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Adam Stanier 1, P. Browning 1, M. Gordovskyy 1, K. McClements 2, M. Gryaznevich 2,3, V.S. Lukin 4 Simulations of magnetic reconnection during merging start-up in the MAST Spherical Tokamak EPS Conference, Espoo, July 2013 MAST 1 Jodrell Bank Centre for Astrophysics, University of Manchester, UK 2 EURATOM/CCFE Fusion Association, Culham Science Centre, UK 3 Present affiliation: Imperial College of Science and Technology, London, UK 4 Space Science Division, Naval Research Laboratory, DC, USA

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

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3 ▶ 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 T e = 1 keV achieved in on ms timescale measured with Thomson Scattering (TS) laser. Merging start-up Pick-up Coil (CCMV20) Thomson Scattering lasers P3 Plasma P3 φ φ Magnetic: B p = 0.1 T, B T = 0.5 T, I T = MA Thermal: T e = T i = 10 eV, n = 5x10 18 m -3, Deuterium Typical start-up parameters: Time (ms) Current (kA.turn)

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4 Merging start-up Time resolution: 0.1 ms. Total time: ~ 7ms.

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5 ▶ Solved with the HiFi framework (eg. Lukin and Linton) with 4 th Order polynomial basis. ▶ Crank-Nicholson time advance (to avoid CFL condition). ▶ 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: Fluid model β T = 4x10 -5 β p = ▶ Initial Lundquist #: S = 2 x 10 4 → Collisional Current Sheet (CS) width: δ SP ~ 1 cm. ▶ Kinetic scales become important when larger than collisional CS width. ▶ Ion skin depth: d i = 15 cm, Electron: d e = 0.25 cm, Larmor radius: ρ i = ρ is = 0.13 cm. for Hyper-resistivity (anomalous electron viscosity). Hyper-resistivity (electron viscosity) Anisotropic heat conduction: ▶ Heat cond.:, ion-stress tensor: ▶ Will vary μ, η and η H in simulations presented.

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6 φ Toroidal (R,φ,Z) Code and initial conditions ▶ Currently no q-profiles of pre-merged flux-ropes (plan for M9 campaign). ▶ Idealised initial conditions using ▶ Solved in 2D Cartesian and toroidal geometry with spectral-element code HiFi. ▶ 4 th 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, I T = 0.27 MA. ▶ Balanced against pinching by B T increase (β p ~ ), individually force-free. ▶ Conducting walls with line-tied vertical flux B v = T. ▶ Radial dependence (1/R) of toroidal field. Grid: ∆R min = 0.5mm ∆Z min = 0.3mm

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7 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 ≤ due to magnetic pressure pile-up. X-point at t = 0 Current Sheet (CS) width: δ = 2.4 cm Grid: ∆Z min = 0.03cm

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8 Nd:YAG TS laser ▶ 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 n e Hall-MHD Simulation: Density 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. 20 t 0 40 t 0 60 t 0 80 t 0

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9 What causes the double peak in density? ▶ MHD: Density peak on inner side, cavitation on outer edge. ▶ Two-fluid: Additional density asymmetry, disappears after merging completion. MHD d i = 0 d i = m η H = High density seperator ▶ 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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10 Cartesian Hall-MHD: Effect of collisions ▶ However, large aspect ratio current sheet is unstable to island formation (for η H ≤ ). ▶ In Cartesian geometry a central island stalls the reconnection. ▶ Stronger B T : weaker density asymmetry. Multiple, shorter wavelength islands. Scan in hyper-resistivity (collisions) Weaker collisionality ▶ η H = ▶ Stable. ▶ ▶ Order 1 density variations in “quadrupole” structure. ▶ Electrons accelerated in low density regions. ▶ Similar to reduced two-fluid reconnection model of Kleva et al ▶ η H = ▶ Island (ejected in toroidal geometry). ▶ η H = ▶ Localised CS: δ = 4.5 mm ρ is = 2.9 cm Grid: ∆R min = 4x10 -4 m, ∆Z min = 2x10 -4 m Grid: ∆R min = 1x10 -4 m, ∆Z min = 4x10 -5 m

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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 >> ρ is ), open X-point (δ < ρ is ) and an intermediate regime that is unstable to island formation (δ ≥ ρ is ). ▶ Resistive MHD simulations in the sloshing regime. ▶ Three regimes in Hall-MHD simulations: collisional (δ >> ρ is ), open X-point (δ < ρ is ) 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 >> ρ is ), open X-point (δ < ρ is ) 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. 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. (Stanier et al. 2013). We have also simulated separate ion and electron temperature profiles.

