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The impact of rotation and anisotropy due to NBI energetic particles in toroidal magnetic confinement M. J. Hole 1, M. Fitzgerald 1, G. von Nessi 1, K. G. McClements 2, J. Svensson 3, R. Kemp 2, L. C. Appel 2, I. Chapman 2, N. Walkden 4 18 th International Stellarator Heliotron Workshop & 10 th Asia Pacific Plasma Theory Conference 29 th January – 3 rd February Res. School of Physics and Engineering, Australian National University, ACT Australia. 2 EURATOM/CCFE Fusion Assoc., Culham Science Centre,Abingdon, Oxon OX14 3DB, UK 3 Max Planck Institute for Plasma Physics, Teilinstitut Greifswald, Germany 4 University of Bath, Bath, UK, BA2 7AY Acknowledgement: R. Andre, PPPL, USA

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Outline Rotation (toroidal and poloidal), anisotropy −Axis-symmetric equilibrium Show large anisotropy for MAST (#18696), p /p || = 1.7 −Impact of anisotropy on stability through changes in q profile Implications on CAE, TAE mode frequencies for MAST #18696 Anisotropy / flow implications for Component Test Facility (CTF) −Preliminary assessment of impact of flow, anisotropy on CTF Probabilistic (Bayesian) inference framework −Provides model validation (equilibrium and mode structure) −Harnessed to infer properties of plasma (e.g. energetic particle pressure) Toroidal field ripple: impact on injected beams −Test particle simulations in presence of toroidal field ripple. −Compute diffusivity due to field ripple KSTAR wave activity during RF and NBI heated discharges

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MHD with rotation & anisotropy Inclusion of anisotropy and flow in equilibrium MHD equations [R. Iacono, et al Phys. Fluids B 2 (8). 1990]

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MHD with rotation & anisotropy Inclusion of anisotropy and flow in equilibrium MHD equations Equilibrium eqn becomes: [R. Iacono, et al Phys. Fluids B 2 (8). 1990] Set of 6 profile constraints Frozen flux gives velocity:

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Reduction to isotropic and equilibrium eqn becomes: Suppose Set of 5 profile constraints Then

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Further reduction Set of 4 profile constraints Toroidal flow only

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Further reduction Set of 4 profile constraints Set of 2 profile constraints Toroidal flow only Static case

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Centrifugal density shift Expression for n( ,R) in presence of flows. Rotation effects can be large for the ST

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“transonic” poloidal flows (~ sound speed c s B /B) G-S & Bernoulli eqns hyperbolic in part of solution domain radial contact discontinuities where MHD fails For isothermal flux surfaces, McClements & Hole show poloidal flow hyperbolicity threshold of G-S equation [PoP17, ] lower for isothermal (cf isentropic) flux surfaces lower for high performance ST plasmas, M s 0.7 Centrifugal density shift Expression for n( ,R) in presence of flows. Two types of closure: isentropic flux surfaces: ( ) = p/ isothermal flux surfaces: T( ) Rotation effects can be large for the ST

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Expected impact of anisotropy If p ⊥ > p ||, an increase will occur in centrifugal shift : [R. Iacono, A. Bondeson, F. Troyon, and R. Gruber, Phys. Fluids B 2 (8). August 1990] Compute p ⊥ and p ||, from moments of distribution function, computed by TRANSP [M J Hole, G von Nessi, M Fitzgerald, K G McClements, J Svensson, PPCF 53 (2011) ] [see V. Pustovitov, PPCF , 2010 and references therein] Infer p ⊥ from diamagnetic current J ⊥ If p || sig. enhanced by beam, p || surfaces distorted and displaced inward relative to flux surfaces Broad pressure profile Peaked pressure profile Parallel pressure contours(solid) Flux surfaces (dashed) [Cooper et al, Nuc. Fus. 20(8), 1980] Small angle b between beam, field p || > p ⊥ Beam orthogonal to field, b = /2 p ⊥ >p ||

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Anisotropy on MAST [M.P. Gryaznevich et al, Nuc. Fus. 48, , 2008.; Lilley et al 35th EPS Conf. Plas.Phys June 2008 ECA Vol.32D, P-1.057] MAST # MW NB heating I p = 0.7MA, n =2.5 TRANSP simulation available Magnetics shows CAEs What is the impact on q profile due to presence of anisotropy and flow? Magnetics

