Symmetry breaking in Ion Temperature Gradient (ITG) mode and Beta-induced

Alfvén Eigenmode (BAE) Zhixin Lu 1

Collaborating with: E. Fable 1, W. Hornsby 1, A. Bottino 1, C. Angioni 1,

X. Wang 1, Ph. Lauber 1, F. Zonca 2 ,

T. Hayward-Schneider 1, I. Chavdaroski 1 , O. Maj 1

1 Max Planck Institut für Plasmaphysik, Garching, Germany

2C.R. ENEA Frascati, Frascati, Italy

8th IAEA Technical Meeting, Theory of Plasma Instabilities

12-14 June 2017, Vienna, Austria

1

Outline

• Motivation

• Symmetry breaking in ITG mode• Physics models and tools

• Results and analysis of symmetry breaking with global effects• Comparison of different methods

• Generalized tilting angle captures global effects and symmetry breaking

• Symmetry breaking in BAE• Physics models and tools

• Results and analysis of symmetry breaking with EP effects• Analytical model gives good estimate for weak coupled case compared with (X)HMGC

• EPs’ non-perturbative effect induces mode structure distortion and symmetry breaking

• Summary

2

Motivation of symmetry breaking studies: ITG \\• ITG (micro-turbulence) problem

• Up-down asymmetry ⟨𝜃⟩, spectrum asymmetry ⟨𝑘||⟩: related to toroidal

momentum transport (residual stress) [Peeters NF11, Angioni NF12, Diamond NF 13]

• Present local simulation only includes radial propagation (tilting angle) [Camenen NF 11; Abdoul

PPCF15] but not turbulence radial gradient induced symmetry breaking [Gurcan POP10]

• Traditional Ballooning formulism (BF) needs to be generalized for mode structure study• BF frequently used since 1970s theoretically [Culham&PPPL&Texas…] and numerically [FULL, GS2, GKW, HD7…]

• Ballooning parameter 𝜃𝑘 is treated as real except few exceptions [Zonca 92, Lu 15, 17]

• Baseline (global, linear) for symmetry breaking studies is needed• Global mode structure symmetry breaking e.g., ⟨𝑘||⟩, is connected to Π𝑟𝜙

𝑅 and is a key ingredient for interpreting intrinsic rotation in simulation [W. Wang PRL 09, 11, Ku NF 09, Kwon NF12]

• Baseline for neoclassical, higher order corrections [Parra, Barnes 11,13…; Hornsby 17] and nonlinear effects, e.g., zonal flow shear

3

Reynolds stress: 𝜐𝑟 𝜐𝜙 = −𝜒𝜙𝜕 𝜐𝜙 /𝜕𝑟+V 𝜐𝜙 + Π𝑟𝜙𝑅 ; … : flux surface average

residual stress Π𝑟𝜙𝑅 can be generated due to mode structure symmetry breaking, e.g., ⟨𝑘||⟩

Mode structure tilting: related to up-down asymmetry

Motivation of symmetry breaking studies: BAE

• Energetic particle driven Alfvén eigenmode problem• From present experiment (NBI, ICRF generated EP) to future burning plasma (alpha particle),

EP energy, anisotropy are different• Understanding of mode structure with kinetic effect, local-global physics etc, helps predict burning

plasma behavior such as EP transport and thermal ion response to mode

• Co- &counter passing particles have different wave-particle resonance and saturationbehavior [X.Wang&Briguglio 16]

• Besides EPs' non-perturbative effect on mode width [Z.Wang, Lin PRL 13], symmetry breaking in mode pattern can also be affected by EP and helps interpret experimental observations• Interaction between EPs and modes is affected by mode structure such as 𝑘|| and in return, mode

patterns reflect EP information (diagnosis)

• Symmetry breaking can be relevant to wave-wave, wave-particle nonlinearity and EP transport [Chen&Zonca RMP16]

4

Analysis for strongly or weakly coupled poloidal harmonics

• Mode Structure Decomposition (MSD) method can handle both cases in principle [Lu&Zonca POP12]

• In reality, MSD is used in ITG problem (strongly coupled case)• Calculate parallel mode structure first; then radial envelope

