3.2 Understanding of plasma characteristics for improved concepts

3.2.1 Analysis of turbulent structures in fusion plasmas

Background and Objectives: Coherent turbulence structures play an important role in the dynamics of turbulence and modify the transport of (passive) scalars and contribute to the preferential concentration of transported particles. In the case of edge turbulence in fusion plasmas, it is important to be able to model the effect of coherent structures on density and electric potential fields. This activity complements software development carried out under activity 1.2.2 and is devoted to theoretical refinements of the DPSD concept, originally applied to hydrodynamic and MHD turbulence, so that it can be used more effectively as a diagnostic under edge plasma conditions.

Work performed in 2012:

(i)  During 2012, simulation results from the ESEL code were analyzed and the one-point turbulence structure tensors were computed. These predicted structure tensors were then compared to the computed edge plasma parameters in order to understand the correlation between them. The analysis of the results in being continued in early 2013 with the aim to develop correlation functions between the one-point turbulence structure tensors and the predicted transport parameters.

3.4 Theory and modelling

3.4.1 Plasma rotation

Background and Objectives: It is proposed to investigate the possibility of ECRH affecting the toroidal rotation in support of the "Study of ECRH induced core toroidal changes" for ASDEX Upgrade. The proposed analytical -numerical tool will be based on a recently developed kinetic and action- space time dependent Fokker-Planck description of RF - plasma interaction. This description involves a time-dependent diffusion tensor (induced by RF) and collisions (at low collisionality regime) which enables one to consider resonant and non-resonant interactions simultaneously. When high frequency RF is present (EC), at the low collisionality regime, it has been found that the distribution function is highly affected in much shorter time scales than the collisional ones. Thus, under these circumstances a complete reformulation of the transport is necessary along with understanding of the apparent (yet to be understood) ECRH induced toroidal rotation. Since the basis of our proposed investigation is purely kinetic, it is expected to reveal possible synergy with the intrinsic (density driven) poloidal rotation: Due to the aforementioned temporal evolution of the EC -electron interaction on the same time scale the distribution (in action space - radial as well as velocity-like) evolves, there is a possibility affecting the radial profile (and the associated gradients). Note that, in this approach, two of the actions are velocity like (roughly and loosely speaking, poloidal and toroidal motion) and one is space-like entity (radial position). The general belief is that changes in the rotation cannot be explained by the increase of the momentum diffusivity. However, this belief is based on the classical approach where the transport equations and coefficients are estimated by expanding (perturbing) a local equilibrium distribution function which is seemingly is only locally and weakly affected by the applied ECRH. This approach is adequate if collisions are the dominant mechanism. In the framework of the new approach the distribution suffers changes as a whole especially near the core where the temperature peaks. Thus, deviations of momentum diffusivity from the conventional one are expected. On the other hand, ions may indirectly affected by the modification of transport coefficients pertinent to the electron-ion collisional interaction. This may indirectly has an impact on the ion distribution function (modified local equilibrium).

Work performed in year 2012 (in co-operation with the Institutes indicated):

