy and kinetic helicity.
Hk(t) disappears in presence of magnetic field
2D
2
E (t )   u 2 dx
H k (t )    2 dx
2
E (t )   u 2  b 2 dx
 
H c (t )   u  b dx
  
H (t )   A    A dx
 


3D
E (t )   u 2 dx
 
H k (t )   u   dx
3D
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
GOY shell model
dun

1 

un2un1  bn2bn1 
 ik n un 1un  2  bn 1bn  2   un 1un 1  bn 1bn 1  
dt
2
4


*
dbn

1 m

un2bn1  un2bn1 
 ik n (1     m )un 1bn  2  bn 1un  2   m un 1bn 1  bn 1un 1  
dt
2
4


E   un  bn
2
Conserved
quantities
H 
bn
n


H c  2 Re   unbn* 
 n

n
 m  1 / 3
Positive definite: 2D case
Khalkidiki, Grece 2003
There is the possibility to introduce
“2D” and “3D” shell models.
H   (1)
k n2
  5 / 4;
The model conserves also a “surrogate” of
magnetic helicity
2
n
2
*
  1 / 2;
n
bn
2
kn
 m  1/ 3
Gledzer, Ohkitamni &
Yamada (1973, 1989) for
the hydrodynamic case.
Non positive definite: 3D case
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
un  un e
 n 2   n1   n  0 mod( 2 )
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
Constraint
Khalkidiki, Grece 2003
i n

unun 3m
unu *n
dun
 i k nun  2un*1  bk n 1un 1un*1  ck n  2un 1un  2
dt
 n 2   n1   n  0 mod( 2 )

This simplifies the spectrum
of correlations.
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
“Old” MHD shell model -1




dun
  k nun21  k n 1unun 1  k nbn21  k n 1bnbn 1 
dt
 k nun 1un  k n 1un21  k nbn 1bn  k n 1bn21
dbn
 k n 1 un 1bn  unbn 1   k n unbn 1  un 1bn 
dt
Conserved quantities
E   un2  bn2
n
H c   unbn
Gloaguen, Leorat, Pouquet,
& Grappin (1986)
Real variables, only nearest
shells involved, one free
parameter.
Desnyansky & Novikov (1974)
for the hydrodynamic analog
n
Main investigations:
1) Transition to chaos in N-mode models (Gloaguen et al., 1986)
2) Time intermittency (Carbone, 1994)
Khalkidiki, Grece 2003
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
“Old” MHD shell model -2
dU n
 k n F  , U n , Bn  k n2U n  f n
dt
dBn
 k nG , U n , Bn  k n2 Bn  g n
dt
U n (t )  u n2
Bn (t )  bn2
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
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Properties 3D model: “dynamo action”
Numerical simulations with:
N = 24 shells; viscosity = 10-8
Time evolution of magnetic energy
K-2/3
time
Khalkidiki, Grece 2003
Constant forcing acting on large-scale:
f4+ = f4- = (1 + i) 10-3
ONLY velocity field is injected
The Kolmogorov spectrum is a fixed
point of the system
E(kn) = <|un|2> / kn
Starting from a seed the
magnetic energy increases
towards a kind of
equipartition with kinetic
energy.
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.
K-4/3
The spectrum expected for 2D kinetic
situation due to a cascade of 2D
hydrodynamical invariant
From the shell
model we have:
H 
n
bn
2
k n2
dH
2
 2  bn
dt
n
H(t) cannot decreases
H(t) – H(0) is bounded
t
4   bn (t ' ) dt '  H (t )  H (0)  H (0)
2
0
n
Khalkidiki, Grece 2003
Convergence for large t only when the
magnetic energy is zero.
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
“Turbulent dynamo” and “anti-dynamo”?
What “turbulent dynamo action” means in the shell model
Magnetic energy 3D
Magnetic energy 2D
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).
We will come back to
this point in the following
Khalkidiki, Grece 2003
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.
un (t )  bn (t )
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)
df
f
    (t )
dt

