PPPL, T169Thursday, 08/08/2019, 10:45 AM

Phase-space theory of the Dimits shift andcross-scale interactions in drift-wave turbulence

Ilya Y. Dodin

Princeton Plasma Physics Laboratory

in collaboration with:

Hongxuan Zhu (Princeton)

Yao Zhou (PPPL)

Daniel E. Ruiz (Sandia)

also thanks to Bill Dorland (U. Maryland) and Alex Schekochihin (U. Oxford)

PPPL Theory Seminar

Princeton, NJ

August 8, 2019

idodin@princeton.edu

PPPL, T169Thursday, 08/08/2019, 10:45 AMIntroduction

Drift-wave (DW) turbulence is ubiquitous in magnetized plasmas. In fusion science,DW turbulence is actively studied because it affects plasma confinement.

DW turbulence can spontaneously generate zonal flows (ZF), which are bandedshear flows with k‖ � 0. ZFs reduce turbulent transport but can be unstable.

2/26

PPPL, T169Thursday, 08/08/2019, 10:45 AMSimple questions are still awaiting simple answers.

Gyrokinetic simulations provide numerical data but basic physics is not entirely clear.

- What determines the ZF saturation/oscillations/merging, amplitudes, scales?

- What determines the ZF stability? How do ZFs suppress turbulence?

- What determines the propagating zonal structures seen in subcritical turbulence?

- How does electron-scale turbulence interact with ion-scale turbulence?...

Analytic modeling is needed to develop robust qualitative understanding.Gyrokinetic calculations are not intuitive. High-level theories can be advantageous.

Figures taken from Zhu et al. (2019) (left) and van Wyk et al. (2016) (right). 3/26

PPPL, T169Thursday, 08/08/2019, 10:45 AMTwo high-level theories of specific effects will be presented.

Part 1: Interactions of electron-scale and ion-scaleturbulence (“cross-scale interactions”)

Why does ITG turbulence suppress ETGturbulence, as seen in gyrokinetic simulations?

Part 2: Stability of zonal flows, nonlinearsuppression of DW turbulence, and the Dimits shift

What determines the stability of ZFsin collisionless and collisional turbulence?Minimal model of the tertiary instability andthe Dimits shift.

Figures taken from Maeyama et al. (2017) (upper) and Dimits et al. (2000) (lower). 4/26

PPPL, T169Thursday, 08/08/2019, 10:45 AMThe bigger project includes many papers not covered in this talk.

I. Y. Dodin, H. Zhu, Y. Zhou, and D. E. Ruiz, Modeling drift-wave turbulence as quantumlike

plasma, in Proceedings of the 46th EPS Conference on Plasma Physics (Milan, Italy, 2019).

H. Zhu, Y. Zhou, and I. Y. Dodin, Nonlinear saturation and oscillations of collisionless zonal

flows, New J. Phys. 21, 063009 (2019).

Y. Zhou, H. Zhu, and I. Y Dodin, Formation of solitary zonal structures via the modulational

instability of drift waves, Plasma Phys. Control. Fusion 61, 075003 (2019).

D. E. Ruiz, M. E. Glinsky, and I. Y. Dodin, Wave kinetic equation for inhomogeneous drift-wave

turbulence beyond the quasilinear approximation, J. Plasma Phys. 85, 905850101 (2019).

H. Zhu, Y. Zhou, and I. Y. Dodin, On the Rayleigh–Kuo criterion for the tertiary instability of

zonal flows, Phys. Plasmas 25, 082121 (2018).

H. Zhu, Y. Zhou, and I. Y. Dodin, On the structure of the drifton phase space and its relation

to the Rayleigh–Kuo criterion of the zonal-flow stability, Phys. Plasmas 25, 072121 (2018).

H. Zhu, Y. Zhou, D. E. Ruiz, and I. Y. Dodin, Wave kinetics of drift-wave turbulence and zonal

flows beyond the ray approximation, Phys. Rev. E 97, 053210 (2018).