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12 Hall-MHD: Cartesian geometry

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13 Resistive MHD ▶ Several studies have shown length-scale ρ is = (T e /m i ) 1/2 /Ω ci important for fast reconnection with strong B T. ▶ Peak reconnection rate in Hall-MHD for CS width > ρ is have (weak) dependence on η H. ▶ η H = is slow during CS formation, but explosive when width drops below ρ is (t=7 t 0 ). Additional: Reconnection rates (eg. Kleva et al , Simakov et al. 2010) t=7 t 0

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14 25 cells across CS width Additional: Numerical grid and convergence N R =360, N Z =540, N P =4 N R =180, N Z =270, N P =4 ▶ Convergence test for simulation with η H = (lowest dissipation scale). ▶ Coarsening by factor of 2 changes peak reconnection rate by only 0.2 %.

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15 Additional slide: q-profile Direction of island ejection depends upon radial positions of O- points and X-point ▶ 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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16 Additional: Resistive MHD sloshing ▶ Increase in B T 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 B R on sheet edge, and reconnection stalls. ▶ Sloshing of flux-ropes, c.f. coalescence instability. S = 10 5, mu = 10 -3, B T = 0.5 T, a=0.6, I plasma = 268 kA (Biskamp & Welter 1980, Knoll and Chacon 2005)

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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).

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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 ~ J over 100 sec. Tokamak Sawtooth crash: Build up (sawtooth period) ~ 100 ms, crash ~ 100 μs.

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

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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 ▶ T e increase from 10 eV to ~100 eV ▶ Central T e peak with ΔR, ΔZ ~1cm. ▶ Oscillations (τ~30 μs) in CCMV20 signal during/after merging. Data taken by T. Yamada (University of Tokyo) and the MAST team. See Ono et al nene TeTe CCMV20 pick-up coil ∂tBZ∂tBZ

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21 Thomson Scattering: Electron heating ▶ 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 τ ie ≈ 0.2 ms * (T e [eV]/ T 0 ) 3/2 T 0 = 10 eV Ruby laser + P6 (vertical position) coils: 2D profile Peaked Case: n = m -3 at 10.0 ms Hollow Case: n = 5 x m -3 at 10.0 ms Ruby TS laser φ

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

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

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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): B T = 0.5 T ▶ Current: ≤ 1.6 MA ▶ Temperature ~ 0.1 – 3 keV ▶ Density: – m -3 ▶ Ion Species: Deuterium ▶ P1 is central solenoid: normal method for current drive. ▶ P3 coils: used for merging-compression start-up. ▶ P4, P5: vertical field, P6: vertical position. R a P1 P3 φ

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25 Hall-MHD: Cartesian geometry ▶ However, large aspect ratio current sheet is unstable to island formation (for η H ≤ ). ▶ In Cartesian geometry a central island stalls the reconnection. ▶ Stronger B T : weaker density asymmetry. Multiple, shorter wavelength islands. Scan in hyper-resistivity (collisions) Weaker collisionality ▶ η H = ▶ Stable ▶ ▶ Order 1 density variations in “quadrupole” structure. ▶ Electrons accelerated in low density regions. ▶ Similar to reduced two-fluid reconnection model of Kleva et al ▶ η H = ▶ Island. ▶ η H = ▶ Localised CS.

