Quantifying ETG Transport In Spherical Tokamaks With Flow Shear

Walter Guttenfelder

Aug. 28, 2009

Overview

• Relevance of ETG to spherical tokamaks

• ST-like nonlinear gyrokinetic simulations (low aspect ratio, shaped, strong E×B shear)

• Convergence of simulations using kinetic ions & electrons with grid size, resolution and boundary conditions using GYRO (comparison with adiabatic ion model)

• Temperature profile predictions using a model transport expression derived from nonlinear simulations

Experimental Example: MAST Thermal Transport• Considering plasmas PNBI≤2 MW, n~3-5×1019 m-3, Te(0)~Ti(0)≤1.5 keV,

B~0.45-0.55 T• χi approaches neoclassical• χe anomalous, a few times electron

gyroBohm

0 0.2 0.4 0.6 0.8 10

5

10

15

r/a

Shot 8500

m2 /s

χi (exp)

χe (exp)

χi (NC)

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

Te (

keV

)

85008505

0 0.2 0.4 0.6 0.8 10

5

10

15

20

25

r/a

χ e / (ρ

e2 v Te/L

Te)

8500

8505

Te

Te2eGB

e Lvρ

=χ

Field et al., IAEA 2004

E×B Shear Rates Strong Compared to Ion Scale Linear Growth Rates

• From [Kinsey et al., 2007], complete suppression expected at γE≈1⋅γlin,max for R/a=1.6, κ=1.5

LCFS,t

t0unit

unit

rE

rr

BB

BE

rq

rqr

ψψ

=ρ∂ρ∂ρ

=

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

=γ

0 0.5 1 1.5 20

0.2

0.4

0.6

kθρ

i

Ti

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

r/a

γ e / (c

s/a)

8500

8505

17661

Experimental E×B shear rateLinear growth rates

γ/ (

c s/a

)

MAST mid-radius parametersUsing GYRO

ETG Is Potentially Relevant• One MAST prediction with TGLF works “well”, ion transport neoclassical

(large E×B shear), electron transport dominated by ETG• ETG model in TGLF calibrated by one non-linear gyrokinetic simulation

Staebler et al. IAEA 2009; Colyer Stutman et al., PRL 2009

Obviously can not account for flat profiles in core of high power discharges, but maybe the edge – when and where is ETG valid?

ETG?

NSTX

ST-ETG Modeling GoalsLinear stability

• Known linear dependence for low beta, modest shaping [Jenko et al. 2001]

• TGLF will hopefully capture modifications at high beta – benchmarking with GS2, GYRO, GKW is underway [motivated by Roach, EPS 2009]

Non-linear transport• Capture dominant scaling in non-linear transport• Validate (or recalibrate?) ETG model in TGLF

What are the dominant non-linear scalings?

( ) ⎟⎠⎞

⎜⎝⎛ κ+⋅ε−⋅⎟⎟

⎠

⎞⎜⎜⎝

⎛+⋅⎟⎟

⎠

⎞⎜⎜⎝

⎛+=⎟⎟

⎠

⎞⎜⎜⎝

⎛drdr3.015.11

qs91.133.1

TTZ1

LR

i

eeff

critTe

Interpreting Published Simulations Assuming a Critical Gradient Formulation

( )0.1

critTeTeTe

Te2e

e LR

LR

LvF ⎟

⎟⎠

⎞⎜⎜⎝

⎛−⋅⎟⎟

⎠

⎞⎜⎜⎝

⎛ ρ⋅⋅⋅⋅=χ

gyroBohm → local effects only

Non-lineareffects

( ) ⎟⎠⎞

⎜⎝⎛ κ+⋅ε−

×⎟⎟⎠

⎞⎜⎜⎝

⎛+⋅⎟⎟

⎠

⎞⎜⎜⎝

⎛+=⎟⎟

⎠

⎞⎜⎜⎝

⎛

drdr3.015.11

qs91.133.1

TTZ1

LR

i

eeff

critTe

Linear threshold scaling[Jenko et al. 2001]

Strong Dependence Of Nonlinear ETG Transport On Magnetic Shear Expected

• Interpreting three published sets of simulations assuming a critical gradient model

