Progress in theoretical understanding of the Dimits shift andthe tertiary instability in drift-wave turbulence

Hongxuan ZhuCoworkers: Yao Zhou and Ilya Dodin

PPPL and Princeton University

Feb 20, 2020

This work was supported by the U.S. DOE through Contract No.

DE-AC02-09CH11466. This work made use of computational support by CoSeC,

through CCP Plasma (Grant No. EP/M022463/1) and HEC Plasma (Grant No.

EP/R029148/1).

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Plasma microturbulence causes anomalous particle and heat transportand sets the core plasma profiles in tokamaks.

Some widely studied electrostatic drift-wave microinstabilities:• ion-temperature gradient mode (ITG),• electron-temperature gradient mode (ETG),• trapped-electron mode,• resistive drift-wave mode near the edge,• entropy mode in Z pinches.

Effects of sheared flows on turbulence are also widely studied.• Parallel/perpendicular flow; toroidal/poloidal flow

These microinstabilities are highly anisotropic: k‖ ∼ R−1 andk⊥ ∼ ρ−1, so E ×B advection is the dominant nonlinearity1:

E‖(∂δf/∂v‖)

V E · ∇δf∼k‖k⊥∼ ρ

R� 1 (1)

(R: tokamak major radius, ρ: gyroradius)

1W. D. Dorland, PhD thesis, Princeton University (1993).2 / 22

E ×B nonlinearity generates zonal flows in the perpendicular direction.

Zonal flows are poloidally symmetric but radially band-like structures.

The process of zonal-flow generation is known as the secondaryinstability (cf. “primary” and “tertiary” instability).

Figure 1: Zonal flows in tokamaks, from GYRO simulations (by Prof. Hammett).

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Zonal flows greatly reduce ITG turbulent transport.

The zonal-flow electrostatic potential is constant on a flux surface, sothe approximation of electron adiabatic response is violated2.• This results in enhanced zonal flows and reduced ITG turbulence level.

Dimits shift: near marginal stability, ITG modes can be completelysuppressed by zonal flows.• It is a shift between linear threshold of primary instabilities (without

zonal flow) and the actual turbulent onset (with zonal flows).

Figure 2: Left: reduced ion heat flux due to zonal flows [Lin et al., Science (1998)].Right: ITG-driven turbulent transport. The threshold is shifted due to zonalflows. [Dimits et al. PoP (2000).]

2Hammett et al. Plasma Phys. Control. Fusion 35, 973 (1993).4 / 22

The shearing paradigm does not explain why the Dimits shift is finite.Dimits regime: zonal flows dominant, and turbulence isinhomogeneous.

The shear decorrelation theory3 assumes constant shear andhomogeneous turbulence, and is only qualitative.

The predator–prey models4 make similar assumptions, and predictinfinite Dimits shift absent collisional damping of the flow.

We have found that5 it is the flow curvature d2U/dx2, rather than theflow shear |dU/dx|, that determines the Dimits shift.(U : flow velocity, x: radial coordinate)

Figure 3: Electrostatic potential in the Dimits regime. From GS2 simulation ofentropy modes in Z pinches. [Kobayashi and Rogers, PoP (2012).]

3H. Biglari, P. H. Diamond, and P. W. Terry, Phys. Fluids. B 2, 1 (1990).4M. A. Malkov, P. H. Diamond, and M. N. Rosenbluth, PoP 8, 5073 (2001).5H. Zhu, Y. Zhou, and I. Y. Dodin, Phys. Rev. Lett. 124, 055002 (2020).

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The tertiary instability (TI) was proposed to explain the Dimits shift.

Zonal flows are subject to tertiary instabilities: they cannot suppressthe turbulence if they are unstable themselves6 (Dimits regime ends).

Understanding of the Dimits shift had been qualitative until St-Onge’srecent analytic theory7:• However, St-Onge (and most other authors) use 4-mode truncation

(4MT) to study the TI, which misses the key features.• Another idea was proposed based on dynamical systems approach8, but

its applicability is also restricted by the 4MT.

