1. Simulations of magnetic reconnection during merging start-up in the MAST Spherical Tokamak
Adam Stanier1, P. Browning1, M. Gordovskyy1,
K. McClements2, M. Gryaznevich2,3, V.S. Lukin4
1Jodrell Bank Centre for Astrophysics, University of Manchester, UK
2EURATOM/CCFE Fusion Association, Culham Science Centre, UK
3Present affiliation: Imperial College of Science and Technology, London, UK
4Space Science Division, Naval Research Laboratory, DC, USA
MAST
So this work being done by a small group of us at Manchester,
Ken McClements and Mikhail Gryaznevich at Culham
and Slava at NRL.
EPS Conference, Espoo, July 2013

2. 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.
Understanding of reconnection in parameter regime relevant to tokamak disruptions.
Solar flare: lifetime of active region ~ 1 week, release ~ 1025 J over 100 sec.
Tokamak Sawtooth crash: Build up (sawtooth period) ~ 100 ms, crash ~ 100 μs.

3. Merging start-up
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.
250
Current (kA.turn) P3
200
Plasma
150
100 50 1 2 3 4 5 6
Time (ms)
Thomson Scattering lasers
Pick-up Coil (CCMV20)
This start-up method was first demonstrated in MAST's predecessor START, and now it is routinely used on MAST. It can be used with or without the central solenoid.
The P3 coils are quickly ramped down causing breakdown and inducing two flux-ropes with toroidal current.
When the plasma current is large enough the mutual attraction between the flux ropes is stronger than that to the P3 coils, and they detach and merge at the midplane through reconnection.
I have labelled here the CCMV20 coil which is one of an array of pick-up coils measuring dBzdt, and TS lasers.
Through merging compression 0.5 MA of plasma current has been obtained, but perhaps more impressively 1keV electron temperatures on a ms timescale. P3
Typical start-up parameters:
Magnetic: Bp = 0.1 T, BT = 0.5 T, IT = MA
Thermal: Te = Ti = 10 eV, n = 5x1018 m-3, Deuterium φ φ

4. Merging start-up Visible light video of the merging.
I hope you can see two plasma rings that clearly detach from the P3 coils before merging.
Here their shape is quite clear, and in the next frame they appear to be merged, so the merge time is on or less than 0.1 ms timescale.
Time resolution: 0.1 ms. Total time: ~ 7ms.

5. Fluid model
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.
βT = 4x10-5
βp = 10-3
Hyper-resistivity
(electron viscosity)
The plasma state during merging compression is not the same is at normal operation. The initial temperature is only 10eV, and with the strong fields this gives very low toroidal and poloidal beta values.
The toroidal beta is actually less than the mass ratio initially.
The Lundquist number is large, and the ion-skin depth is large compared to the smallest measured feature in the data, the central electron temperature peak in TS data. The initial electron skin depth and ion larmor radius are smaller than this value, and we dont include them in this study.
So we have the hall term, and also we use a hyper-resistivity to damp grid scale whistler waves. This also breaks the frozen in condition here.
So with such high lundquist number and low beta these simulations may have some relevence to the Solar Corona.
Heat cond.: , ion-stress tensor:
Will vary μ, η and ηH in simulations presented.
Hyper-resistivity is used to set dissipation scale for Whistler waves.
Can (and does) set diffusion scale here by breaking frozen-in condition.
Physically related to an electron viscosity:
Anisotropic heat conduction:
Hyper-resistivity (anomalous electron viscosity).
for
Solved with the HiFi framework (eg. Lukin and Linton) with 4th Order polynomial basis.
Crank-Nicholson time advance (to avoid CFL condition).

6. 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).
Toroidal
(R,φ,Z)
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 Bv = T.
Radial dependence (1/R) of toroidal field.
Unfortunately there is no magnetic data on the flux tubes prior to merging at the moment, so we start with an idealised model.
We split the total toroidal current between two flux-ropes and estimate their width from visible light photographs.
There is an increase in toroidal field inside these flux-ropes to balance against the pinch force, as the poloidal beta is so low.
We use perfectly conducting walls, and in the toroidal case we add in line-tied vertical field to balance against the radially outwards hoop-force.
In cartesian case we use uniform out-of-plane field but in toroidal we have the proper radial dependence.
Grid:
∆Rmin = 0.5mm
∆Zmin = 0.3mm φ
Currently no q-profiles of pre-merged flux-ropes (plan for M9 campaign).
Idealised initial conditions using

7. Hall-MHD simulation in toroidal geometry
Current Sheet
(CS) width: δ = 2.4 cm
Grid:
∆Zmin = 0.03cm
X-point at t = 0
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.

