Cross Scale Interaction Mechanisms: Suppression of the ETG

Cross Scale Interaction Mechanisms:Suppression of the ETG linear instability by ion

gyroradius scale turbulence

M.R.Hardman1,2, M.Barnes1,2, C.M.Roach2, F. I. Parra1,21Rudolf Peierls Centre for Theoretical Physics, University of Oxford, UK

2 CCFE, Culham Science Centre, Abingdon, Oxon, UK

1 / 26

Outline

I Motivation for studying multi-scale turbulence

I Brief introduction to theory of scale-separated ion scale (IS) and electronscale (ES) turbulence

I Insights gained by simulations probing the effect of IS turbulence on ESinstabilities

2 / 26

Introduction

Anomalous transport is driven by turbulence,

I IS : at scales where kρi . 1

I ES : at scales where kρe . 1 kρi

I N.B. ρe/ρi ∼ (me/mi)1/2 ' 1/60 1

I do all scales matter?

I is cross scale coupling important?

I To answer these questions we take a scale separated approach

I (me/mi)1/2 → 0

3 / 26

Introduction: do all scales matter?

I simulation evidence whereQe ∼ 10QegB ∼ (?)QigB e.g.

Jenko and Dorland (2002)

I recent experimental evidence onNSTX Ren et al. (2017)

I Howard et al. (2016) Fig 3:

4 / 26

Introduction: is cross scale coupling important?

I Fig 2 from Maeyama et al. (2015):

(a) β = 0.04% (b) β = 2.0%

I See also

Maeyama et al. (2017); Howard et al. (2016); Bonanomi et al. (2018)

Gorler and Jenko (2008); Candy et al. (2007); Waltz et al. (2007)

5 / 26

Introduction: a scale separated approach

6 / 26

A Quick Reminder: Scale separation in local δf turbulence

log

log

I scale separation: ρ∗ = ρ/a→ 0 ⇒ f = F + δfI statistical periodicity: 〈δf〉turb = 0I gyro average: 〈·〉|gyroRI orderings:

δf ∼ ρ∗F

∇F ∼ ∇⊥δf ∼ ρ−1∗ ∇‖δf

∂tδf ∼ (vt/a)δf ∼ ρ∗Ωδf

∂tF ∼ ρ3∗ΩF

7 / 26

A Quick Reminder: The local δf Gyrokinetic Equation

The gyrokinetic equation for h = δf + (Zeφ/T )F0:

∂h

∂t+ v‖b · ∇θ

∂h

∂θ+ (vM + vE) · ∇h+ vE · ∇F0 =

ZeF0

T

∂ϕ

∂t, (1)

where,

ϕ = 〈φ〉|gyroR , vE =c

Bb ∧∇ϕ. (2)

Closed by quasi-neutrality,

∑α

Zα

∫d3v|rhα =

∑α

Z2αenα

Tαφ(r). (3)

I electrostatic approximation

I zero equilibrium toroidal rotation

I (h for compactness – we later find g = 〈δf〉|gyroR = h− (Zeϕ/T )F0 is moreconvenient)

8 / 26

Separating IS and ES Turbulence

log

log

I scale separation: ρe/ρi ∼ vti/vte ∼√me/mi → 0, ⇒ δf = δf + δf

I ES statistical periodicity:⟨δf⟩ES

= 0

I orderings:

∇⊥δf ∼ ρ−1i δf,

∂δf

∂t∼vti

aδf

∇⊥δf ∼ ρ−1e δf ,

∂δf

∂t∼vte

aδf.

9 / 26

Separating IS and ES Turbulence: Size of the Fluctuations– Possible impacts of cross-scale interaction

We show in Hardman et al. (2019) that the only ordering which allows saturateddominant balance is

∇F0 ∼ ∇⊥δf ∼ ∇⊥δf , (4)

resulting in the usual gyro-Bohm ordering,

eφ

T∼ ρi∗,

eφ

T∼ ρe∗ (5)

hi

F0i∼

he

F0e∼eφ

T,

he

F0e∼eφ

T,

hi

F0i∼(me

mi

)1/4 eφ

T(6)

(4) ⇒ ES eddies can be driven or suppressed by gradients of the IS distributionfunction

(4) ⇒ ∇φ ∼ ∇φ ⇒ eddy E×B drifts vE×B, are comparable at all scalesCritical balance ⇒ parallel correlation lengths are the same for IS and ES eddies