There needs a transparent theory of the TI.• We show that the 4MT (Fourier-space approach) does not work.

Instead, one should consider drift waves in phase space.• We also show that the TI is not Kelvin–Helmholtz instability.

6B. N. Rogers, W. Dorland, and M. Kotschenreuther, PRL 85, 5336 (2000).7D. A. St-Onge, J. Plasma Phys. 83, 905830504 (2017).8R. A. Kolesnikov and J. A. Krommes, PRL 94, 235002 (2005).

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We use a phase-space approach to study drift waves in zonal flows.

The second-order cumulant expansion (CE2)9 has been used to zonaljets in atmospheres of rotating planets.

(e.g. )

The Wigner–Moyal formulation is the spectral representation of CE2.It is more intuitive, because it works in phase space.10

i∂tw(x, t) = Hw ⇒ ∂tW (x,p, t) = {{HH ,W}}+ [[HA,W ]] (2)

Phase-space dynamics is abundant11. In particular, we studied• Suppression of the Kelvin–Helmholtz instability for large-scale zonal

flows due to electron adiabatic response12,13

• Nonlinear saturation and oscillation of collisionless zonal flows14

• Tertiary instability and Dimits shift15

9J. B. Parker, Ph.D. thesis, Princeton University (2014).10D. E. Ruiz, J. B. Parker, E. L. Shi, and I. Y. Dodin, PoP 23, 122304 (2016).11H. Zhu, Y. Zhou, D. E. Ruiz, and I. Y. Dodin, Phys. Rev. E 97, 053210 (2018).12H. Zhu, Y. Zhou, and I. Y. Dodin, Phys. Plasmas 25, 072121 (2018).13H. Zhu, Y. Zhou, and I. Y. Dodin, Phys. Plasmas 25, 082121 (2018).14H. Zhu, Y. Zhou, and I. Y. Dodin, New J. Phys. 21, 063009 (2019).15H. Zhu, Y. Zhou, and I. Y. Dodin, Phys. Rev. Lett. 124, 055002 (2020).

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Theory of TI and Dimits shift from our phase-space approachModified Hasegawa–Wakatani equation (mHWE)16: edge resistive driftturbulence model, exhibits Dimits shift and transition to turbulence.

(∂t + z ×∇ϕ · ∇)w = κ∂yϕ− Dw, w.= ∇2ϕ− n, (3)

(∂t + z ×∇ϕ · ∇)n = α(ϕ− n)− κ∂yϕ− Dn. (4)

• x: radial, y: poloidal, κ: density gradient, α−1: resistivity, D: viscosity.Zonal flow: U(x, t)

.= ∂x〈ϕ〉, N(x, t)

.= 〈n〉, where 〈. . . 〉 is zonal average.

In the Dimits regime, drift waves are localized, and predator–preyoscillators are observed.

Figure 4: Dimits regime, from Dedalus simulation results, α = 5, κ = 12,

D = 0.1∇4, κeff.= κ−N ′. Blue and red colors show the fluctuating w(x, y).

16R. Numata, R. Ball, and R. L. Dewar, Phys. Plasmas 14, 102312 (2007).8 / 22

Tertiaty instability of a zonal flow (classical problem!)Assume a zonal flow

U(x) = u cos qZx, N(x) = 0 (5)

and consider a perturbation w = Re[ψ(x)ei(pyy−ωt)] on top.Linearized mHWE is

ωψ = Hψ, H = py[U + (κ+ U ′′)ˆp−2]− iD, (6)

ˆp2 .= p2

x + p2y +

iα+ pyκ

iα+ iD + ω − pyU, px

.= −i

d

dx. (7)

Hydrodynamic limit (α→ 0): Orr–Sommerfeld equation. Continuummodes, boundary layer modes, and Kelvin–Helmholtz instability.Plasma (α & 1): potential well and discrete energy level.