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛−⋅⎟⎟

⎠

⎞⎜⎜⎝

⎛ ρ⋅⋅⋅⋅=χ

critTeTeTe

Te2e

e LR

LR

LvF

−1 0 1 2 30

2

4

6

8

10

( ),...sF

s0 0.2 0.4 0.6 0.8 1

−2

0

2

4

6

r/a

mag

netic

she

ar

8505

8500

17661

MAST experimental s

• Nearly all simulations at low safety factor (q=1.4) – want to expand (s,q) parameter space for STs with a consistent set of simulations

ETG Saturation Problems

• Convergence problem in ETG simulations using adiabatic ions for magnetic shear > 0.4 [Nevins et al., 2006]

Unphysical, monotonicspectra

Simulation Issues

• Convergence problem in ETG simulations using adiabatic ions for magnetic shear > 0.4

• Evidently kinetic ions (ETG-ki), strong E×B shear (ETG-ai-exb), or finite electron collisionality can provide long wavelength cut-off/physical saturation mechanism in the simulations[Nevins et al. 2006; Mikkelsen; Candy et al. 2007; Roach et al., 2009]

• Following simulations for MAST-like mid-radius parameters:R/a=1.6, r/a=0.5κ=1.5, δ=0.2q=1.4, s=0.8a/LT=3, a/Ln=1BUTβ=0 (electrostatic), ν=0

Kinetic ions and electrons (mi/me=3600)

Using GYRO but in the local limit(no n, T, ∇n, ∇T profile variation→no adaptive source)

E×B shear → Er profile, ωE(r)Non-periodic, fixed BCs

Ion Scale Transport Suppressed In Local Simulations With Experimental Level of E×B Shear

• Kinetic ions and electrons• 70ρi×70ρi box, 16/64 toroidal/radial modes, 128 velocity space

• Neither simulation well resolved → transport contributions at high kθ not falling off rapidly (or at all)

0 0.2 0.4 0.6 0.8 1 1.2 1.410

−2

10−1

100

101

kθρ

i

Δ(χ/

χGB

s ) / Δ

(kθρ i)

χi

χe

γE=0.9 c

s/a

0 200 400 600 800 1000 1200−2

0

2

4

6

8

10

t (a/cs)

χi (ρ

s2c

s/a)

χe (ρ

s2c

s/a)

γE=0.9 c

s/a

γE = 0.0 c

s/a

Fractional transport spectraγlin,max(kθρs≤1)≈0.3 cs/a ( ) ( )T

2 sink~k ϕθθ αϕχ

10−1

100

101

102

10−4

10−3

10−2

10−1

100

kθρ

i

Δ(χ/

χGB

s )With Reduced Box Size Spectrum Peak Recovered At

Electron Scales

• Transport peak occurs at kθρe≈0.15

−60 −40 −20 0 20 40 60

−0.2

0

0.2

0.4

0.6

0.8

1

Δ (ρe)

C(Δr)L

r≈24ρ

e

C(Δy)L

r≈9ρ

e

Density contours

Correlation functions

L=64→2ρi (120ρe)γE=0.9 cs/a

Fractional χe transport spectra

y (ρ

e)

60

0

6

0

4.1q8.0s

==

Increased Box Size Alone Does Not Lead To Convergence

• Toroidal/radial grid 16/64→128/512• Turbulence characteristics (spectra, correlation lengths) remain relatively

constant

−500 0 5000

0.05

0.1

0.15

0.2

0.25

r (ρe)

ρ s2 c s/a

0 20 40 60

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Δ (ρe)

C(Δr)L

r≈27−33ρ

e

C(Δy) L

y≈9−10ρ

e

Spatial correlation

FWHM

χe radial profile

Lx=Ly≈2-17ρi

Equilibrium Gradient Perturbations Are Significant• Even in local simulations (flat profiles, a/ρe=7500), perturbations grow

(δTe,n=0/Te~0.01) to cancel equilibrium driving gradients

• Need to optimize boundary regions for these electron scale simulations(new territory)

−600 −300 0 300 600−0.015

−0.01

−0.005

0

0.005

0.01

0.015

r (ρe)

L=4

L=16

−600 −300 0 300 6002

2.5

3

3.5

4

r (ρe)