8. Density profiles: Comparison with experiment
Experiment: Nd:Yag ne
5.4 ms
5.5 ms
5.6 ms
5.7 ms
Density measured at R = [0.2, 1.2 m], Z = m.
Typically has double peak at beginning of merging.
Nd:YAG TS laser
Hall-MHD Simulation: Density
20 t0
40 t0
60 t0
80 t0
Simulated density profile has double peak. Outer peak disappears after merging.
What causes the double peak in density?

9. 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.
Toroidal resistive MHD simulation
(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.
MHD: Density peak on inner side, cavitation on outer edge.
Two-fluid: Additional density asymmetry, disappears after merging completion.
MHD di = 0
di = m
ηH = 10-8
High density seperator

10. Cartesian Hall-MHD: Effect of collisions
Scan in hyper-resistivity (collisions)
ηH = 10-6
Stable.
Weaker collisionality
ηH = 10-8
Island (ejected in toroidal geometry).
Grid: ∆Rmin = 4x10-4 m, ∆Zmin = 2x10-4 m
However, there are some differences from the standard picture. The current sheet is extended in the outflow direction, and so is unstable to island formation. Here the magnetic pressure within the island causes the reconnection to stall. The island isn't ejected because of a symmetry that is present in the Hal
Also the reconnection rate, plotted here has some dependence on the dissipation.
ηH = 10-10
Localised CS:
δ = 4.5 mm
ρis = 2.9 cm
Grid: ∆Rmin = 1x10-4 m, ∆Zmin = 4x10-5 m
However, large aspect ratio current sheet is unstable to island formation (for ηH ≤ 10-8).
In Cartesian geometry a central island stalls the reconnection.
Stronger BT: weaker density asymmetry. Multiple, shorter wavelength islands.
Order 1 density variations in “quadrupole” structure.
Electrons accelerated in low density regions.
Similar to reduced two-fluid reconnection model of Kleva et al

11. 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 < ρis) 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.
We have also simulated separate ion and electron temperature profiles.
(Stanier et al ).
Resistive MHD simulations in the sloshing regime.
Three regimes in Hall-MHD simulations: collisional (δ >> ρis), open X-point (δ < ρis) and intermediate
Tilt of outflow jets, and ion temperature profiles in two-fluid case – signature of two-fluid effects with guide field.
Toroidal geometry has little effect on reconnection process (eg. the reconnection rate) but can modify density profiles.
Simulated 1D density profiles show same time evolution as in TS profiles – motivation to look for density “quadroupole” with 2D measurement.
Resistive MHD simulations in the sloshing regime.
Three regimes in Hall-MHD simulations: collisional (δ >> ρis), open X-point (δ < ρis) and intermediate
Tilt of outflow jets, and ion temperature profiles in two-fluid case – signature of two-fluid effects with guide field.
Toroidal geometry has little effect on reconnection process (eg. the reconnection rate) but can modify density profiles.
Simulated 1D density profiles show same time evolution as in TS profiles – motivation to look for density “quadroupole” with 2D measurement.

12. Hall-MHD: Cartesian geometry

13. Additional: Reconnection rates
Resistive MHD
Several studies have shown length-scale ρis = (Te/mi)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 = is slow during CS formation, but explosive when width drops below ρis (t=7 t0).
t=7 t0
(eg. Kleva et al , Simakov et al. 2010)

14. Additional: Numerical grid and convergence
25 cells across CS width
NR=360, NZ=540, NP=4
NR=180, NZ=270, NP=4
Convergence test for simulation with ηH = (lowest dissipation scale).
Coarsening by factor of 2 changes peak reconnection rate by only 0.2 %.

15. Additional slide: q-profile
Direction of island ejection depends upon radial positions of O-points and X-point
Additional slide: q-profile
t=60 midplane
Vacuum field
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.

