bg – ursi school and workshop on waves and turbulence phenomena in space plasmas kiten, july 2006...
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BG – URSI School and Workshop on
Waves and Turbulence Phenomena in Space Plasmas
Kiten, July 2006
Nonlinear effects in charged particle transport in
turbulent magnetic fields
Madalina Vlad
National Institute of Laser, Plasmas and Radiation Physics,
P. O. Box MG-36, Magurele, Bucharest, Romania
.
Outline• The transport of charge particles in stochastic magnetic field: an important problem in
many astrophysical issues (cosmic rays in heliosphere, galactic cosmic rays, Fermi acceleration process, …). Quasiliniar theory is used in these studies, or phenomenological models (Bohm diffusion coef.)
• We have developed a semi-analytical approach that applies to the nonlinear case. • The results are unexpected and rather non-trivial in many situations:
- The diffusion coefficients are completely different of those obtained in quasilinear conditions. - The dependence on the specific parameters is reversed.
• We have concentrated on laboratory magnetized plasmas (fusion experiments): - transport induced by the ExB stochastic drift (including effects of collisions, average flows, parallel
motion,…)- Transport in magnetic turbulence and influence of collisions- Lorentz transport for arbitrary Larmor radius and cyclotron frequency.
• Parameters are completely different in fusion plasmas but, since the transport process depends on dimensionless numbers, some of the results are directly relevant for space plasmas. Certaily, the methods we have developed can be extended to specific particle transport problems in space plasmas.
AimTo present the statistical method and some results on charged particle transport
in magnetic turbulence
Content
1) Introduction- Diffusion by continuous movements- Special case of 2-dimensional divergence free velocity fields
2) Statistical methods
3) Diffusion of magnetic lines and structures formation
4) Particle diffusion and effects of perturbations
5) Larmor radius effects
6) Conclusions
1) Diffusion by continuous movements (generalization of the Brownian diffusion)
Non-linear stochastic equation:
where is a continuous field in each realization. It is statistically described as a stationary and homogeneous stochastic velocity field with Gaussian distribution and known Eulerian correlation (EC):
0)0();),(()(
xttxvdt
txd
),( txv
ccjiij
txfVttxxvtxvtxE
,),(),(),( 2
1111
V
VK c
flfl
c
c
c
,
V the amplitude , the correlation length, the correlation time
The Kubo number :
cc
Kubo number describes the decorrelation due to time variation of the stochastic field.
Similar dimensionless quatities are defined for other decorrelation mechanisms (collisions, average velocity, etc.)
To determine: The statistical properties of the trajectories, MSD, D(t), probability.
)(lim,)(
2
1)(,)(
2
2 tDDdt
txdtDtx
t
• A deterministic equation for each realisation of v(x,t) having a smooth, unique solution x(t)
)),(()0,0()( ttxvvtL jiij
0
0
')'(,')'()( dttLDdttLtD xx
t
xxx ,')'('2)(0
2 t
xx dttLtttx
Lagrangian velocity correlation (LVC) :
• Taylor has shown that:
for integrable LVC, the process is diffusive at large time
superdiffusive (non-integrable LVC) or subdiffusive (LVC with zero integral) transport
in all cases, there is a ballistic regime at small time,
)(lim,2)(: 2 tDDDttxtfor
,1,)(: 2 ttxtfor
tVtDtVtxtfor c2222 )(,)(:
222 /, KVD cccqlc • K<<1, quasilinear regime (fast variation of the stochastic field )
flc
cfl • K>>1, nonlinear regime (slow time variation ) KVVD cccflBfl /, 22 (Bohm diffusion coefficient)
EC LVCE(x,t) L(t)
2V 2V
x t
E(x,t)
c
- Universal scalings, EC function determines only numbers;- Gaussian distribution of the trajectories;- short time memory and coherence.
,')'()( 2
0
VdttLtDt
xx
ANOMALY: Two-dimensional divergence-free stochastic velocity fields (magnetized plasma (magnetic turbulence, ExB stochastic drift), incompressible fluid turbulence, etc.)