D. E. Ruiz, J. B. Parker, E. L. Shi, and I. Y. Dodin, Zonal-flow dynamics from a phase-space

perspective, Phys. Plasmas 23, 122304 (2016).

DOE grant (2017-2020) 5/26

PPPL, T169Thursday, 08/08/2019, 10:45 AMBackground

Gyrokinetic simulations show that low-k ITG turbulence can suppress high-k ETGturbulence. For example, see Maeyama et al., PRL (2015):

Our goal is to explain this effect within the simplest meaningful model.

Maeyama et al. (2017); Howard et al. (2016); Maeyama et al. (2015) . . . 6/26

PPPL, T169Thursday, 08/08/2019, 10:45 AMAs a starting point, consider the Hasegawa–Mima model.

Basic physics of DW turbulence is often studied within the Hasegawa–Mima model:

Btw � tϕ,wu � βBxϕ � 0, w � p∇2 � paqϕ, β � ByN

Electrons respond adiabatically to drift waves (k‖ � 0) and do not respond to zonalflows (ZFs), which are spontaneously-generated banded shear flows with k‖ � 0.

wdw � p∇2 � 1qϕdw, wzf � ∇2ϕzf

Hammett et al. (1993); Krommes and Kim (2000) . . . 7/26

PPPL, T169Thursday, 08/08/2019, 10:45 AMWe further reduce the model using the quasilinear approximation.

The quasilinear approximation is sufficiently accurate to capture basic effects.

average: BtU � Byvxvy � 0, v � pz�∇ϕ, w � p∇2 � 1qϕfluctuations: Btw � UBxw � rβ � pB2yUqsBxϕ � v �∇w � v �∇wloooooooooomoooooooooon

neglected (QL model)

The equation for w can be expressed as a Schrodinger equation for “driftons”:

iBtw � pHw � , pH � pkx pU � pkxpβ � pU 2qp1 � pk2Kq�1, pk .� �i∇

Ruiz et al. (2016); Zhou et al. (2019) 8/26

PPPL, T169Thursday, 08/08/2019, 10:45 AMThe quasilinear HM model captures cross-scale interactions.

kx2{kx1 � 5 kx2{kx1 � 15

1

3

5

enst

roph

ies,

Z

Only high-k included

ZF

High-k

0 20 40 60 800

1

t

ener

gies

,E

ZF

High-k

1

3

5

enst

roph

ies,

Z

Only high-k included

ZF

High-k

0 20 40 60 800

1

2

t

ener

gies

,E ZF

High-k

2

4

6

enst

roph

ies,

Z

Both low-k and high-k included

ZF

Low-k

High-k

0 20 40 60 800

2

4

t

ener

gies

,E

ZF

Low-k

High-k

1

3

5

enst

roph

ies,

Z

Both low-k and high-k included

ZF

Low-kHigh-k

0 20 40 60 800

2

4

t

ener

gies

,E

ZF

Low-k

5 ´ High-k

Zdw.� 1

2

³d2x w2, Zzf

.� 1

2

³dy pU 1q2, Edw � �1

2

³d2x wϕ, Ezf

.� 1

2

³dy U2 9/26

PPPL, T169Thursday, 08/08/2019, 10:45 AMThe main claim

Simulations show that ZFs exhibit substantialmerging in multi-scale turbulence.

Claim: this merging is the cause of the high-kturbulence demise and a generic property of

multi-scale turbulence.

To show this, some theory will be needed:

- general wave-kinetic theory,

- topology of the drifton phase-space,

- approximate closure for U ,

- conditions for ZF merging.

We will also argue that the physics beyond thequasilinear approximation is not very different.

10/26

PPPL, T169Thursday, 08/08/2019, 10:45 AMA statistical theory is constructed by analogy with that in QM.

