Vincenzo Carbone Dipartimento di Fisica, Università della Calabria Rende (CS) – Italy...

Vincenzo CarboneDipartimento di Fisica, Università della Calabria

Rende (CS) – [email protected]

Models for turbulence

Khalkidiki, Grece 2003Khalkidiki, Grece 2003

Vincenzo Carbone Vincenzo Carbone Dipartimento di Fisica, Dipartimento di Fisica,

Università della CalabriaUniversità della Calabria

http://astronomy.swin.edu.au/~pbourke/fractals/lorenz/lorenz11.gif



Università della CalabriaUniversità 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

Second approach

Cannot write equations, just collect experimental data and try to write toy models which can reproduce observations



Università della CalabriaUniversità della Calabria Acknowledgments

Pierluigi Veltri, Annick Pouquet, Angelo Vulpiani, Guido Boffetta, Helène Politano

Roberto Bruno, Vanni Antoni and the whole crew in Padova for experiments on laboratory and solar wind plasmas

Paolo Giuliani (PhD thesis on MHD shell model)Fabio Lepreti (PhD thesis on solar flares)Luca Sorriso (PhD thesis on solar wind turbulence)



Università della CalabriaUniversità della Calabria Turbulence: Solar wind as a wind tunnel

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.

Results from Helios 2

40 50 60 70 80 90 100 110

300

600

900

0.3AU0.7AU

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

0.9AU

0.0

Heliocentric D

istance [AU]

0.5

1.0

Sola

r W

ind

Spee

d [k

m/s

ec]

Helios 2: day of 1976



Università della CalabriaUniversità della Calabria Turbulence in plasmas: laboratory

0 50 100 150 200 250 300 350 400-150

-100

-50

0

Br

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.

Data from RFX (Padua) Italy



Università della CalabriaUniversità della Calabria Turbulence: numerical simulations

High resolution direct numerical simulations of MHD equations. Mainly in 2D configurations.

R 1600Space 10242 collocation points Fluctuations BOTH in space and time



Università della CalabriaUniversità 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



Università della CalabriaUniversità della Calabria “Turbulence”: different examples

Strong defect turbulence in Nematic Liquid Crystal filmsDensity

fluctuations in the early universeoriginate massive objects The Jupiter’s

atmosphere



Università della CalabriaUniversità della Calabria Main features of turbulent flows

1) Randomness in space and time2) Turbulent structures on all scales3) Unpredictability and instability

to very small perturbations

What’s the problem




uPuut

u

2

Nonlinear Dissipative

Incompressible Navier-Stokes equationu velocity fieldP pressure kinematic viscosity

0 u

Turbulence is the result of nonlinear dynamics

4/

2

Bubuz

zPzzt

z

z+ z-

Hydromagnetic flows: the same “structure” of NS equations

Nonlinear interactions happens only between fluctuations propagating in opposite direction with respect to the magnetic field.

Elsasservariables

Fourier analysis




Consider a periodic box of size L, Fourier analysis

nL

k

dxetxzLtkz

etkztxz

xki

k

xki

2

),(),(

),(),(

3

)()(;)(

0)(

)(),(),(

12

0

01

2,1

kek

kike

Bk

Bkike

kek

ketkztkz

Divergenceless condition

0)(

)(),(),(

kek

ketkztkz

3D

2D

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.

),(),(),(),,(),( 2 tkzktqztpzqpkM

t

tkz

kqp

2)(

),(),()(),(

k

kkikM

tztzMt

tz

p

k

pkpkk

Infinite number of modes involved in nonlinear interactions for inviscid flows




In the limit of high R, assuming a Kolmogorov spectrum E(k) ~ k-5/3 dissipation takes place at scale:

Why models for turbulence?

4/93)/( RlLN D

4/3LRlD

Typical values at present reached by high resolution direct simulations

R ~ 103 - 105

Input

Output

Transfer

the # of equations to be solved is proportional to

For space plasmas: R ~ 108 - 1015 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...



Università della CalabriaUniversità della Calabria Two kind of approximations

),(),(),(),,(),( 2 tkzktqztpzqpkM

t

tkz

kqp

modes ofnumberFinite

),(),(),,(),(

tqztpzqpkMt

tkz

),(),(),(),( 2 tkzktkztkzMik

t

tkznnjninijn

n

ji,

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:




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

UL

Re

U L Look here

Landau vs. Ruelle & TakensVincenzo Carbone Vincenzo Carbone

Dipartimento di Fisica, Dipartimento di Fisica, Università della CalabriaUniversità della Calabria


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”.

1) Landau: turbulence appears at the end of an infinite serie of Hopf bifurcations, each adding an incommensurable frequency to the flowThe more frequencies

The more stochasticity

We can understand what “attractor” means, but what about strangeness?

The realm of experimentsVincenzo Carbone Vincenzo Carbone



PRL, 1975

Gollub & Swinney, 1975Vincenzo Carbone Vincenzo Carbone



Incommensurable frequencies cannot coexist

E.N. Lorenz (1963)

The presence of a strange attractor simplifies the description of turbulence




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

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.




“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 turbulenceCambridge Univ. Press.