⇒ ES eddies can be sheared by the IS E ×B drift in the direction parallel to themagnetic field

10 / 26

Separating IS and ES Turbulence: Difficult Points

I electrons at IS – fast electron streaming timescales are removed by the orbitalaverage 〈·〉o – he = 0 (adiabatic response) for non-zonal passing electrons

I ions at ES – non-locality of the gyro average – adiabatic response

I the parallel boundary condition for the ES flux tubes

Gyro-motion

Ion

Electron

11 / 26

Separating IS and ES Turbulence – The Coupled Equations

I IS equations, where the leading-order cross-scale terms are small

∂hi

∂t+ v‖b · ∇θ

∂hi

∂θ+ (vMi + vEi ) · ∇hi + vEi · ∇F0i =

ZieF0i

Ti

∂ϕi∂t

, (7)

∂he

∂t+⟨vMe · ∇α

⟩o ∂he∂α

+⟨vEe · ∇he

⟩o+⟨vEe · ∇F0e

⟩o= −

eF0e

Te

∂ 〈ϕe〉o

∂t, (8)∫

d3v|r(Zihi − he) =

(eZ2i ni

Ti+ene

Te

)φ, (9)

I ES equations, with the new advection and drive terms

∂he

∂t+v‖b·∇θ

∂he

∂θ+(vMe +vEe +vEe )·∇he+vEe ·(∇he+∇F0e) = −

eF0e

Te

∂ϕe

∂t. (10)

−∫d3v|rhe =

(eZ2i ni

Ti+ene

Te

)φ, (11)

12 / 26

Separating IS and ES Turbulence – The Coupled Equations

I IS equations, where the leading-order cross-scale terms are small

∂hi

∂t+ v‖b · ∇θ

∂hi

∂θ+ (vMi + vEi ) · ∇hi + vEi · ∇F0i =

ZieF0i

Ti

∂ϕi∂t

, (7)

∂he

∂t+⟨vMe · ∇α

⟩o ∂he∂α

+⟨vEe · ∇he

⟩o+⟨vEe · ∇F0e

⟩o= −

eF0e

Te

∂ 〈ϕe〉o

∂t, (8)∫

d3v|r(Zihi − he) =

(eZ2i ni

Ti+ene

Te

)φ, (9)

I ES equations, with the new advection and drive terms

∂he

∂t+v‖b·∇θ

∂he

∂θ+(vMe +vEe +vEe )·∇he+vEe ·(∇he+∇F0e) = −

eF0e

Te

∂ϕe

∂t. (10)

−∫d3v|rhe =

(eZ2i ni

Ti+ene

Te

)φ, (11)

N.b. writing (10) in terms of ge and ge shows that the constant in θ piece of vEecan be removed by a rotation ⇒ The parallel-to-the-field variation of vEe matters

12 / 26

The Effect of Cross Scale Interaction on the ES ETG InstabilityI The coupled equations capture the O(1) effects of IS turbulence on ES

fluctuations

I We pick Cyclone Base Case like (CBC) parameters where there is aseparation of scales:

10−1 100 101 102

kyρth,i

10−2

10−1

100

γ /(vth,i/a) at θ0 = 0a/LTi

= 2.3

a/LTi= 1.38

AI

−2 0 2θ0

0.25

0.50

0.75

Kyρ

th,e

γ/(vth,e/a)

-0.063-0.041-0.0190.0040.0260.0490.0710.094

I We simulate the IS turbulence to obtain a sample of vEe and ∇geI We observe the effect on the ES ETG instability:

Strongly driven ETG (a/LTe = 2.3)I persists in the presence of weakly driven (a/LTi

= 1.38) IS turbulenceI is suppressed by strongly driven (a/LTi

= 2.3) IS turbulence

13 / 26

Sampling IS Turbulence with a/LTi = 1.38

0 2000 4000 6000ts/(a/vth,i)

0

5

10

〈 φ2〉xs,ys,θ/(Tρ∗i /e)2 for a/LTi

= 1.38

−50 0 50xs/ρth,i

−50

0

50

y s/ρ

th,i

φ/(Tρ∗i /e) at θ = 0

−6

−4

−2

0

2

4

6

I Saturate ISturbulence

I Calculate ∇geI Calculate vEeI At 6 IS ts times

(blue dashes)

I At 6 radial (xs) × 5binormal (ys) ISpositions (crosses)

14 / 26

Simulations: modification of ES linear physics: CBC a/LTi = 1.38

Top Right: No IS gradients.