The Wigner function W pt, y,kq .� ³d2s e�ik�sxwpt,x� s{2qwpt,x� s{2qy (i.e., the

spectrum of the two-point correlator, or “quasiprobability distribution”) satisfies

BW

Bt� ttHH,W uu � rrHA,W ss,

BU

Bt�

B

By

»d2k

p2πq21

1 � k2K

� kxkyW �1

1 � k2K

HH � kxU �βkx

1 � k2K

�1

2

��U

2,

kx

1 � k2K

��, HA �

1

2

!!U

2,

kx

1 � k2K

))

Geometrical-optics limit: improved wave kinetic equation (iWKE) with new terms:

pL � OpBxBkq � pkLzfq�1

! 1, Aei pL{2

B � 1 � i{2tA,Bu � . . .

BW

Bt� tHH,W u � 2HAW ,

BU

Bt�

B

By

»d2k

p2πq2kxkyW

p1 � k2Kq

2

HH � kxU � kxpβ �U2q{p1 � k

2Kq, HA � �U

3kxky{p1 � k

2Kq

2

pL � t�, �u�� ttA,Buu � 2A sinp pL{2qB

�� rrA,Bss � 2A cosp pL{2qB�� A � B � Aei

pL{2B

Ruiz et al. (2016); Parker (2016) ; cf. Smolyakov and Diamond (1999); Krommes and Kim (2000) . . . 11/26

PPPL, T169Thursday, 08/08/2019, 10:45 AMQuasistatic approximation in the limit U 2 ! β

Let us rewrite the iWKE in the following form using the group velocity vg:

BWBt � B

By pWvgq � BBky

�W

BHH

By� U3

β � U 2Wvg, vg � 2kxky

p1 � k2Kq2pβ � U 2q

By integrating the iWKE over k, one obtains an equation for the drifton density.The term on the right can be neglected compared to ByJ when U 2 ! β.

BtN � ByJ � �JU3{pβ � U 2q, N.� ³

W d2k, J.� ³

Wvg d2k

In this “quasistatic” limit, U becomes a localfunction of N (“equation of state”):

BUBt � B

By�

J

2pβ � U 2q�� ByJ

2β� �BtN

2β

U � �N2β

� const

For quasimonochromatic turbulence as a special case, we discussed this in Zhou et al. (2019) . 12/26

PPPL, T169Thursday, 08/08/2019, 10:45 AMEven at small U , the ZF stability depends on the ZF wavenumber q.

The iWKE is only marginally applicable to ZFformation but can explain it qualitatively.

H � kxp�β � U 2q1 � k2x � k2y

� kxU, U � �N2β

� const�

k2y ! 1 � k2x, q2.� �U 2{U, kx � const

H � kxβ

�k2y2m

� V

� const,

1

m.� 2β2

p1 � k2xq2, V

.��

q2

1 � k2x� 1

N

2

If q2   1 � k2x, driftons reside near minima of V , so the system is stable.

If q2 ¡ 1� k2x, driftons reside near maxima of V . The system can lower the energyby bifurcating to a lower-q state, so it is unstable to ZF merging.

� Here, we assume U2 À β. Unlike in single-scale turbulence, this does not rule out q Á kx. 13/26

PPPL, T169Thursday, 08/08/2019, 10:45 AMZFs have many regimes but not all of them are realized naturally.

In single-scale turbulence, q corresponds to the maximum of γsecondary. Then,there are passing and/or trapped trajectories, and many driftons survive.

q � min k2x?N{β,

a1 � k2x

(, U À Uc1

Low-k waves cause ZFs to merge down to q2 � 1 � k2x. High-k waves becomerunaways and dissipate. Low-k waves remain passing because Uc1 � Uc1pkxq.

Zhu et al. (2019) 14/26

PPPL, T169Thursday, 08/08/2019, 10:45 AMZFs have many regimes but not all of them are realized naturally.

