EURATOM – MEdC Association Days

Magurele, 26-27 November, 2009

Trapping, transport

&

turbulence evolution

Madalina Vlad

Plasma Theory Group

National Institute of Laser, Plasmas and Radiation Physics,

Typical trajectory in the plane perpendicular to the magnetic field generated by the ExB stochastic drift in turbulent plasmas

Random sequence of trapping events and long jumps(completely different of the trajectories in Gaussian stochastic processes)

We have developed in the last decade semi-analytical methods, which are able to determine the statistics of such trajectories

Trapping completely changes trajectory statistics:

• memory effects (long tailes of the correlation of the Lagrangian velocity)

• anomalous transport regimes (diffusion coefficients with different dependence on the parameters, even opposite)

• non-Gaussian distribution of displacements

• high degree of coherence

• trajectory structures

What is the influence of trajectory trapping on the evolution of turbulence ??

Content

1) Test particle statistics in stochastic potential A short review of 10 year work

2) Test modes in turbulent plasmaA more detailed presentation of some results that could trap us for the next 10 years

1) Test particle statistics in stochastic potential We consider a turbulent plasma with given (measured) statistical characteristics of the stochastic potential and study test particle motion

Non-linear stochastic equation:

where is a stationary and homogeneous stochastic velocity field with Gaussian

distribution and known spectrum or Eulerian correlation (EC):

,0)0(),),(()(

xttxvdt

txd

),( txv

,,),(),(),( 21111

cc

txfttxxtxtxE

where is the amplitude , the correlation length, the correlation time.cc

c

cfl

fl

c

c

c VV

VK

,,The Kubo number :

(describes the decorrelation due to time variation of the stochastic field).

B

etxv z

),(

To determine: The statistical properties of the trajectories: the probability of displacements (pdf) P(x,t)

c

cxD

dt

txdtD

dxxtxPtx

)(

,)(

2

1)(

,),()(

22

22

)),(()0,0()( ttxvvtL jiij

,')'()(0t

xx dttLtD ,')'('2)(0

2 t

xx dttLtttx

Lagrangian velocity correlation (LVC) :

Note: -Test particle diffusion coefficients are the same with those obtained from the correlation of velocity and density fluctuation if there is space-time scale separation between fluctuations and the main gradients (due to the zero divergence of the ExB drift);- The characteristics of the turbulence and all the other components of the motion are input data

(A) The invariance of the potential for the static case

(Hamiltonian equation of motion)

• static case ( ): invariance of the potential and permanent trapping on the contour lines;

• slowly varying potential (K > 1): approx. invariance of the potential and temporary trapping

• potential with fast time variation (K < 1): no trapping.

t

ttx

t

ttx

x

ttxttxv

dt

ttxd

ii

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

)),((

K

(B) The statistical invariance of the Lagrangian velocity due to

for stationary and homogeneous turbulence.

0),( txv

),()0),0(()),(( txvPxvPttxvP

There are two strong constraints for the statistical methods:

Semi-analytical statistical methods

The existing analytical methods are not compatible with the conditions (A) and (B). They do not describe trajectory trapping

The decorrelation trajectory method (DTM) 1998

(M.Vlad, F. Spineanu, J. H. Misguich, R. Balescu, “Diffusion with intrinsic trapping in 2-d incompressible stochastic velocity fields”, Physical Review E 58 (1998) 7359)

- DTM is based on a set of simple (deterministic) trajectories determined from the Eulerian correlation EC of the stochastic potential. - main consequences of condition (A) are fulfilled, condition (B) is not; - Main physical result: trapping produces memory effects (long-time correlation of the Lagrangian velocity), subdiffusive transport for static potential, and anomalous diffusion regimes.

The nested subensemble method (NSM) 2004

(M. Vlad, F. Spineanu, “Trajectory structures and transport”, Physical Review E 70 (2004) 056304(14))

- NSM is a systematic expansion based on dividing the space of realizations of the stochastic potential in nested subensembles. NSM yields detailed statistical information.- all consequences of (A) are fulfilled, condition (B) is improved (but not enough in order 2)

- Main physical result: trapping determines coherence in the stochastic motion and quasi-coherent trajectory structures.

The velocity on structure method (VS) 2009

- Both conditions (A) and (B) are respected; - Trajectory structures and the memory effects are confirmed and better described - Main physical result: trapping in a moving potential generates flows.

The probability of displacements for static potential

Time invariant part formed by trajectory structures of different scales (vortical motion)

Moving maximumof trajectories that are not in a structure at that time (radial motion).Decreasing number of free particles.