16. 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)
For comparison I'll quickly mention resistive MHD simulations in Cartesian geometry.
The out-of-plane field is shown here in colour, and you can see the increase inside the flux-ropes to balance the pinch force.
There is also an increase in toroidal field between the plasmoids, only by a few percent but this is sufficient to slow down the initial phase.
The current sheet that forms is initially force free, and extends the width of the flux-ropes. The outflow jets are in the radial direction, but in the second panel they have stopped.
As the front of the plasmoids is compressed, the magnetic pressure on the current sheet edge becomes larger than the driving foces, and the reconnection stalls, the flux-ropes then oscillate and the total merge time is over a hundred compressional Alfven times.
S = 105, mu = 10-3, BT= 0.5 T, a=0.6, Iplasma = 268 kA

17. Hall-MHD: Cartesian geometry
Current sheet and outflow (ion) jets tilt.
O(1) density variations in “quadrupole” structure.
Electrons accelerated in low density regions.
Similar to reduced two-fluid reconnection model of Kleva et al
Cartoon from Kleva et al. (1995).
When two-fluid effects are included the current sheet spreads preferably across one of the seperatrices, appearing to tilt, and the outflow jets are also tilted.
The colour here is the plasma density, which has this quadroupolar structure. The plotted lines are electron drift streamlines. Inside the plasmoids the electron velocity is mainly perpendicular drifts, but around the diffusion region the motion is mostly parallel to the field. The electrons are accelerated in the density cavities and decelerated in the high density regions, so this parallel velocity gradient causes the density compressions.
These features are all in the standard model of collisionless guide field reconnection, where the ions drift across the fieldlines to charge neutralise the electrons in the high density region.

18. Why study reconnection in MAST?
Difficulties in studying magnetic reconnection in nature:
Earth Magnetotail: Detailed in-situ measurements possible only at a few spatial points.
Solar corona: Reconnection can only be inferred from signatures of accelerated particles and plasma heating.
Understanding of reconnection in parameter regime relevant to tokamak disruptions.
Several experiments (mostly) dedicated to the study of reconnection:
TS-3/4 (University of Tokyo), RSX (LANL), 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 during reconnection.
High-resolution Thomson scattering system gives detailed profiles of electron temperature and density.
Solar flare: lifetime of active region ~ 1 week, release ~ 1025 J over 100 sec.
Tokamak Sawtooth crash: Build up (sawtooth period) ~ 100 ms, crash ~ 100 μs.

19. Hall-MHD: Cartesian geometry
x5 BT
Resistive MHD
di = 0
Several studies have shown length-scale ρis important for fast reconnection with strong BT.
Peak reconnection rate in Hall-MHD for CS width > ρis have (weak) dependence on ηH.
ηH = is slow during CS formation, but explosive when width drops below ρis (t=7 t0).
Strong BT suppresses reconnection rate towards collisional limit (→ρis = di = 0).
t=7 t0
(eg. Kleva et al , Simakov et al. 2010)

20. Experimental Data: Thomson Scattering
Nd:YAG laser: 1D radial chord
130 pt Nd:YAG Thomson Scattering system.
Radial chord at midplane (Z = 1.5 cm)
0.1 ms “burst-fire” mode
Double peak feature in density profile
Te increase from 10 eV to ~100 eV
Central Te peak with ΔR, ΔZ ~1cm.
Oscillations (τ~30 μs) in CCMV20 signal during/after merging. ne Te
Most of the data that we have is from a newly upgraded Thomson Scattering system.
These are radial chords in electron density and temperature at the midplane, and this is the signal from that pickup coil on the central post.
The colours here correspond to the times at which the data is taken. At early times there is often a double peaked structure in the density. After the main peak in the Mirnov signal the inner peak remains but the outer peak decreases.
The electron temperature rises from 10 to 100eV, and occasionally we see a central peak that is localised in the radial and axial directions to about 1cm. This is only seen in a small number of shots.
After the main peak in this signal this signal continues to oscillate, I'll come back to this later.
CCMV20 pick-up coil
∂tBZ
Data taken by T. Yamada (University of Tokyo) and the MAST team.
See Ono et al

21. Thomson Scattering: Electron heating
Ruby laser + P6 (vertical position) coils: 2D profile
Peaked Case: n = 1019 m-3 at 10.0 ms
Ruby TS laser
Hollow Case: n = 5 x 1018 m-3 at 10.0 ms φ
Each 2D profile built from several (identical) shots with different vertical shift (P6).
At later time (~5 ms after merging) electron temperature still increasing.
Significant electron heating often occurs after merging in a “hollow” structure.
The 1-10 ms time-scale is in agreement with electron-ion equilibration time
τie ≈ 0.2 ms * (Te [eV]/ T0)3/2 T0 = 10 eV
As well as 1D profiles at the midplane we also have some 2D profiles of electron temperature.
Each of these figures is made up of several shots that are identical apart from the currents in the vertical position coils. So the Thomson Scattering laser is in a fixed position but the whole plasma is shifted vertically.
There are two typical profiles, the peaked case.. at high density, and the hollow case .. at lower density.
temperature continues to increase even after the merging has finished.