In single-scale turbulence, q corresponds to the maximum of γsecondary. Then,there are passing and/or trapped trajectories, and many driftons survive.

q � min k2x?N{β,

a1 � k2x

(, U À Uc1

Low-k waves cause ZFs to merge down to q2 � 1 � k2x. High-k waves becomerunaways and dissipate. Low-k waves remain passing because Uc1 � Uc1pkxq.

Zhu et al. (2019) 15/26

PPPL, T169Thursday, 08/08/2019, 10:45 AMCross-scale interactions in a nutshell

The scale separation persists in the poloidal-k space.

In the radial-k space, the scales are determined by the ZFs, which start at theelectron scale and then merge down to the ion scale.

During the ZF merging, most high-k driftons become runaways and dissipate.

Quasilinear simulations for two different ratios kx2{kx1 16/26

PPPL, T169Thursday, 08/08/2019, 10:45 AMNonlinear HM simulations demonstrate similar behavior.

2

4

6

enst

roph

ies,

Z

Both low-k and high-k included

ZF

Low-kHigh-k

0 20 40 60 800

2

4

t

ener

gies

,E

ZF

Low-k

High-k

1

3

5

enst

roph

ies,

Z

Both low-k and high-k included

ZF

Low-kHigh-k

0 20 40 60 800

2

4

t

ener

gies

,E

ZF

Low-k

5 ´ High-k

kx2{kx1 � 5 kx2{kx1 � 15

Drifton collisions serve as an additional channel though which high-k driftonsbecome runaways. Other than that, the effect remains the same.

Summary: the demise of high-k turbulence via cross-scale interactions isrobustly explained as a phase-space effect in the Hasegawa–Mima model.

17/26

PPPL, T169Thursday, 08/08/2019, 10:45 AMTertiary instability in the conservative Hasegawa–Mima model

In the Hasegawa–Mima model, the drifton Hamiltonian is pseudo-Hermitian,resulting in an instability of the Kelvin–Helmholtz type. (But is it relevant?)

iBtw � pHw, pH � kx pU � kxpβ � pU 2qp1 � k2x � pk2yq�1

γTI � |kxU0|�

1 � 1 � k2xq2

d1 � β2

U20q

4

Zhu et al. (2018)a; Zhu et al. (2018)c ; cf. Kuo (1949); Numata et al. (2007); Kim and Diamond (2002) 18/26

PPPL, T169Thursday, 08/08/2019, 10:45 AMThe dissipative TI is different. Consider the Terry–Horton model...

Btw � tϕ,wu � βByϕ� pDww � p∇2 � pa� ipδqϕpδ � δppkyqloooomoooon

primary instability

, pD � 1 � κ∇2looooooomooooooonfriction & viscosity

The “Hasegawa–Mima” TI mode becomes localized + other localized modes appear.

iBtw � pHw, pH � ky pU � kypβ � pU 2qr1 � pk2x � k2y � iδpkyqs�1� i pD

For this modified Terry–Horton model, see St-Onge (2017) . The function δpkyq can be anything. 19/26

PPPL, T169Thursday, 08/08/2019, 10:45 AMTI modes satisfy the equation of a quantum harmonic oscillator.

The largest growth rates belong to the lowest-order modes. Those are localizedin px, kxq, so the drifton Hamiltonian can be approximated with its Taylor expansion:

BtW � ttHH,W uu � rrHA,W ss ñ truncate H ñ pH � c0 � c1px2 � c2pk2x

This yields an equation of a quantum harmonic oscillator with complex coefficients:��ϑ2 d2

dx2� x2

w � εw ñ wn � Hn

�x?ϑ

e�x

2{2ϑ, εn � p2n� 1qϑ

ϑ.� �i

a2p1 � β{U 2

0q1 � k2y � iδ

, ε.� 2

kyU 20

�ωTI � kyU0 � iD0 � kypβ � U 2

0q1 � k2y � iδ

�

20/26

PPPL, T169Thursday, 08/08/2019, 10:45 AMThe growth rate is obtained explicitly and agrees with simulations.

