Amplification of magnetic fields in core
collapse
Miguel Àngel Aloy Torás, Pablo Cerdá-Durán, Thomas Janka, Ewald
Müller, Martin Obergaulinger, Tomasz Rembiasz
Universitat de València; Max-Planck-Institut für Astrophysik, Garching
INT Program INT 12-2a
Core-Collapse Supernovae: Models and Observable Signals
Nuclear and Neutrino Physics in Stellar Core Collapse
1 / 17
Introduction
Contents
1
Introduction
2
Field amplification mechanisms
3
Summary
2 / 17
Introduction
Progenitor fields
I
magnetic fields need to be strong to have an
effect on SNe
I
But: stellar evolution theory predicts rather weak
fields in the pre-collapse core
→ efficient amplification desired
Meier et al., 1976
3 / 17
Introduction
Magnetic fields and MHD
I
I
I
~2
magnetic energy 12 B
ideal MHD: field lines and flux tubes
frozen into the fluid
Lorentz force (Maxwell stress) consists of
I
I
I
~2
isotropic pressure 21 B
anisotropic tension B i B j
increase the energy by working against
the forces
I
I
compressing the field
stretching and folding field lines
→ estimate for the maximum field energy: ∼
kinetic energy
I
actual amplification may be less
4 / 17
Introduction
Magnetic fields and MHD
I
I
I
~2
magnetic energy 12 B
ideal MHD: field lines and flux tubes
frozen into the fluid
Lorentz force (Maxwell stress) consists of
I
I
I
~2
isotropic pressure 21 B
anisotropic tension B i B j
increase the energy by working against
the forces
I
I
compressing the field
stretching and folding field lines
→ estimate for the maximum field energy: ∼
kinetic energy
I
actual amplification may be less
4 / 17
Introduction
Magnetic fields and MHD
I
I
I
~2
magnetic energy 12 B
ideal MHD: field lines and flux tubes
frozen into the fluid
Lorentz force (Maxwell stress) consists of
I
I
I
~2
isotropic pressure 21 B
anisotropic tension B i B j
increase the energy by working against
the forces
I
I
compressing the field
stretching and folding field lines
→ estimate for the maximum field energy: ∼
kinetic energy
I
actual amplification may be less
4 / 17
Field amplification mechanisms
Contents
1
Introduction
2
Field amplification mechanisms
3
Summary
5 / 17
Field amplification mechanisms
Compression
I
(radial) collapse and accretion compress
the field
I
magnetic flux through a surface is
conserved
→ B ∝ ρ2/3 for a fluid element; energy grows
faster than gravitational
I
no change of field topology
I
core collapse: factor of 103 in field
strength
I
possible saturation: emag ∼ ekin,r is
unrealistic in collapse
I
occurs in every collapse
I
no difficulties in modelling
6 / 17
Field amplification mechanisms
Amplification of Alfvén waves
I
requires an accretion flow decelerated
above the PNS and a (radial) guide field
→ accretion is sub-/super-Alfvénic
inside/outside the Alfvén surface
I
Alfvén waves propagating along the field
are amplified at the Alfvén point
I
waves are finally dissipated there →
additional heating
I
in core collapse: efficient for a limited
parameter range (strong guide field);
strong time variability of the
Alfvén surface may be a problem
I
modelling issues: high resolution,
uncertainties in the dissipation
7 / 17
Field amplification mechanisms
Amplification of Alfvén waves
I
requires an accretion flow decelerated
above the PNS and a (radial) guide field
→ accretion is sub-/super-Alfvénic
inside/outside the Alfvén surface
I
Alfvén waves propagating along the field
are amplified at the Alfvén point
I
waves are finally dissipated there →
additional heating
I
in core collapse: efficient for a limited
parameter range (strong guide field);
strong time variability of the
Alfvén surface may be a problem
I
modelling issues: high resolution,
uncertainties in the dissipation
7 / 17
Field amplification mechanisms
Linear winding
I
~ 6= 0, e.g., differential
~ ·B
works if ∇Ω
rotation and poloidal field
I
creates toroidal field B φ ∝ Ωt
I
feedback: slows down rotation
→ saturation: complete conversion of
differential to rigid rotation
I
core collapse: slow compared to dynamic
times except for rapid rotation and strong
seed field
I
should be present in all rotating cores
I
no difficulties in modelling
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Field amplification mechanisms
The magneto-rotational instability
instability of differentially rotating fluids with weak magnetic field
(simplest) instability criterion ∇$ Ω < 0
I exponential growth ∝ exp Ωt
I starts with coherent channel modes, but leads to turbulence
I feedback: redistribution of angular momentum
→ maximum saturation: conversion of differential to rigid rotation
I
I
Balbus & Hawley (1998)
9 / 17
Field amplification mechanisms