(i)  Our preliminary investigation shows that RF waves (in the LH and EC ranges of frequencies) have a measurable effect on the intrinsic rotation. Presented results [54th APS-DPP, TP8-64 (A. Papadopoulos, Y. Kominis, K. Hizanidis and A. K. Ram) and ANNEX 29 show deviations in the average electron perpendicular and parallel momentum from the standard quasilinear approach to the same problem recently reported. As far as the physics argument behind our approach is concerned, the rational goes as follows: When the high frequency EC is present at the plasma core along with collisions (certainly at low collisionality regime) enables us to consider resonant and non-resonant interactions simultaneously. We have found that the local equilibrium distribution function is highly affected in much shorter time scales than the collisional ones and thus, a complete reformulation of the transport becomes necessary. It is expected that this approach may lead to a better understanding of the ECRH induced toroidal rotation. The numerical tools employed so far (explicit pseudo-spectral with adaptive time-step Runge-Kutta) suffer of numerical instabilities. This renders the numerical calculations tedious since the choices of temporal versus action-space discretization is a critical choice for the optimal run duration and the longest time choice. There is a need of an efficient numerical module which enables transformations back and forth between the action-angle space and velocity space necessary for the collisional part of the operator in the Fokker-Planck equation. We conclude that it is highly possible synergy with the intrinsic (density driven) poloidal rotation to be revealed to exist. This is based on our conjecture that, due to the aforementioned temporal evolution of the EC-electron interaction on the same time scale the local equilibrium distribution function evolves, the radial profile (and the associated gradients) could be affected. Note that, in this approach, two of the actions are velocity like (roughly and loosely speaking, poloidal and toroidal motion) and one is space-like entity (radial position). The general belief is that changes in the rotation cannot be explained by the increase of the momentum diffusivity. However, this belief is based on the classical approach where the transport equations and coefficients are estimated by expanding (perturbing) a local equilibrium distribution function which is seemingly only weakly affected by the applied ECRH. This approach is adequate if collisions are the dominant mechanism. In the framework of our approach the distribution suffers changes as a whole especially near the core where the temperature peaks. Thus, deviations of momentum diffusivity from the conventional one are expected. On the other hand, ions may indirectly affected by the modification of transport coefficients pertinent to the electron-ion collisional interaction. This may indirectly has an impact on the ion distribution function (modified local equilibrium).

3.4.2 Turbulence and Anomalous Transport Phenomena

Background and Objectives: This activity addresses several tasks: In a first part of the activity, we study particle and heat transport in turbulent environments, with the use of realistic models of turbulence and in toroidal topologies, and with particular focus on the edge region, impurities, fast particles, and anomalous transport behavior. Several tools are used in parallel in this study: (1) We perform test-particle simulations in turbulent electromagnetic field environments. (2) A code for the solution of the Fokker-Planck equation is developed, as an alternative tool to the test-particle simulations and the Langevin equation, and moreover for comparison to the ETS code. (3) Random walk models and fractional diffusion equations are developed as tools to describe anomalous transport, with particular aim to describe in parallel the combined diffusion in position and velocity space. (4) Eulerian-Lagrangian CFD models for the study of near-wall motion of particulate matter and its interaction with stochastic turbulent fluid flow in the presence of deterministic/stochastic magnetic fields. In a second part of the activity, we explore particular turbulence models that are based on the concept of Self-Organised Criticality (SOC), where the turbulent plasma is considered as a complex system, in which localized instabilities are relaxed in small-scale diffusive events. SOC is usually modeled in the form of Cellular Automata and the sand-pile analogy is made use of, yet it is a particular aim of our approach to use the natural physical variables and that the SOC models have a consistent physical interpretation. In particular, we develop the Extended Cellular Automaton (X-CA) model for the magnetic field, which is fully MHD compatible, as well as a SOC model for turbulence driven by micro instabilities. In parallel, we build the MHD code MYDAS2, as an alternative to the X-CA model and in order to validate results of the latter. Also, gyrokinetic simulations are performed, with the gyrokinetic code GENE, in order to study turbulence driven by micro-instabilities (ITG, TEM, and ETG mode driven instabilities). The last part of this activity utilizes Hamiltonian methods for obtaining evolution equations of the particle distribution functions under the presence of turbulent or other perturbing modes. These equations describe anomalous transport phenomena more accurately than standard theories, such as the quasi-linear theory, since they incorporate the actual underlying particle dynamics and are not based on commonly used statistical assumptions with questionable validity in realistic cases.

