Nu Numerical stud udy o of the e com omplex insta tabilit ity - PowerPoint PPT Presentation

Nu Numerical stud udy o of the e com omplex insta tabilit ity s structu ture i e in t the c e core o e of the he sawtooth thing ng plas asma i influenced d by E ECRH CRH Y. B. Nam, G. H. Choe, G. S. Yun, H. K. Park, A. Bierwage a Department of physics, POSTECH, Pohang, 790-784, Korea a JAEA, Rokkasho, Aomori, 039-3212, Japan southub@postech.ac.kr KSTAR C Conference ce 2013 13 Lotte Buyeo Re Resort, Buyeo, , Chung ngnam, K Korea Febru bruary 2 26 6 – 27, 7, 2 201 013 1

Abstract The generation and evolution of complex instability structures in the core of the • sawtoothing plasma with electron cyclotron resonance heating (ECRH) have been observed by a 2D electron cyclotron emission imaging (ECE-I) system in the KSTAR tokamak. The instability structure ranges from the conventional m=1 internal kink mode to multiple flux tubes depending on the ECRH injection condition. The structural diversity of the sawtoothing core is presumably caused by the modified current density profile and magnetic shear due to the contributed localized current through ECRH, which is also well known to changing the stability of the m=1 internal kink mode [1]. As a first attempt to numerically reproduce the ECRH-induced structural alteration, a model based on reduce MHD equations in the cylindrical geometry [2] has been investigated. The initial conditions for the numerical simulations have been chosen considering the well- known experimental parameters such as B t , I p , n e , and T e which translate to plasma resistivity η, kinematic viscosity ν, electron skin depth δ e in the simulation to include an extensive set of perturbed q profiles encompassing the entire KSTAR ECRH experiments. For each initial q profile, the temporal evolution of the poloidal magnetic fluxes has been calculated and the 2D mode structures have been compared with the corresponding ECEI images. The correlation between various ECRH condition and formation of additional complex instability structures has been inductively studied. The heat and current source terms from ECRH will be included in the model for future improvements. *Work supported by the NRF Korea under the contract no. 20120005920. • 2

Multiple flux tubes under ECRH Various sawtooth patterns has been observed including formation • & merging of the multiple flux tubes under ECH when – the ECH resonance position is within the q = 1 radius – the ECH injection angle is perpendicular or co-current ECEI images of multiple flux tubes from KSTAR experiments Z(cm) R(cm) ECE 2-D image at t = 3.189845s (#6024) ECE 2-D image at t = 3.498780s (#6024) G. S. Yun et al , Phys. Rev. Lett. 109 09, 145003 (2012) 3

Pattern variation with ECH condition • The generation & control Smaller r, mechanism of multiple flux tubes narrower coverage are not clear • Statistical analysis indicates that – ECH resonance position – Radial coverage of ECH beam affects the overall dynamics Poloidal view of ECH injection • Resear earch o objec ective: e: larger r, wider coverage Numer eric ical ally r reproduce t e the d e dynam amic ics of m multip iple e fluxes es p phen enomen ena with ith Inversion known c cond ndit itio ions ns & & param ameter ers radius ECH resonance position Plasma center G. H. Choe et al , KSTAR Conference 2013, Poster #50 on Feb. 27 th 4

Reduced MHD equations Dynamics of resistive kink and coupled tearing modes in a tokamak • plasma with multiple resonant surfaces Assumptions to gain simplicity on analysis have been made, such as • 0 − 𝛾 limit: pressure and heat conduction terms are omitted – Details of parallel dynamics has been neglected – � ⋅ 𝛼𝛼 + η 2 Ψ − E z 𝜖 t Ψ = −𝚪 𝛼 ⊥ µ 0 2 𝛼 = − 1 2 Ψ + ν m 𝛼 ⊥ 2 𝛼 ⊥ 2 𝛼 Ψ : magnetic flux function ρ m d t 𝛼 ⊥ B ⋅ 𝛼𝛼 ⊥ µ 0 𝛼 : electrostatic potential η : plasma resistivity ρ m : mass density ν m : kinematic viscosity H. R. Strauss, Phys. Fluids. 19, 134 (1976) 5

