Contributions of MEdC Association to the
EFDA Transport Topical Group
2010
EFDA Task
Title of the MEdC contribution
MEdC
Coordinator
Page
WP10-TRA-01-02
Zonal flow generation and particle trapping in the
structure of the turbulent potential.
Madalina Vlad
2
WP10-TRA-01-04
Quasi-coherent precursors and generation of
filaments in the H-mode rotation layer. Study of
the quasi-coherent modes and and of the
stationary state of the Edge Localized Modes
Study of the stationary parallel dynamics and
saturation of the filaments generated in strong
Kelvin-Helmholtz events in the H-mode regime.
Florin Spineanu
5
WP10-TRA-01-05
Stochastic reduced models of different plasma
regimes and description of the transition in term
of global parameters
Gyorgy
Steinbrecher
9
WP10-TRA-03-01
Numerical experiments with low dimensional
kinetic models of the impurity transport
Gyorgy
Steinbrecher
12
WP10-TRA-03-03
Impact of central electron heating on both
electron and ion temperatures, and on particle
and impurity densities
Nicolae
Pometescu
15
WP10-TRA-05-01
Effects of filamentary structures on impurity and
density transport
Study of the turbulent structures and
intermittency by stochastic differential equations
Madalina Vlad
21
Gyorgy
Steinbrecher
24
Test particles simulations for impurities in the
edge region with focus on particle and heat
transport
Marian Negrea
28
WP10-TRA-05-01
WP10-TRA-05-03
1
WP10-TRA-01-02-xx-01/MEdC
Zonal flow generation and particle trapping in the structure of the turbulent potential
EFDA Transport Topical Group
Deliverable under Base- Line Support
Reference:
Topical Group: Transport
Area: Physics of L-H transition
Task: WP10-TRA-01-02-xx-01/MEdC
Document:
Task Title :
Zonal flow generation and particle trapping in the structure of the turbulent
potential
Task Coordinator B. Weyssow
Author(s):
Name and Association(s) of the Author(s) :
Madalina Vlad, Association EURATOM – MEdC Romania
Date:
Distribution list:
21, January 2011
Boris Weyssow (CSU Responsible Officer)
Clemente Angioni (Topical Group Chair)
Content:
We have shown that the main role in the nonlinear evolution of drift type
turbulence is produced by the ion trapping in the structure of the electrostatic
potential that has an average poloidal rotation with the effective diamagnetic
velocity. Ion flows in opposite directions (zonal ion flows) are generated by the
combined effect of trapping and potential motion.
Test modes on turbulent plasmas were studied taking into account the process
of ion trajectory trapping in the structure of the background potential. The case
of drift turbulence was considered, and the frequency ω and the growth rate γ
were determined as functions of the statistical properties of the background
turbulence. The main characteristics of the evolution of the turbulence were
deduced from γ and ω. We have shown that ion trajectory trapping has a
strong and complex influence on the evolution of the turbulence.
2
EFDA Transport Topical Group
Deliverable under Base- Line Support
Reference:
Topical Group: Transport
Area: Physics of L-H transition
Task: WP10-TRA-01-02-xx-01/MEdC
A different physical perspective on the nonlinear evolution of drift
turbulence is obtained. The main role is played by the trapping of the ions in
the stochastic potential that moves with the diamagnetic velocity. The trapped
ions determine the evolution of the turbulence toward large wave lengths (the
inverse cascade). They also determine a slower increase of the amplitude of the
potential fluctuations and the evolution to more ordered states. The influence
of the ion flows produced by the moving potential appears later in the
evolution of the turbulence. The ion flows determine the damping of the small
k modes, the decay of the growth rate and eventually the damping of the drift
modes with any k.The ion flows also determine transitory zonal flow modes
(with wave number on the poloidal direction ky=0 and ω=0) in connection with
the fluctuation of the diamagnetic velocity due to the background turbulence.
Thus, in this perspective, there is no causality connection between the damping
of the drift turbulence and the zonal flow modes. Both processes are produced
by ion trapping in the moving potential, which determines ion flows. The drift
turbulence does not saturate but has an intermittent evolution.
A DC electric field modifies the relative velocity of the ions and of the
potential and consequently changes the value of the parameter that measure
the effect of the nonlinearity for the drift turbulence, which is the ratio of the
amplitude of the stochastic ExB drift and the effective diamagnetic velocity.
This determines the increase or the decrease of the maximum level of potential
fluctuations, depending on the sign of the DC electric field.
Revision No: 1
Changes:
Written by:
Revised by:
Approved by:
Name: Madalina
Name
F. Spineanu
HRU – MEdC
Association
Vlad
3
Objectives
The objective was to study the connection between trajectory trapping in the structure of the stochastic
potential and the evolution of drift turbulence.
The aim was to investigate other possible physical explanations for the generation of zonal flows.
Achievements
The connection between zonal flow mode generation and ion trapping in the structure of the potential
was demonstrated for the drift turbulence in plasmas with constant confining magnetic field
Shortcomings
These fundamental results need to be extended to turbulence characteristics specific to large tokamak
devices.
Conclusions and General perspectives
These results represent the first theoretical (analytical) evaluation of the evolution of the turbulence in
the strongly nonlinear regime and they could bring an important contribution to the understanding of
plasma turbulence.
References (to Conference paper/poster, Meeting, Publication)
F. Spineanu and M. Vlad, “Zonal flow generation and particle trapping in the structure of the turbulent
potential” 3rd EFDA Transport Topical Group meeting and 15th EU-US Transport task Force Meeting,
Cordoba, 2010. P3.22.
