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An Explanation for the High-Beta Runaway: the Non-Zonal Transition M.J. Pueschel many thanks to : P .W. Terry, F. Jenko, D.R. Hatch, W.M. Nevins, T. G orler, D. Told, A.E. White 55th Annual Meeting of the APS Division of Plasma Physics

SLIDE 1

An Explanation for the High-Beta Runaway: the Non-Zonal Transition

M.J. Pueschel

many thanks to: P .W. Terry, F. Jenko, D.R. Hatch, W.M. Nevins, T. G¨

rler, D. Told, A.E. White

55th Annual Meeting of the APS Division of Plasma Physics Denver, November 12, 2013

SLIDE 2

What is the runaway? Zonal Flow Dynamics Field Line Decorrelation

1

What is the runaway?

2

Zonal Flow Dynamics

3

Field Line Decorrelation

M.J. Pueschel The Non-Zonal Transition

SLIDE 3

What is the runaway? Zonal Flow Dynamics Field Line Decorrelation

Gyrokinetics and GENE

Gyrokinetics: eliminate the fast gyrophase from the equations of motion ⇒ significant speed-up ⇒ gyrokinetic Vlasov, field equations GENE: gene.rzg.mpg.de nonlinear gyrokinetic equations radially local and nonlocal modes δf approach general geometry equilibria linear eigenvalue solver collisions, electromagnetic (A, B)

M.J. Pueschel The Non-Zonal Transition

SLIDE 4

What is the runaway? Zonal Flow Dynamics Field Line Decorrelation

Electromagnetic Cyclone

Standard parameter set: Cyclone Base Case (Dimits 2000) ⇒ three linear regimes: ITG, TEM, KBM Also: no sudden changes among subdominant/stable modes Nonlinearly: saturation often only temporary at high β ⇒ turbulent amplitudes tend to grow to extremely large values

M.J. Pueschel The Non-Zonal Transition

SLIDE 5

What is the runaway? Zonal Flow Dynamics Field Line Decorrelation

Nomenclature the high-β runaway is a consequence of the non-zonal transition (NZT)

M.J. Pueschel The Non-Zonal Transition

SLIDE 6

What is the runaway? Zonal Flow Dynamics Field Line Decorrelation

The High-β Runaway

Cyclone Base Case: above βNZT

crit

= 0.8%, no simple saturation See: Nevins APS 2009, Waltz 2010, Pueschel 2013 PRL/PoP simulations with β > βNZT

crit

experience (possibly long) transient saturation, then continue to grow with γITG high-Q regime only relevant for qualitative arguments, no physical meaning (fluxes too high δf approximation) high-Q: nonlinear frequency matches linear, ωNL ≈ ωITG

lin

M.J. Pueschel The Non-Zonal Transition

SLIDE 7

What is the runaway? Zonal Flow Dynamics Field Line Decorrelation

KBM behavior

Kinetic Ballooning Mode: highly detrimental to confinement Does γKBM linger subdominantly near zero at lower β? ⇒ No KBM influence expected below the linear threshold nonlinear βKBM

crit

same as linear (Pueschel 2008, 2010) Waltz 2010: tertiary subcritical KBM excitation possible Pueschel 2013: subcritical KBM not the reason for runaway

M.J. Pueschel The Non-Zonal Transition

SLIDE 8

What is the runaway? Zonal Flow Dynamics Field Line Decorrelation

NZT Occurrences

Robust phenomenon, re- produced by multiple codes: shown: GENE, GYRO, GKW Also: extensive numerical tests underscore physical nature Also occurs for other parame- ter regimes: pure-ITG case (Pueschel 2010), GA-std (Waltz 2010): no runaway observed for TEM turbulence

M.J. Pueschel The Non-Zonal Transition

SLIDE 9

What is the runaway? Zonal Flow Dynamics Field Line Decorrelation

1

What is the runaway?

2

Zonal Flow Dynamics

3

Field Line Decorrelation

M.J. Pueschel The Non-Zonal Transition

SLIDE 10

What is the runaway? Zonal Flow Dynamics Field Line Decorrelation

Zonal Flow Activity

β < βNZT

crit : strong ZFs

β > βNZT

crit : ZFs break up

Zonal flows can no longer saturate ITG ⇒ non-zonal transition (NZT)