• For weakly coupled case (BAE), we use weak coupling formulae here• Radial structure for central poloidal harmonic first

• Theoretical results compared with ORB5, GKW (for ITG) and XHMGC (for BAE) for identifying physics5

ITG BAE

“Generalized ballooning formulism”

Outline

• Motivation

• Symmetry breaking in ITG mode• Physics models and tools

• Results and analysis of symmetry breaking with global effects• Comparison of different methods

• Generalized tilting angle captures global effects and symmetry breaking

• Symmetry breaking in BAE• Physics models and tools

• Results and analysis of symmetry breaking with EP effects• Analytical model gives good estimate for weak coupled case compared with (X)HMGC

• EPs' non-perturbative effect induces mode structure distortion and symmetry breaking

• Summary

6

Part I. Symmetry breaking in ITG mode using mixed approach analysis &

ORB5/GKW simulationCollaborating with: A. Bottino, W. Hornsby, E. Fable, C. Angioni,

Ph. Lauber, F. Zonca, T. Hayward-Schneider

7

More details in Z.X. Lu, et al, Phy. Plasmas, 24, 042502 (2017)

For simulation of ASDEX Upgrade momentum transport, see “Core intrinsic toroidal rotation mechanisms tested against ASDEX Upgrade observations”, W. Hornsby et al

• Mode Structure Decomposition (MSD)：2D 2 coupled 1D problems• 𝛿𝜙 = 2𝜋𝑒𝑖𝑛𝜙−𝑖𝑛𝑞𝜃 𝑝=−∞

∞ 𝐴𝛿 𝜙 𝜃 + 2𝑝𝜋 𝑒−2𝜋𝑖𝑛𝑝𝑞, 𝐴(𝑟) = exp 𝑖∫ 𝑑𝑟𝑛𝑞′𝜃𝑘 𝑟 − 𝑖𝜔𝑡 .

• Parallel equation with 𝛿ℎ obtained by unperturbed orbit integral :

• Implementation: Ritz method using parabolic cylinder functions ℎ𝑙(𝜃)

• Weak form: ∫∞

∞𝒅𝜽𝒉𝒍 𝜽 𝑳𝜹 𝝓 = 𝟎

• Radial envelope equation for 𝐴(𝑟)

Gyrokinetic-Poisson system captures essence of linear ITG

Essence: WKB + asymptotic matching [Lu 2012POP; Chen, Zonca 2016 RMP]; practically: initial value approach is implemented

• Gyrokinetic (GK)-Poisson system:

8

𝑳𝜹 𝝓 ≡ 𝟏 + 𝝉𝒆 𝜹 𝝓 𝜽 + 𝟎

∞

𝒅𝒗∥ 𝟎

∞

𝒅𝒗⊥ −∞

∞

𝒅𝜽′ 𝟐𝝅𝒊𝒗⊥𝝉𝒆

𝒒𝑹

𝒗∥

𝝎 − 𝝎∗𝑻 𝑱𝟎 𝒂 (𝜽′) 𝑭𝟎𝜹 𝝓𝒆𝒙𝒑 𝒊 𝜽′

𝜽

𝒅𝜽"𝝎 − 𝝎𝑫

| 𝒗∥|𝒒𝑹𝑺 = 𝟎

𝝏𝑫𝟎

𝝏𝝎𝟎𝒊

𝝏

𝝏𝒕− 𝝎𝟎 𝑨 +

𝝏𝑫𝟎

𝝏𝝎𝟎−

𝒊

𝒏𝒒′

𝝏

𝝏𝒓− 𝜽𝒌𝟎 𝑨 +

𝟏

𝟐

𝝏𝟐𝑫𝟎

𝝏𝜽𝒌𝟎𝟐 −𝒊

𝝏

𝝏𝒕− 𝝎𝟎

𝟐

−𝒊

𝒏𝒒′

𝝏𝜽𝒌𝟎

𝝏𝒓𝑨 = 𝟎

𝑒𝑛𝑒0

𝑇𝑒𝛿𝜙 = −

𝑒𝑛𝑖0

𝑇𝑖𝛿𝜙 + ∫ 𝑑𝑣3𝐽0 𝑎 𝛿ℎ

Inspired by [FULL, G. Rewoldt; HD7, Dong, Horton]

