Field line breaking andMicrotearing Turbulence

Hauke Doerk

Thanks to F. Jenko and the GENE team

Max-Planck-Institut für Plasmaphysik, Garching

Winter School on dynamicsand turbulent transport in plasmas and conducting fluids,

Les Houches, March 9 2011

A brief history of microtearing research

• 1968: Tearing instability (Furth, Killeen, Rosenbluth)

• 1975: Instability due to ∇Te: µ-tearing (Hazeltine et al.)

• 1980: Model for saturation (Drake et al.)

• 1990: µ-tearing should be stable for realistic tokamakscenarios (Connor et al.)

• 1999: Focus on µ-tearing in plasma edge (Kesner et al.)

• 2003: Linear gyrokinetic simulations (Redi et al., Applegateet al.); Large electron heat transport in spherical tokamakscaused by µ-tearing?

• 2008: µ-tearing modes also found in conventional tokamaks(linear GK, Vermare et al., Told et al.)

• 2010: Nonlinear Gyrokinetic simulatios (Guttenfelder, thiswork)

Scope of this work

Problems

• Existence of microtearing instability in Tokamak geometry

• Electromagnetic heat transport caused by microtearing

• Nonlinear saturation of microtearing turbulence

Strategy

• Linear and nonlinear simulations using GENE

• Examine impact of steeper gradients, collisional effects...

• Comparison to analytical models

The GENE codeGyrokinetic Electromagnetic Numerical Experiment

Solves gyrokinetic equations on fixed grid in 5D phase space(⇒continuum code)

• Comprehensive physics

• Massively parallel

• Open source

http://gene.rzg.mpg.de

Characteristics of µ-tearing modes

Ballooning representation

• Fluctuating electrostaticpotential φ extends alongfield line

• Vector potential A‖ isstrongly localized aroundθ = 0

µ-tearing modes found in

• Spherical tokamaks(NSTX, MAST)

• Conventional tokamaks(ASDEX Upgrade)

• Model geometry: Circular(Lapillonne et al. 2009)

-10 -5 0 5 10

|A‖|,

ℜ(φ)(a.u.)

ballooning angle θp/π

|A‖|ℜ(φ)

Turbulent field fluctuations

Vector potential A‖

• large structures

Electrostatic potential φ

• small scale eddies

Microtearing turbulence is hard to resolve innumerical sumulation

Standard theory for fluctuation amplitude

Drake et al. 1980

• γlin ∼ DMk2⊥

• ⇒ δB/B0 ∼ %e/LTe

• simplified modelunderlying

Challenge

• 384×64×24×32×16grid points inx , y , z, v‖, µ space

• ∼ 25000 CPU hoursper run

0

1

2

3

4

5

6

0 1 2 3 4 5 6(δB

x/B

0)/(ρ

e/R

)

R/LTe

Gene simulationsδB/B0 = ρe/LTe

lowered resolution as a convergencestudy: scatter is moderate

Can gyrokinetic codes be used to prove or refine Drake’smodel?

Influence of temperature gradients

Ions

• R/LTi notimportant

Electrons

• RLTe

= − RTe

∂Te∂x

crucial

•(

RLTe

)crit∼ 1.3

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0 1 2 3 4 5 6 7

γ/(v t

i/R)

R/LTe

(R/LTe)0

(R/LTe)crit

γ(kyρs = 0.05)γ(kyρs = 0.1)

Existence of a critical electron temperaturegradient confirmed

Influence of other parameters, Example: βe

• Bx and γlin aresensitive to βe

• Critical valueβe,crit ∼ 10−3

• Also geometry(q0, s) andcollisionalityplay a role..