),,),(12
txxx
txV
At K > 1 :
Direct numerical simulations show trajectory trapping (eddying)Stochastic potential : ), tx
0),( txV
• K=10• K=1
• K=10• K=1
The trapping appears at K > 1 • is generic (each trajectory is a sequence of trapping events and long jumps) • is coherent (neighboring trajectories are all trapped)
Trapping strongly influences • particle transport the nonlinear regime is completely different of the Bohm diffusion
• statistical properties of the particles non-Gaussian trajectory distributions with
tails, memory effects and high degree of coherence in the stochastic motion
:, cK Permanent trapping - all trajectories wind on the contour lines of ;Subdiffusive behaviour.
:,1 flcK Temporary trapping on the contour lines of ;
1K 01KJ.-D. Reuss, J. H. Misguich, Phys. Rev. E 54, 1857 (1996).
A typical trajectory for large K
Trapping event
Long jump
a)
b)
The two segments a), b) correspond to the same time interval
)),( ttx )),( ttx
)(tx 2)0()( xtx
Properties of the trajectories:
- long jumps when the particle is at
- trapping on the contour lines with large
0)),( ttx
)),( ttx
Content
1) Introduction- Diffusion by continuous movements- Special case of 2-dimensional divergence free velocity fields
2) Statistical methods - Existing methods and their problems - New approach: nested subensemble method, decorrelation trajectory method
3) Diffusion of magnetic lines and structures formation
4) Particle diffusion and effects of perturbations
5) Larmor radius effects
6) Conclusions
2) Statistical methods
We have to determine the LVC knowing the EC of the stochastic field
Average of a stochastic function of a stochastic argument:
Test particle methods or methods for the passively advected density:• Corrsin approximation and its developments • Direct interaction approximation• Renormalization group technique• Estimation based on percolation in stochastic landscapes
)),(()0,0()( ttxvvtL jiij
)(),()0,0()),(()0,0()( txxtxvvxdttxvvtL jijiij
Corrsin approximation :•(1) Gaussian statistics of the trajectories;•(2) factorization of the average (equivalent with performing first the average over the stochastic field at fixed position (Eulerian) and then the average over trajectories).
),(),()(),()0,0()( txPtxExdtxxtxvvxdtL ijjiij
Bohm diffusion coefficient at large K and diffusion in frozen turbulence
Assumption (2) can be eliminated Additional nonlinear terms appear in L(t) depending on ;they are determined by integrating some half-Lagrangian correlations . A closed system of equations is obtained.
),()( txvtx ji
),()),(( yvttxv ji
Bohm diffusion coefficient at large K and diffusion in frozen turbulence
Thus this method fails because it is based on (1).
Assumption (1) can be improved by taking cumulants of 4th order Bohm diffusion coefficient at large K and diffusion in frozen turbulence Thus the trajectories are strongly non-Gaussian
Essential conditions for statistical methods that can describe this special case are:
• to maintain the statistical consequences of the invariance of the potential
• not to rely on the Gaussian assumption for the trajectories.
The trapping process is strongly connected to the invariance of the potential• 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
t
ttx
t
ttx
x
ttxttxv
dt
ttxd
ii
)),(()),(()),(()),((
)),((
This represents a very strong constraint for the statistical methods.
