Simulations of magnetic fields in core collapse on small and large scales

Simulations of magnetic fields in core collapse on small and large scales Miguel Ángel Aloy Torás, Pablo Cerdá-Durán, Thomas Janka, Ewald Müller, Martin Obergaulinger, Tomasz Rembiasz CAMAP, Departament d Astronomia i Astrofísica, Universitat de València, Spain Max-Planck-Institut für Astrophysik, Garching bei München, Bavaria Multi-messengers from Core-Collapse Supernovae, Fukuoka, 5 December 2013 1 / 21

Contents Introduction 1 Introduction 2 Results 3 Summary 2 / 21

Why magnetic fields? Introduction Indications for magnetic fields asymmetric explosions (but: can be explained by hydro alone) pulsar kicks (also non-magnetically possible) pulsar fields, magnetars magnetic fields are no more exotic in SNe than in the sun Scenarios for MHD effects magneto-convection in PNS and hot bubble: small-/large-scale dynamos? magnetic versions of the SASI: amplification, but how much? additional heating, e.g. by Alfvén waves with rotation: winding, MRI, angular-momentum transport, jet formation 3 / 21

Introduction Magnetic fields and MHD magnetic energy 1 2 B 2 ideal MHD: field lines and flux tubes frozen into the fluid Lorentz force (Maxwell stress) consists of isotropic pressure 1 B 2 2 anisotropic tension B i B j increase the energy by work against the forces t e mag + i e mag v i = P mag i v i + b i b j j v i compressing the field stretching and folding field lines estimate for the maximum field energy: kinetic energy actual amplification may be less 4 / 21

Introduction Turbulence and dynamos instabilities such as convection and SASI drive turbulence energy cascades from the large scale at which the instability operates down to dissipation in a Kolmogorov-like cascade 5 / 21

Introduction Turbulence and dynamos dynamo converting turbulent kinetic to magnetic energy by stretching and folding flux tubes small-scale dynamo: amplifies the field only on length scales of turbulent velocity fluctuations Kolmogorov-like spectrum of turbulent magnetic field large-scale dynamo adds an inverse cascade of field to larger length scales, i.e., generates an ordered component. Key ingredient: helicity v v. 5 / 21

Introduction Turbulence and dynamos Cutting the cascades above the physical length scale may distort the results. At least, a good knowledge of the behaviour of the numerical code is necessary. 5 / 21

Introduction Amplification of MHD waves given the velocity of MHD waves and a time scale we need to resolve the propagation distance of the waves with n grid cells even if the time is set by the large-scale motion, this can be prohibitive for slow waves example: the MRI is the weak-field instability of the slow mode, i.e. c slow c s, v the length scale can be estimated from the Alfvén velocity and the rotational period λ MRI = O(c A Ω 1 ) 6 / 21

Introduction Amplification of MHD waves MRI dispersion normalised growth rate k v A Ω cosθ k 10 1 0.1 Buoyant modes Stable modes Alfvén modes 0.8 0.6 0.4 0.2 0.01 0 100 10 1 0.1 0.01 C dashed line: fastest growing mode solid line: boundary between modes branches Definition of symbols C = (N)2 + ( ϖ ϖ Ω 2) 2 Ω 2 N = bouyancy frequency 1 Stable modes short modes are stablised by magnetic tension Alfvén modes fast growth only for finite wave number Bouyant modes appear only for large entropy gradient; fast growth for long modes v A = Alfven velocity k = wave number θ k = angle between k and the vertical 6 / 21

Introduction Amplification of MHD waves MRI dispersion normalised growth rate 10 1 k v A Ω cosθ k 1 0.1 0.01 Buoyant modes Stable modes Alfvén modes 0.8 0.6 0.4 0.2 0 6 / 21

Introduction Global and local models dispersion relation of the MRI: only short modes, λ B, grow rapidly in core collapse, this can be 1 m grid width < λ computationally not feasible high-resolution shearing-box simulations to determine fundamental properties of the MRI use these results to build models that can be coupled to global simulations 7 / 21

