Adam Stanier1, P. Browning1, M. Gordovskyy1,K. McClements2, M. Gryaznevich2,3, V.S. Lukin4

Simulations of magnetic reconnection during merging start-up in the MAST Spherical Tokamak

EPS Conference, Espoo, July 2013

MAST

1Jodrell Bank Centre for Astrophysics, University of Manchester, UK2EURATOM/CCFE Fusion Association, Culham Science Centre, UK3Present affiliation: Imperial College of Science and Technology, London, UK4Space Science Division, Naval Research Laboratory, DC, USA

Why study reconnection in MAST?

▶ Reconnection important energy release mechanism in magnetotail, solar corona.

▶ Can degrade plasma confinement in magnetic fusion energy device.

▶ We can study reconnection in the laboratory under controlled conditions and with many diagnostics.

▶ Several experiments (mostly) dedicated to the study of reconnection:

▶ RSX (LANL), TS-3/4 (University of Tokyo), MRX (Princeton), VTF (MIT)

▶ Merging start-up in the Mega-Ampere Spherical Tokamak is not dedicated, but has stronger magnetic fields and reaches higher temperatures.

▶ High-resolution Thomson scattering system gives detailed profiles of electron temperature and density.

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▶ Merging start-up is an attractive alternative for start-up without central solenoid.

▶ Breakdown and current induction around in-vessel P3 coils.

▶ Flux-ropes merge via reconnection at mid-plane to form single Spherical Tokamak (ST) plasma.

▶ Up to 0.5 MA plasma current obtained.

▶ Up to Te = 1 keV achieved in on ms timescale

measured with Thomson Scattering (TS) laser.

Merging start-up

Th

om

son

Sca

tter

ing

las

ers

P3Plasma

P3

φφ

Magnetic: Bp = 0.1 T, BT = 0.5 T, IT = 0.2 - 0.5 MA

Thermal: Te = Ti = 10 eV, n = 5x1018 m-3, Deuterium

Typical start-up parameters:

0 1 2 3 4 5 6Time (ms)

200

150

100

50

0

250

Cu

rre

nt

(kA

.tu

rn)

4

Merging start-up

Time resolution: 0.1 ms. Total time: ~ 7ms.

5

Fluid model

βT = 4x10-5 βp = 10-3

▶ Initial Lundquist #: S = 2 x 104 → Collisional Current Sheet (CS) width: δSP ~ 1 cm.

▶ Kinetic scales become important when larger than collisional CS width.

▶ Ion skin depth: di = 15 cm, Electron: de = 0.25 cm, Larmor radius: ρi = ρis = 0.13 cm.

Hyper-resistivity(electron viscosity)

▶ Heat cond.: , ion-stress tensor:

▶ Will vary μ, η and ηH in simulations presented.

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φ

Toroidal(R,φ,Z)

Code and initial conditions

▶ Solved in 2D Cartesian and toroidal geometry with spectral-element code HiFi.

▶ 4th Order polynomial basis functions.

▶ Stretched grid: High resolution in current sheet.

▶ Crank-Nicolson (θ = 0.5) time advance.

(Glasser and Tang 2004, Lukin 2008).

▶ Currently no measurements of flux-rope structure – use idealised flux ropes, IT = 0.27 MA.

▶ Balanced against pinching by BT increase (βp ~ 10-3), individually force-free.

▶ Conducting walls with line-tied vertical flux B

v = -0.03 T.

▶ Radial dependence (1/R) of toroidal field.

Grid:∆Rmin = 0.5mm∆Zmin = 0.3mm

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Hall-MHD simulation in toroidal geometry

▶ Final nested flux-surfaces qualitatively similar for Hall-MHD (di=15 cm, shown) and resistive MHD (di=0, not shown).

▶ Resistive MHD runs exhibit flux-rope “sloshing” (eg. Biskamp and Welter 1980), for η ≤ 10-4 due to magnetic pressure pile-up.