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p ll, p , flow from f(E, ) moments [35th EPS 2008; M.K.Lilley et al] [M J Hole, G von Nessi, M Fitzgerald, K G McClements and J Svensson, PPCF 53 (2011) ] p ⊥ /p || ≈ 1.7 r/a=0.25 = toroidal flux See Fitzgerald S13.4

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Impact of anisotropy on equilibrium Impact on configuration computed using FLOW [Guazzotto L, Betti R, Manickam J and Kaye S 2004 PoP11 604–14] <0: p ⊥ /p || ≈ 1.7 =0: p ⊥ /p || = 1

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Impact of anisotropy on equilibrium Impact on configuration computed using FLOW [Guazzotto L, Betti R, Manickam J and Kaye S 2004 PoP11 604–14] <0: p ⊥ /p || ≈ 1.7 =0: p ⊥ /p || = 1 Toroidal rotation does not change q appreciably with M A,φ 0.3 Increase in q 0 ~ 100% for case with anisotropy Low grid resolution of FLOW at core

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How do predicted mode frequencies change due to changes in q produced by anisotropy and flow? Impact of anisotropy on wave modes n=4 n=-10 Adjacent n Doppler shift

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TAE eigenfrequency Frequency scaling with/without anisotropy [Cheng, Liu & Chance. Annals of Physics (1985)] TAE static equil. TAE flowing, anisotropic equil. Illustration

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Known: R 0 = major radius, = ellipticity, n=-10 Inferred: p= poloidal mode no. = |n|q mn Unknown: s= radial mode no. = 0 (guess) 0 = radial localisation of mode ~ a/2 (guess) [Smith et al PoP 10(5), ,2003] CAE eigenfrequency TAE eigenfrequency Frequency scaling with/without anisotropy [Cheng, Liu & Chance. Annals of Physics (1985)] TAE static equil. TAE flowing, anisotropic equil. Illustration

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Known: R 0 = major radius, = ellipticity, n=-10 Inferred: p= poloidal mode no. = |n|q mn Unknown: s= radial mode no. = 0 (guess) 0 = radial localisation of mode ~ a/2 (guess) [Smith et al PoP 10(5), ,2003] CAE eigenfrequency TAE eigenfrequency Static, isotropic equilibrium: q mn (R=R 0 +a/2 ) =1.3 p=13 Frequency scaling with/without anisotropy [Cheng, Liu & Chance. Annals of Physics (1985)] TAE static equil. TAE flowing, anisotropic equil. Illustration aliased

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Known: R 0 = major radius, = ellipticity, n=-10 Inferred: p= poloidal mode no. = |n|q mn Unknown: s= radial mode no. = 0 (guess) 0 = radial localisation of mode ~ a/2 (guess) [Smith et al PoP 10(5), ,2003] CAE eigenfrequency TAE eigenfrequency Static, isotropic equilibrium: q mn (R=R 0 +a/2 ) =1.3 p=13 Flowing, anisotropic equilibrium: q mn (R=R 0 +a/2 ) =1.3 p=15. Frequency scaling with/without anisotropy [Cheng, Liu & Chance. Annals of Physics (1985)] TAE static equil. TAE flowing, anisotropic equil. Illustration aliased

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Outline Rotation (toroidal and poloidal), anisotropy −Axis-symmetric equilibrium Show large anisotropy for MAST (#18696), p /p || = 1.7 −Stability: ideal (static) MHD – due to changes in equilibrium, Implications on CAE, TAE mode frequencies for MAST #18696 Anisotropy / flow implications for Component Test Facility (CTF) −Preliminary assessment of impact of flow, anisotropy on CTF Exploiting probabilistic (Bayesian) inference framework −Enables model validation (equilibrium and mode structure) −Harnessed to infer properties of plasma (e.g. energetic particle pressure) Toroidal field ripple: impact on injected beams −Test particle simulations in presence of toroidal field ripple. −Compute diffusivity due to field ripple KSTAR wave activity during RF and NBI heated discharges

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Component Test Facility CTF requirements [Abdou et al Fus. Techn. 29(1), 1996] neutron power flux ~1-2 MW/m 2, steady state, test area> 10m 2, B>2T, neutron flux> 6MWm 2 /yr stable steady state ST can provide fusion conditions necessary for components testing operated in parallel with DEMO complement and extend the anticipated database from IFMIF [G.M. Voss et al. / Fusion Engineering and Design 83 (2008) 1648–1653 MHD stability without flow: n = 1, 2, 3 stable if q 0 > 2.1 n = 1 external kink stable if R w /a < 2.2 [Hole et al Plasma Phys. Control. Fusion 52 (2010)]

