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

Nu Numerical stud udy o

• f the

e com

• mplex

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. Bierwagea

Department of physics, POSTECH, Pohang, 790-784, Korea

aJAEA, 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

• bserved 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

• n 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 Bt, Ip, ne, and Te which translate to plasma resistivity η, kinematic viscosity ν, electron skin depth δe 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

• G. S. Yun et al, Phys. Rev. Lett. 109

09, 145003 (2012)

R(cm) Z(cm)

ECEI images of multiple flux tubes from KSTAR experiments

ECE 2-D image at t = 3.189845s (#6024) ECE 2-D image at t = 3.498780s (#6024)

3

Pattern variation with ECH condition

• The generation & control

mechanism of multiple flux tubes are not clear

• Statistical analysis indicates that

– ECH resonance position – Radial coverage of ECH beam

affects the overall dynamics

• Resear

earch o

• bjec

ective: e: Numer eric ical ally r reproduce t e the d e dynam amic ics

• f m

multip iple e fluxes es p phen enomen ena with ith known c cond ndit itio ions ns & & param ameter ers

Poloidal view of ECH injection

• G. H. Choe et al, KSTAR Conference 2013, Poster #50 on Feb. 27th

Inversion radius Plasma center ECH resonance position larger r, wider coverage Smaller r, narrower coverage

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

𝜖tΨ = −𝚪 ⋅ 𝛼𝛼 + η µ0 𝛼⊥

2Ψ − Ez

ρmdt𝛼⊥

2𝛼 = − 1

µ0 B ⋅ 𝛼𝛼⊥

2Ψ + νm𝛼⊥ 2𝛼⊥ 2𝛼 Ψ: magnetic flux function

𝛼: electrostatic potential η: plasma resistivity ρm: mass density νm: kinematic viscosity

• H. R. Strauss, Phys. Fluids. 19, 134 (1976)

5

Reduced MHD equations: derivation

6

𝛼 ∙ 𝐅 = ρ ϵ0 −𝛼 × 𝐅 = 𝜖𝐂 𝜖𝜖 𝛼 × 𝐂 = µ0𝐊 𝛼 ∙ 𝐂 = 0

• Maxwell’s equations
• Generalized Ohm’s law

𝐅 + 𝐰 × 𝐂 = η𝐊

• Equation of motion

ρm d𝐰 dt = 𝐊 × 𝐂 − 𝛼𝛼 − νm𝛼 × 𝛼 × 𝐰 + 𝐓p 1 2 3 4 5 6 𝜖𝐂 𝜖𝜖 = 𝛼 × 𝐰 × 𝐂 − η𝛼 × 𝐊 2, 5 7 High-aspect-ratio ordering (𝜁 ≡

𝑏 𝑆0)

Strong axial magnetic field: 𝐂 = B0𝐴 + 𝛼Ψ × 𝐴

• ExB drift is dominant:

𝐰 = 𝐰𝐅×𝑪 = −𝛼𝛼 × 𝐴

• B0

8 9

7

Reduced MHD equations: derivation

𝜖𝐂 𝜖𝜖 = 𝛼 × 𝐰 × 𝐂 − η𝛼 × 𝐊 7

𝐂 = 𝛂 × 𝐁 Surface integration & Stoke’s theorem

𝜖 𝜖𝜖 𝐁 ∙ d𝐦 = 𝐰 × 𝐂 ∙ d𝐦 − η 𝐊 ∙ d𝐦 𝜖𝐁 𝜖𝜖 = 𝐰 × 𝐂 − η𝐊 + 𝛼ϕ 11

Take out ∮ 𝑒𝒎

𝜖Ψ 𝜖𝜖 = −𝐂 ⋅ 𝛼𝛼 + η µ0 𝛼⊥

2Ψ − Ez

Consider only z- component of 𝐁: 𝐁 ∙ 𝐴

• 𝐰 = −𝛼𝛼 × 𝐴
• B0

ϕ = B0𝛼 Jz = − 1 µ0 𝛼⊥

2Ψ

9 12 13 𝐂 = B0𝐴 + 𝛼Ψ × 𝐴

• 8

𝛼 × 𝐂 = µ0𝐊 4

10

Ψ = 𝐁 ∙ 𝐴

• 14

𝐂 = B0𝐴 + 𝛼Ψ × 𝐴

• 8

Additional static E - field

8

Reduced MHD equations: derivation

ρm d𝐰 dt = 𝐊 × 𝐂 − 𝛼𝛼 − νm𝛼 × 𝛼 × 𝐰 + 𝐓p 6

𝛼 ×

ρm d(𝛼 × 𝐰) dt = 𝛼 × {𝐊 × 𝐂 − 𝛼𝛼 − νm𝛼 × 𝛼 × 𝐰 + 𝐓p}

𝐰 = −𝛼𝛼 × 𝐴

• B0

9 Consider the only z- component of 𝛼 × 𝐰

15

ρmdt𝛼⊥

2𝛼 = − 1

µ0 B ⋅ 𝛼𝛼⊥

2Ψ + νm𝛼⊥ 2𝛼⊥ 2𝛼

Jz = − 1 µ0 𝛼⊥

2Ψ

13

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

𝜖tψ = ψ, ϕ − 𝜖ζϕ − SHp

−1 η

j − Eζ 𝜖tu = u, ϕ + j, ψ + 𝜖ζj + ReHp

−1𝛼⊥ 2u

j: axial current density u: vorticity η: plasma resistivity 𝑇𝐼𝐼 : magnetic Reynolds number ReHp: kinematic Reynolds number 𝜐𝐼𝐼 : poloidal Alfven time

𝑔, 𝑕 = 1 𝑠 (𝜖r𝑔𝜖ϑ𝑕 − 𝜖𝑠𝑕𝜖ϑ𝑔)

𝑢 → 𝑢/𝜐𝐼𝐼 (𝜐𝐼𝐼 = 𝜈0𝜍𝑛𝑏/𝐶𝜘) 𝑠 → 𝑠/𝑏 𝜂 ≡ 𝑨/𝑆0 𝜔 =

Ψ 𝑏𝜁𝐶𝑨 (𝜁 = 𝑏 𝑆0)

𝜚 = 𝛼/(𝑏2𝜁 𝜐𝐼𝐼 )

9

• Non-uniform radial mesh
• 580 grid points with finer mesh around

reconnection region (dr ~ 0.6 mm)

• The calculation results converges at least

down to dr ~ 0.1 mm

Fourier representation

• Fourier transform with respect to the periodic coordinates:

(ϑ, ζ) → (m, n)

𝜖tψm,n = ψ, ϕ m,n + inϕm,n − SHp

−1 η

jm,n − Em,n 𝜖tum,n = u, ϕ m,n + j, ψ m,n + injm,n + ReHp

−1𝛼 m,n 2 um,n f r, ϑ, ζ, t = fm,n r, t ei(mϑ−nζ) + c. c.

m,n

ψ t = 0 = 1 2 Ψ0r r − 1 exp ml ϑ∗ + ϑ0l + c. c.

31 l=0

• Initial perturbation: 32 Fourier modes (m = 0~31) with
• Constant amplitude
• Random phase

10

Equilibrium profiles

q r = q0F1(r) 1 + ( r r0 )2µ(r)

1/µ(r)

q−1 = − 1 r d dr ψ0,0 j0,0 = −𝛼⊥

2ψ0,0 = 1

r d dr r2 q

• Magnetic flux function & current density are related to q(r)
• Initial q-profile defines the equilibrium state

r0 = rA m/n q0

µ rA

− 1

−1/(2µ rA )

µ r = µ0 + µ1r2 F1 r = 1 + f1exp {− r − r11 r22

2

}

11

12

• plasma viscosity νm
• plasma resistivity η
• electron skin depth de

Model parameters

Ψ → Ψ + de

2J

• Major radius R0 = 1.8 m,
• Minor radius a = 0.5 m,
• Toroidal magnetic field B0 ~ 2 T,
• Plasma current Ip ~ 600 kA,
• Electron density ne ~ 2.5 × 1019/m3
• Electron temperature Te
• Radial q-profile