Simplified models

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).




Chaotic dynamics from Navier-Stokes equations

fuPuut

u

2

Let us add an external forcing term to restore turbulence

12 2

1 nnnn uuuu

Chaotic dynamics in a deterministic system

0 2 4 6 8 10 12 14 1610-5

10-4

10-3

10-2

10-1

|xn-y

n|

iteration n

0 50 100 150 200 250 300 350 400

-1

0

1

xn 1) Stochastic behaviour

(randomness)2) No predictability

]1,0[

21 2

1

n

nn

u

uu

poor man’s NS equationU. Frisch

map nonlinear )(1 nn uTu




Sensitivity to initial conditions

2

1 21 nn uu 1,2/1

2/1,0

n

n

x

xA transformation leads to the tent map

]1,0[)(

)...()...2()1(.0

ia

iaaax

n

nnnn

A small uncertainty surely will grows in time ! No predictability in finite timesSensitivity of flow to every small perturbations

)1(2

2

)2/(

1

n

n

n

nn

x

xx

xsinu

Numbers written in binary format

)1(1

)1()(1 ia

iaia

n

n

n

Iterates of the tent map lead to the “Bernoulli shift” 1)1(

0)1(

n

n

a

a




Chaotic dynamic leads to stochasticity

)()(... 001 xTxTTTx n

n Apply the map n times

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

As a consequence of the chaoticity, the trajectory of a SINGLE orbit covers ALL the allowed phase space

ENSEMBLETIME

N

n

n

N

ff

dxxfxTfN

0

1

0

0 )())((1

lim The ensemble is generated by the dynamics, from the uniform measure in [0,1].




Dynamics vs. statistics

4000 4010 4020 4030 4040 4050 4060 40701

2

3

4

long

itud

inal

spe

ed (

m/s

ec)

Time (sec)

4000 4100 4200 4300 4400 4500 4600 47001234

0 1000 2000 3000 4000 5000 6000 7000

0246

While the details of turbulent motions are extremely sensitive to triggering disturbances, statistical properties are not (otherwise there would be little significancein the averages!)

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




How to build up shell models (1)

Nn

kk nn

,...,2,10

1) Introduce a logarithmic spacing of

the wave vectors space (shells);

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.

The intershell ratio in general is set equal to = 2.





0

2 1)(2)()( dkkr

sinkrkExurxu

2) Assign to each shell ONLY two dynamical variables;

)()()( tbtutz nnn

These fields take into account the averaged effects of velocitymodes 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

In this way we ruled out the possibility to investigate BOTH spatial and temporal properties of turbulence.

For example the 2-th order moment is related to the usual spectrum

un(t) u(x+r) – u(x)

ppn xurxuu )()(

spacetime




Measurements

In situ satellite measurements of velocity and magnetic field,the sample is transported with the solar wind velocity

SWSW VttVxutxu ),('),(

SW frame

Taylor’s hypothesis: The time dependence of u(x,t) comes from the spatial argument of u’

Satellite frame

The time variation of u at a fixed spatial location (supersonic VSW), are reinterpreted as being a spatial variation of u’.

SWrV





nnnnnnn

nnnnnnn

gbkbuGkdt

db

fukbuFkdt

du

2

2

,,

,,

3) Write a nonlinear equations with couplingsamong variables belonging to local shells;

1,2,

, )()()(

jijninjin

n tztzMikdt

tdz Different shell models have been built up with different coupling terms

4) Fix the coupling coefficients Mij imposing the conservation of ideal invariants.

nnnnnn fzkzFk

dt

dz 2,




Invariants

dxutH

dxutE

k

)(

)( 2

dxAAtH

dxtxztE

)(

),()(2Invariants of the dynamics

in absence of dissipation and forcing:

1) total energy2) cross-helicity3) magnetic helicity

dxAAtH

dxbutH

dxbutE

c

)(

)(

)( 22

dxAtH2

)(2D 3D

In absence of magnetic field only two invariants: kinetic energy and kinetic helicity. Hk(t) disappears in presence of magnetic field

dxtH

dxutE

k2

2

)(

)(

2D

3D




GOY shell model

*

121211112121

*

121211112121

4

1

2)1(

4

1

2

nnnnm

nnnnm

nnnnmnn

nnnnnnnnnnnnnn

bubuubbuubbuikdt

db

bbuubbuubbuuikdt

du

nnnc

nnn

buH

buE

*

22

Re2

3/1;4/5

2

2

m

n n

n

k

bH

The model conserves also a “surrogate” of magnetic helicityConserved

quantities

3/1;2/1

)1(2

m

n n

nn

k

bH

Positive definite: 2D case Non positive definite: 3D case

There is the possibility to introduce “2D” and “3D” shell models.

Gledzer, Ohkitamni & Yamada (1973, 1989) for the hydrodynamic case.




Phase invariance

ninn euu

*

3

nuu

uu

n

mnn

212*

111*

12 nnnnnnnnnn uuckuubkuuki

dt

du

A phase invariance is present in shell models, and this constraints the possible set of stationary correlation functions with a nonzero mean valueGOY shell model is invariant under

Other constrants exists for high order correlations

Modified shell model

Owing to this phase invariance the only quadratic form with a mean value different from zero is

Constraint )2mod(012 nnn

)2mod(012 nnnWith the constraint

This simplifies the spectrum of correlations.