Note: this gives the probability of one-step jumps (fractal diffusion)

dxxtxPxPtxtFDt

txtD ftrB

22

2

)],()([)(),()(

)(

The diffusion coefficient for static potential:

constant at large t

22)( tVtn f

Decay of D(t) due to trapping

Correlation of the Lagrangian velocity with long negative tail that exactly compensates the positive part at small time delay

Subdiffusive transport (and memory !)

Effect of a perturbation = decorrelation mechanism characterized by a time d2V

t

fl

* The LVC is not influenced; Transport coefficient independent onand stable to such perturbations.

dfld

* The negative part of the LVC is cut out;

Anomalous diffusion regime (increased diffusion at stronger decorrelation)

Trajectory trapping No trajectory trapping

• Weak decorrelation mechanism with large decorrelation time

dwhenD

fl

• Strong decorrelation mechanism with small decorrelation timefld

dd VDwhenD 2,

The effect of time variation of the potential (finite K):

Decay of D due to trapping.

The effect of parallel motion: the correlation of the potential decays because particles move out of the correlated zone along the magnetic field.

Effective Kubo number:II

IIfl

fl

IIII

II

IIeff v

KKK

KKK

,,

11 effII KK

The effect of collisions: diffusion of the potential correlation. E(x) is not destroyed but spreaded due to collisional motion.

Effective Kubo number:

02/3 ,

21

D

K

KKeff

No trapping for electrons

BDKFDKD BB

),()(

Effect of an average velocity: V

VV d

d

1dVTrapping exists for and determines “the acceleration” of the free particles (on the opened contour lines), which reach an average velocity

• nonlinear process related to the statistical invariance of the Lagrangian velocity;• not a dynamical effect but determined by the selection of the free particles.

df VV

Trapping effects for fast particles;

Pinch induced by the gradient of the magnetic field;

.....

Always rather surprising nonlinear effects were found in the presence of trapping

The general conclusion of all this studies of test particles stochastic fields:

Trapping determines: memory effects,

anomalous transport,non-Gaussian distribution,

high degree of coherence (trajectory structures).

Trapping exists when K > 1 and if the other components of the motion are weak perturbations (collisions, average flow, …)

...,1/,1/0 VVVD dc

•M. Vlad, F. Spineanu, J.H. Misguich, R. Balescu, “Diffusion with intrinsic trapping in 2-d incompressible velocity fields”, •Physical Review E 58 (1998) 7359-7368;•M. Vlad, F. Spineanu, J.H. Misguich, R. Balescu, “Collisional effects on diffusion scaling laws in electrostatic turbulence”, Physical Review E 61 (2000) 3023-3032•M. Vlad, F. Spineanu, J.H. Misguich, R. Balescu, “Diffusion in biased turbulence”, Physical Review E 63 (2001) 066304•M. Vlad, F. Spineanu, J.H. Misguich, R. Balescu, “Electrostatic turbulence with finite parallel correlation length and radial diffusion”, Nuclear Fusion 42 (2002), 157-164.•M. Vlad, F. Spineanu, J.H. Misguich, R. Balescu, “Reply to ‘Comment on Diffusion in biased turbulence’”, Physical Review E 66 (2002) 038302.•M. Vlad, F. Spineanu, J.H. Misguich, R. Balescu, “Magnetic line trapping and effective transport in stochastic magnetic fields”, Physical Review E 67 (2003) 026406, 1-12.•M. Vlad, F. Spineanu, “Trajectory structures and anomalous transport”, Physica Scripta T107 (2004) 204-208.•M. Vlad, F. Spineanu, “Trajectory structures and transport”, Physical Review E 70 (2004) 056304(14).•M.Vlad, F. Spineanu, J. H. Misguich, J.-D. Reusse, R. Balescu, K. Itoh, S. –I. Itoh, “Lagrangian versus Eulerian correlations and transport scaling”, Plasma Physics and Controlled Fusion 46 (2004) 1051-1063.•M. Vlad, F. Spineanu, “Larmor radius effects on impurity transport in turbulent plasmas”, Plasma Physics and Controlled Fusion 47 (2005) 281-294.•M. Vlad, F. Spineanu, S.-I. Itoh, M. Yagi, K. Itoh , “Turbulent transport of the ions with large Larmor radii”, Plasma Physics and Controlled Fusion 47 (2005) 1015-1029.•M. Vlad, F. Spineanu, S. Benkadda, “Impurity pinch from a ratchet process”, Physical Reviews Letters 96 (2006) 085001.•M. Vlad, F. Spineanu, S. Benkadda, “Collision and plasma rotation effects on ratchet pinch”, Physics of Plasmas 15 (2008) 032306 (9pp).•M. Vlad, F. Spineanu, S. Benkadda, “Turbulent pinch in non-homogeneous confining magnetic field”, Plasma Physics Controlled Fusion 50 (2008) 065007 (12pp).•M. Vlad, F. Spineanu, “Vortical structures of trajectories and transport of tracers advected by turbulent fluids”, Geophysical and Astrophysical Fluids Dynamics 103 (April 2009) 143-161.

