Theory and Modelling

o~óãçåÇ=hçÅÜ=Eqb`I=bojLhjpF r.koch@fz-juelich.de

A substantial fraction of the theoretical activity of the TEC, closely related to experimental re-search, is also contained in the other chapters of this report addressing the corresponding topics.

The present review covers complementary activities.

Transport and Confinement

Transport processes in magnetically confined plasmas

A new theory is developed, aiming to apply the thermodynamic field theory (TFT) to plasmas in strong non-equilibrium conditions. The work is still in a preliminary stage, limited to analysis of a magnetically confined plasma in the nonlinear Classical and Pfirsch-Schlüter regimes. It has been shown that, when a plasma is in a strong non-equilibrium condition, the linear Classical and the Pfirsch-Schlüter transport is corrected by a universal function. The starting point of the theory is the derivation of the entropy productions for ions and electrons in the neoclassical theory when the drift approximation is properly taken into account. The solutions of the thermodynamic field equations and the gauge-invariant solutions are then obtained. The complete set of ionic and electronic nonlinear transport equations in the Banana and the Plateau regimes have also been obtained. In the nonlinear regime, there are many differences between the form of the gauge-invariant solutions for the Classical and Pfirsch-Schlüter transport and for the Plateau and Banana regimes. The gauge-invariant solutions of the thermodynamic field equations for confined plasmas in a generic n-dimensional thermodynamic space have also been obtained. Specific calculations, where the ITER parameters are taken into account, are at present in progress. Finally, a point that has to be made is that a plasma is subjected to the same fundamental rules which a generic thermodynamic system should obey. This suggests the possibility to formulate a coherent and fundamental formalism gov-erning the plasma state like that govgov-erning the solid or liquid state.

When a plasma, being confined in a rectangular box, is submitted to a magnetic field, two different kinds of effects may be observed: the galvano-magnetic effect in presence of an electric current (Hall and Ettingshausen effects) and thermo-magnetic effects in presence of a thermal gradient (Nerst and Righi-Leduc effects). The problem has been analysed by applying the TFT.

Self-sustained oscillations in a plasma-wall system with strongly inhomogeneous diffusion of charged particles

A simple system with hydrogen plasma confined by a magnetic field parallel to the material wall is considered. The charged particles diffusing out of the plasma recombine on the wall and return into

the plasma volume as neutrals, which are ionised by electrons. It is demonstrated that macroscopic self-sustained oscillations are an intrinsic feature of such a system if the diffusion coefficient of charged particles is strongly inhomogeneous in the plasma. The latter can in particular take place because different types of micro-instabilities are responsible for the transport processes in the core and at the edge in tokamak plasmas. Besides, the edge transport is increased significantly by mag-netic field stochastisation. The results can be of interest for understanding Type III ELMs.

Numerical solution of transport equations for plasmas with transport barriers

An approach is proposed to numerically solve transport equations for plasmas with spontaneously arising and arbitrarily located transport barriers regions, characterised by a strongly reduced transfer of energy. The transport equations are written in a form conserving heat flux and are solved nu-merically by using piecewise exact analytical solutions of linear differential equations. Compared to standard methods, this approach allows reducing significantly the number of iterations (Fig. 1) re-quired to obtain a converged solution with a heat conductivity changing abruptly at a critical tem-perature gradient and using large time steps. Computations for the JET tokamak are done. This work is performed in the framework of the EFDA Task Force on Integrated Tokamak Modelling, Integrated Modelling Project 3: Transport code and discharge evolution.

Fig. 1: Number of iterations needed to obtain a stationary converged solution after different number M of time steps.

Modelling of transport and radial profiles in a tokamak with magnetic field stochastisation

A deliberate stochastisation of the magnetic field in tokamaks with Ergodic Divertors (ED) essen-tially modifies the transport properties at the plasma edge by permitting a radial transfer of particles and heat along stochastic field lines and affecting anomalous transport perpendicular to them. Effec-tive transport coefficients are computed to account for the transport modifications, on the basis of a model previously established, which requires transport coefficients perpendicular to the magnetic

field as input. These are determined by different kinds of micro-instabilities including drift Alfvén, drift resistive and drift resistive ballooning unstable modes, and they are calculated according to models found in the literature. This transport model has been included in the 1.5 D transport code RITM. Simulations of an ohmic discharge in TEXTOR with DED were performed. An increase of the transport coefficients was observed in the edge region, leading to a decrease of density and tem-perature at the edge. A decrease of density is also observed in the experiment, but in a deeper re-gion. This is probably due to the onset of a 2/1 MHD mode. Therefore, comparison with other shots where such a mode does not appear is foreseen.