Contents Results 1 Introduction 2 Results 3 Summary 8 / 21

Simulations Results magneto-rotational collapse global simulations of rotating collapse without neutrino transport Newtonian and GR Non-rotating collapse neutrino transport: two-moment scheme Newtonian hydro with pseudo-relativistic gravity 15 M progenitor, only ν e, ν e polytropic models or stellar cores Magneto-rotational instability local boxes shear-periodic boundaries neglecting all microphysics Parasitic instabilities even more idealised local models determine the growth rate of tearing modes and measure the numerical resistivity/viscosity 9 / 21

Compression Results conservation of radial magnetic flux through a surface B ρ 2/3 for a fluid element; energy grows faster than gravitational no change of field topology core collapse: factor of 10 3 in field strength possible saturation: e mag e kin,r is unrealistic in collapse occurs in every collapse (and continues after bounce) log b [G] 13 12 11 10 9 8 7 field estimate 0.0 0.5 1.0 1.5 2.0 2.5 m [M sun] Profiles of field strength during collapse compared to estimate based on flux conservation (deviations come from non-radial geometry) 10 / 21

Convection/SASI Results hydro instabilities develop quickly highly variable amplitude of the instability, flow field field is amplified in thin filaments final field strength depends on the initial field, and amplification is not (necessarily) leading to equipartition weak initial field: no dynamic impact strong field: very different evolution of the post-shock flows, early development of high-entropy bubbles favourable for an explosion Weak initial field: three snapshots 11 / 21

Results Amplification of Alfvén waves Guilet et al. (2011) requires an accretion flow decelerated above the PNS and a (radial) guide field accretion is sub-/super-alfvénic inside/outside the Alfvén surface Alfvén waves propagating along the field are amplified at the Alfvén point waves are finally dissipated there additional heating in core collapse: efficient for a limited parameter range (strong guide field); strong time variability of the Alfvén surface may be a problem modelling issues: high resolution, uncertainties in the dissipation 12 / 21

Results Amplification of Alfvén waves y log b [G] 11.20 13.35 15.50 300 200 100 0 100 200 log b [G] 11.20 13.35 15.50 log A 4 1 2 300 300 200 100 0 x 100 200 300 log A 4 1 2 requires an accretion flow decelerated above the PNS and a (radial) guide field accretion is sub-/super-alfvénic inside/outside the Alfvén surface Alfvén waves propagating along the field are amplified at the Alfvén point waves are finally dissipated there additional heating in core collapse: efficient for a limited parameter range (strong guide field); strong time variability of the Alfvén surface may be a problem modelling issues: high resolution, uncertainties in the dissipation 12 / 21

Results Rotation and magnetic field Cerdá-Durán et al. (2008): MRI channel modes MPA/València: Newtonian/GR simulations of simplified collapse; Princeton, Japan, Basel: more detailed simulations winding of field lines strong field: MRI observable angular-momentum transport jet formation MHD explosions visible in the GW signal 13 / 21

Results The magneto-rotational instability differentially rotating core is unstable if b 0 growth time Ω 1 fastest growing modes λ MRI c A Ω 1 for SN core: λ MRI 6.9 cm ( b 0 ) ( ) 1/2 ρ ( ) Ω 1 10 11 G 2.5 10 13 g cm 3 1900 s 1 usually no the kind of resolution you would like to have in a supernova simulation use local high-resolution models as a guide to learn about the effects you are missing in global models 14 / 21

Local models Results Equations Ideal MHD simplified (hybrid) EOS external gravity no neutrino transport Models initial data: hydrostatic equilibrium axisymmetric and 3d models small (few kilometres) boxes resolution between 1.25 and 40 m shearing-disc boundary conditions Code Finite-volume code high-order MP reconstruction MUSTA Riemann solver constraint transport Questions, old, current, tbd growth in different regimes? termination mechanism? termination amplitude? saturation phase? transfer to global models? 15 / 21