“Old” MHD shell model -1

nnnnnnnnnn

n

nnnnnnnnnn

nnnnnnnnnnn

bubukbubukdt

db

bkbbkukuuk

bbkbkuukukdt

du

11111

2111

2111

112

1112

1

nnnc

nnn

buH

buE 22

Gloaguen, Leorat, Pouquet, & Grappin (1986)

Real variables, only nearest shells involved, one free parameter.

Conserved quantities

Main investigations:

1) Transition to chaos in N-mode models (Gloaguen et al., 1986)

2) Time intermittency (Carbone, 1994)

Desnyansky & Novikov (1974)for the hydrodynamic analog




“Old” MHD shell model -2

2

2

)(

)(

nn

nn

btB

utU

NOT dynamical models. Introduced in order to investigate spectral properties of turbulence, and competitions between the nonlinear energy cascade and some linear instabilities (reconnection,..)

Main investigations:

1) The first model of development of turbulence in solar surges

2) Spectral properties of anisotropic MHD turbulence

Obtained in the framework of closure approximationsEDQNM, Direct Interaction Approximation

nnnnnnn

nnnnnnn

gBkBUGkdt

dB

fUkBUFkdt

dU

2

2

,,

,,

Anticipated results of high resolution numerical simulations




Properties 3D model: “dynamo action”

Numerical simulations with: N = 24 shells; viscosity = 10-8

Constant forcing acting on large-scale:f4

+ = f4- = (1 + i) 10-3

ONLY velocity field is injected

Time evolution of magnetic energy

K-2/3

time

The Kolmogorov spectrum is a fixed point of the system

Starting from a seed the magnetic energy increases towards a kind of equipartition with kinetic energy.

E(kn) = <|un|2> / kn




Properties 2D: “anti-dynamo”

nn

n n

n

bdt

dH

k

bH

2

2

2

2

t

nn HHtHdttb

0

2)0()0()(')'(4

K-4/3

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

H(t) cannot decreasesH(t) – H(0) is bounded

Convergence for large t only when the magnetic energy is zero.

From the shell model we have:




“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).

We will come back to this point in the following

Magnetic energy 3D

Magnetic energy 2D




Dynamical alignment

)()( tbtu nn 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.

The fixed point is destabilized whenwe use a Langevin equation for the external forcing term, with a correlation time τ (eddy-turnover time)

)'()'()(

)(

tttt

tf

dt

df

Time evolution of velocity and magnetic field for the mode n = 7, with constant forcing terms.




Properties: spectrum and flux

11121

1121

42

)()2(

2

)2()Im(

nnnnm

nnnm

nnnm

nnnm

kZZZZZZ

ZZZZZZ

Kolmogorov fixed point of the system.Inertial and dissipative ranges + intermediate range visible in shell models

Numerical simulations with: N = 26 shells; viscosity = 0.5 ∙ 10-9

Flux: an exact relationship which takes the role of the Kolmogorov’s “4/5”-law




Properties: spectrum and flux

11121

1121

42

)()2(

2

)2()Im(

nnnnm

nnnm

nnnm

nnnm

kZZZZZZ

ZZZZZZ

Kolmogorov fixed point of the systemInertial and dissipative ranges + intermediate range visible in shell models

Numerical simulations with: N = 26 shells; viscosity = 0.5 ∙ 10-9

Flux: an exact relationship which takes the role of the Kolmogorov’s “4/5”-law




Evolution of magnetic field spectrum

10-5

10-4

10-3

10-2

10-1

100

101

102

103

104

105

106

107

0.9AU

0.7AU

0.3AU

trace of magnetic field spectral matrix

-1.72

-1.70

-1.67

-1.07

-1.06

-0.89

pow

er d

ensi

ty

frequency

the spectral break moves to lower frequency withincreasing distance from the sun

This was interpreted as an evidence that non-linear interactions are at work producing a turbulent cascade process

1/f1/f5/3




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




Properties: Time intermittency Velocity field Magnetic field

n = 1

n = 9




Fluctuations in plasmas

Increasing scales

Velocity increments at 3 different scalesin the solar wind: Δur = u(t + r) – u(t)

Small scale: STRUCTURES

Large scale: random signal




Phenomenology: fluid-like

Let us consider the dissipation rate for both pseudo-energies (stochastic quantities equality in law!)