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Numerical eigenmodes concentrate near phase-space equilibria.Two most unstable eigenmodes are obtained numerically. We calculatethe Wigner function (Fourier transform of 2-point correlation function):

W (x, px).=

∫ds e−ipxsψ(x+ s/2)ψ∗(x− s/2). (8)

Intuitively, W is the “drifton” distribution function in phase space(x, px).W peaks at equilibria of the Hamiltonian H ≈ H(x, px → px),

∂xH = ∂pxH = 0, ⇒ U ′ = 0, px = 0. (9)

Umax: “trapped” mode; Umin: “runaway” mode

Figure 5: Left: two most unstable eigenmodes plotted together, arrows showzonal-flow velocity. Right: the corresponding W , dashed lines are contours of H.

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Weyl expansion of the Hamiltonian

An operator is mapped to a function using the Weyl transform:

H(x, px)→ H(x, px) (10)

Then, we can Taylor expand the Hamiltonian in phase space:

H(x, px) ≈ H+ λp p2x + λx x

2, H = H(x = 0, px = 0). (11)

Finally, the function is converted back to an operator:

H ≈ H+ λp p2x + λx x

2 (12)

Approximate U as U ≈ U0 + Cx2/2 and use the Weyl expansion:

H = py

(U0 +

κ+ Cp2

0

)− iD0,

λp = −py(κ+ C)p4

0

, λx =pyC2

[1− py(κ+ C)(iα+ pyκ)

p40(iα+ iD0 + ω − pyU0)2

],

p20 = p2

y +iα+ pyκ

iα+ iD0 + ω − pyU0, D0 = Dp(px = 0)

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Tertiary instability as harmonic oscillators

After Weyl expansion:

ωψ = Hψ ≈ (H+ λp p2x + λx x

2)ψ ⇒ −λp ψ′′ + λx x

2ψ = (ω −H)ψ

Eigenmodes are given by Hermite polynomials Hm, where m = 0, 1, ...

ψm = e−x2

2λHm(x/√λ), ωm = H+ (2m+ 1)λxλ, λ =

√λp/λx

The two modes with m = 0 (ground states, C > 0 and C < 0 each) aremost unstable. The growth rates are given by γTI

.= Imω0.

(C .= U ′′ is the flow “curvature” at U ′ = 0.)

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Our approach also applies to TI of other models (e.g. ITG turbulence)

Similar ITG mode localization is observed in slab GS2 simulations.17

Consider the 2-field gyrofluid model used by Rogers et al.18:

∂tn+ z×∇φ · ∇n+ z×∇(τ∇2φ) · ∇T/2 = 0, ∂tT + z×∇φ · ∇T = 0

We found Reω ≈ pyU0, Imω ∼ py√τηC, and mode width

∆x ∼ (τη/C)1/4, consistent with Rogers et al.

The flow U.= φ′ is stabilizing, while the temperature gradient η

.= T ′ is

destabilizing. (The TI is not Kelvin–Helmholtz instability.)

Figure 6: Left: figure from Rogers et al., toroidal geometry. Right: ourdemonstration that U is stabilizing and η is destabilizing, in slab geometry.

17H. Zhu, Y. Zhou, I. Y. Dodin, PRL 124, 055002 (2020), supplemental material.18B. N. Rogers, W. Dorland, and M. Kotschenreuther, PRL 85, 5336 (2000).

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Properties of the TI and its relation to the Dimits shift

The TI growth rates have two parts (remember C = U ′′):

γTI = γPI + ∆γ(C), γPI = pyκ/p20 − iD0,

∆γ = γPI C/κ+ Im(λxλ), ∆γ(C = 0) = 0.

Therefore, the TI is the primary instability modified by the zonal flow

γTI generally decreases with |C|, so the TI is not the Kelvin–Helmholtzinstability:• The Kelvin–Helmholtz instability is suppressed in large-scale zonal flows

(q2Z . 1) due to electron adiabatic response19.

