1. Evening Talk at EFTC-11 MAGNETIC GEOMETRY, PLASMA PROFILES AND STABILITY J W Connor UKAEA/EURATOM Fusion Association, Culham Science Centre, Abingdon, Oxon, OX14 3DB, UK A story about mode structures with the ‘Universal Mode’ providing a ‘Case History’

2. 1. INTRODUCTION The Universal Mode ●Anomalous transport associated with micro-instabilities such as the electron drift wave ●Early theory for a homogeneous plasma slab or cylinder (with dn/dx = constant) showed that in a shear free situation one could always find an unstable mode –driven by electron Landau resonance, with long parallel wavelengths to minimise ion Landau damping –the so-called Universal Mode ●This talk will explore how the stability and mode structure responds to more realistic magnetic geometry and radial profiles –leads to ballooning theory and more recent developments in this topic

3. Geometry Cylinder ●n(r), q(r): ●Fourier analyse:  =  (r)e -i(m  -nz/R) ● ●For electron Landau drive and to minimise ion Landau damping  long parallel wavelength ●Shear mode localised around resonant surface r 0 : m = nq(r 0 )

4. Axisymmetric Torus, B(r,  ) ●  =  (r,  )e in  ●2D (r,  ) – periodic in  ; different poloidal m coupled ●High-n – simplify with eikonal ●Problem is to reconcile this with small k || and periodicity but not periodic! ; shear  q(r)  q(r 0 )

5. Preview of Model Eigenvalue Equation FLRIon Sound Radial variation Toroidicity Electron Eigenvalue due to  * or  E Drive  (  ) -X = nq′ (r – r 0 ) Resonant surfaces Different cases, depending on magnitudes of ,  1,  2

6. 2. SHEARED SLAB/CYLINDER ●Include magnetic shear, s  0, and density profile n(r) ●Eigenvalue equation FLR shear density electron Landau eigenvalue profile drive Potential Q(X) ●Treat  e as perturbation

7. MARSHALL ROSENBLUTH – 1927-2003

8. Low shear (or rapidly varying n(r)) ●Localised in potential well (Krall-Rosenbluth) –Im   e : UNSTABLE Increased shear ●Well becomes ‘hill’: no localised mode –shear stabilisation criterion? Outgoing waves ●But, Pearlstein-Berk realised these are acceptable solutions (wave propagates to ion Landau damping region where it becomes evanescent)! ●Radiative or shear damping

9. ●Balancing against  e gives stability criterion ●‘Confirmed’ by erroneous numerical calculations! Potential ‘hill’ Q Outgoing wave  L s /L n (L s /L n ) 1/2 X

10. Aside ●But later numerical (Ross-Mahajan) and analytic (Tsang-Catto et al; Antonsen, Liu Chen) treatments with full Plasma Dispersion function Z for electrons showed UNIVERSAL MODEL ALWAYS STABLE! ●Traced to fact that Z has a crucial effect on mode structure in narrow region affecting  e  perturbation treatment inadequate!

11. Quasi-modes ●In periodic cylinder k ||  m – nq (r) ●Fix n, different m have resonant surfaces r m : n = q(r m ) ●For large n, m they are only separated by ●Then each radially localised ‘m-mode’ ‘looks the same’ about its own resonant surface ●Each mode has almost same frequency – ie almost degenerate and satisfies

12. JOHN BRYAN TAYLOR

13. ●Roberts and Taylor realised it was possible to superimpose them to form a radially extended mode –the twisted slicing or quasi-mode which maintains k || << k 

14. BORIS KADOMTSEV – 1928-1998

15. 3. TOROIDAL GEOMETRY ●In a torus two new effects arise from inhomogeneous magnetic fields –new drives from trapped electrons (Kadomtsev & Pogutse): DESTABILISES universal mode: trapped particle modes –changes in mode structure from magnetic drifts: affects shear damping ● a vertical drift

