In SCENIC, the equilibrium calculations are fully included in the iterative scheme, and the equilibrium is updated from iteration to iteration. We now want to determine whether this inclusion is important or if it would be sufficient to base the wave and particle iterations, and their saturated solution, on an unchanged equilibrium. The final equilibrium could be constructed, including the hot particle effects in pressure and current, once a converged solution with unchanged equilibrium is obtained. We have repeated the simulations for the LFS case, keeping the equilibrium constant during the complete simulation, but still updating the wave fields with the thermal and hot minority contributions from VENUS. An immediate difference is in the number of iterations needed. Whereas the LFS simulation discussed in the previous sections required twenty-two iterations with a total duration of about two and a half slowing down times, the simulation without the equilibrium evolution had to be iterated thirty times with a total duration of three and a half slowing down times to achieve convergence. But most importantly, the high energy tails of the converged solution are lower, as can be observed in Fig. 12.22(a) Clearly, more minority particles remain thermal and do not contribute to the anisotropic high energy tail. As a result, the effects on the equilibrium are substantially underestimated, as shown in Fig 12.22(b).
We show as an example the values of the ballooning parameter α⊥, which is a factor
of two lower if one omits the equilibrium calculations in the iterative scheme during the simulations. Similar behaviour is visible on all equilibrium relevant quantities: They show the same general features, but the effects due to energetic particles are smaller if we omit the equilibrium evolution in the simulations. As a last example we plot the dielectric tensor element
E
nn, where the colors in Fig. 12.22(c) are to scale with Fig. 12.16(f), allowing for direct comparison. Clearly, the maxima of the dielectric tensor are less localised in poloidal angle if we omit the equilibrium calculations, due to lower pressure anisotropy. To conclude this study, we can state that the equilibrium has to be included in self-consistent computations of ICRH, and has to be part of the self-consistent scheme, as variations of the equilibrium during the simulation alter e.g. the particle orbits, as discussed in Chap. 8.12.6. Inclusion of the equilibrium in the iterative scheme 155 7 2 3 4 5 6 0.9 0 0.3 0.6 log10(E) [eV] energy distribution [au]
with VMEC
32% 39%
without VMEC
(a) Energy distributions f(E)√E. Without the equilibrium evolution in the iterative scheme, the final tails are lower, and more minority particles remain thermal.
(b) The ballooning parameter
α⊥, with the colors to scale with Fig. 12.21(d). The value of this parameter, which is important for global stability, is underestimated by a factor of two if the equilibrium is not updated during the simulation.
(c) The dielectric tensor elementEnn with the colors to scale with
Fig. 12.16(f). The smaller tails have the effect that the dielectric ten- sor maxima are less localised than if the equilibrium is included in the iterative scheme. Moreover, the maxima are closer to the cold resonant layer than in Fig. 12.16(f).
Figure 12.22.: Comparison of the final states for the low field side simulations with and without updating the equilibrium at the end of each iteration. For the simulation without updating, the equilibrium has been re-computed once the simulation including LE- Man and VENUS converged. Clearly, the results are different, with a smaller high energy tail for the case without inclusion of the equilibrium. As a result, equilibrium quantities such as α⊥differ substantially.
13. Conclusions and outlook
13.1. Conclusions
In this thesis, the new code package SCENIC has been designed, tested and applied to relevant cases. The package is the first to include the equilibrium in self-consistent com- putations of wave injection in the Ion Cyclotron Range of Frequencies (ICRF). SCENIC in- cludes the MHD equilibrium code VMEC, the full wave code LEMan and the guiding centre orbit following code VENUS. In all codes, further improvements with respect to other exist- ing numerical tools for ICRF computations have been included. In particular, all codes are capable of full 3D geometries, and include the effects of pressure anisotropy (T⊥ , Tk)
as well as correct treatment of the spatial localisation of the cyclotron resonance. In the equilibrium and wave field computations, this is achieved by using a bi-Maxwellian for the resonant particles, which includes different parallel and perpendicular temperatures, finite orbit widths and the critical magnetic field strength Bc, defining the cyclotron resonance
and allowing for poloidal dependences in pressure and dielectric tensor. In the evolution of the distribution function, represented by following guiding centre orbits, there is no re- striction of an analytical model. Non standard orbits and preferential detrapping, together with the effects of a Doppler broadened resonant layer are included.
As a first application, it was possible to study the effects of shaping and pressure on the toroidal drift frequency of fast particles, which is an important quantity for MHD sta- bility. Here, it was shown how the second radial derivative of the Shafranov shift and the radial derivative of the perpendicular pressure (perpendicular ballooning parameter) influence the drift frequency as a function of pitch angle, demonstrating significant re- duction, or even sign reversal, of the toroidal drift frequency. After extensive testing and starting benchmarking efforts against SELFO, SCENIC has been applied to different Ion Cyclotron Resonance Heating (ICRH) scenarios, including low and high power heating of hydrogen or helium-3 minorities in deuterium background plasmas. By exploring low power cases, the general development of anisotropic pressure has been illustrated. The effects of anisotropy on the dielectric tensor were studied. While the parallel temperature is directly related to the Doppler broadening of the resonant region, the anisotropy (in- creasing perpendicular with constant parallel temperature) can result in a dominant hot dielectric tensor even if the hot particle density is considerably lower than that of the ma- jority. Furthermore, geometric effects (poloidal variations) are important when anisotropy
158 13. Conclusions and outlook
is large. For the same low power simulations, the applicability of the bi-Maxwellian model could be confirmed, by showing both the convergence of the iterative scheme and the difference in the final distribution compared to the non-iterative case. The latter has been achieved in the high power simulations as well, where additionally the effects on the equilibrium could be highlighted. Large local perpendicular pressure affects the shap- ing mainly through enhanced Shafranov shift, and the resulting strong local variations in ballooning parameter and local shear indicate important changes in global stability. More- over, the effect of the RF induced current on the safety factor profile could be quantified. Finally, the RF induced particle pinch could be verified. All of the results shown suggest that the newly developed code package SCENIC can be applied successfully to minority ICRH scenarios, and the inclusion of the equilibrium in the self-consistent computations is a great improvement in the research related to ion cyclotron resonance.