• The TI develops when the zonal flow is weak, and it is suppressed whenthe zonal flow is strong.

The threshold of γTI = 0 is shifted compared to the threshold ofγPI = 0, this provides a generic explanation of the Dimits shift.This shift is determined by the flow curvature, not by the shear.

19H. Zhu, Y. Zhou, and I. Y. Dodin, Phys. Plasmas 25, 082121 (2018).14 / 22

We have explicitly calculated the Dimits shift in the modifiedTerry–Horton model (large-α limit of the mHWE).

In the large-α limit, density transport is small (N ≈ 0). The mHWEreduces to the modified Terry–Horton equation20:

(∂t+z×∇ϕ·∇)w = κ∂yϕ−Dw, w = (∇2−1+iδ)ϕ, δp.= κp2

y/[α(1+p2y)].

We found ∆DS = κc − κlin, κc =D0

py

(1 + p2y)2 + δ2

p

δp − (1 + p2y)√%/2

,

where κlin.= κc|%=0 and %

.= |C|/κ ≈ const.

Figure 7: Our prediction versus numerical simulation results of the modifiedTerry–Horton equation. κ∗ZF is the prediction by St-Onge.

20D. A. St-Onge, J. Plasma Phys. 83, 905830504 (2017).15 / 22

Consider mHWE in general (not-so-large α), TI can also be suppressedby nonzero N .

Tertiary modes can flatten the local density profile, reducingκeff

.= κ−N ′, which in turn suppresses the TI.

Figure 8: These events correspond to predator–prey oscillations betweenEDW and EN

.=

∫N2dx/2 (blue circles).

So in principle, any flow velocity U can be made stable by reducing κeff

at extrema of U .

Therefore, the Dimits shift cannot be predicted from analyticformulation assuming N = 0.

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But a turbulent burst occurs at small enough flow velocity (by viscosity)

A trapped TI mode can generate a nonlinear propagating structure:• This corresponds to the predator–prey oscillation between EDW and EU .• We now have two types of predator–prey oscillations!

Transition to turbulence at large κ is due to frequent turbulent bursts.

Similar structures are seen in a 2D fluid ITG model21.

Figure 9: Left: a turbulent burst event. Right: the critical shear Sc.= qZuc

roughly goes linearly with γTI. But notice that Sc � γ !

21P. G. Ivanov, A. A. Schekochihin, and W. Dorland (private communication).17 / 22

Our theory of the TI also provides an intuitive understanding of thetypical zonal-flow scale

Zonal flow with too small qZ: plasma is almost homogeneous becauseof small |C| ∼ q2

Zu, so primary instability happens, and then secondaryinstability can generated zonal flows with larger qZ.

Zonal flow with too large qZ: the flow amplitude u ∼ |C/q2Z| is small

assuming |C| is bounded by TI threshold. Then, the flow is subject tomerging instability22.

22J. B. Parker and J. A Krommes, New J. Phys. 16, 035006 (2014).18 / 22

Future directions: ETG transport, zonal flow merging instability, andmultiscale turbulence

Zonal flows are believed weaker in ETG turbulence due to ionadiabatic response.• Large ETG transport is related to radially elongated streamers23.• But studies also found that zonal flows may gradually built up with

time and reduce ETG turbulence level24.

A zonal flow with wavenumber qZ can be unstable to a perturbationwith wavenumber qZ/2.• Maybe this is what happens in ETG turbulence?

Also, whether ETG streamers can survive in the presence of large-scaleITG turbulence is still under investigation

23W. Dorland, F. Jenko, M. Kotschenreuther, and B. Rogers, PRL 85, 5579 (2000).24Colyer et al. Plasma Phys. Control. Fusion 59, 055002 (2017).

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Conclusions (and thank you for your attention!)

