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Studies of Nonlinear Two Fluid Tearing Modes inStudies of Nonlinear Two Fluid Tearing Modes inStudies of Nonlinear Two Fluid Tearing Modes inCylindrical Reversed Field Pinches

Studies of Nonlinear Two Fluid Tearing Modes inCylindrical Reversed Field Pinches

V. V. Mirnov, J. R. King and C. R. Sovinec

University of Wisconsin-Madison and Center for Magnetic Self - Organization in Laboratory and Astrophysical Plasmas

2008 US-Japan Workshop “Progress of Multi-Scale Simulation Models”, Nov. 21-22, Dallas, TX

The Madison Symmetric Torus RFP experiment

R= 1 5 m a = 0 5 m I 0 5 MA T≤ 1 keV n ≤ 1013cm-3

2

R= 1.5 m, a = 0.5 m, I 0.5 MA, T≤ 1 keV, n ≤ 1013cm 3

Magnetic configuration of Reversed Field Pinch

• Toroidal field BT is ~ 5 times smaller than same-current tokamaksame current tokamak

• Equilibrium substantially determined by self-generated

3plasma currents

RFP equilibrium - resonant unstable modes exist

k·B = m BP /r – n BT / R = 0 → m/n = q(r) = r BT/ R Bp

Stability depends on (r) = J / B Stability depends on (r) = J|| / BMST Reversed Field Pinch

~0.2

limiter /

co

resonant modes (tearing / resistive kink)

(1 6)q(r)

/ lineronducting m 1 n 2R/a

(1,6)(1,7)

1 6 7 shell

m=1, n 2R/am=0

m=1, n = 6, 7… core modes

4minor radius, r0 ar

3D structure of core and edge modes

3D magnetic perturbations (m=1, n=6 core mode )

magnetic surfaces formagnetic surfaces for m=0, n=1 edge mode

two unstable core modes are nonlinearly coupled to the edge modenonlinearly coupled to the edge mode

k1 7 + k-1,-6 = k0 15

1, 7 1, 6 0,1

Large magnetic energy only occurs with both core and edge reconnection

6

Multi-scales of two fluid MHD: key findings

wT ~ a/S2/5 ρs ~(1/ωci) [(Te +Ti) /m]1/2 a

resistive tearing layer ion-sound gyroradius minor radius

Two-fluid physics + cylindrical field line curvature

Linear tearing mode

Two-fluid physics + cylindrical field line curvature

l th t ( d t ti ) ith t ff tcomplex growth rate (mode rotation) without ω*i,e effects

Nonlinear NIMROD simulations

Small modification of the mean current is enough for nonlinear mode saturation

Hall dynamo is broadened in cylindrical geometry contrary to the fine structure along the separatrix

7observed in 2D slab simulations

Cylindrical RFP configuration brings new effects into play

Radial component of the induction equation

Parallel component with curvature and current gradient effects

Force balance in cross-field b er direction

• Two fluid tearing instability is driven mainly by the electrons.

• Ion motion enters the equations through the plasma compressibility.

• Equilibrium diamagnetic flows are ignored ( force free equilibrium ).

8

q g g ( q )

Tearing equations are simplified on short scales

Coupled equations for two fluid cylindrical tearing mode

Renormalized function B ||

Quasi-linear approach to mean field characteristicspp

(1/en(0)c) <j(1)B(1)>|| - (1/ c) <v(1) B(1)>|| = <E>|| - <j>|| ,Hall dynamo MHD dynamoHall dynamo MHD dynamo

Cross-phase between Br and B|| determines the Hall dynamo

Plasma flow velocities are reduced and broaden to large s scale

• Flow profiles in two limiting cases:(a) s S2/5 = 0.1(MHD) (b) S2/5 = 10 (two fluid)(b) sS2/5 = 10 (two-fluid)

MHD case ideal solution

ideal solution

two-fluidregime

• Amplitudes of the flow velocities in 2F and 1F cases

10

NIMROD is an initial value solver that capable of modeling the two fluid tearing instability

ept

11 BJBVJB Faraday’s / Ohm’s law

enenet

BJ 0

a aday s / O s a

Ampere’s law

V

0 B divergence constraint

WBJVVV

p

t flow evolution

VIVVW 2T

nDntn

V

particle continuitywith artificial diffusivity

VIVVW3

TnpT

tTn

VV

1temperature evolution

11• We set D, , and <<0.