L-H transition modelling

The effect of boundary conditions at the last closed magnetic surface (LCMS) on the formation of the edge transport barrier (ETB) in tokamaks was investigated by means of one-dimensional trans-port calculations for the radial profiles of the main plasma parameters. For a given heating power the L-H transition can be triggered by increasing the density e-folding length or reducing that of the temperature at the LCMS. Such modifications result in a reduction of convective heat losses from the confined volume, which postpones the H-mode onset. The balance between convective and con-ductive components of the heat flux at the edge is determined by the penetration of neutrals through the scrape-off layer, and particularly by the ratio of the distance that neutrals travel before entering the confined plasma to their mean penetration length. Thus, going to the limiter configuration, where the plasma column is very close to the plasma facing components, or to low density plasmas, where the mean free path of neutrals is increased, leads to the increase of the number of neutrals entering the confined plasma and of the fraction of convective to total losses ξ_cond. Figure 2 shows the varia-tion of the L-H threshold power with ξ_cond. Solid symbols represent the power obtained in transport simulations, circles and squares corresponding to different plasma conditions. Open symbols are from the experimental scaling law for several divertor machines at high density. For a low fraction of convective heat losses both coincide. Above 45% the computed power is much larger than the scaling predictions, which is typical for limiter tokamaks or for low density L-H transitions.

Fig. 2: L-H threshold power versus the fraction of the convective heat losses.

MARFE modelling

Operation close to the density limit is very important for ITER. Therefore, theoretical predictions of the development of thermal instabilities and formation of structures like Multi-Facetted Asymmetric Radiation from the Edge (MARFE) near this limit remain one of the main open questions in toka-mak physics. The previous MARFE model describing the formation of poloidal structures with strongly inhomogeneous plasma parameters has been extended by including a more sophisticated description of plasma-wall interaction. In addition to such processes as recycling of hydrogen trals and release of impurities from the wall, a time dependent description of the wall state and neu-tral influx into the plasma has been included. The emission of the neuneu-trals from the wall is de-scribed by a desorption rate coefficient, characterising the time delay between ion loss and neutral re-emission. The model includes continuity, momentum and energy transfer equations and describes the variation of the plasma parameters, averaged over the neutral penetration depth in the radial di-rection, with the poloidal angle. These equations are coupled with the wall particle balance equa-tion. Rough estimates of the instability growth rate have been obtained using linearised equations.

Results show that (i) the critical density is relatively weakly influenced by the presence of the wall but (ii) the growth rate of the thermal instability and the characteristic time of the MARFE devel-opment significantly change with the wall desorption rate (Fig. 3). In addition, the final stationary MARFE radiation belt becomes, in agreement with experimental observations, more elongated in the poloidal direction when the wall desorption process is taken into consideration.

Fig. 3: Characteristic time of MARFE development versus the wall desorption rate.

Test of the paleo-classical transport model on RTP off-axis EC heated discharges The key hypothesis of the paleo-classical transport model, as developed by J.D. Callen, is that in re-sistive, current-carrying plasmas, the electron guiding centres move with the magnetic field line dif-fusion. Hence, its key ingredients are parallel electron heat conduction and magnetic field difdif-fusion.

Near a low order magnetic surface the parallel equilibration length of Te may be a factor of ~10 smaller than elsewhere, leading to a substantial reduction of the paleo-classical electron heat diffu-sivity. Thus, this model yields, in a natural way, zones of reduced transport near low order magnetic surfaces. The paleo-classical transport however strongly decreases with increasing Te (once in the

collisionless regime); it is estimated that above ~1 keV other, turbulent, transport processes become dominant.

This model has been tested on one of the highlights of the former Dutch tokamak RTP: the stepwise reaction of central Te to changes of the power deposition radius (ρdep) of the localised dominant ECRH heating source. These results could be reproduced very well with an empirical transport model, featuring transport barriers in shallow regions near low order magnetic surfaces (q = 1, 4/3, 3/2, 2, etc.), and high thermal transport elsewhere. Figure 4 shows the simulation of this experimen-tal result with the paleo-classical transport model. Because this model underestimates transport for high Te, a simple sawtooth model has been introduced to limit central Te for plasmas with q(0) < 1.