Experimental parameters

τR = µ0a2/η τHp = τR/SHp ν η = τHp µ0a2 = qaR0 B0a2 nemi µ0 = 2π Ip nemi µ0

3

δe = c e miε0 ne B0 = qaBaR0/a Ba = a µ0nemi/τHp Ip = 2πaBa/µ0 ne = miε0c2/e2δe

2

Input

Output

Parameter translation

• Well-known experimental parameters are translated into model parameters

during the calculation

• Parameters corresponding to

KSTAR has been used:

• Order of observed timescales
• Sawtooth: ~ 𝟐𝟐 𝐧𝐧
• Flux tube formation:

~ 2 2 ms ms

• Flux tube coalescence:

200 ~ ~ 300 𝛎𝐧

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

𝑆0 = 1.8 𝑛 𝑏 = 0.5 𝑛 𝐶0 ~ 2 𝑈 𝐽𝐼 ~ 600 𝑙𝑙 (𝐶𝑏 ~ 0.25 𝑈) 𝑜𝑓 ~ 2.8 × 1019/𝑛3 𝜐𝐼𝐼 ~ 7.0 × 10−7𝑡 𝜐𝜃 ~ 0.7 𝑡

Width control Position control

Equilibrium q-profile 𝑠/𝑏 13

0 0.4 0.4

𝑨/𝑏 𝑠/𝑏

Black contours: poloidal flux 𝜔 Colors: current density j + 70 𝜈𝑡 + 140 𝜈𝑡 + 60 𝜈𝑡 + 60 𝜈𝑡

Results #1

Equilibrium q - profile Equilibrium j - profile

ECH resonance position

𝑠/𝑏

Formation of magnetic islands

• n ECH-covered area

14

0 0.4 0.4

𝑨/𝑏 𝑠/𝑏

+ 130 𝜈𝑡 + 40 𝜈𝑡 + 20 𝜈𝑡 + 20 𝜈𝑡

Results #2

Equilibrium q - profile Equilibrium j - profile

ECH resonance position

𝑠/𝑏

Merging islands

Black contours: poloidal flux 𝜔 Colors: current density j

15

0 0.4 0.4

𝑨/𝑏 𝑠/𝑏

+ 130 𝜈𝑡 + 110 𝜈𝑡 + 70 𝜈𝑡 + 310 𝜈𝑡

Results #3

Equilibrium q - profile Equilibrium j - profile

ECH resonance position

𝑠/𝑏

Formation of many small islands on ECH-covered area

Black contours: poloidal flux 𝜔 Colors: current density j

16

Summary and future work

• Multiple flux tubes under ECH have been observed by ECEI on KSTAR
• The nonlinear dynamics of flux tubes under ECH are numerically calculated by sets
• f reduced MHD equations with known KSTAR parameters
• ECH conditions are applied as a local j perturbation on equilibrium j-profile
• Cold magnetic islands are formed on ECH-covered area: heated islands might

appear as hot flux tubes on ∆𝑈

<𝑈> view of ECEI images

• The time scales of calculated dynamics are much faster than those of observations:

Calculation with more conditions should be done to find out the matching case

• Heat & current source term representing the ECH should be added on base

equations in future

• Calculation of pitch angle from 3D ECEI measurements will give information of

varying q~1 surface applicable for the simulation in future

17

supplements

18

Assumptions on RMHD model

• The plasma is fully ionized and quasi-neutral
• Each fluid has a scalar pressure
• Momentum exchange between ions and electrons is assumed

proportional to their relative mean velocities

• Neglect displacement current

v c ≪ 1

• Neglect electron inertia term in Ohm’s law

ω ωpe ≪ v c

• Neglect Hall term in Ohms’ law

ωΩe ωpe

2 ≪

v c 2

• Neglect pressure gradient in Ohm’s law

ω Ωi ≪ v vti 2

• No mass source, constant mass densitySM = 0, ρm = const.
• All transport coefficients are scalars
• Transport coefficients are taken to be independent of time and space
• The resistivity is small and can be used as an expansion parameter

19