Numerical Computation for Two-Fluid Reconnection and Dynamo

Carl Sovinec and Nicholas MurphyUniversity of Wisconsin-Madison

Center for Magnetic Self-OrganizationGeneral Meeting

Chicago, Illinois August 2-4, 2006

ObjectivesThe objective of this study is to understand the influence of two-fluid effects in nonlinear magnetic relaxation events where there is a guide field, such as RFP dynamo and spheromak flux amplification.

Outline

• Introduction--motivation, equations, solution method

• Linear benchmarking--problem set-up, results

• Nonlinear evolution--temporal behavior, two-fluid signatures, dynamo

• Conclusions

Introduction--motivation• Fast reconnection without a guide field is well established in space and other natural settings and has been studied extensively by many groups.

• Two-fluid reconnection with a guide field is expected for the tokamak sawtooth and has been modeled by the CMRS group.

• The MST group has measured correlation of magnetic field and current density fluctuations leading to Hall dynamo effects, i.e.

, at RFP relaxation events.

• Quasilinear calculations by Mirnov, Hegna, and Prager predict strong but very localized Hall dynamo from two-fluid reconnection in the presence of a large guide field.

• Numerical computation is needed to investigate nonlinear conditions in single and multi-helicity states.

ne||

~~ bj ×

Introduction--physical model

⎟⎠⎞

⎜⎝⎛ ∇−×+×−×−∇=

∂∂

epnenet11 BJBVJB η

BJ ×∇=0μ

WBJVVV νρρ ⋅∇+∇−×=⎟⎠

⎞⎜⎝

⎛∇⋅+

∂∂ p

t

( ) nDntn

∇⋅∇=⋅∇+∂∂ V

αααααα χ

γTnpT

tTn

∇⋅∇+⋅∇−=⎟⎠

⎞⎜⎝

⎛∇⋅+

∂∂

−VV

1

Faraday’s / Ohm’s law

low-ω Ampere’s law

flow evolution

particle continuitywith artificial diffusivity

temperature evolution

In this numerical study, we model reconnection and relaxation with a non-ideal Hall-MHD system of equations.

0=⋅∇ B divergence constraint

⎟⎠⎞

⎜⎝⎛ ⋅∇−∇+∇≡ VIVVW

32T • We will set D, χ, and ν <<η/μ0.

Introduction--solution method• We apply the NIMROD code (www.nimrodteam.org) to solve the HMHD equations with a high-order finite-element spatial representation.• The advance temporally staggers velocity from number density, magnetic field, and temperature, and implicit stepping provides stability at large Δt.

( ) ( ) 2/12/1i

2/12/1i 2

121 ++++ ×=ΔΠ⋅∇+ΔΔ−⎟

⎠⎞

⎜⎝⎛ ∇⋅Δ+Δ∇⋅+

ΔΔ jjjjjj tL

tnm BJVVVVVVV

( )jjjjj pnm VVV i2/12/1

i Π⋅∇−∇−∇⋅− ++

( ) ( )2/12/11121 ++++ ∇−⋅⋅−∇=Δ∇−Δ⋅⋅∇+

ΔΔ jjjj nDnnDn

tn VV

( )12/12/11

11

23

21

221

23

++++

++

⋅∇−∇⋅−=

Δ⋅∇+⋅∇Δ+⎟⎠⎞

⎜⎝⎛ Δ∇⋅+

ΔΔ

jjjj

jj

nTTn

TTnTt

Tn

α

α

ααα

αααααα

VV

qVV

( ) 2/12/1 ++ +⋅∇− jj QT αααq

( ) ( ) JBJBJBVBΔ×∇+×Δ+Δ××∇+Δ××∇−

ΔΔ +++ η

211

21

21 2/12/11 jjj

net

( ) ⎥⎦⎤

⎢⎣⎡ +×−∇−××−∇= +++++ 2/12/11

e2/12/11 jjjjj p

neJBVBJ η

Linear BenchmarkingWe benchmark our linear computations over a wide range of parameters against asymptotic results for two-fluid tearing in sheared slabs. [Mirnov, Hegna, and Prager, Phys. Plasmas 11, 4481 (2004)]

