F. Palermo 1 A.Biancalani 1, C.Angioni 1, F.Zonca 2, A.Bottino 1, B.Scott 1, G.D.Conway 1, E.Poli 1 1 Max Planck Institut für Plasmaphysik, Garching, Germany 2 ENEA C. R. Frascati - Via E. Fermi 45, CP 65-00044 Frascati, Italy NumKin, Strasbourg, October 17 th, 2016
Overview Physical context of Geodesic acoustic mode (GAM) Interest in GAM oscillations State-of-the-Art Investigations of GAM dynamics by means of ORB5 Benchmark between ORB5 and GAM linear theory Global gyrokinetic simulations with realistic values The PL-damping mechanism Applications of the PL-mechanism Conclusions and Perspectives
Context: turbulent transport Temperature gradient at the origin of instabilities: ITG, ETG, TEM,TIM Turbulence self-organization Streamers and Turbulence generation Zonal Flow (m,n)=0,0 T 2 T 1 Radial energy turbulent transport Poloidal shear flow localized in the radial direction Diamond P.H. et al 2005 Plasma Phys. Control. Fusion 47 R35 R161 Palermo F. et al 2015 Phys. Plasmas 22 042304
Importance and ubiquity of the zonal flow Zonal Flow in tokamak devices Zonal flows in planetary atmospheres t 1 belt T 2 T 1 t 2 T 1 T 2 t 3 t 4 View of Jupiter from the South Pole Stationary E r with k r Since the Earth s core is believed to be in Zonal Flow is able to shear and distort convective and turbulent cells leading to reduce the transport a turbulent state, it is possible that Zonal flows play a role in the Earth core and in the solar dynamo mechanism.
Oscillations of the zonal flow Geophysical and tokamak zonal flow can experience temporal variations and oscillations caused by several mechanisms Low oscillation frequency due to the KH instability characterized by (m 0, n=0) at ~2 khz Ghizzo A. and Palermo F. 2015 Phys. Plasmas 22 082303 Ghizzo A. and Palermo F. 2015 Phys. Plasmas 22 082304 Geodesic acoustic modes (GAM) due to the curvature effects in tokamak and characterized by (m,n)=0,0 ~20 khz Winsor N. et al 1968 Phys. Plasmas 320 53
GAM in tokamak GAM Zonal Flow oscillations P(m,n)=1,0 E r (m,n)=0,0 Stationary zonal flows suppress turbulence efficiently, GAM oscillations make the action of the zonal flow less effective Miyato N. et al 2004 Phys. Plasmas 11 5557 Curvature effects Energy transfer pathway of GAMs goes from the zonal flow to the pressure perturbation linked to the turbulence Scott B. 2003 Phys. Lett. A 320 53 Interplay between E r and the perpendicular flux tube compression Particles move between different magnetic field lines transporting charge that acts to reverse the E r GAMs can radially propagate the zonal flows as indicated by experimental observations Ido T. et al 2006 Nucl. Fusion 46 512
State-of-the-Art of GAMs 1968-1971 3 self-citations Winsor N. et al (1968) 1971-1997 Fully forgotten after ~ 2000 ~ 390 citations GAM are an important part in the nonlinear turbulent system InteractionTurbulence/ZF/GAM is an important challenge in tokamak physics Linear behaviour is interesting, in order to be able to judge how the turbulence modifies GAM properties Analytical theory of GAMs: well developed in recent years [Zonca et al (2008), Sugama-Watanabe (2006), Gao et al (2008)...] GAM Linear theory is utilized in benchmarking of various gyrokinetic simulation codes Each theory focus on particular condition (FOW, elongation ) and it is valid in a restricted range of parameter values Several aspects also in linear regime have been not considered: temperature, density gradients
Gyrokinetic code ORB5 Particle-in-cell code ORB5 which now includes all extensions made in the NEMORB project The code solves the full-f gyrokinetic Vlasov equation using a particle-in-cell δf method Vlasov equation is coupled to the quasineutrality condition Adiabatic electrons Massively parallelized The time t is normalized to the inverse of ion cyclotron frequency The radial direction is normalized to The potential is normalized with Jolliet S. et al 2007 Comput. Phys. 177 409 Bottino A. et al 2011 Plasma Phys. Control. Fusion 53 124027
GAM theories A detailed gyrokinetic analysis of GAMs with a finite radial wavenumber effect, is utilized in benchmarking ORB5 code. Frequency Landau damping rate from theory Good agreement between theory w/2 order correction and theories with higher order corrections for q<4 At larger values of q, higher order corrections (i.e. accounting for the 4 th harmonic resonance and higher) are necessary for estimating analytically the GAM damping rate.