The magneto-rotational instability
instability of differentially rotating fluids with weak magnetic field
(simplest) instability criterion ∇$ Ω < 0
I exponential growth ∝ exp Ωt
I starts with coherent channel modes, but leads to turbulence
I feedback: redistribution of angular momentum
→ maximum saturation: conversion of differential to rigid rotation
I
I
9 / 17
Field amplification mechanisms
The magneto-rotational instability
I
instability of differentially rotating fluids with weak magnetic field
I
(simplest) instability criterion ∇$ Ω < 0
I
exponential growth ∝ exp Ωt
I
starts with coherent channel modes, but leads to turbulence
I
feedback: redistribution of angular momentum
→ maximum saturation: conversion of differential to rigid rotation
I
main physical issue: saturation level
I
numerical difficulties: high resolution, low numerical diffusion
9 / 17
Field amplification mechanisms
The MRI: no amplification without representation
dispersion relation of the MRI: only short modes, λ ∝ |B|, grow rapidly
I in core collapse, this can be ∼ 1 m
I grid width < λ computationally not feasible
→ high-resolution shearing-box simulations to determine fundamental
properties of the MRI
I use these results to build models that can be coupled to global
simulations
I
10
1
Stable modes
0.8
1
0.6
k·vA
Ω cosθk
0.1
Buoyant
modes
0.4
Alfvén modes
0.01
−100
0.2
0
−10
−1
−0.1
−0.01
C
10 / 17
Field amplification mechanisms
The MRI: no amplification without representation
dispersion relation of the MRI: only short modes, λ ∝ |B|, grow rapidly
I in core collapse, this can be ∼ 1 m
I grid width < λ computationally not feasible
→ high-resolution shearing-box simulations to determine fundamental
properties of the MRI
I use these results to build models that can be coupled to global
simulations
I
10 / 17
Field amplification mechanisms
The MRI: no amplification without representation
dispersion relation of the MRI: only short modes, λ ∝ |B|, grow rapidly
in core collapse, this can be ∼ 1 m
I grid width < λ computationally not feasible
→ high-resolution shearing-box simulations to determine fundamental
properties of the MRI
I use these results to build models that can be coupled to global
simulations
I
I
10 / 17
Field amplification mechanisms
The MRI: no amplification without representation
dispersion relation of the MRI: only short modes, λ ∝ |B|, grow rapidly
I in core collapse, this can be ∼ 1 m
I grid width < λ computationally not feasible
→ high-resolution shearing-box simulations to determine fundamental
properties of the MRI
I use these results to build models that can be coupled to global
simulations
I
10 / 17
Field amplification mechanisms
The MRI: saturation mechanism
I
amplification until emag ∼ ediffrot ?
I
More complicated actually... Saturation may occur at lower amplitude.
11 / 17
Field amplification mechanisms
The MRI: saturation mechanism
I
amplification until emag ∼ ediffrot ?
I
More complicated actually... Saturation may occur at lower amplitude.
11 / 17
Field amplification mechanisms
The MRI: saturation mechanism
I
MRI channel modes are separated by
current sheets and shear layers →
unstable against parasitic instabilities:
Kelvin-Helmholtz and tearing modes
I
parasites grow at rates
∝ BMRI /λ ∝ exp σMRI t/B0 , i.e., faster as
the MRI proceeds
I
at some point, they overtake the MRI and
break the channels down into turbulence
I
weaker field → thinner channels → lower
termination amplitude
11 / 17
Field amplification mechanisms
The MRI: saturation mechanism
I
I
parasites grow at rates
∝ BMRI /λ ∝ exp σMRI t/B0 , i.e., faster as
the MRI proceeds
at some point, they overtake the MRI and
break the channels down into turbulence
weaker field → thinner channels → lower
termination amplitude
bϖ [ 1014 G ]
−60
−20
20
60
0.2
z [ km ]
I
MRI channel modes are separated by
current sheets and shear layers →
unstable against parasitic instabilities:
Kelvin-Helmholtz and tearing modes
0.1
0.0
−0.1
0.2
z [ km ]
I
15.3
15.4
15.5
15.6
ϖ [ km ]
15.7
15.3
15.4
15.5
15.6
ϖ [ km ]
15.7
0.1
0.0
−0.1
jφ [ 1012 G / cm ]
−9
−3
3
9
11 / 17
Field amplification mechanisms
The MRI: saturation mechanism
I
I
parasites grow at rates
∝ BMRI /λ ∝ exp σMRI t/B0 , i.e., faster as
the MRI proceeds
at some point, they overtake the MRI and
break the channels down into turbulence
weaker field → thinner channels → lower
termination amplitude
bϖ [ 1014 G ]
−60
−20
20
60
0.2
z [ km ]
I
MRI channel modes are separated by
current sheets and shear layers →
unstable against parasitic instabilities:
Kelvin-Helmholtz and tearing modes
0.1
0.0
−0.1
0.2
z [ km ]
I
15.3
15.4
15.5
15.6