 (t )  (t ' )   (t  t ' )
Khalkidiki, Grece 2003
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
(2     m )   
Z n 1Z n Z n 1 
2
(   m )   
 (2     m ) Z n Z n1Z n 2 
Z n Z n 1Z n 1  4  k n1
2
 Im(    m ) Z n Z n1Z n 2 
Flux: an exact relationship which takes
the role of the Kolmogorov’s “4/5”-law
Khalkidiki, Grece 2003
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
(2     m )   
Z n 1Z n Z n 1 
2
(   m )   
 (2     m ) Z n Z n1Z n 2 
Z n Z n 1Z n 1  4  k n1
2
 Im(    m ) Z n Z n1Z n 2 
Flux: an exact relationship which takes
the role of the Kolmogorov’s “4/5”-law
Khalkidiki, Grece 2003
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Evolution of magnetic field spectrum
trace of magnetic field spectral matrix
7
10
1/f
-0.89
6
10
-1.06
the spectral break moves to lower
frequency with
increasing distance from the sun
1/f5/3
power density
5
10 -1.07
4
10
-1.72
-1.67
3
10
0.3AU
2
10
-1.70
0.7AU
1
10
0.9AU
0
10 -5
10
Khalkidiki, Grece 2003
-4
10
-3
10
frequency
-2
10
-1
10
This was interpreted as an
evidence that non-linear
interactions are at work
producing a turbulent cascade
process
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
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Properties: Time intermittency
Magnetic field
Velocity field
n=1
n=9
Khalkidiki, Grece 2003
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
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



z  z

 2
r
1/ 3
r 1/k
Khalkidiki, Grece 2003
  ' 3h1
r
Kolmogorov
scaling
ur  r

r
z  z  ur

r

r
q-th order
moments
u 
2
r




z 


NL

 2
r

NL
r
 
z r
The energy transfer rate is
scaling invariant only when
h = 1/3
ur 
q
 Cq  q / 3 r q / 3
 C2 2 / 3r 2 / 3  E (k )  k 5 / 3
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




z 

 2
r

r
T

Since the Alfvèn time in some
case is LESSER than the eddyturnover time, the cascade is
effectively realized in a time T:

z  z 

 2
r
 2
r
r
ur  r
r 1/k
Khalkidiki, Grece 2003
4 h 1
h = 1/4
z  z  ur

r
Kraichnan scaling
1/ 4
The energy transfer rate is
scaling invariant only when
' 


r




 NL   A  c A
Tr   NL
 A 
ur q
q-th order
moments
u 
2
r

r
 Cq' c A 
q/4
rq/4
 C c A  r 1/ 2  E (k )  k 3 / 2
'
2
1/ 2
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




x n   x n dP( x)   x n p( x)dx

Characteristic function
 (k )   eikx p( x)dx  eikx

Through the inverse transform the pdf can be written in terms of
moments, and moments can be obtained through the knowledge of pdf


1
ik  n
ikx
p ( x) 
dke 
x

2 
n!
n 0

n
n
d
 (k )
n
n
x i
dk n k 0
Gaussian process: the 2-th order moment suffices to fully
determine pdf. High-order moments are uniquely defined from
Khalkidiki, Grece 2003
the 2-th order (in this sense energy spectra are interesting!)
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Δur  un
kn ~ 1/r
Anomalous scaling laws
 q
u  kn
q
n
The “structure functions” in the model
ζq = q/3  Kolmogorov scaling
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
Khalkidiki, Grece 2003
Fields play the same role  the same “amount” of intermittency
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Inertial range in real experiments?
S n (r )  u (t  r )  u (t )  r
n
4.5
A linear range is visible
only in the slow solar
8
n=3
n=5
4.0
wind
log Sn(r)
3.5
6
3.0
Magnetic field in
the solar wind.
Helios data.
2.5
2.0
4
Slow wind
Fast wind
Slow wind
Fast wind
1.5
0
1
2
3
4
5
6
0
log r
Khalkidiki, Grece 2003
n
1
2
3
4
5
6
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
S n (r )  S m (r )
In this case we can measure only
n
the RELATIVE scaling exponents
4.5
4.0
n   n /  m
8
n=5
n=3
The range of self-similarity
extends over all the range
covered by the measurements,
BEYOND the “inertial” range
log Sn(r)
3.5
6
3.0
2.5
2.0
4
Slow wind
Fast wind
1.5
3
4
5
Slow wind
Fast wind
6
2
log S4(r)
Khalkidiki, Grece 2003
4
6
For fluid flows, scaling exponents
obtained through ESS coincides
with scaling exponents measured in
the inertial range.
Just a way to get scaling exponents
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Departure from the Kolmogorov’s laws
2.2
2.5
magnetic slow
2.0
Wind-Tunnel data
magnetic fast
1.8
2.0
velocity
1.6
Scaling exponents
Kolmogorov law: n/3
1.4
n/3
1.2
1.0
0.8
0.6
0.4
0.2
1
2
3
4
5
6
n
Solar wind: Intermittency is
stronger for magnetic field than for
velocity field. Scaling for velocity
field coincide with fluid flows
Khalkidiki, Grece 2003
1.5
1.0
0.5
Velocity
Temperature (passive)
0.0
0
1
2
3
4
5
6
7
8
p
Fluid flows: Intermittency
is stronger for passive scalar
Sharp variations of magnetic field
9
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
Khalkidiki, Grece 2003
r/a  normalized distance
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Numerical simulations
Incompressible MHD equations in 2D configurations
 n
r







