NONLINEAR COMPUTATION OF LABORATORY DYNAMOS

DALTON D. SCHNACK

Center for Energy and Space Science

Science Applications International Corp.

San Diego, CA 92121

OUTLINE

• Brief summary of laboratory dynamo• Role of large scale numerical simulations• Examples of simulations of laboratory dynamo

– Field reversal– Discharge sustainment– Confinement scaling

• The need for two-fluid modeling– Issues for numerical implementation

• The NIMROD code– Features and status– Summary

OVERVIEW OF LAB “DYNAMO”

• Laboratory “dynamo” is a nonlinear driven system• Plasma is resistive• Poloidal flux (toroidal current) sustained by applied voltage• Toroidal flux is sustained by a “dynamo”• Mediating dynamics

– Nonlinear behavior of long wavelength MHD instabilities– Driven by resistive evolution of mean fields– Nonlinear evolution converts poloidal flux => toroidal

flux– Like differential rotation

• Finite resistivity makes flux conversion irreversible• No conversion of toroidal flux => poloidal flux

– Poloidal flux supplied by external circuit Is it really a dynamo??

THE RFP MAGNETIC FIELD

• Toroidal Z-pinch• Positive toroidal field in

center• Negative toroidal field at

edge• Near Taylor “relaxed state”

THE NEED FOR A LAB “DYNAMO”

• Axisymmetric (Taylor) model:– After reversal, positive flux is affected by resistive diffusion– Depends on BC:

• Bz(a) = const. < 0 ==> positive flux decays, total flux becomes negative

• Flux = const. ==> reversal is lost • Require 3-D dynamical flux generation mechanism ==> “RFP

Dynamo”

LAB DYNAMO AND RELAXATION

• More in common with relaxation theory (Taylor) than turbulent dynamo theories

• Fully nonlinear, but only a handful of coherent modes• Low dimensionality makes “realistic” computations

possible

Relaxed State

Diffusion

Instabilities

NonlinearRelaxation

RFPDynamo

NEED FOR LARGE SCALE COMPUTATIONS

• Early theories invoked small scale turbulent processes – Unsuccessful in explaining experiments

• Full 3-D nonlinear simulations with realistic BCs required– An example where large scale computations led the way

for theory and experiment

“Zero-D”Relaxation

Theory

Full 3-DNonlinear

Computations

COMPUTATIONAL ISSUES

• Resistive MHD

– S > 104

– Effieiency of spatial representation

• Long time scale evolution

– Efficient implicit time advance

• Importance of boundary conditions

– Applied voltages

– Non-ideal boundaries

– Applied magnetic fields

• Effective diagnostics

• Comparison with experimental data

COMPUTATIONS OF THE RFP DYNAMO

• Toroidal effects not dominant => doubly periodic cylindrical

geometry

• Use resistive MHD model

• External drive (voltage, current, and flux sources)

THE FIRST DYNAMO SIMULATION

• Resistive MHD, S ~ 200, R/a = 1/2• Cylindrical geometry, 14 X 13 X 13 grid!• Shows cyclic oscillations and sustainment of reversed field• The RFP dynamo is very robust• Subsequent calculations are refinements of this work

Sykes and Wesson, Proc. 8th Euopean Conf. On Controlled Fusion and Plasma Physics, Prague, p. 80 (1977)

RESISTIVE MHD

• Some simulations use the force-free model:

€

∂A

∂t= −E, E = −v ×B +

η

S0J, B = ∇ ×A, J = ∇ ×A

∂ρ

∂t= −∇⋅ρv, p = nT

ρ∂v

∂t+ v ⋅∇v

⎛ ⎝ ⎜

⎞ ⎠ ⎟= −

1

2β0∇p + J×B +ν∇2v

∂p

∂t= −∇⋅ pv( ) − γ −1( )p∇ ⋅v +

2 γ −1( )η

β0S0J 2 +

γ −1( )

S0∇ ⋅κ ⋅∇T

€

ρ =const ., ρdv

dt= J×B +ν∇2v

ALGORITHMS

• Staggered finite differences in radius– Preserve annihilation properties of div and curl

operators• Dealiased pseudospectral in periodic coordinates (,z)

€

F (r ,θ ,z ,t ) = Fm,n (r ,t )e i (mθ +nz / R)

m,n

∑• Boundary conditions:

– Conducting wall with applied voltages– Resistive wall– Concentric resistive walls

• Time advance– Leap-frog for waves– Predictor-corrector for advection– “Semi-implicit” for large time steps

TIME ADVANCEMENT ISSUES

• Fundamental differences in algorithm requirements?

– Laboratory dynamo

• A few low order modes

• Relatively long times scales (compared with Alfvén time)

• Dictates implicit methods

– Astrophysical dynamo

• Dominated by advection

• Dictates explicit methods

Is a “common code” appropriate?