Balescu R., Aspects of Anomalous Transport in Plasmas, Institute of Physics Publishing (IoP), Bristol and Philadelphia, 2005.

Many other applications :

Petrisor I., Negrea M., Weyssow B., Physica Scripta 75 (2007) 1.Negrea M., Petrisor I., Balescu R., Physical Review E 70 (2004) 046409.

Balescu R., Physical Review E 68 (2003) 046409.Balescu R., Petrisor I., Negrea M., Plasma Phys. Control. Fusion 47 (2005) 2145.

still much many to be done (fluids, astrophysics,…)

Other people begin to work with DCT method

M. Neuer, K. Spatschek, Phys. Rev. E 74 (2006) 036401 T. Hauff, F. Jenko, Physics of Plasmas 13 (2006) 102309

• We consider a turbulent plasma with given (measured) statistical characteristics of the stochastic potential and study linear test modes

• AIM: to find the effect of trajectory trapping on test modes

The growth rate and frequency of the test modes are determined as function of the statistical characteristics of the background turbulence

2) Test modes in turbulent plasma

The drift (universal) instability in cartezian slab geometry, with constant magnetic field and density gradient. Turbulence is developed, with inverse cascade and vortical structures.

x

• Drift instability in cartezian slab geometry, with constant magnetic field and density gradient in collisionless plasmas.

-The limit of zero Larmor radius of the ions: - stable drift waves - potentials that move with the diamagnetic velocity

z

x

yB

dn0/dx

- The instability is produced by the combined effect of the non-adiabatic response of the electrons and of the finite Larmor radius (FLR) of the ions

0

0**

0

***2

*

2

2

)(

VV

VVVk

vk

Vk

eff

effeffy

Tez

effy

,2

22

00

Lik

dx

dn

Ben

TV 0

0*

0,* Vk y

),(),,(),,( 0*0 zxztVxtzx is the initial condition ]

1,2/4

1

2 **0

2

max

kVVvk

k eff

Tez

y

The formal solution of the Vlasov equation for an initial perturbation with

obtained with the method of the characteristics (integration along trajectories):

),(0 zx

t

ze

yzMze tvzxVdvFdv

T

e

T

tzxexntzxn

0

*0 ),(),()(),,(

1)(),,(

,),(,),(

,),()()(

),,(1)(),,(

0

*2

0

0

zxzxB

zxVdvFvdT

exn

T

tzxexntzxn

tijij

t

yM

i

),,(),,( tzxntzxn ie ),,( tzx

The average propagator and the average effect of the diamagnetic velocity fluctuations are determines using trajectory statistics

zzxtzxzxtx

vd

dz

B

etVx

d

xdz

zd

,;,,;

,)(

0

The background turbulence produces the stochastic ExB drift in the trajectories (characteristics)

))(exp()(exp tixxkid

t

Average over the trajectories

- the characteristic times ordering is used for simplifications: the parallel motion of the ions and the ExB drift for electrons are neglected

The existence of trapping is determined by the amplitude of the ExB drift: it appears when V is larger than the velocity of the potential

VVV dd /The trapping parameter for drift type turbulence is

iIIcfl

eII *

iIIcfl

eII *

QL (no trapping):

NL (trapping) :

• nonlinear effects appear in ion trajectories if • electron trajectories have quasiliniar statistics

VVd

,/4/ *2* VkVdc

dV

The ordering of the characteristic times is

Thus

Vkvk flezeII 2/2,/2

Effects of the background turbulence through the propagator

Turbulent plasma with small amplitude (quasilinear turbulence )

• the trajectories are Gaussian• the propagator is

0

2

0

***2

2

2

2

)(

ii

effeffy

Tez

DkVVVk

vk

The change appears only in the growth rate: damping due to ion diffusion

dVV

ii

f

Dik

ni 2

0

0**

*

2

VV

Vk

eff

effy

Damping of the test modes at large k (resonance broadening - Dupree)

The amplitude of the potential continues to grow and the correlation length is of the order of the Larmor radius

The probability of displacements is non-Gaussian

Time invariant part formed by trapped trajectories (vortical motion)

Moving maximumof trajectories that are not in a structure at that time(radial motion).Decreasing number of free particles.