• The sheared equilibrium component is •• The equilibrium has finite pressure, but it is uniform, so drift effects are not considered.• Computations have walls at , whereas analytically, • Analytically, the perturbed magnetic flux in the outer ideal region has solutions

( )LxBxB yy tanh)(∞

=

⎩⎨⎧

<>

⎥⎦⎤

⎢⎣⎡

⎟⎠⎞

⎜⎝⎛ ±

+00

,tanh11~xx

Lx

kLe kxmψ

yx

LkkLkL

LL /2,22

0π

ψψψ

=−=′−′

=Δ′=

−+

0but , ≠⋅∇<<∞

Vzy BB

Lx >> +∞<<∞− x

A schematic of the parameter space shows where different effectsbecome important. (Based on Mirnov, et al.)

( ) isseieieAs ccdDDdvc Ω≡≡→+≡≡ ρωγγδβ ηη ,,, ,,22

Sta

bilit

yP

aram

eter

Δ'δ

Plasma Pressure (β)

β=(δΔ')2β=(δ/di)2

β=(δ/di)1/2

(δ/ρs)1/31

(δ/di)1/4

MHD

Whistler-mediated

ρs>δ: wavesdecouple fromion fluid--large Δ'

slow wave importantlarge Δ'

ρs>δ: wavesdecouple fromion fluid--small Δ'

larg

e-Δ'

slow wave and Bzdiff. important--small Δ'

smal

l-Δ'

AB

C

D

E

F

G

• The regime most relevant for MST has small Δ’ and large β.

Our numerically computed growth rates are in good agreement withthe analytical dispersion relation, provided that the tearing layer is sufficiently small with respect to the equilibrium scale.

*Computed from the dispersion relation from Mirnov, et al.

ρμη 0S DLBy∞≡• The definition of S used here is

• Resistivity is the only dissipation mechanism in Cases A, B, and B2.• Computations such as Case B are spatially resolved with a 48×6 mesh of bicubic elements, packed such that Δx0 ~ 0.1δ ~ 10-3xmax.• Temporal resolution is achieved at compressional wave-CFL # ~ 106.

Viscous dissipation was anticipated for the nonlinear computations, and a parameter scan shows very little impact on linear growth rates with some broadening of the V profile of the eigenfunction.

x

Bx

Vx

-0.03 -0.02 -0.01 0 0.01 0.02 0.03x

Bx

Vx

-0.03 -0.02 -0.01 0 0.01 0.02 0.03

Eigenfunction profiles for Pm=1 show moderate broadening of Vx relative to a case with Pm=0 over the equilibrium scale L=0.33. ( )ηνμ0Pm ≡

xB

z

-0.03 -0.02 -0.01 0 0.01 0.02 0.03x

Bz

-0.03 -0.02 -0.01 0 0.01 0.02 0.03

Both computations show the quadrupole structure of out-of-plane magnetic field within the reconnection scale δ = 5.7×10-3.

Nonlinear EvolutionNonlinear computations in the small-Δ′ large-β regime have the parameters of linear Case B with moderate additional dissipation for nonlinear numerical stability.• Viscosity: Pm=0.1; particle: D/Dη=0.01; thermal: (γ-1)χ/Dη=0.01• There is an effective nonuniform emf that attempts to maintain the equilibrium current profile against resistive dissipation.• The best-resolved computation so far has a packed 100×40 mesh of biquartic finite elements.

x

y

0 0.05 0.1 0.15 0.20

0.05

0.1

0.15

0.2

Plot of element borders over 1/2 of the domain indicates the multi-scale resolution requirement.

The nonlinear computation progresses from a linear eigenfunctionthrough weakly nonlinear behavior to saturation over approximately 36,000 Alfvén times.

t/τa

Kin

etic

Ene

rgy

0 10000 20000 300000

2E-10

4E-10

6E-10

8E-10

1E-09

1.2E-09

1.4E-09

1.6E-09

1.8E-09

2E-09

2.2E-09

The temporal evolution of kinetic energy shows non-steady behavior that is presently the subject of further numerical checking.

xy

-0.03 -0.02 -0.01 0 0.01 0.02 0.030

0.05

0.1

0.15

0.2

In this case, the resulting magnetic island has a width that is small with respect to L.