The energy transfer rate is scaling invariant only when

h = 1/3

NL

rz

2

The characteristic time (eddy-turnover time) is the time of life of turbulent eddies

3/1rur

r

NL z

r

r

zz rr

2

13' h

Kolmogorov scaling 3/3/ qq

qq

r rCu q-th ordermoments

r 1/k 3/53/23/2

2

2 )( kkErCur

rrr uzz




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

The energy transfer rate is scaling invariant only when

h = 1/4

r

r

T

z2

4/1rur

A

A c

r

r

zz rr

22

14' h

Kraichnan scaling 4/4/' qq

Aqq

r rcCu q-th ordermoments

r 1/k 2/32/12/1'

2

2 )( kkErcCu Ar

A

NLNLrT

Since the Alfvèn time in some case is LESSER than the eddy-turnover time, the cascade is effectively realized in a time T:

rrr uzz




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

dxxpxxdPxx nnn )()(

Through the inverse transform the pdf can be written in terms of moments, and moments can be obtained through the knowledge of pdf n

n

n

ikx xn

ikdkexp

0 !2

1)(

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!)

0

)(

k

n

nnn

dk

kdix

ikxikx edxxpek

)()(Characteristic function




Anomalous scaling laws

A departure from the Kolmogorov law must be attributed to time intermittency in the shell model.

q

n

q

n ku The “structure functions” in the model

Scaling exponents obtained in the range where the flux scales as kn

-1

Fields play the same role the same “amount” of intermittency

The departure from the Kolmogorov law measures the “amount” of intermittency

Δur un

kn ~ 1/rζq = q/3 Kolmogorov scaling




Inertial range in real experiments?

0 1 2 3 4 5 6

1.5

2.0

2.5

3.0

3.5

4.0

4.5n = 3

Slow wind Fast wind

log

Sn(r

)

log r0 1 2 3 4 5 6

4

6

8

Slow wind Fast wind

n = 5

nrturturS n

n

)()()(

A linear range is visible only in the slow solar wind

Magnetic field in the solar wind. Helios data.




Extended self-similarity

mnn

mnnrSrS

/

)()(

The m-th order structure function (m = 3 or m = 4) plays the role of a generalized scale

3 4 5 6

1.5

2.0

2.5

3.0

3.5

4.0

4.5n = 3

Slow wind Fast wind

log

Sn(r

)

log S4(r)2 4 6

4

6

8

Slow wind Fast wind

n = 5

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

Just a way to get scaling exponents

For fluid flows, scaling exponents obtained through ESS coincides with scaling exponents measured in the inertial range.




Departure from the Kolmogorov’s laws

0 1 2 3 4 5 6 7 8 90.0

0.5

1.0

1.5

2.0

2.5

Wind-Tunnel data

Sca

ling

exp

on

en

ts

p

Velocity Temperature (passive)

1 2 3 4 5 60.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

n/

3

n

magnetic slow magnetic fast velocity Kolmogorov law: n/3

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




Magnetic turbulence in laboratory plasma

The departure from the linear scale increases going towards

the wall

Turbulence more intermittent

near the external wall

r/a normalized distance

Similar to edge turbulence in fluid flows




Numerical simulations

Intermittency is different for different fields.

In particular magnetic field more intermittent

than velocity field

Incompressible MHD equations in 2D configurations

0 2 4 6 8

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

scal

ing

expo

nent

s

n

z+

z-

0 2 4 6 8

n

velocity magnetic

High resolutions 10242 points. Averages in both space and time.

n

n r

rxurxurS

)()()(

No Taylor hypothesis when we are dealing with simulations




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 !




Probability distribution functions

0 1 2 30

10

20

30

40

Gaussian

Slow wind Fast wind

Kurt

osis

log r

Fluctuations are stochastic variables, so the structure functions are defined in terms of pdfs:

rr

n

rn udupurS )()(

3)(

)(2

2

4 rS

rSkurtosis

For a gaussian pdf

Anomalous scaling exponents implies that pdfs have also anomalous scalings

The kurtosis increasesas the scale becomes smaller

Fluctuations at small scales increasingly depart from a GAUSSIAN




i.e. the pdfs of normalized fields increments at different scales collapse on the same shape

(self-similarity)

About self-similarity

And let us consider the normalized variables 2/12

r

rr

u

uw

Let the scaling law holds for differences

hr rxurxuu )()(

Then by changing the scale r r, it can be shown that, if h = cost. pdfs at two scales are related

)()( rr wpdfwpdf




Experimental evidences in atmospheric fluid flows

No global self-similarity!

• PDFs are not Gaussians

• PDFs changes with scale

Large scales

Inertial range

Small scales

Departure from self-similarity




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 Idescribe this model.




A multifractal model for pdfs

each describing the statistics in different regions of volume S(h) each of different variance (h)

each weighted by the occurrence of S(h)

dGLrP ),()()( the sum of gaussians of different width

(blue) gives the resulting “stretched” PDF (red)

This is achieved introducing the distribution L() and computing the convolution with a

Gaussian G

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




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)

At each scale they collapse to a GAUSSIAN with different values of .

–0.1 < + < 0.1, = 0.4 0.9 < + < 1.0, = 0.9




A model for the weight of each gaussian

2

2

2

lnexp

21)( 0

L

The parameter ² can be used to characterize the scaling of the shape of the PDFs, that is the intermittency of the field!