(a/L

Te) to

tal

L=4

L=16

⟨Te,n=0⟩θ,t / Te,eq (a/LTe)total = (a/LTe)eq + ⟨a/LTe,n=0⟩θ,t

Treatment Of Simulation Boundary• Fixed boundaries lead to large variation at domain edges (e.g. ∇⋅q)• Damping n=0 components of distribution counteracts relaxation

−400 −200 0 200 4000

0.1

0.2

0.3

r (ρe)

ρ s2 c s/a

Δb=16ρe, νb=1 cs/aΔb=64ρe, νb=1 cs/aΔb=64ρe, νb=6,30 cs/aΔb=64ρe, νb=2 cs/a

0nM

buffer0n

0n FTZeh)RHS(

th

=σ=

=⎟⎟⎠

⎞⎜⎜⎝

⎛δφ+⋅ν−=

∂∂

[ ] )R(h)x()x(FT

ez)x(f sMss

rrrr+δφ−δφ−=δ

σ

σ

Damping of axisymmetric mode in boundary

bufferrr >

Δb~102 ρe > 2⋅Lr seems okay, but will likely depend on magnitude of transport & correlation lengths

χe radial profile

Resulting Convergence Problem Is Mostly Explained By Perturbed Equilibrium Gradient

• Exacerbated by resulting stiff transport

(a/LTe)total = (a/LTe)eq + ⟨a/LTe,n=0⟩θ,t

5.1La

crit,linTe

≈⎟⎟⎠

⎞⎜⎜⎝

⎛

2.6 2.8 30

0.1

0.2

0.3

a/LTe,total

χ e (ρ s2 c s/a

)

L scan

Δb scan

νb scan

Strength Of Zonal Potential Converges And Reaches Steady State

• Radial modes (zonal/axisymmetric) tend to evolve on slower time scales, but they appear to have converged and reached steady state

0 10 20 30 40 500

1

2

3x 10

−3

t (a/cs)

n=0

Σn>02/1

r0n

2 ⎟⎠

⎞⎜⎝

⎛ϕ∑

>

( ) 2/1r

20n=ϕ

⟨ϕ2⟩r1/2ϕn=0(r,t)

Transport Nearly Doubles With Increased Binormal Resolution

• Similar to ETG-ai benchmark paper [Nevins et al. 2006]

• With such strong shear suppression, can reduce binormal width (Lx=2Ly)

• Additional runs demonstrate convergence with radial resolution (dx →1.33dx), velocity space (nvel=128→288), and parallel orbit time (nτ=10→14)

−200 −100 0 100 2000

0.1

0.2

0.3

0.4

0.5

r (ρe)

ρ s2 c s/a

kθρ

e,max=0.74

1.50

1.50 (Lx=2L

y)

100

101

102

10−5

10−4

10−3

10−2

10−1

kθρ

i

Δ(χ e/χ

GB

i) / Δ

(kθρ i)

χe radial profile Fractional transport spectra

Including Toroidal Rotation & Parallel Flow Shear Does Not Affect These ETG Scale Simulations

• At this level of shear and box size, there is little difference when including toroidal flow (Mtor) and parallel flow shear (γp)γE=0.9γp=(q/ε)γE=4.32Mtor=0.6

100

101

102

10−5

10−4

10−3

10−2

10−1

kθρ

i

Δ(χ e/χ

GB

i) / Δ

(kθρ i)

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

kθρ

i

γ (c

s/a)

γp=M=0

γp,M>0

10−2

10−1

100

101

102

0

1

2

3

4

5

6

7

kθρ

i

γ (c

s/a)

γp=M=0

γp,M>0

ψ∂Φ∂

=

γ=⎟⎟⎠

⎞⎜⎜⎝

⎛ψ∂Φ∂

∂∂

=γ

⎟⎟⎠

⎞⎜⎜⎝

⎛ψ∂Φ∂

∂∂

=γ

0

s

0

E00

0p

0E

cRM

rqR

rR

rqr

Ion scale growth rates Electron scale growth rates Fractional transport spectra

Turbulence Characteristics Qualitatively Constant• While quantitative transport has varied considerably (χe=2-8ρe

2vTe/LTe), qualitative turbulence characteristics are largely unchanged

• In the presence of E×B shear, eddies remain elongated (but tilted) with Lr≈3Ly

-150 75 0 75 150r (ρe)

y (ρ

e)