RESULTS

• “Dynamo” mechanism identified

– Cyclic relaxation reproduced

• Source of anomalous loop voltage identified

– Dynamo enhancement in presence of resistive wall

• Sustainment mechanisms investigated

– Feedback

– Profile modification (current drive, helicity injection)

– Time dependent BCs

• Transport

– Dynamo (relaxation) produces stochastic fields

– Balance between Ohmic heating and thermal loss along stochastic field lines

– Obtained confinement scaling laws

DYNAMO/RELAXATION AT FINITE-

SUSTAINMENT

CYCLIC TAYLOR RELAXATION

• Diffusion drives plasma away from preferred state• Nonlinear modes drive it back

DYNAMO IS CHAOTIC

RELAXED PROFILES

Decaying discharge

TRANSPORT DUE TO DYNAMO

0 100

5 10-4

1 10-3

1.5 10-3

2 10-3

2.5 10-3

3 10-3

3.5 10-3

4 10-3

5 10-3 1 10-2 1.5 10-2 2 10-2 2.5 10-2 3 10-2 3.5 10-2 4 10-2

/t τR

3-D nonlinear calculationsincluding:

- Poynting and Ohmic input- Anisotropic heat flux

Parametric studies determine scaling of confinement with global parameters

€

∝ I −0.4, T ∝ I 0.56 , τ E ∝ I 0.34 , δBrms ∝S −0.14

0,001

0,01

0,1

104 105 106

b r

(rms)

<S(0)>

~ S(0) -0.14

SAME DYNAMO IN SPHEROMAK

• Spheromak is RFP with line-tied BCs– RFP: poloidal flux => toroidal flux– Spheromak: toroidal flux => poloidal flux

r

z IPeriodic

BC

Cylindrical RFP

r

z

ILine-tied

BC

Spheromak

“Toroidal”direction

“Toroidal”direction

Sovinec

COMPUTATION OF MRI

• MRI in doubly periodic cylinder

• Lab dynamo: liquid Gallium

S = 2.59

Pr = 1.5 X 10-2

• Rotating and outer boundaries

• m : 0 => 21• n : -21 => 21• Computed with

DEBS code (old technology)

0.0 1.0

1.00

0.80

0.60

0.40

Z

R

= 6.190e-01

STATUS AND DIRECTIONS

• Resistive MHD computations of laboratory (RFP) dynamo are “mature”

– Basic nature and consequences of RFP dynamo elucidated

– “Good” agreement with experiment

• Recent measurements show resistive MHD is not sufficient for details

– Hall dynamo

– Diamagnetic dynamo

• Require electron (2-fluid) dynamics

• FLR effects??

• Neutrals ?? (astrophysical dynamos)

• MRI??

IT’S ALL ABOUT OHM’S LAW

• Two-fluid relaxation theory (Mahajan & Yoshida, Steinhauer & Ishida, Hegna, Mirnov)

– Flows enter on equal footing with magnetic fields

– Flows are ubiquitous in laboratory plasmas

– Importance to astrophysical plasmas??

• Computational difficulty:

– Dispersive waves can limit time step

– Higher order operators

– Require specialized linear algebra packages

€

E = −Vi ×B +ηJ

Alfvén wavesand

tearing modes

1 2 4 3 4 +1

neJ×B

Whistlerwaves

1 2 4 3 4 −

1

ne∇pe

Kinetic Alfvénwaves

1 2 4 3 4

DISPERSIVE WAVES

€

Whistler : ∂2B

∂t 2=

VA2

Ω

⎛

⎝ ⎜

⎞

⎠ ⎟2

b ⋅∇( )2 ∇ × ∇ ×B ω2 = k||2VA

2 1+k2 VA2

Ωi2

⎛

⎝ ⎜

⎞

⎠ ⎟

KAW : ∂2B

∂t 2=

VAVth *

Ω

⎛ ⎝ ⎜

⎞ ⎠ ⎟2

b ⋅∇( )2 ∇ × bb ⋅∇ ×B( ) ω2 = k||2VA

2 1+k⊥2 Vth *

2

Ωi2

⎛

⎝ ⎜

⎞

⎠ ⎟

ω2 ~ k 4 ⇒ Δt ~ Δx 2 for explicit methods

4th order operators

Possibly amenable to semi - implicit treatment

THE NIMROD CODE

• Resistive MHD– Development essentially complete– Thoroughly tested and benchmarked on a variety of problems– Mixed finite-element/pseudo-spectral representation– Semi-implicit formulation allows “arbitrary” time steps

• Two-fluid model– Explicit two-fluid model available– Hall term and electron pressure– Energetic minority ion species

• Semi-implicit two-fluid model being tested– Based on dispersive wave operators– Will allow large time steps

• Several FLR formulations under consideration– Full gyro-viscous stress tensor– Drift-MHD and the gyro-viscous cancellation

http://www.nimrodteam.org

NIMROD FEATURES

• Flexible geometry and boundary conditions– Axially symmetric boundary with arbitrary poloidal (R, Z)

cross-section– Slab with a variety of boundary conditions

• Highly accurate spatial representation– Arbitrary order finite elements in poloidal plane– Pseudo-spectral/FFT in toroidal dimension

• Accuracy for highly anisotropic problems– Highly accurate semi-implicit time advance– Routinely compute with CFL > 104

• Complete MPI implementation• Applications

– Wide variety of fusion problems– Merging flux tubes (reconnection)– Collimation of magnetic field structures (astrophysical jets)

SUMMARY• Laboratory dynamo can be accurately computed

– Closely related to plasma relaxation– Resistive MHD model– Reveal underlying dynamics and consequences– Computations led theory and experiment– Possible because of low dimensionality

• Relevance to other fields?– Solar dynamo?– Formation and disruption of coronal structures– Other quasi-periodic phenomena?

• Extensions– Two-fluid modeling

• Necessary to explain detailed experimental measurements• May account for ubiquitous plasma flows

• NIMROD and DEBS are available for collaborative applications

http://www.nimrodteam.org