W/2 order dispersion relation for GAM GAM frequency Explicit formula in which k r ρ i effects can be neglected for frequency (but important for damping rates): Landau damping Sugama H. and Watanabe T.-H. 2006 J. Plasma Phys. 72 825 Sugama H. and Watanabe T.-H. 2008 J. Plasma Phys. 74 139
W/2 order dispersion relation for GAM GAM frequency Explicit formula in which k r ρ i effects can be neglected for frequency (but important for damping rates): Landau damping Sugama H. and Watanabe T.-H. 2006 J. Plasma Phys. 72 825 Sugama H. and Watanabe T.-H. 2008 J. Plasma Phys. 74 139
Benchmark: simulations and theory Sinusoidal potential perturbation that evolves in a linear electrostatic simulations Landau damping rate k r n T q Results of simulations (points) in agreement with the results of the theory (continuous lines) For q=1 all of modes are damped at the same rate For q>1 the high k r modes are damped faster than low k r modes
GAM in tokamak device The damping is strong for low values of the safety factor and becomes weaker for large values of the safety factor Generic q profile ~ 5 ~ 1 It is expected that GAMs are prevalent for typical values of parameters at the edge of tokamaks Quasi stationary zonal flows become dominant in the plasma core
Benchmark: simulations and theory Results of simulations Theoretical analysis The weak dependency of ω on k r allows us to deduce that the phase velocity and suggests accurate measurements for the group velocity We find that the group velocity is approximately two order of magnitude lower than the phase velocity. This can have important implications in the linear transport of energy by GAM oscillations. As a general remark, we find very good agreement between theory and simulations so that the benchmark can be considered successful Palermo F. et al 2016 Phys. Plasmas (to be submitted)
Simulations with realistic ASDEX values GAM behaviour in an equilibrium obtained from realistic values of parameters Parameter values at the edge of Asdex Upgrade for the shot width Principal equilibrium profiles in which GAM evolves Conway G. D. et al 2008 Plasma Phys. Cont. Fusion 50 055009
Gyrokinetic simulations Simulations performed on the basis of the experimental value Potential perturbation with T i =T e Flat q profile Flat density profile Hyperbolic tangent Spatial grid r Two peaks of the electric field associated to localized potential perturbation The oscillations for ak T = 10 are damped faster in time than the oscillations corresponding to ak T = 0
GAM behaviour in nonuniform T profile In the absence of Landau damping, the perturbed electric-field oscillates at each radial position with the local GAM continuum frequency Initial value code in the absence of Landau damping t 1 t 2 GAM frequency profile In the case of a nonuniform temperature profile, due to the frequency variation at different radial positions, the initial perturbation will generate finer radial structures by the phase-mixing mechanism F. Zonca et al 2008 Europhys. Lett, 83, 35001
Phase mixing of GAM The phase-mixing does not imply a dissipation of energy. In fact, the energy linked to the electric field is proportional to Approximation of the profile ω with a straight line in the middle of radial domain 1 -c c With increasing time, energy is increasingly shifted towards high k r numbers
Phase mixing in the simulations ak T =0 ak T =10 We can appreciate for the case ak T =10 the generation of thin structures of E r along the radial direction at t = 50000 Moreover, the amplitude of these electric field structures is smaller than that of the case ak T = 0 Radial electric field profiles at t=50000
Phase mixing in the Fourier space Range for q and for k r for which the theory of S-W is a good approximation Time evolution of k r modes of E r in the plane (k r,t) for ak T = 0 (left) and for ak T 0 (right) For k T =0, we find several oscillations of GAM with the amplitude of k r that decreases in time due to Landau damping. The k r range is constant in time For k T 0, the k r range is shifted in time and amplitude of the modes decreases faster with respect to the case k T = 0 Palermo F. et al 2016 EPL 115 15001 Palermo F. et al 2016 43rd EPS P1.046
The PL-damping mechanism In the presence of a continuum spectrum the energy of radial wave is shifted in time with k r = (k r0 + βt), in the region in which Landau damping is more and more efficient. The damping rate due to the combined action of phase-mixing and Landau (PL) damping is: Phase mixing Landau damping PL-mechanism This is a particular case of a general mechanism, that we have found, and that can be observed whenever the phase-mixing acts in the presence of a damping effect that depends on the wave number k r This mechanism can play a very important role not only in the plasma fusion domain, but also in other physical contexts Palermo F. et al 2016 EPL 115 15001 Biancalani A., Palermo F. et al 2016 Phys. Plasmas (accepted)
Radial monochromatic perturbation By using monochromatic waves we emphasize the difference between the Landau damping and the PL mechanism High constant T Low constant T T i =T e T gradient Equilibrium for the shot r Strong Landau damping PL damping Weak Landau damping Radially monocromatic perturbation Global evolution of monochromatic waves in the (r,t) plane