𝛿ℎ determined by linearized GK equation

𝒂 = 𝒌⊥𝝆𝒊

Ballooning Formulism

Comparison with ORB5: parameter scan follows the trend (WPC)

• Growth & frequency using theoretical model agree with OBR5 reasonably

• The key point of this work will be the qualitative difference due to the introduction of a complex local parameter 𝜃𝑘, in particular 𝐼𝑚{𝜃𝑘} , as shown later

WPC (wave packet calculation) [Lu, POP12,17]

QT (quasi-toroidal model)HD7: local eigenvalue code [Dong, Horton 92]

9

Toroidal mode #

𝐼𝑚{𝜃𝑘} is needed to describe amplitude variation• Cyclone case for ITG

• 𝜃𝑘 describes phase & amplitude variation of envelope 𝐴 𝑟 = exp{𝑖∫ 𝑑𝑟 𝑛𝑞′𝜃𝑘}

• 𝑅𝑒 𝜃𝑘 describes “tilting” (traditional), 𝐼𝑚{𝜃𝑘} describes amplitude gradient (not considered usually)

• 𝐼𝑚{𝜃𝑘} can be large for linear ITG mode; or in linear stable region during nonlinear stage

𝑹𝒆 𝜽𝒌

ORB5 result: 𝑅𝑒 𝜃𝑘 ~0.5,

𝐼𝑚{𝜃𝑘}~0.5 at𝑟

𝑎~0.3, 0.6

𝑨 𝒓

10

𝑅𝑒{𝜃𝑘}, 𝐼𝑚{𝜃𝑘}: related to decrease and increase of growth rate

• 𝛾 decreases as 𝑅𝑒{𝜃𝑘} increases along 𝐼𝑚{𝜃𝑘} = 0

• 𝛾 increases as 𝐼𝑚{𝜃𝑘} increases along 𝑅𝑒{𝜃𝑘} = 0

• |𝑅𝑒[Ω]| also depends on 𝑅𝑒{𝜃𝑘} and 𝐼𝑚{𝜃𝑘}Growth

11

𝑰𝒎{𝜽

𝒌}

𝑰𝒎{𝜽

𝒌}

𝑹𝒆{𝜽𝒌} 𝑹𝒆{𝜽𝒌}

Frequency

Global eigenvalue and structure can be calculated using local results with complex 𝜃𝑘 correction

• Combination of local solution with 𝜃𝑘leads to global one• Local 𝛾 decreases near 𝑟 ≈ 0.45 (strong

drive) mainly due to 𝑅𝑒{𝜃𝑘};

• 𝛾 increases outside 𝑟 ∈ (0.38,0. 43) mainly due to 𝐼𝑚{𝜃𝑘}

• The “global oriented local simulation” helps local simulation to include global effects, by adding the complex 𝜃𝑘 term

global

Local, 𝜽𝒌 = 𝟎

global Local𝜽𝒌 = 𝟎

Growth

Frequency

[Lu, POP 17]

12

Symmetry breaking ⟨𝑘||⟩ and ⟨𝜃⟩ depend on 𝑅𝑒{𝜃𝑘}, 𝐼𝑚{𝜃𝑘}

• Former conclusion 𝜃 ∼ 𝑅𝑒{𝜃𝑘} can be reproduced in high 𝑠, 𝑘⊥𝜌𝑖 limit [e.g., Romanelli 93]

• For the (cyclone) case 𝑛 = 20, dependence on 𝑅𝑒{𝜃𝑘}, 𝐼𝑚{𝜃𝑘} is more complicated

13

𝑰𝒎{𝜽

𝒌}

𝑰𝒎{𝜽

𝒌}

𝑹𝒆{𝜽𝒌} 𝑹𝒆{𝜽𝒌}

⟨… ⟩: flux surface average; 𝜽 = ∫𝒅𝜽𝜽| 𝝓𝟐/∫𝒅𝜽| 𝝓

𝟐

14

Complex 𝜃𝑘 is necessary for symmetry breaking calculation

• With complex 𝜃𝑘, WPC (theoretical calculation) agrees with ORB5&GKW

• Linear stage, 𝑅𝑒{𝜃𝑘} dominates near drive center; 𝐼𝑚{𝜃𝑘} more important away from drive center