22. Resistive MHD simulation: Cartesian geometry
μ = 5x10-4
η = 5x10-6
μ = 10-3, η = 10-5
μ = 5x10-4
μ = 10-3, η = 10-5
Decreasing η
Decreasing μ
η = 5x10-6
Reconnection rate: at X-point.
Peak: Av:
c.f. Viscous Sweet-Parker scaling: ~ η3/4 μ-1/4 (for μ>>η Park et al. 1984)

23. Hall-MHD: Cartesian geometry
x5 BT
Resistive MHD
di = 0
Several studies have shown length-scale ρis important for fast reconnection with strong BT.
Peak reconnection rate in Hall-MHD for CS width > ρis have (weak) dependence on ηH.
ηH = is slow during CS formation, but explosive when width drops below ρis (t=7 t0).
Strong BT suppresses reconnection rate towards collisional limit (→ρis = di = 0).
t=7 t0
(eg. Kleva et al , Simakov et al. 2010)

24. Mega Ampere Spherical Tokamak (MAST)
Normal Operation
Major radius: R = 0.85 m
Minor radius: a = 0.65 m
R/a = 1.3
Toroidal field (at R): BT= 0.5 T
Current: ≤ 1.6 MA
Temperature ~ 0.1 – 3 keV
Density: 1018 – 1020 m-3
Ion Species: Deuterium P1 a R P3 φ
So I'll start with a quick overview of MAST.
It is a tight aspect ratio, or Spherical Tokamak, with
Major Radius, minor radius.
Toroidal field of .5 Tesla, Mega amp current and a deuterium plasma.
We are now starting the final experimental campaign before a major upgrade.
In this diagram you can see the plasma shape at equilibrium in normal operation, and the in-vessel PF coils used for plasma control.
I will mention the P3 coil often, as its main use is in merging compression.
P1 is central solenoid: normal method for current drive.
P3 coils: used for merging-compression start-up.
P4, P5: vertical field, P6: vertical position.

25. Hall-MHD: Cartesian geometry
Scan in hyper-resistivity (collisions)
ηH = 10-6
Stable
Weaker collisionality
ηH = 10-8
Island.
However, there are some differences from the standard picture. The current sheet is extended in the outflow direction, and so is unstable to island formation. Here the magnetic pressure within the island causes the reconnection to stall. The island isn't ejected because of a symmetry that is present in the Hal
Also the reconnection rate, plotted here has some dependence on the dissipation.
ηH = 10-10
Localised CS.
However, large aspect ratio current sheet is unstable to island formation (for ηH ≤ 10-8).
In Cartesian geometry a central island stalls the reconnection.
Stronger BT: weaker density asymmetry. Multiple, shorter wavelength islands.
Order 1 density variations in “quadrupole” structure.
Electrons accelerated in low density regions.
Similar to reduced two-fluid reconnection model of Kleva et al

26. Fluid model of merging-compression
One temperature formulation:
μ = (Re)-1 = 10-3 (based on μ//i = 10-4 Pa s)
η = 10-5 (based on η// = 3.5 x 10-5 Ω m and BT0)
Use κ//e and κ┴i based on initial n0, T0.
Will vary μ, η and ηH in simulations presented.
But also:
For most of the simulations we use one pressure equation, with ion viscosity based on the initial parallel value. We use parallel electron conductivity, and the cross-field ion conductivity, rather than the true perpendicular value.
At the end I'll show a simulation with seperate ion and electron pressure equations, so the resistive heating only acts on the electron fluid and ion-viscous heating only on the ions.
Electron-ion temperature equilibration time: τie > 0.2 ms, longer than merge time.
→ will discuss results with two-temperature formulation at the end.
μ = 1/Re = 10-3 (based on μ//i = 10-4 Pa s).
η =10-5 (based on η// = 3.5 x 10-5 Ω m, and BT0)
Use κ//e and κ┴i based on initial n0, T0.
Electron-ion equilibration time: τie > 0.2 ms, longer than merge time.
Temperature dependent resistivity (Spitzer parallel).
Use κ//e, κ┴e, κ//i, κ┴i based on initial n0, T0.