• As ² increases, L() is wider then more and more Gaussians of different width are summed and the tails of P(u) become higher

Width (variance) of

the Log-normal distribution

• When ²= 0, L() is a -function centered in 0 so that: Gaussian P(u)




Scaling of ² and relevant parameters

The parameter ² is found to behave as a power-law of the

scale

rr)(

2d MHD

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)




Scaling of ² for solar wind turbulence

solid symbols: fast streams

Magnetic field

Velocity field

open symbols: slow streams

magnetic field is more intermittent than velocity




Scaling of ² for numerical simulations

²max (v) = 0.8²max (b) = 1.1

(b) = 0.8(v) = 0.5




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 energyof the flow and play an important dynamical role.

Examples from Jupiter’s atmosphere

Need for space AND scale analysis




Orthogonal Wavelets decomposition

Let us consider a signal f(x) made by N = 2m samples (being x = 1), and build up a set of functions starting from a “mother” wavelet

)(x Then we generates from this a set of analysing wavelets by DILATIONS and TRANSLATIONS

j

jj

ij

ixx

2

22)( 2/

dxxxfw

xwxf

ijij

j iijij

)()(

)()(

ijij

ij

wdxxf

xfrxfw

22)(

)()(

Scale Position




Local Intermittency Measure

The energy content, at each scale, is not uniformly distributed in space

Liij

ij

w

wmil

2

2

... 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

Gaussian background Structures

Complete signal

l.i.m. smaller than threshold

l.i.m. larger than threshold




In the solar wind

400

500

600

700

52:1252:0051:1251:0050:1250:0049:12

beginning of intermittent event

residuals

Original LIMed

Sol

ar

Win

d S

peed

[km

/se

c]

DoY 1976

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




Waiting times between “structures”

Interesting! the underlying cascade process is NOT POISSONIAN, that is the intermittent (more energetic) bursts are NOT INDEPENDENT (memory)

Solar wind

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.




Power law distribution for waiting times

Turbulent flows share this characteristic. Power law is generated through the chaotic dynamics and must be reproduced by models for turbulence.

Fluid flow Laboratory plasma




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




What kind of intermittent structures ?

Solar wind: tangential discontinuity (current sheet)

Minimum variance analysis around isolated structure allows to identify them

Solar Wind: shock(compressive structures)




Magnetic structures in laboratory plasmas

RFX edge magnetic turbulence: current sheets

Current sheets are naturally produced as coherent, intermittent structures by nonlinear interactions




Intermittent structures in laboratory plasmas

RFX edge turbulence of electrical potential

Structures are potential holes




Dynamics of intermittent structures

Relationship between intermittent structures of edge turbulence and disruptions of the plasma columns at

the center of RFX Time evolution offloating potential

Minima are related to disruptions

Appearence of intermittent structures in the electrostatic turbulence at the edge of the plasmacolumns (vertical lines)

We don’t have explanation for this!




Statistical flares

Dissipation of (turbulent?) magnetic energy

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




Solar flares are impulsive events

Time series of flare eventsHard 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 Nano

flares




Power law statistics of bursts

Total energy, peak energyand (more or less!) lifetime of individual bursts seems to be distributed according to power laws.




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.




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.


Sandpile model

Size, lifetimes and number of sand grains in each avalanche are power law distributed.

Lack of any typical length

Avalanches of all size i.e. FRACTAL PROCESS



• 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.




Sand Pile Model for Solar Flares

Power peak, total energy and duration are power law distributed.

Cellular Automata model for reconnection : Vector field Bi on a 3D lattice

Local slope dBi = Bi -j wj Bi+jWhen | dBi| > some treshold: instability at position i : Field readjusted in the nearby positions so that the grid point i becomes stableThe readjustment can destabilize nearby points producing an avalanche (flare)

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




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 - 2.38 0.03




The origin of power law distribution for waiting times

)()( aTtaTt

aTTh

This suggests to try to fit the WTD with a Lèvy distribution whose characteristic function is ||exp)( zazL

The parameter 0 < 2, for = 2 one recovers the definition of a Gaussian.

A rescaling gives the same statistical properties

The waiting time sequence forms a “temporal point process”, statistically self-similar




The Lèvy function

0

1 )()cos()( zLtzdztP

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.




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




Dissipative bursts in shell model

The energy dissipation rate is intermittent in time.Energy is dissipated through impulsive isolated events (bursts).

n n

nnnn bkukt2222)(




Multifractal structure of dissipation

Coarse-grained dissipation has been generated from simulations

2/

2/

')'()(

dtttt

Moments of dissipation have a scaling law

ppt




Singularity spectrum

pfppt d

)()(

)](min[ fpp

Spectrum of singularities described by the function f(), which represents the fractal dimension of the space where dissipation related to a singularity .

As p is varied we select different singularities from an entire (continuous) spectrum

pp ppf

dp

dp

)()(;)(

From saddle-point we get

Inverse transform




Inside bursts

Through a threshold process we can identify and isolate each dissipative bursts to make statistics




Some different statistics

Let us define some statistics on impulsive events

1) Total energy of bursts

2) Time duration3) Energy of peak

In all cases we found power laws, the scaling exponentsdepend on threshold.