M. Vlad, F. Spineanu, „Nonlinear evolution of drift turbulence: inverse cascade, zonal flows,
intermittency”, XVth International Conference on Plasma Physics and Applications, 1-4 July, Iasi, invited
paper.
M. Vlad, “Nonlinear effects produced by the ExB drift on transport and structure generation in turbulent
tokamak plasmas”, Association Day, 2 July 2010.
M. Vlad, F. Spineanu, „Nonlinear evolution of drift turbulence: inverse cascade, zonal flows,
intermittency”, Romanian Journal of Physics, in print.
M. Vlad, F. Spineanu, „Trapping, anomalous transport and quasi-coherent structures in magnetically
confined plasmas”,
Plasma and Fusion Research 4 (2009) 053-1:8.
4
WP10-TRA-01-04-xx-01/MEdC
Quasi-coherent precursors and generation of filaments in the H-mode rotation layer
EFDA Transport Topical Group
Deliverable under Base- Line Support
Reference:
Topical Group: Transport
Area: L-H power threshold and ELM control techniques
Task: WP10-TRA-01-04-xx-01/MEdC
Document:
Task Title :
Quasi-coherent precursors and generation of filaments in the H-mode rotation
layer. Study of the quasi-coherent modes and and of the stationary state of the
Edge Localized Modes Study of the stationary parallel dynamics and saturation of
the filaments generated in strong Kelvin-Helmholtz events in the H-mode regime.
Task Coordinator B. Weyssow
Author(s):
Name and Association(s) of the Author(s) :
Florin Spineanu, Association EURATOM – MEdC Romania
Date:
Distribution list:
31, December, 2010
Boris Weyssow (CSU Responsible Officer)
Clemente Angioni (Topical Group Chair)
5
EFDA Transport Topical Group
Deliverable under Base- Line Support
Reference:
Topical Group: Transport
Area: L-H power threshold and ELM control techniques
Task: WP10-TRA-01-04-xx-01/MEdC
Content:
We have investigated the transitory process of increase of the parallel current
induced by the centrally advected vorticity in a Kelvin-Helmholtz event as
constrained by the poloidal viscosity of neoclassical origin. This mechanism, the
usual reason for the decay of poloidal rotation in plasma, is efficient due to the
different drifts of electrons and ions in the radial direction, leading to currents that
are subjected to resistive dissipation. We have shown that his mechanism is a
source of force opposing the gradient of the parallel current. This allows to balance
in an efficient way the parallel dynamics and provides an explanation for the
oscillatory precursors of the Edge Localised Modes.
We show that the processes:
1. current filamentation due to nonlinear tearing instability
2. formation of a vortex with an axis common with the current filament
3. increase of the flow of plasma along the axis
are combined in a synergetic way.
We have shown that the palsma evolves to the break-up of the layer with strong
concentration of the density, vorticity and current density into cuasi-singular
structures (filaments). This evolution is purely growing in an axisymmetric
geometry and is governed by an universal instability of the same type as the
Chaplygin gas with anomalous polytropic. The break-up of the current layer and
formation of filaments takes place on very short time scales, governed by the
Alfven speed in the direction of the layer (transversal on B). This is a fast transient
process which in axisymmetry is only limited by the generation of the singular
filaments. The parallel dynamics is arbitrary, as long as the condition that the
electron parallel velocity is much higher than their thermal velocity.
However the parallel dynamics is inhibited by the same mechanism which damps
the poloidal rotation in tokamak, the magnetic pumping. The fast increase of the
current and of the flow velocity along the magnetic field lines during the
6
EFDA Transport Topical Group
Deliverable under Base- Line Support
Reference:
Topical Group: Transport
Area: L-H power threshold and ELM control techniques
Task: WP10-TRA-01-04-xx-01/MEdC
filamentation is accompanied by radial currents (localised to the filaments) due to
curvature drifts which, in a collisional plasma, absorb energy from the flow on a
time scale of the inverse ion-ion collision frequency. As in the usual neoclassical
damping the effect is nonuniform along the line and is effectively equivalent to a
force acting against the increase of the flow in the filaments. This may generate
oscillatory instead of purely growing states. The effect of thermal channeling due to
filamentation contributes to the reduction of the efficiency of the damping
mechanism and the regime can again be purely growing but saturates the filaments
to a finite amplitude.
The precursors to the ELM event are due to the fact that the nonlinear tearing of
the current sheet (coinciding with the vorticity layer) is not purely growing but also
has a oscillatory part.
Revision No: 1
Changes:
Written by:
Name: F.
Revised by:
Name
Spineanu
Approved by:
F. Spineanu
HRU – MEdC
Association
Objectives
The objective was to find a physical model of filamentation which should complement the peelingballoonning instability.
This must provide an explanation of the ELMs.
In addition the identification of the physical reason behind the oscillatory regime (precursors).
Achievements
7
Two important steps have been done:
-
Demonstration that the layer of vorticity/current density is subject to an instability of Chaplygin
– type leading to filamentation
Identification of the poloidal magnetic pumping as the structural change leading to oscillatory
regime instead of filamentation
Shortcomings
The domain of parameters of the Chaplygin-type theory is limitted at present.
Conclusions and General perspectives
The idea is new and requires numerical work to examine the detailed evolution of the oscillation regime.
The subject seems complex (coroborating experimental input from previous works on DIII-D and Alcator
C, etc.) and cannot be considered as closed.
It is however the only alternative to usual model of peeling-ballooning-based explanation.
Since the Resonant Magnetic Perturbations is a major hardware investment, an examination of models
that do not involve stochastic magnetic fields is necessary.
References (to Conference paper/poster, Meeting, Publication)
“ Parallel dynamics during transient filamentation in ELMs”
F. Spineanu and M. Vlad, 3rd EFDA Transport Topical Group meeting and 15th EU-US Transport task Force
Meeting, Cordoba, 2010. P3.22.