Sn (r )  u ( x  r )  u ( x )  
r

High resolutions 10242 points.
Averages in both space and time.
No Taylor hypothesis when
we are dealing with simulations
1.8
scaling exponents
1.6
1.4
1.2
Intermittency is different
for different fields.
1.0
0.8
0.6
0.4
0
2
4
6
n
Khalkidiki, Grece 2003
z
+
z
-
8
velocity
magnetic
0
2
4
n
6
8
In particular magnetic
field more intermittent
than velocity field
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).
Not fully
reliable !
Khalkidiki, Grece 2003
Differences only
for high order
moments.
Probability distribution functions
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria

Fluctuations are stochastic
variables, so the structure functions
are defined in terms of pdfs:
Sn (r ) 
 u  p(u )du
n
r
r
r

For a gaussian pdf
40
Slow wind
Fast wind
kurtosis 
Kurtosis
30
The kurtosis increases
as the scale becomes
smaller
20
10
0
Gaussian
0
1
2
3
log r
Khalkidiki, Grece 2003
S4 (r )
2 3
S2 (r )
Anomalous scaling exponents implies that
pdfs have also anomalous scalings
Fluctuations at small
scales increasingly
depart from a
GAUSSIAN
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
About self-similarity
Let the scaling
law holds for differences
ur  u ( x  r )  u ( x)  r
And let us consider the
normalized variables
wr 
h
u r
u 
2 1/ 2
r
Then by changing the scale r  r, it can be
shown that, if h = cost. pdfs at two scales are related
pdf (wr )  pdf (wr )
i.e. the pdfs of normalized fields increments at
different scales collapse on the same shape
(self-similarity)
Khalkidiki, Grece 2003
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Departure from self-similarity
Experimental evidences in atmospheric fluid flows
• PDFs are not Gaussians
Small scales
• PDFs changes with scale
No global self-similarity!
Inertial range
Large scales
Khalkidiki, Grece 2003
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
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
Plasmas and shell model: the same property
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 u 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()
and computing the convolution with a Gaussian G
Pr ( ) 
L
(

)
G
(

,

)
d


the sum of gaussians of different width 
(blue) gives the resulting “stretched” PDF (red)
Khalkidiki, Grece 2003
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 (x,r)
–0.1 < + < 0.1,  = 0.4
0.9 < + < 1.0,  = 0.9
At each scale they collapse to a GAUSSIAN with different values of .
Khalkidiki, Grece 2003
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
A model for the weight of each gaussian
L( ) 







1 exp 
2 
ln  
2
22



0 




Width (variance)
of the Lognormal
distribution
• When ²= 0, L() is a -function centered in 0 so that: Gaussian P(u)
• As ² increases, L() is wider then more and more Gaussians of
different width are summed and the tails of P(u) 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
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Scaling of ² and relevant parameters
The parameter ² is found to behave
as a power-law of the scale
 (r)  r

To characterize intermittency, only two
parameters are needed, namely:
 ²max, the maximum value of the parameter
² within its scaling range, represents the
strength of intermittency
(the intermittency level at the bottom of the
energy cascade)
 , 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)
Khalkidiki, Grece 2003
2d MHD