• 𝐼𝑚{𝜃𝑘} can be important for ITB, pedestal, non-uniformly heated plasma and turbulence spreading front

𝜽𝒌

||

𝒓/𝒂

MRC: Mapping space-Real space Connection Formula

Global oriented local simulation with complex 𝜽𝒌 captures symmetry breaking and global effect

• Connection between symmetry breaking variables

𝜽𝒌 = ⟨𝒌𝒓⟩/(nq’)+⟨𝜽⟩• 𝑅𝑒{𝜃𝑘} = 𝑅𝑒⟨𝑘𝑟⟩/(nq’)+⟨𝜃⟩

shows connection between tilting

angle, radial propagation and

average 𝜃 (parallel-radial connection)

• Local parameter 𝜃𝑘 should be consistent with global 𝑅𝑒⟨𝑘𝑟⟩/(nq’)+⟨𝜃⟩ in order to include global effect (global-local connection)

𝑰𝒎{𝜽

𝒌}

𝑹𝒆{𝜽

𝒌}

15

• GKW, WPC calculate 𝜽𝒌 directly (in Clebsch coordinates)• ORB5 calculates ⟨𝒌𝒓⟩ and ⟨𝜽⟩ and then construct 𝜽𝒌 according

to global-local connection (GLC)• Agreement supports 𝜽𝒌 = ⟨𝒌𝒓⟩/(nq’)+⟨𝜽⟩

Part II. Symmetry breaking in Beta induced Alfvén Eigenmode (BAE)

Collaborating with: X. Wang, Ph. Lauber, F. Zonca,

O. Maj, T. Hayward-Schneider, I. Chavdaroski

16

More details will be in [Z.X. Lu, X. Wang, Ph. Lauber, F. Zonca, Mode structure symmetry

breaking of energetic particle driven Beta-induced Alfvén Eigenmode], in preparation

Model: Gyrokinetic equation with kinetic thermal ion & EP• Physics model

• Consider 2D mode structure & non-perturbative EP effect in Gyrokinetic Eq. (GKE), Vorticity Eq. (VE), Quasi-Neutrality Eq. (QNE) [Zonca 96]

17

• Approximations for analytical BAE solution• Weak coupling between 𝛿𝜓𝑚 and 𝛿𝜓𝑚±1, 𝛿𝜓𝑚±2… for weak 𝑠, low n (verified posteriori)

• Unperturbed orbit integral is used to get radial equation of central poloidal harmonic

GKE: 𝒗||𝝏∥ − 𝒊(𝝎 − 𝝎𝒅) 𝜹𝑲 = 𝒊𝒆

𝒎𝑸𝑭𝟎 𝑱𝟎 𝒂 𝜹𝝓 − 𝜹𝝍 +

𝝎𝒅

𝝎𝑱𝟎 𝒂 𝜹𝝍 ,

VE: 𝑩𝝏∥𝟏

𝑩

𝒌⊥𝟐

𝒌𝜽𝟐 𝝏∥𝜹𝝍 +

𝝎𝟐

𝒗𝑨𝟐 𝟏 −

𝝎∗𝒑𝒊

𝝎

𝒌⊥𝟐

𝒌𝜽𝟐 𝜹𝝓 +

𝜶

𝒒𝟐𝑹𝟐 𝒈𝜹𝝍 = 𝒔𝟒𝝅𝒆𝒔

𝒄𝟐𝒌𝜽𝟐 𝑱𝟎 𝒂 𝝎𝝎𝒅𝒔𝜹𝑲𝒔 ,

QNE: 𝟏 −𝟏

𝝉𝒆𝜹𝝓 − 𝜹𝝍 + 𝟏 −

𝝎∗𝒑𝒊

𝝎𝒃𝒊𝜹𝝍 =

𝑻𝒊

𝒏𝒆𝑱𝟎 𝒂 𝜹𝑲𝒔 ,

𝒃𝒊 = 𝒌⊥𝟐𝝆𝒕𝒊

𝟐 /𝟐, 𝝆𝒕𝒊 = 𝒄𝒎𝒊𝒗𝒕𝒊/(𝒆𝑩), 𝒗𝒕𝒊 = 𝟐𝑻𝒊/𝒎𝒊, 𝜶 = −𝑹𝟎𝒒𝟐𝜷′, 𝒈 = 𝒄𝒐𝒔𝜽 + 𝒌𝒓/𝒌𝜽 𝒔𝒊𝒏𝜽 .