27. Two-temperature formulation
Electron-ion equilibration time: τie > 0.2 ms, longer than merge time.
Temperature dependent resistivity (Spitzer parallel).
Use κ//e, κ┴e, κ//i, κ┴i based on initial n0, T0.

28. Two-fluid: Toroidal geometry
Direction of island ejection depends upon radial positions of O-points and X-point
Two-fluid: Toroidal geometry
Toroidal geometry breaks symmetry – central island ejected (possible filament?).
Two-fluid merge time comparable to Cartesian case (= 25 τ0 = 10 μs for ηH = 10-8).

29. Fluid model of merging-compression
One temperature formulation:
μ = (Re)-1 = 10-3 (based on μ//i = 10-4 Pa s)
η = 10-5 (based on η// = 3.5 x 10-5 Ω m and BT0)
Use κ//e and κ┴i based on initial n0, T0.
Will vary μ, η and ηH in simulations presented.
Solved in 2D Cartesian and toroidal geometry with the SEL/HiFi framework
Spectral element code with 4th Order polynomial basis functions.
Stretched grid: High resolution in current sheet.
Crank-Nicholson (θ = 0.5) time advance– avoid severe timestep constraint from dispersive waves.
For most of the simulations we use one pressure equation, with ion viscosity based on the initial parallel value. We use parallel electron conductivity, and the cross-field ion conductivity, rather than the true perpendicular value.
At the end I'll show a simulation with seperate ion and electron pressure equations, so the resistive heating only acts on the electron fluid and ion-viscous heating only on the ions.
(Glasser and Tang 2004, Lukin 2008).
Electron-ion equilibration time: τie > 0.2 ms, longer than merge time.
Temperature dependent resistivity (Spitzer parallel).
μ = 1/Re = 10-3 (based on μ//i = 10-4 Pa s).
η =10-5 (based on η// = 3.5 x 10-5 Ω m, and BT0)
Use κ//e and κ┴i based on initial n0, T0.
Use κ//e, κ┴e, κ//i, κ┴i based on initial n0, T0.

30. Two-temperature with hyper-resistive heating
Time: 17.9
Time: 40.9
Time: 45.9
Time: 85.5
Two-temperature with hyper-resistive heating
Time: 20.8
Time: 34.9
Time: 50.0
Time: 65.2
Ion Temperature
Max: 315 eV
Max: 1402 eV
Max: 1944 eV
Max: 2155 eV
Time: 17.9
Time: 20.8
Time: 40.9
Time: 34.9
Time: 45.9
Time: 50.0
Time: 65.2
Electron Temperature
Electron heating on CS edges:
not cospatial with j.
Max: 368 eV
Max: 891 eV
Max: 1304 eV
Max: 890 eV
Hollow and tilted ion temperature profile. Heating due to viscous damping of sheared outflow jets.
Hollow electron temperature profile. Prefers high n seperator.
Max: 53 eV
Max: 70 eV
Max: 81 eV
Max: 105 eV
Max: 223 eV
Max: 2147 eV
Max: 2461 eV
Max: 1433 eV
Time: 85.5
Ion temperature profile “hollow”. Tilt in ion temperature profile.
Electron temperature ~ 100 eV, but no central peak.

31. What is the dominant electron heating mechanism?
Two-temperature formulation:
compressive
resistive
hyper-resistive
Time: 17.9
Time: 40.9
Time: 45.9
Time: 85.5
Electron Temperature
Max: 53 eV
Max: 70 eV
Max: 81 eV
Max: 105 eV
Electron temperatures order of magnitude less without hyper-resistivity.
→resistive and compressive heating are small for these simulations.
Electron heating needs to be better understood: as ρis ~ Te1/2 (fast-reconnection threshold), and η ~ Te-3/2

32. Spatial scales and Hall-MHD reconnection
Sweet-Parker and Petschek are MHD reconnection models (scale-invariant).
However, new physics when CS width δSP = S-1/2 L drops below scales.
Hall Reconnection
Term:
Hall
Electron inertia
Scale:
di =c/ωpi
ρis = vs/Ωci
de =c/ωpe
Ion skin-depth
Ion-sound radius
Below ion skin-depth di ions decouple and electrons carry the magnetic field.
Fast reconnection regime can exist (eg. Shay and Drake 1998).
A “guide” field (here toroidal non-reconnecting component) modifies this picture.
Sweet-Parker and Petschek are MHD reconnection models (scale-invariant).
However, new physics important when collisional CS width

