1. CCFE is the fusion research arm of the United Kingdom Atomic Energy Authority Challenges for Fusion Theory and Explosive Behaviour in Plasmas Steve Cowley, Chris Ham Culham/UKAEA/Imperial Thanks to: Howard Wilson, Sophia Henneberg, Samuli Saarelma, Omar Hurricane, Andy Kirk, Bryan Fong, Mehmet Artun, Colin Roach, James Harrison, Brendan Cowley David Dickinson, MAST team ……

2. Best Performance. JT-60U Internal Transport Barrier Best equivalent Q DT is in a reverse shear ITB plasma. Equivalent Q DT ~1.25 I certainly wish we could reach this value in JET SHEAR Koide et. al. (1998b)

3. Challenges -- paradox We want good confinement. Smaller cheaper reactors? We have found best confinement in ITB type plasmas – high beta, low or reverse central shear. There is a problem with all good confinement regions (ITB, H mode) – they go Bang. Why do these plasmas sometimes explode? And sometimes not I will argue that this is a nonlinear issue. Talk: Introduction to the issues, the open questions and an example of a mechanism.

4. Critical Gradient Pressure profile without Ballooning modes Critical pressure profile. Actual pressure profile. Critical gradient argument – instability pins profile to local marginal stability. Used in solar convection – tokamak ion temperature gradient modes … etc. argument for Ballooning modes due to Connor, Taylor and Turner 1984

5. Critical Gradient There is plenty of evidence for critical gradient behaviour in ITG turbulence. ITBs and pedestals beat the conventional critical gradients through flow shear Shafranov shift etc. stabilization ASDEX, Suttrop et. al. 2001 Typical “core” Value in Asdex, JET etc. Typical value in Internal Transport Barriers

6. Opening up the Tokamak ECE grating polychromator

7. TFTR Supershot disruptions – going Bang Fredrickson et. al. 1996 Close to linear ballooning threshold

8. TFTR ITB disruptions Fredrickson et. al. 1996 Very difficult Measurements. Not seen on other Machines – why?

9. Passing Through Marginal Stability Moving slowly through marginal stability gives only moderate growth. Let t=0 be the time of marginal stability. Approximate amplitude equation, The hybrid evolution time is, With non-linearity we can get fast singular growth. Or slow saturated growth. What does happen and when?

10. Nonlinear Evolution --Bifurcation Slow growth Critical amplitude for nonlinearity

11. Explosive growth – prefers one direction (out or in) Critical amplitude for nonlinearity The energy lanscape. Bifurcation – small amplitude t<0 t>0 No nearby equilibria in explosive direction. A 3 nonlinearity no A 2

12. What do the ballooning modes do?

13. Elliptical flux tube slides out along surface S parting field lines

14. Surface S=0 r 00 Elliptical flux tube slides out along surface S parting field lines

15. r 0 original flux surface of field line Equations for perturbed field line Perturbed Field Lines,

16. Force across the Tube Field lines outside tube effectively unperturbed if Rotated picture Time to equalise pressure along tube Force across tube Unperturbed field and pressure outside tube S. C. Cowley, B. Cowley, S. A. Henneberg and H. R. Wil- son, Proc. R. Soc. A 471 20140913(2015),

17. Force on the Tube We can get this from shape and strength of field “Archimedes” buoyancy force Radial Force Field lines outside tube effectively unperturbed if Rotated picture

18. Generalised s - alpha We take circular flux surfaces with a narrow region with a small pressure jump. This models the internal transport barrier – in this talk we take:

19. Generalised s - alpha Drag evolution yields nonlinear s-alpha equation for. Linearising yields familiar s-alpha force equation to test for linear instability.

20. Generalised s - alpha We also define an nonlinear potential energy for this Drag motion minimises this energy. Always locally Downhill.

21. Stable profile

22. Perturbing a little r 0 = 0.63

23. Perturbing more r 0 = 0.63

24. Nonlinear excitation - Meta-stability The energy landscape. Potential Energy Amplitude r - r 0 r0r0 Potential Energy Amplitude r - r 0 r0r0

25. Equilibrium Field Line Shapes We show the perturbed equilibrium shape for negative theta Solutions are even Red Line: unperturbed line r = r 0 Black line: stable lowest energy state Green line: unstable equilibria

26. r0r0 r0r0 Is this Energetically favourable Only equilibrium here is unperturbed equilibrium

27. Dbeta = fraction Of pressure jump That the flux tube Crosses – 80% ish

28. Conjecture and Challenge I conjecture that the fast disruptions of ITBs are often due to meta-stability of ballooning modes. Triggering happens from noise when system is stable but near linear marginal stability. Results in large flux tube perturbations. ELMs? I have almost no idea about what affects the cross sectional shape of the tube – FLR, secondary instability etc. Simulation of metastable instability is needed. 3D MHD codes should see the behaviour from our analysis. Transport of heat and particles caused by highly displaced tubes needs study too. We need to find non disruptive ITB states.

29. Thank you