2D equation for BAE excited by EP• Vorticity Eq. with EP contribution

18

• With slowing down EP, 𝐹𝑓 = 𝐶0𝛿 𝛬 − 𝛬0 / 𝜖3/2 for 𝜖 < 𝜖𝑏 (EP born energy),

• Glocal effects along 𝑟 & EP anti-Hermitian part are maintained

• 𝝎𝑩𝑨𝑬 (BAE frequency at accumulation point), recovers 𝜏𝑒 + 7/4 𝑣𝑡𝑖/𝑅 result [Zonca96, Lauber 09]

• 𝑲𝑷𝑪𝒇: Kinetic Particle Compression of EP (BAE drive)

𝑩𝝏∥

𝟏

𝑩𝛁⊥

𝟐𝝏∥𝜹𝝍 + 𝛁⊥ ⋅𝝎𝟐 − 𝝎𝑩𝑨𝑬

𝟐

𝒗𝑨𝟐 𝛁⊥𝜹𝝍 + 𝑲𝑷𝑪𝒇 = 𝟎

𝑲𝑷𝑪𝒇 = 𝟐 𝑵𝒇𝒗𝒕𝒇

𝟐

𝒗𝑨𝟐𝑹𝟐

𝝎∗𝒇

𝝎

𝟏−𝚲𝟎𝐁/𝟐 𝟐

𝟏−𝚲𝟎𝐁𝜹𝝍𝒎 𝝈 𝑲𝝈𝜻𝝈

𝟐 𝟏 +𝜻𝝈

𝟐

𝟐 𝟏−𝚲𝟎𝐁 𝝐𝒃𝒍𝒏 𝟏 −

𝟐 𝟏−𝚲𝟎𝐁 𝝐𝒃

𝜻𝝈𝟐 ,

𝝎∗𝒇 = −𝝆𝒇𝒗𝒕𝒇𝒌𝜽

𝟐𝑳𝒏𝒇, 𝑲𝝈 ≈

𝒌⊥𝟐

𝟒𝒆𝒊𝒎𝜽, 𝜻𝝈 =

𝝎𝒒𝑹

𝒗𝒕𝒇 𝒎−𝒏𝒒+𝝈, 𝝐𝒃: normalized EP born energy

𝐾𝑃𝐶𝑓 is obtained analytically

EP drive included

Radial structure and reduction to local solution

• Radial structure equation for central poloidal harmonic 𝛿𝜓𝑚

• Large aspect ratio, uniform thermal ion assumed

19

• 𝑌0 = 0 gives local solution at rational mode surface with kinetic effects of thermal ion and energetic particle

• Solution for 𝛿𝜓𝑚±1, 𝛿𝜓𝑚±2… & correction for 𝛿𝜓𝑚 can be obtained order by order

𝒀𝟐𝝏𝟐

𝝏𝒓𝟐 + 𝒀𝟏𝝏

𝝏𝒓+ 𝒀𝟎 𝜹𝝍𝒎 = 𝟎 ,

𝒀𝟎 𝒓 =𝒎𝟐

𝒓𝟐

𝝎𝟐 − 𝝎𝑩𝑨𝑬𝟐

𝒗𝑨𝟐 − 𝒌∥

𝟐 +𝒌∥𝒌∥

′

r+𝒌∥𝒌∥

′′ +𝑲𝑷𝑪𝒇,𝜽 .