33. Experimental Comparisons
Are CCMV20 oscillations a signature of sloshing?
CCMV20 (experiment)
Pick-up Coil (CCMV20)
O-point
Hall finishes
Resistive
Vertical field
Oscillations in simulated CCMV signal.
Simulated CCMV is too fast (by factor of 3): other PF coils may be important.
Initial peak in Mirnov does not correspond to merge time.
Simulated Mirnov is too fast (by factor of 3): other PF coils may be important.
Double peaked density profile. Evolution similar to Nd:Yag profiles.
Investigate further with 2D ne profiles (shifting plasma with P6 coils). .

34. Energy conservation
Resistive
Hall

35. Additional Slides: Double Null Merging
In vessel coils not desirable in ST power plant: need to be shielded from neutron flux.
An alternative method is Double-Null Merging.
DNM experiments on MAST: Iplasma = 340 kA, Te = 0.5 keV.
Reconstruction of a DNM shot with a multiple magnetic axis equilibrium code (F. Alladio and P. Mocozzi).
DNM with external PF coils has been achieved in UTST (T. Yamada et al. 2010).
The only TS data on DNM is taken at around ms. Not useful for comparison with simulations.
UTST achieved 50 kA plasma current without CS (as of T. Yamada et al).
P. Micozzi (3rd IAEA Technical Meeting on ST, 2005)

36. Overview So this work being done by a small group of us at Manchester,
Mega-Ampere Spherical Tokamak
(MAST)
Introduction
Magnetic reconnection
Merging start-up method in the MAST tokamak:
Relevant experimental data.
Fluid model.
Fluid simulations of merging start-up
Resistive MHD simulations in Cartesian geometry.
Hall-MHD simulations and the effect of toroidal axisymmetric geometry.
Conclusions
Comparison with MAST data.
Future plans for simulations and experiments.
So this work being done by a small group of us at Manchester,
Ken McClements and Mikhail Gryaznevich at Culham
and Slava at NRL.

37. Reconnection Layers in 2D: Classical Picture
Steady-state reconnection: Resistive MHD
Sweet-Parker (slow)
Petschek (fast)
Diffusion region
Diffusion region 2δ 2δ
shocks 2L 2Δ
Classical (Sweet-Parker) width: δSP = S-1/2 L, Rate ~ S-1/2 where S = Lundquist Number.
Fast reconnection (2D):
Rate: weak or no dependence on S (or relevant dissipation scale).
Diffusion region (not always) localised in outflow direction (Δ << L).
Caused by anomalous resistivity, or two-fluid physics.
How do these models fit into the global picture?

38. Additional Slides: University of Tokyo Experiments
TS-3 Merging Parameters
Major radius: R = 0.2 m
Minor radius: a = 0.1 m
Toroidal field (at R): BT= 0.05 T
Te = eV, Ti = eV
Density: 5 x 1019 m-3
βT = 0.08
Ono et al. PPCF 54, 2012
Ion heating

39. Additional Slides: 1 keV electron temperaturene Te
Plasma P3
Central Solenoid
1 keV temperature peak measured by Ruby TS at 12 ms.
Merging-Compression (P3 = 300 kA turn) + Central Solenoid.

40. Magnetic Reconnection
Relaxation of sheared magnetic fields to a lower energy state involving a change in magnetic topology.
Associated with large release of magnetic energy into bulk particle acceleration and plasma heating.
A multi-scale phenomenon:
Magnetic energy and plasma inflows at large scale-length.
Reconnection and primary energy release in localised diffusion regions, or current sheets.
It is easier than in space!
Difficulties in studying magnetic reconnection in astrophysical settings:
Corona: Reconnection can only be inferred from signatures of accelerated particles and plasma heating.
Dedicated reconnection experiments:
MRX (Princeton), TS-3/4 (University of Tokyo), SSX (Swathmore)
MAST is not dedicated, but is closer to coronal plasma conditions and has a high-resolution Thomson scattering diagnostic.