Below: IS gradients from different IS(xs, ys) locations

Moderate suppression

−2 0 2θ0

0.25

0.50

0.75

Kyρ

th,e

γ/(vth,e/a)

-0.063-0.041-0.0190.0040.0260.0490.0710.094

−2 0 2θ0

0.25

0.50

0.75

Kyρ

th,e

γ/(vth,e/a)

-0.11-0.086-0.058-0.031-0.00320.0240.0520.079

−2 0 2θ0

0.25

0.50

0.75

Kyρ

th,e

γ/(vth,e/a)

-0.14-0.11-0.083-0.054-0.0250.00460.0340.063

15 / 26



Below: average ETG growth rate acrossall sampled IS (xs, ys) locations and tstimes

Weak suppression!−2 0 2

θ0

0.25

0.50

0.75

Kyρ

th,e

γ/(vth,e/a)

-0.063-0.041-0.0190.0040.0260.0490.0710.094

−2 0 2θ0

0.25

0.50

0.75

Kyρ

th,e

〈γ〉ts,xs,ys/(vth,e/a)

-0.089-0.065-0.041-0.0170.00680.0310.0550.079

−2 0 2θ0

0.25

0.50

0.75

Kyρ

th,e

σ〈γ〉ts,xs,ys/(vth,e/a)

0.000130.000440.000750.00110.00140.00170.0020.0023

16 / 26

Sampling IS Turbulence with a/LTi = 2.3

0 500 1000 1500ts/(a/vth,i)

0

50

100

〈 φ2〉xs,ys,θ/(Tρ∗i /e)2 for a/LTi

= 2.3

−50 0 50xs/ρth,i

−50

0

50

y s/ρ

th,i

φ/(Tρ∗i /e) at θ = 0

−10

−5

0

5

10

15

I Saturate ISturbulence

I Calculate ∇geI Calculate vEeI At 6 IS ts times

(blue dashes)

I At 6 radial (xs) × 5binormal (ys) ISpositions (crosses)

17 / 26



Below: IS gradients from different IS(xs, ys) locations

Strong suppression!

−2 0 2θ0

0.25

0.50

0.75

Kyρ

th,e

γ/(vth,e/a)

-0.063-0.041-0.0190.0040.0260.0490.0710.094

−2 0 2θ0

0.25

0.50

0.75

Kyρ

th,e

γ/(vth,e/a)

-0.14-0.12-0.1-0.085-0.065-0.045-0.025-0.0052

−2 0 2θ0

0.25

0.50

0.75

Kyρ

th,e

γ/(vth,e/a)

-0.15-0.13-0.1-0.077-0.052-0.027-0.00130.024

18 / 26




Strong suppression!−2 0 2

θ0

0.25

0.50

0.75

Kyρ

th,e

γ/(vth,e/a)

-0.063-0.041-0.0190.0040.0260.0490.0710.094

−2 0 2θ0

0.25

0.50

0.75

Kyρ

th,e


-0.13-0.12-0.098-0.08-0.062-0.044-0.026-0.0078

−2 0 2θ0

0.25

0.50

0.75

Kyρ

th,e


0.000140.00060.00110.00150.0020.00240.00290.0033

19 / 26




INCLUDING ONLY ∇ge (with vEe = 0)

weak suppression! −2 0 2θ0

0.25

0.50

0.75

Kyρ

th,e

γ/(vth,e/a)

-0.063-0.041-0.0190.0040.0260.0490.0710.094

−2 0 2θ0

0.25

0.50

0.75

Kyρ

th,e


-0.068-0.048-0.028-0.00850.0110.0310.0510.07

−2 0 2θ0

0.25

0.50

0.75

Kyρ

th,e


0.000170.000520.000870.00120.00160.00190.00230.0026

20 / 26




INCLUDING ONLY vEe (with ∇ge=0)

Strong suppression! −2 0 2θ0

0.25

0.50

0.75

Kyρ

th,e

γ/(vth,e/a)