M.J. Pueschel The Non-Zonal Transition

SLIDE 11

What is the runaway? Zonal Flow Dynamics Field Line Decorrelation

Secondary Instability

Secondary instability analysis: get zonal flow growth rate Usual procedure: three modes (ITG, zonal flow, sideband) Here: need extended mode structure to resolve linear physics half of all kx connected to linear mode, other half used as sideband freeze linear mode at constant amplitude let nonlinearity channel energy to zonal flow ⇒ no discontinuity near βNZT

crit , no weakened ˆ

γZF, thus zonal flow growth cannot be the cause of the NZT

M.J. Pueschel The Non-Zonal Transition

SLIDE 12

What is the runaway? Zonal Flow Dynamics Field Line Decorrelation

Impact of Magnetic Perturbations

Radial fluctuations break flux surfaces, short-circuit ZFs Rosenbluth-Hinton residual ↔ ZF impact residual with resonant Bx (non-resonant: no impact) magnetic fluctuations erode residual quadratic erosion/“decay”, with tΦ=0 ∝ B−1

x

M.J. Pueschel The Non-Zonal Transition

SLIDE 13

What is the runaway? Zonal Flow Dynamics Field Line Decorrelation

Analytical Theory

Rosenbluth-Hinton with resonant Bx (Terry 2013) ⇒ look at electrons peeling of flux surface due to Bx ∂fe ∂t + v kyA B0 kxfe − v eF0 Te Aky B0 kxΦ = ˆ SΦ

e δ(t)

(electron source replaces nonlinearity: energy pulse at t = 0) After a lengthy calculation, arrive at Φ(t) ΦR = 1 − ne(t = 0)/Φ(t = 0) k2

xρ2 s

1 + 1.6q2

0/ǫ1/2 t

α2t2 With α = Akxkyvth,e/B0 ≪ t−1, get (in normalized units) ΦR − Φ(t) ∝ ΦR mi me Te Ti B2

xt2

M.J. Pueschel The Non-Zonal Transition

SLIDE 14

What is the runaway? Zonal Flow Dynamics Field Line Decorrelation

NZT Energetics

Look at energetics of saturation below, above βNZT

crit :

Ek =

j
dzdvdµ Tj0

Fj0

gjk + qj

Fj0 Tj0 χjk ∗ gjk Nonlinear transfer from ITG, moderated by ZF: β < βNZT

crit : zonal

flows facilitate saturation β > βNZT

crit : little net

impact due to zonal flows

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35

kyρs

−1.0 −0.8 −0.6 −0.4 −0.2 0.0 0.2 0.4

<N >k ′

x ,k ′ y (Z)/| <N >k ′ x ,k ′ y (NZ)|

β =0.7% β =0.9%

M.J. Pueschel The Non-Zonal Transition

SLIDE 15

What is the runaway? Zonal Flow Dynamics Field Line Decorrelation

1

What is the runaway?

2

Zonal Flow Dynamics

3

Field Line Decorrelation

M.J. Pueschel The Non-Zonal Transition

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What is the runaway? Zonal Flow Dynamics Field Line Decorrelation

Field Line Decorrelation

Bx decomposes into (flux-surface-breaking) resonant and into non-resonant part What if field line decorrelates along its poloidal trajectory? ⇒ if half-turn displacement ∆r1/2 exceeds correlation length λBxx, non-resonant part contributes to stochasticity!

M.J. Pueschel The Non-Zonal Transition

SLIDE 17

What is the runaway? Zonal Flow Dynamics Field Line Decorrelation

Other Examples

Other physical parameters confirm decorrelation theory: GA standard case: (direct confirmation) TEM case (Pueschel 2010): (indirect confirmation) Note: density-gradient-driven TEM relies

n zonal flows for transport regulation

⇒ expect no runaway, but could have NZT (diminished zonal flows → higher flux)

M.J. Pueschel The Non-Zonal Transition

SLIDE 18

What is the runaway? Zonal Flow Dynamics Field Line Decorrelation

Gradient Dependencies

NZT very sensitive to background gradients ωT,n = R0/LT,n ⇒ transition quickly disappears at values below CBC, is only observed for large gradients (limit cycle?)

M.J. Pueschel The Non-Zonal Transition

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What is the runaway? Zonal Flow Dynamics Field Line Decorrelation

Summary

new critical β: non-zonal transition of ITG turbulence at β > βNZT

crit , ITG mode no longer saturated by zonal flows

not observed for TEM turbulence, although some quantitative impact possible due to reduced ZFs zonal flows critically weakened by resonant Bx (captured by residual flow simulations, analytical theory) field lines decorrelate from field at half turn, non-resonant Bx becomes flux-surface-breaking βNZT

crit

strongly dependent on background gradients NZT may be related to linear → saturated Ohmic confinement transition (as one among multiple causes) preliminary: simple estimate βNZT