Weak trapping ftrd nnVV ,

• The distribution of displacements is approximated by two Gaussian functions

(one for the trapped particles (with fixed width) and one of diffusive type)

is the size of the structures of trapped trajectories

• the propagator is

tDStxtxGxGtxP iiiftr 2)(),,()(),( 22

ii

iiDik

Ski 222 1)

2

1exp(

iS

)(

2

1exp *

22 VSk ii Effect of ion trajectory structures

0

0**

0

2

0

***2

*

2

2

2

2

)(,

VV

DkVVVk

vkVk

eff

ii

effeffy

Tez

effy

Decrease of the effective diamagnetic velocity

Displacement of the unstable range toward small k due to trajectory structures, decrease of the maximum growth rate and of spectrum width

The amplitude of the turbulence and its correlation length increase.The inverse cascade appears as the shift of the unstable wave numbers

The moving stochastic potential determines ion flows Trapped particles move with and free particles move in opposite direction 0)( tx

)(2 tx

dV

0

dV 'V

f

tr

ftr

n

nVV

tVntVntx

*

*

'

0')(

is large and

has ballistic time-dependence (because of this separation)

Strong trapping ftrd nnVV ,

Generation of ion (zonal) flows in opposite directions with zero total flux

• The distribution of displacements is approximated by two Gaussian functions

• the propagator is

),',(),(),( tVyxGntVyxGntxP ffdtr

iiy

f

y

trii

DikVk

n

Vk

nSki 2

*

2

')exp(

ftr

eff

trii

effeffy

Tez

effy

nnn

nnVV

nDk

nVVVVk

vk

Vk

/

2

2)1(

2

2

2

))((

0

0**

0

02

0

****2

*

The ion flows modify both the growth rate and the effective diamagnetic velocity

The ion flows induced by the moving potential determines the increase of the effective diamagnetic velocity and the decrease of the growth rate

*0*0* 0,, nVVnV effeff

For n 1 the growth rate is negative for all k

The ion (zonal) flows determine the damping of drift modes

It increases with the increase of n, becomes larger than the diamagnetic velocity (first for small k) mode damping

Effects of the background turbulence through the fluctuations of the diamagnetic velocity

• they appear due to the coupling of the non-adiabatic response of the ions to the background potential through ion trajectories

• the fluctuations of the diamagnetic velocity determine a new term in the growth rate of the modes:

),()0,0(,'

)2()2(2

2

2

2

2

))((

'

00

*

0

02

0

****2

xvy

vMMddR

n

VRkk

nDk

nVVVVk

vk

jiijij

t

ij

ijjitr

ii

effeffy

Tez

• it is much more complicated and it was not possible to obtain a simple evaluation;•The component corresponds to modes with (zonal flow modes).11R 02 k

• quasilinear turbulence: all the components of the tensor are zero

• for weak trapping the non-diagonal component appear but only for non-isotropic turbulence;

• for strong trapping the anisotropy induced by particle flows with the moving potential leads to

Thus, zonal flow modes (with ) are generated due to ion flows. As n increases to 1, the asymmetry induced by the ion flows disappears and

The process does not appear as a competition between drift modes and zonal flow modes, but as the effect of the ion flows induced by the moving potential.

Note- These are simplified estimations that do not take into account the strongly non-isotropic

diffusion coefficients and the non-isotropy of the turbulence.

ftrd nnVV ,011 R

ftrd nnVV ,011 R

0,02 k

011 R

Conclusion Trapping determines a strong and complex influence on test modes in turbulent plasmas producing large scale potential cells and zonal flows.

Nonlinear evolution in which ion trapping determines strong changes in the turbulence that are eliminates later. Drift turbulence does not saturate but has a complex intermittent (oscillatory) behaviuor.- the trapped particles that form trajectory structures lead to long correlations lengths;- the potential motion combined with trapping stabilizes and damps the drift modes (first those with small k) and in the same time generates zonal flow modes (with ), which eventually are damped when the ion flows becomes symmetrical (n=1).

0,02 k

The nonlinear growth rate of the test modes on drift turbulence is

),()0,0(,'

)(2

1exp

2

2)1(

2

2

2

))((

'

22

0

0**

*0

02

0

****2

*

xvy

vMMddR

VSk

nnVV

VRkkn

DknVVVVk

vk

Vk

jiijij

t

ij

ii

eff

effijji

trii

effeffy

Tez

effy

),()0,0(,'

)(2

1exp

2

2)1(

2

2

2

))((

'

22

0

0**

*0

02

0

****2

*

xvy

vMMddR

VSk

nnVV

VRkkn

DknVVVVk

vk

Vk

jiijij

t

ij

ii

eff

effijji

trii

effeffy

Tez

effy

(1) Dupree turbulent damping (V>0)

(2) Trajectory structures 00* ;, ftr nnVV

(3) Ion flows ftr nnVV ,*

(3) Fluctuations of *V

Trajectory trapping determines several types of effects; some of them acts simultaneously, others at different stages of the evolution.