-0.063-0.041-0.0190.0040.0260.0490.0710.094

−2 0 2θ0

0.25

0.50

0.75

Kyρ

th,e


-0.13-0.11-0.093-0.074-0.054-0.034-0.0140.0056

−2 0 2θ0

0.25

0.50

0.75

Kyρ

th,e


0.000150.000570.000980.00140.00180.00220.00270.0031

21 / 26

Simulations: A simple model of parallel-to-be-field shear in vEe

I Simplest possible form for vEe withlocal parallel-to-the-field shear(consistent with flux tube ‖ b.c.Beer et al. (1995))

I (13) leads to vEe · kf with linearvariation e.g. (12) for Kx = 0 (andour parameters)

I maximum ETG growth rateγmax(E) shows suppression for all

E 6= 0

I ⇒ Qualitative explanation of ETGbehaviour in the presence of ISturbulence

I suppression when

ωE = 0.4× 0.5× 1.0(vth,ea

)∼

γmax(E = 0) ' 0.1

vEe · kf = ωEθ, (12)

∂φ

∂ys

∣∣∣xs

= −E,∂φ

∂xs

∣∣∣ys

= −sθE, (13)

ωE = 0.4(vth,e

a

)(Kyρth,e)

(E

T/ea

).

(14)

−1 0 1

E/(T/ea)

0.00

0.05

γmax /(vth,e/a)

22 / 26



Below: ETG growth rate with model vEe ;

(left) E = 0.5T/ea (right) E = 1.0T/ea

Strong suppression!

−2 0 2θ0

0.25

0.50

0.75

Kyρ

th,e

γ/(vth,e/a)

-0.063-0.041-0.0190.0040.0260.0490.0710.094

−2 0 2θ0

0.25

0.50

0.75

Kyρ

th,e

γ/(vth,e/a)

-0.14-0.11-0.084-0.054-0.0240.00590.0360.066

−2 0 2θ0

0.25

0.50

0.75

Kyρ

th,e

γ/(vth,e/a)

-0.16-0.13-0.11-0.084-0.059-0.034-0.0090.016

23 / 26

Conclusions

We have derived coupled, scale-separated equations for IS and ES turbulence.

We assumed

I (me/mi)1/2 → 0; space and time separation; no other small parameters

I spatial isotropy – IS l⊥ ∼ ρth,i; ES l⊥ ∼ ρth,eI negligible direct cascade; separation in the fluctuation spectrum

The model

I efficiently captures cross-scale interactions which persist as (me/mi)1/2 → 0

I is simulated in a system of coupled ES flux tubes nested in an IS flux tube

We found thatI strongly driven ETG (a/LTe = 2.3)

I persists in the presence of weakly driven (a/LTi= 1.38) IS turbulence

I is suppressed by strongly driven (a/LTi= 2.3) IS turbulence

I the primary mechanism responsible for the suppression is parallel-to-the-fieldvariation in vEe

I a simple model of local parallel-to-the-field variation in the flow qualitativelyexplains the result

24 / 26

Questions for Future WorkI Can we retain the effect of ES turbulence on IS fluctuations by taking other

parameters to be small?I distance to marginal stabilityI zonal to non-zonal amplitude

I By taking other parameters to be small, can we find scalings where the ESfluctuation amplitude is comparable to the IS fluctuation amplitude?

I What is the perpendicular scale of an ETG streamer? Dorland et al. (2000);Jenko et al. (2000); Jenko and Dorland (2002); Guttenfelder and Candy(2011)

I Is it possible to enforce time scale separation if ETG turbulence saturatesslowly? Colyer et al. (2017); Nakata et al. (2010)

I What is the effect of IS turbulence on non-linear ETG saturation?

I What changes in this picture with electromagnetic fluctuations?

25 / 26

Thank You for Listening!

The authors would like to thank A. A. Schekochihin, P. Dellar, W. Dorland, S. C.Cowley, J. Ball, A. Geraldini, N. Christen, A. Mauriya, P. Ivanov, Y. Kawazura, J.Parisi, M. Abazorius and O. Beeke for useful discussion. M. R. Hardman wouldlike to thank the Wolfgang Pauli Institute for providing a setting for discussionand funding for travel.This work has been carried out within the framework of the EUROfusionConsortium and has received funding from the Euratom research and trainingprogramme 2014-2018 and 2019-2020 under grant agreement No 633053 and fromthe RCUK Energy Programme [grant number EP/P012450/1]. The views andopinions expressed herein do not necessarily reflect those of the EuropeanCommission. The authors acknowledge EUROfusion, the EUROfusion HighPerformance Computer (Marconi-Fusion), the use of ARCHER through thePlasma HEC Consortium EPSRC grant numbers EP/L000237/1 andEP/R029148/1 under the projects e281-gs2, and software support from JosephParker through the Plasma-CCP Network under EPSRC grant numberEP/M022463/1.