8
WP10-TRA-01-05-xx-01/MEdC
Stochastic reduced models of different plasma regimes and description of the transition
in term of global parameters
EFDA Transport Topical Group
Deliverable under Base- Line Support
Reference:
Task Force: Statistical properties of edge turbulent transport
Area: Physics of L-H transition
Task: WP10TRA-01-05-xx-01/MEdC/BS
Document:
Task Title: Stochastic reduced models of different plasma regimes and description
of the transition in term of global parameters
Task Coordinator: Boris Weyssow
Author(s):
Date:
Distribution list:
Gyorgy Steinbrecher, Association EURATOM-MEdC
17 January 2011
Boris Weyssow (CSU Responsible Officer)
Clemente Angioni (Topical Group Chair)
Content:
New class of reduced stochastic models for the simulation of transitions in
stochastic dynamical systems, transitions that can be identified with L-H transitions,
was elaborated. The main part of the model consists of a noise driven stochastic
model, whose aspects were studied in the refs. [1-3]. The main aspect in this class
of models is related to the fact that the mean values of the moments of the output
of random dynamical system (or, in functional analytic terms, the Lp norms [3])
depends in a very sensitive way on the intensity of the driving multiplicative noise
as well as on its correlation times. The heavy tail exponent of the models is related
to that exponent p of the Lp spaces whose norm is finite. An important parameter
of the model is the critical exponent pcrit. The critical exponent pcrit, that is identical
to the heavy tail exponent of the model, depends on the parameters like intensity
of the driving noise, the correlation time of the driving noise. The L-H transition
take place exactly when the critical exponent crosses the value pcrit =1.
9
EFDA Transport Topical Group
Deliverable under Base- Line Support
Reference:
Task Force: Statistical properties of edge turbulent transport
Area: Physics of L-H transition
Task: WP10TRA-01-05-xx-01/MEdC/BS
Interestingly, this transition is exactly the transition from classical Lp spaces (p>1)
to “exotic Lp spaces”, with 0<p<1. We recall that in generic cases of the
distributions, Lp spaces with 0<p<1 are in a sense pathological: their dual is reduced
to the null vector. The pcrit >1 corresponds to the L mode and the pcrit <1 to the H
mode. Observe that in the pcrit <1 case the model exhibits also an increased
intermittency, which can be associated with the increase of the type I ELM activity.
The hysteresis effect will be manifest if the driving noise of the linear model is
generated by a dynamical system associated to the magnetic field line dynamics.
We mention that the results of ref. [3] are obtained under very general
assumptions related to the driving multiplicative noise. In this case the hysteresis is
related in fact to the hysteresis effect of the driving multiplicative noise, produced
by the complex fractal structure, stickiness of the magnetic island chains. This class
of models is a stochastic generalization of the models from refs. [4,5]. Comparative
study was performed for different set of parameters that describe the driving
noise. Numerical method for generation of driving noise with long-range
correlations was elaborated [6].
Revision No: 1
Changes:
Written by:
Revised by:
Gyorgy Steinbrecher
Name
Approved by:
Florin Spineanu
HRU MEdC
Association
Objectives
Elaboration of class of stochastic models of the intermittent events of the edge plasma turbulence.
Characterization of the intermittent events and particle transport on the plasma boundary in term of the
10
singularity structure of the stationary probability density function obtained from the Fokker-Planck
equation associated to the stochastic processes.
Achevements
The possibility to model the L-H transition by noise-induced transition was proven. The transition is
related to the change of the statistical properties of the noise. The L-mode corresponds to the noise
having finite Lp norm with p>1, (recall that these spaces are reflexive, have highly non-trivial dual, by the
decrease of “p” the duals decreases, in the sense of inclusion. The transition goes through the space L1
(that is no more reflexive, its dual is still non-trivial but it is strictly contained in all of the previous dual
spaces). The H mode corresponds to driving parametric noise that is described by non-traditional Lp
spaces with 0<p<1. The dual of these spaces contains only the null function.
Shortcomings
The driving noise that produces the transition is related to linear one- dimensional models. It is
worthwhile to obtain results on the higher dimensional stochastic models.
Conclusions and General perspectives
A class of new stochastic models of transitions that models the L-H transition was elaborated. More
exact results will be obtained by higher dimensional generalization of this class of models.
References (to Conference paper/poster, Meeting, Publication)
[1]. X. Garbet, G. Steinbrecher, “On-off intermittency. Exact results”. Presented at the “Solvay
Workshop <A Tribute to Radu Balescu>”, Brussels, 6-8 March 2008.
[2] Steinbrecher G., Weyssow B., Phys. Rev. Lett. 92, 125003 (2004)
[3] G. Steinbrecher, X. Garbet. B. Weyssow. “Large time behaviour in random multiplicative processes”.
arXiv:1007.0952v1 [math.PR], (2010)
[4] D. A Gates et all, “ Neoclassical islands in COMPASS-D”, Nucl. Fusion 37, (1997), 1593-1606.
[5]. R. J. Buttery et all, “Onset of the neoclassical tearing modes on JET”. Nucl. Fusion 43, (2003) 69-83.
[6] G. Steinbrecher, B. Weyssow “New representation and generation algorithm for fractional Brownian
motion”, Roumanian Journal of Physics, 55,1120-1130, (2010).