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MASTCTF Major/minor radius (m)0.8/ / 0.47 Elongation, Triangularity1.5, , 0.3 Plasma / pressure-driven current (MA)1, 0.38, 3 q 0, q , 102.2, 6.2 Line-avge, n( m -3 ) 220 Core T i, T e (keV)1,118.5, 15 (%), N 6, 520, 3.4 Ion sound speed c s ( 10 5 ms -1 ) NB energy D, D+T (keV)60150 P fusion (thermal + beam) (MW) NBI velocity ( 10 6 ms -1 ) Rotational Mach no (M s ) Critical energy, E crit (keV)19301 Critical velocity, v crit ( 10 6 ms -1 ) Avge neutron wall loading (MWm -2 )-1.6 |B 0 |, vacuum at geometric centre (T) CTF base-line [Cottrell& Kemp, Nuc. Fus. 49 (2009) ; Voss. et al. Fus. Eng. Des. 83 (2008) 1648–1653]

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CTF Equilibrium q 0 2.17 Ballooning mode threshold Rotation profile predicted using = 0.5 i [G.A. Cottrell and R. Kemp, Nucl. Fusion 49 (2009) ] NBI projections - top NBI projections – pol. plane [Hole et al Plasma Phys. Control. Fusion 52 (2010)]

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CTF has weak anisotropy Beams heat off-axis plasma No significant anisotropy p ⊥ ~ p || s 0 1 P (kPa) s 0 1 n (m -3 ) s 0 1 P (kPa) s 0 1 R1: off-axis, 30 R2: on-axis, horizontal R3: off-axis, horizontal parallel perpendicular R2: on-axis R1: off-axis, 30 If beams turned off independently... p ⊥ > p || p ⊥ < p ||

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...but strong rotation, which lowers q 0 flow-stabilised to ideal MHD modes for weak flow (M A 0.17). eigenfunction peak at position of strongest rotation gradient R /c s s flowing equilibrium static equilibrium [Chapman et al, sub PPCF, 2011] N. R. Walkden, I. T. Chapman

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Outline Rotation (toroidal and poloidal), anisotropy −Axis-symmetric equilibrium Show large anisotropy for MAST (#18696), p /p || = 1.7 −Stability: ideal (static) MHD – due to changes in equilibrium, Implications on CAE, TAE mode frequencies for MAST #18696 Compute anisotropy for Component Test Facility (CTF) −Preliminary assessment of impact of flow, anisotropy on CTF stability Probabilistic (Bayesian) inference framework −Provides model validation (equilibrium and mode structure) −Harnessed to infer properties of plasma (e.g. energetic particle pressure) Toroidal field ripple: impact on injected beams −Test particle simulations in presence of toroidal field ripple. −Compute diffusivity due to field ripple KSTAR wave activity during RF and NBI heated discharges

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Inference of energetic physics ANU/CCFE/IPP developed a probabilistic framework based on Bayes’ theorem for validating models for equil. & mode structure Motivation: handle data from multiple diagnostics with strong model dependency provides a validation framework for different equilibrium models: e.g. Two fluid with rotation, multi-fluid, MHD with anisotropy yield uncertainties in inferred physics parameters (e.g. q profile) from models, data, and their uncertainties. Can be inverted : By reducing force-balance model uncertainty to zero, use as a technique to infer physics difficult to experimentally diagnose directly (e.g. Energetic particle pressure)

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Bayesian equilibrium modelling Aims (1)Improve equilibrium reconstruction (2)Validate different physics models Two fluid with rotation [McClements & Thyagaraja Mon. Not. R. Astron. Soc – ] Ideal MHD fluid with rotation [Guazzotto L et al, Phys. Plasmas –14, 2004] Energetic particle resolved multiple-fluid [Hole & Dennis, PPCF , 2003] (3) Infer poorly diagnosed physics parameters MSE signals Flux loops Pick-up coils TS spectra CXRS spectra S12.5 Greg von Nessi see S10.1 Oliver Ford

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Mean in posterior gives flux surfaces MAST D plasma, 3MW NB heating I p = 0.8MA, n =3 Last closed flux surface of MSE& EFIT Current TomographyPoloidal flux surfaces J and surfaces plotted for currents corresponding to the maximum of the posterior [M.J. Hole, G. von Nessi, J. Svensson, L.C. Appel, Nucl. Fusion 51 (2011) ]

