1

A mechanism for the strong excitation of

zonal modes during an edge pedestal

collapse

Hogun Jhang

National Fusion Research Institute, Korea

Collaborators*: R. Singh, S. S. Kim, Helen H Kaang

8th IAEA Technical Meeting on Theory of Plasma Instabilities

June. 12-14, 2017 Vienna, Austria

2

Outline

Introduction

Nonlinear MHD simulations of a pedestal collapse

Model description

Features

Impact of zonal modes

Mechanism for generation and bursts of zonal modes

Linear phase: Coherent nonlinear (NL) interaction

Nonlinear phase: NL instability

Conclusions

3

Introduction

Pedestal collapse dynamics involves rich nonlinear processes

many of which yet to be elucidated.

Strong NL evolution of ideal BMs [Wilson and Cowley PRL 2004]

Turbulence-MHD interaction [Xi et. al. PRL 2014, Lee et. al., PRL 2016]

Pedestal recovery after an ELM

Recent reduced MHD simulations reveal interesting nonlinear

dynamical processes in pedestal collapse:

Strong stochastization of magnetic fields and ensuing energy release [T.

Rhee et. al. NF 2015]

Impact of zonal flows [H Jhang et. al. NF 2017]

4

Inclusion of ZF dynamics (minimal level of nonlinearity) leads to

surprisingly diverse phenomena [H Jhang et. al. NF 2017]:

GAMs are generated due to strong up-down asymmetry of pressure

perturbations (geodesic curvature coupling)

Quasi-periodic bursts after the main ELM likely due to the

destabilization of an instability by strong zonal modes!

Theoretical mechanism for such a strong GAM activity not explored.

This talk:

I. Review important features of pedestal collapse simulations

II. Discuss physics picture for generation and bursts of secondary

activities during/after a pedestal collapse.

5

Model

90 10/ ARVS

1230 10/ HAH VRS

Resistivity:

Hyper-resistivity [Xu, et. al., PRL 2010]

Reduced 3-field MHD equations keeping U00 and P10

Implemented in BOUT++ framework [B Dudson et. al., PPCF 2011]

P

en

1Φ

B

1U 22

0

Geodesic Curvature Coupling (GCC)

6

Simulation features

No sources/sinks Not flux-driven simulations

Monotonic q-profile: ,

Initiate simulations from a strongly unstable initial pressure profile with

a single unstable mode (n=20) Ideal MHD completely stabilized when

t > 65 tA.

7

Field line stochastization

Strong stochastization of magnetic field lines during a pedestal collapse

due to the generation of a series of nonlinearly driven tearing modes

from initially unstable ballooning modes (BMs) [T Rhee et. al. NF 2015]

8

ZF governs dynamics in later stage

After an initial crash, a series of smaller secondary crashes occur in

later stage of a pedestal collapse (i.e. when t 100tA)

effectively prolongs the crash time and enhances eventual

energy loss

9

Geodesic curvature coupling (GCC) responsible

GCC becomes the dominant driver for zonal mode due to the strong

cancellation of Reynolds and Maxwell stresses [Naulin et al PoP 2005]

A similar evolution to the “w/o ZF” case observed when the GCC term in

U00 is neglected.

10

GAM driven by GCC!

|P10|2 is persistent even when t >120 tA.

Quasi-periodic bursts of |P10| resonate

with secondary crashes

Radial propagation of GAM oscillations

< U00 >GCC governs <U00> in later stage

Reynolds stress GCC

11

Linear phase: features and model

Dynamic features are distinct for linear and nonlinear phases

Perform a separate analysis for two phases

Linear phase - features:

No GAM oscillations

P10 ∝ exp(2g0t) (g0: growth rate of the BM): no sign of F00 growth

Coherent NL interaction

Abrupt growth near the

end of the main crash.

2g0

𝜕𝛻⊥2Φ00

𝜕𝑡≃ 2 < 𝐛0 × 𝜿𝟎 ⋅ 𝛻P10 >

𝜕P10

𝜕𝑡≃ − Φ𝑚0±1,𝑛0

, 𝑃𝑚0𝑛0

12

Linear phase: physics picture

A physics picture hypothesized leading to secondary mode

from initial BMs:

𝜕𝑃𝑚0𝑛0

𝜕𝑡= 𝛾𝑷𝒎𝟎𝒏𝟎

= − Φ𝑚0𝑛0, 𝑃0

𝜕𝛻⊥2Φ𝑚0±1,𝑛0

𝜕𝑡= 𝛾𝛻⊥

2Φ𝑚0±1,𝑛0= 2𝐛𝟎 × 𝜿𝟎 ⋅ 𝛻P𝑚0𝑛0

𝜕P10

𝜕𝑡= 2𝛾𝐏𝟏𝟎 ≃ − Φ𝑚0±1𝑛0

, 𝑃𝑚0𝑛0

𝜕𝛻⊥2Φ00

𝜕𝑡= 2𝛾𝛻⊥

2𝚽𝟎𝟎 = 2 < 𝐛𝟎 × 𝜿𝟎 ⋅ 𝛻P10 >

Linear drive

Toroidicity

Coherent NL coupling

GCC

13

Linear phase: P10 grows not F00!

