Transcript Magnetic dynamos through driven turbulence

Magnetic dynamos in accretion disks
Magnetic helicity and
the theory of astrophysical dynamos
Dmitry Shapovalov
JHU, 2006
Outline
- Turbulence and magnetic fields in astrophysics
- Dynamo problem
- How the dynamo works:
- old theory: mean-field dynamo
- new theory: role of magnetic helicity
- How can we learn is it true?
- Results: what we did so far
- Future: what else can be done
Cosmic magnetic fields
Crucial:
stellar and solar activity, star formation, pulsars, accretion disks,
formation and stability of jets, cosmic rays, gamma-ray bursts
Probably crucial:
protoplanetary disks, planetary nebulae, molecular clouds,
supernova remnants
Role is unclear:
stellar evolution, galaxy evolution, structure formation in the
early Universe
Probably unimportant:
planetary evolution
Cosmic magnetic fields
- Polarization of radiation: orientation ||
B
Orion
(Zweibel & Heiles, 1997, Nature, 385, 131)
Cosmic magnetic fields
- Polarization of radiation: orientation ||
- Zeeman splitting: 
B
B
Cosmic magnetic fields
- Polarization of radiation: orientation ||
- Zeeman splitting: 
B
B
- Synchrotron radiation: intensity
B2/ 7, polarization  B
NGC 2997
Cosmic magnetic fields
- Polarization of radiation: orientation ||
- Zeeman splitting: 
B
B
- Synchrotron radiation: intensity
B2/ 7, polarized  B
- Faraday rotation: for lin. polarized waves: RM
Han et al.,
1997, A&A
322, 98
 ne B dl
Cosmic magnetic fields
- Polarization of radiation: orientation ||
- Zeeman splitting: 
B
B
- Synchrotron radiation: intensity
B2/ 7, polarized  B
- Faraday rotation: for lin. polarized waves: RM
 ne B dl
- direct measurements for the Sun, solar wind & planets
TRACE
satellite,
1998-2006
Vieser &
Hensler,
2002
Turbulence
Observed / predicted in:
- convective zones of the stars and planets
- stellar wind and supernova explosions
- interstellar medium, both neutral and ionized
- star forming regions
- accretion disks
- motion of galaxies through IGM
Freyer & Hensler, 2002
Vieser &
Hensler,
2002
Turbulence
Observed / predicted in:
- convective zones of the stars and planets
- stellar wind and supernova explosions
- interstellar medium, both neutral and ionized
- star forming regions
- accretion disks
- motion of galaxies through IGM
Freyer & Hensler, 2002
Origin of the cosmic magnetic fields
1. Origin of the weak initial local or uniform “seed” field
2. Amplification of the seed field
Origin of the cosmic magnetic fields
1. Origin of the weak initial local or uniform “seed” field
2. Amplification of the seed field
Theories range
from fluctuations of hypermagnetic fields during the time of
decoupling of electroweak interations (come together with
baryonic asymmetry of the Universe),
to various “battery effects”, which produce macroscopic seed
fields on a continuing basic up to our time. One example is a
Poynting-Robertson effect:
M.Harwit,
Astrophysical
Concepts
Origin of the cosmic magnetic fields
1. Origin of the weak initial local or uniform “seed” field
2. Amplification of the seed field
Origin of the cosmic magnetic fields
1. Origin of the weak initial local or uniform “seed” field
2. Amplification of the seed field
local (chaotic)
seed field
Small-scale
dynamo
strong intermittent field,
with a scale of the
largest eddies
Origin of the cosmic magnetic fields
1. Origin of the weak initial local or uniform “seed” field
2. Amplification of the seed field
local (chaotic)
seed field
uniform (large-scale)
seed field
Small-scale
dynamo
Large-scale
dynamo
strong intermittent field,
with a scale of the
largest eddies
strong field with a
largest scale available,
EM  EK
Systems with large-scale dynamos
- Earth, Jupiter, some other planets and their satellites
- the Sun
- accretion disks
- some spiral galaxies
- giant molecular clouds
Earth
- turbulence is driven by temperature gradient in
liquid outer core
- large-scale shear is given by Earth rotation
- Re ~ 108, Rm ~ 350, resistive timescale ~ 2 105
years
- B ~ 3 gauss (at CMB), exists for billions of years
Radial component
of the Earth’s field
at core-mantle
boundary (CMB)
G.Rüdiger,
The Magnetic
Universe, 2004
Accretion disks
- protostellar disks
- close binaries
- active galactic nuclei (AGNs)
T Tauri YSO, image by NASA
Illustration, D.Darling
Accretion disks
- protostellar disks
- close binaries
- active galactic nuclei (AGNs)
Illustration, NASA
Accretion disks
- protostellar disks
- close binaries
- active galactic nuclei (AGNs)
Quasar PKS 1127-145, image by Chandra
Illustration, NASA/ M.Weiss
Accretion disks
- large-scale shear is given by Keplerian motion, 
- angular momentum L r1/ 2, i.e. it should be
removed in some way when r  0
- ordinary viscosity is too small
- turbulent viscosity requires turbulence and
even then it will be small
r 3/ 2
Accretion disks
- large-scale shear is given by Keplerian motion, 
- angular momentum L r1/ 2, i.e. it should be
removed in some way when r  0
- ordinary viscosity is too small
- turbulent viscosity requires turbulence and
even then it will be small
- large-scale poloidal magnetic field can remove
angular momentum from the system:
r 3/ 2
Accretion disks
- large-scale shear is given by Keplerian motion
- even if Keplerian flow is stable over radial perturbations, in
presence of vertical magnetic field turbulence can exist via
MRI (magnetorotational instability, Balbus & Hawley, 1991)
- MRI can drive the growth of azimuthal field, i.e. large-scale
seed for a dynamo process
Now we should explain how dynamo works
Dynamo theory
Dynamo theory
Macroscopic magnetohydrodynamic (MHD) framework:
V
 (V )V  (  B)  B  V  P  f
t
B
   (V  B)  B
t
  B  0,
 V  0
V  velocity, B  Alfven velocity  B / 4
scale >> m.f.path, plasma scales (Larmor & Debye radii)
velocity << sound speed (for incompressibility)
Induction equation:
B
   (V  B )  B
t
8
20
Solar flares: Rm 10 ; Galaxies: Rm 10
  0 => magnetic fields are “frozen” into liquid, B can’t
change its topology from small scales to the large ones
=> for dynamos   0
Mean-field electrodynamics (Moffatt, 78; Parker, 79)
In differentially rotating object:
radial seed field is stretched along
the direction of rotation
=> azimuthal field grows
To keep azimuthal field growing one
needs to maintain radial component
in some way: -effect
<.> - “large-scale part”
 - “turbulent e.m.f.”
 t B    ( v  B )       2 B ,
  vb ,
   t B   tM j   tV  ;

