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 ?
First results on test modes on turbulent plasmas show that
trapping has very strong effects on turbulence and that
complex evolution appears beyond the quasi-linear limit.
These results determine a new perspective on two important paradigms
on the nonlinear evolution of drift turbulence:
• inverse cascade;
• zonal flows.
Content
1) Test particle statistics in stochastic potential
A short review of 10 year work
2) Test modes in turbulent plasma
A 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:

dx (t )  
 v ( x (t ), t ),
dt

x (0)  0,

 
  e z
v ( x, t )  
B
 
where v ( x , t ) is a stationary and homogeneous stochastic velocity field with Gaussian
distribution and known spectrum or Eulerian correlation (EC):



 
t 
2  x

E ( x , t )   ( x1 , t1 ) ( x1  x, t1  t )   f  , ,
 c  c 
where

is the amplitude ,
The Kubo number :
c
K
the correlation length,
V c
c
c
the correlation time.
c
c


,  fl 
, V 
 fl
V
c
(describes the decorrelation due to time variation of the stochastic field).
To determine: The statistical properties of the trajectories:
the probability of displacements (pdf) P(x,t)
x 2 (t )   P ( x, t ) x 2 dx,
2
d
x
(t )
1
D(t ) 
,
2
dt
D
x 2 ( c )
c

Lagrangian velocity correlation (LVC) : Lij (t )  vi (0,0)v j ( x (t ), t )
t
x 2 (t )  2 t  t 'Lxx (t ' )dt ',
Note:
0
t
D(t )   Lxx (t ' )dt ',
0
-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
There are two strong constraints for the statistical methods:
(A) The invariance of the potential for the static case
(Hamiltonian equation of motion)





d ( x (t ), t )
 ( x (t ), t )  ( x (t ), t )  ( x (t ), t )
 vi ( x (t ), t )


dt
xi
t
t
•
•
•
static case ( K   ): 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.
 
(B) The statistical invariance of the Lagrangian velocity due to   v ( x , t )  0
 
 
 
Pv ( x(t ), t )  Pv ( x (0), 0)  Pv ( x, t )
for stationary and homogeneous turbulence.
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 maximum
of 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)
The diffusion coefficient for static potential:
D(t ) 
x 2 (t )
t
 DB F (t ),
x 2 (t )   [ Ptr ( x)  Pf ( x, t )] x 2 dx
constant at large t
 n f (t )V 2t 2
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
Trajectory trapping
V2
d
No trajectory trapping
 fl
 fl
t
• Strong decorrelation mechanism with small decorrelation time
D  when  d ,
D  V 2 d
• Weak decorrelation mechanism with large decorrelation time
* The negative part of the LVC is cut out;
Anomalous diffusion regime (increased
diffusion at stronger decorrelation)
D  when  d 
 d   fl
 d   fl
* The LVC is not influenced;
Transport coefficient independent on
and stable to such perturbations.
d
 The effect of time variation of the potential (finite K):
D ( K )  DB F ( K ),

DB 
B
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:
K eff
KK II

,
K  K II
K II  1  K eff  1
 II
II
K II 
,  fl 
 fl
vII
No trapping for electrons
 The effect of collisions: diffusion of the potential correlation.
E(x) is not destroyed but spreaded due to collisional motion.
Effective Kubo number:
K eff
K

,
3/ 2
1  2 K 

D0

 Effect of an average velocity:
Vd
Vd 
V
Trapping exists for Vd  1 and determines “the acceleration” of the free
particles (on the opened contour lines), which reach an average velocity V f  Vd
• nonlinear process related to the statistical invariance of the Lagrangian velocity;
• not a dynamical effect but determined by the selection of the free particles.
 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, …)
  D0 / Vc  1, Vd / V  1, ...
•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
2) Test modes in turbulent plasma
• 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
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
z
• Drift instability in cartezian slab geometry, with constant magnetic
field and density gradient in collisionless plasmas.
B
y
-The limit of zero Larmor radius of the ions:
- stable drift waves   k yV* ,   0
- potentials that move with the diamagnetic velocity


 ( x , z, t )  0 ( x  V*t, z ),


0 ( x , z )
is the initial condition
dn0/dx
]
x
- 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
  k yV*eff
 k y (V*  V* )V*

k z vTe
2  0
2
V*  V*
eff
eff
eff
0
2  0
 k  2  Li 2 
,
0  0 

 2 
T dn0
V* 
en0 B dx
 max 
 k y2
1
k z vTe 2  0  4
V
*
eff

 V* / 2, k  1
The formal solution of the Vlasov equation for an initial perturbation with

0 ( x , z )
obtained with the method of the characteristics
(integration along trajectories):

t



e
e

(
x
,
z
,
t
)
e
e
n ( x , z, t )  n0 ( x) 1 
  dvz FM (v z )  d   V* y  x ( ), z  vz (t   ), 
T
T
0





 e ( x , z , t ) 
i 
n ( x , z , t )  n0 ( x) 1 


T


e
n0 ( x)
T




 
 d v FM (v)  d V* y x     ( ), z, 
t
2
0



  ij
 
 

 j x     ( ), z ,  i   t  x   ( ), z , 
 B





i 
n ( x , z, t )  n ( x , z, t )
e



 ( x, z, t )