X-point at t = 0

Current Sheet(CS) width: δ = 2.4 cm

Grid:∆Zmin = 0.03cm

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Nd:

YA

G T

S la

ser

▶ Simulated density profile has double peak. Outer peak disappears after merging.

▶ What causes the double peak in density?

Density profiles: Comparison with experiment

Experiment: Nd:Yag ne

Hall-MHD Simulation: Density

5.4 ms5.5 ms5.6 ms5.7 ms

▶ Density measured at R = [0.2, 1.2 m], Z = 0.015 m.

▶ Typically has double peak at beginning of merging.

20 t040 t060 t080 t0

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What causes the double peak in density?

▶ Density “quadrupole” in Cartesian Hall-MHD simulation.

▶ High (low) density regions correspond to negative (positive) parallel electron velocity gradients. (see also Kleva et al. 1995).

Cartesian Hall-MHD simulation

▶ Resistive MHD simulation in toroidal geometry has inboard (outboard) density peak (cavity).

▶ Both two-fluid effects and toroidal geometry are needed for double peaked profiles in simulation.

Toroidal resistive MHD simulation

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Cartesian Hall-MHD: Effect of collisions

Scan in hyper-resistivity (collisions)W

eake

r co

llisi

onal

ity

▶ ηH = 10-6

▶ Stable.

▶ ηH = 10-8

▶ Island (ejected in toroidal geometry).

▶ ηH = 10-10

▶ Localised CS:

δ = 4.5 mm

ρis

= 2.9 cm

Grid: ∆Rmin = 4x10-4 m, ∆Zmin = 2x10-4 m

Grid: ∆Rmin = 1x10-4 m, ∆Zmin = 4x10-5 m

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Summary

▶ We use merging start-up in MAST as a magnetic reconnection experiment.

▶ Resistive and Hall-MHD simulations were run in Cartesian and toroidal axisymmetric geometry.

▶ We find MAST-like nested flux-surfaces after merging completion in toroidal geometry.

▶ Simulated Thomson Scattering density profiles evolve as in experiment.

▶ Three regimes in Hall-MHD simulations: collisional (δ >> ρis), open X-point (δ < ρ

is)

and an intermediate regime that is unstable to island formation (δ ≥ ρis).

Future work: Simulations and M9 Campaign (with H. Tanabe and the MAST team)

▶ Measure 2D Ion Temperature profiles, compare with simulations evolving separate ion and electron pressures.

▶ Look for density “quadrupole” with 2D Thomson scattering image.

▶ Compare q-profiles between experiment and simulation.

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Resistive MHD

▶ Several studies have shown length-scale ρis = (Te/m

i)1/2/Ωci important for fast reconnection

with strong BT.

▶ Peak reconnection rate in Hall-MHD for CS width > ρis have (weak) dependence on ηH.

▶ ηH = 10-10 is slow during CS formation, but explosive when width drops below ρis (t=7 t0).

Additional: Reconnection rates

(eg. Kleva et al. 1995., Simakov et al. 2010)

t=7 t0

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25 cells across CS width

Additional: Numerical grid and convergence

NR=360, N

Z=540, N

P=4

NR=180, N

Z=270, N

P=4

▶ Convergence test for simulation with ηH = 10-10 (lowest dissipation scale).

▶ Coarsening by factor of 2 changes peak reconnection rate by only 0.2 %.

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Additional slide: q-profile

▶ Paramagnetic equilibrium (just after merging).

▶ q-profile > 1: Sensible. Should be stable to m=n=1 kink-mode.

▶ Final state current profile qualitatively similar for resistive and Hall-MHD.

Vacuum field

t=60 midplane

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Additional: Resistive MHD sloshing

▶ Increase in BT between flux-ropes slows approach.

▶ Large aspect ratio current-sheet: L >> δ (Sweet-Parker).

▶ Initial low-β sheet: c.f. force-free Harris sheet.

▶ Pile-up of BR on sheet edge, and reconnection stalls.

▶ Sloshing of flux-ropes, c.f. coalescence instability.(Biskamp & Welter 1980, Knoll and Chacon 2005)