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

(Zweibel & Heiles, 1997, Nature, 385, 131)

Orion

Cosmic magnetic fields

- Polarization of radiation: orientation ||

- Zeeman splitting:

B

B

Cosmic magnetic fields

- Polarization of radiation: orientation ||

- Zeeman splitting:

- Synchrotron radiation: intensity , polarization

B

B 2/ 7B B

NGC 2997

Cosmic magnetic fields

- Polarization of radiation: orientation ||

- Zeeman splitting:

- Synchrotron radiation: intensity , polarized

- Faraday rotation: for lin. polarized waves:

B

B 2/ 7B B

eRM n B dl

Han et al., 1997, A&A 322, 98

Cosmic magnetic fields

- Polarization of radiation: orientation ||

- Zeeman splitting:

- Synchrotron radiation: intensity , polarized

- Faraday rotation: for lin. polarized waves:

- direct measurements for the Sun, solar wind & planets

B

B 2/ 7B B

eRM n B dl

TRACE satellite, 1998-2006

TurbulenceObserved / 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

TurbulenceObserved / 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

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

strong intermittent field, with a scale of the

largest eddies

Small-scale dynamo

Origin of the cosmic magnetic fields

1. Origin of the weak initial local or uniform “seed” field

2. Amplification of the seed field

uniform (large-scale) seed field

local (chaotic) seed field

strong intermittent field, with a scale of the

largest eddies

Small-scale dynamo

Large-scale dynamo

strong field with a largest scale available,

M KE E

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 ~ , Rm ~ 350, resistive timescale ~ 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

810 52 10

Accretion disks- protostellar disks

- close binaries

- active galactic nuclei (AGNs)

Illustration, D.Darling

T Tauri YSO, image by NASA

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 , i.e. it should be removed in some way when

- ordinary viscosity is too small

- turbulent viscosity requires turbulence and even then it will be small

3/ 2r 1/ 2L r

0r

Accretion disks- large-scale shear is given by Keplerian motion,

- angular momentum , i.e. it should be removed in some way when

- 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:

3/ 2r 1/ 2L r

0r

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:

( ) ( )

( )

0, 0

, / 4

VV V B B V P f

tB

V B Bt

B V

V velocity B Alfven velocity B

scale >> m.f.path, plasma scales (Larmor & Debye radii)

velocity << sound speed (for incompressibility)

Induction equation:

( )

BV B B

t

Solar flares: ; Galaxies:

=> magnetic fields are “frozen” into liquid, B can’t change its topology from small scales to the large ones => for dynamos

810Rm 2010Rm

0

0

Mean-field electrodynamics

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

(Moffatt, 78; Parker, 79)

( f r o m D . B i s k a m p , M H D T u r b u l e n c e ,2 0 0 3 )

2( ) ,

,

;

t

M Vt t t

t t

B v B B

v b

B j

B j

3 , ,

2 3 ;

K K

M Kt

H H d V

E

t v

<.> - “large-scale part”

- “turbulent e.m.f.”

- kinetic helicity:

- symmetry should be broken in all 3 directions for

- 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)

Kijk i j kH v v

KH 0

Mean-field theory have to be revised

, , 0MH H A B B A A

Magnetic helicity

0 0 0 0( ) ( ) , MaltH A A B B dV B A

- is conserved quantity in MHD:

, , 0MH H A B B A A

Magnetic helicity

HdV

, t H HV B A V J B V B

HJ 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

2

2

( ) ( )

,

t

t h

h

H B v b B A v b

h B v b J

J a B v b B a v

v b v B

2B v b

- transfer of magnetic helicity between scales

Mean-field dynamo depends on the transfer of magnetic helicity between scales

totalH H h A B a b

where

v b t B v b - turbulent e.m.f. (In mean-field treory )

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:

0 0, , ( ,0,0), 1ikyV B V B e k

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|):

- l.-s. shear is maintained const for anisotropy / against dissipative decay

0 0, , ( ,0,0), 1ikyV B V B e k

Results

Energy evolution

sm.scale forcing: kx/k = 1

Ro ~ 1,

timespan ~ 10 e.t._____________

6o

Vo

B 10

0.7

small scale shear

large scale shearRo

k vRo

Energy spectra

sm.scale forcing: kx/k = 1

Ro ~ 1,

timespan ~ 10 e.t.

6o

Vo

B 10

0.7

|k|

|k|

Magnetic energy spectra

Kinetic energy spectra

time

time

Bx By

BzB total

time

|k|

Magnetic field spectra

sm.scale forcing: kx/k = 1

Ro ~ 1,

timespan ~ 10 e.t.

6o

Vo

B 10

0.7

|k|

time

Bx By

B totalBz

Magnetic field spectra

sm.scale forcing: kx/k = 1

| Bo | ~ | Vo |, Ro ~ 10, timespan ~ 10 e.t.

Evolution of large-scale magnetic energy for different initial large-scale fields

timesteps, 5K ~ 1.e.t.

Bo = 0.1

Bo = 0.01

Bo = 0.001

Evolution of large-scale magnetic energy for

60

B 10

timesteps, 5K ~ 1.e.t.

Magnetic helicity spectra

|k|time

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 large-scale shear to the dissipation scale )

All results we obtained so far already fit well into the helicity-based picture of the large-scale dynamo

The end

Energy spectra for different forcing spectral distribution

Here: ky/k = 0.1 other parameters - “real-life” _______________________

No dynamo action when kx/k = 0

|k|

|k|

Magnetic energy spectra

Kinetic energy spectra

time

time

V, B = large scalev, b = small scaleVo, Bo - initial large-scale fields ______________

Vo = Vx ~ exp(iky)Bo = Bx ~ exp(iky)

sm.scale forcing: kx/k = 1

| Bo | << | Vo |, Ro ~ 10 50

6oB 10

Bx By

BzB total

time

|k|

Balance formula

, , ... T TB B b V V v

2t hh v b B j

2 2

( ) )

, ( ), ( )

h h (

h

1 2

1 2

j a B v b a v B V b a V

a b v B V b

( ) )h h ( 1 2j a B v b a v B V b 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.