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Tokamak Physics Jan Mlynář 10. Tokamak plasma instabilities I. Overview of instabilities, operational limits, energy principle, kink instability, sawteeth, Neoclassical Tearing Mode, Resistive Wall Mode, Vertical Displacement Event. Introduction.

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## Tokamak Physics Jan Mlynář 10. Tokamak plasma instabilities I

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**Tokamak Physics**• Jan Mlynář • 10. Tokamak plasma instabilities I Overview of instabilities, operational limits, energy principle, kink instability, sawteeth, Neoclassical Tearing Mode, Resistive Wall Mode, Vertical Displacement Event 1: Úvod, opakování**Introduction**Why instabilities?Because tokamaks will always be pushed to the limits - obviously, something has to stop us at some point. Classification of instabilities: Many possibilities,e.g. according to the scale, amplitude, growth time, quantities involved... ...according to the cause: 1. too steep gradients 2. too high current density 3. fast (non-maxwellian) particles 4. accumulation of impurities ...according to the consequences - major (...fatal... i.e. disruptions) - localised (often repetitive, leading to bursts of losses) - microinstabilities (causing turbulences) ...or, according to the physics involved - MHD (prevailing): ideal or resistive - kinetic (fast particles like fishbones) - radiative (due to low temp. or impurities) Tokamak Physics 10: Tokamak plasma instabilities I 2**Operational Limits (Revision)**Current limit Current limit The current limit is due to the Kink instability. Beta limit (Troyon) Density limit (Greenwald density) The density limit is due to the radiation cooling Greenwald density Tokamak Physics 10: Tokamak plasma instabilities I 3**Overview of instabilities**Tokamak instabilities (the most popular that are often met in practice): • Kink instabilities (external, internal) • Neo-classical tearing mode – NTM • Resistive wall mode – RWM, locked mode • Sawteeth • Ballooning • Toroidal Alfvén Eigenmode (TAE) • Fishbones • Edge Localised Mode (ELM) • Vertical Displacement Event (VDE) • Radiative collaps, MARFE • Microinstabilities e.g. due to - Ion / Electron Temperature Gradient driven mode (ITG, ETG) - Trapped Electron Mode (TEM) Tokamak Physics 10: Tokamak plasma instabilities I 4**Radiative Instabilities**• Radiative collaps: • Radiation Cooling Enhanced radiation • either radiative collapse / or kink of the centre due to the current peaking • Multi-faceted Asymmetric Radiation From the Edge (MARFE) • Influx of the recycling particles on the high-field side • Impurity accumulation • Drop in the central temperature current displacement, reversed shear further drop in the central T, double tearing modes possible Tokamak Physics 10: Tokamak plasma instabilities I 5**Pressure instabilities**Looking for stability condition with respect to the interchange mode Cylindrical plasmas: Suydam criterion shear Toroidal plasmas, large aspect ratio, circular cross-section: Mercier criterion • factor • q>1 makes plasma stable against the interchange Necessary but not sufficient condition Tokamak Physics 11: Tokamak plasma instabilities II 6**Ballooning mode**Localised version of the interchange instability (localised to the low-field side, i.e. the bad curvature side of the toroidal field). Dependent on - min B - shear For an optimum q profile: …consistent with the Troyon limit Negative shear balloning stable pressure gradient length Tokamak Physics 11: Tokamak plasma instabilities II 7**Current instabilities**Electric current in the plasma tokamak is prone to current instabilities Ideal MHD :Kink modes (very fast disruption) - internal (q=1) “Kruskal-Shafranov stability condition” q>1 - external (rational surface outside the edge of the plasma) Kink is stable if q(a)>2 or rather 3 Resistive MHD :Tearing modes (slower, may lead to disruption depending on the wall conductivity) rational surfaces inside the plasma i.e. tearing mode correspond to the increase of the width of magnetic islands Tokamak Physics 10: Tokamak plasma instabilities I 8**Looking for Instabilities**Perturbative methods - nonlinear (unstable can result in saturation) - linear: with stable or unstable solution Energy principle Infinitesimal perturbation of the plasma flux tube Can unveil instability but cannot predict characteristic times Infinitesimal perturbation is self-adjoint • Solution is either waves or instabilities (cannot be combined) Tokamak Physics 10: Tokamak plasma instabilities I 9**Energy principle**Solution of Substitute linearised MHD equations… Integrate …. waves instabilities sonic wave (plasma compression) Alfvén compressional wave (fieldline compression) Alfvén torsional wave (fieldline bending) kink (current driven instabilities) ballooning (pressure driven instabilities) Stability condition: marginal stability (normal vector) Tokamak Physics 10: Tokamak plasma instabilities I 10**Kink instability**The whole plasma column moves; i.e. shape and profile dependent Kink mode (plasma edge) Internal kink is unstable only for n=1, m=1 (i.e. q=1) Tokamak Physics 10: Tokamak plasma instabilities I 11**Kink instability**Courtesy J. Wesson Tokamak Physics 10: Tokamak plasma instabilities I 12**Sawteeth**Relaxation instability, regular in time, probably initiated by the internal kink. Central temperature reaches a critical level temperature collapses. Kadomtsev model – cyclic reconnection of the m=1, n=1 mode. Reliable predictive model (including time, amplitude, position of sawteeth)does not yet exist. Inversion radius – radius where temperature does not change during the collapse. It is good to avoid sawteeth as they can among others seed the NTM instability. For example, the hybrid regime avoids q<1 and, therefore, the sawteeth. Tokamak Physics 10: Tokamak plasma instabilities I 13**Sawteeth – Kadomtsev model**SXR data from TCV (inversion radius in red) Tokamak Physics 10: Tokamak plasma instabilities I 14**Sawteeth – another example**Tokamak Physics 10: Tokamak plasma instabilities I 15**Magnetic reconnection**Tokamak Physics 10: Tokamak plasma instabilities I 16**NTM - Neoclassical Tearing Modes**Reconnection of field lines, forming magnetic islands. Simple theory predicts very bad tokamak confinement.A more detail theory predict saturation of the island width and a minimum width (instability seeding is required, e.g. via sawteeth) Magnetic islands flatten the density no bootstrap current in the region increase of size of the island „universal mode“ (does not exist due to nonlinear effects) Tearing parameter: ( rs is the position of island’s separatrix) instability“: (discontinuous 1st derivative) NTM m=2, n=1 Tokamak Physics 10: Tokamak plasma instabilities I 17**NTM - Neoclassical Tearing Modes**O-point X-point r=r2 2pR r pR Rf r=r1 pr 2pr 0 rq 2pR r=r2 r pR Rf r=r1 pr 2pr 0 Toroidal direction rq Poloidal direction A field line has constant helical angle, mx=mq-nf Courtesy of H. Wilson Toroidal direction Poloidal direction Tokamak Physics 10: Tokamak plasma instabilities I 18**NTM - Neoclassical Tearing Modes**Tokamak Physics 10: Tokamak plasma instabilities I 19**Overlap of the magnetic islands**Tokamak Physics 10: Tokamak plasma instabilities I 20**RWM – Resistive Wall Mode**Kink / edge NTM critically depend on the border condition – is plasma surrounded by an “ideal wall”, or is there “no wall” (vacuum) around? Ideal wall: much higher b can be reached in the stable conditions Real wall: the currents induced in the wall would saturate, i.e. Resistive wall can counteract an instability only temporarily. However, when plasma rotates or when there is active feedback the instability can be damped. Plasma rotation – due to electromagnetic friction between plasma perturbation and the wall the rotation slows down. When it stops, the instability (now locked to the wall, therefore “LOCKED MODE”) grows until plasma collapses in a disruption. Tokamak Physics 10: Tokamak plasma instabilities I 21**Mode locking**dB (a.u.) bq PLH(kW) 160 180 200 220 240 260 time (ms) NTM disappears and bq increases Tokamak Physics 10: Tokamak plasma instabilities I 22**Double tearing mode**DTM can appear due to the off-axis current, i.e. in the advanced regimes, during the plasma start-up or as a consequence of large impurity acummulation. Tokamak Physics 10: Tokamak plasma instabilities I 23**RWM – Resistive Wall Mode**Tokamak Physics 10: Tokamak plasma instabilities I 24**Tearing and kink modes**Tokamak Physics 10: Tokamak plasma instabilities I 25**VDE - Vertical Displacement Event**Very fast (inertial time scale) and therefore dangerous disruption, due to vertical forces on an elongated plasma column. Act with a big force on the vessel and the magnetic coils Stabilisation: conducting shell (lower growth rate), active feedback Halo currents - currents flowing between plasma