11
WP10-TRA-03-01-xx-01/MEdC/BS
Numerical experiments with low dimensional kinetic models of the impurity transport
EFDA Transport Topical Group
Deliverable under Base- Line Support
Reference:
Task Force: TRANSPORT
Area: Particle and impurity transport in standard and advanced tokamak scenarios
Task: WP10-TRA-03-01-xx-01/MEdC/BS
Document:
Task Title: Numerical experiments with low dimensional kinetic models of the
impurity transport
Task Coordinator: Boris Weyssow
Author(s):
Date:
Distribution list:
Content:
Gyorgy Steinbrecher, Association EURATOM-MEdC
17 January 2011
Boris Weyssow (CSU Responsible Officer)
Clemente Angioni (Topical Group Chair)
In the previous stages, the statistical properties of the electrostatic field, generated
with the low dimensional kinetic codes, were studied. It was found, by empirical
numerical simulations, that it is possible to approximate the stochastic electrostatic
field by linear superposition of 2 dimensional generalization of the classical onedimensional self–similar processes. In order to perform test particle transport
simulations in an efficient way a C++ code was written that generate random
electric field with self-similarity and long-range correlation properties. The
theoretical foundations of this algorithm were exposed in Refs. [1,2,3]. The particle
transport induced by a random electrostatic field, generated by this algorithm, with
prescribed statistical properties, extracted from numerical first principle low
dimensional gyrokinetic simulations, was studied by numerical methods. Oversimplified models, were long range temporal correlations are simplified to frozen
turbulence, were studied by the methods of ergodic theory and numerical methods
in ref. [4]. The statistical post processing of the results of the particle transport
simulations was performed by a new class of stable numerical processing method,
elaborated on the basis of the results from ref. [5]. In the framework of this new
class of numerical filtering methods, it is possible to extract information on the
whole spectrum of anomalous diffusion exponents, when the mean square
12
EFDA Transport Topical Group
Deliverable under Base- Line Support
Reference:
Task Force: TRANSPORT
Area: Particle and impurity transport in standard and advanced tokamak scenarios
Task: WP10-TRA-03-01-xx-01/MEdC/BS
displacement of the test particle is represented as
R  (t )
Where
2
2

  t  d ( )
(1)
0
d ( ) is a probability measure.
In eq. (1) the averaging is performed over an ensemble of random
trajectories. The representation (1) is a generalization of the usual asymptotic
large time behaviour in anomalous transport. In the work [5] we proved that
in the cases when we have no apriori information, the reconstruction of the
spectrum of all of the diffusion exponents, given by the nonnegative measure,
is an ill posed, unstable problem. A possibility to stabilize this set of
problems is given by the results in Ref. [5] that proves that it is possible to
reconstruct in weak sense, that means to compute only the mean values of
the probability measure from (1). The results of this analysis on simulated
data shows that the maximal, dominant exponent is not well separated and a
naïve fit give rise to biased determination of the leading exponent. The
dependence of the mean values on the test particle mass was studied.
Revision No: 1
Changes: Y
Written by:
Revised by:
Gyorgy Steinbrecher
Name
Approved by:
F. Spineanu
HRU – MEdC
Association
Objectives
Numerical experiments on the test particle transport, by using the results obtained with first
principle simulations of the plasma turbulence with low-dimensional models.
13
Achievements
New method was elaborated for the study of the particle transport, that allows to study the multi fractal
structure of the anomalous diffusion process. It allows to separate in a stable manner the various
components of the transport process, ranging from ballistic, superdiffusive to subdiffusive components.
Shortcomings
The spectrum of the diffusion exponents in the case of multifractal structure can be estimated by
means of mean values. More exact restriction on this spectrum can be obtained only by deriving more
restrictive stabilizing conditions, like smoothness of the probability measure.
Conclusions and General perspectives
Numerical experiments were performed and the results were processed by new method. This new
method allows a very stable determination of the spectrum of the diffusion exponents, in
contradistinction with the classical methods that find only the lowest and the highest exponent. These
results are useful in the cross check of the results of large-scale simulations of the anomalous transport
in tokamak.
References (to Conference paper/poster, Meeting, Publication)
[1] Steinbrecher G., Weyssow B., Phys. Rev. Lett. 92, 125003 (2004)
[2] G. Steinbrecher, X. Garbet. B. Weyssow. “Large time behaviour in random multiplicative processes”.
ArXiv: 1007.0952v1 [math. PR], (2010)
[3] G. Steinbrecher, B. Weyssow “New representation and generation algorithm for fractional Brownian
motion”, Roumanian Journal of Physics, 55,1120-1130, (2010).
[4] G. Steinbrecher, “Trapping in frozen electrostatic turbulence”, The 15-th International Conference on
Plasma Physics and Applications, CPPA2010, 1-4 July 2010, Iasi, Romania.
[5]. G. Steinbrecher, R. Scorei, V. Cimpoiasu, I. Petrisor “Stable Reconstruction of the T2 Distribution by
Low Resolution NMR Measurement and the Classical Markov and Hausdorff Moment Problem”. Journal
of Magnetic Resonance, 146, 321-334 (2000).
14
WP10-TRA-03-03-xx-01/MEdC
Impact of central electron heating on both electron and ion temperatures, and on particle
and impurity densities.
EFDA Transport Topical Group
Deliverable under Base- Line Support
Reference:
Task Force: Transport
Area: Particle and impurity transport in standard and advanced tokamak scenarios
Task: WP10-TRA-03-03-xx-01/MEdC
Document:
Author(s):
Date:
Distribution list:
Task Title: Impact of central electron heating on both electron and ion temperatures,
and on particle and impurity densities.
Pometescu Nicolae-Ion , MEdC Romania , University of Craiova
14.01.2011
Boris Weyssow (CSU Responsible Officer)
Clemente Angioni (Topical Group Chair)
Content:
In JET, various type of discharges at ITER relevant collisionality had shown that the
density profile of Ni (Z=28) impurity is flattening when ion cyclotron resonance heating
(ICRH) is applied [1]. Influence of ICRH on the impurity density profile was intensively
studied but some dicrepancies between calculated peaking factor and experimental
measurements rest to be explained and clarified.