With this sawtooth model included, the simulation is very good, both qualitatively (sharp transitions between plateaus of nearly constant Te(0)), and quantitatively (yielding a good reproduction of the Te(0) values). Like in the experiment, the sharp transitions of Te(0) in the simulations correspond with the crossing by ρdep of a low order magnetic surface. Further tests, in particular of dynamic scans of ρdep during the discharge, are under way.

Fig. 4: Central electron temperature T_e(0) versus dimensionless ECRH power deposition radius (ρ_dep), as measured in a series of similar discharges in the RTP tokamak: comparison of experiment (blue)

with paleo-classical calculation (orange/red with/without sawtooth model added).

L-H transition and Reynolds stress

Results obtained with probe measurements of flows and zonal flows drive by Reynolds stresses on CASTOR were compared with a turbulence model linked to a one dimensional fluid model describ-ing the electrostatic turbulence and its influence on the background flow. A pseudo-spectral simula-tion code is being developed, in which the turbulence is simulated locally on the basis of the Hase-gawa-Wakatani (HW) equations, completed with magnetic inhomogeneity terms, which describe

the curvature of the magnetic field. The inclusion of these curvature terms in the simulations results in an increased poloidal dependence of the strength of the Reynolds stress and of the diffusion due to the anomalous transport. In the fluid model, the poloidal dependence of the Reynolds stress, is modelled by a fit of numerical results obtained by direct simulation with the turbulence model.

Various sources and sinks like electric currents induced by biasing, Reynolds stress, interaction with neutrals and bulk viscosity are included in the fluid model. Further developments of the turbu-lence code are under investigation: non-periodic boundary conditions, extension to more global ge-ometry and inclusion of a background electric field. With the tools of the simulation code and the recently developed 1D fluid model at hand we are now ready to simulate the influence of the elec-trostatic turbulence on the poloidal velocity in TEXTOR.

A new collisionless Trapped Electron Mode instability in Negative Central Shear plasmas; application to electron Internal Transport Barriers

It was previously shown that the bounce average of the radial components of the trapped electrons and curvature drifts lead to a non-resonant trapped electron mode (TEM) instability in negative magnetic shear ( ) plasmas; the side-bands of the primary oscillation are responsible for unusually large ratios _{e e e}, where _e en-ergy fluxes. This new mechanism is to be contrasted with the resonant instability drive that is asso-ciated with the poloidal components of those drifts for positive magnetic shear. These two comple-mentary mechanisms provide an elegant interpretation of electron internal transport barriers (ITB) in low collisionality negative central shear (NCS) discharges, those being characterised by ex-tremely flat electron temperature profiles inside, and moderately flat profiles outside the steep gra-dient region; the position of the barrier coincides with that of _min. Indeed, the reversed magnetic shear instability explains the anomalous transport in the core, the positive shear instability, the transport at the edge, and the barrier results from both growth rates vanishing when .

Q

q

→0 sˆ

The FULL code has been unable to explain the electron ITB experimental results in the region of negative magnetic shear, leading to postulate that anomalous transport in the core results from tur-bulence spreading. That code, however, is based on the lowest order ballooning representation in which the radial components of the and curvature drifts, and therefore the instability drive for do vanish, being proportional to

gradB

<0

sˆ sinθ. On the more fundamental level, we have also

demonstrated that the solutions of the TEM eigenvalue equation obtained when postulating that the coupling between modes centred on neighbouring rational surfaces is dominant are inconsistent with the prerequisite that the characteristic radial length is larger than the distance between rational surfaces: Coupling due to overlapping, therefore, never is dominant. A major conclusion of our work is that approximations of the basic equations must always be validated by the results they lead to.

Edge modelling and radiation transport

A revised version of the 3D Monte Carlo fluid model EMC3 (IPP Greifswald), coupled to the 3D kinetic neutral code EIRENE (IPP-EIRENE), has been re-installed at FZ Jülich. It was applied to a number of typical TEXTOR-DED 12/4 mode discharge scenarios and compared to dedicated edge diagnostics. The local sensitivity of quantitative 3D results in particular to the plasma equilibrium underlying the magnetic topology in EMC3 has been clearly identified. Various parameter scans have been carried out, such as density scans, and variations of the assumed anomalous cross field

transport coefficients. First steps in extending the code towards impurity transport options (and coupling to the ADAS database) have been carried out.