t B  t j
 t    H K 3,
H K   v   dV ,
 tM  2 E K 3;
(from D.Biskamp, MHD Turbulence,
2003)
H K  ijk vi  j vk
- kinetic helicity:
- symmetry should be broken in all 3 directions for HK  0
- doesn’t depend on any magnetic quantitities (, B)
- “small-scale quantity”: direct cascade
- not a conserved quantity in MHD, even for negligibly small 
- can’t support dynamo for a long time: magnetic back-reaction
cancels all kinetic helicity at large scales (there is no preferred
orientation for spirals)
- -effect contradicts to simulations (Hughes & Cattaneo, 96,
Brandenburg, 01)
Mean-field theory have to be revised
Magnetic helicity
H M  H  A  B,
B   A,
M
Halt
 ( A  A0 )  (B  B0 )dV ,
 A  0
B0   A0
Magnetic helicity
H M  H  A  B,
B   A,
 A  0
-  HdV is conserved quantity in MHD:


 t H    HV  B    A V    J  B,
    V  B 
J H  HV  B    A V  - helicity current
- H is the only integral in 3D, which has inverse cascade:
can’t dissipate at small scales, remains at large ones, where resistivity
is negligible, i.e. exists for a time bigger than dissipative timescale
Htotal  H  h  A  B  a  b