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
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Scaling of ² for numerical simulations
²max (b) = 1.1
 (b) = 0.8
Khalkidiki, Grece 2003
²max (v) = 0.8
 (v) = 0.5
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
Khalkidiki, Grece 2003
Need for space AND scale analysis
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
 (x)
Scale

Position

f ( x)   wij ij ( x)
j   i  

wij 
 f ( x)

Khalkidiki, Grece 2003
Then we generates from this a
set of analysing wavelets by
DILATIONS and TRANSLATIONS
j
x

i
2


 j/2
 ij ( x)  2  

j
 2 
wij  f ( x  r )  f ( x)

ij
( x)dx


f ( x) dx   wij
2
ij
2
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. 
wij
wij
2
2
i
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.i.m. larger
than threshold
l.i.m. smaller
than threshold
Complete signal
Khalkidiki, Grece 2003
Gaussian background
Structures
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
In the solar wind
Original
LIMed
Solar Wind Speed [km/sec]
700
600
residuals
500
beginning of intermittent event
400
49:12
50:00
50:12
51:00
51:12
52:00
52:12
DoY 1976
Khalkidiki, Grece 2003
Point process
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.
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
Solar wind
The turbulent
energy cascade
generates
intermittent
“coherent”
events at small scales.
Interesting! the underlying cascade process is NOT
POISSONIAN, that is the intermittent (more
energetic) bursts are NOT INDEPENDENT (memory)
Khalkidiki, Grece 2003
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
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
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
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
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
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
Khalkidiki, Grece 2003
We don’t have explanation for this!
Appearence of
intermittent
structures in the
electrostatic
turbulence at the
edge of the plasma
columns
(vertical lines)
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
Khalkidiki, Grece 2003
Dissipation of (turbulent?) magnetic energy
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Solar flares are impulsive events
Time series of flare events
Hard X-ray ( > 20
keV):
Intermittent spikes
Duration 1-2 s,
Emax ~ 1027 erg
Numerous smaller
spikes down to 1024
erg (detection limit)
X-ray corona: superposition of a very
large number of flares
Khalkidiki, Grece 2003
Nano flares
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
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
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
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
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
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
Khalkidiki, Grece 2003
P(Δt)  Δt -
  2.38 0.03
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
T  aT
Dt (T )  a h Dt (aT )
This suggests to try to fit the
WTD with a Lèvy distribution
whose characteristic function is
L ( z )  exp  a | z | 
The parameter 0 <   2, for  = 2 one
recovers the definition of a Gaussian.
Khalkidiki, Grece 2003
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
The Lèvy function

P(Dt )   1  dz cos(zDt ) L ( z )
0
For large Dt this function
behaves like a power law
P(Dt)  Dt-(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
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
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Dissipative bursts in shell model
 (t )    k un    k bn
2
n
n
The energy
dissipation rate is
intermittent in
time.
Energy is
dissipated
through impulsive
isolated events
(bursts).
Khalkidiki, Grece 2003
2
2
n
n
2
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
t
Khalkidiki, Grece 2003
p

p
 t ( ) 
 /2
  (t  t ' )dt '
 /2
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
  