and the wall during a VDE disruption. Tokamak Physics 10: Tokamak plasma instabilities I 26**Disruptions**Tokamak Physics 11: Tokamak plasma instabilities II 27**Disruptions**Tokamak Physics 11: Tokamak plasma instabilities II 28**Edge localised mode**Relaxation, irregular instability localised to the edge – to the H-mode pedestal. Historically, ELMs can be classified into three groups Type I – large, giant. Plasma is close to the Ballooning limit. Frequency increases with the heating power. Type II – grassy. Strongly shaped plasmas (close to the double null), high edge pressure. Type III – small. Frequency decreases with the heating power, worse confinement than in type I. Tokamak Physics 11: Tokamak plasma instabilities II 29**Edge localised mode**Tokamak Physics 11: Tokamak plasma instabilities II 30**Edge localised mode**Tokamak Physics 11: Tokamak plasma instabilities II 31**ELM : model**Ballooning-peeling instability: Pedestal grows up to the ballooning limit (pressure driven instability) bootstrap current builds up Current density increases up to the peeling limit (current driven instability, external kink mode) ELM crash Tokamak Physics 11: Tokamak plasma instabilities II 32**ELM : filaments**Tokamak Physics 11: Tokamak plasma instabilities II 33**Edge localised mode**Tokamak Physics 11: Tokamak plasma instabilities II 34**Troubles with ELMs**Tokamak Physics 11: Tokamak plasma instabilities II 35**Possible ELM control**• ELM mitigation or ELM suppression via mg. field ergodisation • ELM pace making using pellets • Others (e.g. vertical position shaking...) DIII-D: ELM suppression with n=3 Tokamak Physics 11: Tokamak plasma instabilities II 36**ELM control**JET n=1 ELM mitigation using EFCC ASDEX-U pellet pace-making Tokamak Physics 11: Tokamak plasma instabilities II 37**Fishbone instability**Resonance interaction between trapped fast particles (toroidal precession of their banana orbits) and m=1 n=1 mode (inverse of Landau damping) Computer model PLT data: SXR, poloidal field fluctuation, neutron intensity Tokamak Physics 11: Tokamak plasma instabilities II 38**Alfvén eigenmodes (AEM)**Alfvén waves: Mg. field lines oscillation in ideal MHD (‘frozen field’, plasma sets the medium elasticity). Can be destabilised by fast particles (difficult for ITER due to the alpha particles) Tokamak Physics 11: Tokamak plasma instabilities II 39**Toroidal Alfvén Eigenmode**In a cylindric plasma, frequency changes continuously with q, so that all shear waves can be strongly damped. In toroidal geometry, due to B ~ 1/R , gaps exist. In the gap, the wave is only weakly damped Toroidal Alfvén Eigenmode (TAE) Tokamak Physics 11: Tokamak plasma instabilities II 40**Alfvén eigenmodes (AEM)**TAE – Toroidal Alfvén Eigenmode RSAE – Reverse Shear AE EAE – Ellipticity AE (second harmonic of TAE) KTAE – Kinetic TAE (effects of finite Larmor radius, causes resonance energy splitting) NAE – Noncircular triangularity AE (third harmonic of TAE) CLM – core localised mode (low shear version of TAE) BAE – beta induced AE (slow, compressional Alfvén wave) EPM – Energetic particle continuum mode Tokamak Physics 11: Tokamak plasma instabilities II 41**AEM – mode coupling**Tokamak Physics 11: Tokamak plasma instabilities II 42**Alfvén eigenmodes (AEM)**See also http://www.ornl.gov/sci/fed/mhd/lmhd.html Tokamak Physics 11: Tokamak plasma instabilities II 43**Microinstabilities**• Microinstabilities are highly localised instabilities(wavelength comparable to the Larmor radius) that can trigger turbulences and therefore cause the anomalous transport. Disussed are in particular • Electron drift wave instability • Ion temperature gradient (ITG or hi) instability • Electron temperature gradient (ETG or he) instability • Trapped electron mode (TEM), Trapped ion mode (TIM) • trapped particles: the parallel dynamics is less important • dissipative or collisionless • Microtearing instability if electromagnetic phenomena are considered Tokamak Physics 11: Tokamak plasma instabilities II 44**Microinstabilities**• From the 2008 IAEA Fusion Energy Conference (Geneva, Xavier Garbet): • consensus that the ion anomalous transport is due to ITG • growing confidence that the electron anomalous transport is due to TEM and ETG • understanding of transport barriers is not satisfactory • there is no understanding of momentum transport Tokamak Physics 11: Tokamak plasma instabilities II 45

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