Eirik Endeve. Simula'ons of SASI, turbulence, and magne'c field amplifica'on

Transcription

1 Simula'ons of SASI, turbulence, and magne'c field amplifica'on INT Program INT- 12-2a: Core- Collapse Supernovae: Models and Observable Signals Eirik Endeve Funding: DoE Office of Advanced Scien'fic Compu'ng Research and Office of Nuclear Physics Compu'ng resources provided by DoE through the INCITE program 1

2 Turbulence and Magne'c Fields from SASI Objec'ves: Does turbulence develop from the SASI? - What are the characteris'cs of SASI- driven turbulence? - Does turbulence result in nonlinear SASI satura'on? (e.g., Guilet et al. 2010, ApJ, 713, 1350) Does SASI- driven turbulence result in magne'c field amplifica'on? - What are the characteris'cs of SASI- driven magne'c field amplifica'on? Do amplified magne'c fields change characteris'cs of the SASI? - Implica'ons for the explosion mechanism of CCSNe? Do SASI + B- fields impact observables associated with CCSNe? - Implica'ons for proto- neutron star magne'c fields? Endeve, Cardall, Budiardja, & Mezzacappa (2010), ApJ, 713, 1219 Endeve, Cardall, Budiardja, Beck, Bejnood, Toedte, Mezzacappa, & Blondin (2012), ApJ, 751, 26 Endeve, Cardall, Budiardja, Mezzacappa, & Blondin (2012), Physica Scripta, in press (arxiv:1203:3748) 2

3 Turbulence and Magne'c Fields from SASI Methods: High- resolu'on magnetohydrodynamic (MHD) simula'ons - Mul'ple models (vary rota'on rate and ini'al field strength) - Grids up to zones - CPU 'me at OLCF provided by DoE through the INCITE program Idealized semi- analy'c ini'al condi'on - Blondin & Mezzacappa (2007), Nature, 445, 58 + weak B- field - Analysis includes Fourier decomposi'on - Kine'c, Magne'c, and Enstrophy spectra Endeve, Cardall, Budiardja, & Mezzacappa (2010), ApJ, 713, 1219 Endeve, Cardall, Budiardja, Beck, Bejnood, Toedte, Mezzacappa, & Blondin (2012), ApJ, 751, 26 Endeve, Cardall, Budiardja, Mezzacappa, & Blondin (2012), Physica Scripta, in press (arxiv:1203:3748) 3

4 t U ijk = 1 V ijk MHD Scheme in GenASiS Finite volume method solves integral form of ideal MHD equa'ons F(U, B) da + S ijk (U, Φ) V ijk t Bx i+ 1 2 jk = 1 A x i+ 1 2 jk A x i+ 1 2 jk E(U, B) dl Second- order semi- discrete finite volume method Second- order in space through slope- limited linear interpola'on HLL- type approximate Riemann solver flux + electric fields Constrained transport for divergence- free B- field update Second- order in 'me through TVD Runge- Kuha 'me integra'on Kurganov et al. (2001), SIAM, J.Sci.Comput., 23, 707 Londrillo & Del Zanna (2004), JCP, 195, 17 Endeve et al. (2012), J. Phys.: Conf. Ser., in press (arxiv:1203:3385) 4

5 MHD Scheme in GenASiS Second- order convergence for smooth problems: Circularly polarized Alfven wave Table 1. L 1 -error and convergence rate for the circularly polarized Alfvén wave test. TW (t f =1) SW(t f =1) TW(t f =5) SW(t f =5) N x L 1 (B z ) Rate L 1 (B z ) Rate L 1 (B z ) Rate L 1 (B z ) Rate E E E E E E E E E E E E E E E E E E E E E E E E First- order convergence for problems with discon'nui'es: Orzag- Tang vortex 5

6 Turbulence from Spiral SASI Modes Supersonic flows contribute mostly to the wings of the PDF for v x Subsonic SASI Spiral modes emerge favorably Turbulence driven by supersonic shear flows Postshock flow: Supersonic driving component +subsonic turbulent (volume filling) component 6