Nonlinear changes to out-of-plane B on the equilibrium scale exceed the quadrupole profile, but current density contours indicate its persistence along the island separatrix.

x

y

-0.03 -0.02 -0.01 0 0.01 0.02 0.030

0.05

0.1

0.15

0.2

5.3E-074.4E-073.5E-072.6E-071.7E-077.5E-08

-1.6E-08-1.1E-07-2.0E-07-2.9E-07-3.8E-07-4.7E-07-5.6E-07

x

y-0.0025 0 0.0025

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0.11100.0

83.366.750.033.316.7

0.0-16.7-33.3-50.0-66.7-83.3

-100.0

Out-of-plane magnetic field on the equilibrium scale at t=18,000 τa. [Beq(0)=1.]

Contours of constant Jy at t=18,000 τanear the X-point show fine structure in the late nonlinear stage.

Correlation of current density and magnetic field fluctuations and flow and magnetic field fluctuations lead to Hall and MHD dynamo effects, respectively.

x

Dyn

amo

Ele

ctric

Fiel

d10-4 10-3 10-2 10-1

-0.0008

-0.0006

-0.0004

-0.0002

0

0.0002

0.0004

0.0006

0.0008

0.001

0.0012-<vxb>||<jxb>||/ne

x

Q.L

.Dyn

amo

Ele

ctric

Fiel

d(a

rb)

10-4 10-3 10-2 10-1

0

10000

20000

30000-<vxb>|| * 104

<jxb>||/ne

Quasilinear dynamo computations, based on the linear eigenfunction, show very strong and localized Hall dynamo, consistent with the analytical prediction. [Average over the y-direction is plotted.]

In nonlinear saturation, the Hall dynamo expands to the equilibrium scale L, and the two dynamos become comparable in magnitude. [For reference, the driving emf is of order 105.]

Examining the dynamo contributions locally in y shows that the two act in phase with respect to y in the quasilinear result; though, the x-scales and magnitudes are very different.

x

y

-0.002 -0.001 0 0.001 0.0020

0.05

0.1

0.15

0.2

8.13E+047.32E+046.51E+045.71E+044.90E+044.09E+043.28E+042.47E+041.66E+048.51E+034.19E+02

-7.67E+03-1.58E+04

x

y

-0.02 -0.01 0 0.01 0.020

0.05

0.1

0.15

0.2

5.70E+005.20E+004.70E+004.20E+003.70E+003.20E+002.70E+002.20E+001.70E+001.20E+007.00E-012.00E-01

-3.00E-01

Contours of local fluctuation-induced Hall electric field, parallel to the equilibrium magnetic field.

Contours of local fluctuation-induced MHD electric field, parallel to the equilibrium magnetic field.

In the nonlinear state, the local Hall electric field acts primarily near the separatrix and remains large in magnitude relative to the MHD contribution.

Contours of local fluctuation-induced Hall electric field. Comparison with the y-average indicates that most of it leads to shaping but not net redistribution of current density with respect to x.

Contours of local fluctuation-induced MHD electric field. The broad profile is expected for current density redistribution.

xy

-0.02 -0.01 0 0.01 0.020

0.05

0.1

0.15

0.2

1.05E-039.31E-048.10E-046.89E-045.69E-044.48E-043.27E-042.07E-048.61E-05

-3.45E-05-1.55E-04-2.76E-04-3.96E-04

x

y

-0.005 0 0.0050

0.05

0.1

0.15

0.2

1.00E-028.33E-036.67E-035.00E-033.33E-031.67E-030.00E+00

-1.67E-03-3.33E-03-5.00E-03-6.67E-03-8.33E-03-1.00E-02

Conclusions• The implicit leapfrog algorithm accurately reproduces two-fluid tearing in slab geometry with a large guide field, at large time-step.• Nonlinear computations require greater spatial resolution to reproduce structure at the island separatrix.• When the mode is at small amplitude, the computations reproduce the quasilinear prediction for Hall and MHD dynamo effects.• With nonlinear saturation, the net Hall dynamo effect broadens and decreases in magnitude, becoming comparable to the net MHD dynamo effect; though, locally the amplitude of the Hall electric field remains far greater.• Nonlinear single mode computations should also be performed with L > ρsand in the other parameter regimes.• Multi-helicity two-fluid computations are needed to understand relaxation through nonlinear interactions among fluctuations.