Work performed in year 2012 (in co-operation with the Institutes indicated):

(i)  During 2012 we dealt with the question of the spatial homogeneity of turbulence, especially the multi-scale turbulence. Our main conclusions from our scientific literature survey are that the assumption of spatial homogeneity is questionable. This might have paramount impact on the models which are based on averaging out the particle gyromotion, especially for energetic ions. Furthermore the standard corrections based on the so-called finite Larmor effects is not enough to remedy the problem which is basically the breakdown of the gyrocenter picture of particle kinetics. In order to test our main concerns we reformulated the latter on the basis of the kinematics of the transverse evolute: this is the orbit of the projection of the center of curvature on the instantaneous transverse plane, that is, the plane transverse to the local total (equilibrium plus turbulent component) magnetic field. In the expressions obtained, the turbulent electric component associated with the temporal evolution of the magnetic component as well as relativistic effects (though, there are negligible for the ions which are the main focus of our investigation). The generalized formula, thus obtained are amenable to comparison with the gyrocenter’s position and Larmor radius. Late in 2012 we initiated numerical testing of the generalized formula for particles moving in an environment where a homogeneous magnetic field and a force-free turbulent magnetic field (of ABC type) are superimposed. Our preliminary results show that even in chaotic magnetic field situations the ion dynamics, though complicated, is not always chaotic. On the contrary, there is robustness associated with their motion, although fast spatial transition may occur during their motion.

(ii)  A numerical module has been developed in FORTRAN90 for the transport of test dust particles in realistic tokamak plasma environment. The code is robust, flexible, and computationally inexpensive, but it includes the essential physics of modelling dust transport. The dust transport module is coupled with edge plasma transport code B2.5-SOLPS, from which the spatial equilibrium distributions of the plasma and the impurity parameters in tokamaks are obtained. The main physical models implemented as sub-modules in the dust transport module are: (1) equations of motion, (2) charging of dust particles, (3) dust collisions, (4) source of dust particles, (5) dust energy balance and ablation, (6) dust breakup, and (7) basic dust properties. Preliminary simulations of carbon dust transport in the realistic tokamak plasma environment based on the results from the B2.5 -SOLPS code coupled with the newly developed dust particle module indicate that the dust particles are generally very mobile and they can penetrate quite deeply into the plasma core. Finally, destruction of the particle happens in very short time and, thus, large part of grain’s material is delivered to the core plasma as impurities (see ANNEX 30). (in cooperation with UoCyprus, MEdC, WP12-ITM, WP12-IPH)

(iii)  The development of a 2-D MHD fluid code for edge plasma has been initiated. It is based on the numerical solution of the fluid transport equations of Braginskii based on Cartesian coordinate system. The development of the MHD fluid code will be continued next year.

(iv)  For the X-CA SOC model, no work was done in the period in subject, since our alternative MHD modeling tool, the MHD code MYDAS, has reached by now an advanced state, which lessens the need for a SOC model, also not least since there is not much evidence for SOC states at the MHD level in fusion devices (contrary to the kinetic level).

(v)  In order to validate the SOC model for ITG mode driven turbulence, a spectral analysis of the heat-fluxes in radial direction was made, with aim the comparison to published results from gyrokinetic (GC) simulations. Contrary to the GC results, the spectra were found to be flat (white noise), which must be attributed to the one dimensional form of the model, where SOC is known to exhibit only a limited number of its standard properties. This made clear the need to extend the SOC model to 2D, which could not yet have been done in the period in subject. Also, the threshold has been made radially dependent, yielding more realistic profiles at the plasma center. Having given priority to validation of the model, the determination of the transport coefficients, for use in the Fokker Planck code CHET1, and the study of transport barriers have not yet been done (related work on gyrokinetic simulations is reported below) For more details, see ANNEX 31 (in collaboration with Uni. Bayreuth).

