Interaction Between MagneticInteraction Between MagneticIslands and ElectrostaticIslands and Electrostatic

TurbulenceTurbulenceF.L. Waelbroeck1,2, F. Militello1,2, R.Fitzpatrick1

1 Institute for Fusion Studies, The University of Texas at Austin2 Center for Multiscale Plasma Dynamics, University of Maryland

OutlineOutline

nn Islands and magnetic confinementIslands and magnetic confinementnn Model for resistive drift waves andModel for resistive drift waves and

electrostatic turbulenceelectrostatic turbulencenn Linear analysisLinear analysisnn Interaction of island and turbulenceInteraction of island and turbulence

Dynamical scales in magneticDynamical scales in magneticfusionfusion

nn Macroinstabilities Macroinstabilities are coherent, globalare coherent, globalmodes with modes with kkrrii << 1 and << 1 and AlfvAlfvéénic nic growthgrowthrates (rates (gg~~kVkVA A ).).

nn Microinstabilities Microinstabilities give rise to turbulencegive rise to turbulencewith with kkrrii~1~1

nn Many important problems involve theMany important problems involve theinteraction of micro and macro-scales: e.g.interaction of micro and macro-scales: e.g.reconnection, zonal-flow generation, andreconnection, zonal-flow generation, andmagnetic islandsmagnetic islands

Magnetic confinement reliesMagnetic confinement relieson closed nested flux surfaceson closed nested flux surfaces

nn Parallel transport isParallel transport ismuch faster thanmuch faster thanperpendicularperpendiculartransport.transport.

nn Good confinementGood confinementrequires that the fieldrequires that the fieldlines lie on fluxlines lie on fluxsurfacessurfaces

Magnetic confinement reliesMagnetic confinement relieson closed nested flux surfaceson closed nested flux surfaces

nn Magnetic islandsMagnetic islandsoccur when the fluxoccur when the fluxsurfaces pinch andsurfaces pinch andreconnect (tear)reconnect (tear)

nn Magnetic islandsMagnetic islandsallow heat to flowallow heat to flowoutwards along fieldoutwards along fieldlines.lines.

Resonant perturbations createResonant perturbations createMagnetic IslandsMagnetic Islands

nn Force balance:Force balance:B B ⋅— J=0 => J=J( J=0 => J=J(yy))

nn Flux determined fromFlux determined fromAmpereAmpere’’s law s law DyDy = J(= J(y)y)

nn Current profile can beCurrent profile can bedetermined fromdetermined from(a) flux conservation(a) flux conservation(b) Ohm(b) Ohm’’s law s law hh<J>=E<J>=E

There are similarities betweenThere are similarities betweenMagnetic Islands and BGK modesMagnetic Islands and BGK modes

nn Vlasov Vlasov steady-state:steady-state:ZZ⋅—f=0 => f = f(E)f=0 => f = f(E)

nn jj determined by determined byPoissonPoisson’’s law: s law: DjDj = = Ú dvdv ff

nn Distribution functionDistribution functiondetermined bydetermined by(a) particle conservation(a) particle conservation(b) (b) collisional collisional diffusiondiffusion

Interaction of magnetic islandsInteraction of magnetic islandswith turbulencewith turbulence

nn Magnetic islands and turbulence coexist in fusionMagnetic islands and turbulence coexist in fusionexperimental devices.experimental devices.

nn The standard approach is to model the effects ofThe standard approach is to model the effects ofturbulence through anomalous dissipativeturbulence through anomalous dissipativeparameters in the island equations. This isparameters in the island equations. This isquestionable when the dimensions of the islandquestionable when the dimensions of the islandand turbulent structures are comparable .and turbulent structures are comparable .

nn We show here the results of a self-consistentWe show here the results of a self-consistentanalysis of their interaction.analysis of their interaction.

Physical Physical ModelModel

nn If theIf theturbulence time scale isturbulence time scale isfaster than evolution offaster than evolution ofthe magnetic field.the magnetic field.

nn The two time scales canThe two time scales canbe separated.be separated.

Electrostatic ApproximationElectrostatic Approximation*/ w<<wwdt

IN THE TIME SCALE OFTURBULENCE ISLAND WIDTH IS ASSUMED FIXED

w = CONST

Reduced Electrostatic ModelReduced Electrostatic ModelThe solutions depend on:The solutions depend on:nn C = 1 C = 1

(edge plasma)(edge plasma)nn D = D = mm

(like the Reynolds(like the Reynolds-1-1 number) number)nn w w

(island width /(island width / larmor larmor radius) radius)nn ww

(through the b.c.)(through the b.c.)

