1. Fyzika tokamaků1: Úvod, opakování1 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

2. Tokamak Physics210: Tokamak plasma instabilities I Why instabilities? Because tokamaks will always be pushed to the limits - obviously, something has to stop us at some point. Introduction 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)

3. Tokamak Physics310: Tokamak plasma instabilities I Current limit Operational Limits (Revision) 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  Current limit Greenwald density

4. Tokamak Physics410: Tokamak plasma instabilities I Tokamak instabilities (the most popular that are often met in practice): Overview of instabilities 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)

5. Tokamak Physics510: Tokamak plasma instabilities I 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

6. Tokamak Physics611: Tokamak plasma instabilities II Pressure instabilities Cylindrical plasmas: Suydam criterion Toroidal plasmas, large aspect ratio, circular cross-section: Mercier criterion Necessary but not sufficient condition Looking for stability condition with respect to the interchange mode shear  factor  q>1 makes plasma stable against the interchange

7. Tokamak Physics711: Tokamak plasma instabilities II Localised version of the interchange instability (localised to the low- field side, i.e. the bad curvature side of the toroidal field). Ballooning mode Dependent on - min B - shear For an optimum q profile: …consistent with the Troyon limit Negative shear  balloning stable pressure gradient length

8. Tokamak Physics810: Tokamak plasma instabilities I Electric current in the plasma  tokamak is prone to current instabilities 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

9. Tokamak Physics910: Tokamak plasma instabilities I Perturbative methods - nonlinear (unstable can result in saturation) - linear: with stable or unstable solution Looking for Instabilities Energy principle Infinitesimal perturbation of the plasma flux tube Can unveil instability but cannot predict characteristic times is self-adjoint  Solution is either waves or instabilities (cannot be combined) Infinitesimal perturbation

10. Tokamak Physics1010: Tokamak plasma instabilities I Energy principle Stability condition: Solution of Substitute linearised MHD equations… Integrate …. (normal vector) sonic wave (plasma compression) Alfvén compressional wave (fieldline compression) Alfvén torsional wave (fieldline bending) kink (current driven instabilities) ballooning (pressure driven instabilities) waves instabilities marginal stability

11. Tokamak Physics1110: Tokamak plasma instabilities I The whole plasma column moves; i.e. shape and profile dependent Kink instability Internal kink is unstable only for n=1, m=1 (i.e. q=1 ) Kink mode (plasma edge)

12. Tokamak Physics1210: Tokamak plasma instabilities I Kink instability Courtesy J. Wesson

13. Tokamak Physics1310: Tokamak plasma instabilities I 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.

14. Tokamak Physics1410: Tokamak plasma instabilities I Sawteeth – Kadomtsev model SXR data from TCV (inversion radius in red)

15. Tokamak Physics1510: Tokamak plasma instabilities I Sawteeth – another example

16. Tokamak Physics1610: Tokamak plasma instabilities I Magnetic reconnection

17. Tokamak Physics1710: Tokamak plasma instabilities I 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) NTM m=2, n=1 Tearing parameter: ( r s is the position of island’s separatrix) instability“: (discontinuous 1st derivative)

18. Tokamak Physics1810: Tokamak plasma instabilities I NTM - Neoclassical Tearing Modes r r=r 1 r=r 2 rr RR 0 rr 2r2r RR 2R2R X-point O-point Poloidal direction Toroidal direction r r=r 1 r=r 2 rr RR 0 rr 2r2r RR 2R2R Poloidal direction Toroidal direction A field line has constant helical angle, m  =m  -n  Courtesy of H. Wilson

19. Tokamak Physics1910: Tokamak plasma instabilities I NTM - Neoclassical Tearing Modes

20. Tokamak Physics2010: Tokamak plasma instabilities I Overlap of the magnetic islands

21. Tokamak Physics2110: Tokamak plasma instabilities I 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  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.

22. Tokamak Physics2210: Tokamak plasma instabilities I Mode locking 160 180 200 220 240 260 time (ms)  B (a.u.)  P LH (kW) NTM disappears and   increases

23. Tokamak Physics2310: Tokamak plasma instabilities I 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.

24. Tokamak Physics2410: Tokamak plasma instabilities I RWM – Resistive Wall Mode

25. Tokamak Physics2510: Tokamak plasma instabilities I Tearing and kink modes

26. Tokamak Physics2610: Tokamak plasma instabilities I 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.

27. Tokamak Physics2711: Tokamak plasma instabilities II Disruptions

28. Tokamak Physics2811: Tokamak plasma instabilities II Disruptions

29. Tokamak Physics2911: Tokamak plasma instabilities II 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.

30. Tokamak Physics3011: Tokamak plasma instabilities II Edge localised mode

31. Tokamak Physics3111: Tokamak plasma instabilities II Edge localised mode

32. Tokamak Physics3211: Tokamak plasma instabilities II 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

33. Tokamak Physics3311: Tokamak plasma instabilities II ELM : filaments

34. Tokamak Physics3411: Tokamak plasma instabilities II Edge localised mode

35. Tokamak Physics3511: Tokamak plasma instabilities II Troubles with ELMs

36. Tokamak Physics3611: Tokamak plasma instabilities II Possible ELM control 1)ELM mitigation or ELM suppression via mg. field ergodisation 2)ELM pace making using pellets 3)Others (e.g. vertical position shaking...) DIII-D: ELM suppression with n=3

37. Tokamak Physics3711: Tokamak plasma instabilities II ELM control JET n=1 ELM mitigation using EFCC ASDEX-U pellet pace-making

38. Tokamak Physics3811: Tokamak plasma instabilities II 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

39. Tokamak Physics3911: Tokamak plasma instabilities II 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) Alfvén eigenmodes (AEM)

40. Tokamak Physics4011: Tokamak plasma instabilities II Toroidal Alfvén Eigenmode In the gap, the wave is only weakly damped  Toroidal Alfvén Eigenmode (TAE) 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.

41. Tokamak Physics4111: Tokamak plasma instabilities II 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

42. Tokamak Physics4211: Tokamak plasma instabilities II AEM – mode coupling

43. Tokamak Physics4311: Tokamak plasma instabilities II Alfvén eigenmodes (AEM) See also http://www.ornl.gov/sci/fed/mhd/lmhd.htmlhttp://www.ornl.gov/sci/fed/mhd/lmhd.html

44. Tokamak Physics4411: Tokamak plasma instabilities II 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  i ) instability Electron temperature gradient (ETG or  e ) 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 Microinstabilities

45. Tokamak Physics4511: Tokamak plasma instabilities II 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