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




Statistics of dissipative events

/

)(

)()()(2/12

t

ttt

• Pdfs of normalized fluctuations of energy released in the MHD shell model, are the same as normalized fluctuations of solar flares energy flux.

flares shell model




Could SOC describes turbulent cascade?*

Kadanoff sand pile Dissipative Kadanoff sand pile

1. PDFs are non gaussian and collapse to a single PDF (fractal)2. Esponential distribution for waiting times: avalanches are INDEPENDENT events

* Apart for the 4/5-law

Simulations of Kadanoff SOC model: Rescaled energy fluctuations at different scales and waiting times at the smallest scale




• 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.

The Running sandpile




Running sandpileSimulations 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 timesHigh Jin gaussian pdfs and power law for waiting times




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




Anomalous transport in laboratory plasmas

Diffusion causes loss of particles, energy, …

In general turbulent fluctuations of electric field enhance loss, the transport is called “anomaloues” since it is due to turbulence.

Perhaps the main cause of disruption of magnetic confinement needed to achieve nuclear fusion.

Anomalous transport A problem with language:

Plasma physics: Transport driven by turbulent fluctuationsPhysics of fluids: Transport with non-Gaussian features




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




“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.




• 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.

Waiting times between transport events




A modifications of sandpile model

Sanchez, Newman, Carreras, PRL 88, 068302 (2002)

Note 2: Correlated input (necessarily!) correlated output (SOC is a linear model)

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)

Introduces correlated input to reproduce power laws in waiting times.Quite trivial!




A “different” sandpile model

Gruzinov, Diamond, Rosenbluth, PRL 89, 255001 (2002)

Modification of output

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




Charged particles diffusion

t

iiii

ii

xutxudttxtx

udt

dx

0

2))0(())'(('2)0()(

ns)correlatio Lagrangian strong(very 1 with ))0(())(( 2)

variance)infinite cunrealistiy (physicall flightsLèvy 1) 2

xuxu

u

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

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!




3D velocity field from shell model

nnn ekk

N

n

xkin

jnj ccetuCtxu n

1

)( ..)(),(

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.)




A different approach

)sin()cos(),( )()()()( txkebtxkeatxu nb

nb

nnn

na

na

nn

)(2

1122n

nnnn kE

kkba

)(2

1 3nnn kEk

A simple model for turbulence with coherent structures at all dynamical scales:

Perhaps there is no need to run a shell model + the equations of motion for a test particle.

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, …




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!!).




Confining turbulence ?

20

0 BB

dt

rd

In astrophysics, turbulent fluctuations are usefulsince they CONFINE cosmic rays within the galaxy

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

2B

BE

dt

rd

E X B drift




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




Random phases

Correlated phases (weak superdiffusion)

Diffusive properties

jumps)"long" make can (particles sionsuperdiffu 1/2 2)

onsubdiffusi 1/2 1)

)t limit the (in

22)0()( tDxtx e

De ~ 1 0.1

De ~ 10-3

~ 0.68

= 0.5




Reduction of particle flux

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.

No barrier




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




Simmetrization of particle flux-2

-5

-4

-3

-2

-1

0

1

2

3

4

5

# c

ross

ings

for

par

ticl

e

-250

-200

-150

-100

-50

0

50

100

150

200

250

# c

ross

ings

per

par

ticl

e

# 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.




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

Time [µs] poloidal mode number

Polo

idal A

ng

le (

°)

before

during

Pascal Devynck et al., 2003

Perhaps crazy people taking more seriously than ourself our “continuous playing” in the realm of tokamak plasma physicists




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.

Galerkin approximation

integers of pair yx

yx

kk

kkL

k

,

,2

kqp

z

kqp

z

tpvtqbtqvtpbkpq

keqptkb

t

tqbtpbtqvtpvkpq

qpeqptkv

t

,,,,2

,

,,,,2

)(,

2

22




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

k

kc

k

k

tkbtA

tkvtkbtkbtkvtH

tkbtkvtE

2

2

22

,

2

1

,,,,2

1

,,2

1

Rugged invariants of motion: they remain invariant in time for each triad of interacting wave vectors which satisfy the condition k = p + q

Simplified N-modes models

yyxx NNNNNNk

L

,;,

2




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.



Università della CalabriaUniversità della Calabria Models vs. Simulations

Main advantages : rugged invariants are conserved in absence of dissipation, true dissipationless runs.

Example:N = 25

Main disadvantages : higher computational times (N2 vs. N log N)




Note: the occurrence of an inverse cascade of magnetic helicity in shell models is yet controversial

2D example: inverse cascade

t = 0

Coarse-grained energy averaged over circular shellsof amplitudes m = (kx

2 + ky2)1/2

Equipartition betweenkinetic and magnetic energy at small scales and dominance of magnetic energy at largest scale



Università della CalabriaUniversità della Calabria2D 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.

k

k

tkbk

tkvkN

22

22

),(

),()(

In the limit μ 0 and N , we found Δ 1.

Equipartition between kinetic and magnetic energy on small scales in the inviscid case.



