1. Congresso del Dipartimento di Fisica Highlights in Physics 2005 11–14 October 2005, Dipartimento di Fisica, Università di Milano Dynamics, heating and transport problems in fusion plasmas F. De Luca *, D. Farina †, A. Jacchia †, H. Maaßberg**, R. Pozzoli*, M. Romé*, and F. Ryter ‡ * Dipartimento di Fisica, Università di Milano †CNR - Istituto di Fisica del Plasma, EURATOM - ENEA - CNR Association, Milano **Max-Planck-Institut für Plasmaphysik, Greifswald, Germany ‡Max-Planck-Institut für Plasmaphysik, Garching, Germany Major radius 2 m Av. plasma radius 0.18 m Magnetic field 2.5 T Modular coils (5 field periods) ECRH, ICRH, NBI, Ohmic heating Left, Isolines of the electron distribution function from the FP computations, in the plane v ||, v ⊥ (normalized to the thermal velocity), in the case of LFS launch. The value N || LFS = 0.34 corresponds to the point of maximal absorbtion along the beam. Also plotted are the level curves of the quasi-linear ECRH diffusion coefficient, D ⊥⊥. The two solid lines delimit the region of trapped particles. The ECRH power density is 10 Wcm −3. Right, The same for the case of HFS launch (N || HFS = -N || LFS ). W7-X Stellarator Left, LFS scenario: two microwave beams of X-mode polarization are launched in the same plane but with opposite signs of N ||. The time schedules of gyrotrons P1 and P2 were chosen to have the total injected power equal to 400 kW. Right, HFS scenario: the same as for LFS, but with O-mode injection. Both beams pass through the plasma without being absorbed, are reflected by the 45˚ corrugated mirrors (which rotate their polarization by 90˚) and re-enter the plasma. For both beams N || HFS  N || LFS. Time evolution of the power of the two microwave beams (the same for both HFS and LFS), and of the loop voltage, U l, obtained for HFS and LFS launch, respectively. The time intervals of pure co- and counter-launch are indicated in the upper plot.. ECCD efficiency, normalized to the value corresponding to a homogeneous magnetic field, as a function of the injected ECRH power, for different values of the electron density: (a) n e = 3 × 10 19 m −3 ; (b) n e = 4 × 10 19 m −3 ; (c) n e = 6 × 10 19 m −3 ; (d) n e = 8 × 10 19 m −3. The open squares and the open triangles correspond to LFS and HFS launch, respectively. ECRH power deposition profile (a) and electron cyclotron driven radial current density profile (b), from ray-tracing computations. The magnetic induction is adjusted to get on-axis deposition in both cases. The toroidal angle of injection at the resonance layer is 19˚, corresponding to the maximum ECCD efficiency. The open squares and the open triangles correspond to LFS and HFS launch, respectively. In these simulations N || HFS = −N || LFS is used. A slightly lower current drive efficiency is obtained, both theoretically and experimentally, in the case of HFS injection, confirming the LFS launch as the preferred mode of operation for ECCD at X2-mode heating, in perspective also for W7- X [1] This tendency is not expected to change at higher ECRH power levels, both for low and high electron densities. The results also confirm the difficulty of finding advanced scenarios with an enhanced ECCD efficiency with ECRH only. [1] M. Romé M., V. Erckmann, H. P. Laqua, H. Maaßberg and N. B. Marushchenko, Plasma Phys. Control. Fusion 45, 783 (2003). The electron cyclotron current drive (ECCD) with extraordinary mode at the second cyclotron harmonic launched from the high-field-side (HFS) has been investigated in the W7-AS Stellarator. Trapped particle effects and quasi-linear effects have been taken into account in linear ray-tracing and quasi-linear bounce-averaged Fokker– Planck (FP) computations of the driven current. The results for both low-field-side (LFS) and HFS injection in W7-AS are compared with the experimental results. W7-AS Stellarator Major radius 5.5 m Av. plasma radius 0.53 m Magnetic field 3 T Modular coils (5 field periods) Plasma pulse length: 30 min with 10 MW ECRH, 10 s with NBI and ICRH Perturbative electron heat transport study in ASDEX-Upgrade & FTU Tokamaks Toroidal field B T = 2.1-2.2 T Plasma current I p ≈ 400 kA q 95 ~ 8 Line Density n e ≈ 2.3 10 19 m -3 Ohmic power P OH ~ 180 kW ECH power P ECH ≈ 800 kW ECH deposition  dep = 0.6 - 0.4 Laser Ablation: Si (4 Hz) ~ 300 kW  dep ≈ 0.9 Cold Pulse propagation speed different in the two sections of the plasma column. But Cold Pulse amplitude increases while approaching rho_dep. Possible explanation: when heat conduction changes in time (and rho) by a factor  we may write T e time evolution as: where the "apparent" time varying source p  ≈ -[  ( ,t) -1] p e0 is proportional to the steady state power! Since  T e increases at each CP p  also increases (more where p e0 is greater). Cold pulses (LBO of Silicon) and MECH–induced temperature perturbations are studied and compared. Steady and perturbative analysis are consistent with critical gradient length driven heat transport: Steady ECH power induces a transition from low to high electron heat transport at the sides of the injection radius. Close to the transition temperature perturbations show a non-linear behaviour consistent with a critical gradient length concept. Electron heat transport in high temperature (multi keV) plasmas is a crucial issue for reactor extrapolation. In a reactor grade plasma, in fact, electron and ion temperature are foreseen to be quite close and energy losses through the electron channel are important. Steady state and “perturbative” heat transport studies allow a quite good theory-experiment comparison [1,2]. The ASDEX-Upgrade Experiment ECH Modulated phase Cold pulses High above threshold Low below threshold Fast Slow MECH Pulse 'propagation' asymmetries at RF switch-on/off Different pulse speeds at RF switch-on /off (Slow at the switch-off) Same behaviour outside rho_dep. (Slow at the switch-on) 'Time of flight' of the heat pulse: data group on different branches. From rho_dep to the core From rho_dep to the boundary Well resolved branches mean: Transport increases outside rho_dep and decreases inside rho_dep at the time of ECH injection. e_transport changes with delay after ECH switch-on/off (~5/6 ms). The change occours in a time shorter than half modulation period i.e. < 15 ms (threshold?) 