0

0.01

0.02

0.03

0.04

0.05

0.001 0.01

γ/(v t

i/R)

βe

βe0

βe(ASDEXupgrade)

kyρs = 0.06

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.002 0.004 0.006 0.008

(Bx/B

)/(ρ

e/L

Te)

βe

Gene simulationDrake’s formula

Drake’s Formula gives rough estimate for the fluctuationamplitude

Magnetic Field Stochastization

weak drive strong drive

Magnetic field fluctuation amplitude ofmicrotearing turbulence determines degree of field

stochastization

Heat Transport in Stochastic Magnetic Fields

Diffusivity model

• χeme = vteDM

• DM = LC(δB/B0)2

(quasilinear result)

• B− correlation lengthLC = qR

Heuristic: random walk

• D = (∆r)2/∆t

• ∆r = (δB/B0)LC

• ∆t = LC/vte

0

0.5

1

1.5

2

2.5

3

3.5

4

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

χe em

/(ρ2 iv t

i/R)

(δBx/B0)/(ρi/R)

modelR/LTe variation

β variation

χeme = 1.36vteqR(δB/B0)2

GENE resultsconfirm simple model (e.g.Liewer

1985) collisionless case(breaks down at weak drive)

Effects of collision rate νe

Collisional regime

• Transition expectedaround νc = 0.01 (linearphysics)

• Lc → λmfp

• λmfp = vte/νe

• χeme = vteλmfp(δB/B0)2

∝ 1/νe expected

Gyrokinetic results

• χeme decreases weaker

than 1/νc

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.1 1 10

γ/(v t

i/R)

collision frequency ν∗

λmfp

2πq0R=1

ν∗0

λmfp

2πq0R=10

γ ∝−νc

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.1 1 10

γ/(v t

i/R)

collision frequency ν∗

λmfp

2πq0R=1

ν∗0

λmfp

2πq0R=10

γ ∝−νc

kyρs = 0.01kyρs = 0.01kyρs = 0.05

0.1

1

1

L‖/qR

collision rate ν∗

model: χeem = L‖vte(δB/B0)

2

Gene simulationγlin, ky = 0.12 (a.u.)∼ 1/νc for νc > 0.01

Nonlinear Saturation of Microtearing Turbulence

Model by Drake ’80

• γlin ∼ vteλmfp (%e/LTe)2 k2⊥

• γNL ∼ −DMk2⊥

• Balance: ⇒ δB/B0 ∼ %e/LTe

Gyrokinetic simulations

• Accessible parameter regime:γlin,max ∼ (R/LTe)2vti/R

• Dissipation scale:kdiss ≈ 0.2/%i

• γNL = −χeme k2

diss

• Balance: ⇒ δB/B0 ∼ %e/LTe

0

1

2

3

4

5

6

0 1 2 3 4 5 6

(δB

x/B

0)/(ρ

e/R

)

R/LTe

Gene simulationsδB/B0 = ρe/LTe

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.1 1

a.u.

kyρi

linear growth rate γlfree energy sources/sinks

magnetic electron heat flux Qeme

Effects of E × B shear flow

Simulation results

• Heat flux is reduced

• It does not seem to vanishup to vE×B ≈ 10γlin

• Low ky remain (unusualcompared to ITG/TEM)

Possible explanation:

• Special microtearingmode structure

0

0.2

0.4

0.6

0.8

1

1.2

0 2 4 6 8 10 12

χem e

/χem e,0

E ×B shearing rate/γmax

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0 0.1 0.2 0.3 0.4 0.5 0.6

heatflux

kyρi

lin. growth rate γL (A.U.)Qe

em ExB rate=0.84×Qe

em ExB rate=0.8Qe

em R/LTe = 4.5

Summary and Conclusions

• Microtearing modes can be unstable in conventionalTokamaks

• Heat transport can be substantial (∼ 1m2/s)

• Drake’s Formula for saturation amplitude is roughlyconfirmed, but parameters other than LTe (βe) are crucial

• Transition to collisional regime is seen

• Further simulations:System size: globalmicrotearing + ITG. . .

• Very recent: compare to ASDEX Upgrade measurements inthe outer core (Wolfrum et al.)

Thank you for your attention!