Università della CalabriaUniversità 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; …)

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..N = 5 N = 7

Some triads which satisfy ki= kj + km

k1 = (1,1)k2 = (2,-1)k3 = (3,0)

N = 3

No chaos here



Università della CalabriaUniversità 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)

312123

231312

132321

312123

231312

132321

9)(9

5)(5

2)(2

9)(3

5)(7

2)(4

BBVVBB

BBVVBB

BVBBVB

BBBVVV

VBBVVV

VBBVVV

Vi(t) = Re[v(ki,t)] Bi(t) = Re[b(ki,t)]

Only real fields

312123

23*13

*12

13*23

*21

312123

23*13

*12

13*23

*21

9)(9

5)(5

2)(2

9)(3

5)(7

2)(4

bbvvbb

bbvvbb

bvbbvb

vbbvvv

vbbvvv

vbbvvv

How a simple model can be interesting without chaos?

No chaos here!



Università della CalabriaUniversità della Calabria Free decay: asymptotic states

E : energyHc : cross-helicityA : magnetic helicity

Starting from any initial condition, the system evolves towards a curvein the parameter space (A/E, 2Hc/E)

Analysis of a wide serie of different numerical simulations on free decay 2D MHD reported by Ting, Mattheus and Montgomery (1986).

1212

22

E

A

E

H c

0.0 0.2 0.4 0.6 0.8 1.0

-1.0

-0.5

0.0

0.5

1.0T = 0

2Hc/

E

A/E0.0 0.2 0.4 0.6 0.8 1.0

-1.0

-0.5

0.0

0.5

1.0

T = 20

2Hc/

E

A/E

0.0 0.2 0.4 0.6 0.8 1.0

-1.0

-0.5

0.0

0.5

1.0T = 40

2Hc/

E

A/E

0.0 0.2 0.4 0.6 0.8 1.0

-1.0

-0.5

0.0

0.5

1.0T = 80

2Hc/

E

A/E

The 3-modes real model seems to reproduce these results.

μ = 0.01

0 AE

0; ubb



Università della CalabriaUniversità della CalabriaSelective decay and dynamical

alignment

Selective decay (SD) due to inverse cascade(large-scale magnetic field)

0 cHE

Variational principle

Extreme points of the curve represents decay of rugged invariants with respect to total energy.

Dynamical alignment (DA) due to approximately equal Decay of energies of alfvènic fluctuations(alignment between velocity and magnetic field)

0.0 0.2 0.4 0.6 0.8 1.0

-1.0

-0.5

0.0

0.5

1.0 2Hc/E

A/E

DA

DA

SDThe curve “… does not represent the locus of the extrema of anything over its entire range of variation”. (Ting et al., 1986)

bu

Laboratoryexperiments

Astrophysics



Università della CalabriaUniversità 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 atime-translation operator

)()(

),(),,()(

ttT

Stkbtkvt

ItT

ISI

t

)()0(

)0(;

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 MHDΦ(0)={v(k,0),b(k,0)} such that b(k,0) = 0.From MHD equations b(k,t) = 0 for each time.

Subspaces in the 3-modes model

312123

231312

132321

312123

231312

132321

9)(9

5)(5

2)(2

9)(3

5)(7

2)(4

BBVVBB

BBVVBB

BVBBVB

BBBVVV

VBBVVV

VBBVVV

ii BV

0;0 ii BV




0;;

0;;

kji

kji

VVB

BBV

Fluid

Alfvènic (fixed point)

Cross-helicity = 0

(B1,V2,B3)

(B1,B2,V3)

(V1,B2,B3)

SubspacesSubspaces due to symmetries can be generalized to the true MHD equation to any N-order truncation

Example (B1,V2,B3)

312123

231312

132321

312123

231312

132321

9)(9

5)(5

2)(2

9)(3

5)(7

2)(4

BBVVBB

BBVVBB

BVBBVB

BBBVVV

VBBVVV

VBBVVV




Cross-helicity = 0

3123

1321

2312

99

22

57

BBVB

BBVB

VBBV

Example of invariant subspace

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

)2/(7|)2(18)2()( 2/12/12 AEAtAEdnAEtV

Stable and unstable subspaces

iiext

iiin

ji

E

E

CI; ΓΦ

CIS

2

2 321

321

;;

;;

VBV

BVB




Stability of subspaces are investigated according to time evolution of distance from a given subspace.

Let Φ(0) and Γ(0) such that Eext « Ein at t = 0.

23

21

22

23

21

22

VVBE

BBVE

ext

in

Let us investigate the time evolution of both Ein and Eext

Example:

D= √Eext is the distance of the point from a given subspace.

Subspace (V1,V2,V3)

0 20 40 60 80 100

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Eext

(Magnetic energy)

Ein (Kinetic energy)

Ener

gies

Time




0 20 40 60 80 10010-4

10-3

10-2

10-1

100

Eext

/Ein

Eext

Ein

Time

dissipation = 0.0 dissipation = 0.01

Stable (no dynamo effect)

Selective dissipationAttractor

Subspace (B1,B2,V3)

0 20 40 60 80 100

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Eext

Ein

Ener

gies

Time




0 20 40 60 80 10010-4

10-3

10-2

10-1

100

Eext

/Ein

Eext

Ein

Time


Stable (Magnetic field on the largest scales)

Selective dissipationAttractor

Subspace (B1,V2,B3)

0 20 40 60 80 100

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Eext

Ein

Ener

gies

Time




0 20 40 60 80 10010-3

10-2

10-1

100

Eext

Eint

Eext

/Eint

Time


Unstable (inverse cascade at work from k3)

The subspace repels all nearest trajectories.