-120

60

0

60

120

−100 −50 0 50 100

−0.2

0

0.2

0.4

0.6

0.8

1

Δ (ρe)

C(Δr)L

r≈26ρ

e

C(Δy)L

y≈9ρ

e

Potential contours

Ltotal≈36ρe

1-D correlation functions

Spectra Are Anisotropic In The Presence Of E×B Shear

• Spectra are anisotropic at high kρe>0.2, contrary to:– ETG simulations without E×B shear [Nevins et al., 2006]

– Multi-scale simulations without E×B shear [Waltz et al., 2007]

10−2

10−1

100

10−10

10−9

10−8

10−7

10−6

kρe

<φ2(kθ)>

<φ2(kr>0)>

<φ2(kr<0)>

log[ϕ2(kθ,kr)] ∫ ϕ dk2

T

Er kk

ωΔγ

≈ θ

Consideration for interpretation of high-k turbulence measurements & synthetic diagnostics?

0 10 20 30 40 500

1

2

3

4x 10

−3

t (a/cs)

n=0

Σn>0

Adiabatic Ion Model Can Be Unreliable Even With E×B Shear

kinetic ion νb=6 cs/aadiabatic ion νb=6 cs/aadiabatic ion νb=60 cs/a

Stronger buffer dampingonly delays onset

0 10 20 30 40 500

0.2

0.4

0.6

0.8

t (a/cs)

ρ s2 c s/a

Strong Radial Mode Develops

ϕn=0(r,t) ϕn=0(r,t)

t=100 (a/cs)

Kinetic ions Adiabatic ions

While The Adiabatic Ion Model Can Be Unreliable It Has Been Used For An Initial Scan in s & q

• Earlier (naïve) simulations for comparable box size and resolution were run for varying magnetic shear and safety factor

• They used different boundary conditions and adaptive source (relatively strong, small perturbed gradients)

• None of the simulations exhibited the transition

Significant Transport Occurs For Window Of s/q• Higher safety factor recovers larger transport• Simple fit to simulations for comparison with experiment

−0.5 0 0.5 1 1.50

2

4

6

8

10

12

s/q

F

q=1.4

q=4.4

SimFit

( )s

q

c

54

32

c

1

)qc1(c)qc1(cq/s1

qcq,sF

++−

+

⋅=

( )

cq ≈ 0.7cs ≈ 1.9

⎟⎟⎠

⎞⎜⎜⎝

⎛−⋅⎟⎟

⎠

⎞⎜⎜⎝

⎛ ρ⋅⋅⋅⋅=χ

critTeTeTe

Te2e

e LR

LR

LvF

84.0

13.1PPPLIFS

i s1q~+

χ −⎟⎟⎠

⎞⎜⎜⎝

⎛<<<<

2s5.08q7.0

For comparison, χi from IFS-PPPL

Model ETG Transport Significant Over Midradius• Sensitive to safety factor profile (and derivative) – experimental

uncertainties?

Decreasing qmin (& larger s/q)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛−⋅⎟⎟

⎠

⎞⎜⎜⎝

⎛ ρ⋅⋅⋅⋅=χ

critTeTeTe

Te2e

e LR

LR

LvF

0 0.2 0.4 0.6 0.8 10

5

0

5Shot 17661

0 0.2 0.4 0.6 0.8 10

5

1

5

r/a

ExpModel

0 0.2 0.4 0.6 0.8 10

5

0

5Shot 8505

0 0.2 0.4 0.6 0.8 10

.5

1

.5

r/a

ExpModel

0 0.2 0.4 0.6 0.8 10

5

10

15

χ e (m

2 /s)

Shot 8500

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

r/a

Te (

keV

)

ExpModel

( ) BC,eelmod

eexpee

elmode

T,qT

T

+χ∇−

=

∫

Direct comparison of χe using experimental profiles

Predicting Te profile using experimental heat flux and transport model

Additional Scaling Expected From Varying Collisionality

• Collisionality in MAST stabilizing to trapped electron contributions• Transport reduced ~25%

• Additional scaling expected with γE and β,∇β

100

101

102

0.0001

0.001

0.01

kθρ

i

Δ(χ e/χ

GB

i)