In the Dimits regime, drift waves are localized at extrema of the zonalflow velocity, so spatial inhomogeneity must be taken into account.

The phase-space approach based on the Wigner–Moyal formulationhelps explain drift-wave dynamics in zonal flows 25,26,27,28,29.

In particular, we studied the tertiary instability and the Dimits shiftbased on the this approach:• The Kelvin–Helmholtz instability is stabilized in large-scale zonal flows.• The TI is the primary instability modified by zonal flow.• The Dimits shift is determined by the flow curvature, not by the shear.• The TI can also generate nonlinear turbulent bursts.

The phase-space approach may be useful for exploring broader physics.

25H. Zhu, Y. Zhou, D. E. Ruiz, and I. Y. Dodin, Phys. Rev. E 97, 053210 (2018).26H. Zhu, Y. Zhou, and I. Y. Dodin, Phys. Plasmas 25, 072121 (2018).27H. Zhu, Y. Zhou, and I. Y. Dodin, Phys. Plasmas 25, 082121 (2018).28H. Zhu, Y. Zhou, and I. Y. Dodin, New J. Phys. 21, 063009 (2019).29H. Zhu, Y. Zhou, and I. Y. Dodin, Phys. Rev. Lett. 124, 055002 (2020).

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Relation between the TI and the Kelvin–Helmholtz instabilityIn the adiabatic limit, α→∞, the TI equation of the mHWE is[

d2

dx2− (1 + p2

y) +pyU

′′

ω − pyU

]ϕ =

κ

U − ω/pyϕ. (13)

KHI of zonal flows in Z-pinches30:(d2

dx2− p2

y +pyU

′′

ω − pyU

)ϕ =

γ2ideal

(U − ω/py)2ϕ. (14)

The difference lies in geometry: in Z pinches a py 6= 0 perturbation canhave p‖ = 0. In tokamaks this is impossible unless on rational surfaces.

Figure 10: Although α = 5 is not infinite, the KHI is already irrelevant.

30P. Ricci, B. N. Rogers, and W. Dorland, Phys. Rev. Lett. 97, 245001 (2006).21 / 22

Two-dimensional gyrofluid model (Rogers et al., PRL (2000).)

∂tn+ z ×∇φ · ∇n+ z ×∇(τ∇2φ) · ∇T/2 = 0, ∂tT + z ×∇φ · ∇T = 0.

τ = Ti/Te, φ is electrostatic potential, T is perpendicular temperature,and n

.= (1− τ∇2)(φ− 〈φ〉)−∇2(φ+ τT/2) is density.

Consider φ = φ0(x) + φ(x)ei(pyy−ωt) and T = T0(x) + T (x)ei(pyy−ωt).Then, n0 = −φ′′0 − τT ′′0 /2, n = (1 + τ p2 + p2)φ+ τ p2T /2, p2 .

= −∇2.

Define U(x).= φ′0, ω(x)

.= ω − pyU , and assume that η

.= T ′0 is a

positive constant. Then, ωn = Hn,

H = pyU+py(U ′′+τηp2/2−pyτηU ′′/2ω)A−1, A = 1+(1+τ)p2−pyητω−1p2/2.

Also assume U ≈ U0 + Cx2/2 and expand the Hamiltonian asH ≈ H+ λp p

2x + λx x

2, with

H = pyU0 +p3yτη

2A0−p2yτηC

2ω0A0, A0

.= 1−

p3yητ

2ω0, ω0

.= ω − pyU0,

λp =pyτη

2A0+p4yτ

2η2

4ω0A20

−p3yτ

2η2C4ω2

0A20

, λx =pyC2

+p7yτ

2η2C8ω2

0A20

−p6yτ

2η2C2

8ω30A

20

−p3yτηC2

4ω20A0

.

Real frequency: Reω ≈ pyU0, growth rate: Imω ≈ 0.51py√τηC, mode

width: ∆x ∼ (τη/C)1/4.

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