Subspace (V1,B2,B3)

0 20 40 60 80 100

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Eext

Ein

Ener

gies

Time




0 20 40 60 80 10010-3

10-2

10-1

100

Eext

/Ein

Eext

Ein

Time


Unstable (inverse cascade at work from k2 and k3)

The subspace repels all trajectories

Attractors and “repellers”




0.0 0.2 0.4 0.6 0.8 1.0

-1.0

-0.5

0.0

0.5

1.0 2Hc/E

A/E

Vi = Bi

Vi = - Bi

(B1,B2,V3)(V1,V2,V3) (V1,B2,B3)

(B1,V2,B3)

11

22

/1

2;

1

1

BVxx

x

E

H

xE

A c

Only one wave vector survive

Attractors drive the system towards

“Repellers” drive the system towards the whole




Do you remember?

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).

The structure of stable and unstable time-invariant subspaces of real MHD are reproduced in the GOY Shell model

Magnetic energy 3D

Magnetic energy 2D



Università della CalabriaUniversità della Calabria Models for low-β plasmas

La

P

P

B

kin

/

zB

yx

zL

a

a/La/L << 1 << 1

ββ << 1 << 1

When

Laboratory plasmasCoronal loops

11

1

110

0

2

RB

Ba

LR




Reduced MHD equations

Incompressible 2D MHD in perpendicular variablesAlfven wave propagation along background magnetic field

Bz

vBvBBv

t

vz

BBBB

Bpvv

t

z

z

2

22

2

yxyx BBtzyxBvvtzyxvyx

,),,,(,,),,,(,,

Total energy and cross-helicity survive. Only two time invariants in ideal RMHD




Simplified models

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.

The cylinder has been divided in Nsez “planes” at fixed zn.




The Galerkin truncature model

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 ?

Both magnetic and kinetic energies accumulates at m = 1. for all z. Equipartition between energies.

Inverse cascade without conservation of A ?




Self-organization in RMHD

T

FFT

Tnmfzmf ,,

Magnetic energy on the wave vectors plane (m,n)

R = 14 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).




Quasi-single helicity states in RFX

Quasi-single helicity states observed in laboratory plasmas in some situation (example RFX).

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).

Spectrum for m = 1 Time evolution of some modes




A Hybrid Shell ModelRMHD equations in the wave vector space perpendicular to

B0 :

),,(),,(),,((),,( 2 txktxtxMtxx

ct imlilmiA kzpkzpz k)kz

p

A shell model in the wave vector space perpendicular to B0

can be derived:

(Hybrid : the space dependence along B0 is kept)




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 motionsv ~ 5 10-4 cA ~ 1 Km/s

Model parameters: L ~ 3 104 Km, R ~ 6, cA ~ 2 103 Km/s




Energy balance

After a transient a statistical equilibrium is reached between incoming flux, outcoming flux and dissipation

Stored Energy

Energy flux

Dissipated Power

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




Energy spectra

A Kolmogorov spectrum is formed mainly on magnetic energy

Magnetic energy dominates with respect to kinetic energy

Magnetic Energy

Kinetic Energy




Statistical analysis of dissipated power

Power laws are recovered on Power peak, burst duration, burst energy and waiting time distributions

The obtained energy range correspond to nanoflare energy range

Power Peak Burst duration

Burst Energy Waiting time




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.




Proper Orthogonal Decomposition

• The field is decomposed as:

• The functions which describe the base are NOT GIVEN A PRIORI (empirical eigenfunctions).

r

0

)()(),(j

jj tatu rr

• 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

2

2),(max

uX

rdgfgfj

ii

3

1

*),(

An inner product is defined




Empirical eigenfunctions

whose kernel is the averaged autocorrelation function.

'),'(),( rrr dtutu

j

jk

m

E 2

Very huge computational efforts !

The maximum is reached through a variational method thus obtaining the integral equation

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

N

jjjN tatu

0

)()(),( rr

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

mn,

)()()(

,, tataMdt

tdamnmnj

jThe coupling coefficients depend on the empirical eigenfunctions




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




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.




Reconstruction of velocity field

The velocity field has been reconstructed using only

J = 0, 1 J = 2, 3




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

Time

Angle




3 POD modesOriginal

Reconstruction with 3 POD modes

Reconstruction with 2 POD modes

Reconstruction with 1 POD mode

periodicities

+ migration

+ stochasticity




Conclusions

Became a “Plasma Physicist”

Deadline for applications:September 28, 2003

Acknowledge Loukas Vlahos and the local organizing committee

Let sand piles evolve …Vincenzo Carbone Vincenzo Carbone






Don’t care about…

AvalanchesAvalanches oror

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria Rende (CS) – Italy...

Documents

Transcript of Vincenzo Carbone Dipartimento di Fisica, Università della Calabria Rende (CS) – Italy...