3/2 n e0 ∂ t T e - . {n e0 [  ( ,t)  e0  T e ]} = p ext +p   Cold pulse propagation Core SLOWSLOW FASTFAST ECH Boundary LBO-Si The FTU experiment: Modulated ECH on I p ramp up (skin effect changes s/q) Theory prediction Above a critical value of |  T e /T e | transport becomes turbulent and strongly increases (T e profiles become “stiff”) Critical value determined by the magnetic structure R Simple “Step-  ” model fits well data in FTU more peaked Step in  HP gives the radial position of critical  T e /T e. Stability parameter s/q associated with the critical  T e /T e from power balance and MHD analysis Tore Supra: from G.T. Hoang et al., Physical Review Letters 87, 125001 (2001) FTU: different methods, similar results Correlation between (  T e /T e ) and the (ETG) stability parameter s/q Steady state and perturbative analysis are consistent with critical gradient length driven electron heat transport. Correlation between critical gradient length and the stability parameter s/q is confirmed; good agreement in different machines with different plasma parameters. Electron heat transport is described by theory! Conclusions [1] A.Jacchia et al. - Nuclear Fusion 45, 40 (2005) [2] S.Cirant et al. - Nuclear Fusion 43, 1384 (2003) Comparison of high-field-side and low-field-side launch ECCD in the W7-AS stellarator Time evolution of the field amplitude A and of the finite time maximum Lyapunov exponent  for different values of the parameter . Hamiltonian chaos arises when  c. Correspondingly, A exhibits an exponential growth followed by nonlinear oscillations. 2D phase space portrait of late time evolution of the particles in an unstable case. The figure on the left represents the solution of the self consistent Vlasov equation. The other two figures represent the solution of the Hamiltonian with N=10 5,  and  central  and  right  The role of dissipation and of radiation damping When a dissipative mechanism is introduced in the system, e.g., radiation damping or frictional effects, a linearly stable system can become unstable and the topological transition can occur. Time evolution of the field amplitude A (left) and of the finite time maximum Lyapunov exponent  (right) for the same value of the parameter  =2>  c corresponding to a linearly stable case, with (  =0.01) and without (  =0) dissipation terms. References [1] T. M. O’Neil, J. Winfrey, and J. H. Malmberg, Phys. Fluids 14, 1204 (1971). [2] H. E. Mynick and A. N. Kaufman, Phys. Fluids 21, 653(1978). [3] J. L. Tennyson, J. D. Meiss, and P. J. Morrison, Physica D 71, 1 (1994). [4] D. Farina, F. Casagrande, U. Colombo, and R. Pozzoli, Phys. Rev. E 49, 1603 (1994). [5] D. Farina and R. Pozzoli, Phys. Rev. E 70, 036407 (2004) 2D phase space portrait of late time evolution of the particles in an unstable case. The figure on the left represents the solution of the self consistent Vlasov equation. The figure on the right represents the solution of the Hamiltonian code. Order-Chaos transition in a paradigmatic plasma system A beam plasma system can be considered a paradigm for the study of plasma-wave interaction and vortex formation in phase space. The analysis is based on a Hamiltonian model which by itself has a broader range of applications ranging from fusion plasmas to free electron laser and interacting stars. The system is initially very close to an equilibrium condition where the field is almost zero, and energy and momentum mostly reside in the particles. It is then followed during its whole evolution, looking for conditions where self-sustained large oscillations of the field are produced. The self-consistent dynamics of the N particles and the electrostatic wave can be described by a time independent Hamiltonian with N degrees of freedom, which depends on a single parameter  where The field amplitude is The corresponding Hamilton equations read Stochasticity of the system and finite time maximum Lyapunov exponent The finite time maximum Lyapunov exponent  measures the evolution of stochasticity of the system: regular dynamics corresponds to     with    while finite asymptotic value indicates occurrence of chaos. Starting from a evanescent perturbation the field amplitude can set to a large value if    c, with  c = 3/2 2/3 ≈ 1.89. This corresponds to a topological transition in 2N dimensional phase space . This corresponds also to a transition to chaos in the system, and formation of holes structures in the corresponding 2D one particle phase space . Conclusions It has been shown that in a generic case of a Hamiltonian system with many degrees of freedom, the transition to a self-consistent nonlinear equilibrium characterized by large amplitude nonlinear oscillations (starting from a perturbed, linearly unstable, equilibrium) is always connected to an increase of stochasticity. This process is related to the evolution of the finite time maximum Lyapunov exponent . It has been also shown that such transition can occur in a otherwise stable system if a weakly dissipative mechanism, as friction, or radiation damping, is taken into account, and that it is marked by a sharp variation of , as in the Hamiltonian case.