γTI � �D0 � Im

�kypβ � U 2

0q � ikyU20

ap1 � β{U 20q{2

1 � k2y � iδ

�

Upper: analytic formula. Lower: numerical simulations + an alternative theory with a fitting parameter (not shown). 21/26

PPPL, T169Thursday, 08/08/2019, 10:45 AMDimits shift

γTI � �D0 � Im

�kypβ �U 2

0q � ikyU20

ap1 � β{U 20q{2

1 � k2y � iδ

�� γ

plinearqprimary � ∆γpU 2

0q

The tertiary instability can be viewed as the primary instability modified by ZFs.

- If γTI   0, turbulence is suppressed; ZFs survive, assuming D acts only on DWs.

- If γTI ¡ 0, the system ends up in a turbulent state.

- Due to ∆γ, the transition to the turbulent state occurs at plasma parametersdifferent from those without ZFs. This is called the Dimits shift.

The figures are taken from Dimits et al. (2000) and St-Onge (2017) . 22/26

PPPL, T169Thursday, 08/08/2019, 10:45 AMOur explicit formula for the Dimits shift agrees with simulations.

We calculate the values of β that correspond to γplinearqprimary � 0 and γTI � 0 using

U 20 � q2Uc1. The difference between these values is the Dimits shift (shaded).

Compared with a related calculation by St-Onge (2017), our model is a betterfit at both large and small δ. For example, it has no spurious cutoff at δ � 2.

βc �Drp1 � k2yq2 � δ2s{kyδ � p1 � k2yq

aU 20{2β

,U 20

β� q2

k2y � 1

St-Onge (2017) used four-mode truncation (somewhat a stretch, also not intuitive) and did not calculate the Dimits shift per se. 23/26

PPPL, T169Thursday, 08/08/2019, 10:45 AMSummary

Studying drift-wave turbulence in phase space requires:

- looking beyond geometrical optics: do not neglect λ{L and U 2{β,

- deriving WME/WKE from first principles: hand-waving leads to errors.

Cross-scale interactions within the Hasegawa–Mima model:

- Zonal flows tend to merge when there are driftons with kθ À q.

- Zonal-flow merging gradually reduces the DW radial scale.

- High-kθ DWs are efficiently dissipated during this process.

Tertiary instabilities and the Dimits shift:

- Dissipation localizes the tertiary modes near the ZF-velocity extrema.

- The growth rate of these modes can be made negative by U 2 ñ Dimits shift.

- An analytic theory is developed within the Terry–Horton model.

- Two-fluid models have additional features.

Drift-wave solitons in subcritical turbulence

24/26

References

[Dimits et al. (2000) ]A. M. Dimits, G. Bateman, M. A. Beer, B. I. Cohen,W. Dorland, G. W. Hammett, C. Kim, J. E. Kinsey,M. Kotschenreuther, A. H. Kritz, L. L. Lao, J. Mandrekas,W. M. Nevins, S. E. Parker, A. J. Redd, D. E. Shumaker,R. Sydora, and J. Weiland, Comparisons and physics basis oftokamak transport models and turbulence simulations, Phys.Plasmas 7, 969 (2000).

[Hammett et al. (1993) ]G. W. Hammett, M. A. Beer, W. Dorland, S. C. Cowley,and S. A. Smith, Developments in the gyrofluid approach totokamak turbulence simulations, Plasma Phys. Control. Fusion35, 973 (1993).

[Howard et al. (2016) ]N. T. Howard, C. Holland, A. E. White, M. Greenwald,J. Candy, and A. J. Creely, Multi-scale gyrokinetic simulations:Comparison with experiment and implications for predictingturbulence and transport, Phys. Plasmas 23, 056109 (2016).

[Kim and Diamond (2002) ]E.-J. Kim and P. H. Diamond, Dynamics of zonal flowsaturation in strong collisionless drift wave turbulence, Phys.Plasmas 9, 4530 (2002).