Equilibrium magnetic configuration of the paramagnetic pinch

(2) λ0=3.3, S=5x103• Force-free equilibrium

• External electric field Ezdetermines λdetermines λ

• Ampere’s law magnetic configuration is determined by one dimensionlessby one dimensionless parameter 0

radius

Robinson's study of paramagnetic equilibrium provides a map of the linear stability

m=1 The mode is growing if stability factor > 0

tearing unstable

stability factor 0

k / λ0=0.6 with λ0=2.7 are chosen initially to provide a mode close tostable initially to provide a mode close to marginal stability with ' =2.λ0

stable

' =16

more unstable modes with ‘=16 are studied at λ0 = 3.3 (k/λ0 remains constant to keep q constant)

k /λ0=0.6constant to keep q0 constant).

D.C.Robinson, Nucl. Fusion (1978)' =2

λ0

The growth rate and mode rotation scales with ρs

MHD λ0=3.3 Electronال skin depth5.00E-02

MHD ال2fl ال2fl Rot

Rat

e δ=0.1

0S=5x103

β=0.1 Pm=0.1 d 1 8 10 2at

e

Electron skin-depth

collisionless part deis small compared t th i ti t

Gro

wth

R de=1.8x10-2

Gro

wth

ra

to the resistive term 5.00E-03 δ=0.16

G

B 0 4 T B 0 2 TS l

0.010 0.100 1.000

ρsIon gyroradius ρs/a

B0=0.4 Tn=1x1019 m-3

β=0.10 51

B0=0.2 Tn=1x1018 m-3

β=0.10 51

Some examples of plasma parameters:

a=0.51 mρs/a=5.8x10-2

a=0.51 mρs/a=1.8x10-1

2F dynamics and curvature effects result in mode rotation without diamagnetic flows

15

The quasilinear MHD dynamo is broadened at large ρs

The amplitude of <v(1) ×B(1)> decreases in 2F case (consistent with the theory)

single fluid ρs = 0.1tie

s / v

Asingle fluid ρs 0.1

ow v

eloc

it

r/a

flo

r/a

B0

mo/

(vA/

c)B

D d

ynam

MH

D

r/a r/a

The magnitude of the quasilinear Hall dynamo varies with ρs and is localized at the rational surface

The Hall dynamo is determined by Re J┴, which is zero in the single fluid limit

single fluid ρs = 0.5ρs = 0.1g s

curr

ents

s

oss-

field

C

ro

r/a r/a r/a

D

and

MH

amos

Hal

ldy

na r/a r/ar/a

Mean field modification takes place after the mode saturation

small equilibrium field reconstruction is enough for nonlinear mode saturation

0 1 2ln E

ρs = 0.10m=0 m=1 m=2

t/τ

ln Emagsingle fluid

t/τa

> >

t/τaa

z(a)

/<B

z> single fluidρs = 0.02ρs = 0.10 θ(

a)/<

Bz>

F =

Bz s

ρs = 0.50

t/τ t/

θ =

Bθ

t/τa t/τa

The saturated magnetic island is wider than the resistive layer

nonlinear saturation linear stagesingle fluid ρs = 0.10 ρs = 0.10

r/a

• unlike the linear cases, the

Z W=0.35

,out-of-phase nonlinear solutions are small

• island width is consistent

W=0.35

with the theory

r/a

Both Hall and MHD dynamos are small in saturated state

single fluid ρs = 0.10 ρs = 0.10nonlinear saturation linear stage

r/a

r/a

The overall shape of the parallel current modification is not sensitive to ρs

• Small modification of the equilibrium current ∆λ 1% is needed for saturation

single fluid ρs = 0.10

The saturated islandThe saturated island reduces the parallel current inside the island, and enhances the parallel a d e a ces t e pa a ecurrent on the outboard side of the island.

r/a

r/a

Fine structure of the slab model is not seen in cylindrical geometry

Slab Geometry

A characteristic of the

Cylindrical geometry

A characteristic of the fine structure is large m,n >>1 harmonics –this is not seen in the λ nonEQY this is not seen in the current calculations –the m=2 mode is small.Jpar

Y

ZX

<JxB>par

Y

R

Y

RX

Summary

Electron-ion decoupling on short scales affects the phases of the eigenfunctions creating a Hall dynamo absent in single fluid MHD.eigenfunctions creating a Hall dynamo absent in single fluid MHD.

Two fluid effects and field line curvature cause a mode rotation even without diamagnetic flows.

Quasilinear analytic predictions are confirmed by NIMROD simulations. Small modification of the mean current is enough for the nonlinear mode

saturation.

The magnitude of the saturated dynamo is small.

In saturated state, the radial profile of the total (MHD + Hall) dynamo is much less sensitive to ρs than during the linear stage.

f Fine structure of the reconnection layer observed in slab simulations is not seen in the cylindrical case.