Theory and simulation comparison: local growth rate follows trend of global simulation

• EP density and q profiles• Uniform EP temperature, varying density

• Low 𝑠 ~0.1 at 𝑟/𝑎 = 0.5; n=2

• Local solution gives reasonable estimate of global BAE eigenvalue from XHMGC

XHMGC

Local theory

Fre

qu

en

cyG

row

th

EP density on axis

Local theory

XHMGC

r/a

𝒏𝒇

𝒒

𝒔

−𝒅 𝒏𝒇/𝒅 𝒓

20

BAE Symmetry Breaking: due to non-perturbative EP contribution

• A simplified 𝐾𝑃𝐶𝑓 term is used for testing EPs’ non-perturbative effect

• As EP drive center 𝑟𝑐𝑓 shifts away from rational mode surface, BAE mode structure symmetry breaks

• Non-perturbative effects of EP ⟹radial structure symm. breaking (preferential radial propagation) &parallel symm. Breaking, i.e., ⟨𝑘||⟩

21

𝑲𝑷𝑪𝒇 = 𝟒𝒊 𝒆𝒙𝒑 − 𝒓 − 𝒓𝒄𝒇 /𝟎. 𝟏𝟐

𝒓𝒄𝒇 = 0.5 (rational mode surface) 𝒓𝒄𝒇 = 0.55

More realistic EP term yields solution similar to XHMGC results

• Analytic results (top & middle) and XHMGC results (bottom) agree on mode pattern distortion

• Central column: Boomerang-like 2D structure (up-down symmetry breaking); tails relatively symmetric

Observed for TAE [ORB5, Biancalani] and high 𝑛 BAE (theory) [Ma, POP 15]

• Left & right columns: asymmetric tails for shifted EP drive (radial symmetry breaking)

XHMGC

analytic

analytic

XHMGC

analytic

analytic

XHMGC

analytic

analytic

𝒓𝒄𝒇=0.42 𝒓𝒄𝒇=0.5, q=m/n 𝒓𝒄𝒇=0.55

22

𝑘𝑟 symmetry breaking: mismatch of Re 𝑘𝑟 = 0 & Im 𝑘𝑟 = 0corresponds to asymmetric radial propagation as EP drive shifts

Mismatch

Match

23

• Im 𝑘𝑟 = 0 overlaps with max. amplitude, since

𝐴 𝑟 ≡ exp 𝑖∫ 𝑘𝑟𝑑𝑟

• For 𝒓𝒄𝒇=0.5, Re 𝑘𝑟 = 0 at Im 𝑘𝑟 = 0location (match)

• For 𝒓𝒄𝒇=0.55, Re 𝑘𝑟 ≠ 0 at Im 𝑘𝑟 = 0location (mismatch), indicating preferential radial propagation

• Complex Gaussian beam 𝐴 𝑟 = exp{−𝜎 𝑟 − 𝑟0

2}describes radial envelope symmetry breaking

• Anti-Hermitian part complex (𝜎, 𝑟0) symmetry breaking in 𝐴 𝑟

𝒓𝒄𝒇=0.5, q=m/n 𝒓𝒄𝒇=0.55

XHMGC XHMGC

Symm. Break. of ⟨𝑘||⟩: volume averaged ⟨𝑘||⟩𝑉 can be generated

24

XHMGCTheory

• The theoretical result captures the key feature of the parallel symmetry breaking• Opposite sign of ⟨𝑘||⟩ on the two sides of EP drive center

• Net ⟨𝑘||⟩𝑉 in the whole volume can be generated as EP drive shifts away from 𝑟/𝑎 = 0.5

Symmetry breaking of 𝑘𝑟 and ⟨𝑘||⟩: radial & parallel asymmetry are intimately coupled

• As EP drive center shifts away from 𝑟/𝑎 = 0.5, the volume averaged ⟨𝑘𝑟⟩ and ⟨𝑘||⟩ both increase in magnitude• Application of finite element method, inclusion of geometric factors, e.g., 1/𝑟, and higher order correction

due to poloidal coupling is expected to improve the agreement of theoretical calculation with XHMGC

25

Volume averaged ⟨𝑘𝑟⟩ and ⟨𝑘||⟩ from theory (left) and XHMGC (right)