)cos(162

],[1

],[],[

],[],[

22

2

2

2

ywx

nC

J

U

nDJnt

n

UJUt

U

+-=

-=

—=

—+=+∂

∂

—+=+∂

∂

y

yj

j

yj

myj

Equations:

With:

Linear AnalysisLinear Analysis

Linear Stability of the Drift WavesLinear Stability of the Drift Waves

w = 0.5

The rotation frequencymesures the amount offlattening.

The larger the island,the larger is the regionof flattening.

nn The equilibriumThe equilibriumradial densityradial densitygradient destabilizesgradient destabilizesresistive drift-wavesresistive drift-waveswhich drive thewhich drive theturbulence.turbulence.

Linear Stability of the Drift WavesLinear Stability of the Drift Waves

w = 1.5

The rotation frequencymesures the amount offlattening.

The larger is the island,the larger is the regionof flattening.

nn The equilibriumThe equilibriumradial densityradial densitygradient destabilizesgradient destabilizesresistive drift-wavesresistive drift-waveswhich which drive thedrive theturbulenceturbulence..

Summary of the linear resultsSummary of the linear results

w = 0.5

1. We clearly see that slow large islands can quench turbulence

2. We find a region of the parameter space where only even modesdrive turbulence: in the following we assume D=m=0.1

w = 3

Nonlinear ResultsNonlinear Results

1st Turbulence Simulation1st Turbulence Simulation

nn Periodicity in the y-Periodicity in the y-direction.direction.

nn No-slip conditionNo-slip conditionat the edgeat the edge

nn Density gradient inDensity gradient inthe the xx-direction-direction

Vorticity in the turbulent phase.

Effect of the turbulenceEffect of the turbulencenn The turbulent currents modify the evolution The turbulent currents modify the evolution of theof the

magnetic islandmagnetic island..nn The The component component in in phase affects phase affects the the widthwidth::

nn and the and the component component in in phase phase quadrature the quadrature the propagationpropagationfrequencyfrequency::

ÚÚ•

•--D= )cos(

16'

82.02

ydyJdxwdt

dw

C turbp

ÚÚ= )sin(16

),(2

ydyJdxLL

wwF turb

yxy w

),( wwpolD=

The The two approachestwo approaches

nn Fixed Fixed Rotation Rotation frequencyfrequency:: w w , , ww FFy y , , DDpolpol

nn Fixed Fixed Force (Force (equals to equals to zero):zero): w w , , FFyy=0=0 ww , , DDpolpol

ÚÚ= )sin(16

),(2

ydyJdxLL

wwF turb

yxy w

Remember!

C = 1

D = m =0.1

Fixed Fixed Rotation - Rotation - QuiescentQuiescent statesstates

nn If odd parity isIf odd parity isenforced for enforced for nn and andjj, , there are nothere are noelectrostaticelectrostaticmicroinstabilitiesmicroinstabilities..

nn The The quiescentquiescentstates states are are taken astaken asa a referencereference for for thetheturbulent statesturbulent states..

w = 1.5

Fixed Rotation - Turbulent statesFixed Rotation - Turbulent states

nn If parity is If parity is notnotenforced,enforced,turbulence canturbulence candevelop whendevelop whenw w > > wwcc((ww).).

nn The differenceThe differencebetween thebetween thecurves gives thecurves gives thenet effect of thenet effect of theturbulence.turbulence.

w = 1.5

wc

Fixed Rotation - Turbulent statesFixed Rotation - Turbulent states

w = 1.5 w = 3w = 0.5

nn Dependence on the island width:Dependence on the island width:

wcwcwc

Free rotationFree rotation –– FFyy=0=0

nn TurbulenceTurbulenceinduces transitionsinduces transitionsto states with ato states with aslower rotationslower rotationfrequency.frequency.

Summary Summary and and ConclusionsConclusionsnn We have investigated the interactionWe have investigated the interaction

between turbulence and magnetic islandsbetween turbulence and magnetic islandsin a self-consistent wayin a self-consistent way

nn For thin islands the turbulence drivesFor thin islands the turbulence drivesisland growth.island growth.

nn The turbulence induces transitions toThe turbulence induces transitions tostates with a slower rotation frequency.states with a slower rotation frequency.

nn For sufficiently large islands the final stateFor sufficiently large islands the final statehas a negative drive.has a negative drive.