(vi)  For the test-particle simulations, an appropriate guiding-centre approximation has been determined and implemented, as well as a collisional term has been added and implemented in the equations of motion (for the Lorentz-force and for the guiding center approximation). For the study of transport in NTM with ECRH, the form of the NTM model has been improved, the determined statistics has been extended to include transport in velocity space, and a criterion has been implemented to select between trapped (banana) and free orbits. So-far, the particles are tracked without ECRH. In parallel, transport in an ETB (pedestal), as modeled by the MHD code MYDAS, has been studied, with so-far very preliminary results (anomalous transport in parallel velocity was found). Preliminary studies of transport in the pedestal have also been done with the gyrokinetic code GENE and are reported below (item ix). The complex problem of electron heat transport under conditions of multi-scale physics could not have been initiated yet. For more details, see ANNEX 32 (in collaboration with, MEdC, Univ. of Craiova).

(vii)     (1) In the Fokker-Planck code CHET, a collision operator for collisions with a Maxwellian background has been implemented and tested successfully. The grid has been modified to impose explicit boundary conditions on the θ= 0, π axis, and thereafter the coordinate system has been changed from spherical to cylindrical. Turning then to quasi-linear wave-particle interaction, (a) the simple test-case of Karney seemed not suited for the pseudo-spectral method, and first trials with the quasi-linear operator showed a need for grid-refinement, so further work on code development is needed (2) No work has been done on using the transport coefficients from the ITG-SOC model. (3) Our basic FP solver has been integrated within the ETS and is being tested, so-far with partial success. Work on the use of and comparison with the Fokker -Planck code RELAX could not yet have been done and will be initiated in 2013. For more details, see ANNEX 33 (In collaboration with Univ. of Bayreuth; ITM IMP3).

(viii)  The fractional transport equation (FTE) was further validated as a tool to model anomalous transport, on the example of the propagation of heat pulses. A parametric study was made and it was found that in order for clear anomalous transport to appear (of ballistic nature), the degree of the fractional derivative must be close to one, else only mildly anomalous phenomena can be modeled. No work has been done on the validation of the FTE approach through test-particle simulations (In collaboration with MEdC, Univ. of Craiova).

(ix)  (1) The MHD code MYDAS was extended from 2 to 3 spatial dimensions, both for cylindrical and toroidal coordinates, and also the energy evolution-equation has been implemented. The resistivity term was generalized to allow for spatially dependent resistivity, a constant viscosity term was introduced, and the equations were normalized. The full 3D, toroidal, visco-resistive, non-linear code MYDAS has been successfully verified with the Method of Manufactured Solutions, and a second verification through code comparison has been initiated. An analytical, 11-free-parameter solution to the Grad-Shafranov equation was implemented that allows to determine various equilibria as initial conditions, and an interface has been developed to import equilibria from the code CHEASE, with which a first trial has been made to dynamically evolve an ETB in global MHD (NTM activity and pedestal formation could not yet have been addressed). For more details, see ANNEX 34. (2) With the gyro-kinetic code GENE, a pure ITG mode case was set up, and the statistics of the heat fluxes has been determined, with aim the comparison to the SOC model for ITG mode driven turbulence. So-far histograms from heat flux time-series have been determined, for varying driving temperature gradient. The histograms show a clear formation of a tail. Also, some first runs have been made for studying transport in an ETB, with only very preliminary results, so-far. Due to the complexity of the problem, the investigation of electron heat transport under conditions of multi-scale physics has not been initiated, yet. For more details, see ANNEX 35 (In collaboration with IPP Garching.)

3.4.3 Discrete kinetic and stochastic models for transport

Background and Objectives: Discrete kinetic simulations may provide an alternative mesoscale type of description of the transport properties of complex physical systems. The main unknown is a particle distribution function, which obeys a suitably chosen kinetic equation, while the bulk quantities of practical interest are obtained by taking moments of the distribution function. The benefit of this description, compared to the more traditional micro- and macro-scopic approaches, is due to the fact that physics, at particle level, may be investigated with moderate computational effort. This activity aims to develop kinetic computer codes capable to simulate in a computationally efficient manner non-linear resistive MHD flows and vacuum systems in fusion devices.