[Krommes and Kim (2000) ]J. A. Krommes and C.-B. Kim, Interactions of disparate scalesin drift-wave turbulence, Phys. Rev. E 62, 8508 (2000).

[Kuo (1949) ]H.-L. Kuo, Dynamic instability of two-dimensional nondivergentflow in a barotropic atmosphere, J. Meteorol. 6, 105 (1949).

[Maeyama et al. (2015) ]S. Maeyama, Y. Idomura, T.-H. Watanabe, M. Nakata,M. Yagi, N. Miyato, A. Ishizawa, and M. Nunami, Cross-scale interactions between electron and ion scale turbulence ina tokamak plasma, Phys. Rev. Lett. 114, 255002 (2015).

[Maeyama et al. (2017) ]S. Maeyama, T.-H. Watanabe, Y. Idomura, M. Nakata,A. Ishizawa, and M. Nunami, Cross-scale interactions betweenturbulence driven by electron and ion temperature gradientsvia sub-ion-scale structures, Nucl. Fusion 57, 066036 (2017).

[Numata et al. (2007) ]R. Numata, R. Ball, and R. L. Dewar, Bifurcation inelectrostatic resistive drift wave turbulence, Phys. Plasmas14, 102312 (2007).

[Parker (2016) ]J. B. Parker, Dynamics of zonal flows: failure of wave-kinetictheory, and new geometrical optics approximations, J. PlasmaPhys. 82, 595820602 (2016).

[Ruiz et al. (2019) ]D. E. Ruiz, M. E. Glinsky, and I. Y. Dodin, Wave kineticequation for inhomogeneous drift-wave turbulence beyond thequasilinear approximation, J. Plasma Phys. 85, 905850101(2019).

[Ruiz et al. (2016) ]D. E. Ruiz, J. B. Parker, E. L. Shi, and I. Y. Dodin, Zonal-flowdynamics from a phase-space perspective, Phys. Plasmas 23,122304 (2016).

[Smolyakov and Diamond (1999) ]A. I. Smolyakov and P. H. Diamond, Generalized actioninvariants for drift waves-zonal flow systems, Phys. Plasmas6, 4410 (1999).

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References

[St-Onge (2017) ]D. A. St-Onge, On non-local energy transfer via zonal flow inthe Dimits shift, J. Plasma Phys. 83, 905830504 (2017).

[van Wyk et al. (2016) ]F. van Wyk, E. G. Highcock, A. A. Schekochihin, C. M.Roach, A. R. Field, and W. Dorland, Transition to subcriticalturbulence in a tokamak plasma, J. Plasma Phys. 82,905820609 (2016).

[Zhou et al. (2019) ]Y. Zhou, H. Zhu, and I. Y. Dodin, Formation of solitaryzonal structures via the modulational instability of drift waves,Plasma Phys. Controlled Fusion 61, 075003 (2019).

[Zhu et al. (2018)a ]H. Zhu, Y. Zhou, and I. Y. Dodin, On the Rayleigh–Kuocriterion for the tertiary instability of zonal flows, Phys. Plasmas25, 082121 (2018).

[Zhu et al. (2018)b ]H. Zhu, Y. Zhou, and I. Y. Dodin, On the structure ofthe drifton phase space and its relation to the Rayleigh–Kuocriterion of the zonal-flow stability, Phys. Plasmas 25, 072121(2018).

[Zhu et al. (2019) ]H. Zhu, Y. Zhou, and I. Y. Dodin, Nonlinear saturation andoscillations of collisionless zonal flows, New J. Phys. 21, 063009(2019).

[Zhu et al. (2018)c ]H. Zhu, Y. Zhou, D. E. Ruiz, and I. Y. Dodin, Wavekinetics of drift-wave turbulence and zonal flows beyond theray approximation, Phys. Rev. E 97, 053210 (2018).

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