Singularity spectrum

 t p   d ( ) p  f ( )  
p
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
 p  min[ p  f ( )]
Inverse transform
 ( p) 
d p
dp
;
f ( )  p ( p)   p
As p is varied we select different
singularities from an entire
(continuous) spectrum
Khalkidiki, Grece 2003
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
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
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
The waiting times
The time between
two bursts is , and
let us calculate the
pdf p(.
WE FOUND A
POWER LAW
Even dissipative
bursts are NOT
INDEPENDENT
Khalkidiki, Grece 2003
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Statistics of dissipative events
  (t )   (t   )   (t )
     (t )
2 1/ 2
    /  
• Pdfs of normalized
fluctuations of energy
released in the MHD shell
model, are the same as
normalized fluctuations of
solar flares energy flux.
Khalkidiki, Grece 2003
flares
shell model
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
1.
2.
3.
Dissipative Kadanoff sand pile
PDFs are non gaussian and collapse to a single PDF (fractal)
Esponential distribution for waiting times:
 avalanches are INDEPENDENT events
Khalkidiki, Grece 2003
* Apart for the 4/5-law
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
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
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.
f-1
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.
This is an interesting property, with profound consequences.
The 1/f spectrum is ubiquitous in nature
Khalkidiki, Grece 2003
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
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
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
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
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
A modifications of sandpile model
Sanchez, Newman, Carreras, PRL 88, 068302 (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
Khalkidiki, Grece 2003
output (SOC is a linear model)
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
A “different” sandpile model
Gruzinov, Diamond, Rosenbluth, PRL 89, 255001 (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
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Charged particles diffusion
dxi
 ui
dt
t
xi (t )  xi (0)
2
 2t  dt ' ui ( x(t ' ))ui ( x(0))
0
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
1) u 2    Lèvy flights (physicall y unrealisti c infinite variance)
2) u ( x( ))u ( x(0))     with   1 (very strong Lagrangian correlatio ns)
Khalkidiki, Grece 2003
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.


kn  kn en
Use an “inverse transform”
N
 

on a shell model (with random
ik n  x
( j)
u j ( x , t )   Cn un (t )e
 c.c.
coefficients Cn) to get a
n 1
velocity and magnetic field.

(e.g. P. Kalliopi & L.V.)
Khalkidiki, Grece 2003

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:
 (a) 
 (b ) 
 
 (a)
 (b )
u ( x , t )   an en cos( k n  x  nt )  bn en sin( k n  x  nt )
n
Amplitudes an and bn are related
to energy spectrum.
k n 1  k n 1
a b 
E (k n )
2
2
n
2
n
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
1 3
n 
k n E (k n )
2
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
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Confining turbulence ?
In astrophysics, turbulent fluctuations are useful
since they CONFINE cosmic rays within the galaxy
Test-particle simulations in
electrostatic turbulence
2D slab geometry B0 = (0,0,B)
  
dr E  B

dt
B2
E X B drift
 
dr

 B0  2
dt
B0
A simple model for electrostatic turbulence with coherent
structures at all dynamical scales
Khalkidiki, Grece 2003
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
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Diffusive properties
x(t )  x(0)
 Det 2
2
(in the limit t  )
1)   1/2
 subdiffusi on
2)   1/2
 superdiffu sion (particles can make "long" jumps)
De ~ 10-3
Correlated phases
(weak superdiffusion)
 = 0.5
De ~ 1  0.1
 ~ 0.68
Khalkidiki, Grece 2003
Random phases
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Reduction of particle flux
No barrier
When the barrier is active we observe a
reduction of the flux of particles
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
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+
NN  # of particles which cross the line
Khalkidiki, Grece 2003
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Simmetrization of particle flux-2
# of crossings for each particle of a line near the border
5
# crossings for particle
4
Without barrier particles
leave the integration region
after some few crossings. The
flux is mainly directed from the
center towards the border.
3
2
1
0
-1
-2
-3
-4
-5
# crossings per particle
250
200
150
100
50
0
-50
-100
-150
-200
-250
Khalkidiki, Grece 2003
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.
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
Pascal Devynck et al., 2003
Perhaps crazy people taking more seriously
than ourself our “continuous playing” in the
realm of tokamak plasma physicists
Khalkidiki, Grece 2003
Poloidal Angle (°)
before
during
Time [µs]
poloidal mode number
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
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
  







 
p  q   ez ( p 2  q 2 )

v k ,t  
v p, t  vq , t   b p, t bq , t  k  2 k x , k y 
 
t
2kpq
L
p  q k
  
k x , k y  pair of integers






 
p  q   ez k 2
b p, t vq, t   bq, t v p, t 
b k ,t  


t
2kpq
p  q k
 
 
The convolution sum involves an infinite set of wave vectors
 
1  
 v k ,t
2 k 
 
1
H c t   
v
k ,t
2 k
 b k, t
1 
At   

2 k  k 2

E t  
 
  2
 b k ,t 

 
 
 
b k ,t  b k ,t v k ,t
2
        
 
Khalkidiki, Grece 2003
2




Rugged invariants of motion:
they remain invariant in time for
each triad of interacting wave
vectors which satisfy the condition
k=p+q
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
L  2

k  N  N   N x , N x ;  N y , N y 

Khalkidiki, Grece 2003
The result is a “Pandora’s box” of different
N-modes models whose dynamics exactly
conserve the rugged invariants.