26 / 26

F. Jenko and W. Dorland. Phys. Rev. Lett., 89:225001, 2002.

Y. Ren, E. Belova, N. Gorelenkov, W. Guttenfelder, S.M. Kaye, E. Mazzucato,J.L. Peterson, D.R. Smith, D. Stutman, K. Tritz, W.X. Wang, H. Yuh, R.E.Bell, C.W. Domier, and B.P. LeBlanc. Nuclear Fusion, 57(7):072002, 2017.

N.T. Howard, C. Holland, A.E. White, M. Greenwald, and J. Candy. NuclearFusion, 56:014004, 2016.

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W. Guttenfelder and J. Candy. Physics of Plasmas, 18:022506, 2011.

G J Colyer, A A Schekochihin, F I Parra, C M Roach, M A Barnes, Y c Ghim,and W Dorland. Plasma Physics and Controlled Fusion, 59:055002, 2017.

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1 / 14


Top Right: ETG growth rate γ(E) for

Kyρth,e = 0.57, θ0 = 0.0.

Below: (left) corresponding eigenmodes(right) corresponding drift coefficients

−1.0 −0.5 0.0 0.5 1.0

E/(T/ea)

−0.05

0.00

0.05

0.10γ /(vth,e/a) at Kyρth,e = 0.57, θ0 = 0

−10 −5 0 5 10θ − θ0

0.0

0.5

1.0|φ|/|φ|max for Kyρth,e=0.57, θ0 =0.00

E/(T/ea)0.00.250.5

−10 −5 0 5 10θ − θ0

−2

0

Drift Freq. for Kyρth,e=0.57, θ0 =0.00

ωMωE E/(T/ea) =0.25ωE E/(T/ea) =0.5

1 / 14

Separating IS and ES Turbulence: Technicalities

I We introduce a fast spatial variable rf and a slow spatial variable rs and thefast and slow times tf , ts

I In the gyrokinetic equation we send,

δf(t, r)→ δf(ts, tf , rs, rf ), ∇ → ∇s +∇f ,∂

∂t→

∂

∂ts+

∂

∂tf, (15)

I then asymptotically expand in the mass ratio (me/mi)1/2

I remembering ∇s ∼ (me/mi)1/2∇f , and ∂/∂ts ∼ (me/mi)

1/2∂/∂tfI explicitly define the ES average,

δf(ts, rs) =⟨δf(ts, tf , rs, rf )

⟩ES=

1

τcA

∫ ts+τc/2

ts−τc/2dtf

∫A,rs

d2rf δf(ts, tf , rs, rf ),

(16)

I We assume that,

δf(ts, tf , rs, rf ) = δf(ts, tf , rs, rf + n∆cxx +m∆cyy), (17)

I This enforces⟨δf⟩ES

= 0.

2 / 14

Splitting the Quasi-Neutrality Relation

I We split the guiding centre into a slow Rs and a fast Rf part.

I R = r− ρ(r), where ρ(r) is the vector gyroradius

I Thus using the periodicity property equation (17) the ES average may betaken over guiding centre or real space coordinates.

I This observation allows us to note that the ES average commutes with thegyro average,⟨

1

2π

∫ 2π

0dγ|Rφ(rs, rf )

⟩ES

=1

2π

∫ 2π

0dγ|R

⟨φ(rs, rf )

⟩ES=

1

2π

∫ 2π

0dγ|Rφ(rs),

(18)The splitting of the quasi neutrality relation follows directly,

∑α

Zα

∫d3v|rhα(Rs) =

∑α

Z2αenα

Tαφ(rs), (19)

∑α

Zα

∫d3v|rhα(Rs,Rf ) =

∑α

Z2αenα

Tαφ(rs, rf ). (20)