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26 One temperature formulation: ▶ μ = (Re) -1 = (based on μ //i = Pa s) ▶ η = (based on η // = 3.5 x Ω m and B T0 ) ▶ Use κ // e and κ ┴ i based on initial n 0, T 0. ▶ Will vary μ, η and η H in simulations presented. ▶ Electron-ion temperature equilibration time: τ ie > 0.2 ms, longer than merge time. → will discuss results with two-temperature formulation at the end. Fluid model of merging-compression ▶ μ = 1/Re = (based on μ //i = Pa s). ▶ η =10 -5 (based on η // = 3.5 x Ω m, and B T0 ) ▶ Use κ // e and κ ┴ i based on initial n 0, T 0. ▶ Electron-ion equilibration time: τ ie > 0.2 ms, longer than merge time. ▶ Temperature dependent resistivity (Spitzer parallel). ▶ Use κ // e, κ ┴ e, κ // i, κ ┴ i based on initial n 0, T 0. But also:

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27 Two-temperature formulation: ▶ Electron-ion equilibration time: τ ie > 0.2 ms, longer than merge time. ▶ Temperature dependent resistivity (Spitzer parallel). Two-temperature formulation ▶ Use κ // e, κ ┴ e, κ // i, κ ┴ i based on initial n 0, T 0.

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28 Two-fluid: Toroidal geometry Direction of island ejection depends upon radial positions of O- points and X-point ▶ Toroidal geometry breaks symmetry – central island ejected (possible filament?). ▶ Two-fluid merge time comparable to Cartesian case (= 25 τ 0 = 10 μs for η H = ).

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29 One temperature formulation: ▶ μ = (Re) -1 = (based on μ //i = Pa s) ▶ η = (based on η // = 3.5 x Ω m and B T0 ) ▶ Use κ // e and κ ┴ i based on initial n 0, T 0. ▶ Will vary μ, η and η H in simulations presented. Fluid model of merging-compression ▶ μ = 1/Re = (based on μ //i = Pa s). ▶ η =10 -5 (based on η // = 3.5 x Ω m, and B T0 ) ▶ Use κ // e and κ ┴ i based on initial n 0, T 0. ▶ Electron-ion equilibration time: τ ie > 0.2 ms, longer than merge time. ▶ Temperature dependent resistivity (Spitzer parallel). ▶ Use κ // e, κ ┴ e, κ // i, κ ┴ i based on initial n 0, T 0. ▶ Solved in 2D Cartesian and toroidal geometry with the SEL/HiFi framework ▶ Spectral element code with 4 th 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. (Glasser and Tang 2004, Lukin 2008).

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30 Two-temperature with hyper-resistive heating Time: 17.9 Time: 40.9 Time: 45.9 Time: 85.5 Max: 223 eVMax: 2147 eVMax: 2461 eVMax: 1433 eV Max: 53 eV Max: 70 eVMax: 81 eVMax: 105 eV ▶ Ion temperature profile “hollow”. Tilt in ion temperature profile. ▶ Electron temperature ~ 100 eV, but no central peak. Ion Temperature Electron Temperature Time: 17.9 Time: 40.9 Time: 45.9 Time: 20.8 Time: 50.0 Time: 65.2 Time: 20.8 Time: 34.9 Time: 50.0 Time: 65.2 Max: 315 eVMax: 1402 eVMax: 1944 eVMax: 2155 eV Max: 368 eVMax: 891 eVMax: 1304 eVMax: 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. Electron heating on CS edges: not cospatial with j. Time: 34.9

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31 Two-temperature formulation: ▶ 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 ρ is ~ T e 1/2 (fast-reconnection threshold), and η ~ T e -3/2 What is the dominant electron heating mechanism? Electron Temperature Time: 17.9 Time: 40.9 Time: 45.9 Time: 85.5 Max: 53 eVMax: 70 eVMax: 81 eVMax: 105 eV resistive hyper-resistive compressive

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32 Spatial scales and Hall-MHD reconnection Hall Reconnection ▶ Sweet-Parker and Petschek are MHD reconnection models (scale-invariant). ▶ However, new physics when CS width δ SP = S -1/2 L drops below scales. ▶ Below ion skin-depth d i 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. Hall Electron inertia Term: Scale: d i =c/ω pi d e =c/ω pe ρ is = v s /Ω ci Ion skin-depthIon-sound radius ▶ Sweet-Parker and Petschek are MHD reconnection models (scale-invariant). ▶ However, new physics important when collisional CS width

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33 Hall finishes Resistive ▶ 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 n e profiles (shifting plasma with P6 coils).. Experimental Comparisons CCMV20 (experiment) Pick-up Coil (CCMV20) Vertical field ▶ Oscillations in simulated CCMV signal. ▶ Simulated CCMV is too fast (by factor of 3): other PF coils may be important. O-point Are CCMV20 oscillations a signature of sloshing?