Vo

lum

e a

vg⟨𝑘

||⟩ V

olu

me

avg⟨𝑘

𝑟 ⟩

Vo

lum

e avg⟨𝑘

𝑟 ⟩ Vo

lum

e a

vg⟨𝑘

||⟩

Conclusion: BAE symmetry breaking

• 𝐼𝑚{𝜃𝑘} is a key ingredient for global solution construction from local solution

• 𝐼𝑚{𝜃𝑘} & 𝑅𝑒{𝜃𝑘} describe radial propagation & intensity gradient

• Symmetry breaking ⟨𝜃⟩ and ⟨𝑘||⟩ depend on 𝐼𝑚{𝜃𝑘} and 𝑅𝑒{𝜃𝑘}

• “Global oriented local simulation” is proposed to include global effects in local simulation

Conclusion: ITG symmetry breaking

• Anti-Hermitian contribution from EP leads to non-perturbative mode structure symmetry breaking

• Radial shift of EP drive leads to the symmetry breaking of BAE mode structure

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Outlook (BAE)

• Nonlinear physics• Intrinsic torque change during nonlinear stage considering spectrum change and zonal flow shearing• Effects of nonlinear terms on mode structure and symmetry breaking

• Study relevant to ASDEX Upgrade experimental parameters• Global effects and nonlinear effects on rotation profile generation using ORB5 and its comparison with GKW

• Scaling of symmetry breaking and intrinsic torque (𝑞, 𝑠, 𝑇′, 𝑇′′, 𝑛′, 𝑛′′…) using mixed approach

Outlook (micro-turbulence)

• Non-perturbative effect of EPs on growth rate and mode structure; parameter dependencies• More realistic parameters will be considered (ITER & CFETR)

• Semi-analytical/numerical estimate of FLR/FOW and nonlinear effects on symmetry breaking as complement to XHMGC/LIGKA/ORB5 and GFLDR

• The effects of symmetry breaking on EP transport and thermal ion response• Symmetry breaking links the BAE physics with the EP transport and thermal ion momentum transport

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Backup

• ITG model and implementation

• BAE local and global solution

• Heuristic radial solution with symmetry breaking

• GFLDR with symmetry breaking

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Vlasov-Poisson with approx. captures essence of linear ITG

• Model: Vlasov-Poisson system for ITG

• || Eq.

• Radial Eq.

Essence: WKB + asymptotic matching [Lu 2012POP; Chen, Zonca 2016 RMP]; practically: initial value approach is implemented

• Implementation: physics oriented• Local: Ritz method (complex space)

• Radial: initial value method• Problem: local global

Inspired by:[FULL, G. Rewoldt],[HD7, Dong, Horton],but different…

• Mode Structure Decomposition (MSD)：2D 2 couple 1D problems

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BAE global solution obtained by solving radial equation

• Generally, local solution can be different than the continuous spectrum

• Global solution can be viewed as a suitable superposition of local solutions

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Global and local eigenvalue of BAE

EP impacts mode structure (non-perturbative effect)• Theoretical derivation

• Complex Gaussian(Weber) solution captures essence

• Heuristic solution constructed

• 𝐴 𝑟 = exp −𝜎 𝑟 − 𝑟02 , 𝜎 ≡ −𝜒.

• 𝑟𝑅, 𝑟𝐼 are related to Hermitian and anti-Hermitian parts of the wave equation.

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GFLDR with symmetry breaking: anti-Hermitian contribution from EP leads to non-perturbative mode structure distortion

• Mode Structure Decomposition: 2D problem cast to 2 coupled 1D problems• Perturbed variable:

• Ampere-Quasi-neutrality-Gyrokinetic system:

• Radial solution connects EP anti-Hermitian contribution with symmetry breaking

⟹

Generalized Fishbone-like Dispersion Relation (FLDR) [Zonca, 96,06]

where Im{𝑟0} leads to 𝑟𝑅& 𝑟𝐼mismatch&radial symm. break.

Similar idea as ITG problem [Lu,POP 2012,2015]

Solution near stationary point

Logic: 𝐼𝑚{𝛿𝑊𝑘} ⟹ 𝐼𝑚{𝑟0} ⟹ 𝐴(𝑟) symm. Break. & || symm. break.

⟹

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