),,),(12
txxx
txv
The subensembles (S) : 0)0,0( vv
,)0,0( 0
Sjiij ttxvvvPvddtL )),((),()( 00000
The LVC is the sum of the contributions of each subensemble (S):
SjiSji ttxvvttxvv )),(()),(()0,0( 0
The invariance property of the Lagrangian potential
The LVC (2-point average) the average Lagrangian velocity (1-point average)
The average Lagrangian potential in (S) can be determined
for frozen turbulence
,)0())0(())(( 0 xtx
Stx ))((
THE DECORRELATION TRAJECTORY METHOD (DTM)
Statistical properties of the Eulerien potential and velocity fields in (S): they are non-stationary and non-homogeneous Gaussian fields, having space-time dependent averages :
2020 //),(),( aEVEvtxtx iiS
S
2020 //),(),( aEVEvtxVtxv jijiS
jSj
0)0,0( S 0),( txS and as ,x
t
0)0,0( vV S 0),( txV S
and as ,x
t
),()0,0(),(,),()0,0(),(,),()0,0(),( txvtxEtxvvtxEtxtxE iijiij
are determined by the EC of the potential as: .,,21
2
1222
2
11 etcExx
EEx
E
),(,),(12
txxx
txV SS
Zero-divergence average velocity in (S)
These subensemble (conditional) averages describe the structure of the correlated zone
ijE
The average Lagrangian velocity in (S)
• Aim: to find an approximation compatible to the invariance of the average Lagrangian potential in (S): (Corrsin approx. in (S) determines an average Lagrangian potential which decays to 0)
)];([);(
)();(
StXVdt
StXd
txStXS
S
tx ))((
));(())(( StXVtxv S
S
The approximation of the decorrelation trajectory method: the fluctuations of the trajectories in (S) are neglected
),(),12
XVXXXdt
Xd SS
An equation is obtained for the average trajectory in (S) (the decorrelation trajectory), which is of Hamiltonian type:
0);0( SX
The LVC and the diffusion coefficientSjiij ttxvvvPvddtL )),((),()( 00000
RESULTS ( isotropic turbulence, factorized EC , ):
)()(')( 2 thtKFVtL ijij
)()(2
tKFKtDc
),(12
exp2
1)(
0 0
22
3 puXpu
ududpF
The function F is :
,/, 00 upvu
t
thdtt0
)'(')(is the decorrelation trajectory in (S) along for the static turbulence
),( puX 0v
h(t) - the time-dependence of the EC of the potential; - describes the decorrelation due to the time variation ofF(t) - determined by the nonlinearity (by the space-dependece ) - describes the trapping in the structure of the stochastic field
)(x
), tx
D(t) results from a competition between trapping and release of the trajectories:
)()(),( thxtxE
- Thus, the decorrelation trajectory method relies on a set of simple, smooth trajectories that are determined from the Eulerian correlation of the stochastic potential. The LVC is determined as a weighted average of these trajectories, with the weighting factor determined analytically. The trajectories are usually determined numerically (calculations of the order of 10s for this case)
-The decorrelation trajectory method is the first order of a systematic expansion, the nested subensemble method (NSM).
The idea is to determine averages not on the whole set of trajectories but to group together trajectories that are similar, to average on them and then to perform averages of these averages. Similar trajectories are obtained by imposing suplementary initial conditions besides x(0)=0.
The subensembles (S1) :0)0,0( vv
,)0,0( 0
THE NESTED SUBENSEMBLE METHOD (NSM)Space of realization (R) = Σ subensembles (S1),
Subensemble (S1) = Σ subensembles (S2), …
The subensembles (S2) : , …. (Sn) 0
2 )0,0(ij
ji xx
• LVC in (S2):
• LVC in (S1):
• LVC in (R):
)2;()),(()),(()0,0( 0
2
0
2StVvttxvvttxvv L
jiSjiSji
)2;()2()),(( 022
012
011
0
1
0 StVSPdddvttxvv LjiSji
)1;()()( 000 StVvSPvddtL Ljiij
)2(),1( SPSP are the probabilities that a realization belongs to (S1) or (S2)
- Systematic expansion based on similarities of the trajectories: inner subensembles in this nested classification contain more similar trajectories. In the limit of large n the trajectories in a (Sn) are almost identical.
Approximation in NSM : neglect the fluctuations of the trajectories in (S2)
Similarity of the trajectories in (S2) [the potential value determines an average size; trajectories are strongly super-determined: 6 supplementary initial conditions contained in the definitions of (S2) and (S1)]
)2);2;(())2;(())(()2;(22
SStXVStXvtxvStV E
SS
L
)2;( StX
is the average trajectory in (S2); It is the solution of the Hamiltonian system:
),2;,)2;(12
SXXX
SXVdt
Xd EE
0)2;0( SX
The Eulerian subensemble averages of the velocity are analytically determined. They are functions of the parameters of (S1), (S2) and of the EC.
)2;),2;( SXSXV EE
Trajectory statistics in (S) is obtained by averages of X(t;S2) with determined weighting factors.
