Fulvio Militello – Princeton, Active MHD control workshop 2006 1

Interaction Between Magnetic

Islands and Electrostatic

TurbulenceF. Militello1,2, F.L. Waelbroeck1, R.Fitzpatrick1, W. Horton1

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

Fulvio Militello – Princeton, Active MHD control workshop 2006 2

Outline

• Basic Principles: from Standard to Reduced

Physical Model

• The Numerical Codes

• Magnetic Island affecting Turbulence

• Turbulence affecting Magnetic Island

• Summary and Conclusions

Fulvio Militello – Princeton, Active MHD control workshop 2006 3

Standard Electromagnetic Model

• A Standard 2D slab, three-field model with curvature

terms with constant Te and cold ions can describe both

magnetic reconnection and turbulence:

Fulvio Militello – Princeton, Active MHD control workshop 2006 4

Parametrization of the Magnetic

Island

• How does electrostatic

turbulence affect theevolution of w and ?

• The method could be

generalized to more

complex island structures:

the detailed shape is not

important, the topology is!

w

Fulvio Militello – Princeton, Active MHD control workshop 2006 5

• If the

turbulence time scale is

faster than evolution of the

magnetic field.

• The two time scale can be

separated.

Electrostatic Approximation

IN THE TIME SCALE OF

TURBULENCE ISLAND WIDTH IS

ASSUMED FIXED

w = CONST

Fulvio Militello – Princeton, Active MHD control workshop 2006 6

Reduced Electrostatic Model

• We use an extended version of the Hasegawa-Wakatani equations in the island frame ofreference.

Fulvio Militello – Princeton, Active MHD control workshop 2006 7

The Island Frame of Reference

• Like in a wind tunnel, we keep the island still while theplasma flows at a velocity, V.

• The island rotation frequency, , is calculated self-consistently.

Laboratory frame of reference:

Island moving at velocity: V= /ky

Island frame of reference

Plasma velocity -V=- /ky

Fulvio Militello – Princeton, Active MHD control workshop 2006 8

Linear Code mode dispersion

relation

• Linear stabilityinvestigated with thelinear code LinHWDF.

• When G>0, theinterchange mode isunstable and drivesthe turbulence.

0.020.020.451

μDGC

Fulvio Militello – Princeton, Active MHD control workshop 2006 9

Nonlinear Code HasWak

• Equations solved using an initial value,

finite-difference (4°-order in space, 2°-

order in time)

• The code employs fully-implicit, multi-grid,

time-stepping algorithm constructed using

PETSc.

• It runs on a two-processor machine at the

IFS and on NERSC.

Fulvio Militello – Princeton, Active MHD control workshop 2006 10

Turbulence Simulation Set Up

• Initial Condition: random turbulence localized

with a Gaussian envelope.

• Adiabatic Fields ( ).

• Turbulent relaxation and driven Turbulence (by

means of interchange instability) investigated.

• Simulations in the island frame of reference,

island rotation frequency calculated using force

balance in poloidal direction.

Fulvio Militello – Princeton, Active MHD control workshop 2006 11

Turbulence Simulation Set Up

• Periodicity in the y-direction.

• G pointingdownward, ElectronDiamagneticvelocity from left toright.

• Density gradient inthe x-direction

Fulvio Militello – Princeton, Active MHD control workshop 2006 12

Turbulent Relaxation

Sheared Magnetic Field Magnetic Island

G=0Vorticity Field

Fulvio Militello – Princeton, Active MHD control workshop 2006 13

Turbulent RelaxationG=0

Vorticity Field

Sh

eare

d F

ield

Mag

neti

c I

sla

nd

Fulvio Militello – Princeton, Active MHD control workshop 2006 14

Driven Interchange Turbulence

Sheared Magnetic Field Magnetic Island

G 0Vorticity Field

Fulvio Militello – Princeton, Active MHD control workshop 2006 15

Driven Interchange Turbulence

Sh

eare

d F

ield

Mag

neti

c I

sla

nd

G 0Vorticity Field

Fulvio Militello – Princeton, Active MHD control workshop 2006 16

Turbulent Particle Flux

• The presence of a magnetic island changes thebehavior of the turbulent diffusion (D

T~10D):

w=0 w=0.4

Fulvio Militello – Princeton, Active MHD control workshop 2006 17

Effect of the turbulence on the

Island I

• The effect of the turbulence on the islandgrowth is mesured by ’T:

Standard Rutherford Equation: w=w( ’,equilibrium,Bootsrtap current)

New term due to electrostatic turbulence,

Fulvio Militello – Princeton, Active MHD control workshop 2006 18

Effect of the turbulence on the

Island II

• The island magnetic field

and the turbulent currents

produce an

electromagnetic force on

the island and induce

island rotation.

• The island rotationfrequancy, , is

determined using the

poloidal momentum

conservation equation.

Fulvio Militello – Princeton, Active MHD control workshop 2006 19

Time evolution of and ’T

• The island velocity matches that of the coherentturbulent structures.

• ’T is positive (destabilizing) and oscillates around aconstant value.

Fulvio Militello – Princeton, Active MHD control workshop 2006 20

Island rotation Frequency, an

interpretation

• Turbulenceproduces zonalflows.

• Coherent structuresmove in the zonalflow field.

• The island is“dragged” by thecoherentstructures.

Fulvio Militello – Princeton, Active MHD control workshop 2006 21

MI affected by Turbulence

• We did a scan of ’T and

for different w.

• In this case, Turbulence

produces a destabilizing

effect and reduces the

poloidal rotation

frequency of the island.

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Summary and Conclusions

• We have investigated the interaction betweenturbulence and magnetic islands in both“directions”

• We find that turbulence is strongly affected by areconnected magnetic field.

• The magnetic islands are destabilized byturbulence and their rotation frequency issignificantly modified.

• Further work is planned to understand how theturbulence and the magnetic island react todifferent set of parameters.