Work performed in year 2012 (in cooperation with KIT, CEA):

(i)  The conductance and the mass flow rate through pipes and piping elements has been continued through pipes of various lengths and cross sections as well as through diverging and converging pipe elements (see ANNEX 36) for various pressure ratios and in a wide range of the reference Knudsen number. In the case of low speed flows linearized kinetic models, such as the BGK, the Shakhov and the Ellipsoidal models are implemented, while in high speed flows results are based mainly on the Direct Simulation Monte Carlo (DSMC) method. In some cases of high speed flows nonlinear kinetic model equations are also used in the simulations. In general these computations are highly demanding and have been performed at the HPC-FF facility at Julich. The developed data base has been sufficiently enlarged in order to provide the required information for the implementation of integrated algorithms for simulating gas distribution systems of any complexity and under any vacuum conditions as well as vacuum devices such as vacuum sensors and pumps. Based on these data simulation of the performance and optimization of the Holweck vacuum pump has been performed (see ANNEX 37).

(ii)  The in-house algorithm for simulating in order to be effective in the simulation of gas piping distribution systems of any complexity has been extended to include pipes of any cross section and length. This has been achieved by developing the required kinetic data base providing the conductance of single pipes and integrating this data base into the algorithm. In particular, for channels with L/Dh > 50 , where L is the length and Dh the hydraulic diameter, the channel is considered as long and the available kinetic conductance results based on the theory of the infinite long channels are applied. For channels of moderate length 10 ≤ L/Dh ≤ 50 the end correction theory is introduced. This theory has been recently successfully implemented to define the fictitious increment length which must be added to the channel length in order to provide accurate results for the conductance by taking into account the channel end effects. Thus, the kinetic data base has been enriched with the values of the increment length in terms of the gas rarefaction. Finally, for short channels, i.e., L/Dh <10 the above theory is not valid and extensive simulations based on non-linear kinetic theory have been executed to provide a complete set of results for the channel conductance in terms of gas rarefaction, pressure difference and channel length. To demonstrate the effectiveness of the algorithm, a similar geometry to the one of the ITER lower port region has been simulated (see ANNEX 20).

(iii)  In various positions in DT fusion reactors, such as at the inlet of the vacuum pumping system, there is rarefied gas flow under the influence of magnetic field. In order to simulate this type of gas flow, the BGK-Boltzmann equation has been implemented with the use of the discrete velocity method. The influence of the magnetic field on the distribution function has been incorporated by a proper modification of the equilibrium distribution function. For the limiting case of continuum fluid, the implementation of the new equilibrium distribution yields the well-known MHD equations. In addition a recently introduced vectorized kinetic type approach has been used for the induction equation. The divergence-free condition for the magnetic field is theoretically guaranteed. A numerical algorithm has been developed and some preliminary results for simple flows such as the Hartmann flow have been recovered. These results will be used for the validation of both the theoretical approach followed and the developed numerical code.

3.4.4 2-D MHD code for plasma trapping in compact magnetic fusion devices, and development of specific modules to calculate rate equations of collisions of the particles for a 2-fluid, 1-D MHD code for Tokamak plasma. Development of an alternative source and neutralizer techniques aiming at increase efficiency of neutral beam heating systems. [Task Agreement HCD-02-02: Neutral Beam Advanced Technologies]