Trajectory statistics in (R) is obtained by averages of X(t;S1) with determined weighting factors.
• Thus, the nested subensemble method (NSM) is a semi-analytical approach based on a set of deterministic trajectories, X(t;S2). They are smooth trajectories determined from the EC of the potential. • NSM fulfils all the statistical constraints determined by the invariance of the Lagrangian potential provides a good statistical description of trapping• NSM appears to have a fast convergence: the decorrelation trajectory method (DTM), based only on (S1) yields close results for D(t)
• NSM provides much more statistical information: the distribution of trajectories and of the distance between neighboring trajectories in the whole set of realizations R and also in each subensemble (S1)
Content
1) Introduction- Diffusion by continuous movements- Special case of 2-dimensional divergence free velocity fields
2) Statistical methods
3) Diffusion of magnetic lines and magnetic structure formation
4) Particle diffusion and effects of perturbations
5) Larmor radius effects
6) Conclusions
3) Diffusion of magnetic lines and magnetic structure formation
zezxB
zxB
dz
txd
),(),()(
0
IIc
zxfVzzxxzxtxE
,),(),(),( 2
1111
VL
L
VK cII
c
II
,
V the amplitude , the correlation length, the parallel correlation length
The magnetic Kubo number :
IIc
where L is the parallel length necessary for which the magnetic line performs a perpendicular displacement of (the equivalent of the time of flight)
The magnetic Kubo number describes the decorrelation due to the variation along z of the stochastic field. If it is not dependent on z, K is infinite, the potential is conserved and the magnetic line spreading must be subdiffusive.
z
x
y
0B B
Average trajectories in the subensembles (S2)
- closed paths (trapped trajectories) - small sizes and periods
- some open paths (free trajectories)
- larger sizes and periods
0Large 00
),2;,12
SXXXdt
Xd E
Statistics of magnetic lines in the subensembles (S)
Average trajectory in (S) : )2;()2();( 022
012
011 StXSPdddStX
0Large
)();(,0);( 21 SStXStX
00 Path along initial velocity with continuous time increase
*
*DTM
NSM
Completely different results obtained with NSM and DTM trajectory fluctuations in (S)
are not negligible but the mixing process is well described in both cases
size of the structure
Magnetic line statistics in the presence of trapping for z-independent fluctuations Magnetic line fluctuations in (S1): dispersion and average
)1;()2;()2()1;( 22022
012
011 StXStXSPdddStd iii
Saturation in a timeat finite size
)1(Ss
0Large 00
Continuous increase (slower than linear)
Magnetic line probability function (pdf) in (S1):
)2;()2();,( 022
012
011 StXxSPdddStxP
Pdf far from Gaussian in both cases
* Saturation and localization * Continuously expanding part(with a velocity > average velocity)
Statistics of the distance between neighboring magnetic lines in (S1):0)0('),()('
xtxtx
St)(2
1
St)(2
2);(1 Std
);(2 Std
Very strong anomalous clump effect
sflcl S 10100)( Richardson law for small time and later slower increase as
Absence of the clump effect
Anomalous clump effect (usually trajectory clumps have a life time of the order of the flight time)
3t
5.1t
3t
Richardson law for the dispersion perpendicular tothe average velocity in (S) and ballistic for the other
st 5.1t
pdf of the distance between neighboring field lines in (S):
* Saturation and localization* Continuously expanding part(with a velocity > average velocity)similar with the pdf of the trajectories
Thus magnetic line structures
are generating in 2-d magnetic turbulence in magnetized plasmas.
They are similar with fluid vortices and represent solenoidal or eddying regions.Their statistical characteristics
(formation ‘time’ , size and dispersion ) are determined as function of the parameters of the subensemble (S1)
A very strong anomalous clump effect characterizes the relative distance between magnetic lines in such structures.
The evolution of magnetic lines in the structures is quasi-coherent.
For z-independent magnetic fluctuations, the size of the structures increase as decreases and go to infinity when .
For z-dependent magnetic fluctuations, structures appear if K>1 and their maximum size depends on K (increases with K).