16. ●Doppler shift:    - k. v D normalgeodesic ●  (r)   (r,  ): two-dimensional ●Eigenvalue equation (absorb  e into  )  ~ 0(1)

17. Suppression of Shear Damping (Taylor) ●Seek periodic solution ●Approximate translational invariance of rational surfaces   2 << 1 ●Model equation: ignore geodesic drift XmXm X m+1  X = 1 Reflect

18. ●Cylindrical limit:  = 0 –independent shear damped Fourier modes ●Strong toroidal coupling limit:  > 1  u m slowly varying ‘functions’ of m –each u m is localised about x = m: no shear damping –decays before  ~ 2 , so periodicity not an issue

19. ●  is a quasi-mode –‘balloons’ in  –‘m’ varies with X –radially extended  determine slowly varying radial envelope A(m) by reintroducing  2 << 1 –seen in gyrokinetic simulations (eg W Lee)

20. ●Arbitrary  Try  2 <<  - one-dimensional equation in  But solution must be periodic in  over 2  - must reconcile with secular terms! ●Solve problem with Ballooning Transformation (Connor, Hastie, Taylor)

21. Aside ●The ballooning transformation was first developed for ideal MHD ballooning modes, leading to the s-  diagram ●Interesting to contrast published and actual diagrams! ●Lack of trust in –numerical computation (confirmed by two-scale analysis for low s) –ability to find access to 2 nd stability (eg raise q 0 )

22. Alternatively (Lee, Van Dam) Translational invariance limit,  2  0  system invariant under X  X + 1, m  m + 1 Bloch Theorem  u m (x) = e imk u(x – m) Fourier Transform of u leads to ballooning equation

23. Return to drift wave: ●Potential Q(  ) -  <  <   no longer need periodic solution! ●Often consider ‘Lowest Order’ equation,  2 =0 –Schrődinger equation with potential Q(  ) with k a parameter,  (  ) a ‘local’ eigenvalue –k usually chosen to be 0 or  (to give most unstable mode) –increasing  removes shear damping; vanishes for  > 4: more UNSTABLE

24. Potential Q 1.0 2.0 3.0 4.0 5.0  0 Marginal Shear damping -0.5 - 1.0 -1.3 ss 0

25. 4. RADIAL MODE STRUCTURE ●‘Higher order’ theory: determines radial envelope A(X) (Taylor, Connor, Wilson) ●Reintroduce  1,2 << 1  k = k(X) –yields radial envelope in WKB approximation –eigenvalue condition Conventional Version:  * has a maximum ●Lowest order theory gives ‘local’ eigenvalue ●Expand  about k 0 where shear damping is minimum

26. Quadratic profile Linear profile ●Implications –mode width –mode localised about X = 0, ie  *max,, but covers many resonant surfaces –spread in k:  k ~ n -1/2 << 1  k  k 0 (minimum shear damping) (a) (b)

27. Limitations 1. Low magnetic shear 2. High velocity shear 3. The edge -unfortunately characteristics of transport barriers! ITBH-mode

28. Low Magnetic Shear, s << 1 ●Two-scale analysis of ballooning equations   ( , u):  periodic equilibrium scale, u = s   ‘averaged’ eqn, independent of k   independent of k! –corresponds to uncoupled Fourier harmonics at each m th surface, localised within X ~ |s|: ie non-overlapping radially ●Recover k-dependence by using these as trial functions in variational approach (Romanelli and Zonca) –exponentially weak contribution –becomes very narrow as |s|  0, or k   i  0

29. ●Estimate anomalous transport:   Lin X M 2 suggests link between low shear and ITBs (Romanelli, Zonca) ●Presence of q min acts as barrier to mode structures Gyrokinetic simulation by Kishimoto s crit s  1

30. Modelling of Impurity Diffusion in JET with D z ~ (  i L) 1/2 exp (-c/s), V = D/R L Lauro-Taroni

31. ●Ballooning mode theory fails for sufficiently low s (or long wavelength) –reverts to weakly coupled Fourier harmonics, amplitude A m, when –spectrum of narrows as mode centre moves towards q min ●In practice, ballooning theory holds to quite low n eg ITG modes with k   I ~ 0(1) largely unaffected by q min