Theoretical model: In the present study was applied the equation for impurity/ion
density perturbation due to ITG modes in plasmas with radio-frequency heating
obtained in Weiland multi-fluid model with trace impurity approximation in [2].
Experimental data inputs: The profiles for Ni-impurity (Z=28) and electron densities
and temperatures are given from shots 69808 and 68383 in JET [3, 4]. The reference
discharge 69808 is without RF power and the discharge 68383 is with the maximum
ICRF power of 8.3 MW in H Minority Heating.
Milestones:
1. To find the eigenfrequency modes for ITG instability in the two shots;
15
EFDA Transport Topical Group
Deliverable under Base- Line Support
Reference:
Task Force: Transport
Area: Particle and impurity transport in standard and advanced tokamak scenarios
Task: WP10-TRA-03-03-xx-01/MEdC
2.
Evaluation of the diffusivity, convective velocity and density peaking factor in
the two shots; Conclusions about the effects of the RF heating in the specific
heating scheme.
Results:
The leading results obtained in the present study was presented in [5].
The influence of the heating on the density perturbation profile is taken into account
by the specific profiles of electron density and electron /impurity temperature profiles
for the two shots.
1. To find the eigenfrequency modes for ITG instability is a dificult task in general
case because of complexity of the dispersion equation. The simplification
adopted here is to neglect the contribution from trapped electrons. This
restrict the study to the inner region of plasma (0<r/a<0.6). Also, the ion
(Hydrogen) temperature profile is assumed the same with NI-impurity
temperature profile. In this case, from the dispersion equation result four
eigenvalues and between them we choose the most unstable mode to
evaluate the diffusivity, convective velocity and normalized impurity density
peaking factor for Ni-impurity. The growth rate (see figure 1) of this
eigenmode in the presence of RF heating becomes smaller than in the case
without RF heating which means a more 'quiet' ITG instability.
2. The diffusivity in the presence of RF heating (see figure 2) is smaller than
diffusivity in absence of RF heating.
16
EFDA Transport Topical Group
Deliverable under Base- Line Support
Reference:
Task Force: Transport
Area: Particle and impurity transport in standard and advanced tokamak scenarios
Task: WP10-TRA-03-03-xx-01/MEdC
Figure 1: Growth rate of ITG mode in discharge #69808 (full line) and discharge
#68383 (dashed line)
Figure 2: Diffusivity in discharge #69808 (full line) and discharge #68383 (dashed line)
3. The presence of RF heating reduce nearly to zero the magnitude of the convective
velocity in the region with 0<r/a<0.3. Hence, in the central region of the plasma
the transport due to ITG instability is practically suppressed. But for r/a>0.4 the
convective velocity is greater than difffusivity (see figure 3).
17
EFDA Transport Topical Group
Deliverable under Base- Line Support
Reference:
Task Force: Transport
Area: Particle and impurity transport in standard and advanced tokamak scenarios
Task: WP10-TRA-03-03-xx-01/MEdC
Figure 3: Convective velocity in discharge #69808 (full line) and discharge #68383
(dashed line)
4. The density peaking factor (see figure 4) in the shot with RF heating is almost
two times greater than the density peaking factor in the shot without RF heating.
Figure 4: Normalized impurity density peaking factor in discharge #69808 (full line)
and discharge #68383 (dashed line)
References
18
EFDA Transport Topical Group
Deliverable under Base- Line Support
Reference:
Task Force: Transport
Area: Particle and impurity transport in standard and advanced tokamak scenarios
Task: WP10-TRA-03-03-xx-01/MEdC
[1] Puiatti M E, Valisa M, Angioni C, Garzotti L, Mantica P, Mattioli M, Carraro L, Coffey I and Sozzi C, 2006,
Phys. Plasmas, 13, 042501
[2] H Nordman, R Singh, T Fülop, L-G Eriksson and all, „Influence of radio frequency ponderomotive force
on anomalous impurity transport in tokamaks”, Phys. Plasmas 15, 042316 (2008)
[3] Carraro L, Angioni C, Giroud C, Puiatti M E, Valisa M, Buratti P, Buttery R, Coffey I, Garzotti L, Van
Eester D, Lauro Taroni L, Lawson K, Lerche E, Pantica P, Mattioli M and Naulin V, 2007, Proceedings of the
34th EPS Conference on Controlled Fusion and Plasma Physics, Warsaw, July 2-July 6, 2007 ECA Vol 31F, O4.028
[4] Valisa M, Angioni C, Carraro L, Coffey I, Predebon I, Puiatti m E, Lauro Taroni L, Alper B, Belo P, Giroud
C, Van Eester D, Lerche E, Naulin V, Tala T, Tsala M and JET-EFDA contributors, 2009, Proceedings of the
36th EPS Conference on Plasma Physics, Sofia, June 29-July 3, 2009 ECA Vol 33E, P-2.169
[5] N. Pometescu, Diffusion coefficient and convective velocity profiles of Ni-impurity in JET plasmas with
ICRH, poster session at “Theory of Fusion Plasmas, Joint Varenna-Lausanne International Workshop”,
Varenna, Italy, August 30 – September 3, 2010; published in J. Phys.: Conf. Ser. 260 012016 (2010) doi:
10.1088/1742-6596/260/1/012016
Revision No: 1
Changes:
Written by:
Revised by:
Name: N. I. Pometescu
Name
Approved by:
Objectives
In JET, various type of discharges at ITER relevant collisionality had shown that the density profile of Ni
(Z=28) impurity is flattening when ion cyclotron resonance heating (ICRH) is applied. Influence of ICRH
on the impurity density profile was intensively studied but some dicrepancies between calculated
peaking factor and experimental measurements rest to be explained and clarified.