)1(SL)1(Ss )1(Sdi
000
The LVC and the running diffusion coefficient in (R)
• Long-time Lagrangian correlation build up in the structures; L(t) has long negative tail of power law type; positive and negative parts compensate.
Subdiffusive spreading of the magnetic lines for z-independent fluctuations
Memory and subdiffusion !
fl
c
t
xatxE
exp
2/1
1),(
22
BDD /
);()()( 101
00 StVvSPvddtL L
Subdiffusive transport
The running diffusion coefficient in (R)
The nested subensemble method(NSM)
The decorrelation trajectory method(DTM)
• The results of the DTM concerning diffusion coefficients are validated ;
• The NSM appears as a fast convergent approach: second order is sufficient for determining the physically interesting statistical quantities.
Diffusive soreading of magnetic lines for z-dependent fluctuations
• frozen turbulence : subdiffusion (continuous lines - )
• time-dependent tubulence : diffusion(dashed lines - )
K
100K
The quasi-linear regime K<1
1,
1,)(
KK
KKKF )38.0(10
KFDKD B)(
-2 -1 0 1 2 3 4 log(K)
0
-1
-2
BD
Dlog
11
)(2
KKD
c2
2
)( KKDc
Non-linear regime K>1 with structures
Diffusion in time-dependent stochastic potential
KVDc
c
cB
2
the Bohm diffusion coefficient
Content
1) Introduction- Diffusion by continuous movements- Special case of 2-dimensional divergence free velocity fields
2) Statistical methods
3) Diffusion of magnetic lines and structures formation
4) Particle diffusion and effects of perturbations
5) Larmor radius effects
6) Conclusions
The z-independent magnetic fluctuations represent an unstable system: any weak perturbation has strong influence on the transport and anomalous regimes are obtained.
fl
BDD /
Particles with velocity v in turbulent magnetic field, with small Larmor radius are transported in the perpendicular direction with the magnetic line:
)()(),()( 2 tvDtDtLvtL pp
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)
Destruction of large structures with
Trajectory structure No trajectory structures and trapping
• Weak decorrelation mechanism with large decorrelation time
dwhenD
fl
• Strong decorrelation mechanism with small decorrelation timefld
dd VDwhenD 2,
dS
Effect of weak collisions
)(
2/,2/),,,()()(),,( 22
tdt
dz
Vtzxvtttzxbdt
xd
II
thIIII
zetzxtzxb
),,(),,(
Complex triple stochastic processe described by four dimensionless parameters:
II
mccIImII
IIII KK
VM ,,,
22
)(),()(),,( 2/12/1 tdt
dztttzxbM
dt
xdIIIIII
,),,(),,( 0 tzxbeBtzxB z
a supplementary stochastic function (multiplicative white noise)
Two types of trapping : • magnetic line trapping on the contour lines of the potential string-like segments of the magnetic lines;• parallel trapping of the particles due to collisions which force them to return along the magnetic lines.Two decorrelation mechanisms :• time variation of the stochastic magnetic field;• the perpendicular collisional velocity.
),,( tzxbK
dz
xdm
cIIcII MDMD ,,,,,, int2
Effect of in the static case :
Minimum of the diffusion coeffcient(’'resonance’' condition)
A very small collisional perpendicular diffusion transforms the subdiffusive transport into a diffusive one with effective diffusion coefficient much larger than
Very strange regime with decreasing D,determined by the combined action of magnetic line trapping and parallel trapping
Thus
long time correlation of the Lagrangian velocity (memory effect)
appears in the magnetic lines for z-independent fluctuations.
Memory determines subdiffusive transport in static potential
and a class of anomalous diffusion regimes
in the presence of a decorrelation mechanism.
Decorrelation can be produced by - z component of the turbulent magnetic field (pitch angle scattering) - weak collisions - average velocity, etc.
Content
1) Introduction- Diffusion by continuous movements- Special case of 2-dimensional divergence free velocity fields
2) Statistical methods
3) Diffusion of magnetic lines and structures formation
4) Particle diffusion and effects of perturbations
5) Larmor radius effects
6) Conclusions
Large Larmor radius of impurity ions and of fusion particles
and
• the guiding center approximation is not adequate and Lorentz force has to be used for particle trajectories
Lorentz transport of test particles in turbulent plasmas
• large Larmor radius effects for ion and impurity transport;• effect of the turbulence on the fast fusion particle.