Analytical and numerical evolution of GAM The analytical evolution is then compared with the results of simulations for several values of ak T and k r Landau damping in agreement with SW expression The over-plotted black line is the analytical envelop of E r for the case ak T = 10 Electric field in the middle of the radial domain as a function of the time for ak T = 0 in which only Landau damping acts (green line) and for ak T = 10 (red line) observed in the simulations is a function of the time
PL-mechanism: time and damping rate The damping rate is calculated at the characteristic time t PL in which the GAM electric field is half of its initial value PL damping rate as a function of the ak T gradient The PL mechanism can play an important role in the suppression of GAM oscillations in the regions characterized by a strong nonuniform temperature profile such as in the H- and I- modes
PL-damping in nonuniform n and q profiles Global evolution of monochromatic waves in the (r, t) plane for k n = 0, k T 0 Global evolution of monochromatic waves in the (r, t) plane for k n 0, k T = 0 Simulations performed with a density gradient different from zero have given results very close to the simulations performed with a flat density profile The gradient of q has a weak influence on the phase-mixing and consequently on γ PL. This aspect has been verified by numerical simulations showing that the main parameter in the PL mechanism is the temperature gradient
The PL damping in L- I- H- regimes L-mode low confinement regime I-mode Improvement low regime H-mode High confinement regime H-mode Similar temperature pedestal H-mode L-mode L-mode is the reference regime in tokamak device characterized by profiles that arise from the heating of the plasma core I-mode with good confinement characteristics with a temperature pedestal while the density profile remains similar to that of the L-mode Ryter, F. et al 1998 P. Phys. and Cont. Fusion 40 725 H-mode with very good confinement provided by a temperature and density pedestal. It is the best regime of interest for ITER Wagner F et al 1982 Phys. Rev. Lett. 49 1408
Physics of GAM in I- H-modes GAM can be studied in particular at the tokamak edge At tokamak edge it is possible to find very large gradient of temperature and density due to several confinement regimes I-mode, H-mode L-mode I-mode H-mode Formation of transport barrier (zonal flow), typically with several centimeters wide Suppression of the turbulence Shot with GAM oscillations in the L-I and I-H transition The temporal sequence is not understood: creation of E r, suppression of turbulence, steepening of density and temperature profiles. Their causal relationship helps to identify the mechanism that triggers the H-mode Cziegler I. et al 2013 Phys. Plasmas 20 055904
Time of drive and Time of damping energy energy background GAM GAM background Growing rate Damping rate Zonca F. and Chen L. 2008 EPL 83 35001 The PL damping rate exceeds the energy transfer rate from the ITG turbulence to the GAM Damping of GAM Amplitude of GAM increases
Estimate of the Time of drive The equation of drive for GAM is based on a preferential three-wave interaction that emerges from the ITG turbulence dynamics. This is in agreement with recent experimental observations on H-L-2A. Isotropic turbulence Zonca F. and Chen L. 2008 EPL 83 35001 Lan T. et al 2008 Plasma Phys. Control. Fusion 51 012001
Parameter values in several regimes Presence of GAM Yes Yes No Table with indicative values of density fluctuations and ratio between temperature and density gradient for the several considered regimes T n Edge T n Edge
Characteristic times as a function of k T L-mode I-mode H-mode ηi = 1 ηi = 5 The times are given in sound unit t s = R 0 /v thi with t s Ω i = 1741.6. For the H mode there is a threshold in k T, above which (Damping of GAM) The competition between the two times opens new possible scenarios close the H-mode transition, such as the intermittent behaviour of the GAM characteristic of the prey/predator dynamics. Moving towards lower values of ηi in the respective ranges, the two red t RB curves are slowly shifted towards higher time values leaving unchanged the results. The PL mechanism is consistent with the observed existence or nonexistence of GAMs in the different confinement regimes
Conclusions and perspectives A New mechanism able to damp very fast GAM oscillation in the presence of a Temperature gradient has been identified This mechanism is due to the combined action of phase mixing and Landau damping. (PL damping) This is a particular case of a general mechanism, that can be observed whenever the phase-mixing acts in the presence of a damping effect that depends on k r Palermo F. et al 2016 EPL 115 15001 The results here discussed represent a useful piece in the complex important jigsaw puzzle of the L-H/L-I transitions The PL mechanism is consistent with the observed existence or nonexistence of GAMs in the different confinement regimes Nonlinear gyrokinetic simulations finalized to investigate the PL- damping in the interaction Turbulence/ZF/GAM are necessary Experimental campaign finalized to find the PL damping effect in the tokamak devices is going to start
Acknowledgements A.Biancalani 1, C.Angioni 1, F.Zonca 2,3, A.Bottino 1, B.Scott 1, G.D.Conway 1, E.Poli 1 P. Manz 1, Z. Qiu 3, R. Bilato 1 and ASDEX Upgrade team and all of you for your attention 1 Max Planck Institut für Plasmaphysik, Garching, Germany 2 ENEA C. R. Frascati - Via E. Fermi 45, CP 65-00044 Frascati, Italy 3 Institute for Fusion Theory and Simulation, Zhejiang University - Hangzhou, PRC Simulations were performed on the IFERC-CSC Helios supercomputer within the framework of the VERIGYRO and the ORBFAST project.