 

dx
 ( x  Vd t )  ez dz


,
 vz
d
B0
d
 



x t; x , z   x , z t; x , z   z
The background turbulence
produces the stochastic ExB drift
in the trajectories (characteristics)
- the characteristic times ordering is used for simplifications: the parallel
motion of the ions and the ExB drift for electrons are neglected
The average propagator and the average effect of the diamagnetic velocity
fluctuations are determines using trajectory statistics

t



 

d exp ik  x ( )  x  exp( i(t   ))
Average over the trajectories
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 Vd
The trapping parameter for drift type turbulence is
Vd  Vd / V
The ordering of the characteristic times is
QL (no trapping):
   *   fl   c  
NL (trapping) :
   fl   *   c  
e
II
 *  c / Vd  4 / k2V* ,
e
II
i
II
i
II
 IIe  2 / kz ve ,  fl  2 / k2V
Thus
• nonlinear effects appear in ion trajectories if
• electron trajectories have quasiliniar statistics
Vd  V
Effects of the background turbulence through the propagator
Turbulent plasma with small amplitude (quasilinear turbulence
• the trajectories are Gaussian
• the propagator is
i
  k yV*eff
V*
eff
V  Vd
nf
  iki 2 Di
0
 V*
2  0
2
k
(
V

V
)
V
2ki Di
 y * * *


k z vTe
2  0
2  0
2
eff
eff
The change appears
only in the growth
rate: damping due to
ion diffusion
)
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
Weak trapping
V  V ,
d
ntr  n f 
The probability of displacements is non-Gaussian
Time invariant part formed by trapped trajectories
(vortical motion)
Moving maximum
of trajectories that
are not in a structure
at that time
(radial motion).
Decreasing number of
free particles.
• The distribution of displacements is approximated by two Gaussian functions
P( x, t )  Gtr ( x )  G f ( x, t ),
xi (t )  Si  2 Di t
2
2
(one for the trapped particles (with fixed width) and one of diffusive type)
Si is the size of the structures of trapped trajectories
• the propagator is
1 2 2
1
  i exp(  ki Si )
2
  ik i 2 Di
 k y (V*  V* )V*
2
2
  k yV* ,  
 ki Di
k z vTe
2  0
2  0
2
eff
eff
eff
V*
eff
0
 V*
2  0
 1 2 2

  exp   ki Si (V* ) 
 2

Decrease of the effective diamagnetic velocity
Effect of ion trajectory structures
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
Strong trapping
V  V ,
d
ntr  n f 
The moving stochastic potential determines ion flows
Trapped particles move with Vd and free particles move in opposite direction
x(t )  0
x (t )  ntrV*t  n f V ' t  0
ntr
V '  V*
nf
Generation of ion (zonal) flows
in opposite directions with zero
total flux
x 2 (t )
0
Vd
V'
is large and
has ballistic time-dependence
(because of this separation)
• The distribution of displacements is approximated by two Gaussian functions
P( x, t )  ntrG ( x, y  Vd t )  n f G f ( x, y  V ' t ),
• the propagator is
 ntr

nf
  i exp( ki Si ) 


2
   k yV*   k yV 'ik i Di 
2
  k yV*eff
2
eff
eff
k
(
V

V
)(
V
 nV* )
 y * *
2  0ntr The ion flows
2
*

 ki Di
modify both the
k z vTe
2  0
2  0 growth rate and
V*
eff
0(1  n )  2n
 V*
2  0
n  ntr / n f
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
eff
*
V
0, n  V 0,0  nV*
eff
*
It increases with the increase of n,
becomes larger than the
diamagnetic velocity (first for
small k)
mode damping
For n
1
the growth rate is negative for all k
The ion (zonal) flows determine the damping of drift modes
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:
 k y (V*  V* )(V*  nV* ) 2 2  0ntr
2V*

 ki Di
 ki k j Rij
2  n(2  0)(2  0)
k z vTe
2  0
2  0
2
t
  '
Rij   d ' 


eff
eff


d M ij  , M ij    vi (0,0) v j  x ( ),  
y
• it is much more complicated and it was not possible to obtain a simple
evaluation;
•The component R11 corresponds to modes with k2  0 (zonal flow modes).
• quasilinear turbulence: all the components of the tensor are zero


 the anisotropy induced by
• for weak trapping V  Vd , ntr  n f the non-diagonal
component appear but only for non-isotropic turbulence; R11  0

• for strong trapping V  Vd , ntr  n f
particle flows with the moving potential leads to
R11  0
Thus, zonal flow modes (with k2  0,   0 ) are generated
due to ion flows. As n increases to 1, the asymmetry induced
by the ion flows disappears and R11  0
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.
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 k2  0,   0 ), which
eventually are damped when the ion flows becomes symmetrical
(n=1).
The nonlinear growth rate of the test modes on drift turbulence is
  k yV*
eff
 k y (V*  V* )(V*  nV* ) 2 2  0ntr
eff
 ki k j RijV*
 ki Di

2  0
2  0
k z vTe
2
V*
eff
eff
eff
0 (1  n )  2n
 V*
2  0
 1 2 2 
  exp   ki Si (V ) 

 2
t
  '
Rij   d ' 




d M ij  , M ij    vi (0,0) v j  x ( ),  
y
Trajectory trapping determines several types of effects; some of them acts
simultaneously, others at different stages of the evolution.
(1) Dupree turbulent damping (V>0)
  k yV*
eff
 k y (V*  V* )(V*  nV* ) 2 2  0ntr
eff

 ki Di
 ki k j RijV*
k z vTe
2  0
2  0
2
V*
eff
eff
0 (1  n )  2n
 V*
2  0
 1 2 2 
  exp   ki Si (V ) 
 2

t
  '
Rij   d ' 


eff
(3) Fluctuations of
V*
(3) Ion flows V  V* , ntr  n f
(2) Trajectory structures
V  V* , ntr  n f ; 0  0 


d M ij  , M ij    vi (0,0) v j x ( ),  
y