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Sampling of posterior gives distribution Distributions generated by sampling, e.g. q profile q profile #24600 Inference of poloidal currents: allow f( ) to be a 4 th -order polynomial in No poloidal currents Bayesian models for TS and CXRS Errors < 5%, but are model dependant

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Integrate force balance yields P energetic Equilibrium Validation: MAST Uses MSE, pick up coils, flux loops Uses TS, CXRS MSE data range Integrate residue P energetic estimate

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Outline Rotation (toroidal and poloidal), anisotropy −Axis-symmetric equilibrium Show large anisotropy for MAST (#18696), p /p || = 1.7 −Stability: ideal (static) MHD – due to changes in equilibrium, Implications on CAE, TAE mode frequencies for MAST #18696 Compute anisotropy for Component Test Facility (CTF) −Preliminary assessment of impact of flow, anisotropy on CTF stability Exploiting probabilistic (Bayesian) inference framework −Enables model validation (equilibrium and mode structure) −Harnessed to infer properties of plasma (e.g. energetic particle pressure) Toroidal field ripple: impact on injected beams −Test particle simulations in presence of toroidal field ripple. −Compute diffusivity due to field ripple KSTAR wave activity during RF and NBI heated discharges

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Toroidal field ripple on beam confinement Motivation: anomalous fast ion diffusion (D b ∼ 0.5 m 2 s −1 ) invoked to match the experimentally observed neutron rate and stored energy [M. Turnyanskiy et al Nucl. Fusion 49, (2009).] [K. G. McClements, M. J. Hole, to be sub. PoP. ]

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Toroidal field ripple on beam confinement Motivation: anomalous fast ion diffusion (D b ∼ 0.5 m 2 s −1 ) invoked to match the experimentally observed neutron rate and stored energy [M. Turnyanskiy et al Nucl. Fusion 49, (2009).] Possible sources of anomalous fast ion diffusion: MHD activity, turbulence or toroidal ripple [K. G. McClements, M. J. Hole, to be sub. PoP. ]

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Toroidal field ripple on beam confinement Motivation: anomalous fast ion diffusion (D b ∼ 0.5 m 2 s −1 ) invoked to match the experimentally observed neutron rate and stored energy [M. Turnyanskiy et al Nucl. Fusion 49, (2009).] Possible sources of anomalous fast ion diffusion: MHD activity, turbulence or toroidal ripple Strategy: Use full orbit code CUlham Energy-conserving orBIT (CUEBIT) to quantify ripple transport of beam ions in MAST [B. Hamilton, K. G. McClements, L. Fletcher, & A. Thyagaraja, Sol. Phys 214, 339 (2003).] CUEBIT modified to include collisions between suprathermal test-particles (hyrdrogenic beam ions) with bulk ions and bulk electrons (both Maxwellian populations) [K. G. McClements, M. J. Hole, to be sub. PoP. ]

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For MAST R coil = 2.09m N = 12, R edge ~1.45m ~1.2%. Comparable to maximum ripple amplitude in ITER, predicted to cause fusion -particle losses of between 2% and 20%, depending on the magnetic shear profile. [S. V. Konovalov, E. Lamzin, K. Tobita, and Yu. Gribov, Proceedings of the 28th EPS Conference on Controlled Fusion and Plasma Physics, edited by C. Silva, C. Varandas and D. Campbell (European Physical Society, Petit-Lancy, 2001), Vol. 25A, p. 613.] MAST toroidal field ripple Assume ripple perturbation to B due to N toroidal coils Toroidal Field coils fully vertical

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Track full orbits of beam ions with -function birth distributions at energies E 1 =65kEv, E 2 =51keV, E 1 /2, E 2 /2, E 1 /3, E 2 /3. beam ion birth positions MAST beam confinement Example collisionless beam ion orbit diatomic ions tri-atomic ions Field = MAST Grad-Shafranov equil. (I p =726 kA, B 0 =0.43T) + vacuum ripple.