3 / 14

Addressing the Non-Locality of the Gyro AverageI Taking the gyro average at fixed guiding centre 〈·〉|gyroR , couples multiple rs

points.I but we aim to find scale separated equations!I Expanding both the slow and the fast spatial variable in Fourier series we

note that,

ϕ(ts, tf ,Rs,Rf ) = 〈φ(ts, tf , rs, rf )〉|gyroR =1

2π

∫ 2π

0dγ|Rφ(ts, tf , rs, rf )

=1

2π

∫ 2π

0dγ|R

∑ks,kf

φks,kfeiks·rseikf ·rf

=∑

ks,kf

φks,kfeiks·Rseikf ·Rf J0(|(ks + kf )|ρ), (21)

for electrons:I |kf |ρe ∼ 1 and |ks|ρe ∼ (me/mi)

1/2

I we can expand the Bessel function to return to a local picture in the slowvariable with O(me/mi)

1/2 error.I We will exploit this in scale separation.

for ions:I |ks|ρi ∼ 1 and |kf |ρi ∼ (me/mi)

−1/2.I we are unable to expand the Bessel functionI we are unable to avoid the coupling of multiple rs in the equations for ions at

ES

4 / 14

Addressing the Non-Locality of the Gyro Average: continuedI we can neglect the ion contribution to ES quasi neutrality

I ion gyroradius >> ES fluctuation scale length → ion can only respond toa large-scale average of ES potential

I J0(|kf |ρi) ∼ (me/mi)1/4 << 1

I Hence,

ϕe(ts, tf ,Rs,Rf ) =∑

ks,kf

φks,kfeiks·Rseikf ·Rf J0(|(ks + kf )|ρ)

= −Te

nee

∑ks,kf

eiks·Rseikf ·Rf J0(|(ks+kf )|ρ)

∫d3v he,ks,kf

J0(|(ks+kf )|ρ) (22)

I now we use that,

J0(|(ks + kf )|ρe) = J0(|kf |ρe) +O

(ks · kfρ2e

dJ0(z)

dz|z=|kf |ρe

), (23)

I exploit that |ks|ρe ∼ (me/mi)1/2 to bring Rs under the velocity integral

I regard Rs as a fixed parameter in the integration, to find,

ϕe(ts, tf ,Rs,Rf ) = −e(∑

ν

Z2νnνe

2

T

)−1∑kf

eikf ·Rf J0(|(ks + kf )|ρ)

×∫d3v|Rs hekf

(Rs)J0(|kf |ρe)(

1 +O(

(me/mi)1/2))

(24)

I we can evaluate quasi-neutrality purely locally in the slow variable.

5 / 14

Splitting the Gyrokinetic EquationI we apply the ES average to the gyrokinetic equationI we neglect terms which are small by (me/mi)

1/2

Ion scale equation:

∂h

∂ts+v‖b·∇θ

∂h

∂θ+(vM+vE)·∇sh+∇s ·

⟨ cBhvE

⟩ES+vE ·∇F0 =

ZeF0

T

∂ϕ

∂ts. (25)

I we subtract the IS equation from the full equation and neglect termsI The electron equation is orbital averaged to remove fasts electron streaming

timescalesI We consistently take he = 0 for non-zonal passing electrons, for whichhe ∼ (me/mi)

1/2

ES equation:

∂h

∂tf+ v‖b ·∇θ

∂h

∂θ+ (vM + vE +vE) ·∇f h+ vE · (∇sh+∇F0) =

ZeF0

T

∂ϕ

∂tf, (26)

wherevE =

c

Bb ∧∇sϕ, vE =

c

Bb ∧∇f ϕ. (27)

Note that,I there are two additional terms on the ES, vE · ∇f h and vE · ∇sh

I there is one new term at the IS, ∇s ·⟨cBhvE

⟩ES

I vE cannot be removed with the boost or a solid body rotation because of theθ dependence of ϕ

6 / 14

Critical balance

Note ∇φ ∼ ∇φ

⇒eddy E×B drifts vE×B, are comparable at all scales

I applying the critical balance argument

I vte/l‖ ∼ τ−1nl ∼ vE×B/l⊥

I vti/l‖ ∼ τ−1nl ∼ vE×B/l⊥

I l‖ ∼ l‖⇒ parallel correlation lengths are the same for IS and ES eddies

⇒ parallel correlation length are set by the system size – distance betweenstabilising inboard midplane regions parallel to the field