Milestones:
1. To find the eigenfrequency modes for ITG instability in the two shots;
2. Evaluation of the diffusivity, convective velocity and density peaking factor in the two shots;
Conclusions about the effects of the RF heating in the specific heating scheme.
Achievements
19
1. The eigenfrequency modes for ITG instability was found by neglecting the contribution from
trapped electrons, which restrict the study to the inner region of plasma (0<r/a<0.6).
2. For the most unstable mode was evaluated the diffusivity, convective velocity and normalized
impurity density peaking factor for Ni-impurity.
Shortcomings
The objectives are accomplished but the conclusions of the study are restricted by some particular
assumptions. In the next actions some of these will be removed to increase the applicability of the
results.
Conclusions and General perspectives
The work was done by neglecting the trapped electrons, simplification which restricts the conclusions.
The study will continue by taking into account the effects of trapped electrons. Also, the conclusions are
based on the multi-fluid model with trace impurity approximation. Application of other theoretical
model and comparative study is preview.
References (to Conference paper/poster, Meeting, Publication)
N. Pometescu, Diffusion coefficient and convective velocity profiles of Ni-impurity in JET
plasmas with ICRH, poster session at “Theory of Fusion Plasmas, Joint Varenna-Lausanne
International Workshop”, Varenna, Italy, August 30 – September 3, 2010; published Diffusion
coefficient and convective velocity profiles of Ni-impurity in JET plasmas with ICRH N I
Pometescu 2010 J. Phys.: Conf. Ser. 260 012016 , doi: 10.1088/1742-6596/260/1/012016
20
WP10-TRA-05-01-xx-01/MEdC
Effects of filamentary structures on impurity and density transport
EFDA Transport Topical Group
Deliverable under Base- Line Support
Reference:
Topical Group: Transport
Area: Statistical properties of the edge turbulent transport
Task: WP10-TRA-05-01-xx-01/MEdC
Document:
Task Title :
Effects of filamentary structures on impurity and density transport
Task Coordinator B. Weyssow
Author(s):
Name and Association(s) of the Author(s) :
Madalina Vlad, Association EURATOM – MEdC Romania
Date:
Distribution list:
Content:
31, December, 2010
Boris Weyssow (CSU Responsible Officer)
Clemente Angioni (Topical Group Chair)
We have proposed a study of the effects of structures on edge plasma transport
based on test particle approach. A complex 3-dimensional model that takes into
account the nonlinear mechanism produced by parallel motion in toroidal
geometry and a Monte Carlo computer code were developed.
We have recently found a new mechanism of impurity accumulation/loss. It is a
nonlinear affect determined by the ratchet pinch velocity [1-3] in toroidal geometry
due to the poloidal motion of the impurity ions induced by the motion along
magnetic lines and by the flows generated by the moving potential. This poloidal
motion determines a time oscillation of the peaking factor due to the variation of
the confining magnetic field on the magnetic surface.
The work scheduled for 2010 consists of the improvement of the
description of the geometrical aspects and of developing the module for
determining the local transport coefficient and the pinch velocity. The latter
includes radialy moving structures.
21
EFDA Transport Topical Group
Deliverable under Base- Line Support
Reference:
Topical Group: Transport
Area: Statistical properties of the edge turbulent transport
Task: WP10-TRA-05-01-xx-01/MEdC
Scale separation between the fundamental diffusion process and the ion
“macroscopic” trajectory permitted to introduce a two step averaging procedure.
The turbulent advection was first averaged taking into account trajectory
trapping in the structure of the stochastic potential (eddying motion). It was shown
that the complex nonlinear process can be described by determining separately the
transport characteristics of the trapped and free ions. The decorrelation trajectory
method was adapted for these calculations. Realistic models for plasma turbulence
were developed. The toroidal geometry is included in the Eulerian correlation of
the stochastic field, which is calculated from the radial, poloidal and toroidal wave
number spectrum. The model includes space anisotropy, dominant poloidal waves,
potential motion with the diamagnetic velocity and filamentary structures with
radial motion. The latter are represented by supplementary term in the Eulerian
correlation.
The second step in the averaging procedure determines the impurity
density from the “macroscopic” trajectory simulation by a Monte Carlo method.
The “macroscopic” trajectories are determined in toroidal geometry taking into
account the turbulence represented by local diffusion coefficients and pinch
velocities. The module for the transport coefficients provides detailed statistical
information on turbulence effects: the fraction of trapped particles, running (time
dependent) coefficients for the trapped and for the free particles. All are functions
of the magnetic surface and of the toroidal position on it.
Revision No: 1
Changes:
Written by:
Revised by:
Approved by:
Name: Madalina
Name
F. Spineanu
HRU – MEdC
Association
Vlad
22
Objectives
The objective was to study the effects of filamentary structures on ion transport in the frame of a
realistic test particle model.
Achievements
A realistic model for determining impurity density was developed. It takes into account the 3dimensional structure of the magnetic field and the complexity of the turbulence (anisotropy, dominant
poloidal waves, potential motion with the diamagnetic velocity and filamentary structures with radial
motion). A computer code was developed.
Shortcomings
A typical run is of the order of 20 hours on PC due to the complexity of the model. The code has to be
optimized and adapted to parallel computing.
Conclusions and General perspectives
These test particle studies complement the self-consistent simulations developed in several
Associations.