32. The Wavenumber Representation ●More general contours of k (X,  ) (Romanelli, Zonca) ●‘Closed’ contours already discussed; ‘passing’ contours sample all k –WKB treatment in X-space still possible –easier to use alternative, but entirely equivalent, Wavenumber Representation (Dewar, Mahajan)

33. ● ● satisfies ballooning eqn on -  <  < , ie not periodic in  ● –  is periodic in  if S(k) is periodic in k: eigenvalue condition Example ●Suppose linear profile:  ●Periodicity of S yields eigenvalue condition

35. Implications ●Re  related to local  * (x): ●Im –k not restricted to near k 0, all k contribute to give an average of the shear damping! –some shear damping restored: more STABLE eg  = 4:  s (0) = - 0.02, ● ●Mode width: (i)  real   X ~  n, or  r ~  a (ii)  complex   X ~ n 1/2  1/2, or  r ~ (  /n) 1/2 a

36. Sheared Radial Electric Fields ●Believed to reduce instability and turbulence – prominent near ITBs ●    - n  E (x) (Doppler Shift); suppose  E =  x ●  mode narrows as d  E /dq increases, reducing estimates of  X and transport ●Are these modes related to conventional ballooning modes? –introduce density profile variation ●Model: ie  has maximum at X = 0

37. ●Wavenumber representation produces quadratic eqn for dS/dk –exp (in S)   (k) - periodic S(k) is Floquet solution of Mathieu eqn: yields eigenvalue Analytic solution for transition region possible (Connor)  continuous evolution from conventional mode to more STABLE ‘passing’ mode ● for large d  E /dq  reverts to Fourier modes! FULL CIRCLE?

38. 5. EXTENSIONS TO BALLOONING THEORY ●Have seen limitations imposed by low magnetic shear and high flow shear ●The presence of a plasma edge clearly breaks translational invariance –have used 2D MHD code to study high-n edge ballooning modes; mode structure resembles ballooning theory ‘prediction’

39. ●Non-linear theory –the ‘twisted slices’ of Roberts and Taylor form a basis for flux-tube gyrokinetic simulations ConventionalST Tokamak  x y

40. –introducing non-linearities into the theory of high-n MHD ballooning modes predicts explosively growing filamentary structures, seen on MAST SimulationExperiment

41. 6. SUMMARY AND CONCLUSIONS ●Have used the story of the ‘Universal Mode’ to illustrate developments in the theory of toroidal mode structures and stability –the universal mode is stable, but plenty of other toroidal modes are available to provide anomalous transport! ●Problems of toroidal periodicity in the presence of magnetic shear resolved by Ballooning theory ●Ballooning theory provides a robust and widely used tool, but its validity can break down for: –Low magnetic shear –Rotation shear –Plasma edge when the higher order theory is considered ●Re-emergence of Fourier modes in the torus for low s and high d  E /dq ●Ballooning theory also provides a basis for some non-linear theories and simulations

42. STABLE UNSTABLE 1960 1970 1980 1990 Universal Mode (Galeev et al) 1963 Outgoing Waves (Pearlstein- Berk 1969) Loss of shear damping in Torus (Taylor) 1976 Toroidally Induced mode (Chen-Cheng) 1980 (Hesketh, Hastie, Taylor) 1979 Time Shear stabilisation (Krall-Rosenbluth) 1965 ‘Correct’ calculation (Tsang, Catto, Ross, Mahajan: Antonsen, Liu Chen) 1978 Steep gradients in torus (Connor, Taylor) 1987 Not at  *MAX (Connor, Taylor, Wilson) 1992, 1996 Low magnetic shear (Romanelli, Zonca 1993; Connor, Hastie, 2003) Velocity shear (Taylor, Wilson; Dewar) 1996-7 2000