 t H  2 B  v  b    B  A  v  b

 t h  2 B  v  b    J h
where
J h  ( a  B ) v  b  B ( a  v )
    v  b ,
    v  B
2B  v  b - transfer of magnetic helicity between scales
  vb
- turbulent e.m.f. (In mean-field treory
t B  v  b )
Mean-field dynamo depends on the transfer
of magnetic helicity between scales
Simulations
General features:
- incompressible 3D MHD
- pseudospectral (E, H - conserved, unlike in spatial code)
- periodic box (H is gauge invariant)
- resolution: from 64^3 to 1024^3
- timescale up to 100 eddy turnover times
- both OpenMP & MPI parallel versions available
Simulations
Dynamo-specific features:
-  =  (to simplify)
- turbulence is driven by external random (gaussian) forcing
- forcing has N components with variable spectral properties
- forcing correlation time is variable
- forcing has both linearly and circularly polarized
components (for helicity injection into the turbulence)
- divF =0
Simulations
Dynamo-specific features:
- turbulence is driven by external random (gaussian) forcing
- forcing has N components with variable spectral properties
- forcing correlation time is variable
- forcing has both linearly and circularly polarized
components (for helicity injection into the turbulence)
- divF =0
- forcing is usually set at some fixed small scale (to simulate
real systems)
Simulations
Dynamo-specific features:
- forcing correlation time is variable
- forcing has both linearly and circularly polarized
components (for helicity injection into the turbulence)
- divF =0
- forcing is usually set at some fixed small scale (to simulate
real systems)
- initial large scale shear and weak seed field:
V , B  V0 , B0 (eiky ,0,0),
k 1
Simulations
Dynamo-specific features:
- forcing has both linearly and circularly polarized
components (for helicity injection into the turbulence)
- divF =0
- forcing is usually set at some fixed small scale (to simulate
real systems)
- initial large scale shear and weak seed field (|Bo| << |Vo|):
V , B  V0 , B0 (eiky ,0,0),
k 1
- l.-s. shear is maintained const for anisotropy / against
dissipative decay
Results
Energy evolution
sm.scale forcing:
kx/k = 1
Bo  106
Vo  0.7
Ro ~ 1,
timespan ~ 10 e.t.
_____________
small scale shear
Ro 
large scale shear
Ro 
kv

Magnetic energy spectra
Energy spectra
sm.scale forcing:
kx/k = 1
Bo  106
Vo  0.7
|k|
time
Ro ~ 1,
Kinetic energy spectra
timespan ~ 10 e.t.
time
|k|
Magnetic field
spectra
Bx
By
time
sm.scale forcing:
kx/k = 1
|k|
Bo  106
Vo  0.7
Ro ~ 1,
Bz
timespan ~ 10 e.t.
B total
Magnetic field
spectra
Bx
By
time
sm.scale forcing:
kx/k = 1
|k|
| Bo | ~ | Vo |,
Ro ~ 10,
Bz
timespan ~ 10 e.t.
B total
Evolution of
large-scale
magnetic energy
for different initial
large-scale fields
Bo = 0.1
Bo = 0.01
Bo = 0.001
timesteps, 5K ~ 1.e.t.
Evolution of
large-scale
magnetic energy
for B 106
0
timesteps, 5K ~ 1.e.t.
Magnetic helicity
spectra
time
|k|
All results we obtained so far already fit well into the
helicity-based picture of the large-scale dynamo
Future
- next big goal is to prove numerically that turbulent e.m.f.
depends on transfer of magnetic helicity between scales
(balance formula)
- then it is interesting to see how different terms in helicity
current influence the dynamo
- code: do subgrid modelling in order to expand dynamic
range even more (to have everything covered: from largescale shear to the dissipation scale )
The end
Magnetic energy spectra
Energy spectra for different
forcing spectral distribution
Here:
ky/k = 0.1
other parameters - “real-life”
_______________________
time
|k|
Kinetic energy spectra
No dynamo action
when kx/k = 0
time
|k|
V, B = large scale
v, b = small scale
Vo, Bo - initial largescale fields
______________
Bx
By
time
|k|
Vo = Vx ~ exp(iky)
Bo = Bx ~ exp(iky)
sm.scale forcing:
kx/k = 1
Bo  106
| Bo | << | Vo |,
Ro ~ 10  50
Bz
B total
Balance formula
 t h  2 v  b  B    jh
BT  B  b, VT  V  v , ...
jh  (a  B )v  b1  a  v B  hV  b( 2  a V )
h  a b ,
 21    (v  B ),
 22    (V  b)
jh  (a  B)v  b1  a  v B  hV  b( 2  a V )
Simulations with real-life parameters
• Ro ~ 1 in accretion disks, Sun’s convection zone.
• Constant large-scale shear (to compensate disipative decay
and to maintain non-uniformity in the system).
• Weak initial magnetic field: |Bo| << |Vo|. We want large
scale shear to help in generation of magnetic field from
some small seed field.
• Forcing correlation time ~ eddy turnover time. Small scale
turbulence is driven by some instability which saturates
when its growth time ~ eddy turnover time.