MRI termination Results equipartition, e mag e diffrot, overestimates the final state MRI is quenched earlier by secondary instabilities linear evolution: channel modes Kelvin-Helmholtz or tearing modes disrupt the MRI channel flows Tomasz Rembiasz (2013): 2d MRI shearing box with viscosity and resistivity. 16 / 21

MRI termination Results Pessah (2010) perturbations (δb, δv) of an MRI mode grow at constant rate σ NRI MRI channel modes are separated by current sheets and shear layers unstable against parasitic instabilities: Kelvin-Helmholtz and tearing modes Tomasz Rembiasz (2013): 2d MRI shearing box with viscosity and resistivity. 16 / 21

MRI termination Results Pessah (2010) parasites grow at rates B MRI /λ exp σ MRI t/b 0, i.e., faster as the MRI proceeds at some point, they overtake the MRI and break the channels down into turbulence MRI growth ends, and final field is fixed Tomasz Rembiasz (2013): 2d MRI shearing box with viscosity and resistivity. 16 / 21

MRI termination Results Rembiasz (2013) can we test this model? in principle yes, but you have to carefully distinguish physics from numerics numerical errors introduce an effective viscosity/resistivity and modify the growth of the parasitic instabilities Tomasz Rembiasz (2013): 2d MRI shearing box with viscosity and resistivity. 16 / 21

Results Numerical resistivity/viscosity viscosity/resistivity... show up in different effects: dampen linear waves measuring damping rate drive tearing modes measure growth rate In all cases, you need to run many simulations with different resolutions in space and time, (numerical schemes), and physical resistivity/viscosity, and you need to know the theoretical growth/dampening rates very well. For tearing modes, the latter is a problem on its own. Viscous damping of a sound wave: profiles and comparison of numerical and theoretical solution; numerical dissipation vs. resolution for different schemes. 17 / 21

Results Numerical resistivity/viscosity viscosity/resistivity... show up in different effects: dampen linear waves measuring damping rate drive tearing modes measure growth rate In all cases, you need to run many simulations with different resolutions in space and time, (numerical schemes), and physical resistivity/viscosity, and you need to know the theoretical growth/dampening rates very well. For tearing modes, the latter is a problem on its own. Field lines before and after reconnection by tearing modes. 17 / 21

Results Numerical resistivity/viscosity viscosity/resistivity... show up in different effects: I dampen linear waves measuring damping rate I drive tearing modes measure growth rate In all cases, you need to run many simulations with different resolutions in space and time, (numerical schemes), and physical resistivity/viscosity, and you need to know the theoretical growth/dampening rates very well. For tearing modes, the latter is a problem on its own. Field lines before and after reconnection by tearing modes. Field lines of tearing modes as visualised in the garden of a Zen temple in Kyoto 17 / 21

Results Numerical resistivity/viscosity viscosity/resistivity... show up in different effects: dampen linear waves measuring damping rate drive tearing modes measure growth rate In all cases, you need to run many simulations with different resolutions in space and time, (numerical schemes), and physical resistivity/viscosity, and you need to know the theoretical growth/dampening rates very well. For tearing modes, the latter is a problem on its own. Growth rate is a complex function of wave number. For clear results, setup a perturbation consisting of a single mode, which is not terribly trivial. 17 / 21

Results Parasitic instabilities of the MRI A 3d MRI box. The parasitic instability is of Kelvin-Helmholtz type. High resolution is required. Comparison with theory is complicated because most approximations made for theoretical growth rates do not hold. Within the uncertainties, you can understand the termination point with the help of parasitic instabilities. 18 / 21

Contents Summary 1 Introduction 2 Results 3 Summary 19 / 21

Summary Summary Scenarios for MHD effects magneto-convection in PNS and hot bubble: small-/large-scale dynamos? magnetic versions of the SASI: amplification, but how much? additional heating, e.g. by Alfvén waves with rotation: winding, MRI, angular-momentum transport, jet formation Prospects fairly likely, requires high-resolution 3d modelling, good ν transport 3d, high-resolution, accurate ν treatment unclear, high-resolution combination of global and local models, still a long way to go for a MRI model 20 / 21