Background and Objectives: The development during the last few years of the 2-D MHD resistive numerical code describing high density and high temperature plasmas in an external applied magnetic field enables to study the operation of a Compact Magnetic Fusion (CMF) device in an open magnetic field topology. The results of these studies concern the optimization of the CMF for neutron production under different initial spatial configurations for both the plasma density distribution in the device and the topology of the external applied mirror-like magnetic field. The different initial spatial plasma density distribution was obtained by introducing nonlinear optical interactions of an ultra-short laser beam with deutereted clusters (see previous annual reports and related annexes). These investigations are common for the CMF and the negative ion production and extraction. The limitations on the maximum value of the applied magnetic field and the plasma trapping time due to losses of the open topology lead us to modify the magnetic field topology in order to improve the plasma trapping introducing a new mode of operation of the CMF. The present investigations concern (a) the introduction, in the code, of an azimuthal current which will modify the local topology of the magnetic field, increase the trapping time of the plasma in the device and improve the neutron flux and (b) continue our investigation on the production acceleration and extraction of a high energy and high current negative ion beam from a Magnetically Insulated Diode (MID) and the neutralization by using laser techniques. The know-how of our team and our collaborators on the generation of high pulsed magnetic fields will be used for the case of the MID and the extraction of the negative ion beam.

Work performed in year 2012:

(i)  The neutron production depends mainly on both the square of the local plasma density and the trapping time in the CMF. In our case the trapping time is limited by the plasma losses near the magnetic mirrors of the device. The results of our collaboration with the teams from the France Laboratory (Ecole Polytechnique) show that there is an upper limit on the value of the external applied magnetic field in the mirror-like topology which is up to 120 Tesla. Higher magnetic fields modify the structure and destroy the coils. In order to optimize the magnetic field topology in the device and increase the trapping time an azimuthal current was introduced in the code which confines the plasma inside a magnetic configuration produced by the superposition of the external applied magnetic field in the mirror-like configuration and the field produced by the introduced azimuthal current. The proposed new configuration is similar to the ASTRON Machine. The current will be generated by a number of coils covering the distance between the mirror-like coils. The induced current in the plasma of the CMF device produced by a fast variation of a relatively lower magnetic field, compared to this produced by the mirror-like coils. The value of the induced current and the pulse duration will allow the numerical study of the spatio-temporal evolution of the final magnetic topology in the device and the plasma trapping in order to compare with the previous studies. The first results show that the trapping time increases by a factor of 20 .

(ii)  The main investigation for this period was the development of a 2-fluid, 1-D code in cylindrical geometry in order to describe the operation of a Magnetically Insulated Diode (MID). Recent investigations (last November) performed in collaboration with the team from the University of Bordeaux using the CELIA laser facility allow to confirm the production of high density negative ions from ultra-short laser beam interaction with clusters for relatively low laser beam energy up to 200 mJ.

(iii)  The code simulates the MID operation based on:
(1) the production of a high density plasma on the cathode by ultra-short laser beam interaction with high density gaze clusters,
(2) the produced plasma on the cathode is capable to generate negative ions,
(3) the application of a high voltage between the anode and the cathode produced by a pulsed power device,
(4) the generation and application of a externally applied magnetic field in order to avoid the closure of the space between the cathode and the anode by the electron current and trapping the electrons near to the cathode surface, and
(5) the negative ion current closure the cathode-anode and produce a high energy and high current beam.
The 2-fluides in the code correspond to the electrons and negative ions population for different initial physical parameters concerning the density and the spatial distribution in the MID (see ANNEX 38).

(iv)  The code describes the acceleration and the extraction of the negative ion current as a function of (a) different physical and geometrical values of the MID and (b) the external magnetic field. The initial values used for the code are: the negative ions density on the cathode is 10¹² cm^-3, the applied accelerating voltage 1.5 MeV, the cathode surface 25 cm², the distance between anode and cathode 3 cm and the external applied magnetic field varies from 1 T to 10 T. The results for the extracted negative ions current is 200 A with a kinetic energy of 1.5 MeV for a magnetic field up to 2 T. The pulse duration of the extracted current beam is 15-20 nsec and the power of the beam is 0.3 GW. The calculation for a photon-neutralizer applied for the above negative ion beam show that a laser beam with 20-22 J and 1.05 μm wavelength enables to neutralize the 95% of the initial beam and produce a high power neutral beam for Tokamak applications. Next improvements will concern investigations on both the increase of the extracted negative ion current and the pulse duration (see ANNEX 38).