References (to Conference paper/poster, Meeting, Publication)
1. M. Vlad, F. Spineanu, S. Benkadda, “Impurity pinch from a ratchet process”, Physical Reviews Letters
96 (2006) 085001.
2. M. Vlad, F. Spineanu, S. Benkadda, “Collision and plasma rotation effects on ratchet pinch”, Physics
of Plasmas 15 (2008) 032306.
3. M. Vlad, F. Spineanu, S. Benkadda, “Turbulent pinch in non-homogeneous confining magnetic field”,
Plasma Physics Controlled Fusion 50 (2008) 065007.
23
WP10-TRA-05-01-xx-01/MEdC
Study of the turbulent structures and intermittency by stochastic differential equations
EFDA Transport Topical Group
Deliverable under Base- Line Support
Reference:
Task Force: Statistical properties of edge turbulent transport
Area: Turbulent structures and intermittency
Task: WP10-TRA-05-01-xx-01/MEdC/BS
Document:
Task Title: Study of the turbulent structures and intermittency by stochastic
differential equations
Task Coordinator: Boris Weyssow
Author(s):
Date:
Distribution list:
Content:
Gyorgy Steinbrecher, Association EURATOM-MEdC
16 January 2011
Boris Weyssow (CSU Responsible Officer)
Clemente Angioni (Topical Group Chair)
In the previous works from Refs. [1, 2, 3] a new methodology for the study of the
noise –driven intermittency was elaborated. In the Ref. [1], for the study of the
intermittent events in the plasma boundary, by using the mathematical results
from the ref. [4], the class of the reduced models studied in the refs. [5, 6] were
generalized adapted. The intermittency in this class o models is related to the
probability large excursions related to the heavy tail.
The mathematical aspects are related to the problem of stochastic linear
approximation in the study of the stability problem of the non-linear partial
differential equations was studied. Explicit results were obtained in the study of the
wave destabilization in Ref. [1].
The problem studied in this work is the stochastic counterpart of the study of large
time behaviour of the linear deterministic differential equations with time
dependent coefficients, or in other term: parametric excitation. In this category of
deterministic linear system the typical interesting phenomena are: parametric
resonance, parametric stabilization of the inverted pendulum. The analogue of
these effects is studied in the framework of the drift waves driven by stochastic
24
EFDA Transport Topical Group
Deliverable under Base- Line Support
Reference:
Task Force: Statistical properties of edge turbulent transport
Area: Turbulent structures and intermittency
Task: WP10-TRA-05-01-xx-01/MEdC/BS
density gradient
The methods from in Refs. [1-6], for the study of the stability under parametric
stochastic perturbations of the physical systems described by non -linear
differential equations, was used in the study of the equations of the density
gradient drift waves.
The problem studied was the stability under parametric random perturbations of
the drift waves. The main goal is to extend the study from Refs. [1, 2] to the
equations of the drift wave turbulence. In this stage we studied the possibility of
stochastic extension of the simplest model. In the Hasegawa-Mima model of the
density gradient drift waves, a random term was added to the constant driving
density gradient. The random density gradient term was modelled by a spatial and
temporal white noise. The physical origin of this stochastic fluctuation of the
density gradient is related to the turbulence that originates from the plasma core.
The methodology used was the Galerkin discretization of the linearized HasegawaMima equation, which contains a random background term. As a result a system of
classical stochastic differential equation of the Ito type was obtained. By the
methods from Refs. [1-3] from these stochastic evolution equations a set of
deterministic linear differential equations were obtained for fixed time spatial
correlation function of the electrostatic potential. These equations were solved by
discrete Fourier transform. and the evolution of the second order moments of the
random electric field were obtained.
The results are the following:
In analogy to the stochastic Klein-Gordon equation, also in the framework of
stochastic Hasegawa-Mima equation there exists a threshold, depending on the
wave vector, for the random density fluctuation intensity, such that the modes with
non-zero wave vector remains stable below this threshold. This threshold
decreases with |k|.
The mode with k=0 is always destabilized. Exact algebraic method was elaborated
for the study of the destabilization.
25
EFDA Transport Topical Group
Deliverable under Base- Line Support
Reference:
Task Force: Statistical properties of edge turbulent transport
Area: Turbulent structures and intermittency
Task: WP10-TRA-05-01-xx-01/MEdC/BS
This stability pattern is a candidate for the explanation of the long-range spatial
correlations of the electric field.
Revision No: 1
Changes:
Written by:
Revised by:
Gyorgy Steinbrecher
Name
Approved by:
Florin Spineanu
HRU MEdC Association
Objectives
Elaboration of class of stochastic models of the intermittent events of the edge plasma turbulence.
Characterization of the intermittent events and particle transport on the plasma boundary in term of the
singularity structure of the stationary probability density function obtained from the Fokker-Planck
equation associated to the stochastic processes.
Achievements
Previously elaborated stochastic linear stability and intermittency analysis was applied to the linearized
stochastic approximation of the Hasegawa-Mima equation. Numerical method for the study of the
stability of a given wave number mode was elaborated, that does not require long Monte-Carlo
estimations. The existence of stability threshold value parametric noise was established. It was proven
that this threshold decreases to zero for low wave numbers.
Shortcomings
The intermittency was studied in simplified models. In the case of multi- dimensional models the
stochastic effects were modelled by white noise. More realistic noise spectrum was used only in onedimensional models.
26
Conclusions and General perspectives
The stochastic linearized versions of non-linear models of the edge plasma turbulence give a sound
qualitative understanding of the intermittence effects.