In order to improve the situation with respect to the underlying MHD equilibrium the EFIT equilib-rium reconstruction code (version from Tore Supra, CEA Cadarache, with iron code model adapted to TEXTOR) has been installed at IPP and magnetic diagnostics have been installed or re-activated at TEXTOR for this purpose.

Typical target heat load patterns as seen in the experiments, and their scaling with the control-parameters for the level of achieved degree of ergodicity, have been satisfactorily reproduced by the 3D edge plasma flow model.

The EIRENE neutral particle code has been further extended, in particular by options facilitating resolution of velocity space effects. These are currently used for post processing of velocity re-solved side-on spectra near the test limiter at TEXTOR. First test runs, on a simplified 1D slab con-figuration, have shown that such computed spectral line shapes are quite sensitive to the ion tem-perature and ion temtem-perature gradients. Similar calculations are presently being undertaken with the full 3D geometry taken into account. By these new options velocity resolved signals can be ob-tained at even lower plasma temperatures, where molecular components (Franck-Condon atoms) start playing a role in the spectra. A new version of the hydrogen molecule collisional radiative model (collaboration with Shinshu University, Nagano), focusing in particular on the energetics of dissociation products, has been installed on the IPP Linux Cluster and is currently being prepared for implementation into the EIRENE code itself.

Radiation transfer (opacity) effects for large, dense divertor plasmas have been investigated by means of the Monte Carlo Code EIRENE. Monte-Carlo radiation transport analysis applications have been carried out for Alcator C-Mod typical model parameters, on the basis of 2D "onion skin model" reconstructed edge plasma flow fields. The experimentally inferred relevance of Lyman line radiation trapping, due to the extremely high densities achievable there, has been confirmed in the model calculations, which show a significant extra ionisation rate in the private flux plasma region, due to the optical pumping effects. Implementation and code verification of Zeeman-splitting into the EIRENE photon gas simulations has been completed. For the Alcator C-Mod model they lead to a 5-10% reduction of opacity effects.

The B2-EIRENE code version as used by the ITER-IT (SOLPS-4.2) team has been further signifi-cantly upgraded in 2005 (to EIRENE-2004). Currently modelling studies with this new tool are car-ried out, jointly with the ITER modellers. In particular low temperature hydrogenic plasma chemis-try effects and neutral gas viscosity in the vacuum system are still being assessed for the ITER di-vertor design. First fully consistent coupling between multifluid plasma transport, neutral particle transport and the non LTE-radiation field have been completed. The results indicate that although the divertor plasma dynamics are significantly modified by radiation trapping effects, the scaling of the basic ITER divertor design criteria (target heat loads, impurity concentration) remain essentially unaffected. Zeeman splitting seems irrelevant for radiation transfer in ITER, despite the large mag-netic field. Due to the high gas densities and large size of the gas cloud the Lyman alpha resonance radiation is trapped to 100%. Electron and ion Stark broadening effects are presently being investi-gated, in collaboration with the University of Marseille.

The EIRENE code has been further extended with options for radiative heat transfer in technical plasma applications (lighting), in collaboration with Philips Research, Aachen. In particular tech-niques for core-saturation have been implemented to eliminate the black-body part of the emission

spectrum in high pressure discharge lamps, this rendering the Monte-Carlo radiation transport simu-lation far more robust.

The EIRENE code has been installed at JET and carefully benchmarked with the local JET NIM-BUS Monte Carlo code. Once this benchmark is completed and full backward compatibility with NIMBUS is achieved, it is planned that EIRENE, being a still actively supported code, will replace the NIMBUS code for most JET applications.

An online tool (http://www.eirene.de/eigen/index.html) for fast analysis of Hydrocarbon catabolism in fusion plasma conditions has been established. The tool provides online graphical presentation of the FZ Jülich Hydrocarbon database, integration of all cross sections into rate coefficients, solution of automatically constructed systems of time dependent rate equations, and determination of S/XB factors for interpretation of spectroscopy on molecular bands resulting from Hydrocarbon plasma

An online tool (http://www.eirene.de/eigen/index.html) for fast analysis of Hydrocarbon catabolism in fusion plasma conditions has been established. The tool provides online graphical presentation of the FZ Jülich Hydrocarbon database, integration of all cross sections into rate coefficients, solution of automatically constructed systems of time dependent rate equations, and determination of S/XB factors for interpretation of spectroscopy on molecular bands resulting from Hydrocarbon plasma