Reduced MHD equations: derivation • Maxwell’s equations 𝛼 ∙ 𝐅 = ρ −𝛼 × 𝐅 = 𝜖𝐂 𝛼 × 𝐂 = µ 0 𝐊 𝜖𝜖 𝛼 ∙ 𝐂 = 0 ϵ 0 1 3 4 2 High-aspect-ratio ordering • Generalized Ohm’s law 𝑏 ( 𝜁 ≡ 𝑆 0 ) 𝐅 + 𝐰 × 𝐂 = η𝐊 5 Strong axial magnetic field: � + 𝛼Ψ × 𝐴 � 𝐂 = B 0 𝐴 8 • Equation of motion ExB drift is dominant: d 𝐰 𝐰 = 𝐰 𝐅 × 𝑪 = −𝛼𝛼 × 𝐴 � dt = 𝐊 × 𝐂 − 𝛼𝛼 − ν m 𝛼 × 𝛼 × 𝐰 + 𝐓 p ρ m 6 B 0 9 𝜖𝐂 2, 5 𝜖𝜖 = 𝛼 × 𝐰 × 𝐂 − η𝛼 × 𝐊 7 6

Reduced MHD equations: derivation Surface integration & Stoke’s theorem 𝜖𝐂 𝜖 𝜖𝜖 = 𝛼 × 𝐰 × 𝐂 − η𝛼 × 𝐊 𝜖𝜖 � 𝐁 ∙ d 𝐦 = � 𝐰 × 𝐂 ∙ d 𝐦 − η � 𝐊 ∙ d 𝐦 7 10 𝐂 = 𝛂 × 𝐁 Take out ∮ 𝑒𝒎 𝐰 = −𝛼𝛼 × 𝐴 � B 0 9 𝜖𝐁 𝜖𝜖 = 𝐰 × 𝐂 − η𝐊 + 𝛼ϕ 11 ϕ = B 0 𝛼 12 𝛼 × 𝐂 = µ 0 𝐊 J z = − 1 4 Consider only z- 2 Ψ 𝛼 ⊥ � 𝐂 = B 0 𝐴 � + 𝛼Ψ × 𝐴 component of 𝐁 : µ 0 13 8 𝐁 ∙ 𝐴 � � 𝐂 = B 0 𝐴 � + 𝛼Ψ × 𝐴 � Ψ = 𝐁 ∙ 𝐴 14 8 Additional static E - field 𝜖Ψ � ⋅ 𝛼𝛼 + η 2 Ψ − E z 𝜖𝜖 = −𝐂 𝛼 ⊥ µ 0 7

Reduced MHD equations: derivation d 𝐰 dt = 𝐊 × 𝐂 − 𝛼𝛼 − ν m 𝛼 × 𝛼 × 𝐰 + 𝐓 p ρ m 6 𝛼 × d( 𝛼 × 𝐰 ) = 𝛼 × { 𝐊 × 𝐂 − 𝛼𝛼 − ν m 𝛼 × 𝛼 × 𝐰 + 𝐓 p } ρ m dt 15 𝐰 = −𝛼𝛼 × 𝐴 � B 0 9 Consider the only z- component of 𝛼 × 𝐰 J z = − 1 2 Ψ 𝛼 ⊥ µ 0 13 2 𝛼 = − 1 2 Ψ + ν m 𝛼 ⊥ 2 𝛼 ⊥ 2 𝛼 ρ m d t 𝛼 ⊥ B ⋅ 𝛼𝛼 ⊥ µ 0 8

Coordinate setup and normalization Cylindrical geometry with right-handed set of coordinates ( 𝑠 , 𝜘 , 𝑨 ) • periodic boundary conditions along the axial coordinate z • Ideally conducting boundary condition at 𝑠 = 𝑏 • −1 η 𝜖 t ψ = ψ , ϕ − 𝜖 ζ ϕ − S Hp � j − E ζ −1 𝛼 ⊥ 2 u 𝜖 t u = u, ϕ + j, ψ + 𝜖 ζ j + Re Hp j: axial current density u: vorticity η : plasma resistivity 𝑇 𝐼𝐼 : magnetic Reynolds number Re Hp : kinematic Reynolds number 𝑔 , 𝑕 = 1 𝑠 ( 𝜖 r 𝑔𝜖 ϑ 𝑕 − 𝜖 𝑠 𝑕𝜖 ϑ 𝑔 ) 𝜐 𝐼𝐼 : poloidal Alfven time 𝑢 → 𝑢 / 𝜐 𝐼𝐼 ( 𝜐 𝐼𝐼 = 𝜈 0 𝜍 𝑛 𝑏 / 𝐶 𝜘 ) • Non-uniform radial mesh 𝑠 → 𝑠 / 𝑏 - 580 grid points with finer mesh around 𝜂 ≡ 𝑨 / 𝑆 0 reconnection region (dr ~ 0.6 mm) Ψ 𝑏 𝜔 = 𝑏𝜁𝐶 𝑨 ( 𝜁 = 𝑆 0 ) • The calculation results converges at least 𝜚 = 𝛼 /( 𝑏 2 𝜁 down to dr ~ 0.1 mm ) 𝜐 𝐼𝐼 9