More precise result can be obtained by generalization of the previous results from refs. [1,5,6]
References (to Conference paper/poster, Meeting, Publication)
[1] G. Steinbrecher, X. Garbet, “Stochastic Linear Instability Analysis”, International Workshop on
“Hamiltonian Approaches to ITER Physics”, CIRM, Marseille, 2-6 November 2009. http://www.cirm.univmrs.fr/web.ang/liste_rencontre/programmes/AbstractsProgRenc395.pdf
[2] V. N. Kuzovkov, W. von Niessen, “Random walk approach to the analytic solution of random systems
with multiplicative noise-The Anderson localization problem”, Physica A 369 (2006), 251-265.
[3]. X. Garbet, G. Steinbrecher, “On-off intermittency. Exact results”.
Workshop <A Tribute to Radu Balescu>”, Brussels, 6-8 March 2008.
Presented at the “Solvay
[4] Steinbrecher, W. T. Shaw. “Quantile Mechanics”, European Journal of Applied Mathematics, 19, 87,
(2008).
[5] Steinbrecher G., Weyssow B., “Generalized Randomly Amplified Linear System Driven by Gaussian
Noise. Extreme Heavy Tail and Algebraic Correlation Decay in Plasma Turbulence”, Physical Review
Letters 92, 125003 (2004).
[6] G. Steinbrecher, X. Garbet. B. Weyssow. “Large time behavior in random multiplicative processes”.
arXiv:1007.0952v1 [math.PR], (2010)
27
WP10-TRA-05-03-xx-01/MEdC
Test particles simulations for impurities in the edge region with focus on particle and heat
transport
EFDA Transport Topical Group (TTG)
Deliverable under Base- Line Support
Reference:
Task Force: Transport
Area: Statistical properties of edge turbulent transport
Task: WP10-TRA-05-03-xx-01/MEdC
Document:
Task Title : Test particles simulations for impurities in the edge region with
focus on particle and heat transport
Task Coordinator: Boris Weyssow
Author(s):
Date:
Distribution list:
Name and Association(s) of the Author(s): Marian Negrea, Association EuratomMEdC, University of Craiova
31.12.2010
Boris Weyssow (CSU Responsible Officer)
Clemente Angioni (Topical Group Chair)
Content:
We investigated ion and impurity transport in turbulent, possibly anisotropic,
magnetic fields. The turbulent magnetic field is modeled as a correlated stochastic
field, with Gaussian distribution function and prescribed spatial auto-correlation
function, superimposed onto a strong background field. The (running) diffusion
coefficients of ions are determined in the three dimensional environment, by using
two alternative methods, the semi-analytical de-correlation trajectory (DCT)
method, and test-particle simulations. In a first step, the results of the test-particle
simulations are compared to and used to validate the results obtained from the
DCT method. For this purpose, a drift approximation was made in slab geometry,
and a relatively good qualitative agreement between the DCT method and the testparticle simulations was found. In a second step, the ion species He, Be, Ne, and W,
all assumed to be fully ionized, are considered in ITER-like conditions, and the
scaling of their diffusivities is determined with respect to varying levels of
turbulence (varying Kubo number), varying degrees of anisotropy of the turbulent
28
EFDA Transport Topical Group (TTG)
Deliverable under Base- Line Support
Reference:
Task Force: Transport
Area: Statistical properties of edge turbulent transport
Task: WP10-TRA-05-03-xx-01/MEdC
structures, and atomic number. In a third step, the test particle simulations are
repeated without drift approximation, by using directly the Lorentz-force, first in
slab-geometry, in order to assess finite Larmor radius effects, and second in
toroidal geometry, to account for geometric effects. It is found that both effects are
important, most prominently the effects due to toroidal geometry, the diffusivities
are over-estimated in slab geometry by an order of magnitude.
With these restrictions in mind, our results on impurity transport can be
summarized as follows:
(i) In the stochastic magneto-static environment we study diffusion is always of a
normal nature,
(ii) the diffusivities increase with increasing level of turbulence,
(iii) the diffusivities of the impurities scale as (mZi )-1/2 with the ion mass, and
(iv) the anisotropy in the turbulent magnetic field alters the diffusivities.
Revision No: 1
Changes:
Written by:
Revised by:
Name: M. Negrea
Name
Approved by:
F. Spineanu
HRU – MEdC
Association
Objectives
The objective was to do numerical simulation of test particle transport and comparison with theoretical
models.
Achievements
29
We investigated the influence of the magnetic Kubo number, the drift Kubo number and stochastic
anisotropy on the diffusion of ions in turbulent, stochastic magnetic fields, applying the decorrelation
trajectory method (DCT) and using test-particle simulations. It turns out that the stochastic drifts
provide a decorrelation mechanism for the particles from the magnetic lines, and trapping effects are
more pronounced, the stronger the magnetic perturbation and the larger the anisotropy in it is. The
diffusion coefficients yielded by test particle simulations are in qualitative agreement with those
obtained from the DCT method.
Shortcomings
We have not focus also on heat transport. This is left for future work.
Conclusions and General perspectives
We intend to reconsider the transport coefficients in future work, by using toroidal geometry and by
directly integrating the Lorentz force, with aim to validate and extend the results presented here.
References (to Conference paper/poster, Meeting, Publication)
Reference to any document we can store with the report
1. Aspects of the Diffusion of Electrons and Ions in Tokamak Plasma, M. Negrea, I. Petrisor, Dana
Constantinescu, Rom. Journ. Phys., Vol. 55, Nos. 9–10, P. 1013–1024, Bucharest, 2010.
2. On the Diffusion of Impurities in Tokamak Plasma, Marian Negrea, Iulian Petrisor, Heinz Isliker,
Athanasios Vogiannou, Loukas Vlahos, Dana Constantinescu, to be published in Rom. Journ.
Phys., Bucharest, 2011.
30