⇒ l‖ ∼ l‖ ∼ a

⇒ ES eddies are long enough to be differentially advected by vE×B

7 / 14

Scaling Work: the Relative Size of the FluctuationsI The usual gyro-Bohm ordering is the only ordering which results in

non-linearly saturated balance

eφ

T∼ ρi∗,

eφ

T∼ ρe∗ (28)

hi

F0i∼

he

F0e∼eφ

T,

he

F0e∼eφ

T,

hi

F0i∼(me

mi

)1/4 eφ

T(29)

I We can show that the following orderings are inconsistent with dominantbalance under our assumptions

I

eφ

T ρi∗ (30)

I

eφ

T ρe∗ (31)

I

eφ

T ρi∗,

eφ

T∼ ρe∗ (32)

I The following ordering is possible only when the ES fluctuations are stabilisedby the IS turbulence

I

eφ

T∼ ρi∗,

eφ

T ρe∗ (33)

8 / 14

Scaling Work: Neglecting Ions at ES

note that:

I J0(kfρi) ∼ (me/mi)1/4

I so: ∫d3v|rhi ∼

(me

mi

)1/4 (memi

)1/4 enφ

T(34)

Ions at ES can be neglected to O((me/mi)

1/2)

in the ES equations!

note that:

I ∇s ·⟨cBhiv

Ei

⟩ES∼ O

((me/mi)v

Ei · hi

)Ions at ES can be neglected to O (me/mi) in the IS equations!

9 / 14

Scaling Work: which multiscale terms do we keep?

The only remaining multiscale terms are in electron species equations:

note that:

I vEe · ∇she ∼ vEe · ∇f he ∼ vEe · ∇f heI IS gradients contribute at O(1) to the ES

I IS perpendicular shear in vEe can be neglected to O((me/mi)1/2) at the ES

I ∇s ·⟨cBhevEe

⟩ES∼ O((me/mi)

1/2vEe · he)

I back reaction contributes at O((me/mi)1/2) to the electron equation at IS

I small and therefore neglected along with the effect of non-zonal passingelectrons

10 / 14

The Parallel Boundary Condition

I ψ: radial, α: field line label,θ: poloidal angle, ζ: toroidal angle

I α(ζ, θ, ψ) = α0 + ζ − q0(ψ)θ = α0 + ζ − q0θ + q′0(ψ − ψ0)θ

I α(ζ, θ + 2π, ψ)− α(ζ, θ, ψ) = −2πq0 − 2πq′0(ψ − ψ0)

A(θ + 2π, α(ζ, θ + 2π, ψ), ψ) = A(θ, α(ζ, θ, ψ), ψ) (35)

Beer et al. (1995)

⇒ b.c. enforces statistical periodicity on a (ψ, ζ) plane

⇒ b.c. couples in α

11 / 14


I A) view along thetoroidal symmetryaxis, of a flux tube,in yellow, withparallel ends inmagenta. The fluxsurface is in grey.

I B) flux tube viewedperpendicular tothe toroidalsymmetry axis.

12 / 14


Notation for ESfluctuations

A(θ+2π , αf , ψf︸︷︷︸ES coords

IS coords︷︸︸︷; αs, ψs )

⇒ ES boundarycondition

A(θ, α(ζ, θ, ψf ), ψf ;α(ζ, θ, ψs), ψs)

= A(θ+2π, α(ζ, θ+2π, ψf ), ψf ;α(ζ, θ+2π, ψs), ψs)(36)

13 / 14

Electrons at IS

−20 −10 0 10 20θ − θ0

0.0

0.1

0.2

0.3|δn′e|/(n0ee|φ|max/T )

MEKEKE – RM

−20 −10 0 10 20θ − θ0

0

1

|δn′i|/(n0ie|φ|max/T )

MEKEKE – RM

10−1 100

kyρth,i

10−3

10−2

10−1

γ /(vth,i/a) at θ0 = 0; aνee/vth,e = 10−2

ME

KE

KE–RM

AE

10−1 100

kyρth,i

10−3

10−2

10−1

γ /(vth,i/a) at θ0 = 0; aνee/vth,e = 10−8

ME

KE

KE–RM

AE

14 / 14

Documents

Cross Scale Interaction Mechanisms: Suppression of the ETG