Evening Talk at EFTC-11

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 r0 : m = nq(r0)

4. Axisymmetric Torus, B(r,) •  =  (r, )ein • 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(r0)

5. Preview of Model Eigenvalue Equation • FLR Ion Sound Radial variation Toroidicity Electron Eigenvalue • due to * or E Drive () • X = nq′ (r – r0) • 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 (wavepropagates to ion Landau damping region where it becomes evanescent)! • Radiative or shear damping

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

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 rm: n = q(rm) • 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 . vD normal geodesic •  (r)  (r,): two-dimensional • Eigenvalue equation (absorb e into )  ~ 0(1)

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

18. Cylindrical limit:  = 0 • independent shear damped Fourier modes • Strong toroidal coupling limit:  > 1  um slowly varying ‘functions’ of m • each um 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 withBallooning 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 2nd stability (eg raise q0)

22. Alternatively (Lee, Van Dam) Translational invariance limit, 2  0  system invariant under X  X + 1 , m  m + 1 Bloch Theorem  um(x) = eimk 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. 0 Marginal s -0.5 Shear damping -1.0 -1.3 Potential Q 0 1.0 2.0 3.0 4.0 5.0 

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 k0 where shear damping is minimum

26. (b) (a) 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  k0 (minimum shear damping)

27. Limitations • Low magnetic shear • High velocity shear • The edge - unfortunately characteristics of transport barriers! • ITB H-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 mth surface, localised within X ~ |s|: ie non-overlapping radially • Recover k-dependence by using these as trial functions in variationalapproach (Romanelli and Zonca) • exponentiallyweak contribution • becomes very narrow as |s|  0, or ki  0

29.  1 s scrit • Estimate anomalous transport: Lin XM2 suggests link between low shear and ITBs (Romanelli, Zonca) • Presence of qmin acts as barrier to mode structures Gyrokinetic simulation by Kishimoto

30. Modelling of Impurity Diffusion in JET with Dz ~ (iL)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 Am, when • spectrum of narrows as mode centre moves towards qmin • In practice, ballooning theory holds to quite low n eg ITG modes with kI ~ 0(1) largely unaffected by qmin

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 k0, 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 ~ n1/2 1/2, or r ~ (/n)1/2a

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. y x  • Non-linear theory • the ‘twisted slices’ of Roberts and Taylor form a basis for flux-tube gyrokinetic simulations • Conventional ST • Tokamak

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

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