Lecture9: Plasma Physics 1 APPH E6101x Columbia University
Last Lecture Force balance (equilibrium) in a magnetized plasma Z-pinch θ-pinch screw-pinch (straight tokamak) Grad-Shafranov Equation - Conservation principles in magnetized plasma ( frozen-in and conservation of particles/flux tubes) - Alfvén waves (without plasma pressure)
MHD ( ) n t + (nu) = 0 ρ m v m t = j B p + ρ m g E + v m B = ηj + 1 ne (j B p e) plus magnetostatics
http://www.sandia.gov/z-machine/
http://www.sandia.gov/z-machine/
PHYSICS OF PLASMAS 22, 021805 (2015) Modelling of edge localised modes and edge localised mode control G. T. A. Huijsmans,1 C. S. Chang,2 N. Ferraro,3 L. Sugiyama,4 F. Waelbroeck,5 X. Q. Xu,6 A. Loarte,1 and S. Futatani7 1 ITER Organization, Route de Vinon sur Verdon, 13067 Saint Paul Lez Durance, France Princeton Plasma Physics Laboratory, Princeton University, Princeton, New Jersey 08543, USA 3 General Atomics, P.O. Box 85608, San Diego, California 92186-5608, USA 4 Laboratory for Nuclear Science, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139-4307, USA 5 Institute for Fusion Studies, University of Texas at Austin, Austin, Texas 78712, USA 6 Lawrence Livermore National Laboratory, Livermore, California 94551, USA 7 021805-6de Lyon, Huijsmans et al.! 021805-8 Huijsmans et al. Ecole Centrale 69130 Ecully, Lyon, France 2 Phys. Plasmas 22, 021805 (2015) points are 7formed leading to multiple stripes in the power (Received 6 August 2014; accepted 3 December 2014; published online January 2015) The formation of magnetic tang deposition profile during the ELM. Figure 4 showsturbation a of the ballooning mod Edge Localised Modes (ELMs) in ITER Q ¼ 10 H-mode plasmas are plot likely to magnetic lead to large tran- during an ELM Poincare of the field structure from inside the original separatr 60 sient heat loads to the divertor. To avoid an ELM induced reduction ofinthe divertor lifetime, simulation an ITER Q ¼ 10 scenario.the ing ELM energy losses depend With the formationusing of filaments and magnetic tangles, large ELM energy losses need to be controlled. In ITER, ELM control is foreseen magnetic and the lifetime of the tang one can identity different loss mechanisms for the thermal avalanche-like process may a field perturbations created by in-vessel coils and the injection of small D2 pellets. ITER plasmas energy during the ELM. The ejection of filaments, due to energy the losses. In this case, the are characterised by low collisionality at a high density (high fraction of the Greenwald density convective motion of a ballooning mode corresponds to conthe pedestal leads to large gradie limit). These parameters cannot simultaneously be achieved vective in current Therefore, energyexperiments. losses of density and energy. The losses from ising additional MHD instabili the extrapolation of the ELM properties and the requirements the forfilaments, ELM control in ITER onare both in parallel losses once outsiderelies plasma, region by the ELMs and movin the development of validated physics models and numerical simulations. In thisand paper, direction along the filament due towe the radial movement dients occur further inwards. T the filament into of thethis first paper wall. The magnetic tangles and observed in simulations, leading describe the modelling of ELMs and ELM control methods inofiter. The aim is not the stochasticity cause a direct connection of field lines from areas. However, this is mostly o a complete review on the subject of ELM and ELM control modelling but rather to describe the inside the plasma to the divertor (or the first wall). This leads tivity simulations, raising the q current status and discuss open issues. [http://dx.doi.org/10.1063/1.4905231] I. INTRODUCTION The ITER scenario to obtain an energy amplification Q ¼ 10 is based on the well-established ELMy H-mode. In this scenario about 20 40% of the energy losses across the plasma boundary are due to Edge Localised Modes (ELMs). ELMs have been identified as MHD instabilities destabilised by the large pressure gradients and the associated bootstrap current in the H-mode pedestal. Each ELM can expel a significant fraction of the plasma energy and density on a typi- to large parallel conductive losses causing mostly a reduction vant also for realistic values of t in the pedestal temperature in the simulations. The relative For more realistic simulatio to include more dive importance of these two loss channels may account for necessary the MHD codes. This includes, for experimental classification of ELMs into so-called convecindependent of the ELM frequency, an increase in the fre62 ary conditions. In steady state, tive and conductive ELMs. Convective ELMs, characterquencyised leads smaller density losses losses per ELM and a temperature reduction in link the convective conditions by to significant and small target the peak heat loads. Anamplitude increaseelms in ELM frequency can bethrough a constant shea changes, are small at high density where PIC simulations76 in 1D show the by parallel conduction is low. Conductive ELMs, large achieved providing an external trigger for the with ELM, like, 4 phase, energy losses and densityoflosses comparable to thepellets convecor csh can vary significant for example, the injection frozen hydrogenic comparisons of 1D fluid simul tive ELMs, occur at low density. The minimum ELM density the application of fast variations ( kicks ) vertical position solutions indicate that the error loss (for both 5 convective and conductive ELMs) would then A reduction of the ELM energy loss to fluxes at the divertor ta of the beplasma. ion heat determined by the ejected filaments. If the conducted 0.7 MJlosses in ITER will related require an magnetic increasetangles, of theoneelm fre- csh in the fluid appr a constant are indeed to the would addition to the divertor boundar correlation withover the amplitude (andelm duration) of the quencyexpect by aafactor of 30 the natural frequency. gime may magneticofperturbation the ELM. This remains to beisverithe estimate the naturalofelm frequency in ITER based also have a strong i
This Lecture - Force balance (equilibrium) in a magnetized plasma - Z-pinch - θ-pinch - screw-pinch (straight tokamak) - Grad-Shafranov Equation Conservation principles in magnetized plasma ( frozen-in and conservation of particles/flux tubes) Alfvén waves (without plasma pressure)
MHD ( ) n t + (nu) = 0 ρ m v m t = j B p + ρ m g E + v m B = ηj + 1 ne (j B p e) plus magnetostatics
Frozen-in Flux The plasma moves along with the magnetic field }{ or The plasma within flux tubes remains invariant B ( = (v m B). }{{}) t (v m B) = (B )v m (v m )B + v m ( B) }{{} v m = 1 ρ m ( ρm ( ) db dt t + (v m )ρ m ) =0 = 1 dρ m ρ m dt = (B )v m + B ρ m dρ m dt (Ohm s Law & Faraday s Law) B( v m ) d dt ( B ρ m ) = ( ) B v m ρ m be considered as the number
Frozen-in Flux ( ) d dt ( B ρ m ) = ( ) B v m ρ m be considered as the number
Alfvén Waves x B B 0 B 1 v phase z vphase B 0 z B 1
FIG. 1. (Color) Theoretical patterns of one component, B y, of the Alfvén wave in the kinetic and inertial regimes. The waves propagate from left to right.
Low-Frequency MHD Dynamics B = µ 0 j ρ m v m t = j B B t = (B )v m (v m )B B( v m )
Linearize B = B 0 + B 1 v m = v 0 + v 1. B B 0 B 1 v phase z = v 1 ρ m t B 1 t = 1 µ 0 ( B 1 ) B 0 = (B 0 )v 1.
Linear Plane Waves
Linear Plane Waves
Shear Alfvén Waves B B 0 B 1 v phase z
Compressional Alfvén Waves x B 0 B = µ 0 j vphase z B 1 ρ m v m t = j B B t = (B )v m (v m )B B( v m )
Next Lecture Chapter 6: Plasma Waves