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launch N 0 = 28,000 test- particles, track orbits, record # particles in plasma N b (t), with /without toroidal ripple Particle “lost” if crosses the last closed flux surface at any point on its trajectory. Field ripple No field ripple N b (t) = N 0 (1 − t/ c ) fit gives c (no ripple) = 237ms c (ripple) = 224ms ~1% loss

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launch N 0 = 28,000 test- particles, track orbits, record # particles in plasma N b (t), with /without toroidal ripple Particle “lost” if crosses the last closed flux surface at any point on its trajectory. D b ripple a 2 / c ripple - a 2 / c no ripple ~ 0.1m 2 s −1 below anomalous fast ion diffusion (D b ∼ 0.5 m 2 s −1 ) invoked in order to match observed neutron rate and stored energy. Field ripple No field ripple N b (t) = N 0 (1 − t/ c ) fit gives c (no ripple) = 237ms c (ripple) = 224ms ~1% loss Anomolous diffusivity due to ripple transport:

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launch N 0 = 28,000 test- particles, track orbits, record # particles in plasma N b (t), with /without toroidal ripple Particle “lost” if crosses the last closed flux surface at any point on its trajectory. D b ripple a 2 / c ripple - a 2 / c no ripple ~ 0.1m 2 s −1 below anomalous fast ion diffusion (D b ∼ 0.5 m 2 s −1 ) invoked in order to match observed neutron rate and stored energy. Ripple insufficient to account for fast ion transport Field ripple No field ripple N b (t) = N 0 (1 − t/ c ) fit gives c (no ripple) = 237ms c (ripple) = 224ms ~1% loss Anomolous diffusivity due to ripple transport:

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launch N 0 = 28,000 test- particles, track orbits, record # particles in plasma N b (t), with /without toroidal ripple Particle “lost” if crosses the last closed flux surface at any point on its trajectory. D b ripple a 2 / c ripple - a 2 / c no ripple ~ 0.1m 2 s −1 below anomalous fast ion diffusion (D b ∼ 0.5 m 2 s −1 ) invoked in order to match observed neutron rate and stored energy. Ripple insufficient to account for fast ion transport Field ripple No field ripple N b (t) = N 0 (1 − t/ c ) fit gives c (no ripple) = 237ms c (ripple) = 224ms ~1% loss R edge = 1.37m; 0.3%: R edge =1.51m; 2.3% ripple-induced power loss: Anomolous diffusivity due to ripple transport:

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Outline Rotation (toroidal and poloidal), anisotropy −Axis-symmetric equilibrium Show large anisotropy for MAST (#18696), p /p || = 1.7 −Stability: ideal (static) MHD – due to changes in equilibrium, Implications on CAE, TAE mode frequencies for MAST #18696 Compute anisotropy for Component Test Facility (CTF) −Preliminary assessment of impact of flow, anisotropy on CTF stability Exploiting probabilistic (Bayesian) inference framework −Enables model validation (equilibrium and mode structure) −Harnessed to infer properties of plasma (e.g. energetic particle pressure) Toroidal field ripple: impact on injected beams −Test particle simulations in presence of toroidal field ripple. −Compute diffusivity due to field ripple KSTAR wave activity during RF and NBI heated discharges

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KSTAR electron fishbone, BAE M. J. Hole, C. M. Ryu, M. H. Woo, J. G. Bak, J. Kim In 2009 evidence for electron driven fishbones at ~5kHz in ECRH plasmas. In 2010, evidence of BAE activity was observed at ~40kHz during NBI. #2181 #4220 CSCAS continuum BAE gap mode MISHKA

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KSTAR aliased TAE, ion fishbone M. J. Hole, C. M. Ryu, K. Toi, M. H. Woo, J. G. Bak, J. Kim (m,n)=(1,1) aliased TAE frequency ion fishbone # 5968

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Summary Anisotropy –Computed anisotropy for MAST #18696, showing p ⊥ /p || ≈ 1.7 –Showed impact on equilibrium largely affects q profile. –Explored impact on plasma wave modes: CAEs and TAEs Bayesian validation framework for equilibrium & mode structure –Provides q profile and uncertainty. –Motivation: validate equilibrium models –Exploited force balance discrepancy to infer P energ Toroidal field ripple: impact on full orbits of injected beams −ripple transport diffusivity below that invoked to match neutron loss, −Toroidal field ripple contributes 1% to fast ion loss. CTF –Computed impact of flow and anisotropy on CTF: flow dominates effect –Separate work: showed A=3 CTF flow-stabilised to ideal MHD modes for weak flow, but unstable to Kelvin-Helmholtz at large flow. KSTAR mode activity: electron, ion fishbone, BAE, TAE

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Ongoing work Bayesian inference: –Parameterise Gaussian core localised pressure to infer p energertic –Use diamagnetic loop to obtain p –Add TRANSP pressure as a constraint ? Anisotropy –EFIT with anisotropy. S13.4 Michael Fitzgerald Analysis of wave activity in MAST, KSTAR and H-1 P1.10 Ziping Rao “ Observations of chaotic orbits in a simple 2D ripple model”.

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