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
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
Note: the occurrence of an inverse
cascade of magnetic helicity in shell
models is yet controversial
Khalkidiki, Grece 2003
Equipartition between
kinetic and magnetic
energy at small scales
and dominance of
magnetic energy at
largest scale
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.
 2
 k v(k , t )
2
D( N ) 
k
 2
 k b(k , t )
2
k
In the limit μ  0 and N  ,
we found Δ  1.
Khalkidiki, Grece 2003
Equipartition between kinetic and magnetic energy on
small scales in the inviscid case.
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
k1 = (1,1)
k2 = (2,-1)
k3 = (3,0)
No chaos here
Khalkidiki, Grece 2003
k1 = (1,1)
k2 = (2,-1)
k3 = (3,0)
k4 = (1,2)
k5 = (0,1)
k1 = (1,1)
k2 = (2,-1)
k3 = (3,0)
k4 = (1,2)
k5 = (0,1)
k6 = (1,0)
k7 = (1,-2)
Etc..
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) No chaos here!
k3 = (3,0)
v1  4(v2*v3  b2*b3 )  2 v1
v2  7(v1*v3  b1*b3 )  5v2
v3  3(v2 v1  b2b1 )  9 v3
b  2(v*b  b*v )  2 b
1
2 3
2 3
1
b2  5(b1*v3  v1*b3 )  5b2
b  9(b v  v b )  9 b
3
2 1
2 1
3
Vi(t) = Re[v(ki,t)]
Bi(t) = Re[b(ki,t)]
Only real fields
V1  4(V2V3  B2 B3 )  2 V1
V  7(V V  B B )  5V
2
1 3
1
3
2
V3  3(V2V1  B2 B1 )  9 B3
B1  2(V2 B3  B2V3 )  2 B1
B  5( B V  V B )  5B
2
1 3
1
3
2
B 3  9( B2V1  V2 B1 )  9 B3
How a simple model can be interesting without chaos?
Khalkidiki, Grece 2003
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Free decay: asymptotic states
T=0
T = 20
1.0
1.0
0.5
0.5
2Hc/E
2Hc/E
0.0
-0.5
0.0
-0.5
-1.0
-1.0
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
A/E
T = 40
1.0
0.8
1.0
Starting from any
initial condition,
the system evolves
towards a curve
in the parameter
space (A/E, 2Hc/E)
T = 80
1.0
0.5
2Hc/E
0.5
2Hc/E
0.6
A/E
0.0
-0.5
Analysis of a wide serie of
different numerical
simulations on free decay 2D
MHD reported by Ting,
Mattheus and Montgomery
(1986).
0.0
-0.5
-1.0
-1.0
0.0
0.2
0.4
0.6
A/E
0.8
1.0
0.0
0.2
0.4
0.6
0.8
A/E
μ = 0.01
The 3-modes real model seems
to reproduce these results.
Khalkidiki, Grece 2003
1.0
2
2
A
 2H c  

  1  2   1
E
 E  
E : energy
Hc : cross-helicity
A : magnetic helicity
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
E  A  0
Selective decay (SD) due to inverse cascade
(large-scale magnetic field)
 
 
Laboratory
  b  b ; u  0
experiments
E  H c  0
1.0
Dynamical alignment (DA) due to approximately equal
Decay of energies of alfvènic fluctuations
(alignment between velocity and magnetic field)
DA
2Hc/E
0.5
SD
0.0
-0.5
-1.0
0.0
0.2
Khalkidiki, Grece 2003
0.4
0.6
DA
0.8
1.0
A/E


u  b
Astrophysics
The curve “… does not represent the locus
of the extrema of anything over its entire
range of variation”. (Ting et al., 1986)
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