ϖ [ km ]
15.7
15.3
15.4
15.5
15.6
ϖ [ km ]
15.7
0.1
0.0
−0.1
jφ [ 1012 G / cm ]
−9
−3
3
9
11 / 17
Field amplification mechanisms
The MRI: saturation mechanism
I
I
parasites grow at rates
∝ BMRI /λ ∝ exp σMRI t/B0 , i.e., faster as
the MRI proceeds
at some point, they overtake the MRI and
break the channels down into turbulence
weaker field → thinner channels → lower
termination amplitude
bϖ [ 1014 G ]
−60
−20
20
60
0.2
z [ km ]
I
MRI channel modes are separated by
current sheets and shear layers →
unstable against parasitic instabilities:
Kelvin-Helmholtz and tearing modes
0.1
0.0
−0.1
0.2
z [ km ]
I
15.3
15.4
15.5
15.6
ϖ [ km ]
15.7
15.3
15.4
15.5
15.6
ϖ [ km ]
15.7
0.1
0.0
−0.1
jφ [ 1012 G / cm ]
−9
−3
3
9
11 / 17
Field amplification mechanisms
The MRI: saturation mechanism
MRI channel modes are separated by
current sheets and shear layers →
unstable against parasitic instabilities:
Kelvin-Helmholtz and tearing modes
I
parasites grow at rates
∝ BMRI /λ ∝ exp σMRI t/B0 , i.e., faster as
the MRI proceeds
I
at some point, they overtake the MRI and
break the channels down into turbulence
I
weaker field → thinner channels → lower
termination amplitude
31
30
log ( emag, Mϖφ [cgs] )
I
29
28
27
15
15.50
16
t [ ms ]
16.50
17
11 / 17
Field amplification mechanisms
The MRI: saturation mechanism
⇒ MRI growth limited by initial field strength?
I
current simulations (T. Rembiasz) try to test the predictions of Pessah
(2010) and focus on how parasites depend on resistivity and viscosity.
Prerequisite: careful determination of numerical resistivity and viscosity.
11 / 17
Field amplification mechanisms
Dynamos driven by hydrodynamic instabilities
I
instabilities such as convection and SASI drive turbulence
I
energy cascades from the large scale at which the instability operates
down to dissipation in a Kolmogorov-like cascade
12 / 17
Field amplification mechanisms
Dynamos driven by hydrodynamic instabilities
→ dynamo converting turbulent kinetic to magnetic energy by stretching and
folding flux tubes
I
I
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 × curl~v .
12 / 17
Field amplification mechanisms
Dynamos driven by hydrodynamic instabilities
→ dynamo converting turbulent kinetic to magnetic energy by stretching and
folding flux tubes
I
I
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 × curl~v .
12 / 17
Field amplification mechanisms
Dynamos driven by hydrodynamic instabilities
→ dynamo converting turbulent kinetic to magnetic energy by stretching and
folding flux tubes
I
I
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 × curl~v .
12 / 17
Field amplification mechanisms
Instabilities: an overview
energy
mechanism
role of ~b
SASI
accretion flow
advective-acoustic
cycle
convection
thermal
buoyant transport
of energy/species
passive;
dynamo
passive;
dynamo
Endeve et al., 2008
turbulent
turbulent
MRI
diff. rotation
magnetic transport
of angular momentum
instability
driver;
turbulent dynamo
13 / 17
Field amplification mechanisms
Dynamos
I
MHD uncertainties: type of dynamo, saturation mechanism and level
I
technical complications: 3d necessary, high resolution and Reynolds
numbers
I
kinematic dynamo: weak fields, back-reaction negligible → mean-field
dynamo theory
~ =∇
~ for a fixed turbulent
~ × (~v × B),
solve the induction equation, ∂t B
velocity field
~ = αB.
~ α parametrises the unknown physics of helical
→ α effect: ∂t B
turbulence.
14 / 17
Field amplification mechanisms
Saturation mechanisms
Will instabilities and turbulence amplify a seed magnetic field to dynamically
relevant strength or will the amplification cease earlier?
I if the properties of the instability depend strongly on the field,
amplification might be limited by the initial field (MRI channel disruption)
I quenching of turbulent dynamos: small-scale field resists further
stretching (emag ∼ ekin locally in k -space), reducing the efficiency of
mean-field dynamos by orders of magnitude.
~ ·B
~ (where B
~ =∇
~ is conserved in ideal MHD.
~ × A),
I magnetic helicity: A
Box simulations indicate that fluxes out of the domain may be important to
avoid catastrophic quenching. Inhomogeneity of cores may provide that.
15 / 17
Summary
Contents
1
Introduction
2
Field amplification mechanisms
3
Summary
16 / 17
Summary
Summary
mechnism
compression
winding
Alfvén waves
MRI
dynamo
requires
infall
diff. rotation
accretion flow
diff. rotation
instabilities
results & issues
amplifies field robustly
works well in rapid rotators
works for rather strong fields
saturation level?
saturation level?
Saturation level
Better understanding of turbulence and dynamos is required!
17 / 17