In the large aspect ratio limit:

𝐏𝟏𝟎 grows with ~exp(2g0t) /g02 growth while 𝚽𝟎𝟎 is negligible

Pc is driven predominantly (unless BMs rotates poloidally)

For large g0, 𝐏𝟏𝟎 does not grow strongly (though fast)

Do not have a big influence on initial stochastization dynamics

When g0 0 with finite fluctuation level (near the end of crash)

𝐏𝐜 exhibits linear growth with strong gradient

provide a seed for GAM generation in NL phase

𝐏𝟏𝟎 ≃ 𝐏𝒄 ≃𝑣𝑡ℎ

2

𝑅

1

𝛾02

𝜕 𝑃𝑚0𝑛0

2

𝜕𝑟exp 2𝛾0𝑡 , 𝐏𝒔 = 0

𝚽𝟎𝟎 ≃ 0

14

Nonlinear phase: features and model

Features:

Persistency of strong levels of 𝐏𝟏𝟎 and 𝚽𝟎𝟎

Distinct GAM oscillations and radial propagation:

Quasi-periodic bursts of 𝐏𝟏𝟎 and 𝚽𝟎𝟎 near GAM frequency

𝚽𝟎𝟎~ expi(qrr−wGt)

Analytic model:

𝜕𝛻⊥2Φ00

𝜕𝑡≃ 2 < 𝐛0 × 𝜿𝟎 ⋅ 𝛻P10 >

𝜕P10

𝜕𝑡≃ − Φ00, 𝑃10 −

10

3𝐛0 × 𝜿𝟎 ⋅ 𝛻Φ00

GAM oscillations

Poloidal advection of up-down asymmetry

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Decomposition of into sine (Ps) and cosine (Pc) components:

𝜕𝐸

𝜕𝑡− 𝜔1P𝑠=0

𝜕P𝑠

𝜕𝑡+

10

3𝜔1𝐸=−𝜔0P𝑐𝐸

𝜕P𝑐

𝜕𝑡=𝜔0P𝑆𝐸

𝐏𝟏𝟎

𝐸 = −𝜌𝑠𝛻𝑟Φ00

𝜔1 =𝑣𝑡ℎ

𝑅0, 𝜔0=

𝑣𝑡ℎ

𝑟

𝜕2𝐸

𝜕𝑡2 + (𝜔𝐺2 +𝑃𝑐0𝜔1𝜔0)𝐸 +

𝜔02

2𝐸3=0

𝜔𝐺 2 =

10

3𝜔1

2

Nonlinear Duffing oscillator!

Nonlinear Duffing oscillator without forcing and damping!

Pc starts to contribute to GAM when poloidal rotation is

present

𝑃𝑐0 = 𝑃𝑐 𝑡 = 0

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General characteristics

Changing variable from E to 𝚽 = 𝚽𝟎𝟎 by assuming 𝐸 = −𝑖𝑞𝑥𝜌𝑠Φ00

𝚽

𝑬

𝜕2Φ

𝜕𝑡2 + (𝜔𝐺2 +𝑃𝑐0𝜔1𝜔0)Φ −

𝜔02

2𝑞𝑥

2𝜌𝑠2Φ=0

Hamiltonian system with a quartic potential

𝑉 Φ =1

2𝛼Φ2 −

1

4𝛽Φ4

Constant of motion:

𝜕

𝜕𝑡(𝑃𝑠

2 + 𝑃𝑐2 −

10

3𝑞𝑥

2Φ002 ) ≡

𝜕𝐽

𝜕𝑡=0

Energy exchange between zonal mode and plasma kinetic energy

Allows unbounded solutions for large 𝒒𝒙

Signature of nonlinear instability!

𝛼 = 𝜔𝐺2 + 𝑃𝑐0𝜔1𝜔0, 𝛽 =

𝑞𝑥2𝜔0

2𝜌𝑠2

2

- 𝟐𝜶/𝜷 𝟐𝜶/𝜷

17

A new nonlinear instability

General solutions Jacobi elliptic functions

Φ00 = Φ0𝐬𝐧 𝛼 −𝛽

2Φ0

2,𝛽Φ0

2

2𝛼 − 𝛽Φ02

Interpretations:

Periodic bursts with modified

frequency

capture simulation features!

Secondary crashes might come from

this explosive nonlinear instability

different mechanism from the

Kelvin-Helmholtz mode ( strongly

suppressed by magnetic shear)

Φ0 = Φ00 𝑡 = 0

Burst at every ~½ of TGAM

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Conclusions

Nonlinear reduced MHD simulations of a pedestal collapse show that

ELM crash dynamics could be divided into two phases:

Linear phase: Ideal MHD governs dynamics

Nonlinear phase: GAMs play an important role secondary crashes

Compound ELMs?!

Analysis shows that:

Up-down pressure anisotropy are generated through coherent NL

interaction during the linear phase, while not the zonal potential.

Strong poloidal convection in NL phase leads to Duffing oscillations of F00

and a new type of NL explosive instability

bursts of F00 and subsequent small crashes

(different from KH instability)