 (t )  v(k , t ), b(k , t )  S
T  (t )   (t   )
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.
I  S ;  (0)  I
Tt  (0)   (t )  I
Khalkidiki, Grece 2003
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.
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Subspaces in the 3-modes model
V1  4(V2V3  B2 B3 )  2 V1
V  7(V V  B B )  5V
2
1 3
1
3
2
V3  3(V2V1  B2 B1 )  9 B3
B1  2(V2 B3  B2V3 )  2 B1
B  5( B V  V B )  5B
2
1 3
1
3
2
B 3  9( B2V1  V2 B1 )  9 B3
Subspaces due to symmetries
can be generalized to the true
MHD equation to any N-order
truncation
Khalkidiki, Grece 2003
Vi  0; Bi  0
Fluid
Vi   Bi
Alfvènic
(fixed point)
V ; B ; B   0
B ;V ;V   0
Cross-helicity = 0
i
j
k
i
j
k
(B1,B2,V3)
(B1,V2,B3)
(V1,B2,B3)
Subspaces
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Example (B1,V2,B3)
V1  4(V2V3  B2 B3 )  2 V1
V  7(V V  B B )  5V
2
1 3
1
3
2
V3  3(V2V1  B2 B1 )  9 B3
B1  2(V2 B3  B2V3 )  2 B1
B  5( B V  V B )  5B
2
1 3
1
3
2
Example of invariant subspace
V2  7 B1 B3  5V2
B1  2V2 B3  2 B1
B  9V B  9 B
3
2
1
Cross-helicity = 0
3
B 3  9( B2V1  V2 B1 )  9 B3
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


V2 (t )  ( E  2 A)1/ 2 dn 18( E  2 A) t | 7 A /( E  2 A)
Khalkidiki, Grece 2003
1/ 2
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.
S  I C
Φi  I; Γ j  C
Ein    i2
i
Eext   i2
i
D= √Eext is the distance
of the point from
a given subspace.
Khalkidiki, Grece 2003
Example:
  B1 ;V2 ; B3  Ein  V  B  B
  V1 ; B2 ;V3 
2
2
2
1
2
3
Eext  B22  V12  V32
Let Φ(0) and Γ(0) such that
Eext « Ein at t = 0.
Let us investigate the time evolution
of both Ein and Eext
Subspace (V1,V2,V3)
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
dissipation = 0.0
dissipation = 0.01
3.0
10
Ein (Kinetic energy)
2.5
10
Energies
2.0
1.5
0
Ein
-1
10
-2
10
-3
10
-4
Eext/Ein
1.0
Eext (Magnetic energy)
0.5
0.0
0
20
40
60
80
Time
Stable
(no dynamo effect)
Khalkidiki, Grece 2003
100
Eext
0
20
40
60
80
100
Time
Selective dissipation
Attractor
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Subspace (B1,B2,V3)
dissipation = 0.0
dissipation = 0.01
3.0
10
Ein
2.5
10
Energies
2.0
1.5
10
0
Ein
-1
-2
Eext/Ein
1.0
Eext
0.5
10
-3
10
-4
0.0
0
20
40
60
80
Time
Stable
(Magnetic field on
the largest scales)
Khalkidiki, Grece 2003
100
Eext
0
20
40
60
80
100
Time
Selective dissipation
Attractor
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Subspace (B1,V2,B3)
dissipation = 0.01
dissipation = 0.0
3.0
10
2.5
Energies
2.0
0
Eext/Eint
Ein
10
Eext
10
-2
10
-3
-1
Eint
1.5
1.0
Eext
0.5
0.0
0
20
40
60
80
100
Time
Unstable (inverse cascade
at work from k3)
Khalkidiki, Grece 2003
0
20
40
60
80
100
Time
The subspace repels all
nearest trajectories.
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Subspace (V1,B2,B3)
dissipation = 0.01
dissipation = 0.0
3.0
Ein
2.5
10
Energies
2.0
10
0
Eext/Ein
-1
Ein
1.5
1.0
Eext
0.5
0.0
0
20
40
60
80
100
Time
Unstable (inverse cascade
at work from k2 and k3)
Khalkidiki, Grece 2003
10
-2
10
-3
Eext
0
20
40
60
80
100
Time
The subspace repels all
trajectories
Attractors and “repellers”
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Vi = Bi
1.0
Only one wave vector survive
A
1