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34 Energy conservation ResistiveHall

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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: I plasma = 340 kA, T e = 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). P. Micozzi (3 rd IAEA Technical Meeting on ST, 2005)

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Overview 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. Mega-Ampere Spherical Tokamak (MAST)

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Reconnection Layers in 2D: Classical Picture ▶ Classical (Sweet-Parker) width: δ SP = 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. Steady-state reconnection: Resistive MHD Petschek (fast) shocks 2Δ 2δ 2L Diffusion region ▶ How do these models fit into the global picture? Sweet-Parker (slow)

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

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39 Additional Slides: 1 keV electron temperature Plasm a P3 Central Solenoid nene TeTe ▶ 1 keV temperature peak measured by Ruby TS at 12 ms. ▶ Merging-Compression (P3 = 300 kA turn) + Central Solenoid.

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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.

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41 ▶ Solved with the HiFi framework (eg. Lukin and Linton) with 4 th Order polynomial basis. ▶ Crank-Nicholson time advance (to avoid CFL condition). Merging-compression as a reconnection experiment Magnetic: Plasma: β T = 4x10 -5 β p = S = δ SP /d i ~ 0.02 (δ SP ~ S -1/2 L) } MAST merging-compression is a high S, low beta reconnection experiment. ▶ 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). ▶ Initially S = 10 5 (η // = 3.5 x Ω m) ▶ Ion skin-depth: d i = 15 cm, electron: d e = 0.25 cm, Ion Larmor radius: ρ i = ρ is = 0.13 cm. c.f. Central peak in electron temperature ~ 1cm. Dimensionless : Beta: Lundquist: MAST merging-compression Corona (Active Region) β = – S = δ SP /di ~ Length-scale: Ion skin- depth: B ~ 0.01 T T e ~ 100 eV, n = m -3 L ~ 10 7 m d i ~ 10 m B p = 0.1 T, B T = 0.5 T T e = eV, n = 5x10 18 m - 3 L = 1 m d i = 15 cm

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42 ▶ Solved with the HiFi framework (eg. Lukin and Linton) with 4 th Order polynomial basis. ▶ Crank-Nicholson time advance (to avoid CFL condition). ▶ 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 Fluid model of merging-compression β T = 4x10 -5 < m e /m i β p = ▶ 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). ▶ Hall term may be important: δ SP /d i ~ 0.02 (δ SP ~ S -1/2 L) ▶ Electron skin-depth: d e = 0.25 cm, Initial Ion Larmor radius: ρ i = ρ is = 0.13 cm. ▶ c.f. Central peak in electron temperature ~ 1cm.

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43 One temperature formulation: ▶ μ = (Re) -1 = (based on μ //i = Pa s). ▶ S =10 5 (η // = 3.5 x Ω m) ▶ Use κ // e and κ ┴ i based on initial n 0, T 0. Two-temperature formulation: ▶ Electron-ion equilibration time: τ ie > 0.2 ms, longer than merge time. ▶ Temperature dependent resistivity (Spitzer parallel). ▶ Use κ // e, κ ┴ e, κ // i, κ ┴ i based on initial n 0, T 0. Fluid model of merging-compression ▶ μ = 1/Re = (based on μ //i = Pa s). ▶ η =10 -5 (based on η // = 3.5 x Ω m, and B T0 ) ▶ Use κ // e and κ ┴ i based on initial n 0, T 0. ▶ Electron-ion equilibration time: τ ie > 0.2 ms, longer than merge time. ▶ Temperature dependent resistivity (Spitzer parallel). ▶ Use κ // e, κ ┴ e, κ // i, κ ┴ i based on initial n 0, T 0.

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