1c
1c
5) Larmor radius effects
• M. Vlad, F. Spineanu, ‘Larmor radius effects on impurity transport in turbulent plasmas’, Plasma Physics and Controlled Fusion 47 (2005) 281.• 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.
5) Larmor radius effects; Lorentz diffusion
Non-linear second order Langevin Equation:
zz ezxBzxBeBB
udt
xdBu
m
q
dt
txd
),(),,(
,,)(
0
2
2
.),/(
)()()(,/)()(
ctBmqB
ttxttut jiji
instantaneous Larmor radius: Guiding center position:
vdt
dz
tz
B
v
dt
td
tz
B
v
dt
td
jj
iji
jij
i
),,()(
),,()(
0
0
z
x
y
0B B
jij
i
jij
i
H
dt
td
H
dt
td
)(
)( )(
2
1, 2
fl
H
Two coupled Hamiltonian systems with the same Hamiltonian function
Invariance property for static potential: energy is conserved
0
)(),(
dt
ttdH
?
• drift transport: invariance of the potential trajectory trapping
Does the invariance of the energy produce trapping in Lorentz transport?
Four dimensionless parameters for the Lorentz transport:
II
IIII
cfl vV
,cIIcV ,,,
The potential is a stochastic field(stationary, homogeneous, with zero average and Gaussian distribution)
vLL
Kc
II ,,
IIcc
ztxfttxxtxtxE
exp),(),(),( 2
1111
Given Eulerian correlation:
statistical description stochastic drift velocity
jiji x
txtxv
),(
),(
Parameters of the stochastic drift velocity:
Magnetic Kubo numberz-dependence of magnetic fluctuations
Normalized Larmor radius (rigidity) particle kinetic energy
Normalized cyclotron frequency
Lorentz transport & drift transport for z-independent magnetic fluctuationsStrong Larmor radius effects
The statistical evolution of the guiding centers is determined mainly by short coherent kicks with period T (the cyclotron motion brings back the particle in the correlated zone).
slower effective drift velocity and larger effective flight time
Subsiffusive transport with the same time decay due to trapping but with an amplification factor in the Lorentz case.
efft
Lorentz transport for z-dependent magnetic fluctuations :the asymptotic diffusion coefficient
The asymptotic diffusion coefficient as function of the Larmor radius at fixed K:
maximum at ‘rezonance condition’ and decay as122 mKK
1K
10K
100K
),( KD
1.2 Gyrokinetic approximation
Equivalent with ExB diffusion in the gyroaveraged Eulerian correlation of the potential
Dependence on Larmor radius in the quasilinear regime
Nonlinear process that determines the increase of and of . eff maxD
1
0
~)( D
D
KDDDKD B00
1 ,~),(
fleff 22
Bm
m
DKDD
K
2),(
2
max
2
1.3 Physical image of the amplified diffusion
Estimation of the maximum diffusion coefficient
2/,,2/~ 0DDVVDVV ceffeff OK
Bfleffeff DDV max2/~ wrong
Strong modification of the potential EC due to gyro-average:the effective correlation length is
B
fleffeff
eff
DD
V
max
22/
Conclusions We have presented the problem of non-linear charged particle transport in magnetic turbulence
We have shown that the relevant parameter is the magnetic Kubo number
For K > 1 strong nonlinear effects appear:• magnetic line structures with high degree of coherence • memory effects that leads to subdiffusive transport in z-independent magnetic fluctuations and to anomalous diffusion coefficients in the presence of a decorrelation mechanism• non-Gaussian distribution of displacements
The decorrelation trajectory method and the nested subensemble method can discribe this process of intrinsic structure formation in magnetic turbulence. They can be adapted to the conditions of space plasma magnetic turbulence (3d fluctuations of the magnetic field, small average field, etc.)These methods give a clear physical image on these nonlinear process.
VL
L
BBK cII
c
II
,
)/( 0