7 Turbulence from Spiral SASI Modes Significant post- shock turbulence at satura'on 100 km Turbulence develops from secondary instabili'es (e.g., KH) in the shear layer induced by the spiral SASI mode Non- radial and turbulent kine'c energy grows rapidly during nonlinear SASI development Radial and non- radial kine'c energy in equipar''on during turbulent, saturated state (similar to turbulence from neutrino- driven convec'on; Murphy et al. arxiv: ) Kine'c energy Total Radial Non- radial Turbulent 7

8 Fourier Decomposi'on of Simula'on Results For any vector field we compute Fourier amplitudes (using FFTW library) ˆf(k) = 1 V L V L f(x) exp(ik x) dv Define the spectral energy density in k- space shell ê(k) = 1 2 ˆf 2 k 2 dω k k-shell 0 êdk = 1 2 V L f 2 dv f(x) = ρ u, u, ω, B ê kin, ê u, ê ω, ê mag (i.e., kine'c energy, specific kine'c energy, enstrophy, and magne'c energy spectra) 8

9 Nonlinear Satura'on of SASI due to Turbulence? Growing power in low- order SASI modes ê u k 5/3 Kine'c energy Total Radial Non- radial Turbulent Quasi- steady state with Kolmogorov spectrum SASI satura'on coincides with development of turbulence Quasi- steady state with Kolmogorov spectrum develops - Accre'on- powered large- scale flows cascade to smaller scales and dissipate (numerically) 9

10 Impact of Numerical Resolu'on Kine'c energy Spectrum extends to larger k- values for higher resolu'on - Numerical dissipa'on causes spectrum to decrease faster than k - 5/3 No impact on satura'on level of kine'c energy (earlier onset of nonlinear SASI for lowest resolu'on model) 10

11 Non- Helical Turbulence from SASI PDF: rela've kine'c helicity Rela've kine'c helicity h kin = u ω u rms ω rms Net kine'c helicity small in SASI simula'ons SASI- driven turbulence: B- field amplificaaon via efficient small- scale dynamo - Similar to convec'vely driven turbulence (Brandenburg et al. 1996, J. Fluid Mech., 306, 325) Kine'c helicity may increase with sufficient differen'al rota'on + h kin PDF(h kin ) dh kin < 10 3 Important for magne'c field amplifica'on! Non- helical turbulence: small- scale dynamo (Haugen et al. 2004) Helical Turbulence: Large- scale dynamo (inverse cascade; Meneguzzi et al. 1981, Brandenburg 2001, Blackman 2012)

12 Turbulent Magne'c Field Amplifica'on Post- shock magne'c energy Total magne'c energy E mag = V Sh B B 2µ 0 dv Growth due to turbulent small- scale dynamo E T mag = k T ê mag dk λ<2π/k T 60 km B- fields grow exponen'ally - B 0 =10 10 G: steady growth, τ 65 ms, B 0 =10 13 G: growth tapers off, E mag 5x10 47 erg - Growth rate underes'mated by numerical simula'ons! 12

13 Magne'c field magnitude 13

14 Satura'on of Magne'c Energy emag/ekin=(va/u)2 Magne'c field displays spa'al and temporal intermihency 14

15 Spectral Evolu'on of Magne'c Energy Turbulent kine'c energy E tur kin = k tur ê kin dk ) ( ) u k tur 2π 2 u 2 ( ) MagnePc spapal scale (resolupon dependent) Turbulent rms velocity ( ) ê 2E tur 1/2 0 mag (k) dk kin λ u rms = km s 1 mag =2π 20 km kê 0 mag (k) dk M Sh ( )( τ λ mag /u rms 5ms 15

16 t ( ) B B 2µ 0 Amplifica'on of Magne'c Fields = 1 µ 0 B [(B ) u (u ) B B ( u)] stretching compression = P u (J B) B- field amplified by ``stretching Oh (1998), Phys. Plasmas, 5, 1636 Growth rate (inverse turnover 'me) ( )( Σ J B 400 s 1 u rms λmag 4000 km s 1 20 km - Mismatch with simula'ons (15 s - 1 )! ) 1 16