Fourier representation Fourier transform with respect to the periodic coordinates: • ( ϑ , ζ ) → (m, n) f r, ϑ , ζ , t = � f m , n r, t e i ( mϑ−nζ ) + c. c. m , n −1 η � j m , n − E m , n 𝜖 t ψ m , n = ψ , ϕ m , n + in ϕ m , n − S Hp 2 u m , n −1 𝛼 𝜖 t u m , n = u, ϕ m , n + j, ψ m , n + inj m , n + Re Hp m , n 31 � t = 0 = 1 ψ 2 � Ψ 0 r r − 1 exp m l ϑ ∗ + ϑ 0l + c. c. l=0 Initial perturbation: 32 Fourier modes (m = 0~31) with • - Constant amplitude - Random phase 10

Equilibrium profiles Magnetic flux function & current density are related to q(r) • Initial q-profile defines the equilibrium state • 1 / µ ( r ) q r = q 0 F 1 (r) 1 + ( r ) 2µ ( r ) r 0 −1 /( 2µ rA ) q −1 = − 1 d µ rA m/n dr ψ 0 , 0 r 0 = r A − 1 r q0 r 2 2 ψ 0 , 0 = 1 d µ r = µ 0 + µ 1 r 2 q j 0 , 0 = −𝛼 ⊥ 2 r dr { − r − r 11 } F 1 r = 1 + f 1 exp r 22 11

Parameter translation Well-known experimental parameters are translated into model parameters • during the calculation Experimental parameters Output B 0 = q a B a R 0 /a Major radius R 0 = 1.8 m, • B a = a µ 0 n e m i / τ Hp Minor radius a = 0.5 m, • I p = 2πa B a / µ 0 Toroidal magnetic field B 0 ~ 2 T, • n e = m i ε 0 c 2 /e 2 δ e 2 Plasma current I p ~ 600 kA, • Electron density n e ~ 2.5 × 10 19 /m 3 • Electron temperature T e • Radial q-profile • τ R = µ 0 a 2 / η τ Hp = τ R /S Hp Input Model parameters ν µ 0 a 2 = q a R 0 τ Hp n e m i = 2π n e m i η = 3 plasma viscosity ν m B 0 a 2 µ 0 I p µ 0 • plasma resistivity η • δ e = c m i ε 0 electron skin depth d e • e n e 2 J Ψ → Ψ + d e 12

Application Effect of ECH has been represented as the local current density perturbation of • Gaussian shape (~ 3% of total current) Position & width of the current bump has been controlled case by case • – no constant heat or current source Parameters corresponding to Position • Equilibrium KSTAR has been used: q-profile control 𝑆 0 = 1.8 𝑛 𝑏 = 0.5 𝑛 𝐶 0 ~ 2 𝑈 𝐽 𝐼 ~ 600 𝑙𝑙 ( 𝐶 𝑏 ~ 0.25 𝑈 ) 𝑜 𝑓 ~ 2.8 × 10 19 / 𝑛 3 𝜐 𝐼𝐼 ~ 7.0 × 10 −7 𝑡 𝜐 𝜃 ~ 0.7 𝑡 Order of observed timescales Width • Sawtooth: ~ 𝟐𝟐 𝐧𝐧 - control - Flux tube formation: ~ 2 2 ms ms - Flux tube coalescence: 200 ~ ~ 300 𝛎𝐧 𝑠 / 𝑏 13

Results #1 Equilibrium Black contours: poloidal flux 𝜔 q - profile Colors: current density j 0.4 + 70 𝜈𝑡 0 Equilibrium j - profile ECH resonance 𝑨 / 𝑏 position 𝑠 / 𝑏 0 0.4 𝑠 / 𝑏 + 140 𝜈𝑡 + 60 𝜈𝑡 + 60 𝜈𝑡 Formation of magnetic islands on ECH-covered area 14