41. } Merging-compression as a reconnection experiment
MAST merging-compression
Corona (Active Region)
Magnetic:
Bp = 0.1 T, BT = 0.5 T
Te = eV, n = 5x1018 m-3
B ~ 0.01 T
Te ~ 100 eV, n = 1015 m-3
Plasma:
Length-scale:
Ion skin-depth:
L = 1 m
di = 15 cm
L ~ 107 m
di ~ 10 m
Dimensionless:
Beta:
Lundquist:
βT = 4x10-5
βp = 10-3
S =
δSP/di ~ (δSP ~ S-1/2L)
β = 10-2 – 10-4
S =
δSP/di ~ 0.1-1
The plasma state during merging compression is not the same is at normal operation. The initial temperature is only 10eV, and with the strong fields this gives very low toroidal and poloidal beta values.
The toroidal beta is actually less than the mass ratio initially.
The Lundquist number is large, and the ion-skin depth is large compared to the smallest measured feature in the data, the central electron temperature peak in TS data. The initial electron skin depth and ion larmor radius are smaller than this value, and we dont include them in this study.
So we have the hall term, and also we use a hyper-resistivity to damp grid scale whistler waves. This also breaks the frozen in condition here.
So with such high lundquist number and low beta these simulations may have some relevence to the Solar Corona.
MAST merging-compression is a high S, low beta reconnection experiment. }
Initially S = 105 (η// = 3.5 x 10-5 Ω m)
Ion skin-depth: di = 15 cm, electron: de = 0.25 cm,
Ion Larmor radius: ρi = ρis = 0.13 cm. c.f. Central peak in electron temperature ~ 1cm.
No study done on breaking frozen-in condition that is applicable for this regime.
(Closest by Ricci et al. with B_T/B_p=5, but no collisions and >m_e/m_i).
Solved with the HiFi framework (eg. Lukin and Linton) with 4th Order polynomial basis.
Crank-Nicholson time advance (to avoid CFL condition).

42. Fluid model of merging-compression
Hall term may be important: δSP/di ~ (δSP ~ S-1/2L)
Electron skin-depth: de = 0.25 cm, Initial Ion Larmor radius: ρi = ρis = 0.13 cm.
c.f. Central peak in electron temperature ~ 1cm.
Fluid model of merging-compression
βT = 4x10-5 < me/mi
βp = 10-3
Hyper-resistivity is used to set dissipation scale for Whistler waves.
Can (and does) set diffusion scale here by breaking frozen-in condition.
Physically related to an electron viscosity: for
The plasma state during merging compression is not the same is at normal operation. The initial temperature is only 10eV, and with the strong fields this gives very low toroidal and poloidal beta values.
The toroidal beta is actually less than the mass ratio initially.
The Lundquist number is large, and the ion-skin depth is large compared to the smallest measured feature in the data, the central electron temperature peak in TS data. The initial electron skin depth and ion larmor radius are smaller than this value, and we dont include them in this study.
So we have the hall term, and also we use a hyper-resistivity to damp grid scale whistler waves. This also breaks the frozen in condition here.
So with such high lundquist number and low beta these simulations may have some relevence to the Solar Corona.
No study done on breaking frozen-in condition that is applicable for this regime.
(Closest by Ricci et al. with B_T/B_p=5, but no collisions and >m_e/m_i).
Solved with the HiFi framework (eg. Lukin and Linton) with 4th Order polynomial basis.
Crank-Nicholson time advance (to avoid CFL condition).

43. Fluid model of merging-compression
One temperature formulation:
μ = 1/Re = 10-3 (based on μ//i = 10-4 Pa s).
η =10-5 (based on η// = 3.5 x 10-5 Ω m, and BT0)
Use κ//e and κ┴i based on initial n0, T0.
Two-temperature formulation:
Electron-ion equilibration time: τie > 0.2 ms, longer than merge time.
Temperature dependent resistivity (Spitzer parallel).
For most of the simulations we use one pressure equation, with ion viscosity based on the initial parallel value. We use parallel electron conductivity, and the cross-field ion conductivity, rather than the true perpendicular value.
At the end I'll show a simulation with seperate ion and electron pressure equations, so the resistive heating only acts on the electron fluid and ion-viscous heating only on the ions.
Use κ//e, κ┴e, κ//i, κ┴i based on initial n0, T0.
Use κ//e, κ┴e, κ//i, κ┴i based on initial n0, T0.
μ = (Re)-1 = 10-3 (based on μ//i = 10-4 Pa s).
S =105 (η// = 3.5 x 10-5 Ω m)
Use κ//e and κ┴i based on initial n0, T0.
Electron-ion equilibration time: τie > 0.2 ms, longer than merge time.
Temperature dependent resistivity (Spitzer parallel).