17 Effects of Finite (Numerical) Conduc'vity t ( ) B B = P u (J B) 1 B (ηj) 2µ 0 µ 0 Finite volume scheme mimics effect of finite conduc'vity to stabilize solu'on near discon'nui'es and sharp gradients Decrease in flux tube thickness ( )( halted by resis've dissipa'on Limits field amplifica'on ( ) Magne'c energy growth rate severely underes'mated [ Σ Σ J B 1 R 1 m ( λmag λ d ) 2 ] ( )( ( B 2 λ rms = B 2 ) 1/ Δx

18 Effects of Finite (Numerical) Conduc'vity HLL electric field: E z n p = α+ x α+ y [ ] E SW n z p + [ ] α+ x α y E NW n z p + [ ] α x α+ y E SE n z p + α x α y ( α + x + αx )( α + y + αy ) [ ] E NE n z p + α + x α x ( α + x + α x α+ y α y ( α + y + α y ) ) ( [B E y ] n p [ B W y ] n p) ( [B ] N n x p [ Bx S ] n ), (A4) p Finite volume scheme mimics effect of finite conduc'vity to stabilize solu'on near discon'nui'es and sharp gradients Contributes significantly to magne'c energy growth rate 18

19 Simula'ons with Different Resolu'ons Double grid resolu'on Magne'c energy increases by a factor Growth rate also sensi've to spa'al resolu'on 19

20 3 of B- fields on Post- shock Flows Impact Rm = urms λ mag /η! 1 " k 1 fmag (k) = e! (ξ) dξ!mag 0 mag E emag /ekin = va2 / u 2 t # B B 2µ0 $ = [! ekin,0! ekin ] /! ekin,0 1 B [ (B ) u (u ) B B ( u) ] µ0 = P u (J B) t # B B 2µ0 $ = P u (J B) 1 B (ηj) µ0 Small- scale B- fields impact small- scale flows No impact on large scale dynamics detected 20

21 Implica'ons for Proto- neutron Star Magne'za'on Inferred PNS B- field [log(b/g)] 15 No RotaPon 14 Slow RotaPon 13 Faster RotaPon SASI may result in PNS B- fields exceeding 1014 G Sensi'vity to numerical resolu'on applies - models with weak ini'al fields saturate due to finite grid 21

22 Summary Nonlinear SASI drives vigorous turbulence below the shock Development of turbulence seems to result in SASI satura'on - Turbulence feeds on power in low- order SASI modes Weak magne'c fields amplified exponen'ally Strong magne'c fields emerge and impact flows on small spa'al scales SASI- induced B- fields not likely to play principal role in the explosion of CCSNe - Kine'c energy of explosion Post- shock kine'c energy Turbulent kine'c energy (only about 10% accessed by B- fields) SASI- induced magne'c fields may contribute nontrivially to PNS magne'za'on - Magne'c energy accreted onto the PNS > erg - Es'mates suggest small scale B- fields in the G range 22

23 Are B- Fields Relevant for Non- rota'ng CCSNe? Idealized simula'ons offer proof of principle for efficient B- field amplifica'on Must follow up with mul'- physics simula'ons - Explicitly include PNS Computa'onal cost increases drama'cally! - Neutrino radia'on transport - Does the spiral SASI mode play a central role? - Other field- amplifica'on mechanisms also at play (α- Ω dynamo, MRI) Effects of finite grid resolu'on represents a challenge for global simula'ons - Magne'c energy growth rates underes'mated - Flux- rope thickness (i.e., field strength) limited by size of grid cells - What happens at larger/realis'c R m (10 16 )? - Does modest rota'on result in helical dynamo and large- scale B- fields? - Can we construct useful local simula'ons with guidance from global simula'ons? Details on evolu'on and impact of magne'c fields in CCSNe remain uncertain 23

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25 Post- Shock Kine'c Energy from 2D Mul'- Physics Runs 1600 km Bruenn et al. (2011), in prepara'on Woosley & Heger (2007), Phys. Rep., 442, ms 180 ms 25