;
E 1 x2
x  V1 / B1
2Hc/E
2H c
x

E
1 x2
0.5
(V1,V2,V3)
(V1,B2,B3)
0.0
(B1,B2,V3)
(B1,V2,B3)
-0.5
-1.0
0.0
Attractors drive the
system towards
Khalkidiki, Grece 2003
0.2
0.4
0.6
Vi = - Bi
0.8
1.0
A/E
“Repellers” drive the
system towards the whole
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Do you remember?
What “turbulent dynamo action” means in the shell model
Magnetic energy 3D
Magnetic energy 2D
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).
The structure of stable
and unstable time-invariant
subspaces of real MHD are
reproduced in the GOY
Shell model
Khalkidiki, Grece 2003
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Pkin

PB
Models for low-β plasmas
L
When
z
a/L

Bz
y
x
a
a/L << 1
β << 1
  10 2  1
L
 1
a
B 1
  1
B0 R
R
Laboratory plasmas
Khalkidiki, Grece 2003
Coronal loops
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria


 t


 t
Reduced MHD equations



 
B  
B
2
  B   B  Bz
 v   v   p 
  v
2 
z






 

v
2
 v    B  B   v  Bz    B
z

2





  

   ,  , v ( x, y, z, t )  vx , v y  , B( x, y, z, t )  Bx , By 
 x y 
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
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
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
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Self-organization in RMHD
f m, z  T FFT

 f m, n T
R = 14
Magnetic energy
on the wave vectors
plane (m,n)
Khalkidiki, Grece 2003
R = 21
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).
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).
Spectrum for m = 1
Time evolution of some modes
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
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
A Hybrid Shell Model
RMHD equations in the wave vector space perpendicular to B0 :
 




2
  c A z i (k, x, t )  p M ilm (k) z l (p, x, t )z  m (k  p, x, t ) k z i (k, x, t )
x 
 t
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
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
v ~ 5 10-4 cA ~ 1 Km/s
Model parameters: L ~ 3 104 Km,
Khalkidiki, Grece 2003
R ~ 6,
cA ~ 2 103 Km/s
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
After a transient a
statistical equilibrium
is reached between
incoming flux,
outcoming flux and
dissipation
The level of
fluctuations inside the
loop is considerably
higher than that
imposed at the lower
loop boundary
Dissipated power
displays a sequence of
spikes
Khalkidiki, Grece 2003
Energy balance
Stored Energy
Energy flux
Dissipated Power
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Energy spectra
Magnetic Energy
Kinetic Energy
A Kolmogorov spectrum is formed
mainly on magnetic energy
Khalkidiki, Grece 2003
Magnetic energy dominates
with respect to kinetic energy
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
Khalkidiki, Grece 2003
The obtained energy range
correspond to nanoflare
energy range
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
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Proper Orthogonal Decomposition
• The field is decomposed as:

u (r, t )   a j (t ) j (r )
r 
j 0
• 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 
max (u,  )
X

Khalkidiki, Grece 2003
2
2
3
( f , g )    f i g i* dr
j 1
An inner product is defined
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Empirical eigenfunctions
The maximum is reached through a variational
method thus obtaining the integral equation
 u(r, t )u(r' , t ) dr'  

whose kernel is the averaged autocorrelation function.
Very huge computational efforts !
2 Ek
  j
m
j
Khalkidiki, Grece 2003
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.
Low-dimensional models
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Through empirical eigenfunctions, we can reconstruct
the field using only a finite number N of modes
N
u N (r, t )   a j (t ) j (r)
j 0
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
da j (t )
dt
Khalkidiki, Grece 2003
  M j ,n,m an (t )am (t )
n,m
The coupling coefficients
depend on the empirical
eigenfunctions
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
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
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Reconstruction of velocity field
The velocity field has been reconstructed using only
J = 0, 1
Khalkidiki, Grece 2003
J = 2, 3
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 (1939-1996) of green coronal
emission line 530.3 nm. For every day 72 values
of intensities from 0 to 355 degrees of position angle
Angle
Time
Khalkidiki, Grece 2003
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
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
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Khalkidiki, Grece 2003
Let sand piles evolve …
Vincenzo Carbone
Dipartimento di Fisica,
Università della Calabria
Don’t care about…
Avalanches
Khalkidiki, Grece 2003
or