10−1

100

101

102

0

2

4

6

kθρ

i

γ (c

s/a)

Linear growth rates Fractional transport spectra

γE

Summary

• Demonstrating convergence of ETG simulations for large magnetic shear, important for spherical tokamak parameters

• Sensitive to simulation boundary (not using spectral flux tube representation)

• Adiabatic ion model with E×B shear can reproduce kinetic ion results, at least in some cases – expect larger disparity for reduced E×B shear (increased drive at ion scales)

• Initial simple transport model seems to recover similar behaviour to TGLF predictions (in one instance)

Strong Dependence Of Nonlinear ETG Transport On Magnetic Shear Expected

• Interpreting three published sets of simulations assuming a critical gradient model

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛−⋅⎟⎟

⎠

⎞⎜⎜⎝

⎛ ρ⋅⋅⋅⋅=χ

critTeTeTe

Te2e

e LR

LR

LvF

−1 0 1 2 30

2

4

6

8

10

Jenko & Dorland 2002ai, EM, α=0, f

t=0

( ),...sF

s

Strong Dependence Of Nonlinear ETG Transport On Magnetic Shear Expected

• Interpreting three published sets of simulations assuming a critical gradient model

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛−⋅⎟⎟

⎠

⎞⎜⎜⎝

⎛ ρ⋅⋅⋅⋅=χ

critTeTeTe

Te2e

e LR

LR

LvF

−1 0 1 2 30

2

4

6

8

10

Nevins et al. 2006ki, ES, α=0, f

t ≠ 0

( ),...sF

s

Strong Dependence Of Nonlinear ETG Transport On Magnetic Shear Expected

• Interpreting three published sets of simulations assuming a critical gradient model

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛−⋅⎟⎟

⎠

⎞⎜⎜⎝

⎛ ρ⋅⋅⋅⋅=χ

critTeTeTe

Te2e

e LR

LR

LvF

−1 0 1 2 30

2

4

6

8

10

Joiner et al. 2006MAST Eq. ψ

N = 0.4

ai/ki, EM, ft ≠ 0

MAST Eq.ψ

N = 0.8

( ),...sF

s

Strong Dependence Of Nonlinear ETG Transport On Magnetic Shear Expected

• Interpreting three published sets of simulations assuming a critical gradient model

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛−⋅⎟⎟

⎠

⎞⎜⎜⎝

⎛ ρ⋅⋅⋅⋅=χ

critTeTeTe

Te2e

e LR

LR

LvF

−1 0 1 2 30

2

4

6

8

10

Jenko (2002), analytichigh shear limit

( ),...sF

s

Strong Dependence Of Nonlinear ETG Transport On Magnetic Shear Expected

• Interpreting three published sets of simulations assuming a critical gradient model

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛−⋅⎟⎟

⎠

⎞⎜⎜⎝

⎛ ρ⋅⋅⋅⋅=χ

critTeTeTe

Te2e

e LR

LR

LvF

−1 0 1 2 30

2

4

6

8

10

( ),...sF

s0 0.2 0.4 0.6 0.8 1

−2

0

2

4

6

r/a

mag

netic

she

ar

8505

8500

17661

MAST experimental s

ST-ETG Modeling GoalsLinear stability

• Known linear dependence for low beta, modest shaping [Jenko et al. 2001]

• TGLF will hopefully capture modifications at high beta – benchmarking with GS2, GYRO, GKW is underway [motivated by Roach, EPS 2009]

0 0.5 1 1.5 20

0.1

0.2

0.3

0.4

0.5

kθρ

i

γ (v

Ti/a

)

GS2

GKW

GYRO

0 10 20 30 400

1

2

3

4

5

kθρ

i

γ (v

Ti/a

)

GS2

GKW

GYRO

−2 0 20

0.5

1

1.5

2

θ (rad)

|B|

−2 0 20

0.5

1

θ (rad)

ω∇B

(−), ωκ (−−)

−2 0 20

50

100

θ (rad)

k⊥2

GS2−EQDSK

GYRO−geq

GeometryVerification

GyrokineticVerification

MAST experimental case, mid-radius, EFIT equilibrium specification

Ion scale growth rates Electron scale growth rates